Volume One of
THE WORLD OF
MATHEMATICS A small library of the literature of mathematics from Ath-mose the Scribe to Al...
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Volume One of
THE WORLD OF
MATHEMATICS A small library of the literature of mathematics from Ath-mose the Scribe to Albert Einstein, presented with commentaries and notes by JAMES R. NEWMAN
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SIMON AND SCHUSTER • NEW YORK
COPYRIGHT
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19S6 BY JAMES R. NEWMAN
ALL RIGHTS RESERVED INCLUDING THE RIGHT OF REPRODUCTION IN WHOLE OR IN PART IN ANY FORM
PUBLISHED BY SIMON AND SCHUSTER. ROCKEFELLER CENTER. 630 FIFTH AVENUE. NEW YORK. NEW YORK 10020
TENTH PRINTING
TO
Ruth, Brooke THB BDIToa wishes to express his aratltude for permission to reprint material from the following sources: Messrs Allen " Unw1n, Ltd., for "Definition of Number," bom Int,.odru:llon to MtII1ulllllllcal Philosophy. by Bertrand Russell. Cambridge Uniwrsity Press and tbe Royal Society for "Isaac Newton," by E. N. Da C. Andrade, and "Newton, the Man." by John Maynard Keynes, from Newton Te,.centenary Celeb,.tIIlons: and for "The Sand Reckoner." from Work. of Archimede.r (with the kind permission of Lady Heath). Harvard UniversitY Press for "Greek Mathematics," from Gnek Mtllhellllllical Wo,.k.r, translated by Ivor Thomas and published by tbe Loeb Classical Ubrary. Professor Otto Koehler for "The Ability of Birds to 'Count,' .. from tbe Bulletin 01 Animal Behavio,., No.9, translated by Professor W. Thorpe. Library of Uvlng Philosophers, Evanston, illinois, for "My Mental Development," by Bertrand Russell, from The Philosophy 01 BenTt1.lld RlLSSell, edited by Paul SchUpp. The Macmillan Company for "Mathematics as an Element in the History of Thought," from Science and the Modem Wo,.ld, by Alfred North Whitehead. ® 1925 by The Macmillan Company. McGraw-Hili Book Company for "The Analyst," by Bishop Berkeley, "On tbe Binomial Theorem for Fractional and Negatlw Exponents," by Sir Isaac Newton. and "The Declaration of the Profit of Arithmeticke," by Robert
AND
Jeff
Recorde, from Sou,.ce Book In MtII1ulllllllcs, by D. E. Smith, «1> 1929 by McGraw-Hili Book Company, Inc.: and for "The Queen of Mathematics," from Mathematic.r, Queen and Se,.VIlll1 01 t1u Sciences, ® 1951 by Eric Temple Bell. Methuen" Company, Ltd., for T1u G,.etll Mo.Jhematician.r, by Herbert Westren Turnbull. National Council of Teachers of Mathematics for "From Numbers to Numerals and From Numerals to Computation," from Numbe,.. and Nume,.als, by David Eugene Smith and Jekuthlel Ginsbul'l. Oxford University Press for "TopololD'," bom Who.J Is Mathematics? by Richard Courant and Herbert Robbins «1> 1941 by Richard Courant. Princeton Unlwnity Press for "DUrer .. a Mathematician," from A lbncht Dunr, by Erwin Panofsky, ® 1943 by Princeton Unlwrslty Press: and for Symnwtry, by Hermann Weyl, «1> .952 by Princeton Uniwrslty Press. St. Mariln's Press for "Calculating Prodilles," from MathemallcaIRecretllionsandEuay.r.by W. W. Rouse Ball. Published by Macmillan and Company, London. Sclentlfjc American for "Projectlw Geo_ etry," by Morris Kline, ® 1955 by Scientific American, Inc.: for "The Rhlnd Papyrus," by James R. Newman, ® 1952 by Scientific American. Inc.: and for "SrinivBSa Ramanujan," by James R. Newman, ® 1948 by Scientific American, Inc Simon and Schuster, Inc., for "Gauss, the Prince of Mathematicians" and "Invariant Twins, Cayley and SylWSler," from Men of Mtllhemalic.r. ® 1937 by E. T. Bell.
LIBRARY GF CONGRESS CATALOG CARD NUMBER: SS-JOO6O. MANUFACTURED IN 11IE UNITED STATBS OF AMERICA. COMPOSmGN BY BROWN BROS. LlNOTYPERS, INC., NEW YORK, N. Y. PRINTED BY MURRAY PRINTING COMPANY. BOUND BY H. WOLFF BOOK MFG. CO., INC.• NEW YGRK. N. Y.
In
th.~$~
day$ 01 conflict IHlween anc;'nt and mod~rn $tudi~$, th.~,.. mUlt IH $o~th.in, to be MIld Jor a study which. did not be,in with. Pythagand will not ~nd with. E;n$t~;n. but is tM oldtm and youn'~$t oj aU. -G. H. HAltDv (A MatMmatician'$ Apolo,y)
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Introduction A PREFACE is both a greeting and a farewell. I have been so long ocell.pied with this book that taking leave of it is dilJicult. It is more than fifteen years since I began gathering the material for an anthology which I hoped would convey something of the diversity, the utility and the beauty of mathematics. A t the beginning it seemed that the task would be neither too arduous nor unduly prolonged, for I was acquainted with the general literature of the subject and I had no intention of compiling a mammoth sourcebook. I soon discovered I was mistaken in my estimate. Popular writings on the nature, uses and history of mathematics did not .,ield the variety I had expected: thus it was necessary to go through an immense technical and scholarly literature to search out examples of mathematical thought which the general reader could follow and enjoy. Digestible essays were hard to find on the foundations and philosophy of mathematics, the relation of mathematics to art and music, the applications of mathematics to social and economic problems. Moreover, while I had not planned to write introductions to the separate selections, it became clear as I proceeded that many of the pieces, though illuminating when read in context, had little meaning when they stood alone. Settings had to be provided. explanations of how the pieces came to be written and of the place they occupied in the growth of mathematical thought. So the work I expected to finish in two years has taken the greater portion of two decades; what was envisaged as a volume of moderate size has assumed dimension, which even a self-indulgent author must acknowledge to be extended. I have tried in this book to show the range of mathematics, the richness of its ideas and multiplicity of its aspects. It presents mathematics as a tool, a language and a map; as a work of art and an end in itself; as a fulfillment of the passion for perfection. It is seen as an object of satire, a subject for humor and a sOll.rce of controversy; os a spu.r to wit and a leaven to the storyteller's imagination; as an activity which has driven men to frenzy and provided them with delight. It appears in broad view as a body of knowledge made by men, yet standing apart and independent of them. In this collection, I hope, will be found material to suit every taste and capacity. Many of the selections are long. I dislike snippets and fragments that tease. Understanding mathematical logic, or the theory of relativity, is not
InlTOd:tu:tlon
an indispensable attribute 01 the cultured mind. But if one wishes to learn anything about these subjects, one must learn something. It is necessary to master the rudiments of the language, to practice a technique, to lollow step by step a characteristic sequence 01 reasoning and to see a problem through Irom beginning to end. The reader who makes this efJort wUl not be disappointed. Some 01 the selections are difficult. But what is surprising is how many can be understood without unusual aptitude or special train~ ing. 01 course those who are bold enough to tackle the more formidable subjects will gain a special reward. There are lew gratifications comparable to that 01 keeping up with a demonstration and attaining the prool. 't is lor each man an act of creation, as il the discovery hatl never been made belore,' it inculcates lolly habits 01 mind. An anthology is a work 01 prejudice. This is no less true when the subject is mathematics than when it is poetry or fiction. Magic squares, lor example, bore me, but I never tire of the theory 01 probability. 1 preler geometry to algebra. physics to chemistry, logic to economics, the mathematics 01 infinity to the theory of numbers. 1 have shunned topics, slelmped some and shown great hospitality to others. I malee no apolo/IY for these prejudices; since I pretend to no mathematical sleill, 1 have felt at liberty to present the mathematics I like. MANY persons have helped to malee this boole. lowe a great deal, more than I can express, to my friend and lormer colleague Robert Hatch lor his editorial counsel. No damp idea got by him, no potted phrase. In substance and style the work bears his marie. My teacher and friend Professor Ernest Nagel has been generous beyond measure not only In giving advice and criticism but in writing specially lor this boole a brilliant essay on symbolic logic. Sam Rosenberg read what I wrote and improved it with his wit. My wi/e, as always, has been wise, patient and encouraging. Dr. Ralph Shaw, Professor 01 Bibliography at Rutgers University and former LIbrarian of the Department of Agriculture, gave invaluable assistance in the preparation of the manuscript. And I wish to thank my publishersI mention especially the contributions 01 Jacle Goodman, Tom Torre Bevans, and Peter Schwed-both for their forbearance in waiting until }956 to publish a book scheduled for 1942, and lor their imagination and skill in the difficult tasle 01 design and manulacture. J. R. N.
Table of Contents
VOLUME ONE PART I:
General Survey
Philip E. B. Jourdain: Commentary
1. The Nature of Mathematics by
PART
2
PHILIP E. B. JOURDAIN
4
n: Historical and Biographical
Commentary
14
75
1. The Great Mathematicians by HERBERT WESTREN TURNBULL Commentary
169
2. The Rhind Papyrus by
JAMES R. NEWMAN
170
Commentary
3. Archimedes
119 by PLUTARCH, VITRUVIUS, TZETZES
Commentary
180
188
4. Greek Mathematics by
IVOR THOMAS
Robert Recorde: Commentary
189
210
5. The Declaration of the Profit of Arithmeticke
212
by ROBERT RECORDE Kepler and Lodge: Commentary
6. Johann Kepler by sm
218 OLIVER LODGE
220
Descartes and A nalytical Geometry: Commentary 235
7. The Geometry by Commentary
RENE DESCARTES
239
254
8. Isaac Newton by E. N. DA C. ANDRADE 9. Newton, the Man by JOHN MAYNARD KEYNES vU
2SS
277
CORI~Rts
Bishop Berkeley and Infinitesimals: Commentary 286
10. The Analyst by
288
BISHOP BERKELEY
Gauss: Commentary
294
11. Gauss, the Prince of Mathematicians by ERIC Cayley and Sylvester: Commentary
TEMPLE BELL
295
340
12. Invariant Twins, Cayley and Sylvester
341
by ERIC TEMPLE BELL Commentary
366
13. Srinivasa Ramanujan by
368
JAMES R. NEWMAN
Bertrand Russell: Commentary
14. My Mental Development by
377
BERTRAND RUSSELL
Alfred North Whitehead: Commentary
381
395
15. Mathematics as an Element in the History of Thought
402
by ALFRED NORTH WHITEHEAD
PART
m: Arithmetic, Numbers and the Art of Counting
Poppy Seeds and Large Numbers: Commentary
1. The Sand Reckoner by
418
420
ARCHIMEDES
The Art of Counting: Commentary 430
2. Counting by LEVI LEONARD CONANT 3. From Numbers to Numerals and From Numerals to Computation by DAVID EUGENE SMITH and JEKUTHIEL GINSBURG
432 442
Idiot Savants: Commentary 465
4. Calculating Prodigies by w. Gifted Birds: Commentary
W. ROUSE BALL
488
5. The Ability of Birds to uCount" by o.
KOEHLER
The Mysteries of Arithmetic: Commentary
6. The Queen of Mathematics by (P
+ PQ)'IIIl/'A: Commentary
467 489
497
ERIC TEMPLE BELL
498
519
7. On the Binomial Theorem for Fractional and Negative Exponents by ISAAC NEWTON 521 The Number Concept: Commentary 525
8. Irrational Numbers by RICHARD DEDEKIND 9. Definition of Number by BERTRAND RUSSELL
528 537
C on'lI""
PART IV:
Mathematics of Space and Motion
William Kingdon CIit1ord: Commentary 546
1. The Exactness of Mathematical Laws by WILLIAM KINGDON CLIFFOI.D 2. The Postulates of the Science of Space by WILLIAM KINGDON CLIFFOI.D 3. On the Space Theory of Matter by WILLIAM KINGDON CLIFFORD A Famous. Problem: Commentary
548 552 568
570
4. The Seven Bridges of KlSnigsberg by LEONHARD EULER 5. 'l'opo)ogy by RICHARD COURANT and HERBERT ROBBINS Durer and the Mathematics of Painting: Commentary
600
6. Durer as a Mathematician by ERWIN PANOFSKY 7. Projective Geometry by MORRIS KLINE Hermann von Helmholtz: Commentary
647
669
9. Symmetry by HERMANN INDEX
603 622
642
8. On the Origin and Significance of Geometrical Axioms by HERMANN VON HELMHOLTZ Commentary
573 581
WEYL
671 follows page 724
PART I
General Survey 1. The Nature of Mathematics by
PHILIP E. B. JOURDAIN
COMMENTARY ON
PHILIP E. B. JOURDAIN HILIP E. B. JOURDAIN (1879-1919), whose little book on the nature of mathematics is here reproduced in its entirety, was a logician, a philosopher and a historian of mathematics. To each of these subjects he brought a fresh outlook and a remarkably penetrating and creative intelligence. He was not yet forty when he died and from adolescence had been afflicted by a terrible paralytic ailment (Friedrick~s ataxia) which gradually tightened its grip upon him. Yet he left behind a body of work that influenced the development of both mathematical logic and the history of science. Jourdain, the son of a Derbyshire vicar, was educated at Cheltenham College and at Cambridge. The few years during which he was able to enjoy the normal pleasures of boyhood-long walks were his special delight-are described in a poignant memoir by his younger sister, Millicent, who suffered from the same hereditary disease. In 1900 the brother and sister went to Heidelberg to seek medical help. While at the hospital he began in earnest his study of the history of mathematics. "We had," wrote Millicent, "what was to be nearly our last bit of walking together here." The treatment was unavailing and when they returned to Eng]and, Jourdain could no longer walk or stand or even hold a pencil without difficulty. Nevertheless, he undertook with great energy and enthusiasm the first of a series of mathematical papers which established his reputation. Among his earlier writings were studies of Lagrange's use of differential equations, the work of Cauchy and Gauss in function theory, and conceptual problems of mathematical physics.! Between 1906 and 1912 he contributed to the Archiv der Mathematik unil Physik a masterly group of papers on the mathematical theory of transfinite numbers, a subject in which he was always deeply interested. In the same period the Quarterly Journal of Mathematics published a group of essays on the development of the theories of mathematical logic and the principles of mathematics. Jourdain was an editor of Isis and the Monist, in whose pages appeared his articles on Leibniz, Napier, Hooke, Newton, Galileo, Poincare and Dedekind. He edited reprints of works by De Morgan, Boole, Georg Cantor, Lagrange, Jacobi, Gauss and Emst Mach; he wrote a brilliant and witty book, The Philosophy of Mr. B*rtr*nd R*ss*ll, dealing with Russell's analysis of the problems of logic and the foundations of mathematics; he took out a patent covering an invention of a "silent engine" (I have been unable to
P
1 Bibliographies of Jourdain's writings appear in Isis. Vol. S, 1923. pp. 134-136, and in the Monist. Vol. 30, 1920. pp. 161-182.
2
Philip E. B. lour4tli"
discover what this machine was) and he wrote poems and short stories which never got published. In 1914, at the height of his powers, he was producing enough "to keep two typists busy all day." The distinctive qualities of Jourdain's thought were its independence and its cutting edge. He was renowned for his broad scholarship in the history and philosophy of science, but he was more than a scholar. Never content with comprehending all that others had said about a problem, he had to work it through in his own way and overcome its difficulties by his own methods. This led him to conclusions peculiarly his own. They are not always satisfactory but they always deserve close attention: Jourdain rarely failed to uncover points overlooked by less subtle and original investigators. The Nature of Mathematics reflects his excellent grasp of the subject, his at times oblique but always rewarding approach to logic and mathematics, his wit and clear expression. He had sharpened his thinking on some of the hardest and most baffling questions of philosophy and had achieved an orderly understanding of them which he was fu1ly capable of imparting to the attentive reader. The book is not a textbook collection of methods and examples, but an explanation of uhow and why these methods grew up." It discusses concepts which are widely used even in elementary arithmetic, geometry and aJgebra-negative numbers, for example--but far from wideJy comprehended. It presents also a careful treatment of "the development of analytical methods and certain examinations of principles." There are at least two other excellent popularizations of mathematics, A. N. Whitehead's celebrated Introduction to Mathematics 2 and the more recent Mathematics for the General Reader by E. C. Titchmarsb. 8 Both books can be recommended strongly,. the first as a characteristic, immensely readable work by one of the greatest of twentieth-century philosophers; the second as a first-class mathematician's lucid, unhurried account of the science of numbers from arithmetic through the calcu1us. Jourdain's book follows a somewhat difterent path of instruction in that it emphasizes the relation between mathematics and logic. It is the peer of the other two studies and has for the anthologist the additional appeal of being unjustly neglected and out of print. "I hope that I shan succeed," says Jourdain in his introduction, "in showing that the process of mathematical discovery is a living and a growing thing." In this attempt he did not fail. :I 3
Oxford University Press, New York, 1948. Hutchinson'S University Library, London, n. d.
Pure mathematics consists entirely of such asseverations as thai, if such and such a propos;tu," is true of anything, then such and such another proposition IS true oflhat thing. It is essential not to discuss whether the first proposition is really true, and nvt to mention what the anything is of which it is supposed to be true. . . . If our hypothesis is about anything and not about some one or more particular things. then OUr deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nOr whether what we are saying is -BERTRAND RUSSELL true.
1
The Nature of Mathematics By PHILIP E. B. JOURDAIN
CONTENTS CHAPTElt
PAGB
INTRODUCTION I.
THE GROWTH OF MATHEMATICAL SCIENCE IN ANCIENT TIMES
II.
THE RISE AND PROGRESS OF MODERN MATHEMATIC8--ALGEBRA
III.
THE RISE AND PROGRESS OF MODERN MATHEMATIC8--ANALYTICAL GEOMETRY AND THE METHOD OF INDIVISIBLES
IV.
VII.
43
THE RISE OF MODERN MATHEMATICS--THE INFINITESIMAL CAL-
53
CULUS VI.
31
THE BEGINNINGS OF THE APPLICATION OF MATHEMATICS TO NATURAL SCIENCE-THE SCIENCE OF DYNAMICS
V.
4 8 19
THE NATURE OF MATHEMATICS
62 67
BIBLIOGRAPHY
71
MODERN VIEWS OF LIMITS AND NUMBERS
INTRODUCTION AN eminent mathematician once remarked that he was never satisfied with his knowledge of a mathematical theory until he could explain it to the next man he met in the street. That is hardly exaggerated; however, we must remember that a satisfactory explanation entails duties on both sides. Anyone of us has the right to ask of a mathematician, "What is the use of mathematics?" Anyone may, I think and will try to show. rightly suppose that a satisfactory answer, if such an answer is anyhow possible, can be given in quite simple terms. Even men of a most abstract science, 4
The Nalllre 0/ MQthenuuics
such as mathematics or philosophy, are chiefly adapted for the ends of ordinary life; when they think, they think, at the bottom, Uke other men. They are often more highly trained, and have a technical facility for thinking that comes partly from practice and partly from the use of the contrivances for correct and rapid thought given by the signs and rules for dealing with them that mathematics and modem logic provide. But there is no real reason why, with patience, an ordinary person should not understand, speaking broadly, what mathematicians do, why they do it, and what, so far as we know at present, mathematics is. Patience, then, is what may rightly be demanded of the inquirer. And this really implies that the question is not merely a rhetorical one-an expression of irritation or scepticism put in the form of a question for the sake of some fancied effect. If Mr. A. dislikes the higher mathematics because he rightly perceives that they will not help him in the grocery business, he asks disgustedly, "What's the use of mathematics?" and does not wait for an answer, but turns his attention to grumbling at the lateness of his dinner. Now, we will admit at once that higher mathematics is of no more use in the grocery trade than the grocery trade is in the navigation of a ship; but that is no reason why we should condemn mathematics as entirely useless. I remember reading a speech made by an eminent surgeon, who wished, laudably enough, to spread the cause of elementary surgical instruction. "The higher mathematics," said he with great satisfaction to himself, "do not help you to bind up a broken leg!?t Obviously they do not; but it is equally obvious that surgery does not help us to add up accounts; .•• or even to think logically, or to accomplish the closely allied feat of seeing a joke. To the question about the use of mathematics we may reply by pointing out two obvious consequences of one of the applications of mathematics! mathematics prevents much loss of Ufe at sea, and increases the commercial prosperity of nations. Only a few men-a few intelligent philosophers and more amateur philosophers who are not highly intelligent-would doubt if these two things were indeed benefits. Still, probably, an of us act as if we thought that they were. Now, I do not mean that mathematicians go about with life-belts or serve behind counters; they do not usually do so. What I mean I will now try to explain. Natural science is occupied very largely with the prevention of waste of the labour of thought and muscle when we want to call up, for some purpose or other, certain facts of experience. Facts are sometimes quite useful. For instance, it is useful for a sailor to know the positions of the stars and sun on the nights and days when he is out of sight of land. Otherwise, he cannot find his whereabouts. Now, some people connected with a national institution publish periodically a Nautical Almanac which contains the positions of stars and other celestial things you see
6
Philip E. B. Jo.,,4IlIn
through telescope~ for every day and night years and years ahead. This Almanac. then, obviously increases the possibilities of trade beyond coasting-trade, and makes travel by ship, when land cannot be sighted, much safer; and there would be no Nautical Almanac if it were not for the science of astronomy; and there would be no practicable science of astronomy if we could not organise the observations we make of sun and moon and stars, and put hundreds of observations in a convenient form and in a little space----in short, if we could not economise our mental or bodily activity by remembering or carrying about two or three little formulre instead of fat books full of details; and, lastly, we could not economise this activity if it were not for mathematics. Just as it is with astronomy, so it is with all other sciences-both those of Nature and mathematical science: the very essence of them is the prevention of waste of the energies of muscle and memory. There are plenty of things in the un known parts of science to work our brains at, and we can only do so efficiently if we organise our thinking properly, and consequently do not waste our energies. The purpose of this little volume is not to give----like a text-book-a collection of mathematical methods and examples, but to do, firstly, what text-books do not do: to show how and why these methods grew up. All these methods are simply means, contrived with the conscious or unconscious end of economy of thought-labour, for the convenient handling of long and complicated chains of reasoning. This reasoning, when applied to foretell natural events, on the basis of the applications of mathematics, as sketched in the fourth chapter. often gives striking results. But the methods of mathematics, though often suggested by natural events, are purely logical. Here the word "logical" means something more than the traditional doctrine consisting of a series of extracts from the science of reasoning, made by the genius of Aristotle and frozen into a hard body of doctrine by the lack of genius of his school. Modern logic is a science which has grown up with mathematics, and, after a period in which it moulded itself on the model of mathematics, has shown that not only_ the reasonings but also conceptions of mathematics are logical in their nature. In this book I shall not pay very much attention to the details of the elementary arithmetic, geometry, and algebra of the many text-books, but shall be concerned with the discussion of those conceptions--such .as that of negative number-which are used and not sufficiently discussed in these books. Then, too, I shall give a somewhat full account of the development of an&lytical methods and certain examinations of principles. I hope that I shall succeed in showing that the process of mathematical discovery is a living and a growing thing. Some mathematiciahs have lived long lives full of calm and unwavering faith-for faith in mathematics, as
Tht Naturt of Mathtmatic$
1
I will show, has always been needed--some have lived short lives full of burning zeal, and so on; and in the faith of mathematicians there has been much error. Now we come to the second object of this book. In the historical part we shall see that the actual reasonings made by mathematicians in building up their methods have often not been in accordance with logical rules. How, then, can we say that the reasonings of mathematics are logical in their nature? The answer is that the one word "mathematics" is habitually used in two senses, and so, as explained in the last chapter, I have distinguished between "mathematics," the methods used to discover certain truths, and "Mathematics," the truths discovered. When we have passed through the stage of finding out, by external evidence or conjecture, how mathematics grew up with problems suggested by natural events, like the faUing of a stone, and then how something very abstract and intangible but very real separated out of these problems, we can tum our attention to the problem of the nature of Mathematics without troubling ourselves any more as to how, historically, it gradually appeared to us quite clearly that there is such a thing at al1 as Mathematics--something which exists apart from its application to natural science. History has an immense value in being suggestive to the investigator, but it is, 10gica1ly speaking, irrelevant. Suppose that you are a mathematician; what you eat wUl have an important inftuence on your discoveries, but you would at once see how absurd it would be to make, say, the momentous discovery that 2 added to 3 makes 5 depend on an orgy of mutton cutlets or bread and jam. The methods of work and daily 1ife of mathematicians, the connecting threads of suggestion that run through their work, and the inftuence on their work of the a1lied work of others, all interest the investigator be-cause these things give him examples of research and suggest new ideas to him; but these reasons are psychological and not logical. But it is as true as it is natura1 that we should find that the best way to become acquainted with new ideas is to study the way in which knowledge about them grew up. This, then, is what we will do in the first place, and it is here that I must bring my own views forward. Briefty stated, they are these. Every great advance in mathematics with which we shall be concerned here has arisen out of the needs shown in natural science or out of the need felt to connect together, in one methodically arranged whole, analogous mathematical processes used to describe different natura1 phenomena. The application of logic to our system of descriptions, which we may make either from the motive of satisfying an intellectual need (often as strong, in its way, as hunger) or with the practical end in view of satisfying ourselves that there are no hidden sources of error that may ultimately lead us astray in calculatinB future or past natural events, leads
Ph"'p E. B. JOunfldll
8
at once to those modem refinements of method that are regarded with disfavour by the old-fashioned mathematicians. In modem times appeared clearly-what had only been vaguely suspected before-the true nature of Mathematics. Of this I will try to give some account, and show that, since mathematics is logical and not psychological in its nature, all those petty questions-sometimes amusing and often tedious--of history, persons, and nations are irrelevant to Mathematics in itself. Mathematics has required centuries of excavation, and the process of excavation is not, of course, and never will be, complete. But we see enough now of what has been excavated clearly to distinguish be. tween it and the tools which have been or are used for excavation. This confUsion, it should be noticed, was never made by the excavators themselves, but only by some of the philosophical onlookers who reftected on what was being done. I hope and expect that our reftections will not lead to this confusion.
CHAPTER I THE GROWTH OF MATHEMATICAL SCIENCE IN ANCIENT TIMES
IN the history of the human race, inventions like those of the wheel, the lever, and the wedge were made very early-judging from the pictures on ancient Egyptian and Assyrian monuments. These inventions were made on the basis of an instinctive and unretJ.ecting knowledge of the processes of nature, and with the sole end of satisfaction of bodily needs. Primitive men had to build huts in order to protect themselves against the weather, and, for this purpose, had to lift and transport heavy weights, and so on. Later, by reftection on such inventions themselves, possibly for the purposes of instruction of the younger members of a tribe or the newly-joined members of a guild, these isolated inventions were classified according to some analogy. Thus we see the same elements occurring in the relation of a wheel to its axle and the relation of the ann of a lever to its fulcrumthe same weights at the same distance from the axle or fulcrum, as the case may be, exert the same power, and we can thus class both instruments together in virtue of an analogy. Here what we call "scientific" classification begins. We can well imagine that this pursuit of science is attractive in itself; besides helping us to communicate facts in a comprehensive, compact, and reasonably connected way, it arouses a purely intellectual interest. It would be foolish to deny the obvious importance to us of our bodily needs; but we must clearly realise two things:-(l) The intellectual need is very strong, and is as much a fact as hunger or thirst; sometimes it is even stronger than bodily needs-Newton, for instance,
TM NalllU of Mathematics
,
often forgot to take food when he was engaged with his discoveries. (2) Practical results of value often follow from the satisfaction of intellectual needs. It was the satisfaction of certain intellectual needs in the cases of Maxwell and Hertz that ultimately led to wireless telegraphy; it was the satisfaction of some of Faraday's intellectual needs that made the dynamo and the electric telegraph possible. But many of the results of strivings after intellectual satisfaction have as yet no obvious bearing on the satisfaction of our bodily needs. However, it is impossible to tell whether or no they will always be barren in this way. This gives us a new point of view from which to consider the question, UWhat is the use of mathematics?" To condemn branches of mathematics because their results cannot obviously be applied to some practical purpose is short-sighted. The formation of science is peculiar to human beings among animals. The lower animals sometimes, but rarely, make isolated discoveries, but never seem to reflect on these inventions in themselves with a view to rational classification in the interest either of the intellect or of the indirect furtherance of practical ends. Perhaps the greatest difference between man and the lower animals is that men are capable of taking circuitous paths for the attainment of their ends, while the lower animals have their minds so filled up with their needs that they try to seize the object they want, or remove that which annoys them, in a direct way. Thus, monkeys often vainly snatch at things they want, while even savage men use catapUlts or snares or the consciously observed properties of Bung stones. The communication of knowledge is the first occasion that compels distinct reBection, as everybody can still observe in himself. Further, that which the old members of a guild mechanically pursue strikes a new member as strange, and thus an impulse is given to fresh reBection and investigation. When we wish to bring to the knowledge of a person any phenomena or processes of nature, we have the choice of two methods: we may allow the person to observe matters for himself, when instruction comes to an end; or, we may describe to him the phenomena in some way, so as to save him the trouble of personally making anew each experiment. To describe an event-like the falling of a stone to the earth-in the most comprehensive and compact manner requires that we should discover what is constant and what is variable in the processes of nature; that we should discover the same law in the moulding of a tear and in the motions of the planets. This is the very essence of nearly all science, and we will return to this point later on. We have thus some idea of what is known as "the economical function of science." This sounds as if science were governed by the same laws as the management of a business; and so, in a way, it is. But whereas the
PlrJllp E. B. loJtrtl4ln
10
aims of a business are not, at least directly, concerned with the satisfaction of intellectual needs, science-including natural science, logic, and mathematics-uses business methods consciously for such ends. The methods are far wider in range, more reasonably thought out, and more intelligently applied than ordinary business methods, but the principle is the same. And this may strike some people as strange, but it is nevertheless true: there appears more and more as time goes on a great and compelling beauty in these business methods of science. The economical function appears most plainly in very ancient and modem science. In the beginning, all economy had in immediate view the satisfaction simply of bodily wants. With the artisan, and still more so with the investigator, the most concise and simplest possible knowledge of a given province of natural phenomena-a knowledge that is attained with the least intellectual expenditure-naturally becomes in itself an aim; but though knowledge was at first a means to an end, yet, when the mental motives connected therewith are once developed and demand their satisfaction, all thought of its original purpose disappears. It is one great object of science to replace, or save the trouble of making, experiments, by the reproduction and anticipation of facts in thought. Memory is handier than experience, and often answers the same purpose. Science is communicated by instruction, in order that one man may profit by the experience of another and be spared the trouble of accumulating it for himself; and thus, to spare the efforts of posterity, the experiences of whole generations are stored up in libraries. And further, yet another function of this economy is the preparation for fresh investigation. 1 The economical character of ancient Greek geometry is not so apparent as that of the modem algebraical sciences. We shall be able to appreciate this fact when we have gained some ideas on the historical development of ancient and modem mathematical studies. The generally accepted account of the origin and early development of geometry is that the ancient Egyptians were obJiged to invent it in order to restore the landmarks which had been destroyed by the periodical inundations of the Nile. These inundations swept away the landmarks in the valley of the river, and, by altering the course of the river, increased or decreased the taxable value of the adjoining lands, rendered a tolerably accurate system of surveying indispensable, and thus led to a systematic study of the subject by the priests. Proclus (412-485 A.D.), who wrote a summary of the early history of geometry, tells this story, which is also told by Herodotus, and observes that it is by no means strange that the invention of the sciences should have originated in practical needs, and that, further, the transition from perception with the senses to reflection, 1
C/. pp. S, 13, IS, 16,42, 71.
TII~ Natll'~
.of Matlumaticf
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and from reflection to knowledge, is to be expected. Indeed, the very name "geometryn-which is derived from two Greek words meaning measurement of the earth-seems to indicate that geometry was not indigenous to Greece, and that it arose from the necessity of surveying. For the Greek geometricians, as we shan see, seem always to have dealt with geometry as an abstract science-to have considered lines and circles and spheres and so on, and not the rough pictures of these abstract ideas that we see in the world around us-and to have sought for propositions which should be absolutely true, and not mere approximations. The name does not therefore refer to this practice. However, the history of mathematics cannot with certainty be traced back to any school or period before that of the Ionian Greeks. It seems that the Egyptians' geometrical knowledge was of a wholly practical nature. For example, the Egyptians were very particular about the exact orientation of their temples; and they had therefore to obtain with accuracy a north and south line, as also an east and west Hne. By observing the points on the horizon where a star rose and set, and taking a plane midway between them, they could obtain a north and south line. To get an east and west line, which had to be drawn at right angles to this, certain people were employed who used a rope ABCD, divided by knots or marks at Band C, so that the lengths AB, BC, CD were in the proportion 3 : 4: S. The length BC was placed along the north and south line, and pegs P and Q inserted at the knots Band C. The piece BA (keeping it stretched all the time) was then rotated round the peg P, and similarly the piece CD was rotated round the peg Q, until the ends A and D coincided; the point thus indicated was marked by a peg R. The result was to form a triangle PQR whose angle at P was a right angle, and the line PR would give an east and west line. A similar method is constantly used at the present time by practical engineers, and by gardeners in marking tennis courts, for measuring a right angle. This method seems also to have been known to the Chinese nearly three thousand years ago, but the Chinese made no serious attempt to classify or extend the few rules of arithmetic or geometry with which they were acquainted, or to explain the causes of the phenomena which they observed. The geometrical theorem of which a particular case is involved in the method just described is well known to readers of the first book of Euclid's Elements. The Egyptians must probably have known that this theorem is true for a right-angled triangle when the sides containing the right angle are equal, for this is obvious if a Boor be paved with tiles of that shape. But these facts cannot be said to show that geometry was then studied as a science. Our real knowledge of the nature of Egyptian geometry depends mainly on the Rhind papyrus. The ancient Egyptian papyrus from the collection of Rhind, which was
12
Philip 6. B. JOllrdtdll
written by an Egyptian priest named Ahmes considerably more than a thousand years before Christ, and which is now in the British Museum, contains a fairly complete applied mathematics, in which the measurement of figures and solids plays the principal part; there are no theorems properly so called; everything is stated in the form of problems, not in general terms, but in distinct numbers. For example: to measure a rectangle the sides of which contain two and ten units of length; to find the surface of a circular area whose diameter is six units. We find also in it indications for the measurement of solids, particularly of pyramids, whole and truncated. The arithmetical problems dealt with in this papyrus-which, by the way, is headed "Directions for knowing all dark things"--contain some very interesting things. In modem language, we should say that the first part deals with the reduction of fractions whose numerators are 2 to a sum of fractions each of whose numerators is 1. Thus %9 is stated to be the sum of %4, %S, ~74, and %s2. Probably Ahmes had no rule for forming the component fractions, and the answers given represent the accumulated experiences of previous writers. In one solitary case, however, he has indicated his method, for, after having asserted that % is the sum of % and %, he added that therefore two-thirds of one-fifth is equal to the sum of a half of a fifth and a sixth of a fifth, that is, to ~o + %0. That so much attention should have been paid to fractions may be explained by the fact that in early times their treatment presented considerable difficulty. The Egyptians and Greeks simplified the problem by reducing a fraction to the sum of several fractions, in each of which the numerator was unity, so that they had to consider only the various denominators: the sole exception to this rule being the fraction %. This remained the Greek practice until the sixth century of our era. The Romans, on the other hand, generally kept the denominator equal to twelve, expressing the fraction (approximately) as so many twelfths. In Ahmes' treatment of multiplication, he seems to have relied on repeated additions. Thus, to multiply a certain number, which we wi.lI denote by the letter "a," by 13, he first multiplied by 2 and got 2a, then he doubled the results and got 4a, then he again doubled the result and got Sa, and lastly he added together a, 4a, and Sa. Now, we have used the sign "a" to stand for any number: not a particular number like 3, but anyone. This is what Ahmes did, and what we learn to do in what we call "algebra." When Ahmes wished to find a number such that it, added to its seventh, makes 19, he symbolised the number by the sign we translate "heap." He had also signs for our "+," "_," and "=".2 Nowadays we can write Ahmes' problem as: Find the number x 2 In this book I shall take great care in distinguishing signs from what they signify. Thus 2 is to be distinguished from "2": by "2" I mean the sign, and the sign written without inverted commas indicates the thing signified. There has been, and is, much confusion, not only with beginners but with eminent mathematicians between a sign
rile Nat",. of "'stilelflQllcs
%
=
19. Ahmes gave the answer in the form 16 + % + ~. 7 We shall find that algebra was hardly touched by those Greeks who made of geometry such an important science, partly, perhaps, because the almost universal use of the abacus 8 rendered it easy for them to add and subtract without any knowledge of theoretical arithmetic. And here we must remember that the principal reason why Ahmes~ arithmetical problems seem so easy to us is because of our use from childhood of the system of notation introduced into Europe by the Arabs, who originaUy obtained it from either the Greeks or the Hindoos. In this system an integral number is denoted by a succession of digits, each digit representing the product of that digit and a power of ten, and the number being equal to the sum of these products. Thus, by means of the local value attached to nine symbols and a symbol for zero, any number in the decimal scale of notation can be expressed. It is important to realise that the long and strenuous work of the most gifted minds was necessary to provide us with simple and expressive notation which, in nearly all parts of mathematics, enables even the less gifted of us to reproduce theorems which needed the greatest genius to discover. Each improvement in notation seems, to the uninitiated, but a small thing: and yet, in a calculation, the pen sometimes seems to be more intelligent than the user. Our notation is an instance of that great spirit of economy which spares waste of labour on what is already systematised, so that all our strength can be concentrated either upon what is known but unsystematised, or upon what is un-
such that x
+-
13
known. Let us now consider the transformation of Egyptian geometry in Greek hands. Thates of Miletus (about 64O-S46 B.C.), who, during the early part of his life, was engaged partly in commerce and partly in public dairs, visited Egypt and first brought this knowledge into Greece. He discovered many things himself, and communicated the beginnings of many to his successors. We cannot form any exact idea as to how Thales presented his geometrical teaching. We infer, however, from Proclus that it consisted of a number of isolated propositions which were not arranged in a logical sequence. but that the proofs were deductive, so that the theorems were not a mere statement of an induction from a large number of special instances, as probably was the case with the Egyptian geometri· and what is signified by it. Many have even maintained that numbers are the sign.r used to represent them. Often, for the sake of brevity, I shall use the word in inverted commas (say "an) as short for "what we call 'a,' n bnt the context will make plain what is meant. 8 The principle of the abacus is that a number is represented by connters in a series of grooves, Or beads strung on parallel wires; as many counters beins put on the first groove as there are units. as many on the second as there are tens, and so on. The rules to be fonowed in addition, subtraction, multiplication, and division are given in various old works on arithmetic.
14
PIIIllp E. B.
'0""""'"
cians. The deductive character which he thus gave to the science is his chief claim to distinction. Pythagoras (born about 580 B.C.) changed geometry into the form of an abstract science, regarding its principles in a purely abstract manner, and investigated its theorems from the immaterial and intellectual point of view. Among the successors of these men, the best known are Archytas of Tarentum (428-347 B.C.), Plato (429348 B.C.), Hippocrates of Chios (born about 470 B.C.), Menaechmus (about 375-325 B.C.), Euclid (about 330-275 B.C.), Archimedes (287212 B.C.), and Apollonius (260-200 B.C.). The only geometry known to the Egyptian priests was that of surfaces, together with a sketch of that of solids, a geometry consisting of the knowledge of the areas contained by some simple plane and solid figures, which they had obtained by actual trial. Thales introduced the ideal of establishing by exact reasoning the relations between the different parts of a figure, so that some of them could be found by means of others in a manner strictly rigorous. This was a phenomenon quite new in the world, and due, in fact, to the abstract spirit of the Greeks. In connection with the new impulse given to geometry, there arose with Thales, moreover, scientific astronomy, also an abstract science, and undoubtedly a Greek creation. The astronomy of the Greeks differs from that of the Orientals in this respect: the astronomy of the latter, which is a1together concrete and empirical, consisted merely in determining the duration of some periods or i.n indicating, by means of a mechanical process, the motions of the sun and planets~ whilst the astronomy of the Greeks aimed at the discovery of the geometrical laws of the motions of the heavenly bodies. Let us consider a simple case. The area of a right-angled field of length 80 yards and breadth 50 yards is 4000 square yards. Other fields which are not rectangular can be approximately measured by mentally dissecting them-a process which often requires great ingenuity and is a familiar problem to land-surveyors. Now, let us suppose that we have a circular field to measure. Imagine from the centre of the circle a large number of radii drawn, and let each radius make equal angles with the next ones on each side of it. By joining the points in succession where the radii meet the circumference of the circle, we get a large number of triangles of equal area, and the sum of the areas of all these triangles gives an approximation to the area of the circle. It is particularly instructive repeatedly to go over this and the following examples mentally, noticing how helpful the abstract ideas we call "straight line," "circle," "radius," "angle," and so on, are. We all of us know them, recognise them, and can easily feel that they are trustworthy and accurate ideas. We feel at home, so to speak, with the idea of a square, say, and can at once give details about it which are exactly true for it, and very nearly true for a field which we know is
very nearly a square. This replacement in thought by an abstract geometrical object economises labour of thinking and imagining by leading us to concentrate our thoughts on that alone which is essential for our pur-
pose. Thales seems to have discovered-and it is a good thing to follow these discoveries on figures made with the help of compasses and ruler-the proof of what may be regarded as the obvious fact that the circle is divided into halves by its diameter, that the aDgles at the base of a triangle with two equal sides-an "isosceles" trian.e--are equal, that all the triangles described in a semi-circle with two of their angular points at the ends of the diameter and the third anywhere on the circumference contain a right angle, and he measured the distance of vessels from the shore, presumably by causing two observers at a known distance apart to measure the two angles formed by themselves and the ship. This last discovery is an application of the fact that a triangle is determined if its base and base angles are given. When Archytas and Menaechmus employed mechanical instruments for solving certain geometrical problems, "Plato," says Plutarch, "inveighed against them with great indignation and persistence as destroying and perverting all the good there is in geometry; for the method absconds from incorporeal and intellectual to sensible things, and besides employs again such bodies as require much vulgar handicraft: in this way mechanics was dissimilated and expelled from geometry, and, being for a long time looked down upon by philosophy, became one of the arts of war." In fact, manual labour was looked down upon by the Greeks, and a sharp distinction was drawn between the slaves, who performed bodily work and really observed nature, and the leisured upper classes, who speculated .and often only knew nature by hearsay. This explains much of the naive, hazy, and dreamy character of ancient natural science. Only seldom did the impulse to make experiments for oneself break through; but when it did, a great progress resulted, as was the case with Archytas and Archimedes. Archimedes, like Plato, held that it was undesirable for a philOSOpher to seek to apply the results of science to any practical use; but, whatever might have been his view of what ought to be the case, he did actually introduce a large number of new inventions. We will not consider further here the development of mathematics with other ancient nations, nor the chief problems investigated by the Greeks; such details may be found in some of the books mentioned in the Bibliography at the end. The object of this chapter is to indicate the nature of the ~ence of geometry, and how certain practical needs gave rise to investigations in which appears an abstract science which was worthy of being cultivated for its own sake, and which incidentally gave rise to advantages of a practical nature.
16
PIIIUp E. B. lourtl4ltl
There are two branches of mathematics which began to be cultivated by the Greeks, and which allow a connection to be formed between the spirits of ancient and modem mathematics. The first is the method of geometrical analysis to which Plato seems to have directed attention. The analytical method of proof begins by assuming that the theorem or problem is solved, and thence deducing some result. If the result be false, the theorem is not true or the problem is incapable of solution: if the result be true, if the steps be reversible, we get (by reversing them) a synthetic proof; but if the steps be not reversible, no conclusion can be drawn. We notice that the leading thought in analysis is that which is fundamental in algebra, and which we have noticed in the case of Abmes: the calculation Or reasoning with an unknown entity, which is denoted by a convention a] sign, as if it were known, and the deduction at last, of some relation which determines what the entity must be. And this brings us to the second branch spoken of: algebra with the later Greeks. Diophantus of Alexandria, who probably lived in the early half of the fourth century after Christ, and probably was the original inventor of an algebra, used letters for unknown quantities in arithmetic and treated arithmetical problems analytically. Juxtaposition of symbols represented what we now write as "+," and "-" and "=" were also represented by symbols. All these symbols are mere abbreviations for words, and perhaps the most important advantage of symbolism-the power it gives of carrying out a complicated chain of reasoning almost mechanically-was not made much of by Diophantus. Here again we come across the economical value of symbolism: it prevents the wearisome expenditure of mental and bodily energy on those processes which can be carried out mechanically. We must remember that this economy both emphasises the unsubjugated-that is to say, unsystematised-problems of science, and has a charm-an eesthetic charm, it would seem--of its own. Lastly, we must mention "incommensurables,,' "loci'" and the beginnings of ''trigonometry.'' Pythagoras was, according to Eudemus and Proclus, the discoverer of "incommensurable quantities." Thus, he is said to have found that the diagonal and the side of a square are "incommensurable." Suppose, for example, that the side of the square is one unit in length; the diagonal is longer than this, but it is not two units in length. The excess of the length of the diagonal over one unit is not an integral submultiple of the unit. And, expressing the matter arithmetically, the remainder that is left over after each division of a remainder into the preceding divisor is not an integral submultiple of the remainder used as divisor. That is to say, the rule given in text-books on arithmetic and algebra for finding the greatest
17
The Nature 0/ MatMmatics
common measure does not come to an end. This rule, when applied to integer numbers, always comes to an end; but, when applied to certain lengths, it does not. Pythagoras proved, then, that if we start with a line of any length, there are other lines whose lengths do not bear to the first length the ratio of one integer to another, no matter if we have all the integers to choose from. Of course, any two fractions have the ratio of two integers to one another. In the above case of the diagonal, if the diagonal were in length some number x of units, we should have x 2 2, and it can be proved that no fraction, when "multiplied"-in the sense to be given in the next chapter-by itself gives 2 exactly, though there are fractions which give this result more and more approximately. On this account, the Greeks drew a sharp distinction between "numbers," and "magnitudes" or "quantities" or measures of lengths. This distinction was gradually blotted out as people saw more and more the advantages of identifying numbers with the measures of lengths. The invention of analytical geometry, described in the third chapter, did most of the blotting out. It is in comparatively modem times that mathematicians have adequately realised the importance of this logically valid distinction made by the Greeks. It is a curious fact that the abandonment of strictly logical thinking should have led to results which transgressed what was then known of logic, but which are now known to be readily interpretable in the terms of what we now know of Logic. This subject will occupy us again in the sixth chapter. The question of loci is connected with geometrical analysis, and is difficult to dissociate from a mental picture of a point in motion. Think of a point under restrictions to move only in some curve. Thus, a point may move so that its distance from a fixed point is constant; the peak of an angle may move so that the arms of the angle pass-slipping-through two fixed points, and the angle is always a right angle. In both cases the moving point keeps on the circumference of a certain circle. This curve is a "locus." It is evident how thinking of the locus a point can describe may help us to solve problems.
=
We have seen that Thales discovered that a triangle is determined if its base and base angles are given. When we have to make a survey of either an earthly country or part of the heavens, for the purpose of mapmaking, we have to measure angles-for example, by turning a sight, like those used on guns, through an angle measured in a circular arc of metal -to fix the relative directions of the stars or points on the earth. Now, for terrestrial measurements, a piece of country is approximately a flat surface, while the heavens are surveyed as if the stars were, as they seem to be, scattered on the inside of a sphere at whose centre we are. Secondly, it is a network of triangles-plane or spherical~f which we
II
PIUUp E. B. lo'lll'dlll:lt
measure the angles and sometimes the sides: for, if the angles of a triangle are known, the proportionality of the sides is known; and this proportionality cannot be concluded from a knowledge of the angles of a rectangle, say. Hipparchus (born about 160 B.C) seems to have invented this practical science of the complete measurement of triangles from certain data, or, as it is called, "trigonometry," and the principles laid down by him were worked out by Ptolemy of Alexandria (died 168 A.D.) and also by the Hindoos and Arabians. Usually, only angles can be measured with accuracy, and so the question arises: given the magnitude of the angles, what can be coneluded as to the kind of proportionality of the sides. Think of a circle described round the centre O. and let AP be the arc of this circle which measures the angle A OP. Notice that the ratio of AP to the radius is the same for the angle A OP whatever value the radius may have. Draw PM perpendicular to OA. Then the figure OPMAP reminds one of a stretched bow, and hence are derived the names "sine of the arc Apt' for the line PM, and "cosine" for OM. Tables of sines and cosines of arcs (or of angles, since the arc fixes the angle if the radius is known) were drawn up, and thus the sides PM and OM could be found in terms of the radius, when the arc was known. It is evident that this contains the essentials for the finding of the proportions of the sides of plane triangles. Spherical trigonometry contains more complicated relations which are directly relevant to the position of an astronomer and his measurements. Mathematics did not progress in the hands of the Romans: perhaps the genius of this people was too practical. Still, it was through Rome that mathematics came into medieval Europe. The Arab mathematical textbooks and the Greek books from Arab translations were introduced into Western Europe by the Moors in the period 1150-1450, and by the end of the thirteenth century the Arabic arithmetic had been fairly introduced into Europe, and was practised by the side of the older arithmetic founded on the work of Boethius (about 475-526), Then came the Renascence. Mathematicians had barely assimilated the knowledge obtained from the Arabs, including their translations of Greek writers, when the refugees who escaped from Constantinople after the fall of the Eastern Empire (1453) brought the original works and the traditions of Greek science into Italy. Thus by the middle of the fifteenth century the chief results of Greek and Arabian mathematics were accessible to European students. The invention of printing about that time rendered the dissemination of discoveries comparatively easy.
19
Tile Nature 01 MatllerruUlC$
CHAPTER II TIlE RISE AND PROGRESS OF MODERN MATHEMATICS-ALGEBRA
MODERN mathematics may be considered to have begun approximately with the seventeenth century. It is well known that the first 1500 years of the Christian era produced, in Western Europe at least, very little knowledge of value in science. The spirit of the Western Europeans showed itself to be different from that of the ancient Greeks, and only slightly less SO from that of the more Easterly nations; and, when Western mathematics began to grow, we can trace clearly the historical beginnings of the use, in a not quite accurate form, of those conceptions-variable and function-which are characteristic of modem mathematics. We may say, in anticipation, that these conceptions, thoroughly analysed by reasoning as they are now, make up the difference of our modem views of Mathematics from, and have caused the likeness of them to, those of the ancient Greeks. The Greeks seem, in short, to have taken up a very similar posi.. tion towards the mathematics of their day to that which logic forces us to take up towards the far more general mathematics of to-day. The generality of character has been attained by the effort to put mathematics more into touch with natural sciences--in particular the science of motion. The main difficulty was that, to reach this end, the way in which mathematicians expressed themselves was illegitimate. Hence philosophers, who Jacked the real sympathy that must inspire all criticism that hopes to be relevant, never could discover any reason for thinking that what the mathematicians said was true, and the world had to wait until the mathematicians began logically to analyse their own conceptions. No body of men ever needed this sympathy more than the mathematicians from the revival of letters down to the middle of the nineteenth century t for no science was less logical than mathematics. The ancient Greeks never used the conception of motion in their systematic works. The idea of a locus seems to imply that some curves could be thought of as generated by moving points; the Greeks discovered some things by helping their imaginations with imaginary moving points, but they never introduced the use of motion into their final proofs. This may have been because the Eleatic school, of which one of the principal representatives was Zeno (495-435 B.C.), invented some exceedingly subtle puzzles to empbasize the difficulty there is in the conception of motion. We shall return in some detail to these puzzles, which have not been appreciated in all the ages from the time of the Greeks till quite modem times. Owing to this lack of subtlety, the conception of variability was freely introduced into mathematics. It was the conceptions of constant,
20
Philip
Eo B. lourdaln
variable, and function, of which we shall, from now on, often have occa· sion to speak, which were generated by ideas of motion, and which, when they were logically purified, have made both modem mathematics and modem logic, to which they were transferred by mathematical logicians-Leibniz. Lambert, Boole, De Morgan, and the numerous successors of Boole and De Morgan from about 1850 onwards-into a science much more general than, but bearing some close analogies with, the ideal of Greek mathematical science. Later on will be found a discussion of what can be meant by a "moving point."
Let us now consider more closely the history of modern mathematics. Modem mathematics, like modem philosophy and like one part-the speculative and not the experimental part--of modem physical science, may be considered to begin with Rene Descartes (1 596--1650). Of course, as we should expect, Descartes had many and worthy predecessors. Perhaps the greatest of them was the French mathematician Fran~ois Viete (l 540-1603), better known by his Latinized name of "Vieta." But it is simpler and shorter to confine our attention to Descartes. Descartes always plumed himself on the independence of his ideas, the breach he made with the old ideas of the Aristotelians, and the great clearness and simplicity with which he described his ideas. But we must not underestimate the part that "ideas in the air" play; and, fUrther, we know now that Descartes' breach with the old order of things was not as great as he thought. Descartes, when describing the effect which his youthful studies had upon him when he came to reflect upon them, said: "I was especially delighted with the mathematics, on account of the certitude and evidence of their reasonings: but 1 had not as yet a precise knowledge of their true use; and, thinking that they but contributed to the advancement of the mechanical arts, 1 was astonished that founda· tions so strong and solid should have had no loftier superstructure reared on them." And again: "Among the branches of philosophy, 1 had, at an earlier period, given some attention to logic, and, among those of the mathematics, to geometri· cal analysis and algebra-three arts or sciences which ought, as 1 con· ceived, to contribute something to my design. But, on examination, 1 found that, as for logic. its syllogisms and the majority of its other precepts are of avail rather in the communication of what we already know, or even in speaking without judgment of things of which we are ignorant, than in the investigation of the unknown: and although this science con· tains indeed a number of correct and very excellent precepts. there are, neverthe1ess, so many others, and these either injurious or superfluous,
Tile Nat",. oj MatllelrUltic$
21
mingled with the former, that it is almost quite as difficult to effect a severance of the true from the false as it is to extract a Diana or a Minerva from a rough block of marble. Then as to the analysis of the ancients and the algebra of the modems: besides that they embrace only matters highJy abstract, and, to appearance, of no use, the former is so exclusively restricted to the consideration of figures that it can exercise the understanding only on condition of greatly fatiguing the imagination; and, in the latter, there is so complete a subjection to certain rules and formulas, that there results an art full of confusion and obscurity, calculated to embarrass, instead of a science fitted to cultivate the mind. By these considerations I was induced to seek some other method which would comorise the advantages of the three and be exempt from their defects. . . • ''The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstrations had Jed me to imagine that all things to the knowledge of which man is competent are mutually connected in the same way, and that there is nothing so far removed from us as to be beyond our reach, or so hidden that we cannot discover it, provided only that we abstain from accepting the false for the true, and always preserve in our thoughts the order necessary for the deduction of one truth from another. And I had little difficulty in determining the objects with which it was necessary to begin, for I was already persuaded that it must be with the simplest and easiest to know. and, considering that, of all those who have hitherto sought truth in the sciences, the mathematicians alone have been able to find any demonstrations, that is, any certain and evident reasons, I did not doubt but that such must have been the rule of their investigations. I resolved to begin, therefore, with the examination of the simplest objects, not anticipating, however, from this any other advantage than that to be found in accustoming my mind to the love and nourishment of truth and to a distaste for all such reasonings as were unsound. But I had no intention on that account of attempting to master all the particular sciences commonly denominated 'mathematics'; but observing that, however different their objects, they all agree in considering only the various relations or proportions subsisting among those objects, I thought it best for my purpose to consider these proportions in the most general form possible, without referring them to any objects in particular, except such as would most facilitate the knowledge of them, and without by any means restricting them to these, that afterwards I might thus be the better able to apply them to every other class of objects to which they are legitimately applicable. Perceiving further that, in order to understand these relations, I should have sometimes to consider them one by one and sometimes only to bear in mind or embrace them in the aggregate, I thought that, in order
22
P1Illip E. B. lollnlllbi
the better to consider them individually, I should view them as subsisting between straight lines, than which I could find no objects more simple or capable of being more distinctly represented to my imagination and senses; and, on the other hand, that, in order to retain them in the memory, or embrace an aggregate of many, I should express them by certain characters the briefest possible. In this way I believed that I could borrow all that was best both in geometrical analysis and in algebra, and correct all the defects of the one by help of the other." Let us, then, consider the characteristics of algebra and geometry. We have seen, when giving an account, in the first chapter, of the works of Ahmes and Diophantus, that mathematicians early saw the advantage of representing an unknown number by a letter or some other sign that may denote various numbers ambiguously, writing down-much as in geometrical analysis-the relations which they bear, by the conditions of the problem, to other numbers, and then considering these relations. If the problem is determinate-that is to say, if there are one or more definite solutions which can be proved to involve only numbers already fixed upon-this consideration leads, by the use of certain rules of calculation, to the determination-actual or approximate--of this solution or solutions. Under certain circumstances, even if there is a solution, depending on a variable, we can find it and express it in a quite general way, by rules, but that need not occupy us here. Suppose that you know my age, but that I do not know yours, but wish to. You might say to me: "I was eight years old when you were born." Then I should think like this. Let x be the (unknown) number of years in your age at this moment and, say, 33 the number of years in my age at this moment; then in essentials your statement can be translated by the equation "x - 8 = 33." The meaning of the signs "_," "=," and "+" are supposed to be known-as indeed they are by most people nowadays quite sufficiently for our present purpose. Now, one of the rules of algebra is that any term can be taken from one side of the sign "=" to the other if only the "+" or "-" belonging to it is changed into ,,_to or "+," as the case may be. Thus, in the present case, we have: "x 33 + 8 41." This absurdly simple case is chosen intentionally. It is essential in mathematics to remember that even apparently insignificant economies of thought add up to make a long and complicated calculation readily performed. This is the case, for example, with the convention introduced by Descartes of using the last letters of the alphabet to denote unknown numbers, and the first letters to denote known ones. This convention is adopted, with a few exceptional cases, by algebraists to-day, and saves much trouble in explaining and in looking foe unknown and known quantities in an equation. Then, again, the signs "+," "-," "=" have great
=
=
23
The Nalure of MalMWUJIlcJ
merits which those unused to long calculations cannot readily understand. Even the saving of space made by writing "xy" for "x X y" ("x multiplied by y") is important, because we can obtain by it a shorter and more readily surveyed formula. Then. too. Descartes made a general practice of writing "powers" or "exponents" as we do now; thus "xS" stands for "xxx" and "x5 " for some less suggestive symbol representing the continued mUltiplication of five x's. One great advantage of this notation is that it makes the explanation of logarithms, which were the great and laborious discovery of John Napier (1550-1617), quite easy. We start from the equation "xmx" xm+"':' Now, if x P = y, and we call p the "logarithm of y to the base x"; in signs: up lo&rY"; the equation from which we started gives, if we denote xm by "u" and x" by "v,n so that m log.ru and n )og".v, that logz (uv) loglt'u + logzv. Thus, if the logarithms of numbers to a given base (say x 10) are tabulated, calculations with large numbers are made less arduous, for addition replaces multiplication, when logarithms are found. Also subtraction of logarithms gives the logarithm of the quotient of two numbers. Let us now shortly consider the history of algebra from Diophantus to Descartes.
=
=
=
=
=
=
The word "algebra" is the European corruption of an Arabic phrase which means restoration and reduction-the first word referring to the fact that the same magnitude may be added to or subtracted from both sides of an equation, and the last word meaning the process of simplification. The science of algebra was brought among the Arabs by Mohammed ben Musa (Mahomet. the son of Moses), better known as Alkarismi, in a work written about 830 A.D., and was certainly derived by him from the Hindoos. The algebra of Alkarismi holds a most important place in the history of mathematics, for we may say that the subsequent Arab and the early medieval works on algebra were founded on it, and also that thrOUgh it the Arabic or Indian system of decimal numeration was introduced into the West. It seems that the Arabs were quick to appreciate the work of others--notably of the Greek masters and of the Hindoo mathematicians--but, like the ancient Chinese and Egyptians, they did not systematically develop a subject to any considerable extent. Algebra was introduced into Italy in 1202 by Leonardo of Pisa (about 1175-1230) in a work based on Alkarismi's treatise, and into England by Robert Record (about 1510-1558) in a book called the Whetstone oj Witte published in 1557. Improvements in the method or notations of algebra were made by Record, Albert Girard ( 1595-1632) , Thomas Harriot (1560-1621), Descartes, and many others.
24
PlIIlIp E. B. lol1n111bi
In arithmetic we use symbols of number. A symbol is any sign for a quantity, which is not the quantity itself. If a man counted his sheep by pebbles. the pebbles would be symbols of the sheep. At the present day, when most of us can read and write, we have acquired the convenient habit of using marks on paper, "I, 2. 3, 4," and so on, instead of such things as pebbles. Our "I + 1" is abbreviated into "2," "2 + 1" is abbreviated into "3," "3 + 1" into "4," and so on. When "1," "2," "3," &c., are used to abbreviate, rather improperly, "1 mile," "2 miles," "3 miles," &c., for instance, they are called signs for concrete numbers. But when we shake off all idea of "1," "2," &c., meaning one, two, &c., of anything in particular, as when we say, "six and four make ten," then the numbers are called abstract numbers. To the latter the learner is first introduced in treatises on arithmetic, and does not always learn to distinguish rightly between the two. Of the operations of arithmetic only addition and subtraction can be performed with concrete numbers, and without speaking of more than one sort of 1. Miles can be added to miles, or taken from miles. Multiplication involves a new sort of 1, 2, 3, &c., standing for repetitions (or times, as they are called). Take 6 miles 5 times. Here are two kinds of units, 1 mile and 1 time. In multiplication, one of the units must be a number of repetitions or times, and to talk of multiplying 6 feet by 3 feet would be absurd. What notion can be formed of 6 feet taken "3 feet" times? In solving the following question, "If 1 yard cost 5 shillings, how much will 12 yards cost?" we do not multiply the 12 yards by the 5 shillings; the process we go through is the following: Since each yard costs 5 shillings, the buyer must put down 5 shillings as often (as many times) as the seller uses a one-yard measure; that is, 5 shillings is taken 12 times. In division we must have the idea either of repetition or of partition, that is, of cutting a quantity into a number of equal parts. "Divide 18 miles by 3 miles" means, find out how many times 3 miles must be repeated to give 18 miles: but "divide 18 miles by 3" means, cut 18 miles into 3 equal parts, and find how many miles are in each part. The symbols of arithmetic have a determinate connection; for instance, 4 is always 2 + 2, whatever the things mentioned may be, miles, feet, acres, &c. In algebra we take symbols for numbers which have no determinate connection. As in arithmetic we draw conclusions about 1, 2, 3, &c., which are equally true of 1 foot, 2 feet, &c., 1 minute, 2 minutes, &c.; so in algebra we reason upon numbers in general, and draw conclusions which are equally true of all numbers. It is true that we also use, in kinds of algebra which have been developed within the last century, letters to represent things other than numbers--for example, classes of individuals with a certain property, such as "homed animals," for logical purposes; or certain geometrical or physical things with directions in space, such as "forces"-and signs like "+" and "-" to represent ways of combination
Th~ NaluT~
01
MtlIh~matlclJ
of the things, which are analogous to, but not identical with, addition and subtraction. If "au denotes "the class of homed anima1s" and "b" denotes "the class of beasts of burden, the sign "ab" has been used to denote "the class of homed beasts of burden." We see that here ab = ba, just as in the multiplication of numbers, and the above operation has been called, partly for this reason, "logical mUltiplication," and denoted in the above way. Here we meet the practice of mathematicians-and of all scientific men--of using words in a wider sense for the sake of some analogy. This habit is a11 the more puzz1ing to many people because mathematicians are often not conscious that they do it, or even talk sometimes as if they thought that they were general ising conceptions instead of words. But. when we talk of a "family tree," we do not indicate a widening of our conception of trees of the roadside. We shall not need to consider these modern algebras, but we shall be constantly meeting what are called the "generalisations of number" and transference of methods to analogous cases. Indeed, it is hardly too much to say that in this lies the very spirit of discovery. An example of this is given by the extension of the word "numbers" to include the names of fractions as well. The occasion for this extension was given by the use of arithmetic to express such quantities as distances. This had been done by Archimedes and many others, and had become the usual method of procedure in the works of the mathematicians of the sixteenth century, and plays a great part in Descartes' work. Mathematicians, ever since they began to apply arithmetic to geometry, became alive to the fact that it was convenient to represent points on a straight line by numbers, and numbers by points on a straight line. What is meant by this may be described as fol1ows. If we choose a unit of length, we can mark off points in a straight line corresponding to 0 units -which means that we select a point. called "the origin," to start from,t unit, 2 units, 3 units, and so on, SO that "the point m,t> as we will call it for short, is at a distance of m units from the origin. Then we can divide up the line and mark points corresponding to the fractions lh, ~, %, ~3, %, or the point between 1 and 2 which is the same distance from 1 as % is from 0, and so on. Now, there is nothing here to distinguish fractions from numbers. Both are treated exactly in the same way; the results of addition, subtraction, multiplication, and division 4 are interpretable, in much the same way as new points whether the "au and "b in Of
tt
4 The operation of what is called, for the sake of analogy, "multiplication" of fractions is defined in the manner indicated in the following example. If % of a yard costs 10 X4 X 7 pence, and we de10d., how much does ~ of a yard cost? The answer is 4X7 1 3X8 fine - - as - "multiplied by" %. by analogy with what would happen if % were 1 3X8 " and 'Jt. were, say, 3.
PIIIU, E. B. I"",..
cea
+ btU
ua
- b," "ab," and so on, stand for numbers or fractions, and
we bave, for example, a + b b + a, ab ba, a (b + c) ab + ac, always. Because of this very strong analogy, mathematicians have called the fractions "numbers" too, and they often speak and write of "generalisations of numbers," of which this is the first example, as if the conception of number were generalised, and not merely the ,",me "number," in virtue of a great and close and important analogy. When once the points of a line were made to represent numbers, there seemed to be no further difficulty in admitting certain "irrational numbers" to correspond to the end-points of the incommensurable lines which bad been discovered by the Greeks. This question will come up again at a later stage: there are necessary discussions of principle involved, but mathematicians did not go at all deeply into questions of principle until fairly modem times. Thus it has happened that, until the last sixty years or so, mathematicians were nearly all bad reasoners, as Swift remarked of the mathematicians of Laputa in Gulliver3 Travels, and were unpardonably hazy about first principles. Often they appealed to a sort of faith. To an intelligent and doubting beginner, an eminent French mathematician of the eighteenth century said: "Go on, and faith will come to you." It is a curious fact that mathematicians have so often arrived at truth by a sort of instinct. Let us now return to our numerical algebra. Take, say, the number 8, and the fraction, which we will now call a "number" also, 'At. Add 1 to both; the greater contains the less exactly 8 times. Now this property is possessed by any number, and not 8 alone. In fact, if we denote the num-
=
=
=
ber we start with by "a," we have, by the rules of algebra,
a+l l/G
+1
=
Q.
This is an instance of a general property of numbers proved by algebra. Algebra contains many rules by which a complicated algebraical expression can be reduced to its simplest terms. Owing to the suggestive and compact notation, we can easily acquire an almost mechanical dexterity in dealing with algebraical symbols. This is what Descartes means when he speaks of algebra as not being a science fitted to cultivate the mind. On the other hand, this art is due to the principle of the economy of thought, and the mechanical aspect becomes, as Descartes foresaw, very valuable if we could use it to solve geometrical problems without the necessity of fatiguing our imaginations by long reasonings on geometrical figures. I have already mentioned that the valuable notation "x-" was due to Descartes. This was published, along with all his other improvements in algebra. in the third part of his Geometry of 1637. I shall speak in the next chapter of the great discovery contained in the first two parts of this
work; here I will resume the improvements in notation and method made by Descartes and his predecesso~ which make the algebraical part of the Geometry very like a modem book 011 algebra. It is still the custom in arithmetic to indicate addition by juxtaposition: thus ''2lh'' means "2 + lh." In algebra, we always. nowadays. indicate addition by the sign "+" and multiplication by juxtaposition or, more rarely, by putting a dot or the sign "x" between the signs of the numbers to be mu1tiplied. Subtraction is indicated by "-". Here we must digress to point out-what is often, owing to confusion of thought, denied in text-books-that, where a" and "b" denote numbers, "a - b" can only denote a number if a is equal to or greater than b. If a is equal to b, the number denoted is zero; there is really no good reason for denying, say, that the numbers of Charles ll.'s foolish sayings and wise deeds are equal, if a weU-known epitaph be true. Here again we meet the strange way in which mathematics has developed. For centuries mathematicians used "negative" and "positive" numbers. and identified "positive" numbers with signless numbers like 1, 2, and 3, without any scruple, just as they used fractionary and irrational "numbers.I t And when logically-minded men objected to these wrong statements, mathematicians simply ignored them or said: "Go on; faith will come to you." And the mathematicians were right, and merely could not give correct reasoDl-or at least always gave wrong ones-for what they did. We have. over again, the fact that criticism of the mathematicians' procedure, if it wishes to be relevant, must be based on thorough sympathy and understanding. It must try to account for the rightness of mathematical views, and bring them into conformity with logic. Mathematicians themselves never found a competent philosophical interpreter, and so nearly all the interesting part of mathematics was left in obscurity until, in the latter half of the nineteenth century, mathematicians themselves began to cultivate phUosophy-or rather lOgic. Thus we must go out of the historical order to explain what "negative numbers" means. First, we must premise that when an aIgebraicaI expression is enclosed in brackets, it signifies that the whole result of that expression stands in the same relation to surrounding symbols as if it were one letter only. Thus, "a - (b - c)" means that from a we are to take b - c, or what is left after taking c from b. It is not, therefore, the same as a - b - c.1n fact we easily find that a - (b - c) is the same as a - b + c. Note also that "(0 + b) (c + d)" means (a + b) multiplied by (c + d). Now, suppose a and b are numbers, and a is greater than b. Let a - b be c. To get c from a, we carry out the operation of taking away b. This operation, which is the fulfilment of the order: "Subtract b," is a "negative number." Mathematicians call it a "number" and denote it by "_btt simply because of analogy: the same rules for calculation hold for "negaU
PId", E. B. lourliUlt
28
tive numbers" and "positive numbers" like "+b," whose meaning is now clear too, as do for our signless numbers; when "addition," "subtraction," &c., are redefined for these operations. The way in which this redefinition must take place is evident when we represent integers, fractions, and positive and negative numbers by points on a straight line. To the right of 0 are the integers and fractions, to the left of 0 are the negative numbers, and to the right of 0 stretch the series of positive numbers, +a coinciding with a and being symmetrically placed with -a as regards O. Also we determine that the operations of what we call "addition," &c., of these new "numbers" must lead to the same results as the former operations of the same name. Thus the same symbol is used in diJferent senses, and we write a + b - b a + 0 (+a) + (+b) + (-b) +a a. This is a remarkable sequence of quick changes. We have used the sign of equality, "=". It means originally, "is the same as." Thus 3 + 1 = 4. But we write, by the above convention, "a +a," and so we sacrifice exactness, which sometimes looks rather pedantic. for the sake of keeping our analogy in view, and for brevity. Let us bear this, at first sight, puzzling but, at second sight, justifiable peculiarity of mathematicians in mind. It has always puzzled intelligent beginners and philosophers. The laws of calculation and convenient symbolism are the things a mathematician thinks of and aims at. He seems to identify different things if they both satisfy the same laws which are important to him, just as a magistrate may think that there is not much diJference between Mr. A., who is red-haired and a tinker and goes to chapel, and Mr. B., who is a brown-haired horse-dealer and goes to church, if both have been found out committing petty larceny. But their respective ministers of religion or wives may still be able to distinguish them. Any two expressions connected by the sign of equality form an Uequation." Here we must notice that the words "Solve the equation x 2 + ax = b" mean: find the value or values of x such that, a and b being given numbers, x 2 + ax becomes b. Thus, if a 2 and b -I, the solution is x -1. As we saw above, Descartes fixed the custom of employing the letters at the beginning of the alphabet to denote known quantities, and those at the end of the alphabet to denote unknown quantities. Thus, in the above example, a and b are some numbers supposed to be given, while x is sought. The question is solved when x is found in terms of a and band fixed numbers (like 1, 2, 3); and so, when to a anCt b are attributed any fixed values, x becomes fixed. The signs "a" and "b" denote ambiguously. not uniquely like "2" does; and "x" does not always denote ambiguously when a and b are fixed. Thus, in the above case, when a = 2, b -I, "x"
=
=
= =
=
=
=
=
=
29
TM NllIure of MllIMmtlltcs
denotes the one negative number -1. What is meant is this: In each member of the class of problems got by giving a and b fixed values independently of one another, there is an unknown x, which mayor may not denote different numbers, which only becomes known when the equation is solved. Consider now the equation ax + by c, where Q, b, and c are known quantities and x and yare unknown. We can find x in terms of a, b, c, and y, or y in terms of a, b, c, and x; but x is only fixed when y is fixed, or y when x is fixed. Here in each case of fixedness of a, b, and c, x is undetermined and "variable," that is to say, it may take any of a whole class of values. Corresponding to each x, one y belongs; and y also is a "variable" depending on the "independent variable" x. The idea of "variability" will be further illustrated in the next chapter; here we will only point out how the notion of what is called by mathematicians the "functional dependence" of y on x comes in. The variable y is said to be a "function" of the variable x if to every value of x corresponds one or more values of y. This use has, to some extent, been adopted in ordinary language. We should be understood if we were to say that the amount of work performed by a horse is a function of the food that he eats. Descartes also adopted the custom-if he did not arrive at it independently-advocated by Harriot of transferring all the terms of an equation to the same side of the sign of equality. Thus, instead of "x = 1," "ax + b c," and "3x2 + g = hx," we write respectively "x - 1 0," "ax + (b - c) = 0," and "3x2 - hx + g = 0." The point of this is that all equations of the same degree in the unknown-we shall have to consider cases of more unknowns than one in the next chapter-that is to say, equations in which the highest power of x (x or x 2 or .xl ••• ) is the same, are easily recognisable. Further, it is convenient to be able to speak of the expression which is equated to as well as of the equation. The equations in which X2, and no higher power of x, appears are called "quadratic" equations-the result of equating a "quadratic" function to 0; those in which .xl, and no higher power, appears are called "cubic"; and so on for equations "of the fourth, fifth . . ." degrees. Now the quadratic equations, 3x2 + g = 0, ax2 + bx + c 0, x 2 - 1 = 0, for example, are different, but the differences are unimportant in comparison with this common property of being of the same degree: all can be solved by modifications of one general method. Here it is convenient again to depart from the historical order and briefly consider the meaning of what are called "imaginary" expressions. If we are given the equation x 2 - 1 = 0, its solutions are evidently x = + 1 or x = -1, for the square roots of +1 are +1 and -1. But if we are given the equation x 2 + 1 = 0, analogy would lead us to write down the y=I. But there is no positive or two solutions x = + v=t and x negative "number" which we have yet come across which, when multi-
=
=
=
°
=
=-
,1IIU" E. B.
}D""""
plied by itself, gives a negative "number." Thus "imaginary numbers" were rejected by Descartes and his followers. Thus X 2 - 1 0 had two solutions, but x 2 + 1 = 0 none; further, .xl + x 2 + X + 1 0 had one solution (x -1), while;xl - x 2 - X + 1 0 had two (x 1, x -1), and xl - 2X2 - X + 2 = 0 had three (x= I, x= -I, x=2). Now, suppose, for a moment, that we can have "imaginary" roots and (y=I) (y 1) -1, and also that we can speak of two roots when the roots are identical in a case like the equation x 2 + 2x + 1 = 0, or (x + 1) 2 = 0, which has two identical roots x = -1. Then, in the above five equations, the first two quadratic ones have two roots each (+ 1, -1, and + .y::r, respectively), and the three cubics have three each (-I, + y -I, - Vi; + I, -1, + I; and + I, - 1, +2 respectively). In the general case, the theorem has been proved that every equation has as many roots as (and not merely "no more than," as Descartes said) its degree has units. For this and for many other reasons like it in enabling theorems to be stated more generally, "imaginary numbers" came to be used almost universally. This was greatly helped by one puzzJing circumstance: true theorems can be discovered by a process of calculation with imaginaries. The case is analogous to that which led mathematicians to introduce and calculate with "negative numbers. For the case of imaginaries, let a, h, c, and d be any numbers, then
=
=
= = = =
=
Y-I
It
(a 2
+ b2 )(e2 + d2) =
+ by=I)(a - by'=f) (c + dy=I) (e - dy-I) = (a + by=T)(e + dy=T) (a
(a-by=I)(e-dy'=f) = [(ac - bd) + y=I(ad + be)) [(ae - bd) - V-I (ad + be)] = (ae - bd)2 + (ad + bC)2.
We get, then, an interesting and easily verifiable theorem on numbers by calculation with imaginaries, and imaginaries disappear from the conclusion. Mathematicians thought, then, that imaginaries, though apparently uninterpretable and even self-contradictory, must have a logic. So they were used with a faith that was almost firm and was only justified much later. Mathematicians indicated their growing security in the use of V-I by writing "i" instead of "y=t" and calling it "the complex unity:' thus denying, by implication, that there is anything really imaginary or impossible or absurd about it. The truth is that ",'" is not uninterpretable. It represents an operation, just as the negative numbers do, but is of a different kind. It is geometrically interpretable also, though not in a straight line, but in a plane. For this we must refer to the Bibliography; but here we must point out that, in this "generalisation of number" again, the words "addition," "multi-
Tlte
Natu~
31
of MtIl1lemtJtlcs
plication:' and so on, do not have exactly the same, but an analogous, meaning to those which they had before, and that "complex numbers" form a domain like a plane in which a line representing the integers, fractions, and irrationals is contained. But we must leave the further development of these questions. It must be realised that the essence of algebra is its generality. In the most general case, every .symbol and every statement of a proposition in algebra is interpretable in terms of certain operations to be undertaken with abstract things such as numbers or classes or propositions. These operations merely express the relations of these things to one another. If the results at any stage of an algebraical process can be interpreted-and this interpretation is often suggested by the symbolism--say, not as operations with operations with integers, but as other operations with integers, they express true propositions. Thus (a + b) 2 = a2 + 2ab + b2 expresses, for example, a relation holding between those operations with integers that we call '1ractjonary numbers," or an analogous relation between integers. The language of algebra is a wonderful instrument for expressing shortly, perspicuously, and suggestively, the exceedingly complicated relations in which abstract things stand to one another. The motive for studying such relations was originally, and is still in many cases, the close analogy of relations between certain abstract things to relations between certain things we see, hear, and touch in the world of actuality round us, and our minds are helped in discovering such analogies by the beautiful picture of algebraical processes made in space of two Or of three dimensions made by the "analytical geometry" of Descartes, described in the next chapter.
CHAPTER III THE RISE AND PROGRESS OF MODERN MATHEMATICS-ANALYTICAL GEOMETRY AND THE METHOD OF INDMSIBLES
WE will now return to the consideration of the first two sections of Descartes' book Geometry of 1637. In Descartes' book we have to glean here and there what we now recognise as the essential points in his new method of treating geometrical questions. These points were not expressly stated by him. I shall, however, try to state them in a small compass. Imagine a curve drawn On a plane surface. This curve may be considered as a picture of an algebraical equation involving x and y in the following way. Choose any point on the curve, and call u x" and "y " the numbers that express the perpendicular distances of this point, in terms of
32
Ph/Up E. B. JDII1'4tWt
a unit of length, from two straight lines (called "axes") drawn at right angles to one another in the plane mentioned. Now, as we move from point to point of the curve, x and y both vary, but there is an unvarying relation which connects x and y, and this relation can be expressed by an algebraical equation caUed "the equation of the curve," and which contains, in germ as it were, aU the properties of the curve considered. This constant relation between x and y is a relation like y2 = 4ax. We must distinguish carefully between a constant relation between variables and a relation between constants. We are always coming across the former kind of relation in mathematics; we call such a relation a "function" of x and y --the word was first used about fifty years after Descartes' Geometry was published. by Leibniz-and write a function of x and y in general as "f(x, y)." In this notation, no hint is given as to any particular relation x and y may bear to each other, and, in such a particular function as y2 - 4ax, we say that "the form of the function is constant," and this is only another way of saying that the relation between x and y is fixed. This may be also explained as follows. If x is fixed, there is fixed one or more values of y. and if y is fixed, there is fixed one or more values of x. Thus the equation ax + by + c = 0 gives one y for each x and one x for each y; the equation y2 - 4ax 0 gives two y's for each x and one x for each y.5 Consider the equation ax + by + c = 0, or, say, the more definite instance x + 2y - 2 = O. Draw axes and mark off points; having fixed on a unit of length, find the point x = 1 on the x-axis, on the perpendicular to this axis measure where the corresponding y, got by substituting x 1 in the above equation, brings us. We find y lh. Take x ¥.I, then y lUI; and so on. We find that all the points on the parallels to the y-axis lie on one straight line. This straight line is determined by the equation x + 2y - 2 0; every point off that straight line is such that its x and y are not connected by the relation x + 2y - 2 0, and every point of it is such that its x and yare connected by the relation x + 2y - 2 O. Similarly we can satisfy ourselves that every point on the circumference of a circle of radius c units of length, described round the point where the axes cross, is such that x 2 + y2 = c2, and every point not on this circumference does not have an x and y -such that the constant relation x 2 + y2 c2 is satisfied for it. There are two points to be noticed in the above general statement. Firstly, I have said that the curve "may be expressed," and so on. By this I mean that it is possible--and not necessarily always true--that the curve
=
=
=
= =
=
=
=
=
II We also denote a function of x by "/(x)" or "F(x)" or "C/>(x):' &c. Here ..,. is a sign fOr "function of," not for a number, just as later we shall find "sin" and "a" and "tI" standing for functions and not numbers. This may be regarded as an extension of the language of early algebra. The equation y = f(x) is in a good form for graphical representation in the manner explained below.
The Nature
Df
Mathematics
may be so considered. We can imagine curves that cannot be represented by a finite algebraical equation. Secondly, about the fundamental lines of reference-the "axes" as they are called. One of these axes we have caned the "x-axis," and the distance measured by the number x is sometimes called "the abscissa"; while the line of length, units which is perpendicular to the end of the abscissa farthest from the origin, and therefore parallel to the other axis ("the ,-axis") is called "the ordinate." The name "ordinate" was used by the ancient Roman surveyors. The lines measured by the numbers x and , are caned the "co-ordinates" of the point determining and determined by them. Sometimes the numbers x and , themselves are called "co-ordinates," and we will adopt that practice here. Sometimes the axes are not chosen at right angles to One another, but it is nearly always far simpler to do so, and in this book we always assume that the axes are rectangular. The whole plane is divided by the axes into four partitions, the co-ordinates are measured from the point--called "the origin"-where the axes cross. Here the interpretation in geometry of the "negative quantities" of algebra-which so often seems so puzzling to inteUigent beginners-gives us a means of avoiding the ambiguity arising from the fact that there would be a point with the same co-ordinates in each quadrant into which the plane is divided. Consider the x-axis. Measure lengths on it from the origin, so that to the origin (0) corresponds the number O. Let OA, measured from left to right along the axis. be the unit of length; then to the point A corresponds the number 1. Then let lengths AB, BC, and so on, aU measured from left to right, be equal to OA in length; to the points B, C, and so on, correspond the numbers 2, 3, and so on. Further to the point that bisects OA let the fraction lh correspond; and so on for the other fractions. In this way half of the x-axis is nearly filled up with points. But there are points, such as the point P, such that OP is the length of the circumference of a circle, say of unit diameter. For picturesqueness, we may imagine this point P got by rolling the circle along the x-axis from 0 through one revolution. The point P will fall a little to the left of the point 3* and a little to the right of the point 3%0, and so on; the point P is not one of the points to which names of fractions have been assigned by the process sketched above. This can be proved rigidly. If it were not true, it would be very easy to "square the circle." There are many other points like this. There is no fraction which, multiplied by itself, gives 2; but there is a length-the diagonal of a square of unit side-which is such that, if we were to assume that a number corre· sponded to every point on OX, it would be a number a such that a2 2. We will return to this important question of the correspondence of points and lines to numbers, and will now briefly recall that "negative numbers" are represented, in Descartes' analytical geometry, on the x-axis, by the
=
P1IIU, B. B. IDf11'f111l1t
points to the left of the origiD, and, on the yeWs, by the points below the origin. This was explained in the second chapter. Algebraical geometry gave us a means of classifying curves. All straight lines determine equations of the first degree between x and y. and all such equations determine straight lines; all equations of the second degree between x and y, that is to say, of the form ax 2
+ bxy + cy2 + dx + ey + f = 0,
determine curves which the ancient Greeks had studied and which result from cutting a solid circular cone, or two equal cones with the same axis, whose only point of contact is formed by the vertices. It is somewhat of a mystery why the Greek geometricians should have pitched upon these particular curves to study, and we can only say that it seems, from the present standpoint, an exceedingly lucky chance. For these "conic sections"--of which, of course, the circle is a particular case-are all the curves, and those only, which are represented by the above equation of the second degree. The three great types of curves--the ''parabola,'' the '·elJipse," and the ~'hyperbola"-all result from the above equation when the coefficients a, h, c, d, e, f satisfy certain special conditions. Thus, the equation of a circle-which is a particular kind of ellipse-is always of the form got from the above equation by putting b 0 and c = a. It may be mentioned that, long after these curves were introduced as sections of a cone, Pappus discovered that they could all be defined in a plane as loci of a point P which moves so that the proportion that the distance of P from a fixed point (S) bears to the perpendicular distance of P (PN) to a fixed straight line is constant. As this proportion is less than equal to or greater than 1, the curve is an ellipse, parabola, or hyperbola, respectively. It will not be expected that a detailed account should here be given of the curves which result from the development of equations of the second or higher degrees between x and y. I will merely again emphasize some points which are, in part, usually neglected or not cJearly stated in textbooks. The letters "a, h, ... x, y," here stand for "numbers" in the extended sense. We have seen in what sense we may, with the mathematicians, speak of fractionary, positive, and negative "numbers," and identify, say, the positive number +2 and the fraction ~~ with the signless integer 2. Well, then, the above letters stand for numbers of that class which includes in this sense the fractionary, irrational, positive and negative numbers, but excludes the imaginary numbers. We can the numbers of this class "real" numbers. The question of irrational numbers will be discussed at greater length in the sixth chapter, but enough has been said to show how they were introduced. In mathematics it has, I think, always happened that conceptions have been used long before they were formally intro-
=
35
The NtlIUn of Mathematics
duced, and used long before this use could be logically justified or its nature clearly explained. The history of mathematics is the history of a faith whose justification has been long delayed, and perhaps is not accomplished even now. These numbers are the measurements of length, in terms of a definite unit, like the inch, of the abscissz and ordinates of certain points. We speak of such points simply by naming their co-ordinates, and say, for example, that "the distance of the point (x, y) from the point (a, b) is the positive square root of (x - a)2 + (y - b)2." Notice that X2, for example, is the length of a line. It is natural to make, as algebraists before Descartes did, x 2 stand primarily for the number of square units in a square whose sides are x units in 1ength, but there is no necessity in this. We shall often use the latter kind of measurement in the fourth and fifth chapters. The equation of a straight line can be made to satisfy two given conditions. We can write the equation in the form
x+E.y+c a a
=0,
b c and thus have two ratios, - and -, that we can determine according to a a
the conditions. The equation ax + by + c = 0 has apparently three "arbitrary constants," as they are called, but we see that this greater generality is only apparent. Now we can so fix these constants that two conditions are fulfiUed by the straight 1ine in question. Thus, suppose that one of these conditions is that the straight Jine should pass through the originthe point (0, 0). This means simply that when x = 0, then y = O. Putting them, x
c
= 0 and y = 0 in the above equation, we get - = 0, and thus one a
of the constants is determined. The other is determined by a new condition that, say, the line also passes through the point (%, 2). Substituting,
c
then, in the above equation, we have, as - = 0 as we know already, a 2b b % + - = 0, whence - = - %. Hence the equation of the line passing Q a through (0, 0) and (%, 2) is x - %y = 0, or y - 6x = O. Instead of having to pass through a certain point, a condition may be, for example, that the perpendicular from the origin on the straight line should be of a certain length, or that the line should make a certain angle with the x-axis, and so on.
PIIIII" E. B. 1011I'IIIIIII
36
Similarly, the circle whose equation is written in the form (x - a)2
+ (y -
b)2
= c2
is of radius c and centre (a, b). It can be determined to pass through any three points, or, say, to have a determined length of radius and position of centre. Fixation of centre is equivalent to two conditions. Thus, suppose the radius is to be of unit length: the above equation is (x - a)2 + (y - b) 2 = 1. Then, if the centre is to be the origin, both a and b are determined to be 0, and this may also be effected by determining that the circle is to pass through the points (%, 0) and (-%, 0), for example. Now, if we are to find the points of intersection of the straight line 2x + 2y 1 and the circle x 2 + y2 1, we seek those points which are common to both curves, that is to say, all the pairs of values of x and y which satisfy both the above equations. Thus we need not trouble about the geometrical picture, but we only have to apply the rules of algebra for finding the values of x and y which satisfy two "simultaneous" equations in x and y. In the above case, if (X, Y) is a point of intersection,
=
we have Y
)(2
+(
=
=
1-2X 2
1 - 2X)2
2
, and therefore, by substitution in the other equation,
= 1. This gives a quadratic equation 8X2 - 4X - 3 = 0
for X, and, by rules, we find that X must be either 1A (1 +.y7) or %(1 - Y7). Hence there are two values of the abscissa which are given when we ask what are the co-ordinates of the points of intersection; and the value of y which corresponds to each of these x's is given by substitution in the equation 2x + 2y = 1. Thus we find again the fact, obvious from a figure, that a straight line cuts [a circle] at two points at most. We can determine the points of intersection of any two curves whose equations can be expressed algebraically, but of course the process is much more complicated in more general cases. Here we will consider an important case of intersection of a straight line. Think of a straight line cutting a circle at two points. Imagine one point fixed and the other point moved up towards the first. The intersecting line approaches more and more to the position of the tangent to the circle at the first point, and, by making the movable point approach the other closely enough, the secant will approach the tangent in position as nearly as we wish. Now, a tangent to a curve at a certain point was defined by the Greeks as a straight line through the point such that between it and
Th~ Na'u'~
01
Malh~mo.lic$
37
the curve no other straight line could be drawn. Note that other curves might be drawn: thus various circles may have the same tangent at a common point on their circumference, but no circle--and no curve met with in elementary mathematics-has more than one tangent at a point. Descartes and many of his followers adopted different forms of definition which really involve the idea of a limit. an idea which appears boldly in the infinitesimal calculus. A tangent is the limit of a secant as the points of intersection approach infinitely near to one another; it is a produced side of the polygon with infinitesimal sides that the curve is supposed to be; it is the direction of motion at an instant of a point moving in the curve considered. The equation got from that of the curve by substituting for y from the equation of the intersecting straight line has, if this straight line is a tangent, two equal roots. In the above case, this equation was quadratic. In the case of a circle, we can easily deduce the well-known property of a tangent of being perpendicular to the radius; and see that this property has no analogue in the case of other curves. We must remember that, just as plane curves determine and are determined by equations with two independent variables x and y, so surfacesspheres, for instance--in three-dimensional space determine and are determined by equations with three independent variables, x, y, and z. Here x, y, and z are the co-ordinates of a point in space; that is to say, the numerical measures of the distances of this point from three fixed planes at right angles to each other. Thus, the equation of a sphere of radius d and centre at (a, b, c) is (x - a)2 + (y - b)2 + (z - C)2 = d2. We may look at analytical geometry from another point of view which we shall find afterwards to be important, and which even now will suggest to us some interesting thoughts. The essence of Descartes' method also appears when we represent loci by the method. Consider a circle; it is the locus of a point (P) which moves in a plane so as to preserve a constant distance from a fixed point (0). Here we may think of P as varying in position, and make up a very striking picture of what we call a variable in mathematics. We must, however, remember that, by what we call a ''variable'' for the sake of picturesqueness, we do not necessarily mean something which varies. Think of the point of a pen as it moves over a sheet of writing paper; it occupies different positions with respect to the paper at different times, and we understandably say that the pen's point moves. But now think of a point in space. A geometrical point-which is not the bit of space occupied by the end of a pen or even an "atom" of matter-is merely a mark of position. We cannot, then, speak of a point moving; the very essence of point is to be position. The motion of a point of space, as distinguished from a point of matter, is a fiction, and is the supposition that a given point can now be one point and now another. Motion, in the ordinary sense, is only possible to matter and not to space.
38
Philip E. B. IDurdtWt
Thus, when we speak of a '''variable position," we are speaking absurdly if we wish our words to be taken literally. But we do not really so wish when we come to think about it. What we are doing is this: we are using a picturesque phrase for the purpose of calling up an easily imagined thought which helps us to visualise roughly a mathematical proposition which can only be described accurately by a prolix process. The ancient Greeks allowed prolixity~ and it was only objected to by the uninitiated. Modern mathematics up to about sixty years ago successfully warred against proJixity; hence the obscurity of its fundamental notions and processes and its great conquests. The great conquests were made by sacrificing very much to analogy: thus, entities like the integer 2, the ratio 2/1, and the real number which is denoted by "2" were identified, as we have seen, because of certain close analogies that they have. This seems to have been the chief reason why the procedure of the mathematicians has been so often condemned by ]ogicians and even by philosophers. In fact. when mathematicians began to try to find out the nature of Mathematics, they had to examine their entities and the methods which they used to deal with them with the minutest care, and hence to look out for the points when the analogies referred to break down, and distinguish between what mathematicians had usually faiJed to distinguish. Then the people who do not mind a bit what Mathematics is, and are only interested in What it does, called these earnest inquirers "pedants" when they should have said "philosophers," and "logic-choppers"-whatever they may be-when they should have said "logicians." We have tried to show why ratios or fractions, and so on, are called "numbers," and apparently said to be something which they are not; we must now try to get at the meaning of the words "constant" and "variable." By means of algebraic formulm, rules for the reconstruction of great numbers-sometimes an infinity--of facts of nature may be expressed very concisely or even embodied in a single expression. The essence of the formula is that it is an expression of a constant rule among variable quantities. These expressions "constant" and "variable" have come down into ordinary language. We say that the number of miles which a certain man walks per day is a "variable quantity"; and we do not mean that, on a particular day, the number was not fixed and definite, but that on different days he walked, general1y speaking, different numbers of miles. When, in mathematics, we speak of a "variable," what we mean is that we are considering a class of definite objects-for instance, the class of men alive at the present moment-and want to say something about anyone of them iodefinitely. Suppose that we say: "If it rains. Mr. A. will take his umbrella out with him"; the letter "A" here is what we call the sign of the "variable." We do not mean that the above proposition is about a variable man. There is no such thing; we say that a man varies in health
The Nllture 01 Mllthemlltics
39
and so in time, but, whether or not such a phrase is strictly correct, the meaning we would have to give the phrase "a variable" in the above sentence is not one and the same man at different periods of his own existence, but one and the same man who is different men in tum. What we mean is that if "A" denotes any man, and not Smith or Jones or Robinson alone, then he takes out his umbrella on certain occasions. The statement is not always true; it depends on A. If "An stands for a bank manager, the statement may be true; if for a tramp or a savage, it probably is not. Instead of "A," we may put "B" or "e" or "X"; the kind of mark on paper does not really matter in the least. But we attach t by convention, certain meanings to certain signs; and so, if we wrote down a mark of exclamation for the sign of a variable, we might be misunderstood and even suspected of trying to be funny. We shall see, in the seventh chapter, the importance of the variable in logic and mathematics. "Laws of nature" express the dependence upon one another of two or more variables. This idea of dependence of variables is fundamental in aU scientific thought, and reaches its most thorough examination in mathematics and logic under the name of "functionality." On this point we must refer back to the second chapter. The ideas of function and variable were not prominent until the time of Descartes, and names for these ideas were not introduced until much later. The conventions of analytical geometry as to the signs of co-ordinates in different quadrants of the plane had an important influence in the transformation of trigonometry from being a mere adjunct to a practical science. In the same notation as that used at the end of the first chapter, AP we may conveniently call the number - , which is the same for all OP PM OM lengths of OP, by the name "u," for short, and define - - and - - as OP OP the "sine of u," and the "cosine of un respectively. Thus "sin u" and "cos "," as we write them for short, stand for numerical functions of u. Considering 0 as the origin of a system of rectangular co-ordinates of x y which OA is the x-axis, so that u measures the angle POA and - and r r are cos " and sin u respectively. Now, even if " becomes so great that POA is successively obtuse, more than two right angles . . . , these definitions can be preserved, if we pay attention to the signs of x and y in the various quadrants. Thus sin u and cos u become separated from geometry, and appear as numerical functions of the variable u, whose values, as we see on reflection, repeat themselves at regular intervals as " becomes larger and larger. Thus, suppose that OP turns about 0 in a direction opposite to that in which the hands of a clock move. In the first quadrant, sin "
PIe'"'' E. B. ImuilD.ln y
x
y
-x
and cos u are - and -; in the second they are - and in the third they r r r r -y -x -y x are and - ; in the fourth they are and -; in the fifth they are r r r r y
x
- and - again; and so on. Trigonometry was separated from geometry r
r
mainly by John Bernoulli and Euler, whom we shall mention later. We will now tum to a different development of mathematics. The ancient Greeks seem to have had something approaching a general method for finding areas of curvilinear figures. Indeed, infinitesimal methods, which al10w indefinitely close approximation, naturally suggest themselves. The determination of the area of any rectiUnear figure can be reduced to that of a rectangle, and can thus be completely effected. But this process of finding areas-this Umethod of quadratures"-failed for areas or volumes bounded by curved lines or surfaces respectively. Then the following considerations were applied. When it is impossible to find the exact solution of a question, it is natural to endeavour to approach to it as nearly as possible by neglecting quantities which embarrass the combinations, if it be foreseen that these quantities which have been neglected cannot, by reason of their small value,. produce more than a trifling error in the resu1t of the calculation. For example, as some properties of curves with respect to areas are with difficulty discovered, it is natural to consider the curves as polygons of a great number of sides. If a regular polygon be supposed to be inscribed in a circle, it is evident that these two figures, although always different. are nevertheless more and more alike according as the number of the sides of the polygon increases. Their perimeters, their areas, the solids formed by their revolving round a given axis, the angles formed by these Jines, and so on, are, if not respectively equal, at any rate so much the nearer approaching to equality as the number of sides becomes increased. Whence, by supposing the number of these sides very great, it will be possible, without any perceptible error, to assign to the circumscribed circle the properties that have been found belonging to the inscribed polygon. Thus, if it is proposed to find the area of a given circle, let us suppose this curve to be a regular polygon of a great number of sides: the area of any regular polygon whatever is equal to the product of its perimeter into the half of the perpendicular drawn from the centre upon one of its sides; hence, the circle being considered as a polygon of a great number of sides, its area ought to equal the product of the circumference into half the radius. Now, this resu1t is exactly true. However, the Greeks, with their taste for strictly correct reasoning, could not allow
TIlII NatuTII 01 Matllllmatics
41
themselves to consider curves as polygons of an "infinity" of sides. They were also influenced by the arguments of Zeno, and thus regarded the use of "infinitesimals" with suspicion. Zeno showed that we meet difficulties if we hold that time and space are infinitely divisible. Of the arguments which he invented to show this, the best known is the puzzle of Achilles and the Tortoise. Zeno argued that, if Achilles ran ten times as fast as a tortoise, yet, if the tortoise has (say) 1000 yards start, it could never be overtaken. For, when Achilles had gone the 1000 yards, the tortoise would still be 100 yards in front of him; by the time he had covered these 100 yards, it would still be 10 yards in front of him; and so on for ever; thus Achilles would get nearer and nearer to the tortoise, but never overtake it. Zeno invented some other subtle puzzles for much the same purpose, and they could only be discussed really satisfactorily by quite modem mathematics. To avoid the use of infinitesimals, Eudoxus (408-355 B.C.) devised a method, exposed by Euclid in the Twelth Book of his Elements and used by Archimedes to demonstrate many of his great discoveries, of verifying results found by the doubtful infinitesimal considerations. When the Greeks wished to discover the area bounded by a curve, they regarded the curve as the fixed boundary to which the inscribed and circumscribed polygons approach continually, and as much as they pleased, according as they increased the number of their sides. Thus they exhausted in some measure the space comprised between these polygons and the curve, and doubtless this gave to this operation the name of "the method of exhaustion." As these polygons terminated by straight lines were known figures, their continual approach to the curve gave an idea of it more and more precise, and "the law of continuity" serving as a guide, the Greeks could eventually arrive at the exact knowledge of its propetties. But it was not sufficient for geometricians to have observed, and, as it were, guessed at these properties; it was necessary to verify them in an unexceptionable way; and this they did by proving that every supposition contrary to the existence of these properties would necessarily lead to some contradiction: thus, after, by infinitesimal considerations, they had found the area (say) of a curvilinear figure to be a, they verified it by proving that, if it is not a, it would yet be greater than the area of some polygon inscribed in the curvilinear figure whose area is palpably greater than that of the polygon. In the seventeenth century, we have a complete contrast with the Grecian spirit. The method of discovery seemed much more important than correctness of demonstration. About the same time as the invention of analytical geometry by Descartes came the invention of a method for finding the areas of surfaces, the positions of the centres of gravity of variously shaped surfaces, and so on. In a book published in 1635, and in certain later works, Bonaventura Cavalieri (1598-1647) gave his "method
42
PlIIllp E. B. 10"'"
of indivisibles" in which the cruder ideas of his predecessors, notably of Kepler (1571-1630), were developed. According to Cavalieri, a line is made up of an infinite number of points, each without magnitude, a surface of an infinite number of lines, each without breadth, and a volume of an infinite number of surfaces, each without thickness. The use of this idea may be illustrated by a single example. Suppose it is required to find the area of a right-angled triangle. Let the base be made up of n points (or indivisibles), and similarly let the side perpendicular to the base be made of na points, then the ordinates at the successive points of the base will contain a, 2a • . . , na points. Therefore the number of points in the areas is a + 2a + ... + na; the sum of which is ¥.a(n 2a + na). Since n is very large, we may neglect ¥.ana, for it is inconsiderable compared with ¥.an 2 a. Hence the area is composed of a number %(na)n of points, and thus the area is measured in square units by multiplying half the linear measure of the altitude by that of the base. The conclusion, we know from other facts, is exactly true. Cavalieri found by this method many areas and volumes and the centres of gravity of many curvilinear figures. It is to be noticed that both Cavalieri and his successors held quite clearly that such a supposition that lines were composed of points was literally absurd, but could be used as a basis for a direct and concise method of abbreviation which replaced with advantage the indirect, tedious, and rigorous methods of the ancient Greeks. The logical difficulties in the principles of this and allied methods were strongly felt and commented on by philosophers--sometimes with intelligence; felt and boldly overcome by mathematicians in their strong and not unreasonable faith; and only satisfactorily solved by mathematicians-not the philosophers-in comparatively modem times. The method of indivisibles-whose use will be shown in the next chapter in an important question of mechanics-is the same in principle as "the integral calculus." The integral calculus grew out of the work of Cavalieri and his successors, among whom the greatest are Roberval (1602-1675), Blaise Pascal (1623-1662), and John Wallis (1616-1703), and mainly consists in the provision of a convenient and suggestive notation for this method. The discovery of the infinitesimal calculus was completed by the discovery that the inverse of the problem of finding the areas of figures enclosed by curves was the problem of drawing tangents to these curves, and the provision of a convenient and suggestive notation for this inverse and simpler method, which was, for certain historical reasons, called "the differential calculus." Both analytical geometry and the infinitesimal calculus are enormously powerful instruments for solving geometrical and physical problems. The secret of their power is that long and complicated reasonings can be written down and used to solve problems almost mechanically. It is the
merest superficiality to despise mathematicians for busying themselves, sometimes even consciously, with the problem of economising thought. The powers of even the most god-like intelligences amongst us are extremely limited, and none of us could get very far in discovering any part whatever of the Truth if we could not make trains of reasoning which we have thought through and verified, very ready for and easy in future appJication by being made as nearly mechanical as possible. In both analytical geometry and the infinitesimal calculus, all the essential properties of very many of the objects dealt with in mathematics, and the essential features of very many of the methods which had previously been devised for dealing with them are, so to speak, packed away in a well-arranged (and therefore readily got at) form, and in an easily usable way.
CHAPTER IV THE BEGINNINGS OP THE APPLICATION OP MATHEMATICS TO NATURAL SCIENCE-THE SCIENCE OP DYNAMICS
1HE end of very much mathematics-and of the work of many eminent men-is the simple andt as far as may be, accurate description of things in the world around us, of which we become conscious through our senses. Among these things, let us consider, say, a particular person's face, and a billiard ball. The appearance to the eye of the ball is obviously much easier to describe than that of the face. We can ca11 up the image--a very accurate one----of a billiard ball in the mind of a person who has never seen it by merely giving the colour and radius. And, unless we are engaged in microscopical investigations, this description is usually enough. The description of a face is a harder matter = unless we are skilful modellers, we cannot do this even approximately; and even a good picture does not attempt literal accuracy, but only conveys a correct impression--often better than a model, say in wax, does. Our ideal in natural science is to build up a working model of the universe out of the sort of ideas that all people carry about with them everywhere "in their heads'" as we say, and to which ideas we appeal when we try to teach mathematics. These ideas are those of number. order. the numerical measures of times and distances. and so on. One reason why we strive after this ideal is a very practical one. If we have a working model of, say. the solar system, we can tell, in a few minutes. what our position with respect to the other planets will be at all sorts of far future times, and can thus predict certain future events. Everybody can see how useful this is: perhaps those persons who see it most clearly are those sailors who use the Nautical Almanac. We cannot make the
Pldl'lI B. B. JOlR'tIGbt
earth tarry in its revolution round its axis in order to give us a longer day for finishing some important piece of work; but, by finding out the unchanging laws concealed in the phenomena of the motions of earth, sun, and stars, the mathematician can construct the model just spoken of. And the mathematician is completely master of his model; he can repeat the occurrences in his universe as often as he likes; something like Joshua, he can make his "sun" stand still, or hasten, in order that he may publish the Nautical Almanac several years ahead of time. Indeed, the "world" with which we have to deal in theoretical or mathematical mechanics is but a mathematical scheme, the function of which it is to imitate, by logical consequences of the properties assigned to it by definition, certain processes of nature as closely as possible. Thus our "dynamical world" may be called a model of reality, and must not be confused with the reality itself. That this model of reality is constructed solely out of logical conceptions will result from our conclusion that mathematics is based on logic, and on logic alone; that such a model is possible is really surprising on relection. The need for completing facts of nature in thought was, no doubt, first felt as a practical need--the need that arises because we feel it convenient to be able to predict certain kinds of future events. Thus, with a purely mathematical model of the solar system, we can teU, with an approximation which depends upon the completeness of the model, the relative positions of the sun, stars, and planets several years ahead of time; this it is that enables us to publish the Nautical Almanac, and makes up to us, in some degree, for our inability ''to grasp this sorry scheme of things entire . . . and remould it nearer to the heart's desire." Now, what is called "mechanics" deals with a very important part of the structure of this model. We spoke of a billiard ball just now. Everybody gets into the way, at an early age, of abstracting from the colour, roughness, and so on, of the ball, and forming for himself the conception of a sphere. A sphere can be exactly described; and so can what we call a "square," a "circle" and an "ellipse," in terms of certain conceptions such as those called "point," "distance," "straight line," and so on. Not so easily describable are certain other things, like a person or an emotion. In the world of moving and what we roughly class as inanimate objects-that is to say, objects whose behaviour is not perceptibly complicated by the phenomena of what we call "life" and "will"-people have sought from very ancient times, and with increasing success, to discover rules for the motions and rest of given systems of objects (such as a lever or a wedge) under given circumstances (pulls, pressures, and so on), Now, this discovery means: the discovery of an ideal, exactly describable motion which should approximate as nearly as possible to a natural motion or class of motions. Thus Galileo (1564-1642) discovered the approxi-
T"~
Naluu of MatumDtics
mate law of bodies falling freely. or on an inclined plane, near the earth's surface; and Newton (1642-1727) the still more accurate law of the motions of any number of bodies under any forces. Let us now try to think clearly of what we mean by such a rule, or, as it is usually called, a "scientific" or "natural law," and why it plays an important part in the arrangement of our knowledge in such a convenient way that we can at once, so to speak, lay our hand on any particular fact the need of which is shown by practical or theoretical circumstances. For this purpose, we will see how Galiloo, in a work published in 1638, attacked the problem of a falling body. Consider a body falling freely to the earth: Galileo tried to find out. not why it fell, but how it fell-that is to say, in what mathematical form the distance fallen through and the velocity attained depends on the time taken in falling and the space fallen through. Freely falJing bodies are followed with more difficulty by the eye the farther they have fallen; their impact on the hand receiving them is, in like measure, sharper; the sound of their striking louder. The velocity accordingly increases with the time elapsed and the space traversed. Thus, the modem inquirer would ask: What function is the number (v) representing the velocity of those (s and t) representing the distance fallen through and the time of faUing? Galileo asked, in his primitive way: Is v proportional to s; or again, is v proportional to 11 Thus he made assumptions, and then ascertained by actual trial the correctness or otherwise 0/ these assumptions. One of Galileo's assumptions was, thus. that the velocity acquired in the descent is proportional to the time of the descent. That is to say. if a body faJls once, and then falls .again during twice as long an interval of time as it first fell, it will attain in the second instance double the velocity it acquired in the first. To find by experiment whether or not this assumption accorded with observed fact&. as it was difficult to prove by any direct means that the velocity acquired was proportional to the time of descent, but easier to investigate by what law the distance increased with the time, Galileo deduced from his assumption the relation that obtained between the distance and the time. This very important deduction he effected as follows. On the straight line OA. let the abscissz OE. ~C. OG, and so on. represent in length various lengths of time elapsed from a certain instant represented by 0, and let the ordinates EF, CD, GR, and so on, corresponding to these abscissz. represent in length the magnitude of the velocities acquired at the time represented by the respective abscissz. We observe now that, by our assumption, O. F. D. R, lie in a straight line OB, and so: (1) At the instant C, at which one-half OC of the time of descent OA has elapsed, the velocity CD is also one-half of the final
PIttUp E. B. lolll'fllll"
velocity AB; (2) If E and G are equally distant in opposite directions on OA from C, the velocity GR exceeds the mean velocity CD by the same amount that the velocity EF falls short of it; and for every instant antecedent to C there exists a corresponding one subsequent to C and equally distant from it. Whatever loss, therefore, as compared with uniform motion with half the final velocity, is suffered in the first half of the motion, such loss is made up in the second half. The distance fallen through we may consequently regard as having been uniformly described with half the final velocity. In symbols. if the number of units of velocity acquired in t units of time is v. and suppose that v is proportional to t. the number s of units of space descended through is proportional to lht2 • In fact, s is given by lhvt. and. as v is proportional to t. s is proportional to %t2 , Now, Galileo verified this relation between sand t experimentally. The motion of free famng was too quick for Galileo to observe accurately with the very imperfect means-such as water·clocks-at his disposal. There were no mechanical clocks at the beginning of the seventeenth century; they were first made possible by the dynamical knowledge of which Galileo laid the foundations. Galileo. then, made the motion slower, so that sand t were big enough to be measured by rather primitive apparatus in which the moving balls ran down grooves in inclined planes. That the spaces traversed by the ball are proportional to the squares of the measures of the times in free descent as well as in motion on an inclined plane, Ga1i1eo verified by experimentally proving that a ball which falls through the height of an inclined plane attains the same final velocity as a ball which falls through its length. This experiment was an ingenious one with a pendulum whose string, when half the swing had been accomplished, caught on a fixed nail so placed that the remaining half of the swing was with a shorter string than the other half. This experiment showed that the bob of the pendulum rose, in virtue of the velocity acquired in its descent, just as high as it had fallen. This fact is in agreement with our instinctive knowledge of natural events; for if a ball which falls down the length of an inclined plane could attain a greater velocity than one which faUs through its height, we should only have to let the body pass with the acquired velocity to another more inclined plane to make it rise to a greater vertical height than that from which it had fallen. Hence we can deduce, from the acceleration on an inclined plane, the acceleration of free descent. for. since the final velocities are the same and s %vt. the lengths of the sides of the inclined plane are simply proportional to the times taken by the ball to pass over them. The motion of falJing that Galileo found actually to exist is, accordingly, a motion of which the velocity increases proportionally to the time. Like Galileo, we have started with the notions familiar to us (through
=
47
Tltt NalUrt of MllllttmatlCJ
the practical arts, for example), such as that of velocity. Let us consider this motion more closely. If a motion is uniform and c feet are travelled over in every second, at the end of t seconds it will have travelled ct feet. Put ct s far short. Then we call the "velocity" of the moving body the distance traversed in a
=
I
unit of time so that it is - units of length per second, the number which is t
the measure of the distance divided by the number which is the measure of the time elapsed. Galileo, now, attained to the conception of a motion in which the velocity increases proportionally to the time. If we draw a diagram and set off, from the origin 0 along the x-axis OA, a series of abscissz which represent the times in length, and erect the corresponding ordinates to represent the velocities, the ends of these ordinates will lie on a line DB, which, in the case of the "uniformly accelerated motion" to which Ga1i1eo attained, is straight. as we have already seen. But if the ordinates represent spaces instead of velocities, the straight line DB becomes a curve. We see the distinction between the "curve of spaces" and "the curve of velocities:' with times as abscissz in both cases. If the velocity is uniform, the curve of spaces is a straight line DB drawn from the origin 0, and the curve of velocities is a straight line paranel to the x-axis. If the velocity is variable, the curve of spaces is never a straight line; but if the motion is uniformly accelerated, the curve of velocities is a straight line like DB. The relations between the curve of spaces, the curve of velocities, and the areas of such curves AOB are, as we shall see, relations which are at once expressible by the "differential and integral calculus"-indeed, it is mainly because of this important illustration of the calculus that the elementary problems of dynamics have been treated here. And the measurement of velocity in the case where the velocity varies from time to time is an illustration of the formation of the fundamental conception of the differential calculus. It may be remarked that the finding of the velocity of a particle at a given instant and the finding of a tangent to a curve at a given point are both of them the same kind of problem-the finding of the "differential quotient" of a function. We will now enter into the matter more in detail. Consider a curve of spaces. If the motion is uniform, the number measuring any increment of the distance divided by the number measuring the corresponding increment of the time gives the same value for the measure of the velocity. But if we were to proceed like this where the velocity is variable, we should obtain widely differing values for the velocity. However, the smaller the increment of the time, the more nearly does the bit of the curve of spaces which corresponds to this increment approach straightness, and hence uniformity of increase (or decrease) of s. Thus, if we denote the increment of t by "l\t,"-where "l\" does not stand
Philip E. B. JOUl4tlin
48
for a number but for the phrase "the increment of,"-and the corresponding increment (or decrement) of s by "~s," we may define the measure ~s
of average velocity in this element of the motion as - . But, however ~t
small ~t is, the line represented by ~s is not, usually at least, quite straight, and the velocity at the instant t. which, in the language of Leibniz's differential calculus, is defined as the quotient of "infinitely small" increments ds and symbolised by - , -the ~'s being replaced by d's when we consider dt "infinitesimals,"-appears to be only defined approximately. We have met this difficulty when considering the method of indivisibles, and will meet it again when considering the infinitesimal calculus, and will only see how it is overcome when we have become familiar with the conception of a "limit." This new notion of velocity includes that of uniform velocity as a particuJar case. In fact, the rules of the infinitesimal calculus allow us to tis conclude, from the equation - = a, where a is some constant, the equadt tion s = at + b. where b is another constant. We must remember that all this was not expressly formulated until about fifty years after Ga1ileo had published his investigations on the motion of falling. If we consider the curve of velocities, uniformly accelerated motion occupies in it exactly the same place as uniform velocity does in the curve of spaces. If we denote by v the numerical measure of the velocity at the end of t units of time, the acceleration, in the notation of the differdv dv ential calculus, is measured by - . and the equation - = h, where h is dt dt some constant, is the equation of uniformly accelerated motion. In Newtonian dynamics, we have to consider variably accelerated motions, and this is where the infinitesimal calculus or some practically equivalent calculus such as Newton's "method of fiuxions" becomes so necessary in theoretical mechanics. We will now consider the curve of spaces for uniformly accelerated motion. On this diagram-the arcs being t and s-we will draw the curve gt2
S=-,
2 where g denotes a constant. Of course, this is the same thing as drawing gx2
the curve y = -
in a plane divided up by the x-axis and the y-axis of 2
49
Descartes. This curve is a parabola passing through the origin. An interesting thing about this curve is that it is the curve that would be described by a body projected obliquely near the surface of the earth if the air did not resist, and is very nearly the path of such a projectile in the resisting atmosphere. A free body, according to Galileo's view, always falls towards the earth with a uniform vertical acceleration measured by the above number g. If we project a body vertically upwards with the initial velocity of c units, its velocity at the end of t units of time is - c + gt units, for if the direction downwards (of g) is reckoned positive, the direction upwards (of c) must be reckoned negative. If we project a body horizontally with the velocity of a units, and neglect the resistance of the air, Galileo recognised that it would describe, in the horizontal direction, a distance of at units in t units of time, while simultaneously it would fall a distance gt2
of - units. The two motions are to be considered as going on independ2
ently of each other. Thus also, oblique projection may be considered as compounded of a horizontal and a vertical projection. In all these cases the path of the projectile is a parabola; in the case of the horizontal projection, its equation in x and y co-ordinates is got from the two equations gt2
x
= at and y =-, 2
gx2
and is thus y = - . 262
Now, suppose that the velocity is neither uniform nor increases uniformly, but is different and increases at a different rate at different points of time. Then in the curve of velocities, the line DB is no longer straight. In the former case, the number s was the number of square units in the area of the triangle AOB. In this case the figure AOB is not a triangle, though we shall find that its area is the s units we seek, although v does not increase uniformly from 0 to A. Notice again that if, on OA, we take points C and E very close together, the little arc DF is very nearly straight, and the figure DGF very nearly a rectilinear triangle. Note that we are only trying, in this, to get a first approximation to the value of s, so that, instead of the continuously changing velocities we know--or think we know-from our daily experience, we are considering a fictitious motion in which the velocity increases (or decreases) so as to be the same as that of the motion thought of at a large number of points at minute and equal distances, and between successive points increases (or decreases) uniformly. Note also that we are assuming (what usually happens with the curves with which we shall have to do) that the arc DF which corresponds to CE becomes as straight as we wish if we take C and E close enough together.
Pili"" E. B. lol174isb1
50
And now let us calculate s approximately. Starting from 0, in the first small interval OH the rectilinear triangle OHK, where HK is the ordinate at H, represents approximately the space described. In the next small interval HL, where the length of HL is equal to that of OH, the space described is represented by the rectilinear figure KHLM. The rectangle KL is the space passed over with the uniform velocity HK in time HL; and the triangle KNM is the space passed over by a motion in which the velocity increases from zero to MN. And so on for other intervals beyond HL. Thus s is ultimately given (approximately) as the number of square units in a polygon which closely approximates to the figure AOB. We must now say a few words about the meaning of the letters in geometrical and mechanical equations which, following Descartes, we use instead of the proportions used by Gameo and even many of his contemporaries and followers. It seems better, when beginning mechanics, to think in proportions, but afterwards, for convenience in dealing with the symbolism of mathematical data, it is better to think in equations. A typical proportion is: Final velocities are to one another as the times; or, in symbols,
"V : V' : : T : T." Here "V" (for example) is just short for "the velocity attained at the end of the period of time" (reckoned from some fixed instant) denoted by "T," and V : V', and T: T', are just numbers (real numbers); and the proportion states the equality of these numbers. Hence the proportion is sometimes written "V : V' = T : T'." If, now, v is the numerical measure,
v merely, of V, v' that of V', and so on, we have v'
t
= - or t'
vI'
= v't.
In the last equation, the letters v and t have a mnemonic significance, as reminding us that we started from velocities and times, but we must carefully avoid the idea that we are "multiplying" (or can do so) velocities by times; what we are doing is multiplying the numerical measures of them. People who write on geometry and mechanics often say inaccurately, simply for shortness, "Let s denote the distance, t the time," and so on; whereas, by a tacit convention, small italics are usually employed to denote numbers. However, in future, for the sake of shortness, I shall do as the writers referred to, and speak of v as "the velocity." Equations in gt2
mechanics, such as "s
= -" are only possible if the left-hand side is of 2
the same kind as the right-hand side: we cannot equate spaces and times, for example.
51
Suppose that we have fixed on the unit of length as one inch and the unit of time as one second. As unit of velocity we might choose the velocity with which, say. a inches are described uniformly in one second. If We did this. we should express the relation between the s units of space passed over by a body with a given velocity (v units) in a given time (t units) as "s = avt"; whereas. if we defined the unit of velocity as the velocity with which the unit of length is travelled over in the unit of time, we should write "s vt." Among the units derived from the fundamental units--such as those of length and time-the simplest possible relations are made to hold. Thus, as the unit of area and the unit of volume, the square and the cube of unit sides are respectively used, the unit of velocity is the uniform rate at which unit of length is travelled over in the unit of time, the unit of acceleration is the gain of unit velocity in unit time, and so on. The derived units depend on the fundamental units, and the junction which a given derived unit is of its fundamental units is called its "dimensions:' Thus the velocity v is got by dividing the length s by the time t. The dimensions of a velocity are written
=
[L]
U[V]
=-:' IT]
and those of an acceleration-denoted "F"[V]
[L]
- - - - .»
U[F] -
[T]
-
[T]2
These equations are merely mnemonic; the letters do not mean numbers. The mnemonic character comes out when we wish to pass from one set of units to another. Thus. if we pass to a unit of length b times greater and one of time c times greater, the acceleration f with the old units is related to that (r> with the new units by the equation
f(:) As the units become greater, [L]
r
=/.
becomes less; and, since the dimensions of
c2
F are - - , the factor - is obviously suggested to us-the symbol "[T]2tt [T]2 b
suggesting a squaring of the number measuring the time. From Galileo's work resulted the conclusion that, where there is no change of velocity in a straight line, there is no force. The state of a body unacted upon by force is uniform rectilinear motion; and rest in a special case of this motion where the velocity is and remains zero. This
52
PlJIlip E. B. Jourdaut
"law of inertia" was exactly opposite to the opinion, derived from Aris~ totle, that force is requisite to keep up a uniform motion, and may be rougbly verified by noticing the behaviour of a body projected with a given velocity and moving under little resistance--as a stone moving on a sheet of ice. Newton and his contemporaries saw how important this law was in the explanation of the motion of a planet-say, about the sun. Think of a simple case, and imagine the orbit to be a circle. The planet tends to move along the tangent with uniform velocity, but the attraction of the sun simultaneously draws the planet towards itself, and the result of this continual combination of two motions is the circular orbit. Newton succeeded in calculating the shapes of the orbits for different laws of attraction, and found that, when attraction varies inyersely as the square of the distance, the shapes are conic sections, as had been observed in the case of our solar system. The problem of the solar system appeared, then, in a mathematical dress; various things move about in space. and this motion is completely described if we know the geometrical relations--distances, positions, and angular distances-between these things at some moment, the velocities at this moment, and the accelerations at every moment. Of course, if we knew all the positions of all the things at all the instants, our description would be complete; it happens that the accelerations are usually simpler to find directly than the positions: thus, in Galileo's case the acceleration was simply constant. Thus, we are given functional relations between these positions and their rates of change. We have to determine the positions from these relations. It is the business of the "method of fluxions" or the "infinitesimal cal~ cuI us" to give methods for finding the relations between variables from relations between their rates of change or between them and these rates. This shows the importance of the calculus in such physical questions. Mathematical physics grew up--perhaps too much so--on the model of theoretical astronomy, its first really extensive conquest. There are signs that mathematical physics is freeing itself from its traditions, but we need not go further into the subject in this place. Roberval devised a method of tangents which is based on Gali1eo's conception of the composition of motions. The tangent is the direction of the resultant motion of a point describing the curve. Newton's method, which is to be dealt with in the fifth chapter, is analogous to this, and the idea of velocity is fundamental in his "method of fluxions."
Th~
NatW'l: 01
Math~matlc$
53
CHAPTER V THE RISE OF MODERN MATHEMATICS -THE INFINITESIMAL CALCULUS
IN the third chapter we have seen that the ancient Greeks were sometimes occupied with the theoretically exact determination of the areas enclosed by curvilinear figures, and that they used the "method of exhaustion," and, to demonstrate the results which they got, an indirect method. We have seen, too, a "method of indivisibles," which was direct and seemed to gain in brevity and efficiency from a certain lack of correctness in expression and perhaps even a small inexactness in thought. We shall find the same merits and demerits-both, especially the merits. intensified -in the "infinitesimal calculus." By the side of researches on quadratures and the finding of volumes and centres of gravity developed the methods of drawing tangents to curves. We have begun to deal with this subject in the third chapter: here we shall illustrate the considerations of Fermat (1601-1665) and Barrow (163Q-1677)-the intellectual descendants of Kepler-by a simple example. Let it be proposed to draw a tangent at a given point P in the circumference of a circle of centre 0 and equation x 2 + y2 = 1. Let us take the circle to be a polygon of a great number of sides; let PQ be one of these sides, and produce it to meet the x-axis at T. Then PT will be the tangent in question. Let the co-ordinates of P be X and Y; those of Q will be X + e and Y + a, where e and a are infinitely small increments, positive or negative. From a figure in which the ordinates and abscissae of P and Q are drawn, so that the ordinate of P is PRo we can see, by a well-known property of triangles, that TR is to RP (or Y) as e is to a. Now, X and Y are related by the equation X2 + Y2 1, and. since Q is also on the locus x 2 + y2 1. we have (X + e)2 + (Y + a)2 1. From the two equations in which X and Y occur, we conclude that 2eX + e2 + 2aY + a2 = 0,
=
=
e andhence-(X a
e
a
=
e
+ -) + Y + - = O. But 2
2
a
-Y(Y + a./ 2 ) hence TR = - - - Y X + 6/ 2
TR
=~;
Now, a and e may be neglected in comparison with X and Y, and thus Y2 we can say that, at any rate very nearly, we have TR = --. But this is X exactly right, for, since TP is at right angles to OP, we know that OR is to RP as PR is to RT. Here X and Yare constant, but we can say that the abscissa of the point where the tangent at any point (say y) of the
PltiUp E. B. lourd4bt
54
yJ
circle cuts the x-axis is given by adding - - to x. x
Thus, we can find tangents by considering the ratios of infinitesimals to one another. The method obviously applies to other curves besides circles; and Barrow's method and nomenclature leads us straight to the notation and nomenclature of Leibniz. Barrow called the triangle PQS, where S is where a parallel to the x-axis through Q meets PR, the "differential triangle," and Leibniz denoted Barrow's a and e by dy and dx (short for the "differential of y" and "the differential of x," so that "dtt does not denote a number but "dx" altogether stands for an "infinitesimal") respectively, and called the collection of rules for working with his signs the "differential calculus." But before the notation of the differential calculus and the rules of it were discovered by Gottfried Wilhelm von Leibniz (1646-1716), the celebrated German philosopher, statesman, and mathematician, he had invented the notation and found some of the rules of the "integral calculus": thus, he had used the now well-known sign "1" or long "s" as short for "the sum of," when considering the sum of an infinity of infinitesimal elements as we do in the method of indivisibles. Suppose that we propose to determine the area included between a certain curve y t(x), the x-axis, and two fixed ordinates whose equations are x a and x b; then, if we make use of the idea and notation of differentials, we notice that the area in question can be written as "fy . dx," the summation extending from x a to x = b. We will not here further concern ourselves about these boundaries. Notice that in the above expression we have put a dot between the "y" and the "dx": this is to indicate that y is to multiply dx. Hitherto we have used juxtaposition to denote multiplication, but here d is written close to x with another end in view; and it is desirable to emphasise the difference between "d" used in the sense of an adjective and "d" used in the sense of a multiplying number, at least until the student can easily tell the difference by the context. If, then, we imagine the abscissa divided into equal infinitesimal parts, each of length dx, corresponding to the constituents called "points" in the method of indivisibles, y. dx is the area of the little rectangle of sides dx and y which stand at the end of the abscissa x. If, now, instead of extending to x = b, the summation extends to the ordinate at the indeterminate or "variable" point x, y • dx becomes a function of x. Now, if we think what must be the differential of this sum-the infinitesimal increment that it gets when the abscissa of length x, which is one of the boundaries, is increased by dx-we see that it must be y . dx. Hence d(fy. dx) y • dx,
=
=
=
= =
55
The Nature 01 Matllemtllics
and hence the sign of "d" destroys, so to speak, the effect of the sign "f', We also have Jdx x, and find that this summation is the inverse process to differentiation. Thus the problems of tangents and quadratures are inverses of one another. This vital discovery seems to have been first made by Barrow without the help of any technical symbolism. The quantity which by its differentiation produces a proposed differential~ is called the "integral" of this differential; since we consider it as having been formed by infinitely small continual additions: each of these additions is what we have named the differential of the increasing quantity, it is a fraction of it: and the sum of all these fractions is the entire quantity which we are in search of. For the same reason we call "integrating" or "taking the sum of" a differential the finding the integral of the sum of all the infinitely small successive additions which form the series, the differential of which, properly speaking, is the general term. It is evident that two variables which constantly remain equal increase the one as much as the other during the same time, and that consequently their differences are equal: and the same holds good even if these two quantities had differed by any quantity whatever when they began to vary; provided that this primitive difference be always the same, their differentials will always be equal. Reciprocally, it is clear that two variables which receive at each instant infinitely small equal additions must also either remain constantly equal to one another, or always differ by the same quantity-that is, the integrals of two differentials which are equal can only differ from each other by a constant quantity. For the same reason, if any two quantities whatever differ in an infinitely sman degree from each other, their differentials will also differ from one another infinitely little: and reciprocally if two differential quantities differ infinitely little from one another, their integrals, putting aside the constant, can also differ but infinitely little one from the other. Now, some of the rules for differentiation are as follows. If y = I(x), dy = f(x + dx} - I(x}, in which higher powers of differentials added to lower ones may be neglected. Thus, if y x2, then dy = (x + dx)2 - x 2 2x . dx + (dx) 2 2x . dx. Here it is well to refer back to the treatment of the problem of tangents at the beginning of this chapter. Again. if y = a. x, where a is constant, dy = a. dx. If y = x. ~ then
=
=
dy = (x
dx
=
=
+ dx)(z + dz)
= y • dz + z . dy;
- x. z
hence dy
= x. dz + z. dx. =
x
If y
=-~
x = y. z, so
z
dx - y. dz
. Since
the integral calculus
Z
is the inverse of the differential calculus. we have at once
Phllt" E. B. JormlGht
=
f2x • dx = X2, fa. dx aJdx, fx • d1. + f1. • dx xz, and so on. More fully, from d(xS) 3x 2 . dx, we conclude, not that fx 2 • dx =~, but that fx 2 . dx = ~ + c, where "e" denotes some con-
=
=
stant depending on the fixed value for x from which the integration starts. Consider a parabola y2
= ax;
then 2y. dy
Thus the area from the origin to the point x is 2y2. dy
2y s
- - - ; thus the area is a
3a
= a . dx,
j
2 y2 . dy
a
2y. dy
or dx
= ---, a
+ c;
2y8
but d - = 3a
+ c, or, since y2 = ax, %x. y + c. To deter-
mine c when we measure the area from 0 to x, we have the area zero when x = 0; hence the above equation gives c = O. This whole result, now quite simple to us, is one of the greatest discoveries of Archimedes. Let us now make a few short reflections on the infinitesimal calculus. First, the extraordinary power of it in dealing with complicated questions lies in that the question is split up ioto an infinity of simpler ones which can all be dealt with at once, thanks to the wonderfully economical fashion in which the calculus, like analytical geometry, deals with variables. Thus, a curvilinear area is split up into rectangular elements, an the rectangles are added together at once when it is observed that integral is the inverse of the easily acquired practice of differentiation. We must never lose sight of the fact that, when we differentiate y or integrate y . dx, we are considering, not a particular x or y, but anyone of an infinity of them. Secondly, we have seen that what in the first place had been regarded but as a simple method of approximation, leads at any rate in certain cases to results perfectly exact. The fact is that the exact results are due to a compensation of errors: the error resulting from the false supposition made, for example, by regarding a curve as a polygon with an infinite number of sides each infinitely small and which when produced is a tangent of the curve, is corrected or compensated for by that which springs from the very processes of the calculus, according to which we retain in differentiation infinitely small quantities of the same order alone. In fact, after having introduced these quantities into the calculation to facilitate the expression of the conditions of the problem, and after having regarded them as absolutely zero in comparison with the proposed quantities, with a view to simplify these equations, in order to banish the errors that they had occasioned, and to obtain a result perfectly exact, there remains but to eliminate these same quantities from the equations where they may still be.
The Natrue
Df
57
Matherruzllcs
But all this cannot be regarded as a strict proof. There are great difficulties in trying to determine what infinitesimals are: at one time they are treated like finite numbers and at another like zeros or as "ghosts of departed quantities," as Bishop Berkeley, the philosopher, called them. Another difficulty is given by differentials "of higher orders than the first." Let us take up again the considerations of the fourth chapter. We
d,
saw that v
=-, and found that s was got by integration: s = Iv. dt. This dt
ds is now an immediate inference, since - dt == ds. Now, let us substitute dt dv for v in - . Here t is the independent variable, and all of the older mathedt maticians treated the elements dt as constant-tbe interval of the independent variable was split up into atoms, so to speak, which themselves were regarded as known, and in terms of which other differentials, ds, dx, dy, were to be determined. Thus dv
d(ll8/ Ii')
----dt dt
1/tit. d(ds)
tJ2s
----==-,
dt2 "d2s" being written for "d(ds) " and "dt2" for U(dt)2". Thus the acceleration was expressed as "the second differential of the space divided by the dt
d2s
d2s
square of dt." If - were constant, say, a, then dt 2 dt
= a . dt;
and, integrat-
ing both sides:
d, - == fa . dt == aldt == at + b. dt
where b is a new constant. Integrating again, we have: at2
s
= alt. dt + bldt =- + bt + c, 2
which is a more general form of Galileo's result. Many complicated probJems which show how far-reaching Galileo's principles are were devised by Leibniz and his school. Thus, the infinitesimal calculus brought about a great advance in our powers of describing nature. And tms advance was mainly due to Leibniz's notation; Leibniz himself attributed all of his mathematical discoveries to his improvements in notation. Those who know something of Leibniz's work know how conscious he was of the suggestive and economical value of a good notation. And the fact that we still use and appreciate Leibniz's
PlUlip E. B. JOlUtIahI
58
Uf' and ud," even though our views as to the principles of the calculus are very different from those of Leibniz and his school, is perhaps the best testimony to the importance of this question of notation. This fact that Leibniz's notations have become permanent is the great reason why I have dealt with his work before the analogous and prior work of Newton. Isaac Newton (1642-1727) undoubtedly arrived at the principles and practice of a method equivalent to the infinitesimal calculus much earlier than Leibniz, and, like Roberval, his conceptions were obtained from the dynamics of Galileo. He considered curves to be described by moving points. If we conceive a moving point as describing a curve, and the curve referred to co-ordinate axes, then the velocity of the moving point can be decomposed into two others parallel to the axes of x and y respectively; these velocities are called the "fluxions" of x and y, and the velocity of the point is the fluxion of the arc. Reciprocally the arc is the "fluent" of the velocity with which it is described. From the given equation of the curve we may seek to determine the relations between the fluxions--and this is equivalent to Leibniz's problem of differentiation;-and reciprocally we may seek the relations between the co-ordinates when we know that between their fluxions, either alone or combined with the co-ordinates them~ selves. This is equivalent to Leibniz's general problem of integration, and is the problem to which we saw, at the end of the fourth chapter, that theoretical astronomy reduces. Newton denoted the fluxion of x by "i/' and the fluxion of the fluxion (the acceleration) of:i by '~x:n It is obvious that this notation becomes awkward when we have to consider fluxions of higher orders; and further, Newton did not indicate by his notation the independent variable considered. Thus
di
"y"
tJ2x
dy dy might possibly mean either - or - . We have i dt dx
dt
dr'x
x= -dt = -dt; but a dot-notation for -dp 2
dx
=-,
would be clumsy and incon-
venient. Newton's notation for the "inverse method of fluxions" was far clumsier even, and far inferior to Leibniz's "f'. The relations between Newton and Leibniz were at first friendly, and each communicated his discoveries to the other with a certain frankness. Later, a long and acrimonious dispute took place between Newton and Leibniz and their respective partisans. Each accused-unjustly, it seemsthe other of plagiarism, and mean suspicions gave rise to meanness of conduct, and this conduct was also helped by what is sometimes called
Tile NaJure Df MaJ1Ierruulc8
"patriotism." Thus, for considerably more than a century, British mathematicians failed to perceive the great superiority of Leibniz's notation. And thus, while the Swiss mathematicians, James Bernoulli (1654-1705), John Bernoulli (1667-1748), and Leonhard Euler (1707-1783), the French mathematicians d'Alembert (1707-1783), Clairaut (1713-1765), Lagrange (1736-1813), Laplace (1749-1827), Legendre (1752-1833), Fourier (1768-1830), and Poisson (1781-1850), and many other Continental mathematicians, were rapidly 6 extending knowledge by using the infinitesimal calculus in all branches of pure and applied mathematics, in England comparatively little progress was made. In fact, it was not until the beginning of the nineteenth century that there was formed, at Cambridge, a Society to introduce and spread the use of Leibniz's notation among British mathematicians: to establish, as it was said, "the principles of pure d-ism in opposition to the dot-age of the university." The difficulties met and not satisfactorily solved by Newton, Leibniz, or their immediate successors, in the principles of the infinitesimal calculus, centre about the conception of a "limit"; and a great part of the meditations of modem mathematicians, such as the Frenchman Cauchy (17891857), the Norwegian Abel (1802-1829), and the German Weierstrass (1815-1897), not to speak of many sti11living, have been devoted to the putting of this conception on a sound logical basis. We have seen that, if y (x
form
+ Ax) 2 -
Xlii
=x
dy 2, -
dx
dy
= 2x. What we do in forming -
is to
dx
, which is readily found to be 2x + Ax, and then
Ax
consider that, as Ax approaches 0 more and more, the above quotient approaches 2x. We express this by saying that the "limit, as h [Ax] approaches 0," is 2x. We do not consider Ax as being a fixed "infinitesimal" or as an absolute zero (which would make the above quotient become indeter-
o minate -), nor need we suppose that the quotient reaches its limit (the
o
state of Ax being 0). What we need to consider is that "Ax" should represent a variable which can take values differing from 0 by as little as we please. That is to say, if we choose any number, however small, there is a value which Ax can take, and which differs from 0 by less than that 6 It is difficult for a mathematician not to think tbat the sudden and brilliant dawn on eighteenth century France of tbe magnificent and apparently all-embracing physics of Newton and the wonderfully powerful mathematical metbod of Leibniz inspired scientific men with the beJief that the goal of all knowJedge was nearly reached and a new era of knowledge quiclc1y striding towards perfection begun; and that this optimism bad indirectly much to do in preparing for the Frencb Revolution.
Philip 11. B. Jo",d41n
60
number. As before, when we speak of a "variable" we mean that we are considering a certain class. When we speak of a "limit," we are considering a certain infinite class. Thus the sequence of an infinity of terms 1, lh, %', %, lAo, and so on, whose law of formation is easily seen, has the limit O. In this case 0 is such that any number greater than it is greater than some term of the sequence, but 0 itself is not greater than any term of the sequence and is not a term of the sequence. A sequence like 1, 1 + lh, 1 + lh + %', 1 + lh + %, + % . . ., has an analogous upper limit 2. A function !(x), as the independent variable x approaches a cer2x tain value, like - as x approaches 0, may have a value (in this case 2, x 2x
though at 0, -
is indeterminate). The question of the limits of a func-
x
tion in general is somewhat complicated, but the most important limit is !(x + Ax) - !(x) dy - - - - - - - as Ax approaches 0; this, if y = f(x), is-. Ax dx That the infinitesimal calculus, with its rather obscure "infinitesimals"dy 1 dx treated like finite numbers when we write - dx = dy and - - = -, and dx dll/ tU dy then, on occasion, neglected-leads so often to correct results is a most remarkable fact, and a fact of which the true explanation only appeared when Cauchy, Gauss (1777-1855), Riemann (1826-1866), and Weierstrass had developed the theory of an extensive and much used class of functions. These functions happen to have properties which make them especially easy to be worked with, and nearly all the functions we habitual1y use in mathematical physics are of this c1ass. A notable thing is that the complex numbers spoken of in the second chapter make this theory to a great extent. Large tracts of mathematics have, of course, not been mentioned here. Thus, there is an elaborate theory of integer numbers to be referred to in a note to the seventh chapter, and a geometry using the conceptions of the ancient Greeks and methods of modenl mathematical thought; and very many men stm regard space-perception as something mathematics deals with. We will return to this soon. Again, algebra has developed and branched off; the study of functions in general and in particular has grown; and soon a list of some of the many great men who have helped in all this would not be very useful. Let us now try to resume what we have seen of the development of mathematics along what seem to be its main lines.
Th~ NQtu,~
0/ Mathematics
61
In the earliest times men were occupied with particular questions--the properties of particular numbers and geometrical properties of particular figures, together with simple mechanical questions. With the Greeks, a more general study of classes of geometrical figures began. But traces of an earlier exception to this study of particulars are afforded by "algebra." In it and its later form symbols-like our present x and y-took the place of numbers, so that, what is a great advance in economy of thought and other labour, a part of calculation could be done with symbols instead of numbers, so that the one result stated, in a manner analogous to that of Greek .geometry, a proposition valid for a whole infinite class of different numbers. The great revolution in mathematical thought brought about by Descartes in 1637 grew out of the application of this general algebra to geometry by the very natural thought of substituting the numben expressing the lengths of straight lines for those lines. Thus a point in a planefor instance-is determined in position by two numbers x and y, or coordinates. Now, as the point in question varies in position, x and y both vary; to every x belongs, in general, one or more y's, and we arrive at the most beautiful idea of a single algebraical equation between x and y representing the whole of a curve-the one "equation of the curve" expressing the general law by which, given any particular x out of an infinity of them, the corresponding y or y's can be found. The problem of drawing a tangent-the limiting position of a secant, when the two meeting points approach indefinitely close to one another-at any point of a curve came into prominence as a result of Descartes' work, and this, together with the allied conceptions of velocity and acceleration "at an instant," which appeared in Galileo's classical investigation, published in 1638, of the law according to which freely falling bodies move, gave rise at length to the powerful and convenient "infinitesimal calculus" of Leibniz and the "method of fluxions" of Newton. Mathematically, the finding of the tangent at the point of a curve. and finding the velocity of a particle describing this curve when it gets to that point, are identical problems. They are expressed as finding the "differential quotient," or the "fluxion" at the point. It is now known to be very probable that the above two methods, which are theoretically-but not practically-the same, were discovered independently; Newton discovered his fint, and Leibniz published his first, in 1684. The finding of the areas of curves and of the shapes of the curves which moving particles describe under given forces showed themselves, in this calculus, as results of the inverse process to that of the direct process which serves to find tangents and the law of attraction to a given point from the datum of the path described by a particle. The direct process is called "differentiation," the invene process "integration ...
62
PIIIII" E. B.
J6fU'dalfl
Newton's fame is chiefty owing to his application of this method to the solution, which, in its broad outlines, he gave of the problem of the motion of the bodies in the solar system, which includes his discovery of the law according to which all matter gravitates towards-is attracted by--other matter. This was given in his Principia of 1687; and for more than a century afterwards mathematicians were occupied in extending and applying the calculus. Then came more modern work, more and more directed towards the putting of mathematical methods on a sound logical basis, and the separation of mathematical processes from the sense-perception of space with which so much in mathematics grew and grows up. Thus trigonometry took its place by algebra as a study of certain mathematical functions, and it began to appear that tbe true business of geometry is to supply beautiful and suggestive pictures of abstract-Uanalytical" or "algebraical" or even "arithmetical," as tbey are caned-processes of matbematics. In tbe next cbapter we sball be concerned witb part of the work of logical examination and reconstruction.
CHAPTER VI MODER.N VIEWS OF LIMITS AND NUMBER.S
LET us try to form a clear idea of the conception wbich showed itself to be fundamental in tbe principles of the infinitesimal calculus, the conception of a limit. Notice that tbe limit of a sequence is a number wbich is already defined. We cannot prove that there is a limit to a sequence unless the limit sought is among the numbers already defined. Thus, in the system of unumbers" -here we must refer back to the second cbapter--consisting of all fractions (or ratios), we can say that the sequence (where 1 and 2 are written for the ratios and *) 1, 1 + lA.!, 1 + lh + tA, ... , has a limit (2), but that the sequence
*
1, 1 + ~o, 1 + ~o +
*00. 1 + ~1(J +
*00
+ ~ooo. . . .,
or 1·4142 . . . , got by extracting the square root of 2 by the known process of decimal arithmetic, has not. In fact, it can be proved that the!'e is no ratio such that it is a limit for the above sequence. If there were, and it were denoted by "x," we would have x 2 = 2. Here we come again to the question of incommensurables and "irrational numbers." The Greeks were quite right in distinguishing so sharply between numbers and magnitudes, and it was a tacit, natural, and unjustified-not, as it happens, incorrect-presup-
position that the series of numbers, completed into the series of what are called "real numbers," exactly corresponds to the series of points on a straight line. The series of points which represents the sequence last named seems undoubtedly to possess a limit; this limiting point was assumed to represent some number, and, since it could not represent an integer or a ratio, it was said to represent an "irrational number," .y2. Another irrational number is that which is represented by the incommensurable ratio of the circumference of a circle to its diameter. This number is denoted by the Greek letter "1f," and its value is nearly 3 '1416. . . . Of course, the process of approximation by decimals never comes to an end. The subject of limits forced itself into a very conspicuous place in the seventeenth and eighteenth centuries owing to the use of infinite series as a means of approximate calculation. I shall distinguish what I call "sequences" and "series." A sequence is a collection-finite or infinite--of numbers; a series is a finite or infinite collection of numbers connected by addition. Sequences and series can be made to correspond in the following way. To the sequence 1, 2, 3, 4, . . . belongs a series of which the terms are got by subtracting, in order, the terms of the sequence from the ones immediately following them, thus: (2 - 1) + (3 - 2) + (4 - 3) + ... = 1 + 1 + 1 + ... ; and from a series the corresponding sequence can be got by making the sum of the first, the first two, the first three, . . . terms the first, second, third . . . term of the sequence respectively. Thus, to the series 1 + 1 + 1 + ... corresponds the sequence 1, 2, 3, . . . Now, if a series has only a finite number of terms, it is possible to find the sum of all the terms; but if the series is unending, we evidently cannot. But in certain cases the corresponding sequence has a limit, and this limit is called by mathematicians, neither unnaturally nor accurately, "the sum to infinity of the series." Thus, the sequence 1, 1 + 'h, 1 + 'h + 1A, ••• has the limit 2, and so the ~um to infinity of the series 1 + 'h + 1A, + l;8 + . . . is 2. Of course, aU series do not have a sum: thus 1 + 1 + 1 + . . . to infinity has not-the terms of the corresponding sequence increase continually beyond all limits. Notice particularly that the terms of a sequence may increase continually, and yet have a limit-those of the above sequence with limit 2 so increase, but not beyond 2, though they do beyond any number less than 2; also notice that the terms of a sequence may increase beyond an limits even if the terms of the corresponding series continually diminish, remaining positive, towards O. The series 1 + 'h + ;3 + :tA + ~ + ... is such a series; the terms of the sequence slowly increase beyond all Jimits, as we see when we reflect that the sums ;3 + :tA, ~ + % + 111 +%, ~ + . . . + 1;16, • • • are aU greater than 'h. It is very important to realise the fact illustrated by this example; for it shows that the conditions under which an infinite
PIIIUp E. B. lorudidn
series has a sum are by no means as simple as they might appear at first sight. The logical scrutiny to which, during the last century, the processes and conceptions of mathematics have been subjected, showed very plainly that it was a sheer assumption that such a process as 1-4142 ••. , though all its terms are less than 2, for example, has any limit at all. When we replace numbers by points on a straight line, we feel fairly sure that there is one point which behaves to the points representing the above sequence in the same sort of way as 2 to the sequence 1, 1 + lh, 1 + ~ + lA., _ • • Now, if our system of numbers is to form a continuum, as a line seems to our thoughts to be; so that we can affinn that our number system is adequate, when we introduce axes in the manner of analytical geometry, to the description of all the phenomena of change of position which take place in our space,« then we must have a number -0 which is the limit of the sequence 1'4142 . . . if 2 is of the series 1 + ~ + lA. + ... , for to every point of a line must correspond a number which is subject to the same rules of calculation as the ratios or integers. Thus we must, to justify from a logical point of view our procedure in the great mathematical methods, show what irrationals are, and define them before we can prove that they are limits. We cannot take a series, whose law is evident, which has no ratio for sum, and yet such that the terms of the corresponding 1 1 sequence all remain less than some fixed number (such as 1 + - + 1 1'2
1
1
1'2'3
1'2-3-4
--+
+. . .,
+
when aU the terms of the corresponding se-
quence are less than 3, for example), and then say that it "defines a limit.» All we can prove is that if such a series has a limit, then, if the terms of its corresponding sequence do not decrease as we read from left to right (as in the preceding example), it cannot have more than one limit. Some mathematicians have simply postulated the irrationals. At the beginning of their discussions they have, tacitly or not, said: "In what follows we will assume that there are such things as fill up kinds of gaps in the system of rationals (or ratios)." Such a gap is shown by this. The rationals less than ~ and those greater than lh form two sets and lh divides them. The rationals x such that x 2 is greater than 2 and those x's such that x 2 is less than 2 form two analogous sets, but there is only an analogue to the dividing number lh if we postulate a number V2. Thus by ., The only kind of change dealt with in the science of mechanics is change of posi. tion, that is, motion. It does not seem to me to be necessary to adopt the doctrine that the complete description of any physical event is of a mechanical event; for it is possible to assign and calculate with numbers of our number-continuum to other varying characteristics (such as temperature) of the state of a body besides position.
The NQlIl" 01 ltIatlutmatlc$
postulation we fill up these subtle gaps in the set of rationals and get a continuous set of real numbers. But we can avoid this postulation if we define "V2" as the name of the class of rationals x such that x 2 is less than 2 and "Ph)" as the name of the class of rationals x such that x is less than lh. Proceeding thus, we arrive at a set of classes, some of which correspond to rationals, as (lh) to lh, but the rest satisfy our need of a set without gaps. There is no reason why we should not say that these classes are the real numbers which include the irrationals. But we must notice that rationals are never real numbers; lh is not (lh), though analogous to it. We have much the same state of things as in the second chapter, where 2, +2 and were distinguished and then deliberately confused because, with the mathematicians, we felt the importance of analogy in calculation. Here again we identify (lh) with lh, and so on. Thus, integers, positive and negative '·numbers," ratios, and real "numbers" are all different things: real numbers are classes, ratios and positive and negative numbers are relations. Integers, as we shaU see, are classes. Very possibly there is a certain arbitrariness about this, but this is unimportant compared with the fact that in modem mathematics we have reduced the definitions of all "numbers" to logical terms. Whether they are classes or relations or propositions or other logical entities is comparatively unimportant. Integers can be defined as certain classes. Mathematicians like Weierstrass stopped before they got as far as this: they reduced the other numbers of analysis to logical developments out of the conception of integer, and thus freed analysis from any remaining trace of the sway of geometry. But it was obvious that integers had to be defined, if possible, in logical terms. It has long been recognised that two collections consist of the same number of objects if, and only if, these collections can be put in such a relation to one another that to every object of each one belongs one and only one object of the other. We must not think that this implies that we have already the idea of the number one. It is true that "one and only one" seems to use this idea. But "the class a has one and only one member" is simply a short way of expressing: "x is a member of a, and if y is also a member of a, then y is identical with x." It is true, also, that we use the idea of the unity or the individuality of the things considered. But this unity is a property of each individual, while the number 1 is a property of a class. If a class of pages of a book is itself. under the name of a "volume," a member of a class of books, the same class of pages has both a number (say 360) and a unity as being itself a member of a class. The relation spoken of above in which two classes possessing the same number stand to one another does not involve counting. Think of the fingers on your hands. If to every finger of each hand belongs, by some process of correspondence, one and only one--remember the above mean-
*
Plltllp E. B. IDWtlldn
ing of this pbrase--of the other, they are said to have "the same number." This is a definition of what "the same number" is to mean for US; the word "number" by itself is to have, as yet, no meaning for us; and, to avoid confusion, we had better replace the phrase "have the same number" by the words "are equivalent." Any other word would, of course, do, but this word happens to be fairly suggestive and customary. Now, if the vari~ able u is any class, "the number of u" is defined as short for the phrase: "the class whose members are classes which are similar to u." Thus the number of u is an entity which is purely logical in its nature. Some people might urge that by "number" they mean something different from this, and that is quite possible. An that is maintained by those who agree to the process sketched above is: (1) Classes of the kind described are iden~ tical in all known arithmetical properties with the undefined things people call "integer numbers"; (2) It is futile to say: "These classes are not num~ bers," if it is not also said what numbers are-that is to say, if "the number of' is not defined in some more satisfactory way. There may be more satisfactory definitions, but this one is a perfectly sound foundation for all mathematics, including the theory not touched upon here of ordi~ nal numbers (denoted by "first," "second," . . . ) which apply to sets arranged in some order, known at present. To iUustrate (1), think of this. Acording to the above definition 2 is the general idea we call "couple." We say: "Mr. and Mrs. A. are a couple"; our definition would ask us to say in agreement with this: "The class consisting of Mr. and Mrs. A. is a member of the class 2." We define "2" as "the class of classes u such that, if x is a u, u lacking x is a I"; the definition of "3" fol1ows that of "2"; and so on. In the same way, we see that the class of fingers on your right hand and the class of fingers on your left hand are each of them members of the class 5. It follows that the classes of the fingers are equivalent in the above sense. Out of the striving of human minds to reproduce conveniently and anticipate the results of experience of geometrical and natural events, mathematics has developed. Its development gave priceless hints to the development of logic, and then it appeared that there is no gap between the science of number and the science of the most general relations of objects of thought. As for geometry and mathematical physics, it becomes possible clearly to separate the logical parts from those parts which formu· late the data of our experience. We have seen that mathematics has often made great strides by sacri· ficing accuracy to analogy. Let us remember that, though mathematics and logic give the highest forms of certainty within the reach of us, the process of mathematical discovery, which is so often confused with what is dis· covered, has led through many doubtful analogies and errors arriving
Tile Nllture 0/ Mlllllemlllics
from the great help of symbolism in making the difficult easy. Fortunately symbolism can also be used for precise and subtle analysis, so that we can say that it can be made to show up the difficulties in what appears easy and even negligible-like 1 + 1 2. This is what much modem fundamental work does.
=
CHAPTER VII THE NATURE OF MATHEMATICS
IN the preceding chapters we have followed the development of certain branches of knowledge which are usually classed together under the name of "mathematical knowledge." These branches of knowledge were never clearly marked off from all other branches of knowledge: thus geometry was sometimes considered as a logical study and sometimes as a natural science-the study of the properties of the space we live in. Still less was there an absolutely clear idea of what it was that this knowledge was about. It had a name-umathematics"-and few except "practical" men and some philosophers doubted that there was something about which things were known in that kind of knowledge called "mathematical." But what it was did not interest very many people, and there was and is a great tendency to think that the question as to what mathematics is could be answered if we only knew all the facts of the development of our mathematical knowledge. It seems to me that this opinion is, to a great extent, due to an ambiguity of language: one word-"mathematics"-is used both for our knowledge of a certain kind and the thing, if such a thing there be, about which this knowledge is. I have distinguished, and will now explicitly distinguish, between "Mathematics," a collection of truths of which we know something, and "mathematics," our knowledge of Mathematics. Thus, we may speak of "Euclid's mathematics" or "Newton's mathematics," and say truly that mathematics has developed and therefore had history; but Mathematics is eternal and unchanging, and therefore has no history-it does not belong, even in part, to Euclid or Newton or anybody else, but is something which is discovered, in the course of time, by human minds. An analogous distinction can be drawn between "Logic" and "logic." The sman initial indicates that we are writing of a psychological process which may lead to Truth; the big initial indicates that we are writing of the entity-the part of Truth-to which this process leads us. The reason why mathematics is important is that Mathematics is not incomprehensible, though it is eternal and unchanging. Grammatical usage makes us use a capital letter even for "rna the-
PIIIU, B. B.
J~1lI'tIIIhI
matics" in the psychological sense when the word begins a sentence, but in this case I have guarded and will guard against ambiguity. That particular function of history which I wish here to emphasise wilJ now, I think, appear. In mathematics we gradually learn, by getting to know some thing about mathematics, to know that there is such a thing as Mathematics. We have, then, glanced al the mathematics of primitive peoples, and have seen that at first discoveries were of isolated properties of abstract things like numbers or geometrical figures, and of abstract relations between concrete things like the relations between the weights and the arms of a lever in eqUilibrium. These properties were, at first, discovered and applied, of course, with the sole object of the satisfaction of bodily needs. With the ancient Greeks comes a change in point of view which perhaps seems to us, with our defective knowledge, as too abrupt. So far as we know, Greek geometry was, from its very beginning, deductive, general, and studied for its own interest and not for any applications to the concrete world it might have. In Egyptian geometry, if a result was stated as universally true, it was probably only held to be so as a result of induction -the conclusion from a great number of particular instances to a general proposition. Thus, if somebody sees a very large number of officials of a certain railway company, and notices that all of them wear red ties, he might conclude that all the officials of that company wear red ties. This might be probably true: it would not be certain: for certainty it would be necessary to know that there was some rule according to which all the officials were compelled to wear red ties. Of course, even then the conclusion would not be certain, since these sort of laws may be broken. Laws of Logic, however, cannot be broken. These laws are not, as they are sometimes said to be, laws of thought; for logic has nothing to do with the way people think, any more than poetry has to do with the food poets must eat to enable them to compose. Somebody might think that 2 and 2 make 5: we know, by a process which rests on the laws of Logic, that they makp, 4. This is a more satisfactory case of induction: Fermat stated that no integral values of x, y, and z can be found such that x" + y" z", if n be an integer greater than 2. This theorem has been proved to be true for n 3, 4, 5, 7, and many other numbers, and there is no reason to doubt that it is true. But to this day no general proof of it has been given. s This, then, is an example of a mathematical proposition which has been reached and stated as probably true by induction. Now, in Greek geometry, propositions were stated and proved by the
=
=
8 This is an example of the "theory of numbers," the study of the properties of integers, to which the chief contributions, perhaps, have been made by Fermat and Gauss.
The Naill" 0/ Malhem4llcI
laws of Logic-he1ped, as we now know, by tacit appeals to the conclusions which common sense draws from the pictorial representation in the mind of geometrical figures-about any triangles, say, or some triangles, and thus not about one or two particular things~ but about an infinity of them. Thus, consider any two triangles ABC and DEF. It he1ps the thinking of most of us to draw pictures of particular triangles, but our conclusions do not hold merely for these triangles. If the sides BA and A C are equal in length to the sides ED and DF respectively, and the angle at A is equal to the angle at D, then BC is equal to EF. This is proved rather imperfectly in the fourth proposition of the first Book of Euclid's Elements. When we examine into and complete the reasonings of geometricians, we find that the conception of space vanishes, and that we are left with logic alone. Philosophers and mathematicians used to think-and some do now-that, in geometry, we had to do, not with the space of ordinary life in which our houses stand and our friends move about, and which certain quaint people say is "annihilatedtt by electric telegraphs or motor cars, but an abstract form of the same thing from which all that is personal or material has disappeared, and only things like distance and order and position have remained. Indeed, some have thought that position did not remain; that, in abstract space, a circle, for example, had no position of its own, but only with respect to other things. Obviously, we can only, in practice, give the position of a thing- with respect to other things-"relatively" and not "absolutely." These Hrelativists" denied that position had any properties which could not be practically discovered. Relativism, in a thought..aut form, seems quite tenable; in a crude form, it seems like excluding the number 2, as distinguished from classes of two things, from notice as a figment of the brain, because it is not visible or tangible like a poker or a bit of radium or a mutton-chop. In fact, a perfected geometry reduces to a series of deductions holding not only for figures in space, but for any abstract things. Spatial figures give a striking illustration of some abstract things; and that is the secret of the interest which analytical geometry has. But it is into algebra that we must now look to discover the nature of Mathematics. We have seen that Egyptian arithmetic was more general than Egyptian geometry: like algebra, by using letters to denote unknown numbers, it began to consider propositions about any numbers. In algebra and algebraical geometry this quickly grew, and then it became possible to treat branches of mathematics in a systematic way and make whole classes of problems subject to the uniform and almost mechanical working of one method. Here we must again recall the economical function of science. At the same time as methods-algebra and analytical geometry and the infinitesimal calculus-grew up from the application of mathematics to
70
Philip E. B. IDWdtdll
natural science, grew up also the new conceptions which influenced the form which mathematics took in the seventeenth, eighteenth, and nineteenth centuries. The ideas of variable and function became more and more prominent. These ideas were brought in by the conception of motion, and, unaffected by the doubts of the few logicians in the ranks of the mathematicians, remained to fructify mathematics. When mathematicians woke up to the necessity of explaining mathematics logically and finding out what Mathematics is, they found that, in mathematics the striving for generality had led, from very early times, to the use of a method of deduction used but not recognised and distinguished from the method usually used by the Aristotelians. I will try to indicate the nature of these methods, and it will be seen how the ideas of variable and function, in a form which does not depend on that particular kind of variability known as motion, come in. A proposition in logic is the kind of thing which is denoted by such a phrase as: "Socrates was a mortal and the husband of a scold." If-and this is the characteristic of modern logic-we notice that the notions of variable and function (correspondence, relation) which appeared first in a special form in mathematics, are fundamental in all the things which are the objects of our thought, we are led to replace the particular conceptions in a proposition by variables, and thus see more clearly the structure of the proposition. Thus: "x is a y and has the relation R to Z, a member of the class u" gives the general form of a multitude of propositions, of which the above is a particular case; the above proposition may be true, but it is not a judgment of logic, but of history or experience. The proposition is false if "Kant" or "Westminster Abbey" is substituted for "Socrates": it is neither if "x," a sign for a variable, is, and then becomes what we call a "propositional function" of x and denote by "cpx" or "!/Ix." If more variables are involved, we have the notation "«p(x,y)," and so on. Relations between propositional functions may be true or false. Thus x is a member of the class a, and a is contained in the class b, together imply that x is a b, is true. Here the implication is true, and we do not say that the functions are. The kind of implication we use in mathematics is of the form: "If «px is true, then !/Ix is true"; that is, any particular value of x which makes «px true also makes !/Ix true. From the perception that, when the notions of variable and function are introduced into logic, as their fundamental character necessitates, all matherllatical methods and all mathematical conceptions can be defined in purely logical terms, leads us to see that Mathematics is only a part of Logic and is the class of all propositions of the form: «p(x,y,z, .•• ) implies, for all values of the variables, ",(x,y,z • .•. ). The structure of the propositional functions involves only such ideas as are fundamental in logic, like implication, class, relation, the relation of a term to a class
71
of which it is a member, and so on. And, of course, Mathematics depends on the notion of Truth. When we say that "1 + 1 = 2," we seem to be making a mathematical statement which does not come under the above definition. But the statement is rather mistakenly written: there is, of course, only one whole class of unit classes, and the notation "1 + 1n makes it look as if there were two. Remembering that 1 is a class of certain classes, what the above proposition means is: If x and yare members of 1, and x differs from y, then x and y together make up a member of 2. At last, then, we arrive at seeing that the nature of Mathematics is independent of us personally and of the world outside, and we can feel that our own discoveries and views do not affect the Truth itself, but only the extent to which we or others see it. Some of us discover things in science, but we do not really create anything in science any more than Columbus created America. Common sense certainly leads us astray when we try to use it for the purposes for which it is not particularly adapted, just as we may cut ourselves and not our beards if we try to shave with a carving knife; but it has the merit of finding no difficulty in agreeing with those philosophers who have succeeded in satisfying themselves of the truth and position of Mathematics. Some philosophers have reached the startling conclusion that Truth is made by men, and that Mathematics is created by mathematicians, and that Columbus created America; but common sense, it is refreshing to think, is at any rate above being flattered by philosophical persuasion that it really occupies a place sometimes reserved for an even more sacred Being.
BIBLIOGRAPHY THE view that science is dominated by the principle of the economy of thought has been, in part,D very thoroughly worked out by Ernst Mach (see especially the translation of his Science of Mechanics, 5th ed., La Salle, Ill., 1942). On the history of mathematics, we may mention W. W. Rouse BaWs books, A Primer of the History of Mathematics (7th ed., 1930), and the fuller Short Account of the History of Mathematics (4th ed., 1908, both published in London by Macmillan), and Karl Fink's Brief History of Mathematics (Chicago and London, 3rd ed., 1910). As text-books of mathematics, De Morgan's books on Arithmetic, Algebra, and Trigonometry are still unsurpassed, and his Trigonometry and Double Algebra contains one of the best discussions of complex numbers, for students, that there is. As De Morgan's books are not all easy to get, 9
C/o above, pp. 5, 11, 13, 15, 16, 42.
n
PAillp B. B. lol11'd1i11n
the reprints of his Elementary Illustrations 0/ the Differential and Integral Calculus and his work On the Study and Difficulties 0/ Mathematics (Chicago and London, 1899 and 1902) may be recommended. Where possible, it is best to read the works of the great mathematicians themselves. For elementary books, Lagrange's Lectures on Elementary Mathematics, of which a translation has been published at Chicago and London (2nd ed., 1901), is the most perfect specimen. The questions dealt with in the fourth chapter are more fully discussed in Mach's Mechanics. An excel1ent collection of methods and problems in graphical arithmetic and algebra, and so on, is contained in H. E. Cobb's book on Elements 0/ Applied Mathematics (Boston and London: Ginn & Co., 1911).
PART II
Historical and Biographical 1. 2. 3. 4. S. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
The Great Mathematicians by HERBERT WESTREN TURNBULL The Rhind Papyrus by JAMES R. NEWMAN Archimedes by PLUTARCH, VITRUVlUS, TZETZES Greek Mathematics by IVOR THOMAS The Declaration of the Profit of Arithmeticke by ROBERT RECORDE Johann Kepler by SIR OLIVER LOOOE The Geometry by RENE DESCARTES Isaac Newton by E. N. DA C. ANDRADE Newton, the Man by JOHN MAYNARD KEYNES The AnaJyst by BISHOP BERKELEY Gauss, the Prince of Mathematicians by ERIC TEMPLE BELL Invariant Twins, Cayley and Sylvester by ERIC TEMPLE BELL Srinivasa Ramanujan by JAMES R. NEWMAN My MentaJ Development by BERTRAND RUSSELL Mathematics as an Element in the History of Thought by ALFRED NORTH WHITEHEAD
COMMENTARY ON
The Great Mathematicians A T the outset of assembling this anthology I decided that I ought to
.t"\.
include a biographical history of the subject. This would provide a setting for the other selections, and also serve as a small reference man· ual for the general reader. It was not easy to find a history which was brief, authoritative, elementary and readable. W. W. R. Ball's A Primer of the History of Mathemotics is a book of merit but rather old· fashioned. J. W. N. Sullivan's The History of Mathematics in Europe; an admirable outline, carries the story only as far as the end of the eighteenth century; I commend this book to your attention. Dirk Struik's A Concise History of Mathematics has solid virtues but is a trifle too advanced for my pur· poses and at times dull. A few French and German books which might have been suitable were not considered because of the labor of translating them. Turnbull's excel1ent little volume, a biographical history, turned out to meet the standard in all respects. It is the story of several great mathematicians, "representatives of their day in this venerable science." "I have tried to show," says Professor Turnbull in his preface, "how a mathematician thinks, how his imagination, as well as his reason, leads him to new aspects of the truth. Occasionally it has been necessary to draw a figure or quote a formula-and in such cases the reader who dislikes them may skip, and gather up the thread undismayed a little further on. Yet I hope that he will not too readily tum aside in despair, but will, with the help of the accompanying comment, find something to admire in these elegant tools of the craft." There is overlapping between this survey and the preceding selection, but the two books are complementary. and the reader who enjoys one will derive no less pleasure from the other. Jourdain makes ideas the heroes of his account while Turnbull devotes a good deal of space to lively sketches of the men who made the ideas. H. W. Turnbull, distinguished for his researches in algebra (determinants, matrices, theory of equations), is Regius professor of mathematics at the University of St. Andrews in Scotland, a Fellow of the Royal Society, and, as demonstrated not only in this volume but in other writings, a gifted simplifier of mathematical ideas.
74
We think 0/ Euclid as 0/ lint! ice; we admire Newton as we admire the Peak 0/ TeneriDe. Even the in tensest Jabors, the most remote triumphs 0/ the abstract intellect, seem to carry us into a region diDerent from our own-to be in a terra incopita 0/ pure reasoning, to cast a chill on human -WALTER BAGEHOT glory. Many small make a great. -CHAUCElt Everything 0/ importance has been said be/ore by somebody who did not discover It. -ALFllED NORTH WHrrBHEAD
1
The Great Mathematicians By HERBERT WESTREN TURNBULL PREFACE
THE usefulness of mathematics in furthering the sciences is commonly acknowledged: but outside the ranks of the experts there is little inquiry into its nature and purpose as a deliberate human activity. Doubtless this is due to the inevitable drawback that mathematical study is saturated with technicalities from beginning to end. Fully conscious of the difficul· ties in the undertaking, I have written this little book in the hope that it will help to reveal something of the spirit of mathematics, without unduly burdening the reader with its intricate symbolism. The story is told of several great mathematicians who are representatives of their day in this venerable science. I have tried to show how a mathematician thinks, how his imagination, as well as his reason, leads him to new aspects of the truth. Occasionally it has been necessary to draw a figure or quote a formula-and in such cases the reader who dislikes them may skip, and gather up the thread undismayed a little further on. Yet I hope that he will not too readily turn aside in despair, but will, with the help of the accompanying comment, find something to admire in these elegant tools of the craft. Naturally in a work of this size the historical account is incomplete: a few references have accordingly been added for further reading. It is pleasant to record my deep obligation to the writers of these and other larger works, and particularly to my college tutor, the late Mr. W. W. Rouse Ball, who first woke my interest in the subject. My sincere thanks are also due to several former and present colleagues in st. Andrews who have made a considerable and illuminating study of mathematics among the Ancients: and to kind friends who have offered many valuable suggestions and criticisms. In preparing the Second Edition I have had the benefit of suggestions 75
76
which friends from time to time have submitted. I am grateful for this means of removing minor blemishes, and for making a few additions. In particular, a date list has been added.
PREFACE TO THIRD EDITION A FEW additions have been made to the earlier chapters and to Chapter VI, which incorporate results of recent discoveries among mathematical inscriptions and manuscripts, particularly those which enlarge our knowledge of the mathematics of Ancient Babylonia and Egypt. I gratefully acknowledge the help derived from reading the Manual 01 Greek Mathematics (1931) by Sir Thomas Heath. It provides a short but masterly account of these developments, for which the scientific world is greatly indebted. H. W. T. December, 1940.
PREFACE TO FOURTH EDITION AT the turn of the half-century it is appropriate to add a postscript to Chapter XI, which brought the story of mathematical development as far as the opening years of the century. What has happened since has followed very directly from the wonderful advances that opened up through the algebraical discoveries of Hamilton, the analytical theories of Weierstrass and the geometrical innovations of Von Staudt, and of their many great contemporaries. One very noteworthy development has been the rise of American mathematics to a place in the front rank, and this has come about with remarkable rapidity and principally through the study of abstract algebra such as was inspired by Peirce. a great American disciple of the Hamiltonian school. Representative of this advance in algebra is Wedderburn who built upon the foundations laid, not only by Peirce. but also by Frobenius in Germany and Cartan in France. Through abandoning the commutative law of multiplication by inventing quaternions, Hamilton had opened the door for the investigation of systems of algebra distinct from the ordinary familiar system. Algebra became algebras just as, through the discovery of non-Euclidean systems, geometry became geometries. This plurality. which had been unsuspected for so long, naturally led to the study of the classification of algebras. It was here that Wedderburn, following a hint dropped by Cartan, attained great success. The matter led to deeper and wider understanding of abstract theory, while at the same time it provided a welcome and fertile medium for the
77
Th~ Gr~41 MQth~m4Ilcltuf$
further developments in quantum mechanics. Simultaneously with this abstract approach to algebra a powerful advance was made in the technique of algebraical manipulation through the discoveries of Frobenius, Schur and A. Young in the theory of groups and of their representations and applications. Similar trends may be seen in arithmetic and analysis where the same plurality is in evidence. Typical of this are the theory of valuation and the recognition of Banach spaces. The axiom of Archimedes (p. 99) is here in jeopardy: which is hardly surprising once the concept of regular equal steps upon a straight line had been broadened by the newer forms of geometry. Arithmetic and ana1ysis were, so to speak, projected and made more abstract. It is remarkable that, with these trends towards generalization in each of the four great branches of pure mathematics, the branches lose something of their distinctive qualities and grow more alike. Whitehead's description of geometry as the science of crossclassification remains profoundly true. The applications of mathematics continue to extend, particularly in logic and in statistics. H. W. T. May. 1951.
CONTENTS CHAPTER
PAGE
PREPACE DATE LIST I.
EARLY BEGINNINGS: THALES, PYTHAGORAS AND THE PYTHAoOREANS
II.
75 78
EUDOXUS AND THE ATHENIAN SCHOOL
79 90
m.
ALEXANDRIA: EUCLID, ARCHIMEDES AND APOLLONIUS
IV.
THE SECOND ALEXANDRIAN SCHOOL: PAPPUS AND DIOPHANTUS
100 108
V.
THE RENAISSANCE: NAPIER AND KEPLER; THE RISE OF ANALYSIS
117
VI.
DESCARTES AND PASCAL: THE EARLY FRENCH GEOMETERS AND THEIR CONTEMPORARIES
VII.
vm. lX. X. XI.
ISAAC NEWTON THE BERNOULLIS AND EULER MACLAURIN AND LAGRANGE GAUSS AND HAMILTON: THE NINETEENTH CENTURY MORE RECENT DEVELOPMENTS
128 140 147 151 156 163
DATE LIST 1 18th Century B.C. 6th " " 5th " 4th
'"
3rd
"
"
"
A.D.
2nd lst 2nd 3rd 6th 7th 12th 16th
"
u
17th
18th
19th
20th
u
Ahmes (11800- ). Thates (640-550), Pythagoras (569-500), Anaxagoras (500-428), Zeno (495-435), Hippocrates (470- ), Democritus (1470- ). Archytas (1 400), Plato (429-348), Eudoxus (408355), Menaechmus (375-325). Euclid (1330-275), Archimedes (287-212), Apollonius (? 262-2(0) . Hipparchus (1160- ) . Menelaus (1 100). Ptolemy (1100-168). Hero (1250), Pappus (1300), Diophantus ( 3201). Arya-Bhata (1530). Brahmagupta (1640). Leonardo of Pisa (1175-1230). Scipio Ferro (1465-1526), Tartaglia (1500-1557), Cardan (1501-1576), Copernicus (1473-1543), Vieta (1540-1603), Napier (1550-1617), Galileo (1564-1642), Kepler (1571-1630), Cavalieri (1598-1647) . Desargues (1593-1662), Descartes (1596-1650), Fermat (1601-1665), Pascal (1623-1662), Wallis (1616-1703), Barrow (1630-1677), Gregory (1638-1675), Newton (1642-1727), Leibniz (1646-1716), Jacob Bernoulli (1654-1705), John Bernoulli (1667-1748). Euler (1707-1783), Demoivre (1667-1754), Taylor (1685-1741), Maclaurin (1698-1746), D'Alembert (1717-1783), Lagrange (1736-1813), Laplace (1749-1827), Cauchy 0759-1857). Gauss (1777-1855), Von Staudt (1798-1867), Abel (1802-1829), Hamilton (1805-1865) , Galois (1811-1832), Riemann (1826-1866), Sy1vester (1814-1897), Cayley (1821-1895), Weierstrass (1815-1897), and many others. Ramanujan (1887-1920), and many living mathematicians.
79
Tlttt GNOI MQlltemtlllcilUfl
CHAPTER I EARLY BEGINNINGS: THALES, PYTHAGORAS AND THE PYTHAGOREANS
TO·DAY with all our accumulated skill in exact measurements, it is a noteworthy event when lines driven through a mountain meet and make a tunnel. How much more wonderful is it that lines, starting at the comers of a perfect square, should be raised at a certain angle and successfully brought to a point, hundreds of feet aloft! For this, and more, is what is meant by the building of a pyramid: and aU this was done by the Egyptians in the remote past, far earlier than the time of Abraham. URfortunately we have no actual record to tell us who first discovered enough mathematics to make the building possible. For it is evident that such gigantic erections needed very accurate plans and models. But many general statements of the rise of mathematics in Egypt are to be found in the writings of Herodotus and other Greek travellers. Of a certain king Sesostris, Herodotus says: 'This king divided the land among all Egyptians so as to give each one a quadrangle of equal size and to draw from each his revenues, by imposing a tax to be levied yearly. But everyone from whose part the river tore anything away, had to go to him to notify what had happened; he then sent overseers who had to measure out how much the land had become smaller, in order that the owner might pay on what was left, in proportion to the entire tax imposed. In this way, it appears to me, geometry originated, which passed thence to Hellas.' Then in the Phaedrus Plato remarks: 'At the Egyptian city of Naucratis there was a famous old god whose name was Theuth; the bird which is caned the Ibis was sacred to him, and he was the inventor of many arts, such as arithmetic and calculation and geometry and astronomy and draughts and dice, but his great discovery was the use of letters.' According to Aristotle, mathematics originated because the priestly class in Egypt had the leisure needful for its study; over two thousand years later exact corroboration of this remark was forthcoming, through the discovery of a papyrus, now treasured in the Rhind collection at the British Museum. This curious document, which was written by the priest Ahmes, who lived before 1700 B.C., is called 'directions for knowing all dark things'; and the work proves to be a collection of problems in geometry and arithmetic. It is much concerned with the reduction of fractions such as 2/(2n + 1) to a sum of fractions each of whose numerators is unity. Even with our improved notation it is a complicated matter to work through such remarkable examples as:
'0
BerN" Weltrn TJmtlndI
There is considerable evidence that the Egyptians made astonishing progress in the science of exact measurements. They had their land surveyors, who were known as rope stretchers, because they used ropeSt with knots or marks at equal intervals, to measure their plots of land. By this simple means they were able to construct right angles; for they knew that three ropes, of lengths three, four, and five units respectively, could be formed into a right-angled triangle. This useful fact was not confined to Egypt: it was certainly known in China and elsewhere. But the Egyptian skill in practical geometry went far beyond the construction of right angles: for it included, besides the angles of a square, the angles of other regular figures such as the pentagon, the hexagon and the heptagon. If we take a pair of compasses, it is very easy to draw a circle and then to cut the circumference into six equal parts. The six points of division form a regular hexagon, the figure so well known as the section of the honey cell. It is a much more difficult problem to cut the circumference into five equal parts, and a very much more difficult problem to cut it into seven equal parts. Yet those who have carefully examined the design of the ancient temples and pyramids of Egypt ten us that these particular figures and angles are there to be seen. Now there are two distinct methods of dealing with geometrical problems~the practical and the theoretical. The Egyptians were champions of the practical, and the Greeks of the theoretical method. For example, as Rober has pointed out, the Egyptians employed a practical rule to determine the angle of a regular heptagon. And although it fell short of theoretical precision, the rule was accurate enough to conceal the error, unless the figure were to be drawn on a grand scale. It would barely be apparent even on a circle of radius 50 feet. Unquestionably the Egyptians were masters of practical geometry; but whether they knew the theory, the underlying reason for their results, is another matter. Did they know that their right-angled triangle, with sides of lengths three, four and five units, contained an exact right angle? Probably they did, and possibJy they knew far more. For, as Professor D'Arcy Thompson has suggested, the very shape of the Great Pyramid indicates a considerable familiarity with that of the regular pentagon. A certain obscure passage in Herodotus can, by the slightest literal emendation, be made to yield excellent sense. It would imply that the area of each triangular face of the Pyramid is equal to the square of the vertical height; and this accords well with the actual facts. If this is so, the ratios of height, slope and base can be expressed in terms of the 'golden section', or of the radius of a circle to the side of the inscribed decagon. In short, there was already a wealth of geometrical and arithmetical results treasured by the priests of Egypt, before the early Greek travellers became acquainted with mathematics. But it was only after the keen imaginative
Tltt! Grt!a' Matltt!matlcian'
81
eye of the Greek fell upon these Egyptian figures that they yielded up their wonderful secrets and disclosed their inner nature. Among these early travellers was THALEs, a rich merchant of MiJetus, who lived from about 640 to 550 B.C. As a man of affairs he was highly successful: his duties as merchant took him to many countries, and his native wit enabled him to learn from the novelties which he saw. To his admiring fellow-countrymen of later generations he was known as one of the Seven Sages of Greece, many Jegends and anecdotes clustering round his name. It is said that Thales was once in charge of some mules, which were burdened with sacks of salt. Whilst crossing a river one of the animals slipped; and the salt consequently dissolving in the water, its load became instantly lighter. Naturally the sagacious beast deliberately submerged itself at the next ford, and continued this trick until Thales hit upon the happy expedient of filling the sack with sponges! This proved an effectual cure. On another occasion, foreseeing an unusually fine crop of olives, Thales took possession of every olive-press in the district; and having made this 'comer', became master of the market and could dictate his own terms. But now, according to one account, as he had proved what could be done, his purpose was achieved. Instead of victimizing his buyers, he magnanimously sold the fruit at a price reasonable enough to have horrified the financier of to-day. Like many another merchant since his time Thales early retired from commerce, but unlike many another he spent his leisure in philosophy and mathematics. He seized on what he had learnt in his travels, particularly from his intercourse with the priests of Egypt; and he was the first to bring out something of the true significance of Egyptian scientific lore. He was both a great mathematician and a great astronomer. Indeed, much of his popular celebrity was due to his successful prediction of a solar eclipse in 585 B.C. Yet it is told of him that in contemplating the stars during an evening walk, he fell into a ditch; whereupon the old woman attending him exclaimed, 'How canst thou know what is doing in the heavens when thou seest not what is at thy feet?' We live so far from these beginnings of a rational wonder at natural things, that we run the risk of missing the true import of results now so very familiar. There are the well-known propositions that a circle is bisected by any diameter, or that the angles at the base of an isosceles triangle are equal, or that the angle in a semicircle is a right angle, or that the sides about equal angles in similar triangles are proportional. These and other 1ike propositions have been ascribed to Thales. Simple as they are, they mark an epoch. They elevate the endless details of Egyptian mensuration to general truths; and in like manner his astronomical results replace what was little more than the making of star catalogues by a genuine science.
82
It has been well remarked that in this geometry of Thales we also have the true source of algebra. For the theorem that the diameter bisects a circle is indeed a true equation; and in his experiment conducted, as Plutarch says, 'so simply. without any fuss or instrument' to determine the height of the Great Pyramid by comparing its shadow with that of a vertical stick, we have the notion of equal ratios, or proportion. The very idea of abstracting all solidity and area from a material shape, such as a square or triangle, and pondering upon it as a pattern of lines, seems to be definitely due to Thales. He also appears to have been the first to suggest the importance of a geometrical locus. or curve traced out by a point moving according to a definite law. He is known as the father of Greek mathematics, astronomy and philosophy. for he combined a practical sagacity with genuine wisdom. It was no mean achievement, in his day, to break through the pagan habit of mind which concentrates on particular cults and places. Thales asserted the existence of the abstract and the more general: these, said he, were worthier of deep study than the intuitive or sensible. Here spoke the philosopher. On the other hand he gave to mankind such practical gifts as the correct number of days in the year, and a convenient means of finding by observation the distance of a ship at sea. Thales summed up his speculations in the philosophical proposition 'All things are water'. And the fact that an things are not water is trivial compared with the importance of his outlook. He saw the field; he asked the right questions; and he initiated the search for underlying law beneath aU that is ephemeral and transient. Thales never forgot the debt that he owed to the priests of Egypt; and when he was an old man he strongly advised his pupil PVTHAOORAS to pay them a visit. Acting upon this advice, Pythagoras travelled and gained a wide experience, which stood him in good stead when at length he settled and gathered round him disciples of his own, and became even more famous than his master. Pythagoras is supposed to have been a native of Samos. belonging like Tha]es to the Ionian colony of Greeks planted on the western shores and islands of what we now call Asia Minor. He lived from about 584 B.C. to 495 B.C. In 529 B.C. he settled at Crotona, a town of the Dorian colony in South Italy, and there he began to lecture upon philosophy and mathematics. His lecture-room was thronged with enthusiastic hearers of all ranks. Many of the upper classes attended, and even women broke a law which forbade them to attend public meetings. and flocked to hear him. Among the most attentive was Theano, the young and beautiful daughter of his host Milo, whom he married. She wrote a biography of her husband. but unfortunately it is lost. So remarkable was the influence of this great master that the more attentive of his pupils gradually formed themselves into a society or brother-
83
TIt~ Gr~at Matlt~maI;ciam
hood. They were known as the Order of the Pythagoreans, and they were soon exercising a great influence far across the Grecian world. This influence was not so much political as religious. Members of the Society shared everything in common, holding the same philosophical beliefs, en~ gaging in the same pursuits, and binding themselves with an oath not to reveal the secrets and teaching of the school. When, for example, Hippasus perished in a shipwreck, was not his fate due to a broken promise? For he had divulged the secret of the sphere with its twelve pentagons! A distinctive badge of the brotherhood was the beautiful star pentagram-a fit symbol of the mathematics which the school discovered. It was also the symbol of health. Indeed. the Pythagoreans were specially interested in the study of medicine. Gradually, as the Society spread, teachings once treasured orally were committed to writing. Thereby a copy of a treatise by Philolaus, we are told, ultimately came into the possession of Plato; probably a highly significant event in the history of mathematics.
FlGURB I
In mathematics the Pythagoreans made very great progress, particularly in the theory of numbers and in the geometry of areas and solids. As it was the generous practice among members of the brotherhood to attribute all credit for each new discovery to Pythagoras himself, we cannot be quite certain about the authorship of each particular theorem. But at any rate in the mathematics which are now to be described, his was the dominating influence. In thinking of these early philosophers we must remember that open air and sunJight and starry nights formed their surroundings-not our grey mists and fettered sunshine. As Pythagoras was learning his mensuration from the priests of Egypt, he would constantly see the keen lines cast by the shadows of the piUars across the pavements. He trod chequered floors with their arrays of alternately coloured squares. His mind was stirred by interesting geometrical truths learnt from his master Thates, his interest in
84
number would lead him to count the squares, and the sight of the long straight shadow falling oblique1y across them would suggest sequences of special squares. It falls maybe across the centre of the first, the fourth. the seventh; the arithmetical progression is suggested. Then again the square is interesting for its size. A fragment of more diverse pattern would demonstrate a larger square enclosing one exactly half its size. A
FIGURE 2
little imaginative thought would reveal. within the larger, a smaller square placed unsymmetrically, and so would lead to the great theorem which somehow or other was early reached by the brotherhood (and some say by Pythagoras himself), that the square on one side of a right-angled triangle is equal to the sum of the squares on the remaining sides. The above figures (Figure 2) actually suggest the proof, but it is quite possible that several different proofs were found, one being by the use of similar triangles. According to one story, when Pythagoras first discovered this fine result. in his exultation he sacrificed an ox! Influenced no doubt by these same orderly patterns, he pictured numbers as having characteristic designs. There were the triangular numbers, one, three, six, ten, and so on, ten being the holy tetractys, another sym-
bol highly revered by the brotherhood. A1so there were the square numbers, each of which could be derived from its predecessor by adding an
.. L-shaped border. Great importance was attached to this border: it was caned a gnomon (~fU1>"" carpenter's rule). Then it was recognized that
85
each odd number, three, five, seven, etc., was a gnomon of a square number. For example. seven is the gnomon of the square of three to make the square of four. Pythagoras was also interested in the more abstract natural objects, and he is said to have discovered the wonderful harmonic progressions in the notes of the musical scale, by finding the relation between the length of a string and the pitch of its vibrating note. Thrilled by his discovery, he saw in numbers the element of all things. To him numbers were no mere attributes: three was not that which is common to three cats or three books or three Graces: but numbers were themselves the stuff out of which all objects we see or handle are made----the rational reality. Let us not judge the doctrine too harshly; it was a great advance on the cruder water philosophy of Thales. So, in geometry, one came to be identified with the point; two with the line, three with the surface, and four with the solid. This is a noteworthy disposition that really is more fruitful than the usual alJocation in Which the line is said to have one, the surface two, and the solid three, dimensions. More whimsical was the attachment of seven to the maiden goddess Athene 'because seven alone within the decade has neither factors nor product'. Five suggested marriage, the union of the first even with the first genuine odd number. One was further identified with reason; two with opinion-a wavering fellow is Two; he does not know his own mind: four with justice, steadfast and square. Very fanciful no doubt: but has not Ramanujan, one of the greatest arithmeticians of our own days, been thought to treat the positive integers as his personal friends? In spite of this exuberance the fact remains, as Aristotle sumS. it up: 'The Pythagoreans first applied themselves to mathematics, a science which they improved; and, penetrated with it, they fancied that the principles of mathematics were the principles of all things'. And a younger contemp(Irary, Eudemus, shrewdly remarked that 'they changed geometry into a liberal science; they diverted arithmetic from the service of commerce'. To Pythagoras we owe the very word mathematics and its doubly twofold branches: Mathematics
I
I
I
The discrete
The continued
1
I
The absolute
I
Arithmetic
I
The relative
I.
MuSIC
I
The stable
I
Geometry
I
I I Astronomy
The moving
86
This classification is the origin of the famous Quadrivium of knowledge. In geometry Pythagoras or his followers developed the theory of spacefilling figures. The more obvious of these must have been very well known. If we think of each piece in such a figure as a unit, the question arises, can we fill a Bat surface with repetitions of these units? It is very likely that this type of inquiry was what first led to the theorem that the three angles of a triangle are together equal to two right angles. The same train of thought also extends naturally to solid geometry, including the conception of regular solids. One of the diagrams (Figure 3) shows six equal triangles filling Bat space round their central point. But five such equilateral triangles can likewise be fitted together, to form a blunted bell-tent-shaped figure, spreading from a central vertex: and now their bases form a regular pentagon. Such a figure is no longer Bat; it makes a solid angle, the comer, in fact, of a regular icosahedron. The process could be repeated by surrounding each vertex of the original triangles with five triangles. Exactly twenty triangles would be needed, no more and no less, and the
®E8 FIGUJtE 3
result woulct be the beautiful figure of the icosahedron of twenty triangles surrounding its twelve vertices in circuits of five. It is remarkable that in solid geometry there are only five such regular figures, and that in the plane there is a very limited number of associations of regular space-filling figures. The three simplest regular solids. including the cube, were known to the Egyptians. But it was given to Pythagoras to discover the remaining two--the dodecahedron with its twelve pentagonal faces, and the icosahedron. Nowadays we so often become acquainted with these regular solids and plane figures only after a long excursion through the intricacies of mensuration and plane geometry that we fail to see their full simplicity and beauty. Another kind of problem that interested Pythagoras was called the method oj application oj areas. His solution is noteworthy because it provided the geometrical equivalent of solving a quadratic equation in algebra. The main problem consisted in drawing, upon a given straight line,
87
a figure that should be the size of one and the shape of another given figure. In the course of the solution one of three things was bound to happen. The base of the constructed figure would either fit, fall short of, or exceed the length of the given straight line. Pythagoras thought it proper to draw attention to these three possibUities; accordingly he introduced the terms parabole. ellipsis and hyperbole. Many years later his nomenclature was adopted by Apollonius. the great student of the conic section, because the same threefold characteristics presented themselves in the construction of the curve. And we who follow Apollonius still call the curve the parabola, the ellipse, or the hyperbola, as the case may be. The same threefold classification underlies the signs =, in arithmetic.
b
c
b Q,
FIGURB 4
Many a time throughout the history of mathematics this classification has proved to be the key to further discoveries. For example, it is closely connected with the theory of irrational numbers; and this brings us to the greatest achievement of Pythagoras, who is credited with discovering the (l1.'AOYOJl) irrational. In other words, he proved that it was not always possible to find a common measure for two given lengths a and b. The practice of measuring one line against another must have been very primitive. Here is a long line a, into which the shorter line b fits three times, with a still shorter piece c left over (Figure 4) . To-day we express this by the equation a = 3b + c, or more generally by a nb + c. If there is no such remainder c, the line b measures a; and a is called a multiple of b. If, however. there is a remainder c, further subdivisiOR might perhaps account for each length a, b, c without remainder: experiment might show, for instance, that in tenths of inches, a 17, b 5, c;;;;;: 2. At one time it was thought that it was always pos.. sible to reduce lengths a and b to such multiples of a smaller length. It appeared to be simply a question of patient subdivision, and sooner or later the desired measure would be found. So the required subdivision, in the present example, is found by measuring b with c. For c fits twice into b with a remainder d; and d fits exactly twice into c without remainder. Consequently d measures c, and also measures b and also a. This is how the numbers 17, 5 and 2 come to be attached to a, b, and c: namely a contains d 17 times. Incidentally this shows how naturally the arithmetical progression arises. For although the original subdivisions, and extremity, of the line a occur at distances 5, 10, 15, 17, measured from the left in quarter inches,
=
=
=
H,,,,.,, W,$h',n TIImb,'"
88
they occur at distances 2, 7, 12, 17, from the right. These numbers form a typical arithmetical progression, with a rhythmical law of succession that alone would be incentive for a Pythagorean to study them further. This reduction of the comparison of a line a with a line b to that of the number 17 with 5, or, speaking more technically, this reduction of the ratio a : b to 17 : 5 would have been agreeable to the Pythagorean. It exactly fitted in with his philosophy; for it helped to reduce space and geometry to pure number. Then came the awkward discovery, by Pythagoras himself, that the reduction was not always possible; that something in geometry eluded whole numbers. We do not know exactly how this discovery of the irrational took place, although two early examples can be cited. First when a is the diagonal and b is the side of a square, no common measure can be found; nor can it be found in a second example, when a line a is divided in golden section into parts b ami c. By this is meant that the ratio of a, the whole line t to the part b is equal to the ratio of b to the other part c. Here c may be fitted once into b with remainder d: and then d may be fitted once into c with a remainder e: and so on. It is not hard to prove that such lengths a, b, c, d t • • • form a geometrical progression without end; and the desired common measure is never to be found. If we prefer algebra to geometry we can verify this as follows. Since it is given that a b + c and also a : b :: b : c, it follows that a(a - b) b2 • This is a quadratic equation for the ratio a : b, whose solution gives the result
=
=
a : b
= y5 + 1 : 2.
The presence of the surd y5 indicates the irrational. The underlying reason why such a problem came to be studied is to be found in the star badge of the brotherhood (p. 83); for every line therein is divided in this golden section. The star has five lines. each cut into three
h
c
FIGURE S
parts, the lengths of which can be taken as a, b, a. As for the ratio of the diagonal to the side of cl square, Aristotle suggests that the Pythagorean proof of its irrationality was substantially as follows: If the ratio of diagonal to side is commensurable, let it be p : q, whete
The Great MalhemalldGIII
p and q are whole numbers prime to one another. Then p and q denote
the number of equal subdivisions in the diagonal and the side of a square respectively. But since the square on the diagonal is double that on the side. it follows that p2 = 2 q2. Hence p2 is an even number, and p itself must be even. Therefore p may be taken to be 2r, p2 to be 4r2, and consequently q2 to be 2r2. This requires q to be even; which is impossible because two numbers p, q, prime to each other cannot both be even. So the original supposition is untenable: no common measure can exist; and the ratio is therefore irrational. This is an interesting early example of an indirect proof or reductio ad absurdum; and as such it is a very important step in the logic of mathematics. We can now sum up the mathematical accomplishments of these early Greek philosophers. They advanced in geometry far enough to cover rougbly our own familiar school course in the subject. They made substantial progress in the theoretical side of arithmetic and algebra. They had a geometrical equivalent for our method of solving quadratic equations; they studied various types of progressions, arithmetical, geometrical and harmonical. In Babylon, Pythagoras is said to have learnt the 'perfect proportion' a+ b
a:-2
2ab
--:b a+b
which involves the arithmetical and harmonical means of two numbers. Indeed, to the Babylonians the Greeks owed many astronomical facts, and the sexagesimal method of counting by sixties in arithmetic. But they lacked our arithmetical notation and such useful abbreviations as are found in the theory of indices. From a present-day standpoint these results may be regarded as elementary: it is otherwise with their discovery of irrational numbers. That will ever rank as a piece of essentially advanced mathematics. As it upset many of the accepted geometrical proofs it came as a 'veritable logical scandal'. Much of the mathematical work in the succeeding era was coloured by the attempt to retrieve the position, and in the end this was triumphantly regained by Eudoxus. Recent investigations of the Rhind Papyrus, the Moscow Papyrus of the Twelfth Egyptian Dynasty, and the Strassburg Cuneiform texts have greatly added to the prestige of Egyptian and Babylonian mathematics. While no general proof has yet been found among these sources, many remarkable ad hoc formulae have come to light, such as the Babylonian solution of complicated quadratic equations dating from 2000 B.C., which O. Neugebauer published in 1929, and an Egyptian approximation to the area of a sphere (equivalent to reckoning.". = 256/81).
Berhrl Welt,.. 7_b""
90
CHAPTER II EUDOXUS AND THE ATHENIAN SCHOOL
A SECOND stage in tbe bistory of mathematics occupied tbe fiftb and fourth centuries B.C.~ and is associated with Athens. For after the wonderful victories at Maratbon and Salamis early in tbe fiftb century, wben tbe Greeks defeated the Persians, Athens rose to a position of pre-eminence. The city became not only the political and commercial, but the intellectual centre of the Grecian world. Her pbi10sopbers congregated from East and Wes4 many of whom were remarkable matbematicians and astronomers. Perhaps the greatest among tbese were Hippocrates, Plato, Eudoxus and Menaechmus; and contemporary with the three latter was Archytas tbe Pythagorean, who lived at Tarentum. Thales and Pythagoras bad laid tbe foundations of geometry and aritbmetic. lbe Athenian school concentrated upon special aspects of tbe superstructure; and, whether by accident or design, found tbemselves embarking upon three great problems: (i) the duplication 01 the cube, or the attempt to find the edge of a cube whose volume is double tbat of a given cube; (ii) the trisection 01 a given angle~ and (iii) the squaring 01 a circle, or, the attempt to find a square wbose area is equal to tbat of a given circle. These problems would naturally present themselves in a systematic study of geometry; while, as years passed and no solutions were forthcoming they would attract increasing attention. Such is their inherent stubbornness that not until the nineteenth century were satisfactory answers to these problems found. Their innocent enunciations are at once an invitation and a paradox. Early attempts to solve tbem led indirectly to results that at first sight seem to involve greater difficulties than the problems themselves. For example, in trying to square the circle Hippocrates discovered that two moon-shaped figures could be drawn wbose areas were together equal to that of a right-angled triangle. Tbis diagram (Figure 6) with its three semicircles described on the respective sides of tbe triangle illustrates his tbeorem. One migbt readily suppose that it would be easier to determine the area of a single circle than that of tbese lunes, or lunules, as they are called, bounded by pairs of circular arcs. Yet such is not the case. In this by-product of the main problem Hippocrates gave tbe first example of a solution in quadratures. By tbis is meant the problem of constructing a rectilinear area equal to an area bounded by one or more curves. lbe sequel to attempts of this kind was the invention of the integral calculus by Archimedes, who lived in the next century. But his first success in tbe method was not concerned with tbe area of a circle, but
91
The Gnat MathemollcltulS
FIGURE 6
with that of a parabola, a curve that had been discovered by Menaechmus in an attempt to duplicate the cube. This shows how very interdependent mathematics had now become with its interplay between branch and branch. All this activity led to the discovery of many other new curves, including the ellipse, the hyperbola, the quadratrix, the conchoid (the shell), the cissoid (the ivy leaf), various spirals, and other curves classed as loci on surfaces. The Greeks now found it useful to adopt a special classification for their problems. calling them plane, solid and linear. Problems were plane if their solution depended only on the use of straight lines and circles. These were of distinctly the Pythagorean type. They were solid if they depended upon conic sections; and they were linear if in addition they depended upon still more complicated curves. This early classification shows true mathematical insight, because later experience has revealed close algebraic and analytic parallels. For example, the plane problem corresponds in algebra to the problem soJuble by quadratic equations. The Greeks quite naturally but vainly supposed that the three famous problems above were soluble by plane methods. It is here that they were wrong: for by soUd or linear methods the problems were not necessarily insoluble. One of the first philosophers to bring the new learning from Ionia to Athens was ANAXAOORAS (1500-428 B.C.), who came from near Smyrna. He is said to have neglected his possessions, which were considerable, in order to deYOte himself to science, and in reply to the question, what was the object of being born, he remarked: 'The investigation of the sun, moon and heaven.' In Athens he shared the varying fortunes of his friend Pericles, the great statesman, and at one time was imprisoned for impiety. This we know from an ancient record which adds that 'while in prison he wrote (or drew) the squaring of the circle', a brief but interesting allusion to the famous problem. Nor has the geometry of the circle suffered unduly from the captivity of its devotees. Centuries later another great chapter was opened, when the Russians flung Poncelet, an officer serving under Napoleon, into prison, where he discovered the circular points at infinity. Anaxagoras, however, was famous chiefly for his work in astronomy.
Herbert Westren Turn:'ull
92
HIPPOCRATES 1 was his younger contemporary, who .:;ame from Chios to Athens about the middle of the fifth century. A lawsuit originally lured him to the city: for he had lost considerable property in an attack by Athenian pirates near Byzantium. Indeed, the tastes of Athenian citizens were varied: they were not all artists, sculptors, statesmen, dramatists, philosophers, or honest seamen, in spite of the wealth there and then as· sembled. After enduring their ridicule first at being cheated and then for hoping to recover his money, the simple·minded Hippocrates gave up the quest, and found his solace in mathematics and phiJosophy. He made several notable advances. He was the first author who is known to have written an account of elementary mathematics; in particu· lar he devoted his attention to properties of the circle. To·day his actual work survives among the theorems of Euclid, although his original book is lost. His chief result is the proof of the statement that circles are to one another in the ratio of the squares on their diameters. This is equivalent to the discovery of the formula 1[r2 for the area of a circle in terms of its radius. It means that a certain number 1[ exists, and is the same for all circles, although his method does not give the actual numerical value of 1[. It is thought that he reached his conclusions by looking upon a circle as the limiting form of a regular polygon, either inscribed or circum· scribed. This was an early instance of the method of exhaustions-a particular use of approximation from below and above to a desired limit. The introduction of the method of exhaustions was an important link in the chain of thought culminating in the work of Eudoxus and Archimedes. It brought the prospect of unravelling the mystery of irrational numbers, that had sorely puzzled the early Pythagoreans, one stage nearer. A second important but perhaps simpler work of Hippocrates was an example of the useful device of reducing one theorem to another. The Pythagoreans already had shown how to find the geometric mean between two magnitUdes by a geometrical construction. They merely drew a square equal to a given rectangle. Hippocrates now showed that to duplicate a cube was tantamount to finding two such geometric means. Put into more familiar algebraic language, if
a:x
=x
and if
a:x
=
b, then x 2
= ab,
= x : y = y : 2a
then x3 = 2a3 • Consequently if a was the length of the edge of the given cube, x would be that of a cube twice its size. But the statement also shows that x is the first of two geometric means between a and 2a. We must, of course, bear in mind that the Greeks had no such convenient algebraic notation as the above. Although they went through the 1
Not the great physician.
93
Tift Grtat MlIIlftmaticians
same reasoning and reached the same conclusions as we can, their statements were prolix, and afforded none of the help which we find in these concise symbols of algebra. It is supposed that the study of the properties of two such means, x and y, between given lengths a and b, led to the discovery of the parabola and hyperbola. As we should say, nowadays, the above continued proportions indicate the equations X2 ay, and xy 2a2 • These equations represent a parabola and a hyperbola: taken together they determine a point of intersection which is the key to the problem. This is an instance of a solid solution for the duplication of the cube. It represents the ripe experience of the Athenian school; for MENAECHMUS (1375-325 B.C.), to whom it is credited, lived a century later than Hippocrates. Where two lines, straight or curved, cross, is a point: where three surfaces meet is a point. The two walls and the ceiling meeting at the corner of a room give a convenient example. But two curved walls, meeting a curved ceiling would also make a corner, and in fact iI1ustrate a truly ingenious method of dealing with this same problem of the cube. The author of this geometrical novelty was ARCHYTAS (1400 B.C.), a contemporary of Menaechmus. This time the problem was reduced to finding the position of a certain point in space: and the point was located as the meeting-place of three surfaces. For one surface Archytas chose that generated by a circle revolving about a fixed tangent as axis. Such a surface can be thought of as a ring, although the hole through the ring is completely stopped up. His other surfaces were more commonplace, a cylinder and a cone. With this unusual choice of surfaces he succeeded in solving the problem. When we bear in mind how little was known, in his day, about solid geometry, this achievement must rank as a gem among mathematical antiquities. Archytas, too, was one of the first to write upon mechanics, and he is said to have been very skilful in making toys and models-a wooden dove which could fly, and a rattle which, as Aristotle says, 'was useful to give to children to occupy them from breaking things about the house (for the young are incapable of keeping still)'. Unlike the majority of mathematicians who lived in this Athenian era, Archytas lived at Tarentum in South Italy. He found time to take a considerable part in the public life of his city, and is known for his enlightened attitude in his treatment of slaves and in the education of children. He was a Pythagorean, and was also in touch with the philosophers of Athens, numbering Plato among his friends. He is said upon one occasion to have used his influence in high quarters to save the life of Plato. Between Crotona and Tarentum upon the shore of the gulf of Southern Italy was the city of Elea: and with each of these places we may associate a great philosopher or mathematician. At Croton a Pythagoras had instituted his lecture-room; nearly two centuries later Archytas made his
=
=
mechanical models at Tarentum. But about midway through the intervening period there lived at Elea the philosopher ZENO. This acutely original thinker played the part of philosophical critic to the mathematicianst and some of his objections to current ideas about motion and the infinitesimal were very subtle indeed. For example, he criticized the infinite geometrical progression by proposing the well-known puzzle of Achilles and the Tortoise. How, asked Zeno, can the swift Achilles overtake the Tortoise if he concedes a handicap? For if Achilles starts at A and the tortoise at B, then when Achilles reaches B the tortoise is at C, and when Achilles reaches C the tortoise is at D. As this description can go on and on, apparently Achilles never overtakes the tortoise. But actua)]y he may do so; and this is a paradox. The point of the inquiry is not when. but how does Achilles overtake the tortoise. Somewhat similar questions were asked by DEMOCRITUS, the great philosopher of Thrace, who was a contemporary of Archytas and Plato. Democritus has long been famous as the originator of the atomic theory, a speculation that was immediately developed by EpiclJrus, and later provided the great theme for the Latin poet Lucretius. It is, however, only quite recently that any mathematical work of Democritus has come to light. This happened in 1906. when Heiberg discovered a lost book of Archimedes entitled the Method. We learn from it that Archimedes re-
FIGURE 7
garded Democritus as the first mathematician to state correctly the formula for the volume of a cone or a pyramid. Each of these volumes was one-third part of a circumscribing cylinder, or prism, standing on the same base. To reach his conclusions. Democritus thought of these solids as built up of innumerable paranel layers. There would be no difficulty in the case of the cylinder, for each ]ayer would be equa1. But for the
cone or pyramid the sizes of layer upon layer would taper off to a point. The appended diagram (Figure 7), showing the elevation of a cone or pyramid, illustrates this tapering of the layers. although the picture that Democritus had in mind consisted of very much thinner layers. He was puzzled by their diminishing sizes. 'Are they equal Or unequal?' he asked, 'for if they are unequal, they will make the cone irregular as having many indentations, like steps, and unevennesses; but, if they are equal, the sec~ lions will be equal, and the cone will appear to have the property of the cYlinder and to be made up of equal, not unequal circles, which is very absurd.' This quotation is striking; for it foreshadows the great constructive work of Archimedes. and, centuries later, that of Cavalieri and Newton. It exhibits the infinitesimal calculus in its infancy. The notion .of stratifi~ cation-that a solid could be thought of as layer upon layer-would occur quite naturally to Democritus, because he was a physicist; it would not so readily have occurred to Pythagoras or Plato with their more algebraic tum of mind which attracted them to the pattern or arrangement of things. But here the acute Greek thought is once more restless. No mere rough and ready approximation will satisfy Democritus: there is discrepancy between stratified pyramid and smoothly finished whole. The deep question of the theory of limits is at issue; but how far he foresaw any solution. we do not know. This brings us to the great arithmetical work at Athens, associated with the names of PLATO (429-348 B.C.) and Euooxus (408-355 B.C.) Among the philosophers of Athens only two were native to the place, Socrates and Plato, master and disciple, both of whom were well read mathematicians. Plato was perhaps an original investigator; but whether this is so or not, he exerted an immense influence on the course that mathematics was to take, by founding and conaucting in Athens his famous Academy. Over the entrance of his lecture room his students read the telJing inscription, 'Let no one destitute of geometry enter my doors'; and it was his earnest wish to give his pupils the finest possible education. A man, said he, should acquire no mere bundle of knowledge, but be trained w see below the surface of things, seeking rather for the eternal reality and the Good behind it all. For this high endeavour the study of mathematics is essential; and numbers, in partiCUlar. must be studied, simply as numbers and not as embodied in anything. They impart a character to nature; for instance, the periods of the heavenly bodies can only be characterized by invoking the use of irrationals. Original1y the Greek word apt(Jp.ol., from which we derive 'arithmetic,' meant the natural numbers, although it was at first questioned whether unity was a number; for 'how can unity, the measure, be a number, the thing measured?' But by including irrationals as numbers Plato made a
great advance: he was in fact dealing with what we nowadays can the positive real numbers. Zero and negative numbers were proposed at a far later date. There is grandeur here in the importance which Plato ascribes to arithmetic for forming the mind: and this is matched by his views on geometry, 'the subjeet which has very ludicrously been called mensuration' ('YEOIp.upta land measuring) but which is really an art, a more than human miracle in the eyes of those who can appreeiate it. In his book, the Timaeus. where he dramatically expounds the views of his hero Timaeus, the Pythagorean, reference is made to the five regular solids and to their supposed significance in nature. The speaker tells how that the four elements earth, air, fire and water have characteristic shapes: the cube is appropriated to earth, the octahedron to air, the sharp pyramid or tetrahedron to fire, and the blunter icosahedron to water, white the Creator used the fifth, the dodeeahedron, for the Universe itself. Is it sophistry, or else a brilliant foretaste of the moleeular theory of our own day? According to Proclus, the late Greek commentator, 'Plato caused mathematics in general and geometry in particular to make a very great advance, by reason of his enthusiasm for them, which of coune is obvious from the way he fiUed his books with mathematical i11ustrations, and everywhere tries to kindle admiration for these subjects, in those who make a pursuit of philosophy.' It is related that to the question, What does God do? Plato replied, 'God always geometrizes.' Among his pupils was a young man of Cnidus, named Euooxus, who came to Athens in great poverty, and, like many another poor student, had a struggle to maintain himself. To relieve his pocket he lodged down by the sea at the Piraeus, and every day used to trudge the dusty miles to Athens. But his genius for astronomy and mathematics attracted attention and finally brought him to a position of eminence. He trave1led and studied in Egypt, Italy and Sicily, meeting Archytas, the geometer, and other men of renown. About 368 B.C., at the age of forty, Eudoxus returned to Athens in company with a considerable following of pupils, about the time when Aristotle, then a lad of seventeen, first crossed the seas to study at the Academy. In astronomy the great work of Eudoxus was his theory of concentric spheres explaining the strange wanderings of the planets; an admirable surmise that went far to fit the observed facts. Like his successor Ptolemy, who Jived many centuries later, and all other astronomers until Kepler, he found in circular motion a satisfactory basis for a complete planetary theory. This was great work; yet it was surpassed by his pure mathematics which touched the zenith of Greek brilliance. For Eudoxus placed the doctrine of irrationals upon a thoroughly sound basis, and so well was his task done that it still continues, fresh as ever, after the great arith-
=
Tht! GrtQJ Malht!mIIIlcians
metical reconstructions of Dedekind and Weierstrass during the nineteenth century. An immediate effect of the work was to restore confidence in the geometrical methods of proportion and to complete the proofs of several important theorems. The method of exhaustions vaguely underlay the results of Democritus upon the volume of a cone and of Hippocrates on the area of a circle. Thanks to Eudoxus this method was fully explained. An endeavour will now be made to indicate in a simple way how this great object was achieved. This study of higher arithmetic at Athens was stimulated by the Pythagorean, Theodorus of Cyrene, who is said to have been P1ato's teacher. For Theodorus discovered many irrationals, y3, yS, y6, y7, y8, yl0, yll, Y12. y13, Y14. ylS, V17, 'at which point,' says Plato, 'for some reason he stopped'. The omissions in the list are obvious: y2 had been discovered by Pythagoras through the ratio of diagonal to side of a square, while y4, Y9. y16 are of course irrelevant. Now it is one thing to discover the existence of an irrational such as y2; it is quite another matter to find a way of approach to the number. It was this second problem that came prominently to the fore: it provided the arithmetical aspect of the method of exhaustions already applied to the circle: and it revealed a wonderful example of ancient arithmetic. We learn the details from a later commentator, Theon of Smyrna. Unhampered by a decimal notation (which here is a positive hindrance. useful as it is in countless other examples), the Greeks set about their task in the following engaging fashion. To approximate to y2 they built 1 1 a ladder of whole numbers. A brief scrutiny of the ladder 2 3 shows how the rungs are devised: 1 + 1 2, 1 + 2 3, S 7 2 + 3 S, 2 + S 7, S + 7 12. and so on. Each rung of 12 17 the ladder consists of two numbers x and y, whose ratio 29 41 approaches nearer and nearer to the ratio 1 : y2, the further etc. down the ladder it is situated. Again, these numbers x and y, at each rung, satisfy the equation
=
=
=
=
=
y2 - 2x2 = +1.
The positive and negative signs are taken at alternate rungs, starting with a negative. For example, at the third rung 7 2 - 2"S2 = -1. As these successive ratios are alternately less than and greater than aU that follow, they nip the elusive limiting ratio 1: y2 between two extremes, like the ends of a closing pair of pincers. They approximate from both sides to the desired irrational: is a little too 1arge, but 1%7 is a little too small. Like pendulum swings of an exhausted clock they die down-but they never actually come to rest. Here again, is the Pythagorean notion of hyperbole and ellipsis; it was regarded as very signifi-
*
cant, and was called by the Greeks the 'dyad' of the 'great and small'. Such a ladder could be constructed for any irrational; and another very pretty instance, which has been pointed out by Professor D'kcy Thompson, is closely connected with the problem of the Golden Section. 1 1 Here the right member of each rung is the sum of the pair on 1 2 the preceding rung, so that the ladder may be extended with 2 3 the greatest ease. In this case the ratios approximate, again 3 5 by the little more and the little less. to the limit y5 + 1 : 2. 5 8 It is found that they provide the arithmetical approach to the etc. golden section of a line AB, namely when C divides AB so that
c FIGURE 8
CB : AC = AC : AB. In fact. AC is roughly :~ of the length AB, but more nearly % of AB; and so on. It is only fair to say that this simplest of all such ladders has not yet been found in the ancient literature, but owing to its intimate connexion with the pentagon, it is difficult to resist the conclusion that the later Pythagoreans were familiar with it. The series 1, 2, 3, 5, 8, . . . was known in mediaeval times to Leonardo of Pisa, surnamed Fibonacci, after whom it is nowadays named. Let us now combine this ladder-arithmetic with the geometry of a divided line. For exampJe, let a line AB be divided at random by C, into lengths a and b, where AC = a, CB = b. Then the question still remains, what is the exact arithmetical meaning of the ratio a : b, whether or not this is irrational? The wonderful answer to this question is what has made Eudoxus so famous. Before considering it, Jet us take as an illustration the strides of two walkers. A tall man A takes a regular stride of length a. and his short friend B takes a stride b. Now suppose that eight strides of A cover the same ground as thirteen of B: in this case the single strides of A and B are in the ratio 13 : 8. The repetition of strides, to make them cover a considerable distance, acts as a magnifying glass and helps in the measurement of the single strides a and b, one against the other. Here we have the point of view adopted by Eudoxus. Let us, says he in effect, multiply our magnitudes a and b, whose ratio is required, and see what happens. Let us, he continues, be able to recognize if a and b are equal, and if not, which is greater. Then if a is the greater, let us secondly be able to find multipJes 2b, 3b, . . . , nb, of the smaller magnitude b; and thirdly, let us always be able to find a mUltiple nb of b which exceeds a. (The tall man may have seven-league boots and the short man may be Tom Thumb. Sooner or later the dwarf will be able to overtake one stride of
his friend!) Few will gainsay the propriety of these mild assumptions: yet their mathematical implications have proved to be very subtle. This third supposition of Eudoxus has been variously credited, but to-day it is known as the Axiom 01 Archimedes. A definition of equal ratios can now be stated. Let a, b, c, d be four given magnitudes. then the ratio a: b is equal to that of c: d, if, whatever equimultiples mat me are chosen and whatever equimultiples nb. nd are chosen,
either ma>nb, mc>nd, (i) or ma nb, me nd, (ii) or ma tioo of mathematkal exercisea and praetk:aI namples., worked out in a syncopated, sometimes cryptic style. The tint section presents a table of the division of 2 by odd numben-from '" to %.1. This convenion was 1)ecesury because tbe Egyptians could operate only with unit fractions and bad tberefore to reduce all othen to tbis form. With the exception of for which the Egyptians bad a special symbol. every fraction had to be expressed as the sum of a series of fractioru having 1 as the numerator. For example, the fraction % wu written u %, % (note they did not use the plus sign), and %1 was expressed as %0, *«. ~u, ~'l1o.
*-
lame. R. NewlJUUl
172
It is remarkable that the Egyptians, who attained so much skill in their arithmetic manipulations, were unable to devise a fresh notation and less cumbersome methods. We are forced to realize how little we understand the circumstances of cultural advance: why societies move-or is it perhaps jump--from one orbit to another of intellectual energy, why the science of Egypt "ran its course on narrow lines" and adhered so rigidly to its clumsy rules. Unit fractions continued in use, side by side with improved methods, even among Greek mathematicians. Archimedes, for instance, wrote %, %, for %', and Hero, %, ¥.t7, %4, ~1 for 3~1. Indeed, as late as the 17th century certain Russian documents are said to have expressed ~ as a "half-half-half-half-half-third." The Rhind Papyrus contains some 85 problems, exhibiting the use of fractions, the solution of simple equations and progressions, the mensuration of areas and volumes. The problems enable us to form a pretty clear notion of what the Egyptians were able to do with numbers. Their arithmetic was essentially additive, meaning that they reduced mUltiplication and division, as children and electronic computers do, to repeated additions and subtractions. The only multiplier they used, with rare exceptions, was 2. They did larger multiplications by successive duplications. Multiplying 19 by 6, for example, the Egyptians would double 19, double the result and add the two products, thus: 1 "2
19 38
" 4
76
Total
6
114
The symbol " is used to designate the sub-multipliers that add up to the total multiplier, in this case 6. The problem 23 times 27 would, in the Rhind, look like this:
"1 "4 "2
8 "16
Total
23
27 54 108
216 432 621
In division the doubling process had to be combined with the use of fractions. One of the problems in the papyrus is "the making of loaves 9 for man 10," meaning the division of 9 loaves among 10 men. This problem is not carried out without pain. Recal1 that except for % the Egyptians had to reduce all fractions to sums of fractions with the numerator 1. The Rhind explains: ''The doing as it occurs: Make thou the multiplication % 1Jb %0 times 10.
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173
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1
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Total loaves 9; it, this is. tt In other words, if one adds the fractions obtained by the indicated multiplications (2 + 8 10), he arrives at 9. The reader understandlbly, may find the demonstration baffling. For one thing, the actual working of the problem is not given. If 10 men are to share 9 loaves, each man, says A'h-mose, is to get %t 1,30 (i.e., 27ho) times 10 loaves; but we have no idea how the figure for each share was arrived at. The answer to the problem ( 27ho, or %0) is given first and then verified, not explained. It may be, in truth, that the author had nothing to explain. that the problem was solved by trial and error-as, it has been suggested, the Egyptians solved all their mathematical problems. An often discussed problem in the Rhind is: "Loaves 100 for man 5, of the 3 above to man 2 those below. What is the difference of share?" Freely translated this reads: "Divide 100 loaves among 5 men in such a way that the shares received shan be in arithmetical progression and that of the sum of the largest three shares shall be equal to the sum of the smallest two. What is the difference of the shares?" This is not as easy to answer as its predecessors, especially when no algebraic symbols or processes are used. The Egyptian method was that of "false position"-a mixture of trial and error and arithmetic proportion. Let us look at the solution in some detail: "Do it thus: Make the difference of the shares 5%. Then the amounts that the five men receive will be 23, 17%, 12, 6%, 1: total 60." Now the assumed difference as we shall see, turns out to be correct. It is the key to the solution. But how did the author come to this disingenuously "assumed" figure? Probably by trial and error. Arnold Buffum Chace, in his definitive study The Rhind Papyrus-from which I have borrowed shamelessly--proposes the following ingenious reconstruction of the operation: Suppose, as a starter, that the difference between the shares were 1. Then the terms of the progression would be 1, 2. 3, 4. 5; the sum of the smallest two would be 3, and of the largest three shares would be 1% (l lA4 Egyptian style). The difference between the two groups (3 minus 1%) would be 1*. or B4. Next, trying 2 as the difference between the successive shares, the progression would be 1, 3, 5, 7, 9. The sum of the two smallest terms would be 4; ;, of the three largest terms would be 3, and the difference between the two sides, 1. The experimenter might then begin to notice that for each increase of I in the assumed common difference, the inequality between the two sides was reduced by
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- --, Part of title page of the papyrus Is reproduced In facsimile. Here the bieratic script reads from top to bottom and riabt to left, It bas been translated: "Accurate reckoning of entering Into thinp. knowledge of existing thlnp all. mystenes , , . secrets all. Now was copied book this in year 33. month four of the inundation aeason [under the majesty of thel King of [Upper and) Lower Egypt. 'A-user-Rc'. endowed with life. in likeness of wrltinp of old made in the time of the King of Upper (and Lower! Egypt. [Ne-ma) 'et-Rc', Lo the scribe
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writes copy tbIa."
Y4, lhs. Very well: to make the two sides equal, apparently he must multiply his increase 1 by as many times as lA, lhs is contained in llA, 12s. That figure is 412. Added to the first assumed difference, 1, it gives 512 as the true common difference. "This process of reasoning is exactly in accordance with Egyptian methods." remarks Chace. Having found the common difference, one must now determine whether the progression fulfills the second requirement of the problem: namely, that the number of loaves shall total 100. In other words, mUltiply the progression whose sum is 60 (see above) by a factor to convert it into 100; the factor, of course, is 1%. This the papyrus does: "As many times as is necessary to multiply 60 to make 100, so many times must these terms be multiplied to make the true series." (Here we see the essence of the method of false position.) When multiplied by 1%, 23 becomes 38%, and the other shares, similarly, become 29%, 20, 10% and 1%. Thus one arrives at the prescribed division of the 100 loaves among 5 men. The author of the papyrus computes the areas of triangles. trapezoids and rectangles and the volumes of cylinders and prisms, and of course the area of a circle. His geometrical results are even more impressive than his arithmetic solutions, though his methods, as far as one can tell, are quite unrelated to the discipline today called geometry. "A cylindrical granary of 9 diameter and height 6. What is the amount of grain that goes
175
The IU"ruI Papynu
into it?" In solving this problem a rule is used for determining the area of a circle which comes to Area = (%d)2, where d denotes the diameter. Matching this against the modem formula, Area fl'r2, gives a value for ff' of 3.16-8 very close approximation to the correct value. The Rhind Papyrus gives the area of a triangle as 1h the base times the length of a line which may be the altitude of the triangle, but, on the other hand--Egyptologists are not sure-may be its side. In an isosceles triangle, tall and with a narrow base, the error resulting from using the side instead of the altitude in computing area would make Jitt1e difference. The three triangle problems in the Rhind "Papyrus involve triang1es of this type, but it is clear that the author had only the haziest notion of what triang1es were like. What he was thinking of was (as one expert conjectures) "a piece of 1and, of a certain width at one end and coming to a point, or at 1east narrower at the other end." Egyptian geometry makes a very respectab1e impression if one considers the information derived not on1y from the Rhind but also from another Egyptian document known as the Moscow Papyrus and from lesser sources. Its attainments, besides those already mentioned, include the correct determination of the area of a hemisphere (some sch01ars, however, dispute this) and the formula for the volume of a truncated ("/s) (a2 + ab + b2 ), where a and b are the lengths of the pyramid, V sides of the square and h is the height. I shou1d like to give one more example taken from the Rhind Papyrus, something by way of a historica1 oddity. Chace offers the foUowing translation of the hard-to-translate Problem 79: "Sum the geometrical progression of five terms, of which the first term is 7 and the multiplier 7. 'The sum according to the rule. Multiply 2801 by 7.
=
=
" 1 " 2 "
4
Total
2801 5602 11204
19607
'The sum by addition houses cats mice spelt (wheat) hekat (half a peck) Total
7 49 343 2401 16807 19607"
This catalogue of miscellany provides a strange little prod to fancy. It has been interpreted thus: In each of 7 houses are 7 cats; each cat kills 7 mice; each mouse would have eaten 7 ears of speJt; each ear of spelt would have produced 7 hekat of grain. Query: How much grain is saved
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gd sw :w·yh;;·kwysp.w3 3 . y 5 • y br. y :w, y mb' kwy ply p: 'b' Go down I times 3, Y3 01 me, ~ 01 me is added to me; return I, filled am I. What ;s the quantity saying it? 1
3
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Problem 36 of the papyrus begIns: "Go down I times 3. ¥.J of me. MI of me IS added to me, return I. filled am I. What IS the quantity saYing It?" The problem IS then solved by the EJ)'ptlan method On these pages IS a faCSimile of the problem as It appears In tbe papyrus. The hieratic scnpt reads from ngbt to left. The characten are reproduced in gray and black (the onglnal papyrus was wntten In red and black) In the middle of the page is a rendenng In hIeroglyphic scnpt. which also reads from nght to left. Beneath each line of hieroglyphs IS a phonetic translatIon The numbers are I1ven ID ArabiC With the Egyptian notation. Each Jine
:i
106 318 318 S30 106 10 318 33 159 63 265
"
212 795 53 636 1060
106
212 5 795
35
53 106 1 3 20 10 :318 636 ., 3 3 13 530 i060 1 2
.
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70 2 100
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530 265 265 1060
of hIeroglyphs and Its translation IS numbered 10 correspond to a lme of the hieratic. At the bottom of the page the phonetiC and numerical translation has been reversed to read from left to right. Beneath each phonetIC expressIOn IS ItS English translation. A dot above a number IndIcates that It IS a fraction With a numerator of one. Two dots above a 3 represent :Y", the only Egyptian fraction WIth a numerator of more than one. Readen who have the desire to trace the entire solution are cautioned that the scribe made several mistakes that are preserved In the vanous transJatioD.S.
...~
178
by the 7 houses t cats? (The author confounds us by not only giving the hekats of grain saved but by adding together the entire heterogeneous lot.) Observe the resemblance of this ancient puzzle to the 18th-century Mother Goose rhyme: "As I was going to St. Ives I met a man with seven wives. Every wife had seven sacks, Every sack had seven cats, Every cat had seven kits. Kits, cats, sacks and wives, How many were there going to St. lves?"
(To this question, unlike the question in the papyrus, the correct answer is "one" or "none," depending on how it is interpreted.) A considerable difference of opinion exists among students of ancient science as to the caliber of Egyptian mathematics. I am not impressed with the contention based partly on comparison with the achievements of other ancient peoples, partly on the wisdom of hindsight, that the Egyptian contribution was negligible, that Egyptian mathematics was consistently primitive and clumsy. The Rhind Papyrus, though it demonstrates the inability of the Egyptians to generalize and their penchant for clinging to cumbersome calculating processes, proves that tbey were remarkably pertinacious in solving everyday problems of arithmetic and mensuration, that they were not devoid of imagination in contriving algebraic puzzles, and that they were uncommonly skillful in making do with the awkward methods they employed. It seems to me that a sound appraisal of Egyptian mathematics depends upon a much broader and deeper understanding of human culture than either Egyptologists or historians of science are wont to recognize. As to the question how Egyptian mathematics compares with Babylonian or Mesopotamian or Greek mathematics, the answer is comparatively easy and comparatively unimportant. What is more to the point is to understand why the Egyptians produced their particular kind of mathematics, to what extent it offers a culture clue, how it can be related to their social and political institutions, to their religious beliefs, their economic practices, their habits of daily living. It is only in these terms that their mathematics can be judged fairly.
COMMENTARY ON
ARCHIMEDES ANY of Archimedes' wonderful writings have survived, but of his life only fragments are known. Heracleides wrote a biography but it has been lost and one must turn to various ancient sources of uneven reliability for such particulars as remain. The Byzantine grammarian Tzetzes records that Archimedes died at the age of seventy-five; since he perished at the fall of Syracuse, it is inferred that he was born about 287 B.C. From the Greek historian Diodorus one learns that he studied mathematics at Alexandria; from Pappus, that he wrote a book on mechanics, On Sphere-making; from Cicero, that he constructed a sphere which imitated the motion of the sun, the moon and the planets (Cicero claims to have seen this miniature planetarium); from Lucian, that he set the Roman ships on fire by an arrangement of concave mirrors or burning glasses; from Ptolemy, that he made many astronomical observations; from the Roman philosopher Macrobius, that he discovered the distances of the planets; from Vitruvius' history of architecture, that he once ran naked through the streets shouting "Eureka"-the occasion being familiar to every schoolboy. From Pappus, again, comes the no less famous We that after Archimedes had solved the problem: ''To move a given weight by a given force:' he declared triumphantly: "Give me a place to stand on, and I can move the earth." Plutarch tells the dramatic story of Archimedes' death: it was a violent conclusion to what had apparently been a quiet and contemplative life. The story appears as an aside in the biography of Marcellus, a Roman general who sacked Syracuse after a two-year siege (the resistance was much prolonged, it is said, by the military machines Archimedes designed for defending the city). Observe the curious working of history: Archimedes' death is familiar because of Plutarch's interest in Marcellus; of Marcellus it is generally remembered only that one of his soldiers murdered Archimedes.
M
179
There Is lUI tutonishing i,,",gination, ev.n in the #I.nce oj m4themDIic.r, ••• We repeat, there WQ8 Jar more imtIgination in the head oj A.rchJm.de.r llum in tlull 0/ Homer. -VOLTADlB
3
Archimedes By PLUTARCH, VITRUVIUS, TZETZES
PLUTARCH,
"Marcellus"
MARCELLUS, incensed by injuries done him by Hippocrates, commander of the Syracusans, (who, to give proof of his good affection to the Carthaginians, and to acquire the tyranny to himself, had killed a number of Romans at Leontini,) besieged and took by force the city of Leontini; yet violated none of the townsmen; only deserters, as many as he took, he subjected to the punishment of the rods and axe. But Hippocrates, sending a report to Syracuse, that Marcellus had put all the adult population to the sword, and then coming upon the Syracusans, who had risen in tumult upon that false report, made himself master of the city. Upon this Marcellus moved with his whole army to Syracuse, and, encamping near the wall, sent ambassadors into the city to relate to the Syracusans the truth of what had been done in Leontini. When these could not prevail by treaty. the whole power being now in the hands of Hippocrates, he proceeded to attack the city both by land and by sea. The land forces were conducted by Appius: Marcellus, with sixty galleys, each with five rows of oars, furnished with all sorts of arms and missiles, and a huge bridge of planks laid upon eight ships chained together, upon which was carried the engine to cast stones and darts, assaulted the walls, relying on the abundance and magnificence of his preparations, and on his own previous glory; all which, however, were, it would seem, but trifles for Archimedes and his machines. These machines he had designed and contrived, not as matters of any importance, but as mere amusements in geometry; in compliance with King Hiero's desire and request, some little time before, that he should reduce to practice some part of his admirable speculations in science, and by accommodating the theoretic truth to sensation and ordinary use, bring it more within the appreciation of people in general. Eudoxus and Archytas had been the first originators of this far-famed and highly prized art of mechanics, which they employed as an elegant illustration 180
III
of geometrical truths, and as a means of sustaining experimentally, to the satisfaction of the senses, conclusions too intricate for proof by words and diagrams. As, for example, to solve the problem, so often required in constructing geometrical figures, given the two extremes, to find the two mean lines of a proportion, both these mathematicians had recourse to the aid of instruments, adapting to their purpose certain curves and sections of lines.1 But what with Plato's indignation at it, and his invectives against it as the mere corruption and annihilation of the one good of geometry,-which was thus shamefully turning its back upon the unembodied objects of pure intelligence to recur to sensation. and to ask help (not to be obtained without base subservience and depravation) from matter; so it was that mechanics came to be separated from geometry, and, repudiated and neglected by philosophers, took its place as a military art. Archimedes, who was a kinsman and friend of King Hiero, wrote to him that with a given force it was possible to move any given weight; and emboldened, as it is said, by the strength of the proof, he averred that, if there were another world and he could go to it, he would move this one. Hiero was amazed and besought him to give a practical demonstration of the problem and show some great object moved by a small force; he thereupon chose a three-masted merchantman among the king's ships which had been hauled ashore with great labour by a large band of men, and after putting on board many men and the usual cargo, sitting some distance away and without any special effort, he pulled gently with his hand at the end of a compound pulley 2 and drew the vessel smoothly and evenly towards himself as though she were running along the surface of the water. Astonished at this, and understanding the power of his art, the king persuaded Archimedes to construct for him engines to be used in every type of siege warfare, some defensive and some offensive; he had not himself used these engines because he spent the greater part of his life remote from war and amid the rites of peace, but now his apparatus proved of great advantage to the Syracusans, and with the apparatus its inventor.' Accordingly, when the Romans attacked them from two elements, the 1 The mt1S0IaMS or mt180Iabium. was the name by which this instrument was commonly known. a W'oAvD'rGD'T'Of. Galen, in Hipp. De Artie. iv.47 uses the same word. Tzetzes speaks of a trjple-pulley device (1'"j T'pcD'rGD'T'1» ~'1X4J'j) in the same connelion, and Oribasius, CoN. mt1d. xlix. 22 mentions the T'plD'r4D'T'O$ as an invention of Archimedes; he says that it was so called because it had three ropes, but Vitruvius says it was thus named because it had three wheels. Athenaeus v. 207 a·b says that a ht11ix was used. Heath, Thtl Works 01 A.rchimt1dt18, p. xx, suggests that the vessel, once started, was kept in motion by the system of pulleys, but the first impulse was given by a machine similar to the lCOXAL4$ described by Pappus viii. ed. Hultsch 1066, 1108 fl., in which a COl-wheel with oblique teeth moves on a cylindrical helix turned by a handle. S Similar stories of Archimedes' part in the defence are told by Polybius viii. S. 1-4i and Livy uiv. 34.
182
PIUltu'C1a. Vi1ruvlru. TulrIU
Syracusans were struck dumb with fear, thinking that nothing would avail against such violence and power. But Archimedes began to work his engines and hurled against the land forces all sorts of missiles and huge masses of stones, which came down with incredible noise and speed; nothing at all could ward off their weight, but they knocked down in heaps those who stood in the way and threw the ranks into disorder. Furthermore, beams were suddenly thrown over the ships from the walls, and some of the ships were sent to the bottom by means of weights fixed to the beams and plunging down from above; others were drawn up by iron claws, or crane-like beaks, attached to the prow and were plunged down on their sterns, or were twisted round and turned about by means of ropes within the city, and dashed against the cliffs set by Nature under the wall and against the rocks, with great destruction of the crews, who were crushed to pieces. Often there was the fearful sight of a ship lifted out of the sea into mid-air and whirled about as it hung there, until the men had been thrown out and shot in aU directions, when it would fall empty upon the walls or slip from the grip that had held it. As for the engine which Marcellus was bringing up from the platform of ships, and which was called sambuca from some resemblance in its shape to the musical instrument,· while it was still some distance away as it was being carried to the wall a stone ten talents in weight was discharged at it, and after this a second and a third; some of these, falling upon it with a great crash and sending up a wave, crushed the base of the engine, shook the framework and dislodged it from the barrier, so that Marcel1us in perplexity sailed away in his ships and passed the word to his land forces to retire. In a council of war it was decided to approach the walls, if they could, while it was still night; for they thought that the ropes used by Archimedes, since they gave a powerful impetus, would send the missiles over their heads and would fail in their object at close quarters since there was no space for the cast. But Archimedes, it seems, had long ago prepared for such a contingency engines adapted to all distances and missiles of short range, and through openings in the wall, small in size but many and continuous, short-ranged engines called scorpions could be trained on objects close at hand without being seen by the enemy. When, therefore, the Romans approached the walls, thinking to escape notice, once again they were met by the impact of many missiles; stones fell down on them almost perpendicularly, the wall shot out arrows at them from all points, and they withdrew to the rear. Here again, when they were drawn up some distance away, missiles flew forth and caught "The ua.p.fl6K'I was a triangular musical instrument with four strings._ Polybius (viii. 6) states that Marcellus had eight quinqueremes in pairs locked together. and on each pair a "sambuca" had been erected; it served as a penthouse for raising soldiers on to the battlements.
A,ch'm~d~s
183
them as they were retiring, and caused much destruction among them; many of the ships, also, were dashed together and they could not retaliate upon the enemy. For Archimedes had made the greater part of his engines under the waH, and the Romans seemed to be fighting against the gods, inasmuch as countless evils were poured upon them from an unseen source. Nevertheless Marcellus escaped, and, twitting his artificers and craftsmen, he said: "Shall we not cease fighting against this geometrical Briareus, who uses our ships like cups to ladle water from the sea, who has whipped our sambuca and driven it off in disgrace, and who outdoes all the hundred-handed monsters of fable in hurling so many missiles against us all at once?" For in reality all the other Syracusans were only a body for Archimedes' apparatus, and his the one soul moving and turning everything: all other weapons lay idle, and the city then used his alone, both for offence and for defence. In the end the Romans became so filled with fear that, if they saw a little piece of rope or of wood projecting over the wall, they cried, uThere it is, Archimedes is training some engine upon us," and Bed; seeing this Marcellus abandoned al1 fighting and assault, and for the future relied on a long siege. Yet Archimedes possessed so lofty a spirit, so profound a sou], and such a wealth of scientific inquiry, that although he had acquired through his inventions a name and reputation for divine rather than human intelligence, he would not deign to leave behind a single writing on such subjects. Regarding the business of mechanics and every utilitarian art as ignoble and vulgar, he gave his zealous devotion only to those subjects whose elegance and subtlety are untrammelled by the necessities of life; these subjects, he held, cannot be compared with any others; in them the subject-matter vies with the demonstration, the former possessing strength and beauty, the latter precision and surpassing power; for it is not possible to find in geometry more difficult and weighty questions treated in simpler and purer terms. Some attribute this to the natural endowments of the man, others think it was the result of exceeding labour that everything done by him appeared to have been done without labour and with ease. For although by his own efforts no one could discover the proof, yet as soon as he learns it, he takes credit that he c6uld have discovered it: so smooth and rapid is the path by which he leads to the conclusion. For these reasons there is no need to disbelieve the stories told about himhow, continually bewitched by some familiar siren dwelling with him, he forgot his food and neglected the care of his body; and how, when he was dragged by main force, as often happened, to the place for bathing and anointing, he would draw geometrical figures in the hearths, and draw lines with his finger in the oil with which his body was anointed, being overcome by great pleasure and in truth inspired of the Muses. And
184
Plutarch, Yltruvlra, Tutu,
though he made many elegant discoveries, he is said to have besought his friends and kinsmen to place on his grave after his death a cylinder enclosing a sphere, with an inscription giving the proportion by which the including solid exceeds the included. 5 Such was Archimedes, who now showed himself, and, so far as lay in him, the city also, invincible. While the siege continued, Marcellus took Megara, one of the earliest founded of the Greek cities in Sicily, and capturing also the camp of Hippocrates at Acilz, killed above eight thousand men, having attacked them whilst they were engaged in forming their fortifications. He overran a great part of Sicily; gained over many towns from the Carthaginians, and overcame all that dared to encounter him. As the siege went on, one Damippus, a Lacedzmonian, putting to sea in a ship from Syracuse, was taken. When the Syracusans much desired to redeem this man, and there were many meetings and treaties about the matter betwixt them and Marcellus, he had opportunity to notice a tower into which a body of men might be secretly introduced, as the wall near to it was not difficult to surmount, and it was itself carelessly guarded. Coming often thither, and entertaining conferences about the release of Damippus, he had pretty wen calculated the height of the tower, and got ladders prepared. The Syracusans celebrated a feast to Diana; this juncture of time, when they were given up entirely to wine and sport, Marcellus laid hold of, and, before the citizens perceived it, not only possessed himself of the tower, but, before the break of day, filled the wall around with soldiers, and made his way into the Hexapylum. The Syracusans now beginning to stir, and to be alarmed at the tumult, he ordered the trumpets everywhere to sound, and thus frightened them all into flight, as if all parts of the city were already won, though the most fortified, and the fairest, and most ample quarter was still ungained. It is called Acradina, and was divided by a wall from the outer city, one part of which they call Neapolis, the other Tycha. Possessing himself of these, Marcellus, about break of day, entered through the Hexapylum, all his officers congratulating him. But looking down from the higher places upon the beautiful and spacious city below, he is said to have wept much, commiserating the calamity that hung over it, when his thoughts represented to him, how dismal and foul the face of the city would in a few hours be, when plundered and sacked by the soldiers. For among the officers of his army there was not one man that durst deny the plunder of the city to the soldiers' demands; nay, many were instant that it should be set on fire and laid level to the ground: but this Marcellus would not listen to. Yet he granted, but with great unwillingness 5 Cicero, when quaestor in Sicily, found this tomb overgrown with vegetation, but stiD bearing the cylinder with the sphere, and he restored it (Tuse. Disp. v. 64-66).
185
and reluctance, that the money and slaves should be made prey; giving orders, at the same time, that none should violate any free person, nor kill, misuse, or make a slave of any of the Syracusans. Though he had used this moderation, he still esteemed the condition of that city to be pitiable, and, even amidst the congratulations and joy, showed his strong feelings of sympathy and commiseration at seeing all the riches accumu· lated during a long felicity, now dissipated in an hour. For it is related, that no less prey and plunder was taken here, than afterward in Carthage. For not long after, they obtained also the plunder of the other parts of the city, which were taken by treachery; leaving nothing untouched but the king's money, which was brought into the public treasury. But what specially grieved Marcellus was the death of Archimedes. For it chanced that he was alone, examining a diagram closely; and having fixed both his mind and his eyes on the object of his inquiry, he perceived neither the inroad of the Romans nor the taking of the city. Suddenly a soldier came up to him and bade him follow to Marcellus, but he would not go until he had finished the problem and worked it out to the demon· stration. Thereupon the soldier became enraged, drew his sword and dispatched him. Others, however, say that the Roman came upon him with drawn sword intending to kill him at once, and that Archimedes, on seeing him, besought and entreated him to wait a little while so that he might not leave the question unfinished and only partly investigated; but the soldier did not understand and slew him. There is also a third story, that as he was carrying to Marcellus some of his mathematical instruments, such as sundials, spheres and angles adjusted to the apparent size of the sun, some soldiers fell in with him and, under the impression that he carried treasure in the box, killed him. What is, however, agreed is that Marcellus was distressed, and turned away from the slayer as from a polluted person, and sought out the relatives of Archimedes to do them honour.
VITllUVIUS,
On Architecture
ARCHIMEDES made many wonderful discoveries of different kinds, but of all these that which I shall now explain seems to exhibit a boundless ingenuity. When Hiero was greatly exalted in the royal power at Syracuse, in return for the success of his policy he determined to set up in a certain shrine a golden crown as a votive offering to the immortal gods. He let out the work for a stipulated payment, and weighed out the exact amount of gold for the contractor. At the appointed time the contractor brought his work skilfully executed for the king's approval, and he seemed to
186
have fuJfilled exactly the requirement about the weight of the crown. Later information was given that gold had been removed and an equal weight of silver added in the making of the crown. Hiero was indignant at this disrespect for himself, and, being unable to discover any means by which he might unmask the fraud, he asked Archimedes to give it his attention. While Archimedes was turning the problem over, he chanced to come to the place of bathing, and there, as he was sitting down in the tub, he noticed that the amount of water which flowed over the tub was equal to the amount by which his body was immersed. This indicated to him a means of solving the problem, and he did not delay, but in his joy leapt out of the tub and, rushing naked towards his home, he cried out with a loud voice that he had found what he sought. For as he ran he repeatedly shouted in Greek, heureka. heureka. Then, following up his discovery, he is said to have made two masses of the same weight as the crown, the one of gold and the other of silver. When he had so done, he filled a large vessel right up to the brim with water, into which he dropped the silver mass. The amount by which it was immersed in the vessel was the amount of water which overflowed. Taking out the mass, he poured back the amount by which the water had been depleted, measuring it with a pint pot, so that as before the water was made level with the brim. In this way he found what weight of silver answered to a certain measure of water. When he had made this test, in like manner he dropped the golden mass into the full vessel Taking it out again, for the same reason he added a measured quantity of water, and found that the deficiency of water was not the same, but less; and the amount by which it was less corresponded with the excess of a mass of silver, having the same weight, over a mass of gold. After filling the vessel again. he then dropped the crown itself into the water, and found that more water overflowed in the case of the crown than in the case of the golden mass of identical weight; and so, from the fact that more water was needed to make up the deficiency in the case of the crown than in the case of the mass, he calculated and detected the mixture of silver with the gold and the contractor's fraud stood revealed. 8 e The method may be thus expressed analytically. Let W be the weight of the crown. and let it be made up of a weight and a weight WIl of silver, so that W = WI + WI. Let the crown displace a volume 11 of water. Let the weight W of gold displace a volume VI of water; then a weight W1
displaces a volume -
w
Let the weight
W
of silver displace a volume
displaces a volume - • lit of water. w 11
of gold
WI
of gold
• 111 of water.
WI
It follows that
WI
=-W1W . + -WIW . VI
111
11.
of water; then a weight
WI
of silver
187
A.rchimedes
TZETZES,
Book of Historie$ 7
ARCHIMEDES the wise, the famous maker of engines, was a Syracusan by race, and worked at geometry till old age, surviving five-and-seventyyears 8; he reduced to his service many mechanical powers, and with his triple-pulley device, using on]y his left hand, he drew a vessel of fifty thousand medimni burden. Once, when MarceJlus, the Roman general, was assaulting Syracuse by land and sea, first by his engines he drew up some merchant-vessels, lifted them up against the wall of Syracuse, and sent them in a heap again to the bottom, crews and all. When Marcellus had withdrawn his ships a little distance, the old man gave all the Syracusans power to lift stones large enough to load a waggon and, hurling them one after the other, to sink the ships. When Marcellus withdrew them a bow-shot, the old man constructed a kind of hexagonal mirror, and at an interval proportionate to the size of the mirror he set similar small mirrors with four edges, moved by Jinks and by a form of hinge, and made it the centre of the sun's beams-its noon-tide beam, whether in summer or in mid-winter. Afterwards, when the beams were reflected in the mirror, a fearful kindling of fire was raised in the ships, and at the distance of a bow-shot he turned them into ashes. In this way did the old man prevail over Marcellus with his weapons. In his Doric dialect, and in its Syracusan variant, he declared: "If I have somewhere to stand, I will move the whole earth with my charistion." 9
-----, W1
so that
VI-V
- :: - - - . W.
V-lit
The lines which follow are an example of the "political" verse which prevailed in ByuDtine times. The name is given to verse composed by accent instead of quantity, with an accent on the last syUable but one, especially an iambic verse of fifteen syllables. The twelfth-century Byuntine pedant, John Tzetzes, preserved in his Book 01 His/oriel a great treasure of literary, historical, theological and scientific detail, but it needs to be used with caution. The work is often called the Chiliades from its arbitrary division by its first edition (N. Gerbel, 1546) into books of 1000 lines eacb-it actually contains 12,674 lines. • As he perished in the sack of Syracuse in 212 B.C., he was therefore born about 287 B.C. \l The instrument is otherwise mentioned by Simplicius (in Aristot. Phys.• ed. DieJs 1110. 2-5) and it is implied that it was used for weighing. As Tzetzes in another place writes of a triple-pulley device in the same connexion, it may be presumed to have been of this nature. 'I
COMMENTARY ON
Greek Mathematics ROM Ivor Thomas' admirable source book in the Loeb Classical Library,1 I have chosen a number of examples of the work of Greek mathematicians, fragments of a tale of "one of the most stupendous achievements in the history of human thought." The selections vary from brief comments and anecdotes to complete proofs, in original form, of some of the beautiful theorems on which all subsequent mathematical science is based. Turnbull's little book contains (pp. 75-168) a lucid survey of Greek mathematics; here are reproduced a few of the classical achievements which he discusses. Among the items below are Euclid's proof of the Pythagorean theorem; Plato's comments in the Republic on the philosophy of mathematics; Euclid's demonstration of the method of exhaustion; Archimedes on the principle of the lever, on the prodigious cattle problem and on the displacement of fluids by solids-whereby he may have discovered the proportions of gold and silver in King Hiero's crown; Eratosthenes on measuring the size of the earth; Diophantus' algebraic epitaph, inscribed on his tombstone; Pappus on isoperimetric figures and on the "geometrical forethought" of bees. I hope these will enable the reader to sense the excitement of great inventions and to imbibe ideas and the method of deduction from some of the foremost of ancient thinkers.
P
1 Selectioru Illustrating the History of Greek MatMmatics. 2 vols.. Cambridge. Mass.• 1939.
188
Except the blind lorces 01 Nature, nothIng moves in this world which u not Greek in its origin. -Sill HENRY JAMES SUMNEIl MAINE Grammarian, rhetorician, geomeUtr. painter. trainer, soothsayer, ropedancer. physician, wjtard-he knows everything. Bid the hungry Greekling go to heaven/ He'll go. -JUVENAL
4
Greek Mathematics By IVOR THOMAS PYTHA.GORAS (DIOGENES LAERTIUS)
HE [Pythagoras] it was who brought geometry to perfection, after Moens had first discovered the beginnings of the elements of that science, as Anticleides says in the second book of his History of Alexander. He adds that Pythagoras specially applied himself to the arithmetical aspect of geometry and he discovered the musical intervals on the monochord; nor did he neglect even medicine. Apollodorus the calculator says that he sacrificed a hecatomb on finding that the square on the hypotenuse of the right-angled triangle is equal to the squares on the sides containing the right angle. And there is an epigram as follows:
As when Pythagoras the famous figure found, For which a sacrifice renowned he brought. (PROCLUS,
on Euclid)
Whatsoever offers a more profitable field of research and contributes to the whole of pbilosophy, we shall make the starting-point of further inquiry, therein imitating the Pytbagoreans, among whom there was prevalent this motto, "A figure and a platform, not a figure and sixpence," by which they implied that the geometry deserving study is that which, at each theorem, sets up a platform for further ascent and lifts the soul on high, instead of allowing it to descend among sensible objects and so fulfil the common needs of mortal men and in this lower aim neglect conversion to things above. (PLUTARCH,
The Epicurean Life)
Pythagoras sacrificed an ox in virtue of his proposition, as Apollodorus saysAs when Pythagoras the famous figure found For wbich the noble sacrifice he brought 189
190
whether it was the theorem that the square on the hypotenuse is equal to the squares on the sides containing the right angle, or the problem about the application of the area. Convivial Questions)
(PLUTARCH,
Among the most geometrical theorems, or rather problems, is thisgiven two figures, to apply a third equal to the one and similar to the other; it was in virtue of this discovery they say Pythagoras sacrificed. This is unquestionably more subtle and elegant than the theorem which he proved that the square on the hypotenuse is equal to the squares on the sides about the right angle. SUM OF THE ANGLES OF A T1uANGLB (PltOCLUS,
on Euclid)
Eudemus the Peripatetic ascribes to the Pythagoreans the discovery of this theorem, that any triangle has its internal angles equal to two right angles. He says they proved the theorem in question after this fashion. Let 4
A
r
8 FIGURE I
ABr be a triangle, and through A let ~ be drawn parallel to Br. Now since Br, ~ are parallel, and the alternate angles are equal, the angle AAB is equal to the angle ABr, and EAr is equal to AI'B. Let BAr be added to both. Then the angles AAB, BAr, rAE, that is, the angles MB, BAB, that
is, two right angles, are equal to the three angles of the triangle. Therefore the three angles of the triangle are equal to two right angles. "PYTHAGORAS'S THEOREM" (EUCLID,
Elements i. 47)
In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. Let ABr be a right-angled triangle having the angle BAr right; I say that the square on Br is equal to the squares on BA, Ar. For let there be described on Br the square BAEr, and on BA, Ar the squares HB, ar [Eucl. i. 46], and through A let AA be drawn parallel to
191
Gruk Mat1sellUJlicI
8
K
FIGURE 2
either Ba or rE, and let Aa, zr be joined.! Then, since each of the angles BAr, BAH is right, it follows that with a straight line BA and at the point A on it, two straight lines Ar, AH, not lying on the same side, make the adjacent angles equal to two right angles; therefore rA is in a straight line with AH [Eucl. i. 14], For the same reasons BA is also in a straight Hne with A8. And since the angle aBr is equal to the angle ZBA, for each is right, Jet the angle ABr be added to each; the whole angle aBA is therefore 1 In this famous "windmill" figure, the lines AA, BK, rz meet in a point. Euclid has nO need to mention this fact, but it was proved by Heron. If AA, the perpendicular from A, meets Br in II, as in the detached portion of the figure here reproduced, the triangles liB A, liAr are similar to the triangle ABr and to one another. It follows from Eucl. EJem. vi. 4 and 17 (which do Dot depend on i. 47) that BAit = BII. Br, and Arlt rll . Br. Therefore BAa + A}'2 :::: Br (BM + rll) Brit,
=
=
The theory of proportion developed in Euclid's sixth book therefore offers a simple method of proving "Pythagoras's Theorem. to This proof, moreover. is of the same type as Euet. Elem. i. 47 inasmuch as it is based on the equaHty of the square on Br to the sum of two rectanl1es. Tbis has suggested that Pythagoras proved the theorem by means of his inadequate theory of proportion, which applied only to commensurable magnitudes. When the incommensurable was discovered, it became necessary to find a new proof independent of proportions. Euclid therefore recast Pythagoras's invalidated proof in the form here given so as to get it into the first book in accordance with his general plan of the Elements. For other methods by which the theorem can be proved. the complete evidence bearing on its reputed discovery by Pythagoras. and the history of the theorem in Egypt, Babylonia, and India, see Heath, Th. Thirleen Books 01 Euclid's Elem.nts. i., pp. 351-366. A. Manual 01 Greek Mathematics, pp. 95-100.
192
A
~r
B M
FIGURE 3
equal to the whole angle ZBr. And since Ml is equal to Dr, and ZB to BA, the two [sides] AB, BA are equal to the two [sides] Br, ZB respectively; and the angle MlA is equal to the angle ZBr. The base AA is therefore equal to the base zr, and the triangle ABA is equal to the triangle ZBr [Bucl. i. 4]. Now the parallelogram BA is double the triangle ABA, for they have the same base BA and are in the same parallels BA, AA [Bucl. i. 41]. And the square HB is double the triangle ZBr, for they have the same base ZB and are in the same parallels ZB, Hr. Therefore the parallelogram BA is equal to the square HB. Similarly, if AR, BK are joined, it can also be proved that the parallelogram rA is equal to the square sr. Therefore the whole square BAEr is equal to the two squares HB, sr. And the square BARr is described on Br, while the squares HB, sr are described on BA, Ar. Therefore the square on the side Br is equal to the squares on the sides BA, Ar. Therefore in right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle; which was to be proved.
PLATO (ARISTOXENUS,
Elements of Harmony)
IT is perhaps well to go through in advance the nature of our inquiry, so that, knowing beforehand the road along which we have to travel, we may have an easier journey, because we will know at what stage we are in, nor shall we harbour to ourselves a false conception of our subject. Such was the condition, as Aristotle often used to tell, of most of the audience who attended Plato's lecture on the Good. Bvery one went there expecting that he would be put in the way of getting one or other of the things accounted good in human life, such as riches or health or strength or, in fine, any extraordinary gift of fortune. But when they found that Plato's arguments were of mathematics and numbers and geometry and astronomy and that in the end he declared the One to be the Good, they
Gruk
193
MGt"~mGtlcs
were altogether taken by surprise. The result was that some of them scoffed at the thing, while others found great fault with it. PHILOSOPHY OF MATHEMATICS (PLATO,
Republic vi. S10)
I think you know that those who deal with geometries and ca1culations and such matters take for granted the odd and the even, figures, three kinds of angles and other things cognate to these in each field of inquiry; assuming these things to be known, they make them hypotheses, and henceforth regard it as unnecessary to give any explanation of them either to themselves or to others, treating them as if they were manifest to all; setting out from these hypotheses, they go at once through the remainder of the argument until they arrive with perfect consistency at the goal to which their inquiry was directed. Yes, he said, I am aware of that. Therefore I think you also know that although they use visible figures and argue about them, they are not thinking about these figures but of those things which the figures represent; thus it is the square in itself and the diameter in itself which are the matter of their arguments, not that which they draw; similarly, when they model or draw objects, which may themselves have images in shadows or in water, they use them in tum as images, endeavouring to see those absolute objects which cannot be seen otherwise than by thought.
EUCLID (STOBAEUS,
Extracts)
SOMEONE who had begun to read geometry with Euclid, when he had learnt the first theorem asked Euclid, "But what advantage shall I get by learning these things?" Euclid called his slave and said, "Give him three~ pence, since he must needs make profit out of what he learns." METHOD OF EXHAUSTION (EUCLID,
Elements xii, 2 2)
Circles are to one another as the squares on the diameters. Let ABrA, EZHa be circles, and BA, za their diameters; I say that, as the circle ABrA is to the circle EZHa, so is the square on BA to the square onZa.
For if the circle ABrA is not to the circle EZRa as the square on BA to the square on Z8, then the square on BA will be to the square on za as :a Eudemus attnbuted the discovery of this important theorem to Hippocrates. Unfortunately we do not know how Hippocrates proved it.
194
the circle ABrA. is to some area either less than the circle EZR8 or greater. Let it first be in that ratio to a lesser area ~. And let the square EZR8 be inscribed in the circle EZR8; then the inscribed square is greater than the half of the circle EZH8, inasmuch as, if through the points E, Z, H, 8 we draw tangents to the circle, the square EZH8 is half the square circumscribed about the circle, and the circle is less than the circumscribed square; so that the inscribed square EZR8 is greater than the half of the circle EZR8. Let the circumferences EZ, ZH, H8, 9E be bisected at the points K, A, M, N, and let EK, KZ, ZA, AU, HM, M8, 8N, NE be joined;
A
E
81---------t 4
~---_Ie
H
l' FIGURE 4
therefore each of the triangles EKZ, ZAH, HM8, 8NE is greater than the half of the segment of the circle about it, inasmuch as, if through the points K, A, M, N we draw tangents to the circle and complete the parallelograms on the straight lines EZ, ZH, H9, 8E, each of the triangles EKZ, ZAH, HM8, 8NE will be half of the parallelogram about it, while the segment about it is less than the parallelogram; so that each of the triangles EKZ, ZAH, HM8, 8NE is greater than the half of the segment of the circle about it. Thus, by bisecting the remaining circumferences and joining straight lines, and doing this continually, we shall leave some segments of the circle which will be less than the excess by which the circle EZH8 exceeds the area ~. For it was proved in the first theorem of the tenth book that, if two unequal magnitudes be set out, and if from the greater there be subtracted a magnitude greater than its half, and from the remainder a magnitude greater than its half, and so on continually, there will be left some magnitude which is less than the lesser magnitude set out. Let such segments be then left, and let the segments of the circle EZH9 on EK, KZ, ZA, AH, HM, M8, 8N, NE be less than the excess by which the circle EZH8 exceeds the area ~. Therefore the remainder, the polygon EKZAHM8N, is greater than the area ~. Let there be inscribed, also, in the circle ABrA. the polygon ASBOrrrA.P similar to the polygon EKZAHM8N; therefore as the square on BA. is to the square on Z8, so is the polygon ASBOrIIAP to the polygon EKZAHM8N. But as the square on BA. is to the square on Z8, so
195
is the circle ABf& to the area I; therefore also as the circle ABr& is to the area I. so is the polygon AEBOmaP to the polygon RKZAHM8N; therefore, alternately, as the circle ABf& is to the polygon in it, so is the area X to the polygon RKZAHM8N. Now the circle ABr& is greater than the polygon in it; therefore the area I also is greater than the polygon EKZAHM8N. But it is also less; which is impossible. Therefore it is not true that, as is the square on B& to the square on Z8, so is the circle ABr& to some area less than the circle RZH8. Similarly we shall prove that neither is it-true that. as the square on Z8 is to the square on B&, so is the circle EZB8 to some area Jess than the circle ABr&. I say now that neither is the circle ABr& towards some area greater than the circle EZH8 as the square on B& is to the square on Z8. For, if possible, let it be in that ratio to some greater area I. Therefore, inversely, as the square on Z8 is to the square on &B, so is the area X to the circle ABf&. But as the area X is to the circle ABr&, so is the circle EZB8 to some area Jess than the circle ABr&; therefore also, as the square on Z8 is to the square on B&, so is the circle EZH8 -to some area less than the circle ABr&; which was proved impossible. Therefore it is not true that, as the square on B& is to the square on Z8, so is the circle ABf& to some area greater than the circle EZB8. And it was proved not to be in that relation to a less area; therefore as the square on B& is to the square on Z8, so is the circle ABr& to the circle EZH8. Therefore circles are to one another as the squares on the diameters; which was to be proved.
ARCHIMEDES PJuNCIPLE OF THE LEVER (ARCHIMEDES,
On Plane Equilibriums)
Proposition 6
COMMENSURABLE magnitudes balance at distances reciprocally proportional to their weights. Let A, B be commensurable magnitudes with centres [of gravity] A, B, and let R& be any distance, and let A: B
= M: rE;
it is required to prove that the centre of gravity of the magnitude composed of both A, B is f. Since
A:B=M:I'E,
and A is commensurate with B, therefore r& is commensurate with rE,
196
1'110'
"'' (IIItIU
that is, a straight line with a straight line [Eucl. x. 11]; so that Ert rA have a common measure. Let it be N, and let ~ 11K be each equal to sr, and let EA be equal to Ar. Then since AH rE, it follows that Ar EH; so that ABE = H. Therefore AH = 2M and HK = 2rE; so that N measures both AH and HK, since it measures their halves [Eucl. x. 12]. And since
=
=
A : B = Ar : rE,
while
Ar: rE =AH: HK-
for each is double of the othertherefore
A:B=AH:HK.
Now let Z be the same part of A as N is of AH;
B
A
FIGUU 5
then And therefore, ex aequo,
= AH =
AH: N
KH:
KH: N
A: Z. B: A;
= B: Z;
[Eucl. v., Def. 5 [Eucl. v. 7, coroll. [Eucl. v. 22
therefore Z is the same part of B as N is of KH. Now A was proved to be a multiple of z; therefore Z is a common measure of A, B. Therefore, if AH is divided into segments equal to N and A into segments equal to z, the segments in AH equal in magnitude to N will be equal in number to the segments of A equal to Z. It follows that, if there be placed on each of the segments in AH a magnitude equal to z, having its centre of gravity at the middle of the segment, the sum of the magnitudes wiJI be equal to A, and the centre of gravity of the figure compounded of them all will be E; for they are even in number, and the numbers on either side of E will be equal because AE BE.
=
197
Greek MlltMmlIIiC$
Similarly it may be proved that, if a magnitude equal to z be placed on each of the segments [equal to N] in KH, having its centre of gravity at the middle of the segment, the sum of the magnitudes will be equal to B, and the centre of gravity of the figure compounded of them all will be A. Therefore A may be regarded as placed at E, and B at A. But they will be a set of magnitudes lying on a straight line, equal one to another, with their centres of gravity at equal intervals, and even in number; it is therefore clear that the centre of gravity of the magnitUde compounded of them aU is the point of bisection of the line containing the centres [of gravity] of the middle magnitudes. And since AE = rA and Er = AK, therefore AI' IX; so that the centre of gravity of the magnitude compounded of them all is the point r. Therefore if A is placed at E and B at A, they will balance about r.
=
INDETERMINATE ANALYSIS: THE CATTLE PROBLEM (ARCHIMEDES (1),
Cattle Problem S)
A problem which Archimedes solved in epigrams, and which he communicated to students of such matters at Alexandria in a letter to Eratosthenes of Cyrene. If thou art diligent and wise, 0 stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields of the Thrinacian isle of Sicily, divided into four herds of different colours, one milk white, another a glossy black, the third yellow and the last dappled. In each herd were bulls, mighty in number according to these proportions: Understand, stranger, that the white buns were equal to a half and a third of the black together with the who1e of the yellow, while the black were equal to the fourth part of the dappled and a fifth, together with, once more, the whole of the yellow. Observe furtber that the remaining bulls, the dappled, were equal to a sixth part of the white and a seventh, together with all the yellow. These were the proportions of the cows: The white were precisely equal to the third part and a fourth of the whole herd of the black; while the black were equal to the fourth part once more of the dappled and with it a fifth part, when all, including the bulls, went to pasture together. Now the dappled in four parts 4 were equal in number to a fifth part and a sixth of the yellow herd. Finally the yellow were in number equal to a sixth part and a seventh of the white herd. If thou canst accurately tell, 0 stranger, the number of cattle of the Sun, giving separately the number of well-fed 3 It is unlikely that the epigram itself, first edited by G. E. Lessing in 1773, is the work of Archimedes, but there is ample evidence from antiquity that he studied the actual problem. The most important papers bearing on the subject are A. Amthor, Zeitschrift fur Math. u. Physik (Hist.-litt. AbtheHung), xxv. (1880), pp. 153-171, supplementing an article by B. Krumbiegel (pp. 121-136) on the authenticity of the problem. See also T. L. Heath, The Works of Archimedes, pp. 319-326. 4 i.e. a fifth and a sixth both of the males and of the females.
198
1110' Tllorruu
bulls and again the number of females according to each colour, thou wouldst not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise. But come, understand also all these conditions regarding the cows of the Sun. When the white bulls mingled their number with the black, they stood firm, equal in depth and breadth,15 and the plains of Thrinacia, stretching far in all ways. were filled with their multitude. Again, when the yellow and the dappled bulls were gathered into one herd they stood in such a manner that their number, beginning from one, grew slowly greater till it completed a triangular figure, there being no bulls of other colours in their midst nor none of them lacking. If thou art able, 0 stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and and knowing that thou hast been adjudged perfect in this species of wisdom. 8 5 At a first glance this would appear to mean that the sum of the number of white and black bulls is a square, but this makes the solution of the problem intolerably difficult. There is, however, an easier interpretation. If the bulls are packed together so as to form a square figure, their number need not be a square, since each bull is longer than it is broad. The simplified condition is that the sum of the number of white and black bulls shall be a rectangle. elf x, x are the numbers of white bulls and cows respectively, Y, y " ,. U black " t; It Z. Z .. .. Of yel10w .. Of ,. W, w " " .. dappled u t. .. the first part of the epigram states that
(a)
X=(1h+lh) Y+Z Y = (% + J,f;) W + Z W= (~+ lIT) X + Z
:c= (lh + %)(Y
(6)
•
+ y)
(1) (2) (3)
(4)
y= (% + lA;)(W + w) w = (lA; + lk) (Z + z) z;=(lk+Vr)(X+x)
(5)
(6) (7)
The second part of the epigram states that X + Y = a rectangular number Z + W = a triangular number
•
(8)
·
(9)
This was solved by 1. F. Wurm, and the solution is given by A. Amthor, Zeltschri/t Jur Math. u. Physik. (Rist.-lilt. Abtheilung) , xxv. (1880), pp. 153-171, and by Heath, The Wo,ks 0/ Archimedes, pp. 319-326. For reasons of space, only the results can be noted here. Equations (1) to (7) give the fonowing as the values of the unknowns in terms of an unknown integer n: X= Y= Z= W=
10366482n 7460514n 4149387n 7358060n
x=7206360n y = 4893246n z 5439213n w= 3515820n.
=
We have now to find a value of n such that equation (9) is also satisfied-equation (8) will then be simultaneously satisfied. Equation (9) means that p(p
+ 1)
Z+W=---, 2
199
SoLID IMMERSED IN A FLUID (ARCIDMEDES,
On Floating Bodies)
Solids heavier than a fluid will, if placed in the fluid, sink to the bottom, and they will be lighter [if weighed] in the fluid by the weight of a volume of the fluid equal to the volume of the solid. 7 That they will sink to the bottom is manifest; for the parts of the fluid under them are under greater pressure than the parts lying evenly with them, since it is postulated that the solid is heavier than water; that they will be lighter, as aforesaid will be [thus] proved. Let A be any magnitude heavier than the fluid, let the weight of the magnitude A be B + r, and let the weight of fluid baving the same volume as A be B. It is required to prove that in the fluid the magnitude A will have a weight equal to r. ~
8
A
A
r
FIGURE 6
wbere p is some positive integer, or
(4149387
p(p
+ 7358060)n
+ 1) 2
2471 • 4657n =
i.e.
pep
+ 1)
.
2 This is found to be satisfied by and the final solution is
n
=3
3
.4349,
=
x = 1217263415886
x 846192410280 y = 574579625058 z 638688708099 w 412838131860
=
Y 876035935422 Z = 487233469701 W= 864005479380
=
=
and the total is 5916837175686. If equation (8) is taken to be tbat X + Y a square number. the solution is much more arduous; Amtbor found tbat in this case, ~
=
W= 1598 . wbere means that there are 206541 more digits to follow, and tbe wbole number of cattle 7766 . Merely to write out tbe eight numbers, Amthor calculates, would require a volume of 660 pages, so we may reasonably doubt wbetber the problem was really framed in this more difficult fonn, or, jf it were, whether Arcbimedes solved it. T Or, as we should say, "lighter by the weigbt of fluid displaced."
=
II1D' l'hDmlU
2GO
For let there be taken any magnitude- A lighter than the same volume of the fluid such that the weight of the magnitude A is equal to the weight B~ while the weight of the fluid having the same. volume as the magnitude b. is equal to the weight B + r. Then if we combine tbe magnitudes A, A, the combined magnitude will be equal to the weight of the same volume of the fluid; for the weight of the combined magnitudes is equal to the weight (B + r) + B, while the weight of the fluid having the same volume as both the magnitudes is equal to the same weight. Therefore, if the [combined] magnitudes are placed in the fluid, they will balance the fluid" and will move neither upwards nor downwards; for this reason the magnitude A will move downwards, and will be subject to the same force as that by which the magnitude A is thrust upwards, and since A is lighter than the fluid it will be thrust upwards by a force equal to the weight r; for it has been proved that when solid magnitudes lighter than the fluid are forcibly immersed in the fluid. they will be thrust upwards by a force equal to the difference in weight between the magnitude and an equal volume of the fluid [Prop. 6]. But the fluid having the same volume as A is heavier than the magnitude A by the weight r; it is therefore plain that the magnitude A will be borne upwards by a force equal to r.8
APOLLONIUS OF PERGA CONSTRUCTION OF THE SECTIONS (APOLLONIUS,
Conics)
Proposition 7 9 IF a cone be cut by a plane through the axis, and i/ it be also cut by another plane cutting the plane containing the base 0/ the cone in a straight line perpendicular to the base 0/ the axial triangle,lO or to the base produced, a section will be made on the sur/ace 0/ the cone by the cutting plane, and straight lines drawn in it parallel to the straight line perpendicu· 8 This proposition suggests a method, alternative to that given by Vitruvius, whereby Archimedes may have discovered the proportions of gold and silver in King Hiero's crown. Let W be the weight of the crown, and let W t and W$ be the weights of gold and silver in it respectively, so that w::: WI + W 2• Take a weight W of gold and weigh it in a ftuid, and let the loss of weight be P a• Then the loss of weight when a weight WI of gold is weighed in the fluid, and WI
consequently the weight of fluid displaced, wi11 be -
_ PI_
W
Now take a weight of W of silver and weigh it in the fluid, and let the loss of weight be Pi' Then the loss of weight when a weight W: of silver is weighed in the WI
fluid, and consequently the weight of fluid displaced, will be - . P<Jo w
201
lar to the base of the axial triangle will meet the common section of the cutting plane and the axial triangle and. if produced to the other part of the section. will be bisected by it; if the cone be right. the straight line in the base will be perpendicular to the common section of the cutting plane and the axial triangle; but if it be scalene, it will not in general be perpendicular. but only when the plane through the axis is perpendicular to the base of the cone. Let there be a cone whose vertex is the point A and whose base is the circle Dr, and let it be cut by a plane through the axis, and let the section so made be the triangle ABr. Now let it be cut by another plane cutting the plane containing the circle Dr in a straight line AE which is either perpendicular to Br or to Br produced, and let the section made on the surface of the cone be 4ZE 11; then the common section of the cutting plane
A
A
......-1-...- -......
FlGUltB 7
FlGUltB 8
Finally. weigh the crown itself in tbe fluid, and let tbe loss of weight, and consequently the weight of fluid displaced, be P. WI
Wi
W
W
It follows that - . P1 + -
. PI = P, WI
whence
P:.-P
-=--
W. P -PI This proposition defines a conic section in tbe most general way with reference to any diameter. It is only mucb later in the work that the principal axes are introduced as diameters at right angles to their ordinates. The proposition is an excellent example of the generality of Apollonius's methods. ApoJlonius followed rigorously the Euclidean fOnD of proof. In consequence his general enunciations are extremely long and often can be made tolerable in an English rendering only by splitting them up; but, though Ap0110nius seems to have taken a ma1icious p1easure in their length, they are formed on a perfect logical pattern witbout a superfluous word. 10 Lit. "the triangle throUgh the axis." 11 This applies only to the ftrst two of the figures given in the M$S. (Le. figures 7. 8, above). 9
and of the triangle ADr is ZH. Let any point 8 be taken on AZE, and through 8 let 8K be drawn parallel to .I1E. I say that 8K intersects ZH and, if produced to the other part of the section .I1ZE, it will be bisected by the straight line ZH. For since the cone, whose vertex is the point A and base the circle Dr, is cut by a plane through the axis and the section so made is the triangle ADr, and there has been taken any point 8 on the surface, not being on a side of the triangle ADr, and .I1H is perpendicular to Dr, therefore the straight line drawn through 8 parallel to .I1H, that is 8K, will meet the triangle ADr and, if produced to the other part of the surface, will be bisected by the triangle. Therefore, since the straight line drawn through 8 parallel to .I1E meets the triangle ADr and is in the plane containing the section .I1ZE, it will fall upon the common section of the cutting plane and the triangle ABr. But the common section of those planes is ZH; therefore the straight line drawn through 8 parallel to .I1E will meet ZH; and if it be
A
A
B~--------~~--~I
FlGUIlE ,
FlGUIlE 10
produced to the other part of the section .I1ZE it will be bisected by the straight line ZH. Now the cone is right, or the axial triangle ABr is perpendicular to the circle Dr, or neither. First, let the cone be right; then the triangle ADr will be perpendicular to the circle Dr [Eucl. xi. 18]. Then since the plane ADr is perpendicular to the plane Dr, and .I1E is drawn in one of the planes perpendicular to their common section Dr, therefore .I1E is perpendicular to the triangle ADr [Eucl. xi. Def. 4]; and therefore it is perpendicular to all the straight Jines in the triangle ADr which meet it [Eucl. xi. Def. 3}. Therefore it is perpendicu1ar to ZH. Now let the cone be not right. Then, if the axial triangle is perpendicular to the circle Dr, we may similarly show that .I1E is perpendicular to ZH. Now let the axial triangle ABr be not perpendicular to the .circle Dr. I say that neither is .I1E perpendicular to ZH. For if it is possible, let it be; now
203
G,~~k Malh~matlcs
it is also perpendicular to Br; therefore dE is perpendicular to both Br, ZH. And therefore it is perpendicular to the plane through Br, ZH [Eucl. xi. 4]. But the plane through Br, HZ is ABr; and therefore dE is perpendicular to the triangle ABr. Therefore all the planes through it are perpendicular to the triangle ABr [Eucl. xi. 18]. But one of the planes through dE is the circle Br; therefore the circle Br is perpendicular to the triangle ABr. Therefore the triangle ABr is perpendicular to the circle Br; which is contrary to hypothesis. Therefore dE is not perpendicular to ZH.
Corollary From this it is clear that ZH is a diameter of the section dZE, inasmuch as it bisects the straight lines drawn parallel to the given straight line dE, and also that parallels can be bisected by the diameter ZH without being perpendicular to it. FUNDAMENTAL PROPERTIES (APOLLONIUS,
Conics)
Proposition 11
Let a cone be cut by a plane through the axis, and let it be also cut by another plane cutting the base of the cone in a straight line perpendicular to the base of the axial triangle. and further let the diameter of the section be parallel to one side of the axial triangle; then if any straight line be drawn from the section of the cone parallel to the common section of the cutting plane and the base of the cone as far as the diameter of the section. its square will be equal to the rectangle bounded by the intercept made by it on the diameter in the direction of the vertex of the section and a certain other straight line; this straight line will bear the same ratio to the intercept between the angle of the cone and the vertex of the segment as the square on the base of the axial triangle bears to the rectangle bounded by the remaining two sides of the triangle; and let such a section be called a parabola. For let there be a cone whose vertex is the point A and whose base is the circle Br, and let it be cut by a plane through the axis, and let the section so made be the triangle ABr, and let it be cut by another plane cutting the base of the cone in the straight line dE perpendicular to Br, and let the section so made on the surface of the cone be dZE, and let ZH. the diameter of the section, be parallel to Ar, one side of the axial triangle and from the point Z let ze be drawn perpendicular to ZH, and let Br2 : BA . Ar = ze : ZA. and let any point K be taken at random on the section, and through K let KA be drawn parallel to dE. I say that KA2
=
ez. ZA. For let MN be drawn through A parallel to Br; but KA is parallel to AE;
lYOr 7110_
A
~------~~~----4r
FIGUItE II
therefore the plane through KA, MN is parallel to the plane through Br, ~ [Eucl. xi. 15], that is to the base of the cone. Therefore the plane through KA, MN is a circle, whose diameter is MN [Prop. 4]. And KA is perpendicular to MN, since ~ is perpendicuJar to Br [Eucl. xi. 10]; therefore And since whi1e therefore But and
MA • AN
=
KA2.
Br2 : BA • AI' = 8Z : ZA, Br2 : BA • .AI' (Br : rA) (Br : BA),
=
8Z: ZA
=
Br : rA
= MN : NA
Br : BA
(Br : rA)(rB : BA).
= MA: AZ,
=
MN : MA
=AM:MZ
=
[Eud. vi. 4
NA : ZA.
[ibid. [Eucl. vi. 2
Therefore 8Z : ZA = (MA : AZ) (NA : ZA). But (MA : AZ) (AN : ZA) MA . AN : AZ • ZA. Therefore 8Z : ZA MA . AN : AZ • ZA. But 8Z : ZA = 8Z • ZA : AZ • ZA, by taking a common height ZA; therefore MA • AN : AZ • ZA 8Z • ZA : AZ • ZA. [Euc1. v. 9 Therefore MA • AN = 8Z • ZA. But MA. AN = KA2; KA2 = 8Z. ZA. and therefore
= = =
Let such a section be called a parabola. and let 8Z be called the parameter of the ordinates to the diameter ZH, and Jet it also be called the erect side (latus rectum) .12 111
A parabola (""Ap4PO~~) because the square on the ordinate KA is applied to the parameter 9Z in the form of the rectangle 9Z. ZA, and is
(1I'ApAPA~~i.,)
205
G1uk Mat1t~mtI"cl
ERATOSTHENES MEASUREMENT OF THE EARTH (CLEOMEDES,18
On the Circular Motion of the Heaven Bodies)
SUCH then is Posidonius's method of investigating the size of the earth, but Eratosthenes' method depends on a geometrical argument, and gives the impression of being more obscure. What he says will, however, become clear if the following assumptions are made. Let us suppose, in this case also, first that Syene and Alexandria lie under the same meridian circle; secondly, that the distance between the two cities is 5000 stades; and thirdly, that the rays sent down from different parts of the sun upon different parts of the earth are parallel; for the geometers proceed on this assumption. Fourthly. let us assume that, as is proved by the geometers, straight lines faUing on parallel straight lines make the alternate angles equal, and fifthly, that the arcs subtended by equal angles are similar, that is, have the same proportion and the same ratio to their proper circlesthis also being proved by the geometers. For whenever arcs of circles are subtended by equal angles, if anyone of these is (say) one-tenth of its proper circle, an the remaining arcs will be tenth parts of their proper circles. Anyone who has mastered these facts will have no difficulty in understanding the method of Eratosthenes, which is as follows. Syene and Alexandria, he asserts, are under the same meridian. Since meridian circles are great circles in the universe, the circles on the earth which lie under them are necessarily great circles also. Therefore, of whatever size this method shows the circle on the earth through Syene and Alexandria to be, this will be the size of the great circle on the earth. He then asserts, as is indeed the case, that Syene lies under the summer tropic. Therefore, whenever the sun, being in the Crab at the summer solstice, is exactly in the middle of the heavens, the pointers of the sundials necessarily throw no shadows, the sun being in the exact vertical line above them; and this is said to be true over a space 300 stades in diameter. But in Alexandria at the same hour the pointers of the sundials throw shadows, because this city lies farther to the north than Syene. As the two cities lie under the exactly equal to this rectangle. It was Apollonius's most distinctive achievement to have based his treatment of the conic sections on the Pythagorean theory of the application 01 areas (frflptlfjDA;, .,.~ .. x"pl.,,,). 13 Cleomedes probab1y wrote about the middle of the first century B.C. His handbook De molu c;rcuhul COrpOTum caelestium is largely based on Posidonius.
IYor rifolfllU
same meridian great circle, if we draw an arc frum the extremity of the shadow of the pointer to the base of the pointer of the sundial in Alexan· dria, the arc will be a segment of a great circle in the bowl of the sundial, since the bowl lies under the great circle. If then we conceive straight lines produced in order from each of the pointers through the earth, they will meet at the centre of the earth. Now since the sundial at Syene is vertically under the sun, if we conceive a straight line drawn from the sun to the top of the pointer of the sundial, the line stretching from the sun to the centre of the earth will be one straight line. If now we conceive another straight line drawn upwards from the extremity of the shadow of the pointer of the sundial in Alexandria, through the top of the pointer to the sun, this straight line and the aforesaid straight line will be paraIJel, being straight iines drawn through from different parts of the sun to different parts of the earth. Now on these parallel straight lines there falls the straight line drawn from the centre of the earth to the pointer at Alexandria, so that it makes the alternate angles equal; one of these is formed at the centre of the earth by the intersection of the straight lines drawn from the sundials to the centre of the earth; the other is at the intersection of the top of the pointer in Alexandria and the straight line drawn from the extremity of its shadow to the sun through the point where it meets the pointer. Now this latter angle subtends the arc carried round from the extremity of the shadow of the pointer to its base, while the angle at the centre of the earth subtends the arc stretching from Syene to Alexandria. But the arcs are similar sin~e they are subtended by equal angles. Whatever ratio, therefore, the arc in the bowl of the sundial has
FIGURE 12
to its proper circle, the arc reaching from Syene to Alexandria has the same ratio. But the arc in the bowl is found to be the fiftieth part of its proper circle. Therefore the distance from Syene to Alexandria must necessarily be a fiftieth part of the great circle of the earth. And this distance
is 5000 stades. Therefore the whole great circle is 250000 stades. Such is the method of Eratosthenes.14 DIOPHANTUS (Palatine Anthology lG xiv)
THIS tomb bolds Diophantus. Ah, what a marvel! And the tomb tens scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindJed for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! 1atebegotten and miserable child, when he had reached the measure of half his father's Ufe, the chi11 grave took him. After consoling his grief by this science of numbers for four y.ears, he reached the end of his 1ife.16 REVIVAL OF GEOMETRY: PAPPUS ISOPERIMETkIC FIGURES 1'1 (PAPPUS,
Collection)
THOUGH God has given to men, most excellent Megethion, the best and most perfect understanding of wisdom and mathematics, He has allotted 14 The figure on p. 206 will help to elucidate Cleomedes. S is Syrene and A Alexandria; the centre of the earth is O. The sun's rays at the two places are represented by the broken straiBht lines. If fl be the angle made by the sun's rays with the pointer of the sundial at Alexandria (OA produced), the angle SOA is also equal to fl, Or one-fiftieth of four riglu angles. The arc SA is known to be 5000 stades and it follows that the whole circumference of the earth must be 250000 stades. ill There are in the Anthology 46 epigrams which are algebraical problems. Most of them were collected by Metrodorus, a grammarian who lived about A..D. 500, but their origin is obviously much earlier and many belong to a type described by Plato and the schoJiast to the Charmides. Problems in indeterminate analysis solved before the time of Diophantus include the Pythagorean and Platonic methods of finding numbers representing the sides of riBht-angled triangles, the methods (also Pythagorean) of finding "side- and diameternumbers," Archimedes' Callie Problem and Heron's problems. 16 If x was his age at death, then
= =
~ + lh2X + Vrx + 5 + 1h.x + 4 x, whence x 84. 1'7 The whole of Book v. in Pappus's Collection is devoted to isoperimetry. The first section foDows dosely the exposition of Zenodorus as given by Theon, except that Pappus includes the proposition that of all circular segments having the same circumference the semicircle is the grealest. The second section compares the volumes of solids whose surfaces are equal, and is followed by a digression, on the semi-regular solids discovered by Archimedes. After some propositions on the liRes of Archimedes' De sph. et cyl., Pappus final1y proves that of regular solids having equal surfaces. that is greatest which has most faces. The introduction, here cited, on the sagacity of bees is rightly praised by Heath as an example of the good style of the Greek mathematicians when freed from the restraints of technical language.
bor TllomtU
208
a partial share to some of the unreasoning creatures as well. To men, as being endowed with reason, He granted that they should do everything in the light of reason and demonstration, but to the other unreasoning creatures He gave only this gift, that each of them should, in accordance with a certain natural forethought, obtain so much as is needful for supporting life. This instinct may be observed to exist in many other species of creatures, but it is specially marked among bees. Their good order and their obedience to the queens who rule in their commonwealths are truly admirable, but much more admirable still is their emulation, their cleanliness in the gathering of honey, and the forethought and domestic care they give to its protection. Believing themselves, no doubt, to be entrusted with the task of bringing from the gods to the more cultured part of mankind a share of ambrosia in this form, they do not think it proper to pour it carelessly into earth or wood or any other unseemly and irregular material, but, coUecting the fairest parts of the sweetest flowers growing on the earth, from them they prepare for the reception of the honey the vessels called honeycombs, [with cells] all equal, similar and adjacent, and hexagonal in form. That they have contrived this in accordance with a certain geometrical forethought we may thus infer. They would necessarily think that the figures must all be adjacent one to another and have their sides common, in order that nothing else might fall into the interstices and so defile their work. Now there are only three rectilineal figures which would satisfy the condition, I mean regular figures which are equilateral and equiangular, inasmuch as irregular figures would be displeasing to the bees. For equilateral triangles and squares and hexagons can lie adjacent to one another and have their sides in common without irregular interstices. For the space about the same point can be filled by six equilateral triangles and six angles, of which each is %. right angle, or by four squares and four right angles, or by three hexagons and three angles of a hexagon, of which each is 1 right angle. But three pentagons would not suffice to fill the space about the same point, and four would be more than sufficient; for three angles of the pentagon are less than four right angles (inasmuch as each angle is 1~ . right angle), and four angles are greater than four right angles. Nor can three heptagons be placed about the same point so as to have their sides adjacent to each other; for three angles of a heptagon are greater than four right angles (inasmuch as each is 1 right angle). And the same argument can be applied even more to polygons with a greater number of angles. There being, then, three figures capable by themselves of filling up the space around the same point, the triangle, the square and the hexagon, the bees in their wisdom chose for their work that which has the most angles, perceiving that it would hold more honey than either of the two others.
*.
*.
Greek Malllemtl,lcJ
209
Bees, then, know just this fact which is useful to them, that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material in constructing each. But we, claiming a greater share in wisdom than the bees, will investigate a somewhat wider problem, namely that, 0/ all equilateral and equiangular plane figures having an equal perimeter, that which has the greater number of angles is always greater, and the greatest of them all is the circle having its perimeter equal to them.
COMMENTARY ON
ROBERT RECORDE UOBERT RECORDE, born in Wales in 1510, taught mathematics at .1'... Oxford and Cambridge, got his M.D. degree at the latter university in 1545, became physician to Edward VI and Queen Mary, served for a time as "Comptroller of Mines and Monies" in Ireland and died in the King's Bench Prison, Soutbwark, where he was confined for debt--some historians hint at a darker offense-in 1558. Recorde published a number of mathematical works, chiefly in the then not uncommon form of dialogue between master and scholar. The most popular and influential of these treatises were The Ground 01 Artes (1540), and The Whetstone of Witte (1557), "containying extraction of Rootes: The Cossik.e practise, with the rule of Equation: and the woorkes of Surde Numbers." England Jagged behind Italy and France in the publication of mathematical books.l By the close of the fifteenth century Italy alone had printed more than 200 treatises on mathematics, whilst it was not until 1522 that the first book dealing exclusively with arithmetic L-the "erudite but dunH De Arte Supputandi of Cuthbert Tonstall-appeared in Great Britain. The Ground 01 Artes (the selections below were taken from its edition of 1646) was an immensely successful commercial arithmetic. It went through at least eighteen editions in the sixteenth century and a dozen more in the seventeenth. Its influence, and that of The Whetstone 01 Witte, is at least partly explained by the economic development-manufacture and commerce--of England in the reign of Elizabeth. "Never (says D. E. Smith) was there a better opportunity for a commercial arithmetic, and never was the opportunity more successfully met." Recorde's book discusses operations with arabic numerals as well as computation with counters, proportion, the "golden rule" of three, fractions and "allegation" and "contains the usual commercial topics which European countries north of the Alps had derived from Italy." It is hard to understand how anyone couJd learn from a dialogue delivered in the "formal" language of this primer. On second thought, it is perhaps just as remarkable that any of us profited from the average arithmetic texts visited on pupils as recently as twenty-five years ago. Recorde, whose work was Uentaird upon the People, ratified and sign'd by the approbation 1 For the source of these data see David Eugene Smith, RtJnz Arithmetica. Boston, 1908; also, the same author's standard History 0/ Mathematics, Boston and New York, 1923. :21 The. first printed English book containing a reference to arithmetic is an anonymous translation from the French, The Mirrow 0/ the World or Thymage 01 the same, issued in 1480 from the Caxton press. Chapter 10 of this book begins: "And after of Arsmetrike and whereof it proceedeth." See Smith, RtJra Arilhmetica.
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of Time," 8 was at least modest in his claims for it. He wrote in his preface: "And if any man obiect, that other books haue bene written of Arithmetike alreadie so sufficiently, that I needed not now to put penne to the booke, except I will codemne other mens writings: to them I answere. That as I codemne no mans diligence, so I know that no man can satisfie euery man, and therefore like as many do esteeme greatly other bookes, so I doubt not but some will like this my booke aboue any other English Arithmetike hitherto written, & namely such as shal lacke instructers, for whose sake I haue plain-Iy set forth the exaples, as no book (that I haue seene) hath hitherto: which thing shan be great case to the rude readers." .. 3 A comment by Thomas Willsford in his 1662 edition of The Ground 01 Aries. .. From the 1594 edition; see Smith. RJuo Arithmetico. p. 216.
Someone who 1uul begun to read geometry with Eucll4. when Jut had letll'Mtl the first proposition. asked Eucll4. "But what shall I get by ktunlng these things'" wlutreupon Euclid called his slave and ,al4 "Give him three-pence --STOBABUS since he must make gain out 01 what he learns," There stili remain three studies suitable lor lree man, Arithmetic is OM 01 tMm. -PUTO
5
The Declaration of the Profit of Arithmeticke By ROBERT RECORDE TO THE LOVING READERS. THE PREFACE OF MR. ROBERT RECORD
SORE oft times have I lamented with my self the unfortunate condition of EngJand t seeing so many great Clerks to arise in sundry other parts of the world, and so few to appear in this our Nation: whereas for pregnancy of naturall wit (I think) few Nations do excell Englishmen: But I cannot impute the cause to any other thing. then to be contempt, or misregard of learning. For as Englishmen are inferiour to no men in mother wit. so they passe aU men in vain pleasures, to which they may attain with great pain and labour: and are as slack to any never so great commodity; if there hang of it any painfull study or travelsome labour. Howbeit, yet all men are not of that sort, though the most part be, the more pity it is: but of them that are so glad, not onely with painfull study, and studious pain to attain learning, but also with as great study and pain to communicate their learning to other. and make all England (if it might be) partakers of the same; the most part are such. that unneath they can support their own necessary charges, so that they are not able to bear any charges in doing of that good, that else they desire to do. But a greater cause of lamentation is this. that when learned men have taken pains to do things for the aid of the unlearned. scarce they shall be allowed for their weI-doing, but derided and scorned, and so utterly discouraged to take in hand any like enterprise again. The following is "The declaration of the profit of Arithmeticke" and constitutes the first ten pages of the text. It may be said to represent the influence of this text upon establishing for a long period what educators at present speak of as "the objectives" of elementary arithmetic. 212
Tile DeclllrlJllon 01 the Profil of A.rlthmetlcke
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A DIALOGUE BETWEEN THE MASTER AND THE SCHOLAR: TEACHING THE ART AND USE OF ARITHMEnCK WITH PEN. THE SCHOLAR SPEAKETH.
SIR, such is your authority in mine estimation, that I am content to consent to your saying, and to receive it as truth, though I see none other reason that doth lead me thereunto: whereas else in mine own conceit it appeareth but vain, to bestow any time privately in learning of that thing, that every chi/de may, and doth learn at all times and hours, when he doth any thing himself alone, and much more when he talketh or reasoneth with others. Master. Lo, this is the fashion and chance of all them that seek to defend their blinde ignorance, that when they think they have made strong reason for themselves, then have they proved quite contrary. For if numbring be so common (as you grant it to be) that no man can do anything alone, and much lesse talk or bargain with other, but he shall still have to do with number: this proveth not number to be contemptible and vile, but rather right excellent and of high reputation, sith it is the ground of all mens affairs, in that without it no tale can be told, no communication without it can be continued, no bargaining without it can duely be ended, or no businesse that man hath, justly completed. These commodities, if there were none other, are sufficient to approve the worthinesse of number. But there are other innumerable, farre passing an these, which declare number to exceed all praise. Wherefore in all great works are Clerks so much desired? Wherefore are Auditors so richly fed? What causeth Geometricians so highly to be enhaunsed? Why are Astronomers so greatly advanced? Because that by number such things they finde, which else would farre excell mans minde. Scholar. Verily, sir, if it bee so, that these men by numbring, their cunning do attain, at whose great works most men do wonder, then I see wen I was much deceived, and numbring is a more cunning thing then I took it to be. Master. If number were so vile a thing as you did esteem it, then need it not to be used so much in mens communication. Exclude number, and answer to this question: How many years old are you? Scholar. Mum. Master. How many dayes in a weeke? How many weeks in a year? What lands hath your Father? How many men doth hee keep? How long is it since you came from him to me? Scholar. Mum. Master. So that if number want, you answer all by Mummes: How many miles to London? Scholar. A poak full of plums.
~14
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Master. Why, thus you may see, what rule number beareth, and that if number bee lacking it maketh men dumbJ so that to most questions they must answer Mum. Scholar. This is the cause, sir, that I judged it so vile, because it is so common in talking every while: Nor plenty is not dainty, as the common saying is. Master. No, nor store is no sore, perceive you this? The more common that the thing is, being needfully required, the better is the thing, and the more to be desired. But in numbring, as some of it is light and plain, so the most part is difficu1t, and not easie to attain. The easier part serveth all men in common, and the other requireth some learning. Wherefore as without numbring a man can do almost nothing, so with the help of it, you may attain to aU things. Scholar. Yes, sir, why then it were best to learn the Art of numbring, first of all other learning, and then a man need learn no more, if all other come with it. Master. Nay not so: but if it be first learned, then shall a man be able (I mean) to learn, perceive, and attain to other Sciences; which without it he could never get. Scholar. I perceive by your former words, that Astronomy and Geometry depend much on the help of numbring: but that other Sciences, as Musick, Physick, Law, Grammer, and such like. have any help of Arithmetick, I perceive not. Master. I may perceive your great Clerk-linesse by the ordering of your Sciences: but I will let that passe now, because it toucheth not the matter that 1 intend, and I will shew you how Arithmetick doth profit in all these somewhat grosly, according to your small understanding. omitting other reasons more substantiall. First (as you reckon them) Musick hath not onely great help of Arithmetic, but is made, and hath his perfectnesse of it: for all Musick standeth by number and proportion: And in Physick, beside the calculation of critical I dayes, with other things, which I omit, how can any man judge the pulse rightly, that is ignorant of the proportion of numbers? And so for the Law, it is plain, that the man that is ignorant of Arithmetick, is neither meet to be a Judge, neither an Advocate. nor yet a Proctor. For how can hee well understand another mans cause, appertaining to distribution of goods, or other debts, or of summes of money, if he be ignorant of Arithmetick? This oftentimes causeth right to bee hindered, when the Judge either delighteth not to hear of a matter that hee perceiveth not, or cannot judge for lack of understanding: this commeth by ignorance of Arithmetick. Now, as for Grammer, me thinketh you would not doubt in what it needeth number, sith you have learned that Nouns of an sorts, Pronouns,
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of
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Profit of
Aritll"..tlck~
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Verbs, and Participles are distinct diversly by numbers: besides the variety of Nouns of Numbers, and Adverbs. And if you take away number from Grammer, then is all the quantity of Syllables lost. And many other ways doth number help Grammer. Whereby were all kindes of Meeters found and made? was it not by number? But how need full Arithmetick is to all parts of Philosophy, they may soon see, that do read either Aristotle, Plato, or any other Philosophers writings. For all their examples almost, and their probations, depend of Arithmetick. It is the saying of Aristotle, that hee that is ignorant of Arithmetick, is meet for no Science. And Plato his Master wrote a little sentence over his Schoolhouse door, Let none enter in hither (quoth he) that is ignorant of Geometry. Seeing hee would have al1 his Scholars expert in Geometry, much rather he would the same in Arithmetic~ with· out which Geometry cannot stand. And how needfull Arithmetick is to Divinity, it appeareth, seeing so many Doctors gather so great mysteries out of number, and so much do write of it. And if I should go about to write all the commodities of Arithmetick in civill acts, as in governance of Common-weales in time of peace, and in due provision & order of Armies, in time of war, for numbering of the Host, summing of their wages, provision of victuals, viewing of Artillery, with other Armour; beside the cunningest point of al1, for casting of ground, for encamping of men, with such other 1ike: And how many wayes also Arithmetick is conducible for all private Weales, of Lords and all Possessioners, of Merchants, and all other occupiers, and generally for all estates of men, besides Auditors, Treasurers, Receivers, Stewards, Bailiffes, and such Uke. whose Offices without Arithmetick are nothing: If I should (I say) particularly repeat all such commodities of the noble Science of Arithmetick, it were enough to make a very great book. Scholar. No, no sir, you shall not need: For I doubt not, but this, that you have said, were enough to perswade any man to think this Art to be right excellent and good, and so necessary for man, that (as I think now) so much as a man lacketh of it, so much hee lacketh of his sense and wit. Master. What, are you so farre changed since. by hearing these few commodities in generall: by likelihood you would be farre changed if you knew all the particular Commodities. Scholar. I beseech you Sir, reserve those Commodities that rest yet behinde unto their place more convenient: and if yee will bee so good as to utter at this time this excellent treasure, so that I may be somewhat inriched thereby, if ever I shan be able, I will requite your pain. Master. I am very glad of your request, and will do it speedily. sith that to Jearn it you bee so ready.
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Robert Reco'"
Scholar. And I to your authority my wit do subdue; whatsoever you say, I take it for true. Master. That is too much; and meet for no man to bee beleeved in all things, without shewing of reason. Though I might of my Scholar some credence require, yet except I shew reason, I do it not desire. But now sith you are so earnestly set this Art to attaine, best it is to omit no time, lest some other passion coole this great heat, and then you leave off before you see the end. Scholar. Though many there bee so unconstant of mind, that flitter and turn with every winde, which often begin, and never come to tbe end, I am none of this sort, as I trust you partly know. For by my good will what I once begin, till I have it fully ended, I would never bUn. Master. So have I found you hitherto indeed, and I trust you will increase rather then go back. For, better it were never to assay, then to shrink and flie in the mid way: But I trust you will not do so; therefore tell mee briefly: What call you the Science that you desire so greatly. Scholar: Why sir, you know. Master. That maketh no matter, I would hear whether you know, and therefore I ask you. For great rebuke it were to have studied a Science, and yet cannot tell how it is named. Scholar. Some call it Arsemetrick, and some Augrime. Master. And what do these names betoken? Scholar. That, if it please you, of you would I learn. Master. Both names are corruptly written: Arsemetrick for Arithmetick, as the Greeks call it, and Augrime for Algorisme, as the Arabians found it: which both betoken the Science of Numbring: for Arithmos in Greek is called Number: and of it commeth Arithmetick, the Art of Numbring. So that Arithmetick is a Science or Art teaching the manner and use of Numbring: This Art may be wrought diversly, with Pen or with Counters. But I will first shew you the working with the Pen, and then the other in order. Scholar. This I will remember. But how many things are to bee learned to attain this Art fully? Master. There are reckoned commonly seven parts or works of it. Numeration, Addition, Subtraction, Multiplication, Division, Progression, and Extraction of roots: to these some men adde Duplication, Triplation, and Mediation. But as for these three last they are contained under the other seven. For Duplication, and Triplation are contained under Multiplication; as it shall appear in their place: And Mediation is contained under Division, as I will declare in his place also. Scholar. Yet then there remain the first seven kinds of Numbring. Master. So there doth: Howbeit if I shall speak exactly of the parts of Numbring, I must make but five of them: for Progression is a com-
Tire Declortltion of tire Profit of Aril',,"eticke
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pound operation of Addition, Multiplication and Division. And so is the Extractions of roots. But it is no harme to name them as kindes severa]), seeing they appear to have some severall working. For it forceth not so much to contend for the number of them, as for the due knowledge and practising of them. Scholar. Then you will that I shall name them as seven kindes distinct. But now I desire you to instruct mee in the use of each of them. Master. So I will, but it must be done in order: for you may not learn the last so soon as the first, but you must learn them in that order, as I did rehearse them, if you will learn them speedily, and well. Scholar. Even as you please. Then to begin; Numeration is the first in order: what shall I do with it? Master. First, you must know what the thing is, and then after learn the use of the same.
COMMENTARY ON
KEPLER and LODGE OHANN KEPLER (1571-1630) is usually considered the founder of physical astronomy. Copernicus conceived the heliocentric theoryreviving Pythagorean beliefs-and worked it out in his famous book De Revolulionibus O,bium Coeleslium; Tycho Brahe invented and improved astronomical instruments, and by his wonderful skill in observation introduced undreamed-of accuracy into celestial measurements; Oalileo contributed the telescope, the discovery of new stars and nebulae, the support and diffusion of Copernican ideas in his brilliant writings. Kepler ranks foremost as the mathematician of the sky. Though a devout Lutheran, Kepler was also inclined toward Pythagorean mysticism. He was number-intoxicated, a variety of religious experience not restricted to the weak-minded. Fortunately he not only loved numbers but knew how to handle them. His fanatical search for simple mathematical harmonies in the physical universe produced some silly ideas but also three great laws. The first was that the planets move in ellipses with the sun in one focus. Before he made this discovery it was believed that the planets, being perfect creations of God, followed the most perfect of orbits, namely circles. The second law was that the line joining sun and planet (the radius vector) sweeps out equal areas in equal times. The third Jaw was published in 1618, nine years after the other two. It connected the times and distances of the planets: "The square of the time of revolution of each planet is proportional to the cube of its mean distance from the sun." The discovery of this beautifully simple relationship threw Kepler into a picturesque but justifiable rapture: n • • • What sixteen years ago, I urged as a thing to be sought . . . for which I have devoted the best part of my life . . . at length I have brought to light, and recognized its truth beyond my most sanguine expectations. . . . Nothing holds me; I will indulge my sacred fury; I will triumph over mankind by the honest confession that I have stolen the golden vases of the Egyptians to build up a tabernacle for my Ood far away from the confines of Egypt." The story of Kepler's life and work is briefly and attractively told in the selection which follows. The teller is Sir Oliver Lodge, distinguished for his physical researches and famous as a student of psychical phenomena. Lodge was born in England in 1851 and died in 1940. His father was a prosperous business man who decided when Oliver was fourteen that he should leave school and learn the routine of a pottery-material supply agency. This important work kept him busy for the next eight years.
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Thanks to a materna] aunt, who insisted that he be permitted to visit her occasionally in London, he was able to attend classes in chemistry and geology, and was fortunate enough to hear John Tyndall's lectures on heat. After he had won a scholarship, his father grudgingly allowed him to complete his education at the Royal College of Science. While a graduate student, Lodge did research in electricity, on thermal conductivity and on the foundations of mechanics. Later, when appointed to the chair of physics at Liverpool, he performed important experiments in connection with the ether concept, to which he was devoted. He also investigated electromagnetic radiation and contributed to the development of wireless. It has been said that his work was the "major influence in making spark telegraphy possible." 1 Lodge won many honors and held high postsfor example, Principal of Birmingham University, 1900-1919 and President of the British Association, 1913. The public, however, came to know of him less because of these attainments than because of his psychical research and his pronounced opinions on the survival of the mind after death. He was a pioneer practitioner of thought transference, and attracted world-wide attention by his inquiries into the powers of the famous mediums "Mrs. Piper" and the unlikely "Eusapia Pa]ladino,H a lady "telekinesis" expert. (Miss Palladino could move things by thinking about them, at least so Sir Oliver believed. Unfortunately, she was caught at one seance assisting her thoughts with her hands.) Lodge was a flne scientist, a most estimable man, and an able expositor; in his spiritualistic beliefs he was inclined to be something of a goose. The essay on Kepler is from a book anyone interested in science will enjoy reading. It is called Pioneers of Science, and is based on a course of astronomy lectures Lodge gave at University College, Liverpool, in 1887. 1 Article on Lodge, Dictionary oj National Biography; supplemental volume 193140. See also The Proceedings oj the (London) Physical Society. Vol. 53; Obituary Notices oj Fellows 0/ the Royal Society. Vol. 3.
Our Souls, whose /«ultie8 CQII compreheml The wondrous A.rchitecture 0/ the world: And measure every wand'ring planet'! course, Still climbing alter knowledge infinite. -CHlliSTOPHEk MAkLOWE
(Conque!t! 0/ Tamburlaine)
The die is cast; 1 have written my book; ;t will be read either in the present age or by pMterily. it mailers not which; it may well await a reader, since God has waited six thousand yean lor an interpreter 0/ hi! word!. -JOHANN KEPLER
6
Johann Kepler By SIR OLIVER LODGE KEPLER AND THE LAWS OP PLANETARY MOTION
IT is difficult to imagine a stronger contrast between two men engaged in the same branch of science than exists between Tycho Brahe, the subject of the last lecture, and Kepler, our subject on the present occasion. The one, rich, noble, vigorous, passionate, strong in mechanical ingenuity and experimentaJ skill, but not above the average in theoretical and mathematical power. The other, poor, sickly, devoid of experimental gifts, and unfitted by nature for accurate observation, but strong almost beyond competition in speculative subtlety and innate mathematical perception. The one is the complement of the other; and from the fact of their following each other so closely arose the most surprising benefits to science. The outward life of Kepler is to a large extent a mere record of poverty and misfortune. I shan only sketch in its broad features, so that we may have more time to attend to his work. He was born (so his biographer assures us) in longitude 29 0 7', latitude 48 0 54', on the 21st of December, 1511. His parents seem to have been of fair condition, but by reason, it is said, of his becoming surety for a friend, the father lost all his slender income, and was reduced to keeping a tavern. Young John Kepler was thereupon taken from school, and employed as pot-boy between the ages of nine and twelve. He was a sickly lad, subject to violent illnesses from the cradle, so that his life was frequently despaired of. Ultimately he was sent to a monastic school and thence to the University of Tiibingen, where he graduated second on the list. Meanwhile home affairs had gone to rack and ruin. His father abandoned the home, and later died abroad. The mother quarrelled with all 220
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her relations, including her son John; who was therefore glad to get away as soon as possible. All his connection with astronomy up to this time had been the hearing the Copernican theory expounded in University lectures, and defending it in a college debating society. An astronomical lectureship at Oraz happening to offer itself, he was urged to take it, and agreed to do so, though stipulating that it should not debar him from some more brilliant profession when there was a chance. For astronomy in those days seems to have ranked as a minor science, like mineralogy or meteorology now. It had little of the special dignity with which the labours of Kepler himself were destined so greatly to aid in endowing it. Well, he speedily became a thorough Copernican. and as he had a most singularly restless and inquisitive mind, full of appreciation of everything relating to number and magnitude-was a born speculator and thinker just as Mozart was a bom musician, or Bidder a born calculator-he was agitated by questions such as these: Why are there exactly six planets? Is there any connection between their orbital distances, or between their orbits and the times of describing them? These things tormented him, and he thought about them day and night. It is characteristic of the spirit of the times-this questioning why there should be six planets. Nowadays, we should simply record the fact and look out for a seventh. Then, some occult property of the number six was groped for, such as that it was equal to 1 + 2 + 3 and likewise equal to 1 X 2 X 3, and so on. Many fine reasons had been given for the seven planets of the Ptolemaic system, but for the six planets of the Copernican system the reasons were not so cogent. Again, with respect to their successive distances from the sun. some law would seem to regulate their distance, but it was not known. (Parenthetically I may remark that it is not known even now: a crude empirical statement known as Bode's law is all that has been discovered.)l 1 [Write down the series O. 3, 6, 12, 24, 48 &c.; add 4 to each, and divide by 10. The resulting series gives the approximate mean distances of the planets from the
sun in astronomical units. A.ctual Bode Distance Mean Distance 0.4 0.39 0.7 0.72 Earth 1.0 1.00 Mars 1.6 1.52 2.8 Jupiter 5.2 S.20 Saturn 10.0 9.S3 Uranus 19.6 19.19 Neptune 38.8 30.07 39.S Pluto 77.2 This is the law discovered by the German astronomer Johann Elen Bode (17471826) in 1772. Its failure in the case of Neptune and Pluto, planets found after Pltmet Mercury Venus
Once more, the further the planet the slower it moved; there seemed to be some law connecting speed and distance. This a1so Kepler made continual attempts to discover. One of his ideas concerning the Jaw of the successive distances was based on the inscription of a triangle in a circle. If you inscribe in a circle a large number of equilateral trianRles, they envelop another circle bearing a definite ratio to the first: these might do for the orbits of two planets
FIGtJllE J-orblts of some of the planets drawn to scale: mowing the gap between Mars and Jupiter.
(see Figure 2). Then try inscribing and circumscribing squares, hexa· gons, and other figures, and see if the circles thus defined would correspond to the several planetary orbits. But they would not give any satisfactory result. Brooding over this disappointment, the idea of trying solid figures suddenly strikes him. "What have plane figures to do with the celestial orbits?" he cries out; "inscribe the regular solids." And thenbrilliant idea-he remembers that there are but five. Euclid had shown that there could be only five regular solids. 2 The number evidently corresponds to the gaps between the six planets. The reason of there being only six seems to be attained. This coincidence assures him he is on the right track, and with great enthusiasm and hope he "represents the earth's orbit by a sphere as the norm and measure of all"; round it he circumscribes a dodecahedron, and puts another sphere round that, which is approximately the orbit of Mars; round that, again, a tetrahedron, the Bode's time, has led most astronomers to conclude that the "law" is a "purely empirical relationship, more in the realm of coincidence than an actual physical law." Besid~ not being a law, it was not in fact discovered by Bode but by the German mathematician J. D. Titius (1729-1796). Ed.] 2 The proof is easy. and ought to occur in books on solid geometry. By a "regular" solid is meant one with all its faces, edges, angles, &c., absolutely alike: it is of these perfectly symmetrical bodies that there are only five. Crystalline forms are very numerous.
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FIGURE 2-Many-sided polyaon or approximate circle enveloped by Straiaht Jines. as for lDstaace by a DUmber of equilateral trianrles. No Internal circle has been drawn.
comers of which mark the sphere of the orbit of Jupiter; round that sphere. again. he places a cube, which roughly gives the orbit of Saturn. On the other hand. he inscribes in the sphere of the earth's orbit an icosahedron; and inside the sphere determined by that, an octahedron; which figures he takes to inclose the sphere of Venus and of Mercury respectively. The imagined discovery is purely fictitious and accidental. First of all, eight planets are now known; and secondly, their real distances agree only very approximately with Kepler's hypothesis. Nevertheless, tile idea gave him great delight. He says:-uThe intense pleasure I have received from this discovery can never be told in words. I regretted no more the time wasted; I tired of no labour; I shunned no toil of reckoning, days and nights spent in calculation, until I could see whether my hypothesis would agree with the orbits of Copernicus, or whether my joy was to vanish into air." He then went on to speculate as to the cause of the planets' motion. The old idea was that they were carried round by angels or celestial intelligences. Kepler tried to establish some propelling force emanating from the sun, like the spokes of a windmill. This first book of his brought him into notice, and served as an introduction to Tycho and to Galileo.
Sir Oil.... lAdtt
FlGUltE l-Frameworks with inscribed and circumscribed spheres. representing the ftft rep!.
soUds distributed as Kepler supposed them to be amone the planetary orbits.
Tycho Brahe was at this time at Prague under the patronage of the Emperor Rudolph; and as he was known to have by far the best planetary observations of any man living, Kepler wrote to him to know if he might come and examine them so as to perfect his theory. Tycho immediately replied, "Come, not as a stranger, but as a very welcome friend; come and share in my observations with such instruments as I have with me, and as a dearly beloved associate." After this visit, Tycho wrote again, offering him the post of mathematical assistant, which after hesitation was accepted. Part of the hesitation Kepler expresses by saying that "for observations his sight was dull, and for mechanical operations his hand was awkward. He suffered much from weak eyes, and dare not expose himself to night air." In all this he was, of course, the antipodes of Tycho, but in mathematical skill he was greatly his superior. On his way to Prague he was seized with one of his periodical illnesses, and all his means were exhausted by the time he could set forward again, so that he had to apply for help to Tycho. It is clear, indeed, that for some time now he subsisted entirely on the bounty of Tycho, and he expresses himself most deeply grateful for all the kindness he received from that noble and distinguished man, the head of the scientific world at that date. To illustrate Tycho's kindness and generosity, I must read you a letter written to him by Kepler. It seems that Kepler, on one of his absences from Prague, driven half mad with poverty and trouble, fell fou] of Tycho, whom he thought to be behaving badly in money matters to him and his family, and wrote him a violent letter full of reproaches and insults.
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Tycho's secretary replied quietly enough, pointing out the groundlessness and ingratitude of the accusation. Kepler repents instantly, and replies:"MOST NOBLE TYCHO,n (these are the words of his letter), "how shan I enumerate or rightly estimate your benefits conferred on me? For two months you have liberally and gratuitously maintained me, and my whole family; you have provided for all my wishes; you have done me every possible kindness; you have communicated to me everything you hold most dear; no one, by word or deed, has intentionally injured me in anything; in short, not to your children, your wife, or yourself have you shown more indulgence than to me. This hein, so, as I am anxious to put on record, I cannot reflect without consternation that I should have been so given up by God to my own intemperance as to shut my eyes on all these benefits; that. instead of modest and respectful gratitude, I should indulge for three weeks in continual moroseness towards all your family, in headlong passion and the utmost insolence towards yourself, who possess so many claims on my veneration, from your noble family. your extraordinary learning, and distinguished reputation. Whatever I have said or written against the person, the fame. the honour, and the learning of your excel1ency; or whatever, in any other way, I have injuriously spoken or written (if they admit no other more favourable interpretation), as, to my grief, I have spoken and written many things, and more than I can remember; all and everything I recant, and freely and honestly declare and profess to be groundless, false, and incapable of proof."
Tycho accepted the apology tbus heartily rendered, and the temporary breacb was permanently healed. In 1601, Kepler was appointed "Imperial mathematician," to assist Tycho in his calculations. The Emperor Rudolpb did a good piece of work in tbus maintaining these two eminent men, but it is quite clear that it was as astrologers that be valued them; and an be cared for in tbe planetary motions was limited to tbeir supposed effect on his own and his kingdom's destiny. He seems to have been politically a weak and superstitious prince, who was letting his kingdom get into hopeless confusion, and entangling bimself in aU manner of political complications. While Bohemia suffered, however. the world has benefited at his hands; and tbe tables upon which Tycho was now engaged are welJ called the Rudolphine tables. These tables of planetary motion Tycbo had always regarded as the main work of his life; but he died before tbey were finished, and on his death-bed he intrusted the completion of them to Kepler, who loyally undertook their charge. The Imperial funds were by this time, however. so taxed by wars and other difficulties that the tables could only be proceeded with very slowly, a staff of calculators being out of the question. In fact. Kepler could not get even his own salary paid: he got orders. and promises, and drafts on estates for it; but when the time came for them to be honoured they were worthless, and he had no power to enforce his claims.
So everything but brooding had to be abandoned as too expensive. and he proceeded to study optics. He gave a very accurate explanation of the action of the human eye, and made many hypotheses, some of them shrewd and close to the mark, concerning the law of refraction of light in dense media: but though several minor points of interest turned up, nothing of the first magnitude came out of this long research. The true law of refraction was discovered some years after by a Dutch professor, Willebrod Snell, and by Descartes. We must now devote a little time to the main work of Kepler's life. All the time he had been at Prague he had been making a severe study of the motion of the planet Mars, analyzing minutely Tycho's books of observations, in order to find out, if possible, the true theory of his motion. Aristotle had taught that circular motion was the only perfect and natural motion, and that the heavenly bodies therefore necessarily moved in circles. So firmly had this idea become rooted in men's minds, that no one ever seems to have contemplated the possibility of its being false or meaningless. When Hipparchus and others found that, as a matter of fact, the planets did not revolve in simple circles, they did not try other curves, as we should at once do now, but they tried combinations of circles. The small circle carried by a bigger one was called an Epicycle. The carrying circle was called the Deferent. If for any reason the earth had to be placed out of the centre, the main planetary orbit was called an Excentric. and so on. But although the planetary paths might be roughly represented by a combination of circles, their speeds could not, on the hypothesis of uniform motion in each circle round the earth as a fixed body. Hence was introduced the idea of an Equant, i.e., an arbitrary point, not the earth, about which the speed might be uniform. Copernicus, by making the sun the centre, had been able to simplify a good deal of this, and to abolish the equant. But now that Kepler had the accurate observations of Tycho to refer to, he found immense difficulty in obtaining the true positions of the planets for long together on any such theory. He specially attacked the motion of the planet Mars, because that was sufficiently rapid in its changes for a considerable collection of data to have accumulated with respect to it. He tried all manner of circular orbits for the earth and for Mars, placing them in aU sorts of aspects with respect to the sun. The problem to be solved was to choose such an orbit and such a law of speed, for both the earth and Mars, that a line joining them, produced out to the stars, should always mark correctly the apparent position of Mars as seen from the earth. He had to arrange the size of the orbits that suited best, then the positions of their centres, both being
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supposed excentric with respect to the sun; but he could not get any such arrangement to work with uniform motion about the sun. So he reintroduced the equant, and thus had another variable at his disposal-in fact, two, for he had an equant for the earth and another for Mars, getting a pattern of the kind suggested in Figure 4. The equants might divide the line in any arbitrary ratio. All sorts of combinations had to be tried, the relative positions of the earth and Mars to be worked out for each, and compared with Tycho's recorded observations. It was easy to get them to agree for a short time, but sooner or later a discrepancy showed itself.
FIGURE 4-8 represents the sun: Ee, the centre of the earth'. orbit. to be placed as beat suited; MC. the same for Mus. EE. the eanb's equant. or point about whidl the eanb uni-
formly re"YOlved (i.e •• the point determining the law of speed about the sun), likewlae to be placed anywhere. but supposed 10 be in the line joining S to Ee: ME. the same
thinl for Mus; with ? ME for
aD
altematift b)ll)Olhesfs that perhaps Mara' equant
was on line Jolnlnl EC wllb MC.
I need not say that all these attempts and gropings, thus briefly summarized, entailed enormous labour, and required not only great pertinacity, but a most singularly constituted mind, that could thus continue groping in the dark without a possible ray of theory to illuminate its search. Grope he did, however, with unexampled dil!gence. At length he hit upon a point that seemed nearly right. He thought he had found the truth; but no, before long the position of the planet, as cal· culated, and as recorded by Tycho, differed by eight minutes of arc, or about one.eighth of a degree. Could the observation be wrong by this small amount? No, he had known Tycho, and knew that he was never wrong eight minutes in an observation. So he set out the whole weary way again, and said that with those eight minutes he would yet find out the law of the universe. He proceeded to see if by making the planet librate, or the plane of its orbit tilt up and down, anything could be done. He was rewarded by finding that at any rate the plane of the orbit did not tilt up and down: it was fixed, and this was a simplification on Copernicus's theory. It is not an absolute fixture, but the changes are very small. At last he thought of giving up the idea of uniform circular motion, and of trying vtuying circular motion, say inversely as its distance from the
SI, 0""'" ~
228
FIGURE 5--Excentric circle supposed to be divided lOto equal are8ll. The sun. S. being placed at a selected point. it
W811
possible to represent the varying speed of a planet by saying
that it mOYed from A to B. from B to C. and
80
on. in equal times.
sun. To simplify calculation, he divided the orbit into triangles, and tried if making the triangles equal would do. A great piece of luck, they did beautifully: the rate of description of areas (not arcs) is uniform. Over this discovery he greatly rejoices. He feels as though he had been carrying on a war against the planet and had triumphed; but his gratulation was premature. Before long fresh little errors appeared, and grew in importance. Thus he announces it himself:"While thus triumphing over Mars, and preparing for him, as for one already vanquished, tabular prisons and equated excentric fetters, it is buzzed here and there that the victory is vain, and that the war is raging anew as violently as before. For the enemy left at home a despised captive has burst all the chains of the equations, and broken forth from the prisons of the tables." Still, a part of the truth had been gained, and was not to be abandoned any more. The law of speed was fixed: that which is now known as his second law. But what about the shape of the orbit-Was it after all possible that Aristotle, and every philosopher since Aristotle, had been wrong? that circular motion was not the perfect and natural motion, but that planets might move in some other closed curve? Suppose he tried an oval. Well, there are a great variety of ovals, and several were tried: with the result that they could be made to answer better than a circle, but still were not right. Now, however, the geometrical and mathematical difficulties of calculation, which before had been tedious and oppressive, threatened to become overwhelming; and it is with a rising sense of despondency that Kepler sees his six years' unremitting labour leading deeper and deeper into complication. One most disheartening circumstance appeared, viz. that when he made
the circuit oval his law of equable description of areas broke down. That seemed to require the circular orbit, and yet no circular orbit was quite accurate. While thinking and pondering for weeks and months over this new dilemma and complication of difficulties, tin his brain reeled~ an accidental ray of light broke upon him in a way not now intelligible, or barely intelligible. Half the extreme breadth intercepted between the circle and oval was 4~OO.00f) of the radius, and he remembered that the "optical inequality" of Mars was also about 4~OO.00f). This coincidence, in his own words, woke him out of sleep; and for some reason or other impelled him instantly to try making the planet oscillate in the diameter of its epicycle instead of revolve round it-a singular idea, but Copernicus had had a similar one to explain the motions of Mercury. Away he started through his calculations again. A long course of work night and day was rewarded by finding that he was now able to hit off the
FlGURB 6-Mode of drawing an eWpie. Tbe two pins F are the foci.
motions better than before; but what a singularly complicated motion it was. Could it be expressed no more simply? Yes, the curve so described by the planet is a comparatively simple one: it is a special kind of ovalthe ellipse. Strange that he had not thought of it before. It was a famous curve, for the Greek geometers had studied it as one of the sections of a cone, but it was not so well known in Kepler's time. The fact that the planets move in it has raised it to the first importance, and it is familiar enough to us now. But did it satisfy the law of speed? Could the rate of description of areas be uniform with it? Well, he tried the ellipse, and to his inexpressible delight he found that it did satisfy the condition of equable description of ar~ if the sun was in one focus. So, moving the planet in a selected ellipse, with the sun in one focus, at a speed given by the equable area description, its position agreed with Tycho's observations
SI,. 011".,. Lod.-
230
within the limits of the error of experiment. Mars was finally conquered, and remains in his prison-house to this day. The orbit was found. In a paroxysm of delight Kepler celebrates his victory by a triumphant figure, sketched actually on his geometrical diagram-the diagram which proves that the law of equable description of areas can hold good with an ellipse. Below is a tracing of it.
FIGURE 7
Such is a crude and bald sketch of the steps by which Kepler rose to his great generalizations-the two laws which have immortalized his name. All the complications of epicycle, equant, deferent, excentric, and the like, were swept at once away, and an orbit of striking and beautiful properties substituted. Well might he be called, as he was, "the legislator," or law interpreter, "of the heavens." He concludes his book on the motions of Mars with a half comic appeal to the Emperor to provide him with the sinews of war for an attack on Mars's relations-father Jupiter, brother Mercury, and the rest-but the death of his unhappy patron in 1612 put an end to all these schemes, and reduced Kepler to the utmost misery. While at Prague his salary was in continual arrear, and it was with difficulty that he could provide sustenance for his family. He had been there eleven years, but they had been hard years of poverty, and he could leave without regret were it not that he should have to leave Tycho's instruments and observations behind him. While he was hesitating what best to do, and reduced to the verge of despair, his wife, who had long been suffering from low spirits and despondency, and his three children, were taken iII; one of the sons died of small-pox, and the wife eleven days after of low fever and epilepsy. No money could be got at Prague, so after a short time he accepted a pro-
231
Johann K,pl,r
p
FIGURE 8--1f S is the sun. a planet or comet moves from P to Pl. from Pll to Pal and from p, to P6 in tbe same time. if the shaded areas are equal.
fessorship at Linz, and withdrew with his two quite young remaining children. He provided for himself now partly by publishing a prophesying almanack, a sort of Zadkiel arrangement-a thing which he despised, but the support of which he could not afford to do without. He is continually attacking and throwing sarcasm at astrology, but it was the only thing for which people would pay him, and on it after a fashion he lived. We do not find that his circumstances were ever prosperous, and though 8,000 crowns were due to him from Bohemia he could not manage to get them paid, About this time occurred a singular interruption to his work. His old mother, of whose fierce temper something has already been indicated, had been engaged in a law-suit for some years near their old home in Wilrtemberg. A change of judge having in process of time occurred, the defendant saw his way to turn the tables on the old lady by accusing her of sorcery. She was sent to prison, and condemned to the torture, with the usual intelligent idea of extracting a "voluntary" confession. Kepler had to hurry from Linz to interpose. He succeeded in saving her from the torture, but she remained in prison for a year or so. Her spirit, however, was unbroken, for no sooner was she released than she commenced a fresh action against her accuser. But fresh trouble was averted by the death of the poor old dame at the age of nearly eighty. This narration renders the unflagging energy shown by her son in his mathematical wrestlings less surprising. Interspersed with these domestic troubles, and with harassing and unsuccessful attempts to get his rights, he still brooded over his old problem of some possible connection between the distances of the planets from the sun and their times of revolution, i.e. the length of their years. It might well have been that there was no connection, that it was purely imaginary, like his old idea of the law of the successive distances of the
232
Sir Oliver
£0*
planets, and like so many others of the guesses and fancies which he entertained and spent his energies in probing. But fortunately this time there was a connection, and he lived to have the joy of discovering it. The connection is this, that if one compares the distance of the different planets from the sun with the length of time they take to go round him, the cube of the respective distances is proportional to the square of the corresponding times. In other words, the ratio of ,s to T2 for every planet is the same. Or, again, the length of a planet's year depends on the %th power of its distance from the sun. Or, once more, the speed of each planet in its orbit is as the inverse square-root of its distance from the sun. The product of the distance into the square of the speed is the same for each planet. This fact (however stated) is called Kepler's third law. It welds the planets together, and shows them to be one system. His rapture on detecting the law was unbounded, and he breaks out into an exulting rhapsody:"What I prophesied two-and-twenty years ago, as soon as I discovered the five solids among the heavenly orbits-what I firmly believed long before I had seen Ptolemy's Harmonics-what I had promised my friends in the title of this book, which I named before I was sure of my discovery -what sixteen years ago, I urged as a thing to be sought-that for which I joined Tycho Brahe, for which I settled in Prague, for which I have devoted the best part of my life to astronomical contemplations, at length I have brought to light, and recognized its truth beyond my most sanguine expectations. It is not eighteen months since I got the first glimpse of light, three months since the dawn, very few days since the unveiled sun, most admirable to gaze upon, burst upon me. Nothing holds me; I will indulge my sacred fury; I will triumph over mankind by the honest confession that I have stolen the golden vases of the Egyptians to build up a tabernacle for my God far away from the confines of Egypt. If you forgive me, I rejoice; if you are angry, I can bear it; the die is cast, the book is written, to be read either now or by posterity, I care not which; it may well wait a century for a reader, as God has waited six thousand years for an observer." Soon after this great work his third book appeared: it was an epitome of the Copernican theory. a clear and fairly popular exposition of it, which had the honour of being at once suppressed and placed on the list of books prohibited by the Church. side by side with the work of Copernicus himself, De Revolutionibus Orbium Coelestium. This honour, however, gave Kepler no satisfaction-it rather occasioned him dismay, especially as it deprived him of all pecuniary benefit. and made it almost impossible for him to get a publisher to undertake another book. Still he worked on at the Rudolphine tables of Tycho, and ultimately,
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with some small help from Vienna, completed them; but he could not get the means to print them. He applied to the Court till he was sick of applying: they lay idle four years. At last he determined to pay for the type himself. What he paid it with, God knows, but he did pay it, and he did bring out the tables, and so was faithful to the behest of his friend. This great publication marks an era in astronomy. They were the first really accurate tables which navigators ever possessed; they were the precursors of our present Nautical Almanack. After this, the Grand Duke of Tuscany sent Kepler a golden chain, which is interesting inasmuch as it must really have come from Galileo, who was in high favour at the Italian Court at this time. Once more Kepler made a determined attempt to get his arrears of salary paid, and rescue himself and family from their bitter poverty. He travelled to Prague on purpose, attended the imperial meeting. and pleaded his own cause, but it was all fruitless; and exhausted by the journey, weakened by over-study, and disheartened by the failure, he caught a fever, and died in his fifty-ninth year. His body was buried at Ratisbon, and a century ago a proposal was made to erect a marble monument to his memory, but nothing was done. It matters little one way or the other whether Germany, having almost refused him bread during his life, should, a century and a half after his death, offer him a stone. The contiguity of the lives of Kepler and Tycho furnishes a moral too obvious to need pointing out. What Kepler might have achieved had he been relieved of those ghastly struggles for subsistence one cannot tell, but this much is clear, that had Tycho been subjected to the same misfortune, instead of being born rich and being assisted by generous and enlightened patrons, he could have accomplished very little. His instruments, his observatory-the tools by which he did his work-would have been impossible for him. Frederick and Sophia of Denmark, and Rudolph of Bohemia. are therefore to be remembered as co-workers with him. Kepler, with his ill-health and inferior physical energy, was unable to command the like advantages. Much, nevertheless, he did; more one cannot but feel he might have done had he been properly helped. Besides, the world would have been free from the reproach of accepting the fruits of his bright genius while condemning the worker to a life of misery, relieved only by the beauty of his own thoughts and the ecstasy awakened in him by the harmony and precision of Nature. Concerning the method. of Kepler. the mode by which he made his discoveries, we must remember that he gives us an account of all the steps, unsuccessful as well as successful, by which he travelled. He maps out his route like a traveller. In fact he compares himself to Columbus or Magellan, voyaging into unknown lands, and recording his Wandering route. This being remembered, it will be found that his methods do not differ so utterly from those used by other philosophers in like case. His imagination
was perhaps more luxuriant and was allowed freer play than most men's, but it was nevertheless always controlled by rigid examination and cdmparison of hypotheses with fact. Brewster says of him:-uArdent, restless, burning to distinguish himself by discovery, he attempted everything; and once having obtained a glimpse of a clue, no labour was too hard in following or verifying it. A few of his attempts succeeded-a multitude failed. Those which failed seem to us now fanciful, those which succeeded appear to us sublime. But his methods were the same. When in search of what really existed he sometimes found it; when in pursuit of a chimzra he could not but fail; but in either case he displayed the same great qualities, and that obstinate perseverance which must conquer all difficulties except those really insurmountable." To realize what he did for astronomy, it is necessary for us now to consider some science still in its inf~y. Astronomy is so clear and so thoroughly explored now, that it is difficult to put oneself into a contemporary attitude. But take some other science still barely developed: meteorology, for instance. The science of the weather, the succession of winds and rain, sunshine and frost, clouds and fog, is now very much in the condition of astronomy before Kepler. We have passed through the stage of ascribing atmospheric disturbances-thunderstorms, cyclones, earthquakes, and the like--to supernatural agency; we have had our Copernican era: not perhaps brought about by a single individual, but still achieved. Something of the laws of cyclone and anticyclone are known, and rude weather predictions across the Atlantic are roughly possible. Barometers and thermometers and anemometers, and all their tribe, represent the astronomical instruments in the island of Huen; and our numerous meteorological observatories, with their continual record of events, represent the work of Tycho Brahe. Observation is heaped on observation; tables are compiled; volumes are filled with data; the hours of sunshine are recorded, the fall of rain, the moisture in the air, the kind of clouds, the temperature--millions of facts; but where is the Kepler to study and brood over them? Where is the man to spend his life in evolving the beginnings of law and order from the midst of all this chaos? Perhaps as a man he may not come, but his era will come. Through this stage the science must pass, ere it is ready for the commanding intellect of a Newton. But what a work it will be for the man, whoever he be that undertakes it-a fearful monotonous grind of calculation, hypothesis, hypothesis, calculation, a desperate and groping endeavour to reconcile theories with facts. A life of such labour, crowned by three brilliant discoveries, the world owes (and too late recognizes its obligation) to the harshly treated German genius, Kepler.
COMMENTARY ON
DESCARTES and Analytical Geometry D
ENe DESCARTES (1596-1650) came from a noble fami1y settled ~ since the fourteenth century in southern Touraine. His father was a councilor of the parliament of Brittany and reasonably well off. Descartes inherited from him enough money to support a life of study and travel. He was educated, from 1604 to 1612, at the Jesuit college at La Fleche, where he not only received a good training in the humanities and mathematics but was treated with exquisite consideration. He was aJlowed by Father Charlet, the rector and evidently a sensible man, to lie in bed in the morning-"a habit which he maintained aU his life, and which he regarded as above aJl conducive to inte11ectual profit and comfort. 1 After leaving school, Descartes took a tum at social life in Paris, but he found this tiresome and soon shut himself up in lodgings in the Faubourg Saint Germain to study mathematics. In 1621 he entered the army of Prince Maurice of Nassau. Descartes said that in his youth he liked war, attributing this taste "to a certain animal heat in his liver, which cooled down in the course of time." 2 Whether or not this is true, being a soldier, though occasionally dangerous, was not an arduous occupation. It provided leisure for meditation as well as an opportunity to see something of other countries. In 1619 Descartes enlisted in the Bavarian Army and during the winter of that year had a major philosophical inspiration which he reports in the Discourse on Method. Bertrand Russell gives a witty description of the circumstances: "The weather being cold, he got into a stove 8 in the morning, and stayed there all day meditating; by his own account, his philosophy was half finished when he came out, but this need not be accepted too literally. Socrates used to meditate aU day in the snow, but Descartes' mind only worked when he was warm." 4 After a few years of fighting, interrupted by trips to Italy and Paris, Descartes retired to Holland where for twenty years, 1629 to 1649, he spent his time on science and philosophy. The freedom of thought permitted him in his work was characteristic of HoJland in the seventeenth century. It was only slightly marred by the interference of ··Protestant bigots." for the French ambassador and the Prince of Orange protected It
11. P. Mahafty, Descartes, Philadelphia, 1881, p. 12. • Mahafty, op. Cil., p. 20. a Descartes says it was a stove (polle). but the authors of the standard translation, Elizabeth S. Haldane and G. R. T. Ross, write "shut up alone in a stove-heated room." With Russell, I prefer "stove," "Bertrand Russell, If History oj Western Philosophy, New York, 1945, p. 558. 235
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Editor" Co".,."t
him from the malevolence of his assailants. At one time, booksellers were forbidden to print or sell any of his works and he was even haled before the magistrates to answer charges by the theologians of Utrecht and Leyden that he was an atheist, a "vagabond" and a profligate; but the excitement passed without serious consequences. Nevertheless, when Queen Christina of Sweden invited him in 1649 to adorn her court, he had had enough of controversy and of Holland and he accepted her offer. It is said that a "presentiment" of death came over him as he prepared for the journey. Stockholm, when he arrived there in October, was a cold and disagreeable spot and the Queen, a yeamer for wisdom, shattered the unhappy man's lifelong routine by requiring him to instruct her daily at five in the morning. Descartes withstood the weather and the sovereign for only four months; on February 1 he came down with pneumonia and ten days later he died. Descartes has been called the father of modem philosophy, perhaps because he attempted to build a new system of thought from the ground up, emphasized the use of logic and scientific method, and was "profoundly affected in his outlook by the new physics and astronomy." Undoubtedly he had great infiuence on philosophy; in recent years, however, critics have depreciated his originality by pointing out how much he owed to the scholastics. His contribution to mathematics was of enormous importance. He is usually considered the inventor of analytical geometry, but this notion is "historically inadequate" because the subject did not spring fullarmed from Descartes' head. Ii The study of curves by means of their equations, defined as the "essence" of analytic geometry, was known to the Greeks and "was the basis of their study of the conic sections." 6 Menaechmus, the tutor of Alexander the Great, is reputed to have made this discovery. Among Descartes' other predecessors were the French theologian Nicole Ores me, whose system of "latitudes and longitudes" roughly foreshadowed "the use of co-ordinates in the graphical representation of arbitrary functions," 1 and Fran~ois Viete, the sixteenth-century counselor to the King of France, whose improvements in notation substantially facilitated the development of algebra. The most formidable claimant to Descartes' title of inventor of analytic geometry was his famous contemporary, Pierre Fermat, who enriched every branch of mathematics known in his time, founded the modem theory of numbers and significantly advanced the study of probability. The truth seems to be that each man, simultaneously and independently, carried the subject far beyond where any a For an interesting, popular and authoritative account of the development of this branch of mathematics see Carl B. Boyer, "The Invention of Analytic Geometry," Scientific American, January 1949. A more advanced and comprehensive, but no less readable survey appears in J. L. Coolidge's excellent treatise, A History of Geometrical Methods, Oxford, 1940, pp. 117 el seq. 8 See J. L. Coolidge, cited in the preceding note, p. 119. T Boyer, op. cit., p. 42.
Descartes and ANalytical Georrut'",
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earlier mathematician had taken it. Thus was enacted a prelude to the classic example of simultaneous discovery. Newton and Leibniz at work on the calculus. Fermat may have preceded Descartes in stating problems of maxima and minima; but Descartes went far past Fermat in the use of symbols, in "arithmetizingn analytic geometry. in extending it to equations of higher degree. The fixing of a point's position in the plane by assigning two numbers, co-ordinates, giving its distance from two lines perpendicular to each other, was entirely Descartes' invention. s Fermat's little treatise of eight folio pages appeared in 1679, half a century after it had been composed; Descartes' La Geometrie was published as an appendix to the Discourse in 1673. He made other contributions to mathematics but this was by far the most important. He did not, it should be added, dream up the idea of co-ordinates while lying in bed at La Fleche. This agreeable fable (which I myself have on other occasions repeated) was the concoction of one Daniel Lipstorpius, a LUbeck professor who wrote a life of Descartes in the style of Parson Weems. I will not go so far as to say that men cannot have great thoughts in bed; only that the sixteen-year-old Descartes, having studied mathematics for a few months, was not up to conceiving the co-ordinates during a morning reverie. The following selection is a facsimile, with translation, of the first eight pages of the first edition of La Geometrie. It is written in a "contemptuous vein." Descartes was evidently more interested in showing what he knew than in instructing beginners. He concludes the work with an ironic paragraph: "I hope that posterity will judge me kindly, not only as to the things which I have explained, but also as to those which I have intentionally omitted so as to leave to others the pleasure of discovery!' 8 Coolidge, op. cit., p. 126; Boyer, op. cit., p. 43. "He (Descanes) contemptuously rejected the idea that the only curves we should consider were plane and solid loci, that is to say lines, circles and conics, and maintained that we were at libeny to make use of any smooth curve which has a recognizable equation. . . . He also introduced exponents, except in the case of the square. and other algebraic simplifications, developing an algebra that was infinitely more manageable than that of Vieta and Fermat." Coolidge, op. cit., p. 126.
2,7
G E 0 MET R I E. LI VRE PREMIER. V'S prohl,{mls '1u"(Jn plul con{l,";rt rAIls .y tmplttJer qUI d,s ,,,,clts &' d,s IIgnls droltls. s~~a 0
u s res Vroblefmes de Geometrie fe peUUt'Dt facilement reduire a tels tennes, qu'iln'eft heroio par aprc!s que de conooi· ftre la loogenr de quelques ligoes droitcs, pour Ies cooitruire. Et comme toote .. Arlthmetique 0'eft compofc!e, que CammEr: de quatre 00 cinq operatioos t qui foot l'Addition, Ia ~Act"l Soufttattion,la Multiplicatioo. la Diuifion t 8[ rBxtra- thmeci... aion des raaoes, qu· on peut prendre pour Vile efpece l~~~~r;e de Diui600 : AiDfi o-at·oo autre chofe a faite eo Geo- auxope. toue1lant Ies I'Ignes qu•00 cherche, pour Ies pre- catloDSdc metrle Geomeparer a eftre connues" que leur en adioufter d·autres , OU cnc. eo ofter, Oubieo eo ayaot vne, que Ie oommer.ay rvolte!· poor la rapporter d'aotant mieux aux oombres , & qUi peut ordinairetnent eftre prlfe a difcretioo, pUIS en ayant encore deux aut res , eo trouuer voe quatriefme qui foit al'vne de ces deux, comme l'autre eft arvnite, ce qui eft Ie mefme que la Multiplication; oubien en trouuer vne quatriefme, qui foit a r vne de ces deux comme l'vnitc Pp eft J
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It U ',"pol8ible ""t to lui lIirnd at the tMU,ht 01 the emotions 01 men III certabr historic moment.r of _vemure II1UI discovery-Columbru when he fint
.fdW
the We.rtem .rluRe, PiVlrro when he stared at the Pacific Ocean,
Fnmklln when the el«tric .rplUk came 1'0'" the .rt,ing 01 his kite. Gallko when he lint tumed hb tele.rcope to the heavens. Such moment.r an tdJo rranted to .rtudent.r In the ab.rtract re,1onI 01 tMu,ht. and hi,h among them mast be placed the mo,nin, when Ducarte.r my In bed and invented the rnnhDd 01 co-ordinate ,eometry. -ALJI'REI) NoIt11l WHI11iHJW) There are .rome men who are counted ,reat becarue they rep,e.rent the actuality of their own a,e, and mirror it til It is. Such an one was Yollalre. of wMm it wa.s epi,rammatically sllid: "he expressed everybody's tMUlhts better than anybody," But there are other men who IlItain greatnen becarue they embody the potentiality of their own day and magically re/lect the luture. They express the thoughts which will be everybody's two or three -THOMAS HUXLEY centu,ie.r after them. Such an one Wtll Descartes,
7
The Geometry By RENE DESCARTES
BOOK I PR.OBLEMS THE CONSTRUCTION OP WHICH REQUIRES ONLY STRAIGHT LINES AND CIRCLES
ANY problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction. Just as arithmetic consists of only four or five operations, namely, addition, subtraction, multiplication, division and the extraction of roots, which may be considered a kind of division, so in geometry, to find required lines it is merely necessary to .add or subtract other 239
1ines~
Ren~
Descartes
298 :t A. G EO MET a II • eft a rautre, ce qui eft Ie me(me que la DiuUion; ou enfin sroooervne,ou deux ,Oll plufieurs moyennes proportionDeHes cDtrc 1'"Ditc5, 8c qoelque autre ligne; ce qui eft Ie meCme que tirer la I'acine quarreel on cubique,&c. Et ie De craindray pas d"introduirc ces termes d'Aritbmeti. que en Ia Geometric J aSin de me rendre plus iute!. ligibile• •aMulciSoit p'ar exemple IlicatioD. A Br~vuitc5, & qu·il faille multiplier B 0 par B C, ien'ayqu'aioindre les poiDI A Be C, puis tirer DE para11ele a C A, -----!.---L.---~B at: BE eft Ie produit de D A cete Multiplicatioo. La DiYiOubieDS'il faut diuifer BE par BD, ayant ioint les ROD. poins B Be D , ie tire A C paraUele a DB, Be B C eft Ie 'E produit de eete diumon. l anaaioDdcla Ou stll faut titer la racine ~cille nuanee de G H .. ie lu.y ad. ,aarrce. " " l " ioufte en ligne droite F G, qui eft i·vnite(, &: dioifant P H H en deux parties efgales au point K, du centre K ie pre Ie cercle FIll. puis elleuaot do point G vne ligne droite imques a I, a angles droits fur F H J c·eft GIla racine chcrchc!e. Ie ne dis riea icy de la racine cobique. ny des autres, aeaofc que ifen parleray plus commodcmeDt cy
apr6s. Commit .... !Cut
'a ....: ....
r.
4Y~ ,QUDent
._I!.-
on n•a pas befo°10 d e tracer aIlW ccs 1gne
241
The Geometry
or else, taking one line which I shall call unity in order to relate it as dosely as possible to numbers, l and which can in general be chosen arbitrarily, and ha\'ing given two other lines, to find a fourth line which shall be to one of the given lines as the other is to unity (which is the same as
multiplication); or, again, to find a fourth line which is to one of the given Jines as unity is to the other (which is equivalent to division); or, final1y, to find one, two, or several mean proportionals between unity and some other line (which is the same as extracting the square root, cube root, etc., of the given line). 2 And I shall not hesitate to introduce these arithmetical terms into geometry, for the sake of greater dearness. For example, let AB be taken as unity, and let it be required to mUltiply BD by BC. I have only to join the points A and C, and draw DE parallel to CA; then BE is the product of BD and BC. If it be required to divide BE by BD, I join E and D, and draw AC
parallel to DE; then BC is the result of the division. If the square root of OH is desired, I add, along the same straight line,
FO equal to unity; then, bisecting PH at K, I describe the circle FIH about K as a center, and draw from 0 a perpendicular and extend it to I, and 01 is the required root. I do not speak here of cube root, or other roots, since I shall speak more conveniently of them later. Often it is not necessary thus to draw the lines on paper, but it is sufficient to designate each by a single letter. Thus, to add the lines BD and OH, I call one a and the other b, and write a + b. Then a - b will india cate that b is subtracted from a; ab that a is multiplied by b; - that a is b
divided by b; aa or a2 that a is muJtiplied by itse]f;
as
that this result is
1 In general, the translation runs page for palle with the facing original. On account of figures and footnotes, however, this plan is occasionally varied, but not in such a way as to cause the reader any serious inconvenience. a While in arithmetic the only exact roots obtainable are those of perfect powers, in geometry a length can be found which will represent exactly the square root of a given line, even though this line be not commensurable with unity. Of other roots, Descartes speaks later.
242
L I V REP R £ M J E It. 2.9' gnes fur Ie papicr, & it fuBiO: de les defigner par quelques vfer de lettres, charcune par vne [eule. Comme pour adioufter chiffcCSCD la ligne B D a G H, ie nomme l'vne /I &= fautre IJ.& etcris ~~~mc .. a+ h; Et a-- b,pourfouftraire bd'.; Et.b,pourlesmultiplierl'vnepar rautre;Et :,pourdiuifer/lpar9JEt lIa.
,
~
pour multipliera par foymefme; Htll, pour Ie multiplier encore vne fois par /I ) & ainG a l'infioi • Ht
Oll tI,
i.
.~
11 .Q + b) poor tirer la racine quarrc!e d' ",/ Y
+
J
&
IS
~
+ " ; Et ~
,
C. tI-- b + ab ", pour tirer Ia racine cubiqoe d'. ·.11 a b", & ainfi des autres. &
I
,
Oil il eft a rcmarquer qoe par /I ou b ou femblables~ ie ne con~oy ordinairement que des lignes toutes fimpies, encore que pour me feruir des noms vfitc!s en I'At. gebre, ie les nomme des quarrc!s ou des eu bes, eke. 11 eft auffy a remarquer que toutes Ies parties d'vne mefme ligne.fe doiuent ordinairement exprimer par au. tant de dimenfions I'vne que rautre.lorfque rvnitdn'eft J
point determinc!e enla queftion, comme icy II en con-. I
tient autant qu" ." b ou b dont fe compofe la Iigne que ..
.".r
J
,
fay nomm~e y C. /I -- b + .IJ IJ: mais que ce n-eft: pas de mefme lorfque I-vnit~ eft ddtermin~e J a caufo qu'eUe peut eftre foukntendue par tout ou il y a trop 011 t~op peu de dimen6ons: com me s·it faut tirer fa racine eubique de /lab ....·" , it faut penfer que la quantit6 IIl1bb eft diuifeevne fois parl'vllitc1 &:quel'autreqoao. tite ~ eft multipliee deux fois par la mcfme. p p 2. Au
243
multiplied by a, and SO on, indefinitely.8 Again, if I wish to extract the square root of a2 + b 2 , I write yQ2 + bl ; if I wish to extract the cube root of a3 - b8 + ab2 , I write ~ - b8 + ab!, and similarly for other roots:' Here it must be observed that by a 2, b8 , and similar expressions, I ordinarily mean onJy simple lines, which, however, I name squares. cubes, etc., so that I may make use of the terms employed in algebra. Ii It should also be noted that an parts of a single line should always be expressed by the same number of dimensions. provided unity is not determined by the conditions of the problem. Thus, 0,8 contains as many dimensions as ab2 or b8 , these being the component parts of the line which I have called V'a3 - b8
+ ab 2•
It is not, however, the same thing when
unity is determined, because unity can always be understood, even where there are too many or too few dimensions; thus, if it be required to extract the cube root of a2b2 - b, we must consider the quantity a2b2 divided once by unity, and the quantity b multiplied twice by unity.6 Finally, so that we may be sure to remember the names of these lines, a separate list should always be made as often as names are assigned or changed. For example, we may write, AB OH
= 1, that is AB is equal to 1; 1
= a, BD = b, and so on.
Descartes uses as, at. alii, ae. and so on, to represent the respective powers of a, but he uses both QG and as without distinction. For example, he often has aabb, but 3
3ai
he also uses - . 4b8 " Descartes writes: ycJl8 1)3 'f abb. a At the time this was written, tP was commonly considered to mean the surface of a square whose side is a. and b3 to mean the volume of a cube whose side is b; while b", b lli, • • • were uninteUigible as geometric forms. Descartes here says that as does not have this meaning. but means the line obtained by constructing a third proportional to 1 and a. and so on. o Descartes seems to say that each term must be of the third degree, and that therefore we must conceive of both tPb2 and b as reduced to the proper dimension . ., Descartes writes, AB ex: 1. He seems to have been the first to use this symbol. Among the few writers who foUowed him., was Hudde (1633-1704). It is very commonly supposed that ex: is a ligature representing the first two letters (or dipththong) of "ZQuare." See, for example, M. Aubry's note in W. W. R. Ball's Recrlations MatWmatlques el Problemes des Temps Anciens el Modemes. French edition, Paris, 1909, Part BI. p. 164.
Relll DesctU'tes
300
LA GBOMIT .. I ••
re
AD refte alin de ne pas maDquer a fauuenir des noms de ces lignes, il en faut toufiours Caire vn regiftre £epar' J l mefure qu'on les pofe ou qu'on les change, ercriuallt paresemple. A B :0 I . c'eft a dire, A B cfgal a t. GH~.
ii, ~c. ~.mmec Ainfi vODlant rafoudre quelque problefma, on doit d"a~fra:;e- bordle confiderercomme delia &it, &: donner des noms ECf..aati6s a toutes les lignes, qui femblent necea-aires pour Ie con. ~:~t ~~;.1truirc~ aua-y bien aceUes qui fODt incoDDues J qu·aux foadrc les autres. Puis fans confidercr aucane di1Ference entre ces problct: Ii .. . . . on dOlt . par couru Ja d'Jw.d: mes. gnes CODDues, & IDconnues, culteS ,felon r ordre qui monftre Ie plus naturellemenc de tous en qU'cHe forte eUes dependent mutuellemeDt. tes vnes des autres, iufques a ce qu"on ait trouuc moyeo d'exprimcr vne mefme quandtden deux fasons: ee qui fe nomme vue Equation; car les termes de l'vne de ees deux&sonsfontefgaux aceux de rauue. Et on doit trouuer aarant de telles Equations,qu'on a fuppofc! de lignCSJt qui eftoientinconoues. Oubien s"it De s'en trouue pas rant,&: que oonobftanton n·omettcriende ce qui eft defireeo Ia queftion,cela tefmoigoe qU'elle n-eft pas entierement determinee. Ht lors on peut prendre a diferedon des lignes connues) pour toutes lesioeoonues auf:. qu·elles necorrefpond aucune Equation. Aprds celas'ij eo refte encore plufieurs , il fa faut feruir par ordre de chafcune des Equations qui reftent aua-y , foit en la coofiderant toute feuJ,,roit en la comparant auee Ids aurres, pour cspliquer ehafcune de eeslignes inCODnUeS, &' &ire aiou BO
2)
The Geomel1'7
24S
If, then, we wish to solve any problem, we first suppose the solution
already effected,8 and give names to all the Hnes thnt seem needful for its construction.-to those that are unknown as well as to those that are known. Then, making no distinction between known and unknown Jines, we must unravel the difficulty in any way that shows most naturally the relations between these Jines. until we find it possible to express a single quantity in two ways.9 This will constitute an equation, since the tenns of one of these two expressions are together equal to the tenns of the other. We must find as many such equations as there are supposed to be unknown lines; but if, after considering everything involved, so many cannot be found, it is evident that the question is not entirely detennined. In such a case we may choose arbitrarily lines of known length for each unknown line to which there corresponds no equation. If there are several equations, we must use each in order, either con-
sidering it alone or comparing it with the others. so as to obtain a value for each of the unknown lines; and so we must combine them until there remains a single unknown line 10 which is equal to some known )ine, or whose square, cube, fourth power, fifth power, sixth power, etc., is equal to the sum or difference of two or more quantities, one of which is known, 8 This plan, as is well known, goes back to Plato. It appears in the work of Pappus as follows: "In analysis we suppose that which is required to be already obtained. and consider its connections and antecedents, going back until we reach either something already known (given in the hypothesis), or eJse some fundamental principle (axiom or postulate) of mathematics." Pappi Alexandrini Collectiones quae supersunt e lib,is manu scriptis edidlt Latina interpellalione et commentariis instruxit Fredericw Hultsch. Berlin, 1876-1878; vol II, p. 635 (hereafter referred to as Pappus). See also Commandinus, Pappi Alexandrlni Mathematicae Collectiones, Bologna, 1588, with later editions. Pappus of Alexandria was a Greek mathematician who lived about 300 A.D. His most important work is a mathematical treatise in eight books, of which the first and part of the second are lost. This was made known to modem scholars by Commandinus. The work exerted a happy influence on the revival of geometry in the seventeenth century. Pappus was not himself a mathematician of the first rank, but he preserved for the world many extracts or analyses of lost works. and by his commentaries added to their interest. 9 That is, we must solve the resulting simultaneous equations. 10 That is, a line represented by x, x a, x 3 , xt, ...•
Reltl Descartes
L I V 1l B P R. !
JO r ainu en Ies demellane , qu'il eten demeure qU'vne feule, efgaleaquelql1eautrc, quifoitcoooue, oubien dont Ie quarrc!, ou Ie cube, ou Ie quarrede quarre, ou Ie furfoli. de,oulcquarredecube,&c.foitefgal a ce, qui fe produift par r addition, au fouftraCtion de deux au plufieurs aueres. quantites , dont l'vne foit connue J & les autres foieot compofees de quelques moyennes proponion. nelles entre l'vnite, & ce quarre, ou cube, ou quarrd de quarre,Bec. multiplie'es par d' autres connues. Ce que fte... {eris en cete forte. t:xl b. ou :I.
!{:D-" !{ J
~~
+ bOt ou L
M Jill.
. S
+" ~ +6"~-·e. ou '" , '" t:o • t f - l(+ d. &c.
C'eft a dire, i. que ie preDs pour la quantitd inconnuc" eftefgaleab, oulequarre de 1:, eft: efgal au quarre de II moins II multiplic par~: ou Ie cube de t eft efg~ ill rnu ltipliepar Ie quarrc de (plus Ie quarre de bmulriplic! pat 1:,moins Ie cube de c. &' ainfi des autres. Et on peut touliours reduire ainfi toutes les quantitc!s inconnues avne feule.lorfque Ie ProbIefme fe peut conftruire par des cerclds Be des lignes droites, ou aulfy par des feCtions coniqaes,ou mefme par quelque aotreligne. qui De foit que d'vD ou deux degrc!s plus compofee- ~lajs· ie ne m'areft'e point a expliquer cecy plus en detail, a caufe que ie voas ofterois Ie plaifir de I'apprendre de vous mefme, Be rvtilit4! de cultiuervoftrc e{prie en VQUS y excrceant, qui eft a mOD auis la pribcipale,qu·on puilf'e 'P P 3 tirer
TM GltOtM,ry
247
while the others consist of mean proportionals between unity and this square, or cube, or fourth power, etc., multiplied by other known lines. I may express this as follows:
z
=b,
or
Z2
= -az + b
or
ZS
= az2 + b2z -
or
Z4
2,
=azs -
cSz
c3,
+ d4, etc.
That is, z, which I take for the unknown quantity, is equal to b; or, the square of z is equal to the square of b diminished by a multiplied by z; or, the cube of z is equal to a multiplied by the square of z, plus the square of b multiplied by z, diminished by the cube of c; and similarly for the others. Thus, all the unknown quantities can be expressed in terms of a single quantity,ll whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth.12 But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as of the ad· vantage of training your mind by working over it, which is in my opinion the principal benefit to be derived from this science. Because, I find nothing here so difficult that it cannot be worked out by anyone at all familiar with ordinary geometry and with algebra, who will consider carefully all that is set forth in this treatise. IS 11
See line 20 on the opposite page,
Literally, "Only one or two degrees greater," In the Introduction to the 1637 edition of La Glometrie, Descartes made the following remark; "In my previous writings I have tried to make my meaning clear to everybody; but I doubt if this treatise will be read by anyone not familiar with the books on geometry. and so I have thought it superfluous to repeat demonstrations contained in them." See Oeuvres de Descartes, edited by Charles Adam and Paul Tannery, Paris, 1897-1910. vol. VI, p. 368. In a letter written to Mersenne in 1637 Descartes says: "I do not enjoy speaking in praise of myself, but since few people can understand my geometry, and since you wish me to give you my opinion of it, I think it well to say that it is all I could hope for, and that in La Dioptriqlle and Lea MefeOTeS, I have only tried to persuade people that my method is better than 12
18
Relll Descartes
248
30 .1
LA GROKETltIB. tner ae cete fcieuce.. Awry qoe ie n y remarque rieo de
fi diflicile, que ceux qui CeroDt VB peu verfc!s en la Geometrie commUDe, & en rAlgebre, Be qui prendront garde a tout ce qui eft en ce traite, ne puitrent tfouuer. C'eftpourquoyieme contenteray icy de vous auertit" que pourvd queen demeHanr ces EquatioDs on ne manque point a fc remir de toutes le5 diui6ons, qui ferant poBibles, on aura infalliblement les plus fimples termes,aufquels Ia queftion puHfe eftre redoite. ~e:!s Ht que 1i eUe peDt eftre refolue par Ja Geometric ordiprobl&' naire. c'eft a dire" en nefeferuant que de lignes droites me. plaGS Be circulaires tracc!es fur vuc fuper6cie plate" lorfque Ja demiere Equation aura efttS entierementd6o.1efic!e.i1n·y reftera tont au plus qu·vn quarre inconnu, efgal a ce qoi (e produift de r Addition, au fouftraGtion de fa racine muttiplie'e par quelquc quantitc cannue J Be C:te quelque aua:e quantiteauKy connue ComHt lars cere racine. ou ligne inCODDue Ce trouuc ayfccmcut its Cc rcfol- mente Car fi fay par exemple uear. " ;~....... ~::D (I ~ +6b 0 ..·.... .., iefai~ Ie triangle reaanI" \ gle N LM"doQt Ie co.. jp ftc!L M eft edfgal a b racine quarree c Ja quan.. \ " tit~ CODDoe b0" & fau'-....·L .....-;.:.;;~-----~M trc L N eft i (J 1a moi. tic de r autre qoantitc conoue, qui eftoit multiplicfe par ~ que ie fuppofe eftre 1a ligne inconnue. puis prolopgeant M N la baze ae ce triangle, J
The Geometry
249
I shall therefore content myself with the statement that if the student, in solving these equations, does not fail to make use of division wherever possible, he wilt surely reach the simplest terms to which the problem can be reduced.
And if it can be solved by ordinary geometry, that is, by the use of straight lines and circles traced on a plane surface, when the last equation shall have been entirely solved there will remain at most only the square of an unknown quantity, equal to the product of its root by some known quantity, increased or diminished by some other quantity also known.14 Then this root or unknown line can easily be found. For example, if I have
Z2
= az + b2,lfS I construct a right triang1e NLM with one side LM,
the ordinary one. I have proved this in my geometry, for in the beginning I have solved a question which, according to Pappus, could not be solved by any of the ancient geometers. "Moreover, what I have given in the second book on the nature and properties of curved lines. and the method of examining them, is, it seems to me, as far beyond the treatment in the ordinary geometry, as the rhetoric of Cicero is beyond the a, b, c of children. . • . "As to the suggestion that what I have written could easily have been gotten from Vieta. the very fact that my treatise is hard to understand is due to my attempt to put nothing in it that I believed to be known either by him or by anyone else. . . • J begin the ruJes of my aJgebra with what Vieta wrote at the very end of his book, De emendatione aequationum. • . . Thus, I begin where he left off:' Oeuvres de Descartes, publiees par Victor Cousin, Paris, 1824, Vol. VI, p. 294. In another Jetter to Mersenne, written April 20, 1646, Descartes writes as follows: "I have omitted a number of things that might have made it (the geometry) clearer, but I did this intentionally, and would not have it otherwise. The only suggestions that have been made concerning changes in it are in regard to rendering it clearer to readers, but most of these are so malicious that I am completely disgusted with them." Cousin, Vol. IX, p. 553. In a Jetter to the Princess Elizabeth, Descartes says: "In the solution of a geometrical problem I take care, as far as possible, to use as lines of reference parallel lines or lines at right angles; and I use no theorems except those which assert that the sides of similar triangles are proportional, and that in a right triangle the square of the hypotenuse is equal to the sum of the squares of the sides. I do not hesitate to introduce several unknown quantities, so as to reduce the question to such terms that it shall depend only on these two theorems." Cousin, Vol. IX, p. 143. 14 That is, an expression of the form :z2 = az ± b. "Esgal a ce qui se produit de l'Addition, ou soustraction de sa racine multiplee par quelque quantite connue, & de quelque autre quantite aussy connue," as it appears in line 14, opposite page. 1 ~ Descartes proposes to show how a quadratic may be solved geometricaUy.
250
30 3 angle, infquesaO, en (ortequ'N 0 foitetgaJeaNL,. Ia toute OM eft tlalignc cherchee. Bt eUc s'exprime en cete forte I, I V REP REM 1! R.
t3>ja+riall+ ob.
¥.!at the line MR will not meet the circle and both roots wi11 be imaginary. Also, since RM. QM LM2, ZIZt b a, and RM + QM == Zt + ls == a. 19 It win be seen that Descartes considers only three types of the quadratic equation in z. namely, Zli + az - b 2 O. Zli - az - b' 0, and ~ - til + b' O. It thus appears that he has not heen able to free himself from the old traditions to the extent of generalizing the meaning of the coefficients, - as negative and fractional as well as positive. He does not consider the type Z2 + az + b2 = 0, because it has no positive roots. 20 "Qu'iJs n'ont point eu la vraye methode pour les trouuer toutes."
=
=
=
=
=
COMMENTARY ON
ISAAC NEWTON N Christmas Day, 1642. the year Galileo died, there was born in the Manor House of Woolsthorpe-by-Colsterworth a male infant so tiny that, as his mother told him in later years, he might have been put into a quart mug, and so fraH that he had to wear "a bolster around his neck to support his head." This unfortunate creature was entered in the parish register as "Isaac, sonne of Isaac and Hanna Newton." There is no record that the wise men honored the occasion, yet this child was to alter the thought and habit of the world. The Royal Society of London. over which Newton presided for almost a quarter century, planned to celebrate the tercentenary of his birth in 1942. Strangely, this was to be the first international event in Newton's honor since one during his lifetime at which he was elected a foreign associate of the French Academy of Sciences. Postponed because of the war, the celebration was finally held at London and Cambridge in July of 1946. With representatives of thirty-five nations attending, it was an international gathering such as had rarely been convoked even before passports, iron curtains and the congealing effects of "security" persuaded many a traveler to stay home. The addresses delivered at celebrations are rarely worth remembering. This occasion proved an exception. A high standard was maintained in lectures by (among others) the English physicist E. N. da Costa Andrade, by the mathematician H. W. Turnbull (whose little biographical book appears elsewhere in this volume), by Niels Bohr of Denmark (UNewton's Principles and Modem Atomic Mechanics"), by the French mathematician Jacques Hadamard, by Lord Keynes (who had died and whose paper was read by his brother, Geoffrey Keynes). I have selected two of the lectures. The first, by Andrade, is a lucid survey of Newton·s vast achievement; the second is a sensitive, brilliant delineation of Newton the man by Lord Keynes. I reproduce a fragment of Keynes' eloquence lest you be tempted, having read the comprehensive lecture that precedes it, to pass over his wonderful appreciation. "Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind which looked out on the visible and intel1ectual world with the same eyes as those who began to build our intellectual inheritance rather less than 10,000 years ago."
O
254
Nt!Wton did not shew the calise of the apple falling, but he shewed a similitude between the apple and the stars. --SIR
D'ARCY WENTWORTH THOMPSON
Great Nature's well-set clock in pieces took.
-WILLIAM CowpEIt.
Where the statue stood Of Newton, with his prism and silent face, The marble index 01 a mind forever Voyaging through strange seas of thought alone. -WILLIAM WORDSWORTH
8
Isaac Newton By E. N. DA C. ANDRADE
FROM time to time in the history of mankind a man arises who is of universal significance, whose work changes the current of human thought or of human experience, so that all that comes after him bears evidence of his spirit. Such a man was Shakespeare, such a man was Beethoven, such a man was Newton, and, of the three, his kingdom is the most widespread. The poet's fun greatness is for those to whom his native language is familiar-he can be translated, but in translation his glory is diminished. The musician also expresses himself in an idiom that is limited-the music of the West means little to the East. To-day, natural science is the one universal learning; the language of science is understood by the initiated in every quarter of the globe. and the great leaders of science are reverenced by thoughtful men wherever the pursuit of knowledge is deemed desirable. In his Ufetime the achievements of Newton were celebrated throughout civilized Europe: to-day we have in this room learned men from the five continents, who are gathered together to do honour to his name. Isaac Newton was not one of those precocious figures, like Blaise Pascal and Evariste Galois and William Rowan Hamilton, who seemed singled out for greatness at a schoolboy's age. Apparently he did well in his later years at school, but the little that is recorded of his childhood is derived from schoolmates of his who were questioned in their old age, when their recollections may have been coloured by Newton's fame. He certainly was fond of making things with his hands and of copying pictures and passages from books. The verses attributed to him were taken by him from the Eikon Basilike, where they appear under a picture of Charles I which he copied: the extensive notes on drawing and painting which he made as a schoolboy were al1 taken from John Bate's Mysteries of Art and Nature, which also contains a description and picture of a wooden waterclock 2SS
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E. N. DG C. Jtlldnule
which corresponds closely to the waterclock which Stukeley tens us that Newton made. He likewise made sundials, but while there is fair evidence that he was clever and ingenious with his hands I do not think it can be claimed that, as a boy, he showed any greater signs of genius than the ordinary run of mechanically minded boys. He did well enough at school, and badly enough as a farmer, to be sent to Cambridge in 1661, at the age of eighteen, probably with the intention that he should eventually enter the church, which was the best way for an educated man with influence to make his ways in those days. Trinity College could be counted upon to supply the influence. There is little known about his first two years at the University, but, proceeding, we cannot celebrate Newton's birth without bestowing a pious word of praise on Isaac Barrow, an excellent mathematician who was Lucasian Professor at the time. Newton was about twenty.-one when he came under his influence, and it was Barrow who first recognized the young man's genius, encouraged his mathematics and directed his attention to optics. By 1669 Barrow had such faith in Newton that, when he published his Lectiones Opticae, he turned to Newton for help. Newton took his degree at Cambridge early in 1665. In the autumn of that year the great plague, which was raging in London, caused the University to close, and Newton went back to live at the isolated little house at Woolsthorpe where he was born in 1642. Here he spent most of his time until the spring of 1667, when the University reopened and he returned, being then twenty-four. Newton is throughout his life an enigmatic figure, but nothing is more extraordinary than his development in the period from 1663 to this spring of 1667, that is, during the Woolsthorpe period and the time at Cambridge immediately preceding it. Newton was always so secretive that we cannot say definitely that at the earlier date he had done nothing extraordinary, but we can say that there is no evidence that he had done so. By the time of his return to Cambridge it is tolerably certain that he had already firmly laid the foundations of his work in the three great fields with which his name is for ever associatedthe calculus, the nature of white light, and universal gravitation and its consequences. The binomial theorem, which was fundamental for his early mathematical work, he discovered about 1664. Fontenelle justly observes that we may apply to Newton what Lucan said of the Nile, whose source was unknown to the ancient-that it has not been permitted to mankind to see the Nile feeble and taking its rise as a tiny stream. I cannot here forbear to quote Newton's own words about this golden period of his achievement, although they must be familiar to many of you. They are taken from a memorandum in the Portsmouth Collection, probably written when he was about seventy-three years old:
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In the same year [1666] I began to think of gravity extending to the orb of the moon, and having found out how to estimate the force with which a globe revolving within a sphere presses the surface of the sphere, from Kepler's rule of the periodical times of the planets being in a sesquialterate proportion of their distances from the centres of their orbs [sesquialterate means one and a half times, or, as we say, the square of the years are as the cubes of the orbits] I deduced that the forces which keep the planets in their orbs must be reciprocally as the squares of their distances from the centres about which they revolve: and thereby compared the force requisite to keep the moon in her orb with the force of gravity at the surface of the earth, and found them answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since. In Newton's own words, then, even when he was writing the Principia he was not bending his mind so intensely to science as he did in the short period at Woolsthorpe which ended when he was twenty-four. In those great two years he had to the full two priceless gifts which no one enjoys to-day, full leisure and quiet. Leisure and quiet do not produce a Newton, but without them even a Newton is unlikely to bring to ripeness the fruits of his genius. It is hard for us to realize the powers of abstraction necessary for that great mind to break away from all previous theories of the celestial motions, and to formulate his new mechanics of the universe. Action at a distance, gravity extending into the remote depths of space are conceptions so familiar to us that we are apt to forget their difficulties. Einstein's innovations were less revolutionary to his time than Newton's were to his. Remember that the ruling scheme of the universe was Descartes's great theory of vortices, vortices of subtle particles which swept the planets along. Subtle particles were a familiar conception: the picture which the vortices offered was easy to apprehend and had behind it the overwhelming authority of Descartes, which was so great that it continued in France until long after Newton's death. Descartes's scheme was a pictorial fancy: Newton's was an abstract mathematical machinery. Even great men of science found it hard to understand. lean Bernoulli, who died long after Newton, never accepted it, but remained a Cartesian to his death. There are two stories about the work at Woolsthorpe to which I may perhaps allude. The one is that the faU of an apple from a tree there Jed Newton to the view that the earth was pulling at the apple. There was, however, nothing radically new about the conception that the earth exercised an attractive force on bodies near its surface. What really happened, according to Stukeley's report of a conversation with Newton in his old age, was that, when he was thinking of what pull could hold the moon in its path, the fan of the apple put it into his head that it might be the same gravitational pull, suitably diminished by distance, as acted on the apple.
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The other story is that, finding a small discrepancy between the force required to keep the moon in her orbit and the gravitational pull, calculated from that at the earth's surface by the application of the inverse square law, Newton put the work aside, and that the cause of this discrepancy was that Newton took a wrong value of the earth's radius. I think we may take it that this was not the cause. To find the pull on the moon from that on bodies at the earth's surface it is necessary-it is essentialto show that the gravitational pull of a spherical earth is the same as it would be if the whole mass were concentrated at the centre. This is by no means obvious: in fact, it is true only for an inverse square law of attraction and for no other. We know that it gave Newton some trouble to prove this point, which is the subject of a Proposition in the Principia. This seems to be the reason that, having found that 'it answered pretty nearly', Newton turned to other things. An essential link in the argument was missing, and this, it seems, was not produced by him until 1685. No special pleading, however, is required to account for the fact that Newton published no account of this work on gravitation at the time. He never felt any strong desire to bring his work before the world, and later, as we shall see, had a positive aversion from doing so. In particular, he had a horror of controversy. In any case, all that he did to record his great mathematical discoveries, including the generalized binomial theorem and the new calculus of fluxions and inverse fluxions, as he called the differential and integral calculus, was to write a manuscript account under the title De Analysi per Aequationes Numero Terminorum Infinitas, which he gave to Barrow in 1669. This was not published until 1711. It contained an account of the work done about four years earlier: Newton produced it to Barrow only when he was told of Mercator's published calculation, in 1668, of the area of a hyperbola. We shall, no doubt, hear more of this in the following lectures on Newton's mathematical work. In 1669 Barrow resigned his chair of mathematics, the Lucasian chair, to Newton. In the same year he published his lectures on Jight, in which he thanks Newton warmly for suggestions and for revising the proof sheets. There is, however, in this book no suggestion of any of Newton's great discoveries in optics, although from his own pen we know that he had already made many of them. No, there is little need to seek special reasons why Newton withheld pUblication in any particular case. In mathematics, in particular, he never published anything except under strong persuasion. I am not attempting any systematic history of Newton's life-I am more concerned to try to put the man and his achievements before you-but it is necessary if we are to understand Newton to refer to the happenings of the next few years. In the very year 1669 he started to lecture at Cam bridge on optics, describing his discoveries: written copies of these
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lectures were deposited by him, at the time that they were read, in the archives of the University, and were published in 1729, some sixty years after their delivery, under the title Lectiones Opticae, quite a different work from the famous Opticks. The Lectlones were in Latin: an English translation of the first part only was published in 1728. When I was in Russia last year M. Vavilov told me that he was producing a Russian translation of the whole work. Apparently his lectures did not attract large audiences--his amanuensis, Humphrey Newton, referring to 'when he read in the schools' (or lectured, as we now say) writes ·when he had no auditors he commonly returned .. !. We know practically nothing of his life at that time, but it seems certain that in 1668, after having tried to grind non-spherical lenses, he had convinced himself that chromatic aber· ration would prevent the construction of a satisfactory refracting telescope and had made a re1lecting telescope. This was seen and discussed in Cambridge: in 1671 he constructed a second reflecting telescope and, in response to urgent requests, sent it to the Royal Society. He was then twenty-nine, and had accomplished a body of scientific work such as no one before or since had done at that age, but, although Barrow was convinced of his exceptional genius, he was unknown to the scientific world at large. He had published nothing, and certainly felt no urge to publish. If I insist upon this it is because it affords an important clue to his character and to subsequent disputes about priority. The telescope aroused the greatest interest at the Royal Society and Newton wrote to Oldenburg, the Secretary, At the reading of your letter I was surprised to see so much care taken about securing an invention to me, of which I have hitherto had so little value. And therefore since the Royal Society is pleased to think it worth the patronizing, I must acknowledge it deserves much more of them for that, than of me, who, had not the communication of it been desired, might have let it still remain in private as it hath already done some years. . . . I am very sensible of the honour done me by the Bishop of Sarum [Seth Ward, himself a famous mathematician} in proposing me candidate, and which I hope will be further conferred upon me by my election into the Society. And if so, I shall endeavour to testify my gratitude by communicating what my' poor and soUtary endeavours can effect towards the promoting your phiJosophical designs. Stimulated by the interest of the Royal Society, Newton communicated his first paper to be pUblished. Optics was, perhaps, Newton's favourite study-his Opticks was the last considerable scientific work that he produced, and his most daring specuJations were put forward in the Queries appended to this work, to which he added in subsequent editions. At any rate, except, perhaps, for a few words which he wrote in the Principia after having shown that Descartes's vortices could not possibly give a true account of the solar system, it is the only subject on which he ever betrayed any enthusiasm or warmth of feeling. On 18 January 1671/2, he
wrote to Oldenburg that he would send the Royal Society an account of a philosophical discovery 'being in my judgement the oddest, if not the most considerable detection, which hath hitherto been made in the operation of nature'. The paper in question is dated 19 February. In it Newton describes how he formed the spectrum by refracting with a prism a beam of sunlight from a circular hole, a very careful series of experiments leading up to the famous experimentum crucis. In this experiment, having formed a spectrum he isolated a blue beam by a hole in a screen and then refracted it with a second prism. He found that the blue beam remained blue and was refracted more than was a red beam similarly isolated, or in his own words, 'Light consists of Rays differently refrangible'. The spectrum is formed because white light contains mixed in it monochromatic--or in Newton's term, homogeneal-lights of aU kinds, which are merely separated out because they are differently refrangible. This was in the most direct opposition to the views of his time, which were that white light was homogenea], and that coloured bodies changed its nature in various ways, which I have not time to describe. It is not always easy to follow what Newton's predecessors meant, as they used the language of the Aristotelian philosophy. Grimaldi and others asked: 'Was light a substance or an accident?'-they always wanted to know what light was. Newton's design was, as he later said in his Optics, 'Not to explain the Properties of Light by Hypotheses but to propose and prove them by Reason and Experiment'. I shall later say a word as to what Newton meant by hypotheses. In this first paper Newton points out that, with the object glass of a telescope such as was used in his time, dispersion causes insuperable indistinctness of focus, and, having proved that there was no dispersion by reflexion, he then made his reflecting telescope. Remember that he made his own alloy, and himself cast his mirror, and polished it, and that in all these matters he made great advances. He never published any details on the composition and casting of the metal for his mirrors, but after his death a paper was found giving particulars. His alloy was 12 parts of copper, 4 of tin and 1 of white arsenic, or much like modem speculum metal. The description in his Opticks of this work on chromatic aberration, although it was not published until 1704, derives from this time. This paper of Newton's was so revolutionary in its time that it led to much misunderstanding and also to attacks on the unknown young man from those of established reputations. Ignatius Pardies, Franciscus Linus, Gascoigne and Lucas offered objections, mostly trivial; Newton replied in detail, and convinced Pardies, a temperate and courteous critic. Linus was a troublesome pedant, who also attacked Boyle. The trouble with Lucas, which centred on the length of the spectrum, may have been due to different dispersive powers of the glasses used by the two observers.
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The effect on Newton was, however, deplorable. In his own words to Leibnitz a few years later: 'I was so persecuted with discussions arising from the publication of my theory of light, that I blamed my own imprudence for parting with so substantial a blessing as my quiet to run after a shadow.' The most troublesome and acutest critics were, however, Huygens and Hooke. Newton always admired Huygens-in his old age he told Pember~ ton that he thought Huygens the most elegant of any mathematical writer of modem times, and the most just imitator of the ancients, which Newton considered the highest praise. Huygens, however, was chiefly concerned about questions that Newton did not attempt to answer: even if Newton was right, he said, 'there would still remain the great difficulty of explaining, by mechanical principles, in what consists this diversity of colour'. Hooke was concerned to defend his own views that white light was a pulse which was disturbed by the refraction in some way that he did not make clear. Both critics were themselves experimenters of the highest class, and could not but express admiration of Newton's experiments. The criticisms were dealt with by Newton, who was at some pains to explain what he was after. These details need not concern us, but it is necessary to refer to Hooke, as his opposition had a great influence on Newton's life. If there had been a wise and temperate inftuence, a man who appreciated them both and was concerned to reconcile them, aU might have been well, for both men thought highly of one another at bottom. Newton had studied Hooke's Micrographia carefully, and quoted it in his early work, although he unjustifiably omitted all reference to it in the Opticks. Instead of a reconciling influence there was Oldenburg, who hated Hooke and made it his business to promote dissension between him and Newton. In a very temperate letter to Newton, Hooke, no doubt with Oldenburg in view, spoke of 'two hard.t~yield contenders •.. put together by the ears by other's hands and incentives'. Newton says in a letter to Oldenburg about this time: 'Pray present my service to Mr Hooke, for I suppose there is nothing but misapprehension in what has lately happened.' I do not believe for a moment that Oldenburg did convey this compliment: as More says, Newton was continually being urged and nagged by Oldenburg to justify himself. Hooke died in 1703 and Newton did not publish his Opticks until 1704, although practically all the work that it described had been completed much earlier. It seems that he waited for Hooke's death to avoid criticism. To me the misunderstandings between these two great Englishmen is one of the saddest features of the scientific history of the time. Newton sent a further important communication on light to the Royal Society in 1675. In this he gives a theory of light, ostensibly to satisfy those critics who wanted a machinery by which the experimental effects might be produced. His words are very characteristic of his attitude to
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speculative tbeory. 'Because I bave observed tbe beads of some great vir~ tuosos to run mucb upon hypotheses . . . for tbis reason I have here tbought fit to send you a description of the circumstances of this hypoth~ esis, as much tending to the illustration of the papers I herewith send you.' Here he supposes space to be filled with a subtle etber, the for~ runner of the ethers that ruled in the nineteenth century. He would not have light a wave motion, because of sharp shadows, although later he described a series of acute experiments on diffraction phenomena, first discovered by Grimaldi. Light he considered to be 'possibly' a stream of small swift corpuscles of various sizes, red being the largest and violet the smallest: when these corpuscles strike a boundary between two media, such as a glass-air surface, they put the ether, which is denser in the glass, into a vibrating motion, which travels faster than the ray and, striking any second surface, puts it in a state to reftect or transmit according to the pbase. The same alternation of transmission and reftexion takes places at the surface first struck, owing to the vibrations of tbe denser ether. This is Newton's theory of 'fits'; fits of easy reftexion and easy transmission. It can easily be seen how he could account for interference effects-Newton's rings and soap~bubble colours--on this basis. Newton's rings had been observed by Hooke much earlier, but Newton showed that a much greater number could be observed in monochromatic light and explained in detail the colours obtained with white light. Diffraction he accounted for later by condensation of the ether at solid boundaries. It is unjust to Newton to call his a pure corpuscular theory and to contrast it with Huygens's so-called wave theory. Huygens's theory was nothing like a modem wave theory. In his famous 'construction' the only part of the wavelet that was effective was the pole, the point where it touched the envelope. The particular waves, or wavelets, were incapable of causing light. He thus gave his wavelets point or particle properties. Again, the pulses followed one another quite irregularly. He had no conception of a transverse wave. Newton's theory bad many points of resemblance to the most recent theory, although, be sure, I am not claiming that he really anticipated modem developments. Nevertheless, the coexisting particles and waves and the supervelocity of the phase wave that accompanied the particles cannot but make us think of tbe developments of to-day. His explanation of interference phenomena was essentially a wave explanation. As Michelson, something of an autbority in these matters, says: 'It is true that Newton's explanation of the "colours of thin films" is no longer accepted but the fact remains that he did actuaUy measure the quantity which is now designated as the wave length and showed that every spectral colour is cbaracterized by a definite wave length.' MicheJson was writing before wave mechanics came in: he might have said more if he had written later.
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In his Opt;cks, one of the supreme productions of the human mind, which, as I have said, did not appear until he was sixty-one years old, Newton sets down with beautiful precision and elegance the substances of these early papers, and much beside. The proof that a precise refrangibility is an unalterable characteristic of monochromatic light, and that sunlight is a mixture of lights of all colours; the explanation of the rainbow; the revolutionary discussion of natural colours, with the clear distinction between addition and subtraction colours; the explanation of colour mixture; the investigation of the colours of thin films, such as those displayed by soap bubbles and by Newton's rings, with the explanation to which I have referred; some acute experiments on diffraction-these things are some of the treasures contained in the Opticks. There are appended to the book certain 'Queries' which contain some of Newton's most daring speculations, some of them probably made during his old age, since they appear in the second edition of 1717 and not in the first of 1704. One of them concerns the phenomena exhibited by Iceland spar-what we now can polarization pbenomena--to explain which he has to endow the particles with 'sides', That is, he assumes them to be different in different directions transverse to their direction of travel. Thus, in explaining interference colours he introduced, as I have pointed out, an element of periodicity and practically found a wavelength-in explaining polarization he assumed a transverse property. It is strange that he came so near to transverse waves. He cited sharp shadows as proving that light was not a simple wave motion, but his own experiments on diffraction showed him that shadows were not always sharp. Strange, not that he missed the wave theory of light, if he had been so wedded to his corpuscles as he is usually represented to be, but strange that he had himself supplied al1 the elements of a wave theory and just failed to weld them together, In any case, we can observe nothing about the nature of light in empty space: we only know of its interaction with matter, and in considering this interaction Newton introduced the wave properties. As an example of the close-packed and original detail of the book, I may point out that in some twelve pages Newton compares quantitatively the lack of precise focus due to chromatic and to spherical aberration, discusses the circle of least confusion and the relative brilliance of the different colours, and points out that the errors due to spherical aberration are as the cube of the aperture, while those due to chromatic aberration are directly proportional to the aperture. He describes carefully how to grind metallic mirrors; his description is precise and would hold as good technique to-day with only slight modification, He gives, for instance, what I believe to be the first account of the use of pitch for optical polishing, in the modem manner. Mr. Frank Twyman tells me that he has never found any earlier mention of the use of pitch. He describes how to make
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a reft.ecting telescope? in which he uses a right-angled prism as a reflector. which I believe to be the first mention of this device. All this in twelve pages. In the second edition he adds: 'If the Theory of making Telescopes could at length be fully brought into practice, yet there would be certain Bounds beyond which Telescopes could not perform. For the Air through which we look upon the Stars, is in a perpetual Tremor.' And after some discussion he concludes: 'The only remedy is a most serene and quiet Air, such as may perhaps be found on the tops of the highest Mountains above the grosser Clouds.' I would also add that Newton quite clearly realized the advantage of a slit over a bole in bis spectral work, and discusses the point, and also the use of a triangular aperture, which, he points out, gives a pure spectrum on one side of the strip and an overlapping spectrum on the other. Let us now consider another aspect of Newton's work and for tbe purpose return to 1675. Round about this time Newton had done the bulk of his optical work and had on several occasions expressed a great distaste for science, and especially for publication. I could give many examples. but I wiU content myself with one of the most notable, when he wrote to Hooke in 1679: 'But yet my affection to philosophy being worn out, so that I am almost as little concerned about it as one tradesman uses to be about anotber man's trade or a countryman about learning, I must acknowledge myself averse from spending that time in writing about it which I think I can spend otherwise more to my own content and the good of others: and I hope neither you nor any body else will blame for this averseness.' So far he had published nothing about mechanics or gravity, but he expressed this distaste for science at a time when he was soon to be spurred, cajoled, importuned into writing his greatest work, the Principia. I say 'greatest work., for such everyone must admit it to be, but in reading the Opticks I still feel that Newton wrote it with more creative joy, that the beautiful presentation represents him at the height of his pleasure in shaping. The story of the events tbat led to the writing of the Principia has often been told. Hooke, a genius second to Newton but to few others, had come to the firm conclusion that the motion of the planets could be explained on the basis of an inverse square law of attraction, but could not prove it. The matter had been discussed by him with Christopher Wren and with Halley. Newton was by now known as a mathematician, and Halley visited him at Cambridge to ask his opinion in the matter. Newton told the inquirer that he had already proved that the path of a body under a central attraction acting according to the inverse square Jaw would be an ellipse; Jones says that he had done so in 1676-7. Later he sent Halley two different proofs. This incident apparently roused Newton from his distaste for science and revived the old fire. After writing a little treatise De M DIu
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Corporum. Newton, at the instigation of Halley, who undertook to see the book printed at his own charge, began to write the Principia. There were the usual troubles. Hooke, who was in the irksome position of having been convinced of the truth of the inverse square law-although much later than Newton's unpublished work at Woolsthorpe-and having published his conviction, without being able to support it with a proof, claimed that Newton had the notion from him, though apparently all that he wanted was some mention in the preface of Newton's book. Newton, irritated beyond bounds at Hooke's contention, wrote that he would suppress the third book of the Principia, which is the crown of the work and contains the celestial mechanics. ·Philosophy is such an impertinently litigious Lady, that a man had as good be engaged in lawsuits, as have to do with her. I found it SO formerly, and now I am no sooner come near her again, but she gives me warning.' Halley persuaded him not to mutilate the work. The Principia eventually came forth in 1687. The composition of this book, which changed the face of science, had taken about eighteen months. The book is not an easy one to .read, either now or then, a fact which Newton recognized. Writing to Gilbert Clarke in the year of its publication he called it 'a hard book', and in the work itself he teUs the reader that even a good mathematician may find many of the propositions difficult, and advises him as to what he can skip in the preceding parts if he wants to read the third book. The mathematical machinery is on the lines of strict classical geometry-Pemberton, who saw much of the Master in his old age, since he edited the third edition of the Principia, reports that Newton always expressed great admiration for the geometers of ancient Greece and censured himself for not following them more closely than he did. Many modems have suggested that he obtained his results by other methods and then threw them into geometrical form: this may be so, but it may equaUy be argued that he was so familiar with the methods of c1assical geometry that he could use them with a facility unknown to-day. In any case, Whewell has expressed the situation with great force: Nobody since Newton has been able to use geometrical methods to the same extent for the like purposes; and as we read the Principia we feel as when we are in an ancient armoury where the weapons are of gigantic size; and as we look at them we marvel what manner of man he was who could use as a weapon what we can scarcely lift as a burden. In the first book Newton enunciates his laws of motion, with due acknowledgements to Galileo, and Jays down his mechanical foundations. inclUding the composition of forces, clearly formulated for the first time. He proceeds with extreme caution where fundamentals are concerned, and typically refuses to discuss underlying causes not susceptible to observation and experiment. For example, he says, 'For I here design only to
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give a Mathematical notion of those forces, without considering their Physical causes and seats'; and a little later, he says again, Wherefore, the reader is not to imagine that by those words, I any where take upon me to define the kind, or tbe manner of any Action, the causes or the physical reason thereof, or that I attribute Forces, in a true and Pbysical sense, to certain centres (which are only Mathematical points); when at any time I happen to speak of centres as attracting, or as endued with attractive powers.! Compare this with bis words in a letter to Bentley some six years after the first publication of the Principia: 'You sometimes speak of gravity as essential and inherent to matter. Pray do not ascribe that notion to me; for the cause of gravity is what I do not pretend to know, and therefore would take some time to consider of it.' This and many other passages, both in the Principia and tbe Opticks, make clear what Newton meant when he wrote 'Hypotheses non mgo'. That be made hypotheses, in tbe modern sense, is abundantly clear by the fact that certain passages in the Principia are boldly headed 'Hypotbesis'-be was prepared to make a quantitative assumption from which exact laws could be worked out and compared witb observation. He would not, bowever, willingly speculate beyond the limits where quantitative confirmation could be sought from Nature. The ~nse of the famous words is better given by his remark to Conduitt in his eigbty~tbird year, 'I do not deal in conjectures" or in his first sentence of the Opticks, 'My design in this Book is not to explain tbe Properties of Light by Hypotheses, but to propose and prove them by Reason and Experiments'. Very characteristic is the way in which he introduces certain optical theories: 'They who are averse from assenting to any new Discoveries, but such as they can explain by an Hypothesis, may for the present suppose. . . .' If I seem to labour the point, and to have left the Principia to wander in other fields, it is because the spirit was a new one in science. and because it animated the whole of Newton's work. It is particularly clearly expressed in the famous Rules of Reasoning in Philosophy which usher in the third book of the Principia. To return to the first book, in it the laws of simple orbits. in particular Kepler's laws, are deduced from the inverse square law, and it is demonstrated that, if every point attracts every other point with an inverse square Jaw. the attractive mass of a bomogeneous sphere may be considered as concentrated at the centre. The vast generalization that every mass point attracts every other mass point with an inverse square law is typical of Newton's genius. It is clear tbat he was brougbt to it by considering the particular case of the moon's motion. He solved the problem of the motion of two bodies mutually gravitating. He worked out experimentally the Jaws of impact of two bodies. In general, this book of the Principia is the first 1
Andrew Motte's translation, as with the other passages quoted from the Principia.
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text-book of theoretical mechanics, written in the modern spirit, and it is practically all original. The second book is devoted to motion in a resisting medium and is the first treatment of the motion of real fluids. He considers at length the motion of bodies of various shapes through a resisting liquid. taking the resistance to vary as the square of the velocity. Among other problems propounded is that of the solid of revolution of least resistance, 'which I conceive may be of use in the building of ships'. This difficult problem has been discussed very learnedly by A. R. Forsyth. Newton gives no details as to how he reached his conclusions, but it is clear from this, and from his solution of the brachistochrone problem~ that he must have invented the general methods of the calculus of variations. Incidentally, he was also familiar with the calculus of finite differences. Particularly important is the mathematical treatment of wave motion, the first ever given. Newton strongly insists upon the way in which a wave passing through a hole spreads out--or is diffracted, as we now say-which was his chief reason for rejecting a direct wave theory of light. He deduces the fundamental law that the velocity is expressed by the square root of the elasticity divided by the density. In this book also is laid down the law for the behaviour of real fluids from which the term 'Newtonian viscosity' is derived. The whole book constitutes the first text-book of mathematical physics, and of hydrodynamics in particular, and it is embellished by an account of many experiments, carried out with the highest degree of precision and skill. The third book of the Principia is the crown of the work. Like the third movement of a supreme symphony it opens with a recapitulation of previous themes and short statement of the new theme. In the preceding books I have laid down the principles of philosophy; principles not philosophical but mathematical. ..• These principles are, the laws and conditions of certain motions, and powers or forces, which chiefly have respect to philosophy. But lest they should have appeared of themselves dry and barren, I have illustrated them here and there, with some philosophical scholiums, givin, an account of such things . . . as the density and the resistance of bodies, spaces void of all bodies~ and the motion of light and sounds. It remains, that from the same principles, I now demonstrate the frame of the System of the World. In this third book he establishes the movements of the satellites round their planets and of the planets round the sun on the basis of universal gravitation; he also shows how to find the masses of the planets in terms of the earth's mass. The density of the earth he estimated at between five and six times that of water, the accepted figure to-day being almost exactly 5' 5. From this he calculated the masses of the sun and of planets which have satellites, a feat which Adam Smith considered to be 'above the reach of human reason and experience', But he went much further
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than this. He accounted quantitatively for the flattened figure of the earth and calculated the ellipticity to be lhao, as against the most recent figure of ¥l97. It was of La Condamine's measurement of the arc at the equator after Newton's death, which verified Newton's calculation, that Voltaire wrote to La Condamine: Vous avez trouve par de long ennuis Ce que Newton trouva sans sortir de chez lui. He showed that while the gravitational pun on a sphere acts as if concentrated at the centre, the pull on a spheroid does not, and from this he calculated the conical motion of the earth's axis known as the precession of the equinoxes, which Sir George Airy held to be his most amazing achievement. Incidentally, he discussed the variation of gravitational acceleration over the earth's surface. He worked out the main irregularities of the moon's motion due to the pull of the sun. He laid the foundation of all sound work on the theory of the tides. He was the first to establish the orbits of comets and to show that they, too, were moving under the sun's attraction, so that their return could be calculated. These are a few of the astonishing feats recorded in the third book, feats which would have affirmed Newton as a sublime genius if he had never done anything else. Throughout the Principia there is clear evidence of Newton's wonderful skill as an experimenter. As a comparatively small matter I may cite his proof that gravitational mass and inertial mass are the same, a trifle that is often overlooked. He took two exactly similar spherical wooden boxes as pendulum bobs, so as to make the air resistance the same, filled them with different substances-gold, si1ver, lead, glass, sand, salt, wood, water, wheat-and showed that their time periods were equal. His pendulum work on damping in different circumstances is extraordinary for the attention to essential detail and for its discrimination 'being not very sol1icitous for an accurate calculus, in an experiment that was not very accurate'. His optical work is not his only experimental triumph. Let me now in a few words run through the remaining events in Newton's life. After the composition of the Principia, Newton, possibly exhausted by the strain of producing this prodigious work, appears to have abandoned science for a time. There were many distractions. His friend Henry More died. James II came into conflict with the authorities at Cambridge, and Newton became active in the defence of the University. I take from a letter which he wrote on this occasion a sentence that does him credit as a man: 'An honest courage in these matters will secure all, having law on our sides.' In 1688 James fled the country and Newton was elected M.P. for Cambridge in the so-called Convention Parliament. This brought him much to London. In 1689 his mother died: he was an affec-
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tionate son and sat up an night with her during her last illness. Whether as a result of these shocks and changes or no, he seems to have been much disturbed about this time, and to have sought an administrative post. He was despondent and appears to have been mainly employed on theological matters. In 1693 he fell into a profound melancholy, with long periods of sleeplessness, during which he thought that aU his friends had turned against him. Apparently this attack lasted only some months, and it is a gross exaggeration to refer to this as 'Newton's madness'. Newton's general state certainly alarmed his friends, but the derangement was not more than an exaggeration of his usual suspicious nature and it was not permanent. The story of his little dog Diamond having knocked over a candle and burned papers, the loss of which drove him to madness, is an absurd story first made current in 178~apparently Newton never had a dog, there is no satisfactory evidence of a really serious fire of Newton's manuscripts, and about the time in question it seems clear that Newton was in one of his periods of distaste for science. He conducted some correspondence with Flamsteed about the moon's motion in 1694, but in 1696 came an event which changed his whole mode of life. He was made Warden of the Mint, from which office he passed to be Master, the chief post, in 1699. This post was no sinecure, for Newton's friend Charles Montague, later Lord Halifax, was putting through a great recoin age scheme and Newton was actively involVed in it. He took his work very seriously and rendered valuable service to his country. There have been many regrets that such a mind should be distracted from science by an administrative post. It seems clear, however, that Newton wanted such a post and that, while he retained his prodigious powers, he no longer felt inclined to devote his main energies to science. He published his Opticks, he was interested in the later editions of the Principia. but he entrusted the bringing out of the second to Roger Cotes and of the third to Henry Pemberton, although he supplied certain material. I have already referred to the queries appended to the Opticks. which remind me somewhat of Beethoven's last quartets. The Master's main work done, he seems reaching forward tentatively to new regions which he will not have time to explore. Newton in his old age occupied a unique position. His scientific reputation was unrivalled throughout the learned world. From 1703 to his death he held undisputed sway in the Royal Society as President. He was a national figure: when in 1705 Queen Anne knighted him she awarded an honour never before, I believe, conferred for services to science. For, strangely enough, it was not as Master of the Mint that he was knighted: Conduitt tells us that the Queen, 'the Minerva of her age', thought it a happiness to have lived at the same time as, and to have knoWn, so great a man. He held an honourable and lucrative post. His niece, who lived
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on terms of close friendship with Montague-about the exact nature of which friendship there has been unseemly speculation-was a toast of the town and secured him, should he wish it, the entry into fashionable society. When he died in 1727 the greatest honours were accorded to him: his body lay in state in the Jerusalem Chamber, his pall was supported by the Lord High Chancellor, two dukes and three earls--which meant something in those days--and the place allotted for his monument had been previously refused to the greatest of our nobility. It was a great occasionthe first and the last time that national honours of this kind have been accorded to a man of science or, I believe, to any figure in the world of thought, learning, or art in England. But then those were extraordinary times. Newton was by no means a perfect character, and it is doing science no service to pretend that he was. He was easily irritated, as a man who had wearied himself with such prodigious and concentrated effort might well be. Whiston, who had the greatest admiration for his powers, said of him that 'He was of the most fearful, cautious, and suspicious temper, that ever I knew', but it may with truth be said that Whiston had quarrelled with him. So had Flamsteed-and with some cause-who said that he was 'insidious, ambitious, and excessively covetous of praise, and impatient of contradiction'. This is undoubtedly the exaggeration of one who himself was of no angelic temperament, but John Locke was a firm friend, a man of admirable character, and he wrote 'he is a nice'--that is, difficult and over-precise-'man to deal with, and a little too apt to raise in himself suspicions where there is no ground'. This evident suspicious element in his nature, which also was a characteristic of Flamsteed and Hooke, is his worst blemish. At the same time, he was kind and helpful to young men, and many authentic stories are told of his generosity. In any case, such imperfections of character are not inconsistent with high performance, even in spiritual matters, as anyone who has studied the Church Fathers must admit. One aspect of this strangeness, this almost morbid sensitiveness, was his abnormal dread of controversy, to which I have already alluded. 'There is nothing I desire to avoid in matters of philosophy more than contention, nor any kind of contention more than one in print', he wrote to Hooke, who, jUdging by his writings, was not averse from printed dispute. Even stronger are his words to Oldenburg: 'I see I have made myself a slave to philosophy, but if I get free of Mr. Linus's business, I will resolutely bid adieu to it eternally. excepting what I do for my private satisfaction, or leave to come out after me; for I see a man must either resolve to put out nothing new. or to become a slave to defend it.' Phrases such as 'and signify, but not from me' and 'pray keep this letter private to yourself are common: when he gave Barrow permission to send some of
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his early mathematical work to John Collins he stipulated that his name should be withheld. He intended to publish his Optical Lectures in 1671, but was deterred by horror of controversy. Not one of his mathematical writings was voluntarily given to the world by himself. At the same time, when not worried or irritated he was modest about his achievements. Some of his phrases, as when he says that if he is in London he might possibly supply a vacant week or two at the Royal Society 'with something by me, but that's not worth mentioning'. may have been the language of compliment, but when he said that if he had seen further, it was by standing on the shoulders of giants, it sounds like genuine humility, against which, however. we must set that he would never make acknowledgements to anyone, like Hooke, with whom he had quarrelled, even when they were plainly due. His aims were so high, the problems which he wished to solve so general and so difficult, his scope so wide that I, for one, am sure that he was sincere when, shortly before his death, he said: 'I do not know what I may appear to the world; but to myself. I seem to have been only like a boy, playing on the seashore, and diverting myself, in now and then finding a smoother pebble or a prettier shell than ordinary, while the great ocean of truth lay all undiscovered before me.' Evidence can be cited for the view that Newton was most modest or most overweening: the truth is that he was a very complex character. Further, he could with perfect consistency be modest about his performance in respect to his aims and completely confident in it when viewed from the standpoint of his contemporaries. Many able men will tolerate self-criticism, but revolt against the criticism of men puffed up by place or exalted by supporters and by the fashionable schools. No estimate of Newton would be balanced without some reference to the mystical element in his nature. This raises the question of his work in chemistry. Newton devoted probably a8.much time and effort to alchemy and chemistry, which were one study in his time, as he did to the physical sciences. His library was well stocked with the standard alchemical and mystical books, such as Agrippa De Occulta Philosophia. Dirrius De Transmutatione Metallorum, Fame and Confession 0/ the Rosie Cross, Geber The Philosopher's Stone, Kerkringius Currus Triumphalis Basil;; Valentini, Libavius Alchymia. eight books by Raymond LuUy, five by Maier, four by Paracelsus, The Marrow 0/ Alchemy, The Musaeum Hermeticum and so on, mostly copiously annotated. In most of these books the mystical element is prominent. In many of them the results of experiment were expressed in the allegorical language ridiculed by Goethe in Faust: The Lion Red, bold wooer, bolder mate, In tepid bath was to the Lily married, And both were then by open fire-flame straight From one bride-chamber to another harried.
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Thus in due time the Youthful Queen, inside The glass retort, in motley colours hovered: This was the medicine; the patients died, And no one thought of asking who recovered. Man's spiritual life, death and resurrection, were paralleled in the chemical changes of the material world. The lines of thought, the spirit and the language of these toilers in the early chemical laboratories have passed, and it is hard without prolonged study to disentangle meanings, if any, veiled in allegory, tinged with prophecy, coloured with religious belief and clouded with charlatanism in many cases. There is correspondence between Newton and Boyle about a mercury that grows hot with gold-not every mercury obtained by extraction will do this, says Boyle, so that he did not mean mercury as We know it. What did he mean? Newton, no doubt, was deeply interested in chemical operations-what was he seeking? Humphrey Newton, his amanuensis, tells us of the period 1685-90, during which the Principia was written, that, especiaJly at spring and the fall of the leaf, he used to employ periods of about six weeks in his elaboratory, the fire scarce going out either by day or night. 'What his aim might be', says this faithful, but not overbright, assistant, 'I was not able to penetrate, but his pains, his diligence at these times made me think he aimed at something beyond the reach of human art and industry! There are some extraordinary passages in Newton's writings on this subject. I give you two about the transmutation of metals, from a letter written in 1676, a period when he was mainly concerned with chemical operations. I cannot hope to convince the sceptical that Newton had some power of prophecy or special vision, had some inkling of atomic power, but I do say that they do not read to me as if all that he meant was that the manufacture of gold would upset world trade-'Because the way by which mercury may be so impregnated, has been thought fit to be concealed by others that have known it, and therefore may possibly be an inlet to something more noble, not to be communicated without immense danger to the world, if there should be any verity in the Hermetic writers', and a little further on 'there being other things beside the transmutation of metals (if those great pretenders brag not ) '-the word pretender had no offensive sense in those days, any more than professor has now'which none but they understand'. In pondering what these passages may import, consider the no greater reticence with which he speaks of his optical discoveries in the letter of 23 February 1668/9. He published nothing on chemistry except a short but very significant note De Natura Acidorum. which I should like to have time to discuss, and certain 'queries' appended to the Opticks, which must have been the fruit of mature consideration, since they do not appear in the first edition of 1704. In the last query he sketches a theory of chemistry in terms of attractive forces which seems to me to be an immense advance on the old
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picture of hooked atoms, a conception 'which is begging the question', says Newton. Have not smal1 Particles of Bodies certain Powers, Virtues or Forces, by which they act at a distance, not only upon the Rays of Light for reflecting refracting and inflecting them, but also upon one another, for producing a great part of the Pluenomena of Nature. . . . How these Attractions may be perform'd I do not here consider. . . . The Attractions of Gravity, Magnetism and Electricity, reach to very sensible distances, and so have been observed by vulgar Eyes, and there may be others which reach to so small distances as hitherto escape Observation, and perhaps electrical Attraction may reach to such small distances, even without being excited by Friction. This, surely, is a distinct foreshadowing of modern chemical theory. We have small particles-atoms and molecules---endowed with inherent attractive forces of an electrical nature, by the agency of which they act on light and combine with one another. What I have quoted is but a small part of this 'query', which I should discuss at length did time permit. In the query before it he asks whether gross bodies and light are not convertible into one another. Surely here we have a great seer as well as the greatest man of science who ever lived. Newton's chemistry, then, was not all alchemy, but there are some half million words of manuscript on alchemical subjects in the Portsmouth CoJlection which have never been digested, let alone published. How much of this matter is copied out of books I cannot find: what there is of worth among these writings no one knows. It may be that Newton never found the great truth for which he was seeking, it may be that much of it is of ]jttle value, like his Chronology of the A ncient Kingdoms Amended. That it is exclusively mystical I do not believe---that there is a mystical element seems certain. I hope that one day some profound student-no one less will suffice---will study this mass of papers, among which there may well be matter of prime interest. We need not seek for any special reason why he never published the results of his chemical investigations-it was only by chance that the Principia was published. The mystical element in Newton is abundantly clear. He was a close student of the mystic Jacob Boehme, from whose works he copied large extracts. Strange passages occur in Newton's letters, such as 'but it is plain to me by the fountain that I draw it from, though I will not undertake to prove it to others'. Whiston says: 'Sir Isaac, in mathematics, could sometimes see almost by intuition, even without demonstration. . . . And when he did but propose conjectures in natural philosophy, he almost always knew them to be true at the same time', yet adds caustic comment on Newton's Chronology which shows him to have been anything but a blind worshipper. As an example of his mathematical prescience I may cite his rule for the discovery of imaginary roots of equations, which was not finally proved until by Sylvester in 1865. The psychology of inspira-
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tion is always difficult-the psychology of unique genius is beyond us. If I may, without arrogance, I will use myself the words of Emmanuel Kant: 'I am not aware that anybody has ever perceived in me an inclination to the marvellous, or a weakness tending to credulity': nevertheless, I feel that Newton derived his knowledge by something more like a direct contact with the unknown sources that surround us, with the world of mystery, than has been vouchsafed to any other man of science. A mixture of mysticism and natural science is not unexampled--Swedenborg has important achievements in geology, physiology and engineering to his credit. In any case, it is not honest to neglect half of a man's intellectual life because to do so makes the other half easier to explain. I do not propose to deal with Newton's religious views, although, if you want a picture of the complete man, you must remember that theological questions occupied much of his attention. Archbishop Tenison said to him, 'You know more divinity than all of us put together', and Locke wrote 'divinity too, and his great knowledge in the Scriptures, wherein I know few his equals', He is supposed to have been unsound on the Trinity, in fact to have been tainted with the doctrine of Arianism. The works of the Church Fathers were prominent in his library. His two books the Chronology of Ancient Kingdoms Amended and Observations upon the Prophecies of Daniel and the Apocalypse of St John probably cost him as much effort as the Principia. There were over 1,300,000 words in manuscript on theology in the Portsmouth papers, according to my estimate from the catalogue. Of his immense and very valuable work as Master of the Mint I will also forbear to speak. But I must insist that these activities emphasize the point that he spent a comparatively small part of his long life, probably only ten years or so in all, on the work that has made his name famous throughout the civilized world. For long periods he was indifferent to science-we might even justify the expression that he had a distaste for it -but he never lost his powers. In 1696, shortly after his first appointment to the Mint, when he received one afternoon Bernoulli's problem, set as a challenge to 'the acutest mathematicians in the world', he solved it before going to bed. The problem, I may remind you, was that of the brachistochrone, or line of quickest descent, and required for its solution the calculus of variations. Later, in 1716, when Leibnitz set a problem 'for the purpose of feeling the pulse of the EngJish analysts~, he likewise solved it in a few hours. He took an active part in the production of the second edition of the Principia, which appeared in 1713, and showed his wonted powers. It is clear that he had what we now call a nervous breakdown in 1693, showing to an exaggerated degree the irritability and suspicion which often characterized his actions, but it was nothing like madness and he made a complete recovery. Nothing could be easier to refute than
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Biot's contention that after this breakdown his intellect was permanently impaired. At our celebrations to-day, at which so many great and learned men are gathered to do honour to Newton's name, I see in imagination shadowy figures, each bearing his tribute. I see the austere form of Leibnitz, reluctan~ perhaps in view of the great strife that subsequently arose about the invention of the calculus, I see him, whom we made a Fellow of our Society two hundred and seventy-three years ago, repeating what he wrote: 7aking mathematics from the beginning of the world to the time when Newton Jived, what he has done is much the better part.' I see a small band of the greatest men of science that France has producedLaplace, who wrote that VJbe Principia is pre-eminent above any other production of human genius': Lagrange, who frequently asserted that Newton was the greatest genius that ever existed: Biot, who said of the monumental inscription in Westminster Abbey, which runs CLet mortals rejoice that such and so great an ornament of the human race has existed', tha~ though true in speaking of Newton, it can be applied to no one else: Arago, declaring The efforts of the great philosopher were always superhuman: the questions which he did not solve were incapable of solution in his time'. Gauss is here, who uses the term cla1'us for some scholars, and clarissimus for others, but applies summus to Newton and to Newton alone. Boltzmann and Ernst Mach come forward to represent their generation of mathematical physicists, Boltzmann declaring that the Principia is the first and greatest work ever written on theoretical physics, Mach saying in 1901: ~An that has been accomplished in mechanics since his day has been a deductive, formal and mathematical development of mechanics on the basis of Newton's laws.' And lest it be supposed that the coming of relativity has lessened the awe and admiration with which the great leaders of scientific thought have always regarded him. let me call on Eddington, our much lamented brother, to repeat his words 70 suppose that Newton's great scientific reputation is tossing up and down on these latter-day revolutions is to confuse science with omniscience'; let me take from Einstein's lips his words on Newton: 6Nature to him was an open book, whose letters he could read without effort. In one person he combined the experimenter, the theorist, the mechanic and, not least, the artist in expression.' I think, however, that Einstein is perhaps incorrect in using the words 'without effort'. Newton's own words are eloquent on this point. He told one inquirer that he made his discoveries 'By always thinking unto them', and, in a more communicative mood than usual, said: 61 keep the subject constantly before me and wait till the first dawnings open little by little into the full light.' I would rather say that Newton was capable of greater sustained mental effort than any man, before or
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I have endeavoured within the limits of time which custom and your endurance impose to give some account of the life, but more particularly of the achievements, of this extraordinary man. If any of you think that I have claimed too much for him, tum, I beg you, not so much to what has been written of him but to what he himself wrote and consider his works against the background of his time, which was, nevertheless, a time as fertile in discovery and as filled with great men as any of which record is made. Consider them, too, in the light of subsequent advances. Ponder the scope of his performance---the invention of the calculus, the establishment of the fundamental features of physical optics and the explanation of celestial mechanics and terrestrial perturbations and tides in terms of the laws of motion and of universal gravitation. Remark the clarity with which he laid down the principles which have since governed fertile scientific work. Remember that he was as supreme in experiment as he was in theory, and, when he chose, as in the Opticks. unrivalled as an expositor. Stand back a little, and range the times in order. The face of science changes, theories fail and rise again transformed. Achievement such as Newton's is not lessened by the great advances of the subsequent centuries: it survives in undiminished strength and beauty the strange and formidable shaping of our own times. From being the preoccupation of a few curious spirits science has grown to be a universal study, on the fruits of which peace among people and the prosperity of nations depend, but the great principles enunciated by Newton and their orderly development by him remain as the foundations of the discipline and as a shining example of the exalted power of the human mind.
I should like to know if any man could have laughed if he had seen Sir Isaac Newton rolling in the mud. -SIDNEY SMITH
If we evolved a race of Isaac Newtons, that would not be progress. For the price Newton had to pay for being a supreme intellect was that he was incapable of friendship, love, fatherhood, and many other de.rirable things. As a man he was a failure; as a monster he was superb.-ALoous HUXLEY
9
Newton, the Man By JOHN MAYNARD KEYNES
IT is with some diffidence that I try to speak to you in his own home of Newton as he was himself. I have long been a student of the records and had the intention to put my impressions into writing to be ready for Christmas Day 1942, the tercentenary of his birth. The war has deprived me both of leisure to treat adequately so great a theme and of opportunity to consult my library and my papers and to verify my impressions. So if the brief study which I shall lay before you to-day is more perfunctory than it should be, I hope you will excuse me. One other preliminary matter. I believe that Newton was different from the conventional picture of him. But I do not believe he was less great. He was less ordinary, more extraordinary, than the nineteenth century cared to make him out. Geniuses are very peculiar. Let no one here suppose that my object to-day is to lessen, by describing, Cambridge's greatest son. I am trying rather to see him as his own friends and contemporaries saw him. And they without exception regarded him as one of the greatest 01 men. In the eighteenth century and since, Newton came to be thought of as the first and greatest of the modem age of scientists, a rationalist, one who taught us to think on the lines of cold and un tinctured reason. I do not see him in this Jight. I do not think that anyone who has pored over the contents of that box which he packed up when he finally left Cambridge in 1696 and which, though partly dispersed, have come down to us, can see him like that. Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind which looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual inheritance rather less than 10,000 years ago. Isaac Newton, a posthumous child born with no father on Christmas Day, 1642, was the last wonderchild to whom the Magi could do sincere and appropriate homage. Had there been time, I should have liked to read to you the contemporary record of the child Newton. For, though it is well known to his 277
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biographers, it has never been published in extenso, without comment, just as it stands. Here, indeed, is the makings of a legend of the young magician, a most joyous picture of the opening mind of genius free from the uneasiness, the melancholy and nervous agitation of the young man and student. For in vulgar modem terms Newton was profoundly neurotic of a not unfamiliar type, but-I should say from the records-a most extreme example. His deepest instincts were occult, esoteric, semantic-with profound shrinking from the world, a paralyzing fear of exposing his thoughts, his beliefs, his discoveries in all nakedness to the inspection and criticism of the world. 'Of the most feadul, cautious and suspicious temper that I ever knew', said Whiston, his successor in the Lucasian Chair. The too well-known conflicts and ignoble quarrels with Hooke, Flamsteed, Leibnitz are only too clear an evidence of this. Like all his type he was wholly aloof from women. He parted with and published nothing except under the extreme pressure of friends. Until the second phase of his life, he was a wrapt, consecrated solitary, pursuing his studies by intense introspection with a mental endurance perhaps never equalled. I believe that the clue to his mind is to be found in his unusual powers of continuous concentrated introspection. A case can be made out, as it also can with Descartes, for regarding him as an accomplished experimentalist. Nothing can be more charming than the tales of his mechanical contrivances when he was a boy. There are his telescopes and his optical experiments. These were essential accomplishments, part of his unequalled all-round technique, but not, I am sure, his peculiar gift, especially amongst his contemporaries. His peculiar gift was the power of holding continuously in his mind a purely mental problem until he had seen straight through it. I fancy his pre-eminence is due to his muscles of intuition being the strongest and most enduring with which a man has ever been gifted. Anyone who has ever attempted pure scientific or philosophical thought knows how one can hold a problem momentarily in one's mind and apply all one's powers of concentration to piercing through it, and how it will dissolve and escape and you find tbat what you are surveying is a blank. I believe that Newton could hold a problem in his mind for hours and days and weeks until it surrendered to him its secret. Then being a supreme mathematical technician he could dress it up, how you will, for purposes of exposition, but it was his intuition which was pre-eminently extraordinary-'so happy in his conjectures', said de Morgan, 'as to seem to know more than he could possibly have any means of proving'. The proofs, for what they are worth, were, as I have said, dressed up afterwards-they were not the instrument of discovery. There is the story of how he informed Halley of one of his most fundamental discoveries of planetary motion. 'Yes,' replied Hatley, 'but how do
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you know that? Have you proved it?' Newton was taken aback-'Why, I've known it for years', he replied. 'If you'll give me a few days, I'll certainly find you a proof of it'-as in due course he did. Again, there is some evidence that Newton in preparing the Principia was held up almost to the last moment by lack of proof that you could treat a solid sphere as though all its mass was concentrated at the centre, and only bit on the proof a year before publication. But this was a truth which he had known for certain and had always assumed for many years. Certainly there can be no doubt that the peculiar geometrical form in which the exposition of the Principia is dressed up bears no resemblance at all to the mental processes by which Newton actually arrived at his conclusions. His experiments were always, I suspect, a means, not of discovery, but always of verifying what he knew already. Why do I call him a magician? Because he looked on the whole universe and all that is in it as a riddle. as a secret which could be read by applying pure thought to certain evidence, certain mystic clues which God had laid about the world to allow a sort of philosopher's treasure hunt to the esoteric brotherhood. He believed that these clues were to be found partly in the evidence of the heavens and in the constitution of elements (and that is what gives the false suggestion of his being an experimental natura) philosopher), but also partly in certain papers and traditions handed down by the brethren in an unbroken chain back to the original cryptic revelation in Babylonia. He regarded the universe as a cryptogram set by the Almighty-just as he himself wrapt the discovery of the calculus in a cryptogram when he communicated with Leibnitz. By pure thought, by concentration of mind, the riddle, he believed, would be revealed to the initiate. He did read the riddle of the heavens. And he believed that by the same powers of his introspective imagination he would read the riddle of the Godhead, the riddle of past and future events divinely fore-ordained, the riddle of the elements and their constitution from an original undifferentiated first matter, the riddle of health and of immortality. All would be revealed to him if only he could persevere to the end, uninterrupted, by himself, no one coming into the room, reading, copying, testing-all by himself, no interruption for God's sake, no disclosure, no discordant breakings in or criticism, with fear and shrinking as he assailed these halfordained, half-forbidden things, creeping back into the bosom of the Godhead as into his mother's womb. 'Voyaging through strange seas of thought alone', not as Charles Lamb 'a fellow who believed nothing unless it was as clear as the three sides of a triangle'. And so he continued for some twenty-five years. In 1687, when he was forty-five years old, the Principia was published.
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John Mayruud Keyn"
Here in Trinity it is right that I should give you an account of how he lived amongst you during these years of his greatest achievement. The east end of the Chapel projects farther eastwards than the Great Gate. In the second half of the seventeenth century there was a walled garden in the free space between Trinity Street and the building which joins the Great Gate to the Chapel. The south wall ran out from the turret of the Gate to a distance overlapping the Chapel by at least the width of the present pavement. Thus the garden was of modest but reasonable size. This was Newton's garden. He had the FeUow's set of rooms between the Porter's Lodge and the Chapel-that, I suppose, now occupied by Professor Broad. The garden was reached by a stairway which was attached to a veranda raised on wooden pillars projecting into the garden from the range of buildings. At the top of this stairway stood his telescope-not to be confused with the observatory erected on the top of the Great Gate during Newton's lifetime (but after he had left Cambridge) for the use of Roger Cotes and Newton's successor, Whiston. This wooden erection was, I think, demolished by Whewell in 1856 and replaced by the stone bay of Professor Broad's bedroom. At the Chapel end of the garden was a small two-storied building, also of wood, which was his elaboratory. When he decided to prepare the Principia for publication he engaged a young kinsman, Humphrey Newton, to act as his amanuensis (the MS. of the Principia, as it went to the press, is clearly in the hand of Humphrey). Humphrey remained with him for five years-from 1684 to 1689. When Newton died Humphrey's son-in-law Conduitt wrote to him for his reminiscences, and among the papers I have is Humphrey's reply. During these twenty-five years of intense study mathematics and astronomy were only a part, and perhaps not the most absorbing, of his occupations. Our record of these is almost wholly confined to the papers which he kept and put in his box when he left Trinity for London. Let me give some brief indications of their subject. They are enormously voluminous-I should say that upwards of 1,000,000 words in his handwriting still survive. They have, beyond doubt, no substantial value whatever except as a fascinating sidelight on the mind of our greatest genius. Let me not exaggerate through reaction against the other Newton myth which has been so sedulously created for the last two hundred years. There was extreme method in his madness. All his unpublished works on esoteric and theological matters are marked by careful learning, accurate method and extreme sobriety of statement. They are just as sane as the Principia, if their whole matter and purpose were not magical. They were nearly all composed during the same twenty-five years of his mathematical studies. They fall into several groups. Very early in life Newton abandoned orthodox belief in the Trinity.
Newton, the Man
281
At this time the Socinians were an important Arian sect amongst intellectual circles. It may be that Newton fell under Socinian influences, but I think not. He was rather a Judaic monotheist of the school of Maimonides. He arrived at this conclusion, not on so-to-speak rational or sceptical grounds, but entirely on the interpretation of ancient authority. He was persuaded that the revealed documents give no support to the Trinitarian doctrines which were due to late falsifications. The revealed God was one God. But this was a dreadful secret which Newton was at desperate pains to conceal all his life. It was the reason why he refused Holy Orders, and therefore had to obtain a special dispensation to hold his Fellowship and Lucasian Chair and could not be Master of Trinity. Even the Toleration Act of 1689 excepted anti-Trinitarians. Some rumours there were, but not at the dangerous dates when he was a young Fellow of Trinity. In the main the secret died with him. But it was revealed in many writings in his big box. After his death Bishop Horsley was asked to inspect the box with a view to publication. He saw the contents with horror and slammed the lid. A hundred years later Sir David Brewster looked into the box. He covered up the traces with carefully selected extracts and some straight fibbing. His latest biographer, Mr More, has been more candid. Newton's extensive anti-Trinitarian pamphlets are, in my judgement, the most interesting of his unpublished papers. Apart from his more serious affirmation of belief, I have a completed pamphlet showing up what Newton thought of the extreme dishonesty and falsification of records for which St Athanasius was responsible, in particular for his putting about the false calumny that Arius died in a privy. The victory of the Trinitarians in England in the latter half of the seventeenth century was not only as complete, but also as extraordinary, as St Athanasius's original triumph. There is good reason for thinking that Locke was a Unitarian. I have seen it argued that Milton was. It is a blot on Newton's record that he did not murmur a word when Whiston, his successor in the Lucasian Chair, was thrown out of his professorship and out of the University for publicly avowing opinions which Newton himself had secretly held for upwards of fifty years past. That he held this heresy was a further aggravation of his silence and secrecy and inwardness of disposition. Another large section is concerned with all branches of apocalyptic writings from which he sought to deduce the secret truths of the Universe -the measurements of Solomon's Temple, the Book of David, the Book of Revelations, an enormous volume of work of which some part was published in his later days. Along with this are hundreds of pages of Church History and the like, designed to discover the truth of tradition. A large section, judging by the handwriting amongst the earliest, relates
282
101ln MaYlUU'd K,y".s
to alchemy-transmutation, the philosopher's stone, the elixir of life. The scope and character of these papers have been hushed up, or at least minimized, by nearly all those who have inspected them. About 1650 there was a considerable group in London, round the publisher Cooper, who during the next twenty years revived interest not only in the English alchemists of the fifteenth century, but also in translations of the medieval and post-medieval alchemists. There is an unusual number of manuscripts of the early English alchemists in the libraries of Cambridge. It may be that there was some continuous esoteric tradition within the University which sprang into activity again in the twenty years from 1650 to 1670. At any rate, Newton was clearly an unbridled addict. It is this with which he was occupied 'about 6 weeks at spring and 6 at the fall when the fire in the elaboratory scarcely went out' at the very years when he was composing the Principia -and about this he told Humphrey Newton not a word. Moreover, he was almost entirely concerned, not in serious experiment, but in trying to read the riddle of tradition, to find meaning in cryptic verses, to imitate the alleged but largely imaginary experiments of the initiates of past centuries. Newton has left behind him a vast mass of records of these studies. I believe that the greater part are translations and copies made by him of existing books and manuscripts. But there are also extensive records of experiments. I have glanced through a great quantity of thisat least 100,000 words, I should say. It is utterly impossible to deny that it is wholly magical and wholly devoid of scientific value; and also impossible not to admit that Newton devoted years of work to it. Some time it might be interesting, but not useful, for some student better equipped and more idle than I to work out Newton's exact relationship to the tradition and MSS. of his time. In these mixed and extraordinary studies, with one foot in the Middle Ages and one foot treading a path for modern science, Newton spent the first phase of his life, the period of life in Trinity when he did all his real work. Now let me pass to the second phase. After the pUblication of the Principia there is a complete change in his habit and way of life. I believe that his friends, above aU Halifax, came to the conclusion that he must be rooted out of the life he was leading at Trinity which must soon lead to decay of mind and health. Broadly speaking, of his own motion or under persuasion, he abandons his studies. He takes up University business, represents the University in Parliament; his friends are busy trying to get a dignified and remunerative job for him -the Provostship of King's, the Mastership of Charterhouse, the Controllership of the Mint. Newton could not be Master of Trinity because he was a Unitarian and so not in Holy Orders. He was rejected as Provost of King's for the more
Nl!wto". till! MtlPf
283
prosaic reason that he was not an Etonian. Newton took this rejection very iI1 and prepared a long legalistic brief, which I possess, giving reasons why it was not unlawful for him to be accepted as Provost. But, as ill-luck had it, Newton's nomination for the Provostship came at the moment when King's had decided to fight against the right of Crown nomination, a struggle in which the ColJege was successfu1. Newton was well qualified for any of these offices. It must not be inferred from his introspection, his absent-mindedness, his secrecy and his solitude that he lacked aptitude for affairs when he chose to exercise it. There are many records to prove his very great capacity. Read, for example, his correspondence with Dr Covell, the Vice-Chancellor when, as the University's representative in Parliament, he had to deal with the delicate question of the oaths after the revolution of 1688. With Pepys and Lowndes he became one of the greatest and most efficient of our civil servants. He was a very successful investor of funds, surmounting the crisis of the South Sea Bubble, and died a rich man. He possessed in exceptional degree almost every kind of intellectual aptitUde-lawyer, historian, theologian, not less than mathematician, physicist, astronomer. And when the turn of his life came and he put his books of magic back into the box, it was easy for him to drop the seventeenth century behind him and to evolve into the eighteenth-century figure which is the traditional Newton. Nevertheless, the move on the part of his friends to change his 1ife came almost too late. In 1689 his mother, to whom he was deeply attached, died. Somewhere about his fiftieth birthday on Christmas Day 1692, he suffered what we should now term a severe nervous breakdown. Melancholia, sleeplessness, fears of persecution-he writes to Pepys and to Locke and no doubt to others letters which lead them to think that his mind is deranged. He lost, in his own words, the 'former consistency of his mind'. He never again concentrated after the old fashion or did any fresh work. The breakdown probably Jasted nearly two years, and from it emerged, slightly 'gaga', but still, no doubt, with one of the most powerful minds of England. the Sir Isaac Newton of tradition. In 1696 his friends were finally successful in digging him out of Cambridge, and for more than another twenty years he reigned in London as the most famous man of his age, of Europe, and-as his powers gradually waned and his affability increased-perhaps of all time, so it seemed to his contemporaries. He set up house with his niece Catharine Barton, who was beyond reasonable doubt the mistress of his old and loyal friend Charles Montague, Earl of Halifax and Chancellor of the Exchequer, who had been one of Newton's intimate friends when he was an undergraduate at Trinity. Catharine was reputed to be one of the most brilliant and charming
284
John Maynard KeyJUs
women in the London of Congreve, Swift and Pope. She is celebrated, not least for the broadness of her stories, in Swift's Journal to Stella. Newton puts on rather too much weight for his moderate height. 'When he rode in his coach one ann would be out of his coach on one side and the other on the other.' His pink face, beneath a mass of snow-white hair, which 'when his peruke was off was a venerable sight', is increasingly both benevolent and majestic. One night in Trinity after Hall he is knighted by Queen Anne. For nearly twenty-four years he reigns as President of the Royal Society. He becomes one of the principal sights of London for all visiting inteHectual foreigners, whom he entertains handsomely. He Jiked to have clever young men about him to edit new editions of the Principia-and sometimes merely plausible ones as in the case of Facio de Duillier. Magic was quite forgotten. He has become the Sage and Monarch of the Age of Reason. The Sir Isaac Newton of orthodox tradition-the eighteenth·century Sir Isaac, so remote from the child magician born in the first half of the seventeenth century-was being built up. Voltaire returning from his trip to London was able to report of Sir Isaac-' 'twas his peculiar feJicity, not only to be born in a country of liberty, but in an Age when all scholastic impertinences were banished from the World. Reason alone was cultivated and Mankind cou'd only be his Pupil, not his Enemy! Newton, whose secret heresies and scholastic superstitions it had been the study of a lifetime to conceal! But he never concentrated, never recovered 'the former consistency of his mind', 'He spoke very little in company.' 'He had something rather languid in his look and manner,' And he looked very seldom, I expect, into the chest where, when he left Cambridge, he had packed a1l the evidences of what had occupied and so absorbed his intense and flaming spirit in his rooms and his garden and his elaboratory between the Great Gate and Chapel. But he did not destroy them. They remained in the box to shock profoundly any eighteenth- or nineteenth-century prying eyes. They became the possession of Catharine Barton and then of her daughter, the Countess of Portsmouth. So Newton's chest. with many hundreds of thousands of words of his unpublished writings, came to contain the 'Portsmouth Papers'. In 1888 the mathematical portion was given to the University Library at Cambridge. They have been indexed, but they have never been edited. The rest, a very large collection, were dispersed in the auction room in 1936 by Catharine Barton's descendant. the present Lord Lymington. Disturbed by this impiety, I managed gradually to reassemble about half of them. including nearly the whole of the biographical portion, that is, the 'Conduitt Papers', in order to bring them to Cambridge which I hope
N~wloll. th~
Mall
285
they will never leave. The greater part of the rest were snatched out of my reach by a syndicate which hoped to sell them at a high price, probably in America, on the occasion of the recent tercentenary. As one broods over these queer collections, it seems easier to understand-with an understanding which is not, I hope, distorted in the other direction-this strange spirit, who was tempted by the Devil to believe at the time when within these walls he was solving so much, that he could reach all the secrets of God and Nature by the pure power of mindCopernicus and Faustus in one.
COMMENTARY ON
BISHOP BERKELEY and Infinitesimals NE of the great polemics of philosophy is a tract called The Analyst, issued in 1734 by the famous Irish metaphysician Bishop Berkeley. The object of the attack was the new calculus, especially the concept of the "fixed infinitesimal," as set forth by Isaac Newton in the Principia, in an appendix to the Opticks, and in other writings. Berkeley, though not a mathematician, made a number of extremely effective points in exposing the wea k and confused conceptual foundations of the subject. Since he was an acute thinker and a brilliant writer, his arguments provoked controversy among mathematicians and led to the clarification of central ideas underlying the new system of analysis. The Analyst has been described as marking "a turning point in the history of mathematical thought in Great Britain." 1 To appreciate the significance of Berkeley's tract the reader should refer to the discussions, in an earlier part of this volume, of the invention of the calculus (pp. 53-62) and of the work of Newton and Leibniz (pp. 140146 and 255-276). The Analyst, a book of 104 pages, is addressed "to an infidel mathematician." The gentleman so designated is generaJ]y supposed to have been Newton's friend, the astronomer Edmund Haney.2 Halley financed the pUbJication of the Principia and helped to prepare it for the press. It is said that he also persuaded a friend of Berkeley'S of the "inconceivability of the doctrines of Christianity"; the Bishop thereupon set out to demonstrate that the great innovation of fluxions was neither clearer nor more securely grounded than the tenets of theology.s He did not in this work "deny the utility of the new devices nor the validity of the results obtained. He merely asserted, with some show of justice, that mathematicians had given no legitimate arguments for their procedure. having used inductive instead of deductive reasoning." -l It was not difficult to make ridiculous the concept of "evanescent terms" (the "ghosts of departed quantities," Berkeley caned them); to expose the absurdity of the infinitesimal, a quantity greater than zero. yet so small that no multiple of it attains a measurable size; to suggest that the calculus, though based upon false and contradictory notions, yielded correct results by a "compensation of errors" ("by virtue of a twofold mistake. you arrive, though
O
1 Florian Cajori, A History 0/ the Conception 0/ Limits and Fluxions in Great Britain/From Newton to Woodhouse,' Chicago, 1919, p. 89. 2 For fUMher biographical details about Halley see p. 14] 8. a Florian Cajori, A History 0/ Mathematics; New York, second edition. 1919. p. 218. 4 Carl B. Boyer, The Concepts 0/ the Calculus; New York, 1939, p. 225.
286
Bishop Berkeley and In(fnitesillf(JII
287
not at science, yet at the truth"). But it is difficult to understand why Berkeley should have thought he could restore faith in religion by proving that mathematicians were often as muddleheaded as theologians. Whatever his purposes and whether or not he was sincere--both points have been disputed-Berkeley threw Ught on contradictions and thus promoted sounder definitions of crucial concepts. Among the writers who helped achieve this objective were the Englishman Benjamin Robins G and the able Scotch mathematician Colin Maclaurin.' The rigorous formulation of the calculus based upon the limit concept (and wholly banishing the infinitesimal) waited upon the labors of the French mathematician Augustin Cauchy in the first half of the nineteenth century. It was brought to "logical exactitude" in the second half of the century by the noted German analyst Karl Weierstrass, who "constructed a purely formal arithmetic basis for analysis, quite independent of all geometric intuition." l' The excerpt from The Analyst is worth reading for its literary quality, if for no other reason. Berkeley (1685-1753) was a writer and a philosopher of exceptional force, and a formidable controversialist. To be sure, his doctrines are perhaps less well remembered than the attempt of Samuel Iohnson to refute them by kicking at a stone. The bishop based his philosophy of immaterialism on the argument that the "absolute existence" of sensible things is a meaningless phrase, since the term "existence" when applied to sensible things necessarily implies a relation to perception.s To critics and scoffers he replied (in his A Defense of Free-Thinking in Mathematics, a response to an attack on The Analyst), "My aim is truth; my reasons I have given. Confute them if you can, but think not to overbear me with either authorities or harsh words." Berkeley was a manysided man, practical as well as prophetic. His full life included an i11-fated project to found a university in Bermuda, a two-and-a-half-year sojourn in America during which he made generous gifts to Harvard and Yale, an active career in Irish politics and a successfu1 one in the church, a passionate advocacy of tarwater as a universal medicament, high achievement as a critic and essayist, extensive phi10sophical writings, and at least one serious poem (it contains the famous line, "Westward the course of empire takes its way:' often ascribed to Rudyard Kipling). S Benjamin Robins. A Discourse Concerning the Nature and Certainty 0/ Sir Isaac Newton's Method 0/ Fluxions and 0/ Prime and Ultimate Ratios, London. 1735; also other papers in his Mathematical Tracts. London, 1761. IS CoJin Maclaurin, A Treatise 0/ Fluxions, 2 voJs., Edinburgh, 1742. '7 Carl Boyer, op. cit., p. 284. Boyer's careful study is recommended to anyone with an appetite for more details about the long and fascinating record of this branch of mathematical thought. 8 "It is interesting to notice that just as the arch-materialist Hobbes, being unable to conceive of lines without thickness, denied them to geometry, so also Berkeley, the extreme idealist, wishes to exclude from mathematics the 'inconceivable' idea of instantaneous velocity. This is in keeping with Berkeley"s early sensationalism which led him to think of geometry as an applied science dealing with finite magnitudes which are composed of indivisible 'minima sensibilia.''' Carl Boyer. op. cit., p. 227.
Lo,d Mansfield once gave the following advice to the newly-appointed governor 01 a West India bland. "There is no dif/iculty in deciding a caseonly hear both sides patiently, then consider what you think justice requires, and decide accordingly; but never give your reasons, for your judgement will probably be right, but your reasons will certainly be wrong." -LoRD MANSPIELD
How easy it is to call rogue and villain, and that wittily, but how hard to make a man appear a 1001, a blockhead or a knave, without using any of those opprobrious terms. . . . This is the mastery of that noble trade, which yet no master can teach to his apprentice. -JOHN DRYDEN (Discourse Concerning the Original and Progress of Satire)
10 The Analyst By BISHOP BERKELEY A DISCOURSB ADDRBSSED TO AN INFIDBL MATHEMATICIAN
THOUGH I am a stranger to your person, yet I am not, Sir, a stranger to the reputation you have acquired in that branch of learning which hath been your peculiar study; nor to the authority that you therefore assume in things foreign to your profession; nor to the abuse that you, and too many more of the like character, are known to make of such undue authority, to the misleading of unwary persons in matters of the highest concernment, and whereof your mathematical knowledge can by no means qualify you to be a competent judge . . . Whereas then it is supposed that you apprehend more distinctly, consider more closely, infer more justly, and conclude more accurately than other men, and that you are therefore less religious because more judicious, I shall claim the privilege of a Freethinker; and take the 1iberty to inquire into the object, principles, and method of demonstration admitted by the mathematicians of the present age, with the same freedom that you presume to treat the principles and mysteries of Religion; to the end that all men may see what right you have to lead, or what encouragement others have to follow you . . . The Method of Fluxions is the general key by help whereof the modem mathematicians unlock the secrets of Geometry, and consequently of Nature. And, as it is that which hath enabled them so remarkably to outgo the ancients in discovering theorems and solving problems, the exercise and application thereof is become the main if not the sole employment of all those who -in this age pass for profound geometers. But whether this method be clear or obscure, consistent or repugnant, demonstrative or 288
289
precarious, as I shall inquire with the utmost impartiality, so I submit my inquiry to your own judgment, and that of every candid reader.-Lines are supposed to be generated 1 by the motion of points, planes by the motion of lines, and solids by the motion of planes. And whereas quantities generated in equal times are greater or lesser according to the greater or lesser velocity wherewith they increase and are generated. a method hath been found to determine quantities from the velocities of their generating motions. And such velocities are called fluxions: and the quantities generated are called flowing quantities. These fluxions are said to be nearly as the increments of the flowing quantities, generated in the least equal particles of time; and to be accurately in the first proportion of the nascent, or in the last of the evanescent increments. Sometimes, instead of velocities, the momentaneous increments or decrements of undetermined flowing quantities are considered, under the appellation of moments. By moments we are not to understand finite particles. These are said not to be moments, but quantities generated from moments, which last are only the nascent principles of finite quantities. It is said that the minutest errors are not to be neglected in mathematics: that the fluxions are celerities, not proportional to the finite increments, though ever so small; but only to the moments or nascent increments, whereof the proportion alone, and not the magnitude, is considered. And of the aforesaid fluxions there be other fluxions, which fluxions of fluxions are called second fluxions. And the fluxions of these second fluxions are called third fluxions: and so on, fourth, fifth, sixth, etc., ad infinitum. Now, as our Sense is strained and puzzled with the perception of objects extremely minute, even so the Imagination, which faculty derives from sense, is very much strained and puzzled to frame clear ideas of the least particles of time, or the 1east increments generated therein: and much more so to comprehend the moments, or those increments of the flowing quantities in statu nascenti, in their very first origin or beginning to exist, before they become finite particles. And it seems still more difficult to conceive the abstracted velocities of such nascent imperfect entities. But the velocities of the velocities-the second, third, four, and fifth velocities, etc.--exceed, if I mistake not, all human understanding. The further the mind analyseth and pursueth these fugitive ideas the more it is lost and bewildered; the objects, at first fleeting and minute, soon vanishing out of sight. Certainly, in any sense, a second or third fluxion seems an obscure Mystery. The incipient celerity of an incipient celerity, the nascent augment of a nascent augment, i. e., of a thing which hath no magnitude-take it in what light you please, the clear conception of it will, if I mistake not, be found impossible; whether it be so or no I appeal to the trial of every thinking reader. I
In/rod. ad Quadraturam Curvarum.
290
And if a second fluxion be inconceivable, what are we to think of third, fourth, fifth fluxions, and 80 on without end? • . . All these points, I say, are supposed and believed by certain rigorous exactors of evidence in religion, men who pretend to believe no further than they can see. That men who have been conversant only about clear points should with difficulty admit obscure ones might not seem altogether unaccountable. But he who can digest a second or third fluxion, a second or third difference, need not, methinks, be squeamish about any point in divinity .•. Nothing is easier than to devise expressions or notations for fluxions and infinitesimals of the first, second, third, fourth, and subseq~e~~ orders, ....... proceeding in the same regular form without end or limit x. x. x. x. etc. or dx. ddx. dddx. ddddx. etc. These expressions, indeed, are clear and distinct, and the mind finds no difficulty in conceiving them to be continued beyond any assignable bounds. But if we remove the veil and look underneath, if, laying aside the expressions, we set ourselves attentively to consider the things themselves which are supposed to be expressed or marked thereby, we shall discover much emptiness, darkness, and confusion; nay, if I mistake not, direct impossibilities and contradictions. Whether this be the case or no, every thinking reader is entreated to examine and judge for himself . . . This is given for demonstration. 2 Suppose the product or rectangle AB increased by continual motion: and that the momentaneous increments of the sides A and B are a and b. When the sides A and B were deficient, or lesser by one half of their moments, the rectangle was
A - %a
X B -lhb, i. e., AB -lhaB - lhbA
+ ¥lab.
And as soon as the sides A and B are increased by the other two halves of their moments, the rectangle becomes A
+ %a X B + lhb or AB + lhaB + lhbA + 1f4ab.
From the latter rectangle subduct the former, and the remaining difference will be aB + bA. Therefore the increment of the rectangle generated by the entire increments a and b is aB + bA. Q. E. D. But it is plain that the direct and true method to obtain the moment or increment of the rectangle AB, is to take the sides as increased by their whole increments, and so multiply them together, A + a by B + b, the product whereof AB + aB + bA + ab is the augmented rectangle; whence, if we subduct A B the remainder aB + bA + ab will be the true increment of the rectangle, exceeding that which was obtained by the former illegitimate and indirect method by the quantity abo And this holds universally by the quantities a and b be what they will, big or little, finite or infinitesiII
Philosophiae Naturalis Principia Mathematica, Lib. II, tem. 2.
291
The A",,'y.rl
mal, increments, moments, or velocities. Nor will it avail to say that ab is a quantity exceedingly small: since we are told that in rebus mathematicis errores quam minim; non sunt contemnendi 8 • • • But, as there seems to have been some inward scruple or consciousness of defect in the foregoing demonstration, and as this finding the fluxion of a given power is a point of primary importance, it hath therefore been judged proper to demonstrate the same in a different manner, independent of the foregoing demonstration. But whether this method be more legiti· mate and conclusive than the former, I proceed now to examine; and in order thereto sball premise the following lemma:-"If, with a view to demonstrate any proposition, a certain point is supposed, by virtue of which certain other points are attained; and such supposed point be itself afterwards destroyed or rejected by a contrary supposition; in that case, all the other points attained thereby, and consequent thereupon, must also be destroyed and rejected, so as from thenceforward to be no more supposed or applied in the demonstration."" This is so plain as to need no proof. Now, the other method of obtaining a rule to find the fluxion of any power is as follows. Let the quantity x flow uniformly, and be it proposed to find the fluxion of x". In the same time that x by flowing becomes x + 0, the power X- becomes x + 01", i. e., by the method of infinite series x"
nn-n
+ nox.. - 1 +
2
and the increments o
oox"-2 + &c.,
and nox"-l
+
OOx"-2
+ &tc.
2
are one to another as 1 to nx"-l
nn-n
+
nn-n Ox"-2
2
+ &c.
:. Introd. ad Quadraturam Curvarum. 4 [Berkeley's lemma was rejected as invalid by James Jurin and some other mathematical writers. ne first mathematician to acknowledge openly the validity of Berkeley's lemma was Robert Woodhouse in his Pr;nciple~ 01 A.nalytical Calculation. Cambridge, 1803. p. XII. Instructive, in this connection, is a passage in A. N. Whitehead's Introduction to Mathematics, New York and London, 1911, p. 227. Whitehead does not mention Berkeley's lemma and probably did not have it in mind. Nevertheless, Whitehead advances an argument which is essentially the equivalent of Berkeley's, though expressed in different terms. When discussing the difference-quotient (x + h)- - x 2 - - - - - . Whitehead says: "In reading over the Newtonian method of stateh
ment, it is tempting to seek simplicity by saying that 2.r + b is 2.r, when b is zero. But thi~ will not do; Jor it thereby abolishes the interval/rom x to x + b, over which the average increase was calculated. The problem is, how to keep an interval of length b over which to caJculate the average increase, and at the same time to treat b as if it were zero. Newton did this by the conception of a limit, and we now proceed to give Weierstrass's explanation of its real meaning."] [The above note, and the two which follow are taken from Florian Cajori's edition of this excerpt in D. E. Smith, A. Source Book in Malhemotics, New York, 1929. ED.]
292
Let now the increments vanish, and their last proportion will be 1 to nr-- 1 • But it should seem that this reasoning is not fair or conclusive. For when it is said, let the increments vanish, i. e., let the increments be nothing, or let there be no increments, the former supposition that the increments were something, or that there were increments, is destroyed, and yet a consequence of that supposition, i. e., an expression got by virtue thereof, is retained. Which, by the foregoing lemma, is a false way of reasoning. Certainly when we suppose the increments to vanish, we must suppose their proportions, their expressions, and everything else derived from the supposition of their existence, to vanish with them . . . I have no controversy about your conclusions, but only about your logic and method: how you demonstrate? what objects you are conversant with, and whether you conceive them clearly? what principles you proceed upon; how sound they may be; and how you apply them? . . . Now, I observe. in the first place, that the conclusion comes out right, not because the rejected square of dy was infinitely small, but because this error was compensated by another contrary and equal error I') • • • The great author of the method of :fluxions felt this difficulty, and therefore he gave in to those nice abstractions and geometrical metaphysics without which he saw nothing could be done on the received principles: and what in the way of demonstration he hath done with them the reader will judge, It must, indeed, be acknowledged that he used :fluxions, like the scaffold of a building, as things to be laid aside or got rid of as soon as finite lines were found proportional to them. But then these finite exponents are found by the help of :fluxions. Whatever therefore is got by such exponents and proportions is to be ascribed to :fluxions: which must therefore be previously understood. And what are these :fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities . . .? You may possibly hope to evade the force of all that hath been said, and to screen false principles and inconsistent reasonings, by a general pretence that these objections and remarks are metaphysical, But this is a vain pretence. For the plain sense and truth of what is advanced in the foregoing remarks, I appeal to the understanding of every unprejudiced intel1igent reader . . . And, to the end that you may more clearly comprehend the force and design of the foregoing remarks, and pursue them still farther in your own meditations, I shall subjoin the following Queries:II [Berkeley explains that the calculus of Leibniz leads from false principles to correct results by a "Compensation of errors," The same explanation was advanced again later by Maclaurin, Lagrange, and, independently, by L. N. M. Carnot in his RI· flexions sur la metaphysique du calcul infinitesimal, 1797.]
Th~
A nalJls'
2.93
Query 1. Whether the object of geometry be not the proportions of assignable extensions? And whether there be any need of considering quantities either infinitely great or infinitely small? . • . Qu. 4. Whether men may properly be said to proceed in a scientific method, without clearly conceiving the object they are conversant about, the end proposed, and the method by which it is pursued? . . . Qu. 8. Whether the notions of absolute time, absolute place, and absoJute motion be not most abstracteJy metaphysical? Whether it be possible for us to measure, compute, or know them? • • • Qu. 16. Whether certain maxims do not pass current among analysts which are shocking to good sense? And whether the common assumption, that a finite quantity divided by nothing is infinite. be not of this number? I • • • Qu. 31. Where there are no increments, whether there can be any ratio of increments? Whether nothings can be considered as proportional to real quantities? Or whether to talk of their proportions be not to talk nonsense? Also in what sense we are to understand the proportion of a surface to a line, of an area to an ordinate? And whether species or numbers, though properly expressing quantities which are not homogeneous, may yet be said to express their proportion to each other? . . . Qu. 54. Whether the same things which are now done by infinites may not be done by finite quantities? And whether this would not be a great relief to the imaginations and understandings of mathematical men? . . . Qu. 63. Whether such mathematicians as cry out against mysteries have ever examined their own principles? Qu. 64. Whether mathematicians, who are so delicate in religious points, are strictly scrupulous in their own science? Whether they do not submit to authority, take things upon trust, and believe points inconceivable? Whether they have not their mysteries, and what is more, their repugnances and contradictions? . . . 8 [The earliest exclusion of division by zero in ordinary elementary algebra, on the ground of its being a procedure that is inadmissib1e according to reasoning based on the fundamental assumptions of this algebra, was made in 1828, by Martin Ohm, in his Yel'such eines vollkommen coruequenten Systems del' Mathematik, Vol. I, p. 112. In 1872, Robert Grassmann took the same position. But not until about 1881 was the necessity of excluding division by zero explained in elementary school books on algebra.]
COMMENTARY ON
GAUSS AUSS is often referred to as tbe Prince of Matbematics, a not very belpful designation. No one would dispute tbat Arcbimedest Gauss and Newton are in a special class among matbematicians. Eacb of the three is a tremendous, incredible figure towering above even tbe most eminent of bis contemporaries, and tbe attempt to assign rank among tbem is silly. The royal image is furtber confused by calling aritbmetic tbe Queen of Mathematics-for wbicb Gauss bimself was responsible--and mathematics the Queen of tbe Sciences. Where this leaves the Prince is not wholly clear. Gauss had a long, productive and interesting life, but a full-scale account of it bas not yet been written, not even by a German pedant. 1 Mathematicians apparently frigbten biograpbers, and Gauss is a formidable subject. He made so many outstanding contributions to mathematics, mathematical physics and other applied branches of tbe science that a book describing his work would need to be longer, so Bell conjectures, than a similar treatise on Newton. Considering tbe material that had to be covered, the chapter Bell devotes to Gauss in bis Men of M athematics is a skillful summary. It is a creative essay from which the average reader can gain a sound appreciation of Gauss's role in the development of mathematics and scientific thougbt. The chapter is reproduced below. Eric Temple Ben was born in 1883 in Aberdeen, Scotland. He studied at the University of London, came to tbe United States at tbe beginning of the century, got his Pb.D. in mathematics from Columbia University in 1912, taught at the University of Washington, and since 1926 has been professor of mathematics at the California Institute of Technology. Dr. Bell, now an American citizen, is a former president of the Mathematical Association of America and a former vice-president of the American Association for the Advancement of Science. He has won many honors for mathematical research and is a member of tbe National Academy of Science. I count at least twenty-one books by Bell, besides numerous matbematical articles. His books include several items of science fiction which are simply painful, a study of numerology, an able bistorical survey for experts, The Development of Mathematics, and his biograpbical collection. Men of Mathematics. Bell is a lively. stimulating writer, inoffensively crotchety and opin ionated, with a good sense of historical circumstancet a fine impatience with humbug. a sound grasp of the entire mathematical scene. and a gift for clear and orderly explanation.
G
1 An authoritative German source is Heinrich Mack, C. F. Gauss und Die Seinen, E. Applehaus und Camp., Brunswick, 1927.
294
As yet a child, nor yet a 1001 to lame, llisped in numbers, for the numbers came. --ALEXANDER POPE
11 The Prince of Mathematicians By ERIC TEMPLE BELL The further elaboration and development of systematic arithmetic, like nearly everything else which the mathematics of our [nineteenth] century has produced in the way of original scientific ideas, is knit to Gauss. --LEOPOLD KRONECKER
ARCHIMEDES, Newton, and Gauss, these three, are in a class by themselves among the great mathematicians, and it is not for ordinary mortals to attempt to range them in order of merit. All three started tidal waves in both pure and applied mathematics: Archimedes esteemed his pure mathematics more highly than its applications; Newton appears to have found the chief justification for his mathematical inventions in the scientific uses to which he put them, while Gauss declared that it was all one to him whether he worked on the pure or the applied side. Nevertheless Gauss crowned the higher arithmetic, in his day the least practical of mathematical studies, the Queen of all. The lineage of Gauss, Prince of Mathematicians, was anything but royal. The son of poor parents, he was born in a miserable cottage at Brunswick (Braunschweig), Germany, on April 30, 1777. His paternal grandfather was a poor peasant. In 1740 this grandfather settled in Brunswick, where he drudged out a meager existence as a gardener. The second of his three sons, Gerhard Diederich. born in 1744, became the father of Gauss. Beyond that unique honor Gerhard's life of hard labor as a gardener, canal tender, and bricklayer was without distinction of any kind. The picture we get of Gauss' father is that of an upright, scrupulously honest, uncouth man whose harshness to his sons sometimes bordered on brutality. His speech was rough and his hand heavy. Honesty and persistence gradually won him some measure of comfort, but his circumstances were never easy. It is not surprising that such a man did everything in his power to thwart his young son and prevent him from acquiring an education suited to his abilities. Had the father prevailed, the gifted boy would have followed one of the family trades, and it was only by a series of happy accidents that Gauss was saved from becoming a gardener or a bricklayer. As a child he was respectful and obedient, and although he 295
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Eric Ti!mpli! Bi!ll
never criticized his poor father in later life, he made it plain that he had never felt any real affection for him. Gerhard died in 1806. By that time the son he had done his best to discourage had accomplished immortal work. On his mother's side Gauss was indeed fortunate. Dorothea Benz's father was a stonecutter who died at the age of thirty of tuberculosis, the result of unsanitary working conditions in his trade, leaving two children, Dorothea and her younger brother Friederich. Here the line of descent of Gauss' genius becomes evident. Condemned by economic disabilities to the trade of weaving, Friederich was a highly intelligent, genial man whose keen and restless mind foraged for itself in fields far from his livelihood. In his trade Friederich quickly made a repu· tation as a weaver of the finest damasks, an art which he mastered wholly by himself. Finding a kindred mind in his sister's child, the clever uncle Friederich sharpened his wits on those of the young genius and did what he could to rouse the boy's quick logic by his own quizzical observations and somewhat mocking philosophy of life. Friederich knew what he was doing; Gauss at the time probably did not. But Gauss had a photographic memory which retained the impres· sions of his infancy and childhood unblurred to his dying day. Looking back as a grown man on what Friederich had done for him, and re. membering the prolific mind which a premature death had robbed of its chance of fruition, Gauss lamented that "a born genius was lost in him." Dorothea moved to Brunswick in 1769. At the age of thirty four (in 1776) she married Gauss" father. The following year her son was born. His full baptismal name was Johann Friederich Carl Gauss. In later life he signed his masterpieces simply Carl Friedrich Gauss. If a great genius was lost in Friederich Benz his name survives in that of his grateful nephew. Gauss' mother was a forthright woman of strong character, sharp intellect, and humorous good sense. Her son was her pride from the day of his birth to her own death at the age of ninety seven. When the "wonder child" of two, whose astounding intel1igence impressed all who watched his phenomenal development as something not of this earth, maintained and even surpassed the promise of his infancy as he grew to boyhood, Dorothea Gauss took her boy's part and defeated her obstinate husband in his campaign to keep his son as ignorant as himself. Dorothea hoped and expected great things of her son. That she may sometimes have doubted whether her dreams were to be realized is shown by her hesitant questioning of those in a position to judge her son's abilities. Thus, when Gauss was nineteen, she asked his mathematical friend Wolfgang Bolyai whether Gauss would ever amount to anything.
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When Bolyai exclaimed "The greatest mathematician in Europe!" she burst into tears. The last twenty two years of her life were spent in her son's house, and for the last four she was totally blind. Gauss himself cared little if anything for fame; his triumphs were his mother's life.! There was always the completest understanding between them, and Gauss repaid her courageous protection of his early years by giving her a serene old age. When she went blind he would allow no one but himself to wait on her, and he nursed her in her long last iUness. She died on April 19, 1839. Of the many accidents which might have robbed Archimedes and Newton of their mathematical peer, Gauss himself recalled one from his earliest childhood. A spring freshet had filled the canal which ran by the family cottage to overflowing. Playing near the water, Gauss was swept in and nearly drowned. But for the lucky chance that a laborer happened to be about his life would have ended then and there. In all the history of mathematics there is nothing approaching the precocity of Gauss as a child. It is not known when Archimedes first gave evidence of genius. Newton's earliest manifestations of the highest mathematical talent may well have passed unnoticed. Although it seems incredible, Gauss showed his caliber before he was three years old. One Saturday Gerhard Gauss was making out the weekly payroll for the laborers under his charge, unaware that his young son was following the proceedings with critical attention. Coming to the end of his long computations, Gerhard was startled to hear the little boy pipe up, "Father, the reckoning is wrong. it should be . . ." A check of the account showed that the figure named by Gauss was correct. Before this the boy had teased the pronunciations of the letters of the alphabet out of his parents and their friends and had taught himself to read. Nobody had shown him anything about arithmetic, although presumably he had picked up the meanings of the digits 1. 2, . . . along with the alphabet. In later life he loved to joke that he knew how to reckon before he could talk. A prodigious power for involved mental calculations remained with him all his life. Shortly after his seventh birthday Gauss entered his first school, a squalid relic of the Middle Ages run by a virile brute, one BUttner, whose idea of teaching the hundred or so boys in his charge was to thrash them into such a state of terrified stupidity that they forgot their own names. 1 The legend of Gauss' relations to his parents has still to be authenticated. Although, as will be seen later. the mol her stood by her son, the father opposed him; and, as was customary then (usually, also, now), in a German household, the father had the last word.-I allude later to legends from Jiving persons who had known members of the Gauss family, particularly in respect to Gauss' treatment of his sons. These allusions refer to first-hand evidence; but I do not vouch for them, as the people were very old.
Em:
T~".pk B~ll
More of the good old days for which sentimental reactionaries long. It was in this hell-hole that Gauss found his fortune. Nothing extraordinary happened during the first two years. Then. in his tenth year, Gauss was admitted to the class in arithmetic. As it was the beginning class none of the boys had ever heard of an arithmetical progression. It was easy then for the heroic BUttner to give out a Jong problem in addition whose answer he could find by a formula in a few seconds. The problem was of the following sort, 81297 + 81495 + 81693 + ... + 100899, where the step from one number to the next is the same all along (here 198). and a given number of terms (here 100) are to be added. It was the custom of the school for the boy who first got the answer to lay his slate on the table; the next laid his slate on top of the first. and so on. BUttner had barely finished stating the problem when Gauss flung his slate on the table: "There it ]jes," he said-"Ligget set "in his pleasant dialect. Then, for the ensuing hour, while the other boys toiled, he sat with his hands folded, favored now and then by a sarcastic glance from Buttner, who imagined the youngest pupil in the class was just another blockhead. At the end of the period BUttner looked over the slates. On Gauss' slate there appeared but a single number. To the end of his days Gauss loved to teU how the one number he had written was the correct answer and how all the others were wrong. Gauss had not been shown the trick for doing such problems rapidly. It is very ordinary once it is known, but for a boy of ten to find it instantaneously by himself is not so ordinary. This opened the door through which Gauss passed on to immortality. BUttner was so astonished at what the boy of ten had done without instruction that he promptly redeemed himself and to at Jeast one of his pupils became a humane teacher. Out of his own pocket he paid for the best textbook on arithmetic obtainable and presented it to Gauss. The boy ftashed through the book. "He is beyond me, It BUttner said; "I can teach him nothing more." By himself BUttner could probably not have done much for the young genius. But by a lucky chance the schoolmaster had an assistant, Johann Martin Bartels (1769-1836), a young man with a passion for mathematics, whose duty it was to heJp the beginners in writing and cut their quill pens for them. Between the assistant of seventeen and the pupil of ten there sprang up a warm friendship which lasted out Bartels' life. They studied together, helping one another over difficulties and amplifying the proofs in their common textbook on algebra and the rudiments of analysis. Out of this early work developed one of the dominating interests of Gauss' career. He quickly mastered the binomial theorem,
(l
n
n(n - 1)
1
lx2
+ x)" = 1 + -x +
n(n - 1)(n - 2)
+ ------XS + . . . , lx2x3
in which n is not necessarily a positive integer, but may be any number. If n is not a positive integer, the series on the right is infinite (nonterminating). and in order to state when this series is actually equal to (1 + x)ft, it is mandatory to investigate what restrictions must be imposed upon x and n in order that the infinite series shan converge to a definite, finite limit. Thus, if x = -2, and n = -1, we get the absurdity that (1-2)-1, which is (_1)-1 or 1/(-1), or finally -I, is equal to 1 + 2 + 22 + 21 +. . . and so on ad infinitum; that is, -1 is equal to the "infinite number" 1 + 2 + 4 + 8 +. . " which is nonsense. Before young Gauss asked himself whether infinite series converge and really do enable us to calculate the mathematical expressions (functions) they are used to represent, the older analysts had not seriously troubled themselves to explain the mysteries (and nonsense) arising from an uncritical use of infinite processes. Gauss' early encounter with the binomial theorem inspired him to some of his greatest work and he became the first of the "rigorists." A proof of the binomial theorem when n is not an integer greater than zero is even today beyond the range of an elementary textbook. Dissatisfied with what he and Bartels found in their book, Gauss made a proof. This initiated him to mathematical analysis. The very essence of analysis is the correct use of infinite processes. The work thus well begun was to change the whole aspect of mathematics. Newton, Leibniz, Euler, Lagrange, Laplace--all great analysts for their times--had practically no conception of what is now acceptable as a proof involving infinite processes. The first to see clearly that a "proof' which may Jead to absurdities like "minus 1 equals infinity" is no proof at all, was Gauss. Even if in some cases a formula gives consistent results, it has no place in mathematics until the precise conditions under which it win continue to yield consistency have been determined. The rigor which Gauss imposed on analysis gradual1y overshadowed the whole of mathematics, both in his own habits and in those of his contemporaries--Abel, Cauchy-and his successors--Weierstrass, Dedekind, and mathematics after Gauss became a tota11y different thing from the mathematics of Newton, Euler, and Lagrange. In the constructive sense Gauss was a revolutionist. Before his schooling was over the same critical spirit which left him dissatisfied with the binomial theorem had caused him to question the demonstrations of elementary geometry. At the age of twelve he Was already looking askance at the foundations of Euclidean geometry; by sixteen he had caught his tint glimpse of a geometry other than Euclid's. A year later he had begun a searching criticism of the proofs in the theory of numbers which had
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Eric T"mpl" Bell
satisfied his predecessors and had set himself the extraordinarily difficult task of filling up the gaps and completing what had been only half done. Arithmetic, the field of his earliest triumphs, became his favorite study and the locus of his masterpiece. To his sure feeling for what constitutes proof Gauss added a prolific mathematical inventiveness that has never been surpassed. The combination was unbeatable. Bartels did more for Gauss than to induct him into the mysteries of algebra. The young teacher was acquainted with some of the influential men of Brunswick. He now made it his business to interest these men in his find. They in turn, favorably impressed by the obvious genius of Gauss, brought him to the attention of Carl Wilhelm Ferdinand, Duke of Brunswick. The Duke received Gauss for the first time in 1791. Gauss was then fourteen. The boy's modesty and awkward shyness won the heart of the generous Duke. Gauss left with the assurance that his education wou1d be continued. The following year (February, 1792) Gauss matriculated at the Collegium Carolinum in Brunswick. The Duke paid the bil1s and he continued to pay them till Gauss' education was finished. Before entering the Caroline College at the age of fifteen, Gauss had made great headway in the classical languages by priva·te study and help from older friends, thus precipitating a crisis in his career. To his crassly practical father the study of ancient languages was the height of folly. Dorothea Gauss put up a fight for her boy, won, and the Duke subsidized a two-years' course at the Gymnasium. There Gauss' lightning mastery of the classics astonished teachers and students alike. Gauss himself was strongly attracted to philological studies, but fortunately for science he was presently to find a more compel1ing attraction in mathematics. On entering college Gauss was already master of the supple Latin in which many of his greatest works are written. It is an ever-to-beregretted calamity that even the example of Gauss was powerless against the tides of bigoted nationalism which swept over Europe after the French Revolution and the downfall of Napoleon. Instead of the easy Latin which sufficed for Euler and Gauss, and which any student can master in a few weeks, scientific workers must now acquire a reading knowledge of two or three languages in addition to their own. Gauss resisted as long as he could, but even he had to submit when his astronomical friends in Germany pressed him to write some of his astronomical works in German. Gauss studied at the Caroline College for three years, during which he mastered the more important works of Euler, Lagrange and, above all, Newton's Principia. The highest praise one great man can get is from another in his own class. Gauss never lowered the estimate which as a
The
Princ~
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of MGthemGticians
boy of seventeen he had formed of Newton. Others-Euler, Laplace, Lagrange, Legendre-appear in the flowing Latin of Gauss with the complimentary clariss;mus; Newton is summus. While still at the college Gauss had begun those researches in the higher arithmetic which were to make him immortal. His prodigious powers of calculation now came into play. Going directly to the numbers themselves he experimented with them, discovering by induction recondite general theorems whose proofs were to cost even him an effort. In this way he rediscovered "the gem of arithmetic," "theorema aureum," which Euler also had come upon inductively, which is known as the law of quadratic reciprocity, and which he was to be the first to prove. (Legendre's at· tempted proof slurs over a crux.) The whole investigation originated in a simple question which many beginners in arithmetic ask themselves: How many digits are there in the period of a repeating decimal? To get some light on the problem Gauss calculated the decimal representations of all the fractions lin for n 1 to 1000. He did not find the treasure he was seeking, but something infinitely greater-the law of quadratic reciprocity. As this is quite simply stated we shall describe it, introducing at the same time one of the revolutionary improvements in arithmetical nomenclature and notation which Gauss invented, that of congruence. All numbers in what follows are integers (common whole numbers). If the difference (a - b or b - a) of two numbers a, b is exactly divisible by the number m, we say that a, b are congruent with respect to the modulus m, or simply congruent modulo m, and we symbolize this by writing a == b (mod m). Thus 100 2 (mod 7), 3S 2 (mod
=
=
=
11 ).
The advantage of this scheme is that it recalls the way we write algebraic equations, traps the somewhat elusive notion of arithmetical divisibility in a compact notation, and suggests that we try to carry over to arithmetic (which is much harder than algebra) some of the manipulations that lead to interesting results in algebra. For example we can "add" equations, and we find that congruences also can be "added," provided the modulus is the same in all, to give other congruences. Let x denote an unknown number, rand m given numbers, of which r is not divisible by m. Is there a number x such that x2
=r (mod m)?
If there is, r is called a quadratic residue 01 m, if not, a quadratic non-
residue 01 m. If r is a quadratic residue of m, then it must be possible to find at least one x whose square when divided by m leaves the remainder r; if r is a quadratic non·residue of m, then there is no x whose square when divided
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T~mpl~
Bell
by m leaves the remainder r. These are immediate consequences of the preceding definitions. To illustrate: is 13 a quadratic residue of 11? If so, it must be possible to solve the congruence
x2 == 13 (mod 11) Trying 1, 2, 3, . . . , we find that x = 8, 25, 42, 59, . . . are solutions (8 2 64 3 X 11 + 13; 25 2 625 36 X 11 + 13; etc.,) so that 13 is a quadratic residue of 11. But there is no solution of x 2 5 (mod 11), so 5 is a quadratic non-residue of 17. It is now natural to ask what are the quadratic residues and non· residues of a given m? Namely, given m in x 2 r (mod m), what num· bers r can appear and what numbers r cannot appear as x runs through all the numbers 1, 2, 3, . . . 1 Without much difficulty it can be shown that it is sufficient to answer the question when both r and m are restricted to be primes. So we restate the problem: If p is a given prime, what primes q will make the congruence x 2 q (mod p) solvable? This is asking altogether too much in the present state of arithmetic. However, the situation is not utterly hopeless. There is a beautiful "reciprocity" between the pair of congruences
= =
=
=
=
=
=
x2
=q
(mod p), x 2 == p (mod q),
in which both of p, q are primes: both congruences are solvable, or both are unsolvable. unless both of p, q leave the remainder 3 when divided by 4, in which case one of the congruences is solvable and the other is not. This is the law of quadratic reciprocity. It was not easy to prove. In fact it baffled Euler and Legendre. Gauss gave the first proof at the age of nineteen. As this reciprocity is of fundamental importance in the higher arithmetic and in many advanced parts af algebra, Gauss turned it over and over in his mind for many years, seeking to find its taproot, until in all he had given six distinct proofs, one of which depends upon the straightedge and compass construction of regular polygons. A numerical illustration will illuminate the statement of the law. First, take p 5, q 13. Since both of 5, 13 leave the remainder 1 on division by 4, both of x 2 13 (mod 5), x 2 5 (mod 13) must be solvable, or neither is solvable. The latter is the case for this pair. For p 13, q 11, both of which leave the remainder 1 on division by 4, we get ,%2 11 (mod 13), x 2 13 (mod 17), and both, or neither again must be solvable. The former is the case here: the first congruence has the solutions x 2, 15, 28, . . . ; the second has the solutions x 8, 25, 42, . . . . There remains to be tested only the case when both of p, q leave the remainder 3 on division by 4. Take p 11, q 19. Then, according
=
=
=
=
=
=
=
=
=
=
=
=
303
=
=
to the law, precisely one of x 2 19 (mod 11), x 2 11 (mod 19) must be solvable. The first congruence has no solution; the second bas the solutions, 7, 26, 45, . • . . The mere discovery of such a law was a notable achievement. That it was first proved by a boy of nineteen will suggest to anyone who tries to prove it that Gauss was more than merely competent in mathematics. When Gauss left the Caroline College in October, 1795 at the age of eighteen to enter the University of Gottingen he was still undecided whether to follow mathematics or philology as his life work. He had already invented (when he was eighteen) the method of "least squares," which today is indispensable in geodetic surveying, in the reduction of observations and indeed in all work where the "most probable" value of anything that is measured is to be inferred from a large number of measurements. (The most probable value is furnished by making the sum of the squares of the "residuals"-rougbly, divergences from assumed exactness-a minimum.) Gauss shares this honor with Legendre who published the method independently in 1806. This work was the beginning of Gauss' interest in the theory of errors of observation. The Gaussian law of normal distribution of errors and its accompanying bell-shaped curve is familiar today to all who handle statistics, from high-minded intelligence testers to unscrupulous market manipulators. March 30, 1796, marks the turning point in Gauss' career. On that day, exactly a month before his twentieth year opened, Gauss definitely decided in favor of mathematics. The study of languages was to remain a lifelong hobby, but philology lost Gauss forever on that memorable day in March. The regular polygon of seventeen sides was the die whose lucky faU induced Gauss to cross his Rubicon. 2 The same day Gauss began to keep his It "Before leaving 'Fermat's numbers' 2" + 1 we shaH glance ahead to the last decade of the eighteenth century where these mysterious numbers were partly responsible for one of the two or three most important events in all the long history of mathematics. For some time a young man in his eighteenth year had been hesitating -according to the tradition-whether to devote his superb talents to mathematics or to philology. He was equally gifted in both. What decided him was a beautiful discovery in connection with a simple problem in elementary geometry familiar to every schoolboy. uA regular polygon of n sides has all its n sides equal and aU its n angles equal. The ancient Greeks early found out how to construct regular polygons of 3, 4, 5, 6, 8, 10 and 15 sides by the use of straightedge and compass alone, and it is an easy matter. with the same implements, to construct from a regular polygon having a given number of sides another regular polygon having twice that number of sides. The next step then would be to seek straightedge and compass constructions for regular polygons of 7, 9, 11, 13•... sides. Many sought, but failed to find, because such constructions are impossible. only they did not know it. After an interval of over 2200 years the young man hesitating between mathematics and philology took the next step--a long one-forward . •,As has been indicated it is sufficient to consider only polygons having an odd number of sides. The young man proved that a straightedge and compass construction of a regular polygon having an odd number of sides is possible when, and only when. that number is either a prime Fermat number (that is a prime of the form 2" + I), or
Eric T,mpll BIll
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scientific diary (Notizen-journa/). It is one of most precious documents in the history of mathematics. The first entry records his great discovery. The diary came into scientific circulation only in 1898 t forty three years after the death of Gauss when the Royal Society of Gottingen borrowed it from a grandson of Gauss for critical study. It consists of nineteen small octavo pages and contains 146 extremely brief statements of discoveries or results of calculations, the last of which is dated July 9, 1814. A facsimile reproduction was published in 1917 in the tenth volume (part 1) of Gauss' collected works, together with an exhaustive analysis of its contents by several expert editors. Not all of Gauss' discoveries in the prolific period from 1796 to 1814 by any means are noted. But many of those that are jotted down suffice to establish Gauss' priority in fields-elliptic functions, for instance--where some of his contemporaries refused to believe he had preceded them. (Recall that Gauss was born in 1777.) Things were buried for years or decades in this diary that would have made half a dozen great reputations had they been published promptly. Some were never made public during Gauss' lifetime, and he never c1aimed in anything he himself printed to have anticipated others when they caught up with him. But the record stands. He did anticipate some who doubted the word of his friends. These anticipations were not mere trivialities. Some of them became major fields of nineteenth century mathematics. A few of the entries indicate that the diary was a strictly private affair of its author's. Thus for Iuly 10, 1796, there is the entry t
EYPHKA! num
= A + A + A.
Translated, this echoes Archimedes' exultant "Eureka!" and states that every positive integer is the sum of three triangular numbers-such a number is one of the sequence 0, 1, 3, 6, 10, 15, ... where each (after 0) is of the form ¥.zn (n + 1), n being any positive integer. Another way of saying the same thing is that every number of the form 8n + 3 is a sum of three odd squares: 3 = 12 + 12 + 12; 11 = 12 + 12 + 32; 19 = 12 + 32 + 32 , etc. It is not easy to prove this from scratch. Less intelligible is the cryptic entry for October 11, 1796, "Vicimus GEGAN." What dragon had Gauss conquered this time? Or what giant had he overcome on April 8, 1799, when he boxes REV. GALEN up in is made up by multiplying together different Fermat primes. Thus the construction is possible for 3, 5; or 15 sides as the Greeks knew, but not for 7, 9, 11 or 13 sides, and is also possible for 17 or 257 or 65537 or-for what the next prime in the Fermat sequence 3, 5, 17, 257, 65537, ..• may be, if there is one-nobody yet (1936) knows-and the construction is also possible for 3 X 17, or 5 X 257 X 65537 sides, and so on. It was this discovery, announced on June 1, 1796, but made on March 30th, which induced the young man to choose mathematics instead of philology as his life work. His name was Gauss." I From the chapter on Fermat in Bell's Men of Mathematics.]
Tift Prinu of Mdlhtmdlicfdns
30S
a neat rectangle? Although the meaning of these is lost forever the re~ maining 144 are for the most part clear enough. One in particular is of the first importance: the entry for March 19, 1797. shows that Gauss had already discovered the double periodicity of certain elliptic functions. He was then not quite twenty. Again. a later entry shows that Gauss had recognized the double periodicity in the general case. This discovery of itself, had he published it. would have made him famous. But he never published it. Why did Gauss hold back the great things he discovered? This is easier to explain than his genius-if we accept his own simple statements, which will be reported presently. A more romantic version is the story told by W. W. R. BaH in his well-known history of mathematics. According to this, Gauss submitted his first masterpiece, the Disquisitiones Arlthmeticae, to the French Academy of Sciences, only to have it rejected with a sneer. The undeserved humiliation hurt Gauss so deeply that he resolved thenceforth to pubHsh only what anyone would admit was above criticism in both matter and form. There is nothing in this defamatory legend. It was disproved once for all in 1935, when the officers of the French Academy ascertained by an exhaustive search of the permanent records that the Disquisitiones was never even submitted to the Academy, much less rejected. Speaking for himself Gauss said that he undertook his scientific works only in response to the deepest promptings of his nature, and it was a wholly secondary consideration to him whether they were ever pubHshed for the instruction of others. Another statement which Gauss once made to a friend explains both his diary and his slowness in publication. He declared that such an overwhelming horde of new ideas stormed his mind before he was twenty that he could hardly control them and had time to record but a small fraction. The diary contains only the final brief statements of the outcome of elaborate investigations, some of which occupied him for weeks. Contemplating as a youth the close, unbreakable chains of synthetic proofs in which Archimedes and Newton had tamed their inspirations, Gauss resolved to foHow their great example and leave after him only finished works of art, severely perfect, to which nothing could be added and from which nothing could be taken away without disfiguring the whole. The work itse1f must stand forth, complete, simple, and convincing. with no trace remaining of the labor by which it had been achieved. A cathedral is not a cathedral, he said, tiJI the Jast scaffolding is down and out of sight. Working with this ideal before him, Gauss preferred to polish one masterpiece several times rather than to publish the broad outliftes of many as he might easily have done. His seal, a tree with but few fruits. bore the motto Pauca sed matura (Few, but ripe). The fruits of this striving after perfection were indeed ripe but not
Eric Templ4 Be"
306
always easily digestible. All traces of the steps by which the goal had been attained having been obliterated, it was not easy for the followers of Gauss to rediscover the road he had travelled. Consequently some of his works had to wait for highly gifted interpreters before mathematicians in general could understand them, see their significance for unsolved problems, and go ahead. His own contemporaries begged him to relax his frigid perfection so that mathematics might advance more rapidly, but Gauss never relaxed. Not till long after his death was it known how much of nineteenth-century mathematics Gauss had foreseen and anticipated before the year 1800, Had he divulged what he knew it is quite possible that mathematics would now be half a century or more ahead of where it is, Abel and Jacobi could have begun where Gauss left off, instead of expending much of their finest effort rediscovering things Gauss knew before they were born, and the creators of non-Euclidean geometry could have turned their genius to other things. Of himself Gauss said that he was "all mathematician," This does him an injustice unless it is remembered that ·'mathematician" in his day included also what would now be termed a mathematical physicist. Indeed his second motto 8 Thou. nature, art my goddess; to thy laws My services are bound . .. ,
truly sums up his life of devotion to mathematics and the physical sciences of his time. The "all mathematician" aspect of him is to be understood only in the sense that he did not scatter his magnificent endowment broadcast over all fields where he might have reaped abundantly, as he blamed Leibniz for doing, but cultivated his greatest gift to perfection. The three years (October, 1795-5eptember, 1798) at the University of Gottingen were the most prolific in Gauss' life. Owing to the generosity of the Duke Ferdinand the young man did not have to worry about finances. He lost himself in his work, making but few friends. One of these, Wolfgang Bolyai, "the rarest spirit I ever knew/' as Gauss described him, was to become a friend for life. The course of this friendship and its importance in the history of non-Euclidean geometry is too tong to be told here; Wolfgang's son Johann was to retrace practically the same path that Gauss had followed to the creation of a non-Euclidean geometry, in entire ignorance that his father's old friend had anticipated him. The ideas which had overwhelmed Gauss since his seventeenth year were now caught-partJy-and reduced to order. Since 1795 he had been meditating a great work on the theory of numbers. This now took definite shape, and by 1798 the Disquisi,;ones Arithmeticae (Arithmetical Researches) was practically completed. a Shakespeare's King Lear, Act I, Scene II,. 1-2, with the essential change of "laws"
for :>:>. c o~ 0 00. QlliIIgtlll" milll4. CCCCb:>:>:J..... CCCCI:>:>:>:).1 006 00 0, Dtdl'
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The preceding from the work of a Swiss scholar, Freigius, published in 158~ shows the forms of the Roman numerals recognized in his
time. The Roman numerals were commonly used in bookkeeping in E~ pean countries until the eighteenth century, although our modem numerals were generally known in Europe at least as early as the year 1000. In 1300 the use of our numerals was forbidden in the banks of cenain European cities, and in commercial documents. The argument was that they were more easily forged or falsified than Roman numerals; since, for example, the 0 could be changed into a 6 or a 9 by a single stroke. When books began to be printed, however, they made rapid progress, although the Roman numerals continued in use in some schools until about 1600, and in commercial bookkeeping for another century. One reason why the Roman numerals were preferred in bookkeeping was that it is easier to add and subtract with them than with our modem numerals. This may be seen in these two cases: Subtraction
Addition DCCLXXVII CCXVI DCCCCLXXXXIlI
(777) (216)
DCCLXXVII cc X VI
(993)
D
L X I
(777) (216) (561)
In such work as this it is unnecessary to learn any addition or subtraction facts; simply V and V make X, CC + CC cecc, and so on. The only advantage of our numerals in addition and subtraction is that ours are easier to write. As to multiplication and division. however, our numerals are far superior. The ancient Romans used to perform these operations by the use of counters. The Hebrews used their alphabet in writing numerals in the same way as the Greeks; that is, the first ten letters represented the first ten numbers, as shown below.
=
"tonl''''!l:l N 10
9
8 7 6 5
4 3
2
I
The letters and numerals here shown are arranged from right to Jeft, this being the way of writing used by the Hebrews. Another interesting set of alphabetic numerals was used by the Goths, a people first known in Poland and Germany, who later conquered a considerable part of Europe. These numerals are for the most part of Greek origin. They are shown in the illustration at the top of the next page. Just below to the left is a page from a Bible translated into Gothic by Bishop
UUiJu in the fourth century. On its left margin is • row of numerala. lbesl: numerals. mlaraed. are shown in the small picture to the riahL H you compare these with the alphabetic numerals in the pictUTe above you can easily read them. 'The fint is 300 + 40 + 3, or 143. The tecOOd is 144, and the third is 14S. We DOW come to the numerals that are used in Europe and the Amer-
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After they began to appear in books there were few changes of any significance, the chief ones being in the numerals for four and five. 2 Even at the present time the forms of our numerals frequently change in the attempt to find which kind is the most easily read. For example, consider the following specimens and decide which is the easiest for you to read: 1234567890
1234567890
2 See Smith and Karpinski, The Hindu-Arabic Numerals (Boston, 1911); Sir G. F. Hill. The Development oj Arabic Numerals in Europe (Oxford. 1915); D. E. Smith. History oj Mathematics (2 voL, Boston, 1923, 1925).
The names ror large numben have also chanaed rrom time to time. For example, the word million seems not to have been used before the thirteenlh century. II means "II hiS thousand," mill~ being the Latin for thousand, .and -on meanina (in lIalian) hi,. The word started in Italy, was taken over by France in the fifteenth century or earlier, and was thereafter used In England. Until the seventeenth century, people generally spoke of "a thousand thowand" rather than of "a million," and they do so today in certain parts of Europe. "Billion" is a relatively new word. It comes from the Italian. and is first found as bimlllicm, bi/ioni. and byllion. It originally meant a million mil· lion. and in England it is generally still SO understood, athough the Amer· ican use of the term to mean a thousand million has come to be generally understood of late )'9fS, largely because of the enormous numbers now in use in financial transactions which concern all countries. The larger numbers have names like: trillions, quadrillions, quintillions, and so on, but Ihc:se are seldom wed. Besides the written numerals which have bc:en described. finger numer· als were used during many centuries and by many peoples. The ancient Greeks and Romans wed them as did the Europeans of the Middle Ages: and the Asiatics in later times. Even today they arc: not infrequently used for barpining in the market places. Indeed, "counting on the fingers" and even multiplying and dividing by these means are known in certain countries, but not commonly in Western EUrope and the American cootinc:o.ts.
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The preceding illustration shows a few of the finger symbols as they appear in a manuscript written in Spain in the thirteenth century. The finger symbols below are from a book printed in the sixteenth century, three hundred years after the Spanish manuscript mentioned above. Each represents only part of the system. It may be observed, how-
Finaer symbola from an anthmeuc printed in Germany in 1532.
ever, that the 500 (d) is the same in both illustrations, as are also the 600 ( dc), 700 (dcc), 800 (dccc), and 900 (dcccc). There is a similar correspondence in the other numerals not given in these partial illustrations.
FROM NUMERALS TO COMPUTATION When people first began to use numbers they knew only one way to work with them; that was, ~o count. Little by little they found out how to add, subtract, and multiply; but this was slow work and in some countries special devices were iJlvented to make computation easier, especially in dealing with large nurttbers. The Romans used a counting table, or abacus, in which units, fives, tens, and so on were represented by beads which could be moved in grooves, as shown in this illustration. They caIled these beads calculi, which is the plural of calculus, or pebble. We see here the origin of our word "calculate." Since the syHable calc means lime, and marble is a kind of limestone, we see that a calculus was a small piece of marble, probably much like those used in playing marbles today. Sometimes, as in the Chinese abacus described below, the calculi slid along on rods. This kind of abacus is caBed a suan-pan, and it
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is used today in a1l parts of China. In the: one shown in the followin,8 illustration tbe beads are arranaed so as to make the number 21,091. each bead at the top represenbng five of the different orders (units, tens, and so 00) .
"""'" ......
The Japanae use • 5imil.r instrument known as the soroban. The Chinese and Japanese can add and subtract on the abacus. or countiDg board, much more rapidly than we can with pencil and paper. but they Clnnot multiply and divide 811 quickly 811 we can. In the sorohon here shown, beginnins in the twelfth column from the right, the number represented is 90,278. The other columns are not being used. In Russia there is stiU used a type of abacus known as the s'choty, and
Japanese abac:us.
not long ago a similar one was used in Turkey (the coulba) and also in Armenia (the choreb). Now, to come nearer home, you have often bought things over a "counter" in a store, but did you know that the "counter" teUs part of the story of addition and subtraction? Let us see what this story is. We have already seen that Roman numerals, It V, X, and so on, were in common use in Europe for nearly two thousand yean. It was difficult, however, to write large numbers with these numerals. For example, 98,549 might be written in this way: IxxxxviiiMDXLVIllI. There were other ways of writing this number, but they were equally clumsy. The merchants therefore invented an easier method of expressing large numbers. They drew lines on a board. with spaces between the lines, and used disks (smalJ circular pieces like checkers) to count with. On the lowest line there might be from one to four disks, each disk having the value of 1. 500's 100 to 400-----------50's 10 to 40 5's 1 to 4 A disk in the space above had the value of 5, and this combined with the disks on the line below could give 6, 7, 8. or 9. In this illustration 5 + 3 is represented. Larger numbers were handled in the same way on the upper lines and spaces. Sometimes the counters were slid along the rods .
•
• ••
1000's 500's lOO's 50's 10's 5's Ps
• ••• •
••••• •••• Now look at this figure. There are four disks on the thousand's line, none in the five hundred's space, and so on; that is, you have 4000 + no 500's + 100 + 50 + 40 + 5 + 3, or 4198.
F,om Num""", to Numuall Ilnd F,om NUm6"a's
10
Compulallon
459
You may care to see how such a counting board, with its counters, looked in one of the oldest English arithmetics about four hundred years ago. Maybe you would like to read the old English words as they were printed in the book. The problem is to add 2659 and 8342. The two num· ben are not written but are expressed by counters. The star is put on the board so that the eye may easily see where the thousands come, just as we write a comma in a number like 2,659 to show that the left-hand figure is 2 thousand. (This is usually omitted in a date like 1937, and in many other cases, especially where there are numbers of only four figures.)
pqe from Roben Record's GTOrmd 01 Artts, ponted nearly lour hundred yean 880.
Because these disks were used in counting they were called counters, and the board was sometimes called a counter board. When the European countries gave up using counters of this kind (quite generally four hundred years ago) they called the boards used in the shops and banks "counters," and this name has since been commonly used for the bench on which goods are shown in stores. The expression "counting house" is still used in some places to designate the room in which accounts are kept. One reason for using the counters was that paper was not generally known in Europe until about the eleventh century. Boards covered with a thin coat of wax had been used from the time of the Greeks and Romans, more than a thousand years before. On these it was possible to scratch numbers and words, erasing them by smoothing the wax with a spoon-shaped eraser, but it was very slow work. Slates were used in some
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l~kUlhi~1
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parts of Europe, but usable slate quarries were not common and therefore slates could not readily be used elsewhere. When blackboards were first made, chalk was not always easily found, and so written addition was not so common as addition by counters. When slates, blackboards, and paper all came into use, people added about as we do now. Since all careful computers "check" their work by adding from the bottom up and then from the top down so as to find any mistakes, pupils today add both ways, and there is no reason for teaching addition in only one direction. Subtraction was done on the counting boards in much the same way as addition. The numbers were represented by counters and were taken away as the problem required. The terms "carry" and "borrow" had more meaning than at present, because a counter was actually lifted up and carried to the next place. If one was borrowed from the next place, it was actually paid back. Today we learn the multiplication facts just as we learn to read words. If we need to use 7 X 8, we simply think "56," just as we think "cat" when we see the word CAT. Formerly, however, the "multiplication table" was first written down and then learned as a whole. On the following page are two of these tables from one of the oldest printed arithmetics, a German book of 1489 by Johann Widman. You may wish to see how they were arranged and how to find, in each table, the product of 8 X 9. You may also like to see how multiplication looked in 1478, and to find the meaning of the four cases on the right. It will be easier for you if
8III••1t. From an arithmetic printed in Treviso. Italy. in From Joh.ann Widman's Arithmetic: of 1489.
1478.
From Numlurs to Numertll, and From Numerals
10
461
CompullUion
you are told that in each one the problem is to multiply 934 by 314, the product being 293,276, the comma not being used at that time. Do you see how it was found? Below left another example of multiplication reproduced from a manuscript of the fifteenth century showing how the product of 456,789 X 987,654 was found. The product of these figures is here shown to be 451,149,483,006. The picture looks something like a grating, and so this
'J
3JJ t1~JJ(12
t444 t~
was sometimes caned the "grating method." You may care to see if you can multiply 345 by 678 using this method. You may also care to try mUltiplying a 4-figure number by (say) a 3-figure number. If you understand the above i1Iustration, it is not difficult to make such mUltiplication. Division was rarely used in ancient times except where the divisor was very small. Indeed, at the present time it is not often needed in comparisoh with multiplication, and it is far more rarely employed than addition and subtraction. On the abacus it was often done by subtraction; that is, to find how many times 37 is contained in 74, we see that 74 - 37 37, and 37 - 37 0, so that 37 is contained twice in 74. Above right another way of dividing that was the most common of any in the fifteenth century. It shows the division of 1728 by 144. As fast as the numbers had been used they were scratched out, and so this was often caned the "scratch method." The number 1728 was first written; then 144 was written below it. Since the first figure in the quotient is 1, the numbers 144 and 172 are scratched out, and 144 was again written be]ow. The remainders are written above. Divide 1728 by 144 as you would do it today, and compare your method with this. Our present method, often called "long division," began to be used in the fifteenth century. It first appeared in print in Calandri's arithmetic, pUblished in Florence, Italy, in 1491, a year before Columbus discovered America. The first example shown gives the division of 53A97 by 83, the
=
=
462
D"."d EUleNe Smith and Jek"thlel Glrub"T,
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result being 6444:-~. The second example is the division of % by 60, the result being lhoo. The other three are 137* -;- 12
= 111lh..
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= 160
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You may be interested in foHowing each step in these examples from this very old arithmetic. This interesting picture below is from a book that was well known about four hundred years ago. The book was first printed in 1503 and it shows two styles of computing at that time-the counters and the numerals. The number on the counting board at the right is 1241. The one at the left represents the attempted division of 1234 by 97, unsuccessful because decimal fractions such as we know were not yet invented. A method of computing by counters on a board seems to have been invented by the Indians of South America before the advent of the European discoverers, although our first evidence of the fact appeared in a Spanish work of 1590. In this a Jesuit priest, Joseph de Acosta, tells us: "In order to effect a very difficult computation for which an able calculator would require pen and ink . . . these Indians [of Peru] made use of their kernels [sus granos] of wheat. They place one here, three somewhere else and eight I know not where. They move one kernel here and three there and the fact is that they are able to complete their computation without making the smallest mistake. As a matter of fact, they are better
From Numbers to NUlfl4rals tlltd From
Num~ttlls
to CompUtation
463
at calculating what each one is due to payor give than we should be with pen and ink." 8 In South America, apparently long before the European conquerors arrived, the natives of Peru and other countries used knotted cords for keeping accounts. These were called quipus, and were used to record the results found on the counting table. How old this use of the quipu may be, together with some kind of abacus, we do not know. A manuscript written in Spanish by a Peruvian Indian, Don Felipe Huaman Poma de Ayala, between 1583 and 1613 has recently been discovered, and is now in the Royal Library at Copenhagen. It contains a large number of pen-and-ink sketches. One of these is here reproduced from a bootlet published in 1931.8 This portrays the accountant and treasurer (Cotador maior i tezorero) of the Inca, holding a quipu. In the lower left-hand comer is a counting board with counters and holes for pebbles or kernels. The use of knots to designate numbers is found in Gennany in connection with the number of measures of grain in a sack. Menninger shows the shapes of the knots and their numerical meaning as follows:
1 ~
s
= a., then is c> 0.; hence c belongs to the c1ass A2 and consequently also to the class U 2 , and since at the same time fJ> c, then fJ also belongs to the same class 11 2 , because every number in III is less than every number c in 11 2 , Hence every number fJ different from 0. belongs to the class 111 or to the class 112 according as fJ < 0. or fJ> 0.;
535
I'MtiolltJI Numbers
consequently (1 itself is either the greatest number in i!I1 1 or the least number in i!I12 , i. e., (1 is one and obviously the only number by which the separation of R into the cJasses i!I1 1 , iIl.t 2 is produced. Which was to be proved. OPERATIONS WITH REAL NUMBERS
To reduce any operation with two real numbers (1, /3 to operations with rational numbers, it is only necessary from the cuts (AI' A 2 ). (B 17 B 2 ) produced by the numbers (1 and /3 in the system R to define the cut (CJ, C 2 ) which is to correspond to the result of the operation, "Y' I confine myself here to the discussion of the simplest case, that of addition. If c is any rational number. we put it into the class C h provided there are two numbers one at in Al and one h t in Bl such that their sum al + hI > c; all other rational numbers shall be put into the class C2' This separation of all rational numbers into the two classes C h C 2 evidently forms a cut, since every number Cl in C1 is less than every number C2 in C 2 • If both (1 and /3 are rational, then every number Cl contained in C1 is < (1 + /3, because aJ < (1, hI < /3, and therefore al + hi < (1 + /3; further, if there were contained in C2 a number C2 < (1 + /3, hence (1 + /3 ::: C2 + p, where p is a positive rational number, then we should have C2
=
«(1
-lhp)
+ (/3 -lhp),
which contradicts the definition of the number c 2 , because (1 - '%p is a number in A hand /3 - '%p a number in B]; consequently every number C2 contained in C2 is (1 + /3. Therefore in this case the cut (C l' C 2) is produced by the sum (1 + /3. Thus we shall not violate the definition which holds in the arithmetic of rational numbers if in an cases we understand by the sum (1 + /3 of any two real numbers (1, /3 that number y by which the cut (CI , C2 ) is produced. Further, if only one of the two numbers (1, /3 is rational, e. g., (1, it is eas} to see that it makes no difference with the sum y = (1 + /3 whether the number (1 is put into the class A 1 or into the class A 2• Just as addition is defined, so can the other operations of the so-called elementary arithmetic be defined, viz., the formation of differences, products, quotients. powers, roots, logarithms. and in this way we arrive at real proofs of theorems (as, e. g., yI2 . 0 y"O), which to the best of my knowledge have never been established before. The excessive length that is to be feared in the definitions of the more complicated operations is partly inherent in the nature of the subject but can for the most part be avoided. Very useful in this connection is the notion of an interval, i. e., a system A of rational numbers possessing the fonowing characteristic property: if a and a' are numbers of the system A, then are aU rational numbers lying between a and a' contained in A. The system R of all rational numbers. and also the two classes of any cut are intervals. If there
=
Sl6
RIchard DtlMldrrd
exist a rational number al which is less and a rational number a2 which is greater than every number of the interval A, then A is caned a finite interval; there then exist infinitely many numbers in the same condition as a 1 and infinitely many in the same condition as a2; the whole domain R breaks up into three parts AI' A, A 2 and there enter two perfectly definite rational or irrational numbers 0.1' 0.2 which may be caned respectively the lower and upper (or the less and greater) limits of the interval; the lower limit at is determined by the cut for which the system A 1 forms the first class and the upper 0.2 by the cut for which the system A2 forms the second class. Of every rational or irrational number a lying between 0.1 and 0.2 it may be said that it lies within the interval A. If all numbers of an interval A are also numbers of an interval B, then A is called a portion of B. Still lengthier considerations seem to loom up when we attempt to adapt the numerous theorems of the arithmetic of rational numbers [as, e. g., the theorem (a + b)c = ac + bc] to any real numbers. This, however, is not the case. It is easy to see that it al1 reduces to showing that the arithmetic operations possess a certain continuity. What I mean by this statement may be expressed in the form of a general theorem: "If the number A is the result of an operation performed on the numbers a, f3, 'Y, • . • and A lies within the interval L, then intervals A. B, C, . . . can be taken within which lie the numbers a, f3, 'Y, . • . such that the result of the same operation in which the numbers a, f3, 'Y. • • • are replaced by arbitrary numbers of the intervals A, B, C, . . . is always a number lying within the interval L. u The forbidding clumsiness, however, which marks the statement of such a theorem convinces us that something must be brought in as an aid to expression; this is, in fact, attained in the most satisfactory way by introducing the ideas of variable magnitudes, functions, limiting values, and it would be best to base the definitions of even the simplest arithmetic operations upon these ideas. a matter which, however, cannot be carried further here.
Science, being human enquiry, can hear no answer except an answer couched somehow in human tones. Primitive man stood in the mountains and shouted against a cliO; the echo brought back his own voice, and he believed in a disembodied spirit. The scientist of to-day stands counting out loud in the face of the unknown. Numbers come back to him--and he believes in the Great Mathematician. -RICHARD HUGHES
9
Definition of Number By BERTRAND RUSSELL
[For a commentary on, and an autobiographical article by, Bertrand Russell, see pages 377-391.]
THE question "What is a number?" is one which has been often asked, but has only been correctly answered in our own time. The answer was given by Frege in 1884, in his Grundlagen der Arithmetik. 1 Although this book is quite short, not difficult, and of the very highest importance, it attracted almost no attention, and the definition of number which it contains remained practically unknown until it was rediscovered by the present author in 1901. In seeking a definition of number, the first thing to be clear about is what we may call the grammar of our inquiry. Many philosophers, when attempting to define number, are real1y setting to work to define plurality, which is quite a different thing. Number is what is characteristic of numbers, as man is what is characteristic of men. A plurality is not an instance of number, but of some particular number. A trio of men, for example, is an instance of the number 3, and the number 3 is an instance of number; but the trio is not an instance of number. This' point may seem elementary and scarcely worth mentioning; yet it has proved too subt1e for the philosophers, with few exceptions. A particular number is not identical with any collection of terms having that number: the number 3 is not identical with the trio consisting of Brown, Jones. and Robinson. The number 3 is something which all trios have in common, and which distinguishes them from other collections. A number is something that characterises certain collections. namely, those that have that number. Instead of speaking of a "col1ection," we shaH as a rule speak of a "class," or sometimes a "set." Other words used in mathematics for the same thing are "aggregate" and "manifold." We shaH have much to say later on about classes. For the present, we will say as little as possible. But there are some remarks that must be made immediately. A class or collection may be defined in two ways that at first sight J The same answer is given more fully and with more development in his G rundgesetze der Arithmetik, vol. i., 1893.
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seem quite distinct. We may enumerate its members, as when we say, "The collection I mean is Brown, Jones, and Robinson." Or we may mention a defining property, as when we speak of "mankind" or "the inhabitants of London." The definition which enumerates is called a definition by "extension," and the one which mentions a defining property is called a definition by "intension." Of these two kinds of definition, the one by intension is logically more fundamental. This is shown by two considerations: (1) that the extensional definition can always be reduced to an intensional one; (2) that the intensional one often cannot even theoretically be reduced to the extensional one. Each of these points needs a word of explanation. ( 1) Brown, Jones, and Robinson all of them possess a certain property which is possessed by nothing else in the whole universe, namely, the property of being eitber Brown or Jones or Robinson. This property can be used to give a definition by intension of the class consisting of Brown and Jones and Robinson. Consider such a formula as "x is Brown or x is Jones or x is Robinson." This formula will be true for just three x's. namely, Brown and Jones and Robinson. In this respect it resembles a cubic equation with its three roots. It may be taken as assigning a property common to the members of the class consisting of these tbree men, and peculiar to them. A similar treatment can obviously be applied to any otber class given in extension. (2) It is obvious that in practice we can often know a great deal about a class without being able to enumerate its members. No one man could actually enumerate all men, or even all the inhabitants of London, yet a great deal is known about each of these classes. This is enough to show that definition by extension is not necessary to knowledge about a class. But when we come to consider infinite classes, we find that enumeration is not even theoretically possible for beings who only live for a finite time. We cannot enumerate all tbe natural numbers: they are O. 1, 2, 3, and so on. At some point we must content ourselves with "and so on." We cannot enumerate all fractions or all irrational numbers, or al1 of any other infinite collection. Thus our knowledge in regard to all such collections can only be derived from a definition by intension. These remarks are relevant. when we are seeking the definition of number, in three different ways. In the first place, numbers themselves form an infinite collection, and cannot therefore be defined by enumeration. In the second place, the collections having a given number of terms themselves presumably form an infinite collection: it is to be presumed. for example, that there are an infinite collection of trios in the world, for if this were not the case the total number of tbings in the world would be finite, which, though possible, seems unlikely. In the third place,
Defi"ltio" of NtufIbIt,
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we wish to define "number" in such a way that infinite numbers may be possible; thus we must be able to speak of the number of terms in an infinite collection, and such a collection must be defined by intension, i.e. by a property common to all its members and peculiar to them. For many purposes, a class and a defining characteristic of it are practicaUy interchangeable. The vital difference between the two consists in the fact that there is only one class having a given set of members, whereas there are always many different characteristics by which a given class may be defined. Men may be defined as featherless bipeds, or as rational animals, or (more correctly) by the traits by which Swift delineates the Yahoos. It is this fact that a defining characteristic is never unique which makes classes useful; otherwise we cou1d be content with the properties common and pecuHar to their members. Anyone of these properties can be used in place of the class whenever uniqueness is not important. Returning now to the definition of number, it is clear that number is a way of bringing together certain collections, namely, those that have a given number of terms. We can suppose all couples in one bundle, all trios in another, and so on. In this way we obtain various bundles of collections, each bundle consisting of all the collections that have a certain number of terms. Each bundle is a class whose members are collections, i.e. classes; thus each is a class of classes. The bundle consisting of all couples, for example, is a class of classes: each couple is a class with two members, and the Whole bundle of couples is a class with an infinite number of members, each of which is a class of two members. How shall we decide whether two collections are to belong to the same bundle? The answer that suggests itself is: "Find out how many members each has, and put them in the same bundle if they have the same number of members." But this presupposes that we have defined numbers, and that we know how to discover how many terms a collection has. We are so used to the operation of counting that such a presupposition might easily pass unnoticed. In fact, however, counting, though familiar, is logically a very complex operation; moreover it is only available, as a means of discovering how many terms a collection has, when the collection is finite. Our definition of number must not assume in advance that all numbers are finite; and we cannot in any case, without a vicious circle, use counting to define numbers, because numbers are used in counting. We need, therefore, some other method of deciding when two collections have the same number of terms. In actual fact, it is simpler logically to find out whether two collections have the same number of terms than it is to define what that number is. An illustration will make this clear. If there were no polygamy or polyan-
540
dry anywhere in the world, it is clear that the number of husbands living at any moment would be exactly the same as the number of wives. We do not need a census to assure Us of this, nor do we need to know what is the actual number of husbands and of wives. We know the number must be the same in both collections, because each husband has one wife and each wife has one husband. The relation of husband and wife is what is called "one..one." , A relation is said to be "one..one" when, if x has the relation in question to y, no other term x' has the same relation to y, and x does not have the same relation to any term y' other than y. When only the first of these two conditions is fulfilled, the relation is called "one-many"; when only the second is fulfilled, it is caUed "many-one." It should be observed that the number 1 is not used in these definitions. In Christian countries, the relation of husband to wife is one-one; in Mahometan countries it is one-many; in Tibet it is many-one. The relation of father to son is one-many; that of son to father is many-one, but that of eldest son to father is one-one. If n is any number, the relation of n to n + 1 is one-one; so is the relation of n to 2n or to 3n. When we are considering only positive numbers, the relation of n to n 2 is one-one; but when negative numbers are admitted, it becomes two-one, since nand -n have the same square. These instances should suffice to make clear the notions of one-one, one-many, and many-one relations, which play a great part in the principles of mathematics, not only in relation to the definition of numbers, but in many other connections. Two classes are said to be "similar" when there is a one-one relation which correlates the terms of the one class each with one term of the other class, in the same mannel in which the relation of marriage correlates hu~bands with wives. A few preliminary definitions will help us to state this definition more precisely. The class of those terms that have a given relation to something or other is called the domain of that relation: thus fathers are the domain of the relation of father to child, husbands are the domain of the relation of husband to wife, wives are the domain of the relation of wife to husband, and husbands and wives together are the domain of the relation of marriage. The relation of wife to husband is called the converse of the relation of husband to wife. SimiJarly less is the converse of greater, later is the converse of earlier, and so on. GeneraJly. the converse of a given relation is that relation which ho1ds between y and x whenever the given relation holds between x and y. The converse domain of a relation is the domain of its converse: thus the class of wives is the converse domain of the relation of husband to wife. We may now state our definition of similarity as follows:One class is said to be "similar" to another when there is a one-one
Definition 01 Number
541
relation of which the one class is the domain, while the other is the converse domain. It is easy to prove (l) that every class is similar to itse1f, (2) that if a class a is similar to a class {3, then (3 is similar to a, (3) that if a is similar to fJ and fJ to 'Y. then a is similar to y. A relation is said to be reflexive when it possesses the first of these properties, symmetrical when it possesses the second, and transitive when it possesses the third. It is obvious that a relation which is symmetrical and transitive must be reflexive throughout its domain. Relations which possess these properties are an important kind, and it is worth while to note that similarity is one of this kind of relations. It is obvious to common sense that two finite classes have the same number of terms if they are similar, but not otherwise. The act of counting consists in establishing a one-one correlation between the set of objects counted and the natural numbers (excluding 0) that are used up in the process. Accordingly common sense concludes that there are as many objects in the set to be counted as there are numbers up to the last number used in the counting. And we also know that, so long as we confine ourselves to finite numbers. there are just n numbers from I up to n. Hence it follows that the last number used in counting a collection is the number of terms in the collection, provided the collection is finite. But this result, besides being only applicable to finite collections, depends upon and assumes the fact that two classes which are similar have the same number of terms; for what we do when we count (say) 10 objects is to show that the set of these objects is similar to the set of numbers 1 to 10. The notion of similarity is logically presupposed in the operation of counting, and is logically simpler though less familiar. In counting, it is necessary to take the objects counted in a certain order, as first, second, third, etc., but order is not of the essence of number: it is an irrelevant addition, an unnecessary complication from the logical point of view. The notion of similarity does not demand an order: for example, we saw that the number of husbands is the same as the number of wives, without having to establish an order of precedence among them. The notion of similarity also does not require that the classes which are similar should be finite. Take, for example, the natural numbers (excluding 0) on the one hand, and the fractions which have 1 for their numerator on the other hand: it is obvious that we can correlate 2 with lh. 3 with ¥.i, and so on, thus proving that the two classes are similar. We may thus use the notion of "similarity" to decide when two collections are to belong to the same bundle, in the sense in which we were asking this question earlier in this chapter. We want to make one bundle containing the class that has no members: this will be for the number O.
542
Then we want a bundle of all the classes that have one member: this will be for the number 1. Then, for the number 2, we want a bundle consisting of all couples; then one of all trios; and so on. Given any collection, we can define the bundle it is to belong to as being the class of all those collections that are "similar" to it. It is very easy to see that if (for example) a collection has three members, tbe class of all those collections that are similar to it will be the class of trios. And whatever number of terms a collection may have, those collections that are "similar" to it will have the same num&er of terms. We may take this as a definition of "having the same number of terms." It is obvious that it gives results conformable to usage so long as we confine. ourselves to finite collections. So far we bave not suggested anything in the slightest degree paradoxical. But when we come to the actual definition of numbers we cannot avoid what must at first sight seem a paradox, though this impression will soon wear off. We naturally think that the class of couples (for example) is something different from tbe number 2. But there is no doubt about the class of couples: it is indubitable and not difficult to define, whereas the number 2, in any other sense, is a metaphysical entity about which we can never feel sure tbat it exists or that we have tracked it down. It is therefore more prudent to content ourselves with the class of couples. whicb we are sure of, tban to bunt for a problematical number 2 wbich must always remain elusive. Accordingly we set up the following definition:The number 0/ a class is the class of all those classes that are similar to it. Thus the number ot a couple will be the class of all couples. In fact, the class of all couples will be tbe number 2, according to our definition. At tbe expense of a little oddity, tbis definition secures definiteness and indubitableness; and it is not difficult to prove that numbers so defined have all the properties that we expect numbers to have. We may now go on to define numbers in general as anyone of the bundles into whicb similarity collects classes. A number will be a set of classes sucb as that any two are similar to each other, and DOne outside the set are similar to any inside tbe set. In otber words, a number (in genera]) is any collection whicb is the number of one of its members; or, more simply still: A number is anything which is the number 0/ some class. Sucb a definition bas a verbal appearance of being circuJar. but in fact it is not. We define "the number of a given class" witbout using the notion of num~r in general; therefore we may define number in general in terms of "the number of a given class" without committing any 10gical error.
Definition of NUlftber
Definitions of this sort are in fact very common. The class of fathers, for example, would have to be defined by first defining what it is to be the father of somebody; then the class of fathers will be all those who are somebody's father. Similarly if we want to define square numbers (say), we must first define what we mean by saying that one number is the square of another. and then define square numbers as those that are the squares of other numbers. This kind of procedure is very common, and it is important to realise that it is legitimate and even often necessary.
PART IV
Mathematics of Space and Motion 1. The Exactness of Mathematical Laws by WILLIAM KINGDON CLIFFORD
2. The Postulates of the Science of Space by WILLIAM KINGDON CLIFFORD
3. On the Space Theory of Matter by WILLIAM KINGDON CLIFFORD
4. 5. 6. 7. 8.
The Seven Bridges of Konigsberg by LEONHARD EULER Topology by RJCHARD COURANT and HERBERT ROBBINS Diirer as a Mathematician by ERWIN PANOFSKY Projective Geometry by MORRIS KLINI On the Origin and Significance of Geometrical Axiom~
by HERMANN VON HELMHOLTZ 9. Symmetry by HERMANN WEYL
COMMENTARY ON
WILLIAM KINGDON CLIFFORD ILLIAM KINGDON CLIFFORD was born at Exeter (England) in 1845 and died of tuberculosis in Madeira at the age of thirtyfour. In a tragicaHy short life, with no more than fifteen working years, he enlarged scientific thought by a series of contributions as original as they were fertile, as far-reaching as they were lucid. Clifford was one of the distinguished mathematicians of his century and a philosopher of considerable power. His mathematical work had a prophetic quality; his philosophical expression was rational and humane; he had a gift of clarity, as Bertrand Russell has said, "that comes of profound and orderly understanding by virtue of which principles become luminous and deductions look easy." 1 Clifford's mathematical interests lay principally in geometry. It was there that he did important work, that his "mathematical intuition appeared at its best." The most exciting mathematical advance of the nineteenth century was in non-Euclidean geometry. For more than two thousand years the system perfected by Euclid had occupied a position of absolute authority. The rules he had laid down for geometric relations in space were assumed to be as inviolate as the multiplication table. Space obeyed Euclid, and Euclid obeyed space. This sovereignty was undermined by the investigations of several outstanding mathematicians, among them Gauss, Lobachevsky, Bolyai, Helmholtz, Riemann and Clifford. They decided, after analyzing its foundations, that while Euclid's geometry was unimpeachable as a system of ideal space and as an exercise in logic comparable to a game played with formal rules, its validity as regards actual space should be tested, not by mathematics, but by observation. Other geometries deduced from postulates differing from those framed by EucUd--especially his parallel postulate-were not only logically possible, but might turn out better suited to describe regions of space not normally accessible to our senses. The conclusion was disturbing but fruitful, for it extended enormously the horizons of mathematics. Clifford a11ied himself firmly with the view that applied geometry is an experimental science, a proper part of physics. It is from the success of this chaHenge to established beliefs that the new concepts of space, time, energy and matter underlying modem physics have evolved. Besides his original researches, CHfford translated Riemann's epoch-making inaugural dissertation, On the Hypotheses That Lie at the Bases of Geom-
W
1 Bertrand Russell, in a preface to Clifford's The Common Sense of the Exact Sciences, Newly Edited, with an Introduction, by James R. Newman; New York, 1946. p. V.
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WiIIi_ Ki ...do .. Clillo,d
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etry, and delivered a number of lectures to general audiences in which
various scientific and philosophical ideas are beautifully explained. The lectures illustrate Clifford's singular power for making hard concepts understandable. He spoke extemporaneously and with ease; he liked his audience and appreciated both their appetite for knowledge and their limitations; he mastered the subject and refused to conceal or gloss over its complexities; he approached it from an angle that made its features distinct. The selections which follow have a common theme. The first, taken from "On the Aims and Instruments of Scientific Thought," 2 considers the exactness of mathematical laws, the uniformity of nature, the assumption of exactness in geometric relations, the revolution in thought brought about by the non-Euclidean heretics. The second excerpt is from a group of lectures, "The Philosophy of the Pure Sciences." S It deals in some detail with the postulates of Euclid's geometry which determine its peculiar characteristics, and how changes in these postulates lead to other geometries, primarily Lobachevsky's and Riemann's. The third excerpt is an abstract (I have given all of it) of a remarkable contribution to the Cambridge Philosophical Society "On the Space Theory of Matter." This paper, written three years before Clifford's death when he was already gravely ill, is in a sense the crown of his reflections on geometry. It is hard to realize that it was published forty years before Einstein announced his theory of gravitation. 2 3
Given before the British Association, 1872. Given before the Royal Institution, 1873.
The cowboy, have a way 01 trussing up a ,teer or a pug1UlCioru bronco which fixe, the brute '0 that it can neither move nor think. Thi:t is the -ERIC TEMPLB BELL hog-tie, and it is what Euclid did to geometry. And lor mathematical science, he that doubts their certainty hath need of a dose 0/ hellebore. -JOSEPH GUNVlLL
1
The Exactness of Mathematical Laws By WILLIAM KINGDON CLIFFORD
WHEN a student is first introduced to those sciences which have come under the dominion of mathematics, a new and wonderful aspect of Nature bursts upon his view. He has been accustomed to regard things as essentially more or less vague. All the facts that he has hitherto known have been expressed qualitatively, with a little aDowance for error on either side. Things which are let go fall to the ground. A very observant man may know also that they fall faster as they go along. But our student is shown that, after falling for one second in a vacuum, a body is going at the rate of thirty-two feet per second, that after faDing for two seconds it is going twice as fast, after going two and a half seconds two and a half times as fast. If he makes the experiment, and finds a single inch per second too much or too little in the rate, one of two things must have happened: either the law of falling bodies has been wrongly stated, or the experiment is not accurate--there is some mistake. He finds reason to think that the latter is always the case; the more carefully he goes to work, the more of the error turns out to belong to the experiment. Again, he may know that water consists of two gases, oxygen and hydrogen, combined; but he now learns that two pints of steam at a temperature of 150 Centigrade will always make two pints of hydrogen and one pint of oxygen at the same temperature, all of them being pressed as much as the atmosphere is pressed. If he makes the experiment and gets rather more or less than a pint of oxygen, is the law disproved? No; the steam was impure, or there was some mistake. Myriads of anaJyses attest the law of combining volumes; the more carefully they are made, the more nearly they coincide with it. The aspects of the faces of a crystal are connected together by a geometrical law, by which, four of them being given, the rest can be found. The place of a planet at a given time is calculated by the law of gravitation; if it is half a second wrong, the fault is in the instrument, the observer, the clock, or the law; now t the 0
548
The EmclMU of Mathe1Jftlliad Lawl
more observations are made, the more of this fault is brought bome to the instrument, the observer, and the clock. It is no wonder, then, that our student, contemplating these and many like instances, should be led to say, 'I have been shortsighted; but I have now put on the spectacles of science which Nature had prepared for my eyes; I see that things have definite outlines, that the world is ruled by exact and rigid mathematical laws; lea, crV, (JEOf;, 'YE(Dp,ETPEif;.' It is our business to consider whether he is right in so concluding. Is the uniformity of Nature absolutely exact, or only more exact than our experiments? At this point we have to make a very important distinction. There are two ways in which a law may be inaccurate. The first way is exemplified by the law of Galileo that a body falling in vacuo acquires equal increase in velocity in equal times. No matter how many feet per second it is going, after an interval of a second it will be going thirty-two more feet per second. We now know that this rate of increase is not exactly the same at different heights, that it depends upon the distance of the body from the centre of the earth; so that the law is only approximate; instead of the increase of velocity being exactly equal in equal times, it itself increases very slowly as the body falls. We know also that this variation of the law from the truth is too small to be perceived by direct observation on the change of velocity. But suppose we have invented means for observing this, and have verified that the increase of velocity is inversely as the squared distance from the earth's centre. Still the law is not accurate; for the earth does not attract accurately towards her centre, and the direction of attraction is continually varying with the motion of the sea; the body will not even fall in a straight line. The sun and the planets, too, especially the moon, will produce deviations; yet the sum of all these errors will escape our new process of observation, by being a great deal smaller than the necessary errors of that observation. But when these again have been allowed for, there is still the inftuence of the stars. In this case, however, we only give up one exact law for another. It may still be held that if the effect of every particle of matter in the universe on the falling body were calculated according to the law of gravitation, the body would move exactly as this calculation required. And if it were objected that the body must be slightly magnetic or diamagnetic, while there are magnets not an infinite way off; that a very minute repulsion, even at sensible distances, accompanies the attraction; it might be replied that these phenomena are themselves subject to exact laws, and that when all the laws have been taken into account, the actual motion will exactly correspond with the calculated motion. I suppose there is hardly a physical student (unless he has specially considered the matter) who would not at once assent to the statement I have just made; that if we knew all about it, Nature would be found
550
WilliG". Kln,don ClIfJord
universally subject to exact numerical laws. But let us just consider for another moment what this means. The word 'exact' has a practical and a theoretical meaning. When a grocer weighs you out a certain quantity of sugar very carefully, and says it is exactly a pound, he means that the difference between the mass of the sugar and that of the pound weight he employs is too small to be detected by his scales. If a chemist had made a special investigation, wishing to be as accurate as he could, and told you this was exactly a pound of sugar, he would mean that the mass of the sugar differed from that of a certain standard piece of platinum by a quantity too small to be detected by his means of weighing, which are a thousandfold more accurate than the grocer's. But what would a mathematician mean, if he made the same statement? He would mean this. Suppose the mass of the standard pound to be represented by a length, say a foot, measured on a certain line; so that half a pound would be represented by six inches, and so on. And let the difference between the mass of the sugar and that of the standard pound be drawn upon the same line to the same scale. Then, if that difference were magnified an infinite number of times, it would still be invisible. This is the theoretical meaning of exactness; the practical meaning is only very close approximation; how close, depends upon the circumstances. The knowledge then of an exact law in the theoretical sense would be equivalent to an infinite observation. I do not say that such knowledge is impossible to man; but I do say that it would be absolutely different in kind from any knowledge that we now possess. I shall be told, no doubt, that we do possess a great deal of knowledge of this kind, in the form of geometry and mechanics; and that it is just the example of these sciences that has led men to look for exactness in other quarters. If this had been said to me in the last century, I should not have known what to reply. But it happens that about the beginning of the present century the foundations of geometry were criticised independently by two mathematicians, Lobatschewsky 1 and the immortal Gauss; 2 whose results have been extended and generalized more recently by Riemann 3 and Helmholtz." And the conclusion to which these investigations lead is that, although the assumptions which were very properly made by the ancient geometers are practically exact-that is to say, more exact than experiment can be--for such finite things as we have to deal with, and such portions of space as we can reach; yet the truth of them 1 Ge'ometrische Untersuchungen zur Theor;e der Parallellinien. Berlin. 1840. Translated by Houel. Gauthier-ViUars, 1866. 2 Letter to Schumacher, Nov. 28, 1846 (refers to 1792). 3 Ueber die Hypothesen welche der Geometrie zu Grunde liegen. Gottingen, Abhandl., 1866-7. Translated by Houel in Annal; di Matematica, Milan. vol. iii. 4 The Axioms oj Geometry, Academy, vol. i. p. 128 (a popular exposition). [This lecture. under the title "Origin and Significance of Geometrical Axioms," appears on pp. 647-668.-£D.J
Th~
E)(acI1lt!SS 01 Malhemlllical Laws
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for very much larger things. or very much smaller things. OT parts of space which are at present beyond our reach. is a matter to be decided by experiment, when its powers are considerably increased. I want to make as clear as possible the real state of tbis question at present, because it is often supposed to be a question of words or metaphysics, whereas it is a very distinct and simple question of fact. I am supposed to know then that the three angles of a rectilinear triangle are exactly equal to two right angles. Now suppose that three points are taken in space, distant from one another as far as the Sun is from a. Centauri, and that the shortest distances between these points are drawn so as to form a triangle. And suppose the angles of this triangle to be very accurately measured and added togetber; this can at present be done so accurately that the error shall certainly be less tban one minute, less therefore than the fivetbousandth part of a right angle. Then I do not know that this sum would differ at all from two right angles; but also I do not know tbat tbe difference would be less than ten degrees, or tbe ninth part of a right angle. a And I have reasons for not knowing. This example is exceedingly important as sbowing tbe connexion between exactness and universality. It is found tbat tbe deviation if it exists must be nearly proportional to tbe area of the triangle. So that the error in the case of a triangle wbose sides are a mile long would be obtained by dividing that in the case I have just been considering by four hundred quadrillions; the result must be a quantity inconceivably small, which no experiment could detect. But between this inconceivably small error and no error at a11, tbere is fixed an enormous gulf; tbe gulf between practical and theoretical exactness, and, wbat is even more important, the gulf between wbat is practically universal and wbat is theoretically universal. I say tbat a law is practically universal wbicb is more exact tban experiment for a11 cases that might be got at by such experiments as we can make. We assume this kind of universality, and we find that it pays us to assume it. But a law would be tbeoretical1y universal if it were true of all cases wbatever; and this is what we do not know of any law at a11. 15 Assuming that parallax observations prove the deviation less than half a second for a triangle whose vertex is at the star and base a diameter of the earth's orbit.
Nothing puwes me more than time and space: and yet nothing troublel me less, tIS I never think about them. -CHARLES LAMB (Letter to T. Manning)
2
The Postulates of the Science of Space By WILLIAM KINGDON CLIFFORD
IN my first lecture I said that, out of the pictures which are all that we can really see, we imagine a world of solid things; and that this world is constructed so as to fulfil a certain code of rules, some called axioms, and some called definitions, and some called postulates, and some assumed in the course of demonstration, but all laid down in one form or another in Euclid's Elements of Geometry. It is this code of rules that we have to consider to--day. I do not, however, propose to take this book that I have mentioned, and to examine one after another the rules as Euclid has laid them down or unconsciously assumed them; notwithstanding that many things might be said in favour of such a course. This book has been for nearly twenty-two centuries the encouragement and guide of that scientific thought which is one thing with the progress of man from a worse to a better state. The encouragement; for it contained a body of knowledge that was really known and could be relied on, and that moreover was growing in extent and application. For even at the time this book was written-shortly after the foundation of the Alexandrian Museum-Mathematic was no longer the merely ideal science of the Platonic school, but had started on her career of conquest over the whole world of Phenomena. The guide; for the aim of every scientific student of every subject was to bring his knowledge of that subject into a form as perfect as that which geometry had attained. Far up on the great mountain of Truth, which all the sciences hope to scale, the foremost of that sacred sisterhood was seen, beckoning to the rest to follow her. And hence she was called, in the dialect of the Pythagoreans, 'the purifier of the reasonable souL' Being thus in itse1f at once the inspiration and the aspiration of scientific thought, this Book of Euclid's has had a history as chequered as that of human progress itself. It embodied and systematized the truest results of the search after truth that was made by Greek, Egyptian, and Hindu. It presided for nearly eight centuries over that promise of light and right that was made by the civilized Aryan races on the Mediterranean shores; that promise, whose abeyance for nearly as long an interval is so full of warning and of sadness for ourselves. It went into exile along 552
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with the intellectual activity and the goodness of Europe. It was taught, and commented upon, and illustrated, and supplemented, by Arab and Nestorian, in the Universities of Bagdad and of Cordova. From these it was brought back into barbaric Europe by terrified students who dared tell hardly any other thing of what they had learned among the Saracens. Translated from Arabic into Latin, it passed into the schools of Europe, spun out with additional cases for every possible variation of the figure, and bristHng with words which had sounded to Greek ears like the babbling of birds in a hedge. At length the Greek text appeared and was translated; and, like other Greek authors, Euclid became an authority. There had not yet arisen in Europe 'that fruitful faculty,' as Mr. Winwood Reade calls it, 'with which kindred spirits contemplate each other's works; which not only takes, but gives; which produces from whatever it receives; which embraces to wrestle, and wrestles to embrace.' Yet it was coming; and though that criticism of first principles which Aristotle and Ptolemy and Galen underwent waited longer in Euclid's case than in theirs, it came for him at last. What VesaHus was to GaJen, what Copernicus was to Ptolemy, that was Lobatchewsky to Euclid. There is, indeed, a some· what instructive parallel between the last two cases. Copernicus and Lobatchewsky were both of Slavic origin. Each of them has brought about a revolution in scientific ideas so great that it can only be compared with that wrought by the other. And the reason of the transcendent importance of these two changes is that they are changes in the conception of the Cosmos. Before the time of Copernicus, men knew aU about the Universe. They could tell you in the schools, pat off by heart, all that it was, and what it had been, and what it would be. There was the flat ea~ with the blue vault of heaven resting on it like the dome of a cathedral, and the bright cold stars stuck into it; while the sun and planets moved in crystal spheres between. Or, among the better informed, the earth was a globe in the centre of the universe, heaven a sphere concentric with it; intermediate machinery as before. At any rate, if there was anything beyond heaven, it was a void space that needed no further description. The history of all this could be traced back to a certain definite time, when it began; behind that was a changeless eternity that needed no further history. Its future could be predicted in general terms as far forward as a certain epoch, about the precise determination of which there were, indeed, differences among the learned. But after that would come again a changeless eternity, which was fully accounted for and described. But in any case the Universe was a known thing. Now the enormous effect of the Copernican system, and of the astronomical dig. coveries that have fol1owed it, is that, in place of this knowledge of a little, which was called knowledge of the Universe, of Eternity and Immensity, we have now got knowledge of a great deal more; but we only
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call it the knowledge of Here and Now. We can tell a great deal about the solar system; but, after all, it is our house, and not the city. We can tell something about the star-system to which our sun belongs; but, after all, it is our star-system, and not the Universe. We are talking about Here with the consciousness of a There beyond it, which we may know some time, but do not at all know now. And though the nebular hypothesis tells us a great deal about the history of the solar system, and traces it back for a period compared with which the old measure of the duration of the Universe from beginning to end is not a second to a century, yet we do not call this the history of eternity. We may put it all together and call it Now, with the consciousness of a Then before it, in which things were happening that may have left records; but we have not yet read them. lbis, then, was the change effected by Copernicus in the idea of the Universe. But there was left another to be made. For the laws of space and motion, that we are presently going to examine, implied an infinite space and an infinite duration, about whose properties as space and time everything was accurately known. The very constitution of those parts of it which are at an infinite distance from us, 'geometry upon the plane at infinity,' is just as well known, if the Euclidean assumptions are true, as the geometry of any portion of this room. In this infinite and thoroughly well-known space the Universe is situated during at least some portion of an infinite and thoroughly well-known time. So that here we have real knowledge of something at least that concerns the Cosmos; something that is true throughout the Immensities and the Eternities. That something Lobatchewsky and his successors have taken away. The geometer of to-day knows nothing about the nature of actually existing space at an infinite distance; he knows nothing about the properties of this present space in a past or a future eternity. He knows, indeed, that the laws assumed by Euclid are true with an accuracy that no direct experiment can approach, not only in this place where we are, but in places at a distance from us that no astronomer has conceived; but he knows this as of Here and Now; beyond his range is a There and Then of which he knows nothing at present, but may ultimately come to know more. So, you see, there is a real parallel between the work of Copernicus and his successors on the one hand, and the work of Lobatchewsky and his successors on the other. In both of these the knowledge of Immensity and Eternity is replaced by knowledge of Here and Now. And in virtue of these two revolutions the idea of the Universe, the Macrocosm. the An, as subject of human knowledge, and therefore of human interest, has fallen to pieces. It will now, I think, be clear to you why it will not do to take for our present consideration the postulates of geometry as Euclid has laid them down. While they were all certainly true, there might be substituted for
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them some other group of equivalent propositions; and the choice of the particular set of statements that should be used as the groundwork of the science was to a certain extent arbitrary. being only guided by convenience of exposition. But from the moment that the actual truth of these assumptions becomes doubtful, they fall of themselves into a necessary order and classification; for we then begin to see which of them may be true independently of the others. And for the purpose of criticizing the evidence for them, it is essential that this natural order should be taken; for I think you will see presently that any other order would bring hopeless confusion into the discussion. Space is divided into parts in many ways. If we consider any material thing, space is at once divided into the part where that thing is and the part where it is not. The water in this glass, for example, makes a distinction between the space where it is and the space where it is not. Now, in order to get from one of these to the other you must cross the sur/ace of the water; this surface is the boundary of the space where the water is which separates it from the space where it is not. Every thing, considered as occupying a portion of space, has a surface which separates the space where it is from the space where it is not. But, again, a surface may be divided into parts in various ways. Part of the surface of this water is against the air, and part is against the glass. If you travel over the surface from one of these parts to the other, you have to cross the line which divides them; it is this circular edge where water, air, and glass meet. Every part of a surface is separated from the other parts by a line which bounds it. But now suppose, further, that this glass had been so constructed that the part towards you was blue and the part towards me was white, as it is now. Then this line, dividing two parts of the surface of the water, would itseJf be divided into two parts; there would be a part where it was against the blue glass, and a part where it was against the white glass. If you travel in thought along that line, so as to get from one of these two parts to the other. you have to cross a point which separates them, and is the boundary between them. Every part of a line is separated from the other parts by points which bound it. So we may say altogetherThe boundary of a so1id (Le., of a part of space) is a surface. The boundary of a part of a surface is a line. The boundaries of a part of a line are points. And we are only settling the meanings in which words are to be used. But here we may make an observation which is true of al1 space that we are acquainted with: it is that the process ends here. There are no parts of a point which are separated from one another by the next link in the series. This is also indicated by the reverse process. For I shall now suppose this point-the last thing that we got to-to
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move round the tumbler so as to trace out the line, or edge, where air, water, and glass meet. In this way I get a series of points, one after another; a series of such a nature that, starting from anyone of them, only two changes are possible that will keep it within the series: it must go forwards or it must go backwards, and each of these is perfectly definite. The line may then be regarded as an aggregate of points. Now let us imagine. further, a change to take place in this line, which is nearly a circle. Let us suppose it to contract towards the centre of the circle. until it becomes indefinitely small, and disappears. In so doing it will trace out the upper surface of the water. the part of the surface where it is in contact with the air. In this way we shall get a series of circles one after another-a series of such a nature that, starting from anyone of them, only two changes are possible that will keep it within the series: it must expand or it must contract. This series, therefore, of circles. is just similar to the series of points that make one circle; and just as the line is regarded as an aggregate of points, so we may regard this surface as an aggregate of lines. But this surface is also in another sense an aggregate of points. in being an aggregate of aggregates of points. But, starting from a point in the surface, more than two changes are possible that will keep it within the surface, for it may move in any direction. The surface, then, is an aggregate of points of a different kind from the line. We speak of the line as a point-aggregate of one dimension, because, starting from one point, there are only two possible directions of change; so that the line can be traced out in one motion. In the same way, a surface is a lineaggregate of one dimension, because it can be traced out by one motion of the line; but it is a point-aggregate of two dimensions, because, in order to build it up of points, we have first to aggregate points into a line, and then lines into a surface. It requires two motions of a point to trace it out. Lastly, let us suppose this upper surface of the water to move downwards, remaining always horizontal till it becomes the under surface. In so doing it will trace out the part of space occupied by the water. We shall thus get a series of surfaces one after another. precisely analogous to the series of points which make a line, and the series of lines which make a surface. The piece of solid space is an aggregate of surfaces. and an aggregate of the same kind as the line is of points; it is a surfaceaggregate of one dimension. But at the same time it is a line-aggregate of two dimensions. and a point-aggregate of three dimensions. For if you consider a particular line which has gone to make this solid, a circle partly contracted and part of the way down. there are more than two opposite changes which it can undergo. For it can ascend or descend. or expand or contract, or do both together in any proportion. It has just as great a variety of changes as a point in a surface. And the piece of space is called
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a point-aggregate of three dimensions, because it takes three distinct motions to get it from a point. We must first aggregate points into a line, then lines into a surface, then surfaces into a solid. At this step it is clear, again. that the process must stop in all the space we know of. For it is not possible to move that piece of space in such a way as to change every point in it. When we moved our Jine or our surface, the new line or surface contained no point whatever that was in the old one; we started with one aggregate of points, and by moving it we got an entirely new aggregate, all the points of which were new. But this cannot be done with the soJid; so that the process is at an end. We arrive, then, at the result that space is oj three dimensions. Is this, then, one of the postulates of the science of space? No; it is not. The science of space, as we have it, deals with relations of distance existing in a certain space of three dimensions, but it does not at all require us to assume that no relations of distance are possible in aggregates of more than three dimensions. The fact that there are only three dimensions does regu1ate the number of books that we write, and the parts of the subject that we study: but it is not itself a postulate of the science. We investigate a certain space of three dimensions. on the hypothesis that it has certain elementary properties; and it is the assumptions of these elementary properties that are the real postulates of the science of space. To these I now psoceed. The first of them is concerned with points, and with the relation of space to them. We spoke of a Jine as an aggregate of points. Now there are two kinds of aggregates, which are called respectively continuous and discrete. If you consider this line, the boundary of part of the surface of the water, you wiD find yourself believing that between any two points of it you can put more points of division. and between any two of these more again, and so on; and you do not believe there can be any end to the process. We may express that by saying you beJieve that between any two points of the line there is an infinite number of other points. But now here is an aggregate of marbles, which, regarded as an aggregate, has many characters of resemblance with the aggregate of points. It is a series of marbles, one after anotner; and if we take into account the relations of nextness or contiguity which they possess, then there are only two changes possibJe from one of them as we travel along the series: we must go to the next in front, or to the next behind. But yet it is not true that between any two of them here is an infinite number of other marbles; between these two, for example, there are only three. There, then, is a distinction at once between the two kinds of aggregates. But there is another, which was pointed out by Aristotle in his Physics and made the basis of a definition of continuity. I have here a row of two different kinds of marbles, some white and some black. This aggregate is divided
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into two parts, as we formerly supposed the line to be. In the case of the line the boundary between the two parts is a point which is the element of which the line is an aggregate. In this case before us, a marble is the element; but here we cannot say that the boundary between the two parts is a marble. The boundary of the white parts is a white marble. and the boundary of the black parts is a black marble; these two adjacent parts have different boundaries. Similarly, if instead of arranging my marbles in a series, I spread them out on a surface. I may have this aggregate divided into two portions-a white portion and a black portion; but the boundary of the white portion is a row of white marbles, and the boundary of the black portion is a row of black marbles. And lastly, if I made a heap of white marbles, and put black marbles on the top of them. I should ha.e a discrete aggregate of three dimensions divided into two parts: the boundary of the white part would be a layer of white marbles. and the boundary of the black part would be a layer of black marbles. In all these cases of discrete aggregates. when they are divided into two parts, the two adjacent parts have different boundaries. But if you come to consider an aggregate that you believe to be continuous, you will see that you think of two adjacent parts as having the same boundary. What is the boundary between water and air here'! Is it water'! No; for there would stm have to be a boundary to divide that water from the air. For the same reason it cannot be air. I do not want you at present to think of the actual physical facts by the aid of any molecular theories; I want you only to think of what appears to be, in order to understand clearly a conception that we all have. Suppose the things actually in contact. If. however much we magnified them, they still appeared to be thoroughly homogeneous, the water filling up a certain space. the air an adjacent space; if this held good indefinitely through all degrees of conceivable magnifying, then we could not say that the surface of the water was a layer of water and the surface of air a layer of air; we should have to say that the same surface was the surface of both of them. and was itself neither one nor the other-that this surface occupied no space at all. Accordingly, Aristotle defined the continuous as that of which two adjacent parts have the same boundary; and the discontinuous or discrete as that of which two adjacent parts have direct boundaries. 1 Now the first postulate of the science of space is that space is a continuous aggregate of points, and not a discrete aggregate. And this postulate-which I shall call the postulate of continuity-is really involved in I Phys. Ausc. V. 3, p. 227, ed. Bekker. To ~ O'lIHX~! '0''''' pl" S7r~P Ex0IJ.06" "", Xhw 5' ~l"II' O'lI"~X~! STII" TII~TO 'YE"."TII' Kill b TO EKIITEpOll 7rEpC1S oIs cl7rTO"TlIl, Kill wO'7r~P O''IIJ.IIlJtfl TOVJtOIJ.II O'lIvEX'1TIIl. TovTO 5' OVX ol6" T'f 5110'" 6"TOl" ~lPCIl TO'" 100XclTOl". A little further on he makes the important remark that on the hypothesis of continuity a line is not made up of points in the same way that a whole is made up of parts, VI. I, p. 231. •A5liJtIlTO" I d5llllpETWJt dJICIl Tl O'lI"~XEf, oto" 'YpC1IJ.IJ.y,,, EK O'T''YlJ.o,,,.
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those three of the six 2 postulates of Euclid for which Robert Simson has retained the name of postulate. You will see, on a little reflection, that a discrete aggregate of points could not be so arranged that any two of them should be relatively situated to one another in exactly the same manner, so that any two points might be joined by a straight line which should always bear the same definite relation to them. And the same difficUlty occurs in regard to the other two postulates. But perhaps the most cOllclusive way of showing that this postulate is really assumed by Euclid is. to adduce the proposition he proves, that every finite straight line may ~ bisected. Now this could not be the case if it consisted of an odd number of separate points. As the first of the postulates of the science of space, then, we must reckon this postulate of Continuity; accordiDR to which two adjacent portions of space, or of a surface, or of a line, have the same boundary, viz.-a surface, a line, or a point; and between every two points on a line there is an infinite number of intermediate points. The next postulate is that of Elementary Flatness. You know that if you get hold of a small piece of a very large circle, it seems to you nearly straight. So, if you were to take any curved line, and magnify it very much, confining your attention to a small piece of it, that piece would seem straighter to you than the curve did before it was magnified. At least, you can easily conceive a curve possessing this property, that the more you magnify it, the straighter it gets. Such a curve would possess the property of elementary ftatness. In the same way, if you perceive a portion of the surface of a very large sphere, such as the earth, it appears to you to be flat. If, then, you take a sphere of say a foot diameter, and magnify it mol'e' and more, you will find that the more you magnify it the Ratter it gets. And you may easily suppose that this process would go on indefinitely; that the curvature would become less and less the more the surface was magnified. Any curved surface which is such that the more you magnify it the flatter it gets, is said to possess the property of elementary flatness. But if every succeeding power of our imaginary microscope disclosed new wrinkles and inequaHties without end, then we should say that the surface did not possess the property of elementary ftatness. But how am I to explain how sond space can have this property of elementary flatness? Shalt I leave it as a mere analogy, and say that it is the same kind of property as this of the curve and surface, only in three dimensions instead of one or two? I think I can get a 1iute nearer to it than that; at aU events I will try. If we start to go out from a point on a surface, there is a certain choice of directions in which we may go. These directions make certain angles with one another. We may suppose a certain direction to start with, and 2 See De Morgan, in Smith's Diet. of Biography and Mythology, Art. Euclid; and in the English Cyclopzdia, Art. Axiom.
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then gradually alter that by turning it round the point: we find thus a single series of directions in which we may start from the point. According to our first postulate, it is a continuous series of directions. Now when I speak of a direction from the point, I mean a direction of starting; I say nothing about the subsequent path. Two different paths may have the same direction at starting; in this case they will touch at the point; and there is an obvious difference between two paths which touch and two paths which meet and form an angle. Here, then, is an aggregate of directions, and they can be changed ioto one another. Moreover, the changes by which they pass into one another have magnitude, they constitute distancerelations; and the amount of change necessary to tum one of them into another is called the angle between them. It is involved in this postulate that we are considering, that angles can be compared in respect of magnitude. But this is not all. If we go on changing a direction of start, it will, after a certain amount of turning, come round into itself again, and be the same direction. On every surface which has the property of elementary flatness, the amount of turning necessary to take a direction an round into its first position is the same for all points of the surface. I will now show you a surface which at one point of it has not this property. I take this circle of paper from which a sector has been cut out, and bend it round so as to join the edges; in this way I form a surface which is called a cone. Now on all points of this surface but one, the law of elementary flatness holds good. At the vertex of the cone, however, notwithstanding that there is an aggregate of directions in which you may start, such that by continuously changing one of them you may get it round into its original position, yet the whole amount of change necessary to effect this is not the same at the vertex as it is at any other point of the surface. And this you can see at once when I unroll it; for only part of the directions in the plane have been incJuded in the cone. At this point of the cone, then, it does not possess the property of elementary flatness; and no amount of magnifying would ever make a cone seem flat at its vertex. To apply this to solid space, we must notice that here also there is a choice of directions in which you may go out from any point; but it is a much greater choice than a surface gives you. Whereas in a surface the aggregate of directions is only of one dimension, in solid space it is of two dimensions. But here also there are distance-relations, and the aggregate of directions may be divided into parts which have quantity. For example, the directions which start from the vertex of this cone are divided into those which go inside the cone, and those which go outside the cone. The part of the aggregate which is inside the cone is called a soHd angle. Now in those spaces of three dimensions which have the property of elementary flatness, the whole amount of solid angle round one point is equal to the whole amount round another point. Although the space need
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not be exactly similar to itself in aU parts, yet the aggregate of directions round one point is exactly similar to the aggregate of directions round another point, if the space has the property of elementary flatness. How does Euclid assume this postulate of Elementary Flatness? In his fourth postulate he has expressed it so simply and clearly that you will wonder how anybody could make all this fuss. He says, 'All right angles are equal.' Why could I not have adopted this at once, and saved a great deal of trouble? Because it assumes the knowledge of a surface possessing the property of elementary flatness in all its points. Unless such a surface is first made out to exist, and the definition of a right angle is restricted to lines drawn upon it-for there is no necest\ity for the word straight in that definition-the postulate in Euclid's form is obviously not true. I can make two lines cross at the vertex of a cone so that the four adjacent angles shall be equal. and yet not one of them equal to a right angle. I pass on to the third postulate of the science of space-the postulate of Superposition. According to this postulate a body can be moved about in space without altering its size or shape. This seems obvious enough, but it is worth while to examine a little closely into the meaning of it. We must define what we mean by size and by shape. When we say that a body can be moved about without altering its size, we mean that it can be so moved as to keep unaltered the length of all the lines in it. This postulate therefore involves that lines can be compared in respect of magnitude, or that they have a length independent of position; precisely as the former one involved the comparison of angular magnitudes. And when we say that a body can be moved about without altering its shape, we mean that it can be so moved as to keep unaltered all the angles in it. It is not necessary to make mention of the motion of a body, although that is the easiest way of expressing and of conceiving this postulate; but we may, if we like. express it entirely in terms which belong to space, and that we should do in this way. Suppose a figure to have been constructed in some portion of space; say that a triangle has been drawn whose sides are the shortest distances between its angular points. Then if in any other portion of space two points are taken whose shortest distance is equal to a side of the triangle, and at one of them an angle is made equal to one of the angles adjacent to that side, and a line of shortest distance drawn equal to the corresponding side of the original triangle, the distance from the extremity of this to the other of the two points will be equal to the third side of the original triangle, and the two will be equal in aU respects; or generally, if a figure has been constructed anywhere, another figure, with all its lines and all its angles equal to the corresponding lines and angles of the first. can be constructed anywhere else. Now this is exactly what is meant by the principle of superposition employed by Euclid to prove
William KI.,40. CII60rd
the proposition that I have just mentioned. And we may state it again in this short form-AU parts of space are exactly alike. But this postulate carries with it a most important consequence. It enables us to make a pair of most fundamental definitions-those of the plane and of the straight line. In order to explain how these come out of it when it is granted, and how they cannot be made when it is not granted, I must here say something more about the nature of the postulate itself, which might otherwise have been left until we come to criticize it. We have stated the postulate as referring to solid space. But a similar property may exist in surfaces. Here, for instance, is part of the surface of a sphere. If I draw any figure I Uke upon this, I can suppose it to be moved about in any way upon the sphere, without alteration of its size or shape. If a figure has been drawn on any part of the surface of a sphere, a figure equal to it in all respects may be drawn on any other part of the surface. Now I say that this property belongs to the surface itself, is a part of its own internal economy, and does not depend in any way upon its relation to space of three dimensions. For I can pull it about and bend it in aU manner of ways, so as altogether to alter its relation to solid space; and yet, if I do not stretch it or tear it, I make no difference whatever in the length of any lines upon it, or in the size of any angles upon it. S I do not in any way alter the figures drawn upon it, or the possibility of drawing figures upon it, so far as their relations with the surface itself are concerned. This property of the surface, then, could be ascertained by people who lived entirely in it, and were absolutely ignorant of a third dimension. As a point-aggregate of two dimensions, it has in itself properties determining the distance-relations of the points upon it, which are absolutely independent of the existence of any points which are not upon it. Now here is a surface which has not that property. You observe that it is not of the same shape all over, and that some parts of it are more curved than other parts. If you drew a figure upon this surface, and then tried to move it about, you would find that it was impossible to do so without altering the size and shape of the figure. Some parts of it would have to expand, some to contract, the lengths of the lines could not all be kept the same, the angles would not hit off together. And this property of the surface--that its parts are different from one another-is a property of the surface itself, a part of its internal economy, absolutely independent of any relations it may have with space outside of it. For, as with the SThis figure was made of linen, starched upon a spherical surface, and taken oft when dry. That mentioned in the next paragraph was similarly stretched upon tbe irregular surface of the head of a bust. For durability these models should be made of two thicknesses of linen starched together in such a way that the fibres of one bisect the angles between the fibres of the other, and the edge should be bound by a thin slip of paper. They wiD then retain their curvature unaltered for a long time.
other one, I can pull it about in all sorts of ways, and, so long as I do not stretch it or tear it, I make no alteration in the length of lines drawn upon it or in the size of the angles. Here, then, is an intrinsic difference between these two surfaces, as sur· faces. They are both point-aggregates of two dimensions; but the points in them have certain relations of distance (distance measured always on the surface). and these relations of distance are not the same in one case as they are in the other. The supposed people Jiving in the surface and having no idea of a third dimension might, without suspecting that third dimension at all, make a very accurate determination of the nature of their locus in quo. If the people who lived on the surface of the sphere were to measure the angles of a triangle, they would find them to exceed two right angles by a quantity proportional to the area of the triangle. This excess of the angles above two right angles, being divided by the area of the triangle, would be found to give exactly the same quotient at all parts of the sphere. That quotient is called the curvature of the wrface; and we say that a sphere is a surface of uniform curvature. But if the people living on this irregular surface were to do the same thing, they would not find quite the same result. The sum of the angles would, indeed, differ from two right angles, but sometimes in excess, and sometimes in defect, according to the part of the surface where they were. And though for small triangles in any one neighbourhood the excess or defect would be nearly proportional to the area of the triangle, yet the quotient obtained by dividing this excess or defect by the area of the triangle would vary from one part of the surface to another. In other words, the curvature of this surface varies from point to point; it is sometimes positive, sometimes negative, sometimes nothing at all. But now comes the important difference. When I speak of a triangle, what do I suppose the sides of that triangle to be? If I take two points near enough together upon a surface, and stretch a string between them, that string will take up a certain definite position upon the surface, marking the line of shortest distance from one point to the other. Such a line is called a geodesic line. It is a line determined by the intrinsic properties of the surface, and not by its relations with external space. The line would still be the shortest line, however the surface were pulled about without stretching or tearing. A geodesic line may be pro. duced, when a piece of it is given; for we may take one of the points, and, keeping the string stretched, make it go round in a sort of circle until the other end has turned through two right angles. The new position will then be a prolongation of the same geodesic line. In speaking of a triangle, then, I meant a triangle whose sides are geodesic lines. But in the case of a spherical surface---or, more generally, of
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a surface of constant curvature-these geodesic lines have another and most important property. They are straight, so far as the surface is concerned. On this surface a figure may be moved about without altering its size or shape. It is possible, therefore, to draw a line which shall be of the same shape all along and on both sides. That is to say, if you take a piece of the surface on one side of such a line, you may slide it all along the line and it will fit; and you may turn it round and apply it to the other side, and it will fit there also. This is Leibnitz's definition of a straight line, and, you see, it has no meaning except in the case of a surface of constant curvature, a surface all parts of which are alike. Now let us consider the corresponding things in solid space. In this also we may have geodesic lines; namely, lines formed by stretching a string between two points. But we may also have geodesic surfaces; and they are produced in this manner. Suppose we have a point on a surface, and this surface possesses the property of elementary flatness. Then among all the directions of starting from the point, there are some which start in the surface, and do not make an angle with it. Let an these be prolonged into geodesics; then we may imagine one of these geodesics to travel round and coincide with an the others in tum. In so doing it will trace out a surface which is called a geodesic surface. Now in the particular case where a space of three dimensions has the property of superposition, or is all over alike, these geodesic surfaces are planes. That is to say, since the space is all over alike, these surfaces are also of the same shape all over and on both sides; which is Leibnitz's definition of a plane. If you take a piece of space on one side of such a plane, partly bounded by the plane, you may slide it all over the plane, and it will fit; and you may turn it round and apply it to the other side, and it wiJ] fit there also. Now it is clear that this definition will have no meaning unless the third postulate be granted. So we may say that when the postulate of Superposition is true, then there are planes and straight lines; and they are defined as being of the same shape throughout and on both sides. It is found that the whole geometry of a space of three dimensions is known when we know the curvature of three geodesic surfaces at every point. The third postulate requires that the curvature of aU geodesic surfaces should be everywhere equal to the same quantity. I pass to the fourth postulate, which I call the postulate of Similarity. According to this postulate, any figure may be magnified or diminished in any degree without altering its shape. If any figure has been constructed in one part of space, it may be reconstructed to any scale whatever in any other part of space, so that no one of the angles shan be altered though all the le~gths of lines will of course be altered. This seems to be a sufficiently obvious induction from experience; for we have all frequently seen different sizes of the same shape; and it has the advantage of embodying
Tlul Postulates 01 tlul Science 01 SPlIce
""
the fifth and sixth of Euclid's postulates in a single principle, which bears a great resemblance in form to that of Superposition, and may be used in the same manner. It is easy to show that it involves the two postulates of Euclid: 'Two straight lines cannot enclose a space,' and 'Lines in one plane which never meet make equal angles with every other Hne.' This fourth postulate is equivalent to the assumption that the constant curvature of the geodesic surfaces is zero; or the third and fourth may be put together, and we shaH then say that the three curvatures of space are all of them zero at every point. The supposition made by Lobatchewsky was, that the three first postulates were true, but not the fourth. Of the two Euclidean postulates included in this, he admitted one, viz., that two straight lines cannot enclose a space, or that two lines which once diverge go on diverging for ever. But he left out the postulate about parallels, which may be stated in this form. If through a point outside of a straight )jne there be drawn another, indefinitely produced both ways; and if we turn this second one round so as to make the point of intersection travel along the first line, then at the very instant that this point of intersection disappears at one end it will reappear at the other, and there is only one position in which the Jines do not intersect. Lobatchewsky supposed, instead, that there was a finite angle through which the second line must be turned after the point of intersection had disappeared at one end, before it reappeared at the other. For all positions of the second line within tbis angle there is then no intersection. In the two limiting positiOns, when the lines have just done meeting at one end, and when they are just going to meet at the other, they are called parallel; so that two lines can be drawn through a fixed point parallel to a given straight line. The angle between these two depends in a certai n way upon the distance of the point from the line. The sum of the angles of a triangle is less than two right angles by a quantity proportional to the area of the triangle. The whole of this geometry is worked out in the style of Euclid, and the most interesting conclusions are arrived at; particularly in the theory of solid space. in which a surface turns up which is not plane relatively to that space, but which, for purposes of drawing figures upon it, is identical with the Euclidean plane. It was Riemann, however. who first accomplished the task of analysing all the assumptions of geometry, and showing which of them were independent. This very disentangling and separation of them is sufficient to deprive them for the geometer of their exactness and necessity; for the process by which it is effected consists in showing the possibility of conceiving these suppositions one by one to be untrue; whereby it is clearly made out how much is supposed. But it may be worth while to state formally the case for and against them. When it is maintained that we know these postulates to be universally
S66
William Kin,don Cliflord
true, in virtue of certain deliverances of our consciousness, it is implied that these deliverances could not exist, except upon the supposition that the postulates are true. If it can be shown, then, from experience that our consciousness would te)] us exactly the same things if the postulates are not true, the ground of their validity will be taken away. But this is a very easy thing to show. That same faculty which tells you that space is continuous tells you that this water is continuous, and that the motion perceived in a wheel of life is continuous. Now we happen to know that if we could magnify this water as much again as the best microscopes can magnify it. we should perceive its granular structure. And what happens in a wheel of life is discovered by stopping the machine. Even apart, then, from our know1· edge of the way nerves act in carrying messages, it appears that we have no means of knowing anything more about an aggregate than that it is too fine·grained for us to perceive its discontinuity, if it has any. Nor can we, in general, receive a conception as positive knowledge which is itse1f founded merely upon inaction. For the conception of a continuous thing is of that which looks just the same however much you magnify it. We may conceive the magnifying to go on to a certain extent without change, and then, as it were, leave it going on, without taking the trouble to doubt about the changes that may ensue. In regard to the second postulate, we have merely to point to the example of polished surfaces. The smoothest surface that can be made is the one most completely covered with the minutest ruts and furrows. Yet geometrical constructions can be made with extreme accuracy upon such a surface, on the supposition that it is an exact plane. If, therefore, the sharp points, edges, and furrows of space are only small enough, there will be nothing to hinder our conviction of its elementary flatness. It has even been remarked by Riemann that we must not shrink from this suppa. sition if it is found useful in explaining physical phenomena. The first two postulates may therefore be doubted on the side of the very small. We may put the third and fourth together, and doubt them on the side of the very great. For if the property of elementary flatness exist on the average, the deviations from it being, as we have supposed, too sma)) to be perceived, then, whatever were the true nature of space, we should have exactly the conceptions of it which we now have, if only the regions we can get at were small in comparison with the areas of curvature. If we suppose the curvature to vary in an irregular manner, the effect- of it might be very considerable in a triangle formed by the nearest fixed stars; but if we suppose it approximately uniform to the limit of telescopic reach, it will be restricted to very much narrower limits. I can· not perhaps do better than conc1ude by describing to you as well as I can
The PosrulaUs 01 thi! Sci('nct! of Spact!
567
what is the nature of things on the supposition that the curvature of aU space is nearly uniform and positive. In this case the Universe, as known, becomes again a valid conception; for the extent of space is a finite number of cubic miles. 4 And this comes about in a curious way. If you were to start in any direction whatever, and move in that direction in a perfect straight Jine according to the definition of Leibnitz; after travel1ing a most prodigious distance, to which the parallactic unit-200,OOO times the diameter of the earth's orbitwould be only a few steps, you would arrive at-this place. Only. if you had started upwards, you would appear from below. Now, one of two things would be true. Either, when you had got haU-way on your jour· ney, you came to a place that is opposite to this, and which you must have gone through, whatever direction you started in; or else all paths you could have taken diverge entirely from each other till they meet again at this place. In tbe former case, every two straight lines in a plane meet in two points, in the latter they meet only in one. Upon this supposition of a positive curvature, the whole of geometry is far more complete and interesting; the principle of duality, instead of half breaking down over metric relations, applies to all propositions without exception. In fact, I do not mind confessing that I personally have often found relief from the dreary infinities of homaloidal space in the consoJing hope that, after all. this other may be the true state of things. 4 The assumptions here made about the Zusammenhang of space are the simplest ones, but even the finite extent does not follow necessarily from uniform positive curvature; as Riemann seems to have supposed.
Matter exists only as attraction and repulsion--attraction and repulsion are matter. -EDGAR ALLAN POE (Eureka) These our actors, As 1 foretold you, were all spirits and Are melted into air, into thin air.
-SHAKESPEARE
(The Tempest)
Common sense starts with the notion that there ;s mailer where we can get sensations of touch, but not elsewhere. Then it gets puzzled by wind, breath, clouds, etc., whence it is led 10 the conception 01 "spirU"-l speak etymo· /o~ical/y. After "spirit" has been rep/aced by "gas," there is a further stage, that 01 the aether. -BERTRAND RUSSELL
3
On the Space Theory of Matter By WILLIAM KINGDON CLIFFORD (ABSTRACT)
RIEMANN has shown that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very smal1 portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space. I wish here to indicate a manner in which these speculations may be applied to the investigation of physical phenomena. I hold in fact (1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them. (2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave. (3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial. 568
On thtt 8pactt Thttor, of Maller
(4) That in the physical world nothing else takes place but this variation, subject (possibly) to the Jaw of continuity. I am endeavouring in a general way to explain the laws of double refraction on this hypothesis, but have not yet arrived at any results sufficiently decisive to be communicated.
COMMENTARY ON
A Famous Problem OPOLOGY is the geometry of distortion. It deals with fundamental geometric properties that are unaffected when we stretch, twist or otherwise change an object's size and shape. It studies linear figures, surfaces or solids; anything from pretzels and knots to networks and maps. Another name for topology is analysis situs: analysis of position. Unlike the geometries of Euclid, Lobachevsky, Riemann and others, which measure lengths and angles and are therefore called metric, topology is a non metric or nonquantitative geometry. Its propositions hold as well for objects made of rubber as for the rigid figures encountered in metric geometry. Topology seems a queer subject; it delves into strange implausible shapes and its propositions are either childishly obvious (that is, until you try to prove them) or so difficult and abstract that not even a topologist can explain their intuitive meaning. But topology is no queerer than the physical world as we now interpret it. A world made up entirely of erratic electrical gyrations in curved space requires a bizarre mathematics to do it justice. Euclidian geometry, despite its familiar appearance, is a little too bizarre for this world; it is concerned with wholly fictitious objectsperfectly rigid figures and bodies which suffer no change when moved about. Topology starts from the sound premise that there are no rigid objects, that everything in the world is a little askew, and is further deformed when its position is altered. The aim is to find the elements of order in this disorder, the permanence in this impermanence. Mathematicians use the word trans/ormation to describe changes of position, size or shape, and the word invariant to describe the properties unaffected by these chang~s.l In ordinary geometry metric properties are said to be invariant under the transformation of motion. Motion is assumed to have no distorting effect; my pen retains its dimensions as it moves over the paper, this book neither shrinks nor expands as the reader turns its pages. In topology the problem is to find the geometric properties invariant under distorting transformations. If a triangle is stretched into a circle, which of its geometric properties are retained? Is the hole
T
1 The concept of transformation has a family likeness to the concepts of relatio1l and function and is of the greatest importance in almost all branches of mathematics and logic. It has its root "in the power we have, when given ,my two objects of thought. to aS50ciale either of them with the other" and its special meaning in each of the branches of formal reasoning where it is used-algebra, geometry, group theory, )ogic and so on-is derived from this basic idea For a full discussion of tra1lsformalioll, im ariallf and related terms, see Cassius J. Keyser, Mathematical Philosophy, New York, 1922 (Keyser's book is for "educated laymen"). also, pp. 15351537 I Introduction to Keyser piece on groupsl
570
571
.. FlUPlolUJ Problem
"inside" Or "outside" the doughnut? How can the hole be removed? What is a knot? Can a surface be constructed which has only one side? Can a cylinder with a hole through it be squeezed into a sphere? Is it possible to make a bottle with no edges, no inside and no outside? These are examples of topological questions. Topology got under way as a full·f1edged branch of geometry in the nineteenth century. Vorstudien ZUT Topoiogie, published in 1847 by the German mathematician Listing, was the first systematic treatise in the field. Its origins, however, go back to major discoveries made by Descartes and Euler. Both had observed (Descartes in 1640, Euler in 1752) a fundamentaJ relationship between the vertices, edges and faces of a simple polyhedron. Euler expressed this important geometric fact in a famous formula
V-E+F=2 where V stands for vertices, E for edges, F for faces. He also solved the celebrated problem of the Konigsberg Bridges in a memoir which must be regarded as one of the foundation stones of topology. The material appearing below consists of a translation of Euler's memoir on the seven bridges, and an admirable survey of representative topological problems taken from the book What Is Mathematics? by Richard Courant and Herbert Robbins. Dr. Courant, a leading contemporary mathematician, is known for his work in function theory and for his fascinating researches on the mathematics of soap-films. From 1920 to 1933 he was director of the Mathematical Institute of Gottingen, where the great David Hilbert long ruled. Courant is now head of the department of mathematics at New York University; his collaborator, Dr. Robbins, is on the mathematics faculty of Columbia University. Leonhard Euler (1707-1783) is the most eminent of the scientists born in Switzerland. He enriched mathematics in almost every department, and his energy was at least as remarkable as his genius. "Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind." 2 He worked with such facility that it is said he dashed off memoirs in the half-hour between the first and second calls to dinner.s It has been estimated that sixty to eighty large quarto volumes will be needed for his collected works. He also had thirteen children. The details of the life of this Defoe of mathematics are well set forth in Turnbull's little book (see pp. 148-151); here I need only mention the circumstances of the memoir on the Konigsberg puzzle. The problem-to cross the seven bridges in a continuous walk without recrossing any of them-was reII The French astronomer and physicist Arago, as quoted in E. T. Bell. Mtm of Mathematics. New York. 1937, p. 139. a E. T. Bell, op. cit., p. 146.
S72.
EtUtDr's CDmment
garded as a small amusement of the Konigsberg townsfolk. Euler, however, discovered an important scientific principle concealed in the puzzle:~ He presented his simple and ingenious solution to the Russian Academy at St. Petersburg in 1735. His method was to replace the land areas by points and the bridges by lines connecting these points. The points are cal1ed vertices; a vertex is called odd or even according as the number of lines leading from it are odd or even. The entire configuration is a graph; the problem of crossing the bridges reduces to that of traversing the graph with one continuous sweep of the pencil without lifting it from the paper. Euler discovered that this can be done if the graph has only even vertices. If the graph contains no more than two odd vertices, it may be traversed in one journey but it is not possible to return to the starting point. The general principle is that if the graph contains 2n odd vertices, where n is any integer, it will require exactly n distinct journeys to traverse it. Thus began a "vast and intricate theory," still young and growing, yet already one of the great forces of modern mathematics. 5 4 See Moritz Cantor, Vorlesungen iiber Geschichte der Mathematik; Leipzig, 1901, second edition; vol. III, p. 552. :s For an entertaining and instructive account of topology, supplementary to the discussion by Courant and Robbins, See the article "Topology," by Albert S. Tucker and Herbert W. Bailey, Jr., in Scientific American, January 1950.
It is a pleQSQnt surprise to him [the pure mathematician] and an added problem il he finds that the arts can use his calculations, or that the senses can verily them, much as il a composer lound that the sailors could heave better when singing his songs. -GEORGE SANTAYANA
4
The Seven Bridges of Konigsberg By LEONHARD EULER
1. THE branch of geometry that deals with magnitudes has been zealously studied throughout the past, but there is another branch that has been almost unknown up to now; Leibnitz spoke of it first, calling it the' "geometry of position" (geometria situs). This branch of geometry deals with relations dependent on position alone, and investigates the properties of position; it does not take magnitudes into consideration, nor does it involve calcuJatibn with quantities. But as yet no satisfactory definition has been given of the problems that belong to this geometry of position or of the method to be used in solving them. Recently there was announced a problem that, while it certainly seemed to belong to geometry, was nevertheless so designed that it did not can for the determination of a magnitude, nor could it be solved by quantitative calculation; consequently I did not hesitate to assign it to the geometry of position, especially since the solution required only the consideration of position, calculation being of no use. In this paper I shall give an account of the method that I discovered for solving this type of prob1em, which may serve as an example of the geometry of position.
FIQURE 1
573
574
~,dEule,
2. The problem, which I understand is quite well known, is stated as follows: In the town of Konigsberg in Prussia there is an island A, called "Kneiphof," with the two branches of the river (Pregel) flowing around it, as shown in Figure 1. There are seven bridges, a, b, c, d, e, f and g, crossing the two branches. The question is whether a person can plan a walk in such a way that he will cross each of these bridges once but not more than once. I was told that while some denied the possibi lily of doing this and others were in doubt, there were none who maintained that it was actually possible. On the basis of the above I formulated the following very general problem for myself: Given any configuration of the river and the branches into which it may divide, as well as any number of bridges, to determine whether or not it is possible to cross each bridge exactly once. 3. The particular problem of the seven bridges of Konigsberg could be solved by carefully tabulating all possible paths, thereby ascertaining by inspection which of them, if any, met the requirement. This method of solution, however, is too tedious and too difficult because of the large number of possible combinations, and in other problems where many more bridges are involved it could not be used at all. When the analysis is undertaken in the manner just described it yields a great many details that are irrelevant to the problem; undoubtedly this is the reason the method is so onerous. Hence I discarded it and searched for another more restricted in its scope; namely. a method which would show only whether a journey satisfying the prescribed condition could in the first instance be discovered; such an approach. I believed. would be much simpler. 4. My entire method rests on the appropriate and convenient way in which I denote the crossing of bridges, in that I use capital letters, A, B, C, D, to designate the various land areas that are separated from one another by the river. Thus when a person goes from area A to area B across bridge a or b, I denote this crossing by the letters AB, the first of which designates the area whence he came, the second the area where he arrives after crossing the bridge. If the traveller then crosses from B over bridge f into 0, this crossing is denoted by the letters BO; the two crossings AB and BO performed in succession I denote simply by the three letters ABD, since the middle letter B designates the area into which the first crossing leads as well as the area out of which the second crossing leads. 5. Similarly, if the traveller proceeds from D across bridge g into C, I designate these three successive crossings by the four letters ABDC. These four letters signify that the traveller who was originally in A crossed over into B, then to O. and finally to C; and since these areas are separated from one another by the river the traveller must necessarily have
S7S
crossed three bridges. The crossing of four bridges wilf be represented by five letters, and if the traveller crosses an arbitrary number of bridges his journey wiD be described by a number of letters that is one greater than the number of bridges. For example, eight letters are needed to denote the crossing of seven bridges. 6. With this method I pay no attention to which bridges are used; that is to say, if the crossing from one area to another can be made by way of several bridges it makes no difference which one is used, so long as it leads to the desired area. Thus if a route could be laid out over the seven Konigsberg bridges so that each bridge were crossed once and only once, we would be able to describe this route by using eight letters, and in this series of letters the combination AB (or BA) would have to occur twice, since there are two bridges a and b, connecting the regions A and B; similarly the combination AC would occur twice, while the combinations AD, BD, and CD would each occur once. 7. Our question is now reduced to whether from the four letters A, B, C, and D a series of eight letters can be formed in which all the combinations just mentioned occur the required number of times. Before making the effort, however, of trying to find such an arrangement we do well to consider whether its existence is even theoretically possible or not. For if it could be shown that such an arrangement is in fact impossible, then the effort expended on finding it would be wasted. Therefore I have sought for a rule that would determine without difficulty as regards this and all similar questions, whether the required arrangement of letters is feasible. 8. For the purpose of finding such a rule I take a single region A into which an arbitrary number of bridges, a, b, c, d, etc., leads (Figure 2).
FIGURE 2
Of these bridges I first consider only a. If the traveller crosses this bridge he must either have been in A before crossing or have reached A after crossing. so. that according to the above method of denotation the letter A will appear exactly once. If there are three bridges, a, b, c, leading to A and the traveller crosses all three, then the letter A will occur twice in the expression for his route, whether it begins at A or not. And if there are five bridges leading to A the expression for a route that crosses them all will contain the letter A three times. If the number of bridges is
576
odd, increase it by one, and take half the sum; the quotient represents the number of times the letter A appears. 9. Let us now return to the Konigsberg problem (Figure 1). Since there are five bridges, a, b, c, d, e, leading to (and from) island A, the letter A must occur three times in the expression describing the route. The letter B must occur twice, since three bridges lead to B; similarly D and C must each occur twice. That is to say, the series of eight letters that represents the crossing of the seven bridges must contain A three times and B, C and D each twice; but this is quite impossible with a series of eight letters. Thus it is apparent that a crossing of the seven bridges of Konigs.berg in the manner required cannot be effected. 10. Using this method we are always able, whenever the number of bridges leading to a particular region is odd, to determine whether it is possible, in a journey, to cross each bridge exactly once. Such a route exists if the number of bridges plus one is equal to the sum of the numbers that indicate how often each individual letter must occur. On the other hand, if this sum is greater than the number of bridges plus one, as it is in our example, then the desired route cannot be constructed. The rule that I gave (section 8) for determining from the number of bridges that lead to A how often the letter A will occur in the route description is independent of whether these bridges all come from a single region B, as in Figure 2, or from several regions, because I am considering only the region A, and attempting to determine how often the letter A must
occur. 11. When the number of bridges leading to A is even. we must take into account whether the route begins in A or not. For example, if there are two bridges that lead to A and the route starts from A. then the letter A will occur twice, once to indicate the departure from A by one of the bridges and a second time to indicate the return to A by the other bridge. However, jf the traveller starts his journey in another region, the letter A will occur only once, since by my method of description the single occurrence of A indicates an entrance into as well as a departure from A. 12. Suppose, as in our case, there are four bridges leading into the region A, and the route is to begin at A. The letter A will then occur three times in the expression for the whole route, while if the journey had started in another region. A would occur only twice. With six bridges leading to A the letter A will occur four times if A is the starting point, otherwise only three times. In general. if the number of bridges is even, the number of occurrences of the letter A, when the starting region is not A, wil1 be half the number of the bridges; one more than half, when the route starts from A. 13. Every route must, of course, start in some one region, thus from the number of bridges that lead to each region I determine the number
577
The Sevtn Brld,es 01 Koni,sbtr,
of times that the corresponding letter wi11 occur in the expression for the entire route as fol1ows: When the number of the bridges is odd I increase it by one and divide by two; when the number is even I simply divide it by two. Then if the sum of the resulting numbers is equal to the actual number of bridges pi us one, the journey can be accomplished, though it must start in a region approached by an odd number of bridges. But if the sum is one less than the number of bridges plus one, the journey is feasible jf its starting point is a region approached by an even number of bridges, for in that case the sum is again increased by one. 14. My procedure for determining whether in any given system of rivers and bridges it is possible to cross each bridge exactly once is as fol1ows: 1. First I designate the individual regions separated from one another by the water as A, B, C, etc. 2. I take the total number of bridges, increase it by one, and write the resulting number uppermost. 3. Under this number I write the letters A, D, C, etc .• and opposite each of these I note the number of bridges that lead to that particular region. 4. I place an asterisk next the letters that have even numbers opposite them. 5. Opposite each even number I write the half of that number and opposite each odd number I write half of the sum formed by that number plus one. 6. I add up the Jast column of numbers. If the sum is one less than, or equal to the number written at the top. I conclude that the required jou rney can be made. But it must be noted that when the sum is one less than the number at the top, the route must start from a region marked with an asterisk. And in the other case, when these two numbers are equal. it must start from a region that does not have an asterisk. For the Konigsberg problem I would set up the tabulation as follows: Number of bridges 7, giving 8 (= 7 A,
B, C,
D.
5 3 3 3
+ 1) bridges 3 2 2 2
The last column now adds up to more than 8, and hence the required journey cannot be made. 15. Let us take an example of two islands, with four rivers forming the surrounding water, as shown in Figure 3. Fifteen bridges, marked a, b, c, d, etc., cross the water around the islands and the adjoining rivers; the question is whether a journey can be arranged that will pass over all the bridges, but not over any of them more than once. 1. I begin by marking all the regions that are separated from one another by the water with the letters A, D, C, D, E, F-there are six of them. 2. I take the number of bridges-i5-add one and write this number-l6-uppermost. 3. I write the letters A, B, C, etc. in a column and opposite each tetter I
uonhard Euler
578
FIGURE 3
A*, B* , C* , D, E, F*,
8 4 4 3 5
6
16 4 2
2 2 3 3 16
write the number of bridges connecting with that region, e.g .• 8 bridges for A, 4 for B, etc. 4. The letters that have even numbers opposite them I mark with an asterisk. 5. In a third column I write the half of each corresponding even number, or, if the number is odd, ] add one to it, and put down half the sum. 6. Finally I add the numbers in the third column and get ] 6 as the sum. This is the same as the number 16 that appears above, and hence it follows that the journey can be effected if it begins in regions D or E, whose symbols have no asterisk. The following expression represents such a route: EaFbBcFdAeFfCgAhCiDkAmEnApBoEID. Here I have also indicated, by small letters between the capitals, which bridges are crossed. 16. By this method we can easily determine, even in cases of considerable complexity, whether a single crossing of each of the bridges in sequence is actually possible. But I should now like to give another and much simpler method. which follows quite easily from the preceding.
The Stlytllt BrldBtls 01 Ki)lliBsbuB
579
after a few preliminary remarks. In the first place, I note that the sum of all the numbers of bridges to each region, that are written down in the second column opposite the letters A, B, C, etc., is necessarily double the actual number of bridges. The reason is that in the tabulation of the bridges leading to the various regions each bridge is counted twice, once for each of the two regions that it connects. 17. From this observation it fol1ows that the sum of the numbers in the second column must be an even number, since half of it represents the actual number of bridges. Hence it is impossible for exactly one of these numbers (indicating how many bridges connect with each region) to be odd, or, for that matter, three or five. etc. In other words, if any of the numbers opposite the letters A. B, C, etc., are odd, an even number of them must be odd. In the Konigsberg problem, for instance, all four of the numbers opposite the letters A, B, C. D were odd, as explained in section 14, whi1e in the example just given (section 15) only two of the numbers were odd, namely those opposite D and E. 18. Since the sum of the numbers opposite A, B, C, etc .• is double the number of bridges, it is clear that if this sum is increased by two and then divided by 2 the result will be the number written at the top. When all the numbers in the second column are even, and the half of each is written down in the third column, the total of this column will be one less than the number at the top. In that case it wil1 always be possible to cross all the bridges. For in whatever region the journey begins, there will be an even number of bridges leading to it, which is the requirement. In the Konigsberg problem we could, for instance, arrange matters so that each bridge is crossed twice, which is equivalent to dividing each bridge into two, whence the number of bridges leading to each region would be even. 19. Further, when only two of the numbers opposite the letters are odd, and the others even, the required route is possible provided it begins in a region approached by an odd number of bridges. We take half of each even number. and Hkewise half of each odd number after adding one, as our procedure requires; the sum of these halves will then be one greater than the number of bridges. and hence equal to the number written at the top. Similarly, where four. six, or eight, etc., of the numbers in the second column are odd it is evident that the sum of the numbers in the third column will be one, two, three, etc., greater than the top number, as the case may be, and hence the desired journey is impossible. 20. Thus for any configuration that may arise the easiest way of determining whether a single crossing of all the bridges is possible is to apply the folJowing rules:
If there are more than two regions which are approached by an odd
number of bridges, no route satisfying the required conditions can be found. If, bowever, there are only two regions with an odd number of approach bridges the required journey can be completed provided it originates in one of the regions. If, finally, there is no region with an odd number of approacb bridges, the required journey can be effected, no matter where it begins. These rules so1ve completely the problem initially proposed. 21. After we bave determined that a route actually exists we are left with the question how to find it. To this end the following rule will serve: Wherever possible we menta1ly eliminate any two bridges that connect the same two regions; this usually reduced the number of bridges considerably. Then-and tbis should not be difficult-we proceed to trace the required route across the remaining bridges. The pattern of this route. once we bave found it, will not be substantially affected by the restoration of the bridges wbich were first eliminated from consideration-as a little thought will show; therefore I do not think I need say more about finding the routes themselves.
1 must go in and out.
5
-BERNARD SHAW
(Heartbreak House)
Topology By RICHARD COURANT and HERBERT ROBBINS EULER'S FORMULA FOR POLYHEDRA
ALTHOUGH the study of polyhedra held a central place in Greek geometry, it remained for Descartes and Euler to discover the following fact: In a simple polyhedron let V denote the number of vertices, E the number of edges, and F the number of faces; then always (1)
V-E+F=2.
By a polyhedron is meant a solid whose surface consists of a number of polygonal faces. In the case of the regular solids, all the polygons are congruent and all the angles at vertices are equal. A polyhedron is simple if there are no "holes" in it, so that its surface can be deformed continuously into the surface of a sphere. Figure 2 shows a simple polyhedron which is not regular, while Figure 3 shows a polyhedron which is not simple. The reader should check the fact that Euler's formula holds for the simple polyhedra of Figures 1 and 2, but does not hold for the polyhedron of Figure 3. To prove Euler's formula, let us imagine the given simple polyhedron to be hollow, with a surface made of thin rubber. Then if we cut out one of the faces of the hollow polyhedron, we can deform the remaining surface until it stretches out fiat on a plane. Of course, the areas of the faces and the angles between the edges of the polyhedron will have changed in this process. But the network of vertices and edges in the plane will contain the same number of vertices and edges as did the original polyhedron, while the number of polygons wi11 be one less than in the original polyhedron, since one face was removed. We shall now show that for the plane network, V - E + F = 1, so that, if the removed face is counted, the result is V - E + F = 2 for the original polyhedron. First we "triangulate" the plane network in the following way: In some polygon of the network which is not already a triangle we draw a diagonal. The effect of this is to increase both E and F by 1, thus preS81
582
Richard Courallt alld Herbert Robbilfs
FIGURE i-The regular polyhedra.
serving the value of V - E + F. We continue drawing diagonals joining pairs of points (Figure 4) until the figure consists entirely of triangles, as it must eventually. In the triangulated network, V - E + F has the value that it had before the division into triangles, since the drawing of diag· onals has not changed it. Some of the triangles have edges on the boundary of the pJane network. Of these some, such as ABC, have only one edge on ,the boundary, whiJe other triangles may have two edges on the boundary. We take any boundary triangle and remove that part of it which does not also belong to some other triangle. Thus, from A BC we remove the edge AC and the face, leaving the vertices A, B, C and the two edges AB and BC; while from DEF we remove the face, the two
583
Topology
FIGURE 2-A simple polyhedron. Y - E
... .....
9 - 18
+
11 -= 2 .
......... ,1
...I .....
.........
1
.J ",,""
,. . , ..........
"" ,...
,
,
I
,
" " .... ......
;'
.... I....
, ........... I
l.,.""
"" ""
+F=
,I)I.... .....
"" ""
.....
... .... ....
+ F ... 16 - 32 + 16 - O. edges DF and FE, and the vertex F. The removal of a triangle of type A Be decreases E and F by I, while V is unaffected, so that V - E + F remains the same. The removal of a triangle of type DEF decreases V by 1, E by 2, and F by 1, so that V - E + F again remains the same. By a properly chosen sequence of these operations we can remove triangles with edges on the boundary (which changes with each removal), until finaJly only one triangle remains, with its three edges, three vertices, and one face. For this simple network, V - E + F 3 - 3 + 1 1. But we FIGURE 3-A non-simple polyhedron. Y - E
=
=
Richard COfU'(lnl and Hl!rbul Robbins
584
........
----~rR
\ \
\
\ \
/
\
/
FIGURE 4-Proof of Euler's theorem.
have seen that by constantly erasing triangles V - E + F was not altered. Therefore in the original plane network V - E + F must equal 1 also, and thus equals 1 for the polyhedron with one face missing. We conclude that V - E + F :: 2 for the complete polyhedron. This completes the proof of Euler's formula. On the basis of Euler's formula it is easy to show that there are no more than five regular polyhedra. For suppose that a regular polyhedron has F faces, each of which is an n-sided regular polygon, and that r edges meet at each vertex. Counting edges by faces and vertices, we see that
(2)
nF=2E;
for each edge belongs to two faces. and hence is counted twice in the product nF; moreover, rV::2E,
(3)
since each edge has two vertices. Hence from (1) we obtain the equation 2E 2E -+--E=2 n
r
or 111
1
2
E
-+-=-+-.
(4)
n
r
We know to begin with that n > and r > 3, since a polygon must have at least three sides, and at least three sides must meet at each polyhedral angle. But nand r cannot both be greater than three. for then the left hand side of equation (4) could not exceed %. which is impossible for any positive value of E. Therefore, let us see what values r may have when n 3, and what values n may have when r 3. The totality of polyhedra given by these two cases gives the number of possible regular polyhedra.
=
=
TopoloD
For n
=3, equation (4) becomes 111 - - - =-; T 6 E
r can thus equal 3, 4, or 5. (6, or any greater number, is obviously excluded, since liE is always positive.) For these values of nand r we get E 6, 12, or 30, corresponding respectively to the tetrahedron, octahedron, and icosahedron. Likewise, for r = 3 we obtain the equation
=
111
---=-, n
6
E
from which it follows that n =.:: 3, 4, or 5, and E = 6, 12, or 30, respectively. These values correspond respectively to the tetrahedron, cube. and dodecahedron. Substituting these values for n, T, and E in equations (2) and (3), we obtain the numbers of vertices and faces in the corresponding polyhedra. TOPOLOGICAL PROPERTIES OF FIGURES TOPOLOGICAL PROPERTIES
We have proved that the Euler formula holds for any simple polyhedron. But the range of validity of this formula goes far beyond the polyhedra of elementary geometry, with their flat faces and straight edges; the proof just given would apply equally well to a simple polyhedron with curved faces and edges, or to any subdivision of the surface of a sphere into regions bounded by curved arcs. Moreover, if we imagine the surface of the polyhedron or of the sphere to be made out of a thin sheet of rubber, the Euler formula will still hold if the surface is deformed by bending and stretching the rubber into any other shape, so long as the rubber is not tom in the process. For the formula is concerned only with the numbers of the vertices, edges, and faces, and not with lengths, areas. straightness, cross-ratios, or any of the usual concepts of elementary or projective geometry. We recall that elementary geometry deals with the magnitUdes (length, angle, and area) that are unchanged by the rigid motions, while projective geometry deals with the concepts (point, line, incidence, and cross-ratio) which are unchanged by the stiU larger group of projective transformations. But the rigid motions and the projections are both very special cases of what are called topological transformations: a topological transformation of one geometrical figure A into another figure A' is given by any correspondence
between the points p of A and the points pi of A' which has the following two properties:
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Richard CONrant and
H~rbert
Robbins
1. The correspondence is biunique. This means that to each point p of A corresponds just one point pi of A', and conversely.
2. The correspondence is continuous in both directions. This means that if we take any two points p, q of A and move p so that the distance between it and q approaches zero, then the distance between the corresponding points p', q' of A' will also approach zero, and conversely. Any property of a geometrical figure A that holds as well for every figure into which A may be transformed by a topological transformation is called a topological property of A, and topology is the branch of geometry which deals only with the topological properties of figures. Imagine a figure to be copied "free-hand" by a conscientious but inexpert draftsman who makes straight lines curved and alters angles, distances and areas; then, although the metric and projective properties of the original figure would be lost, its topological properties would remain the same. The most intuitive examples of general topological transformations are the deformations. Imagine a figure such as a sphere or a triangle to be made from or drawn upon a thin sheet of rubber, which is then stretched and twisted in any manner without tearing it and without bringing distinct points into actual coincidence. (Bringing distinct points into coincidence would violate condition 1. Tearing the sheet of rubber would violate condition 2, since two points of the original figure which tend toward coincidence from opposite sides of a line along which the sheet is torn would not tend towards coincidence in the torn figure.) The final position of the figure wiJ] then be a topological image of the original. A triangle can be deformed into any other triangle or into a circle or an ellipse, and hence these figures have exactly the same topological properties. But one
FIGURE S-Topologically equivalent surfaces
FIGURE 6-Topologically non-equivalent surfaces
cannot deform a circle into a line segment, nor the surface of a sphere into the surface of an inner tube. The general concept of topological transformation is wider than the concept of deformation. For example. if a figure is cut during a deformation and the edges of the cut sewn together after the deformation in exact1y the same way as before. the process still defines a topological transformation of the original figure, although it is not a deformation. Thus the two curves of Figure 12 1 are topologically equivalent to each other or to a circle, since they may be cut, untwisted, and the cut sewn up. But it is impossible to deform one curve into the other or into a circle without first cutting the curve. Topological properties of figures (such as are given by Euler's theorem and others to be discussed in this section) are of the greatest interest and importance in many mathematical investigations. They are in a sense the deepest and most fundamental of all geometrical properties, since they persist under the most drastic changes of shape. CONNECTIVITY
As another exampJe of two figures that are not topologically equivalent we may consider the plane domains of Figure 7. The first of these consists
II
b
FIGURE 7-Simple and double conneclivit)'.
of all points interior to a circle, while the second consists of all points contained between two concentric circles. Any closed curve lying in the domain a can be continuously deformed or "shrunk" down to a single point within the domain. A domain with this property is said to be simply connected. The domain b is not simply connected. For example, a circle concentric with the two boundary circles and midway between them cannot be shrunk to a single point within the domain, since during this process the curve would necessarily pass over the center of the circles, which is not a point of the domain. A domain which is not simply connected is said to be multiply connected. ]f the multiply connected domain I
[See p. 592, ED.]
588
Richard Courant and Herbert Robbins
FIGURE 8---Cutting a doubly connected domain to make it simply connected,
b is cut along a radius, as in Figure 8, the resulting domain is simply connected. More generally, we can construct domains with two, three, or more "holes," such as the domain of Figure 9. In order to convert this domain into a simply connected domain, two cuts are necessary. If n - 1 non-
FIGURE 9-Reductlon of a triply connected domain
intersecting cuts from boundary to boundary are needed to convert a given multiply connected domain D into a simply connected domain, the domain D is said to be n-tuply connected. The degree of connectivity of a domain in the plane is an important topological invariant of the domain. OTHER EXAMPLES OF TOPOLOGICAL THEOREMS THE JORDAN CURVE THEOREM
A simple closed curve (one that does not intersect itself) is drawn in the plane. What property of this figure persists even if the plane is
.589
regarded as a sheet of rubber that can be deformed in any way? The length of the curve and the area that it encloses can be changed by a deformation. But there is a topological property of the configuration which is so simple that it may seem trivial: A simple closed curve C in the plane divides the plane into exactly two domains, an inside and an outside. By this is meant that the points of the plane fall into two classesA, the outside of the curve, and B, the inside-such that any pair of points of the same class can be joined by a curve which does not cross C, while any curve joining a pair of points belonging to different classes must cross C. This statement is obviously true for a circle or an el1ipse, but the selfevidence fades a little if one contemplates a complicated curve like the twisted polygon in Figure 10.
FIGURE to-Which poinlS of the plane are inside this polygon?
This theorem was first stated by Camil1e Jordan (1838-1922) in his famous Cours d'Analyse, from which a whole generation of mathematicians learned the modem concept of rigor in analysis. Strangely enough, the proof given by Jordan was neither short nor simple, and the surprise was even greater when it turned out that Jordan's proof was invalid and that considerable effort was necessary to fin the gaps in his reasoning. The first rigorous proofs of the theorem were quite complicated and hard to understand, even for many well-trained mathematicians. Only recently have comparatively simple proofs been found. One reason for the difficulty lies in the generality of the concept of "simple closed curve," which is not restricted to the class of polygons or "smooth" curves, but includes all curves which are topological images of a circle. On the other hand, many concepts such as "inside," Uoutside," etc., which are so clear to the intuition. must be made precise before a rigorous proof is possible. It is
590
Richard Courant ,,"d Herbert RobbllU
of the highest theoretical importance to analyze such concepts in their fullest generality, and much of modem topology is devoted to this task. But one should never forget that in the great majority of cases that arise from the study of concrete geometrical phenomena it is quite beside the point to work with concepts whose extreme generality creates uAnecessary difficulties. As a matter of fact, the Jordan curve theorem is quite simple to prove for the reasonably well-behaved curves, such as polygons or curves with continuously turning tangents. which occur in most important problems. THE POUR COLOR PROBLEM
From the example of the Jordan curve theorem one might suppose that topology is concerned with providing rigorous proofs for the sort of obvious assertions that no sane person would doubt. On the contrary. there are many topological questions, some of them quite simple in form, to which the intuition gives no satisfactory answer. An example of this kind is the renowned "four color problem."
FlGU:R.E II-Colorfng a map.
In coloring a geographical map it is customary to give different colors to any two countries that have a portion of their boundary in common. It has been found empirically that any map, no matter how many countries it contains nor how they are situated, can be so colored by using only four different colors. It is easy to see that no smaller number of colors will suffice for all cases. Figure 11 shows an island in the sea that certainly cannot be properly colored with less than four colors. since it contains four countries. each of which touches the other three. The fact that no map has yet been found whose coloring requires more than four colors suggests the following mathematical theorem = For any subdivision Of the plane into non-overlapping regions, it i.~ always possible to mark the regions with one of the numbers 1. 2. 3, 4 in such a way that no two adjacent regions receive the same number. By "adjacent"
T41pol410
'91
regions we mean regions with a whole segment of boundary in common; two regions which meet at a single point only or at a finite number of points (such as the states of Colorado and Arizona) will not be called adjacent. since no confusion would arise if they were colored with the same color. The problem of proving this theorem seems to have been first proposed by Moebius in 1840, later by DeMorgan in 1850. and again by Cayley in 1878. A uproof' was published by Kempe in 1879, but in 1890 Heawood found an error in Kempe's reasoning. By a revision of Kempe's proof. Heawood was able to show tbat five colors are always sufficient. Despite the efforts of many famous mathematicians, the matter essentially rests with this more modest result: It has been proved that five colors suffice for all maps and it is conjectured that four will likewise suffice. But, as in the case of the famous Fermat theorem neither a proof of this conjecture nOr an example contradicting it has been produced, and it remains one of the great unsolved problems in mathematics. The four color theorem has indeed been proved for an maps containing less than thirty-eight regions. In view of this fact it appears that even if the general theorem is false it cannot be disproved by any very simple example. In the four color problem the maps may be drawn either in the plane or on the surface of a sphere. The two cases are equivalent: any map on the sphere may be represented on the plane by boring a small hole through the interior of one of the regions A and deforming the resulting surface until it is flat, as in the proof of Euler's theorem. The resulting map in the plane will be that of an "island" consisting of the remaining regions, surrounded by a "sea" consisting of the region A. Conversely. by a reversal of this process, any map in the plane may be represented on the sphere. We may therefore confine ourselves to maps on the sphere. Furthermore, since deformations of the regions and their boundary lines do not affect the problem, we may suppose that the boundary of each region is a simple closed polygon composed of circular arcs. Even thus "regularized," the problem remains unsolved; the difficulties here, unlike those involved in the lordan curve theorem, do not reside in the generality of the concepts of region and curve. A remarkable fact connected with the four color problem is that for surfaces more complicated than the plane or the sphere the corresponding theorems have actually been proved, so that, paradoxically enough. the analysis of more compJicated geometrical surfaces appears in this respect to be easier than that of the simplest cases. For example, on the surface of a torus (see Figure 5), whose shape is that of a doughnut or an inflated inner tube, it has been shown that any map may be colored by using seven colors, while maps may be constructed containing seven regions. each of which touches the other six.
Richard Courant tlnd Herbert Robbins
592
KNOTS
As a final example it may be pointed out that the study of knots pre-sents difficult mathematical problems of a topological character. A knot is formed by first looping and interlacing a piece of string and then joining the ends together. The resulting closed curve represents a geometrical figure that remains essentially the same even if it is deformed by pu1ling or twisting without breaking the string. But how is it possible to give an intrinsic characterization that will distinguish a knotted closed curve in space from an unknotted curve such as the circle? The answer is by no means simple, and stiH Jess so is the complete mathematical analysis of the various kinds of knots and the differences between them. Even for the simplest case this has proved to be a sizable task. Consider the two trefoil knots shown in Figure 12. These two knots are completely symmet· rical "mirror images" of one another, and are topologica11y equivalent, but they are not congruent. The problem arises whether it is possible to deform one of these knots into the other in a continuous way. The answer is in the negative, but the proof of this fact requires considerably more knowledge of the technique of topology and group theory than can be presented here.
FIGURE I2-Topologically equivalent knots that are not deformable Into one another.
THE TOPOLOGICAL CLASSIFICATION OF SURFACES THE GENUS OF A SURFACE
Many simple but important topological facts arise in the study of twa. dimensional surfaces. For example, let us compare the surface of a sphere with that of a torus. It is clear from Figure 13 that the two surfaces differ in a fundamental way: on the sphere, as in the plane, every simple closed curve such as C separates the surface into two parts. But on the torus there exist closed curves such as C' that do not separate the surface into two parts. To say that C separates the sphere into two parts means that if the sphere is cut along C it will faU into two distinct and unconnected pieces, or, what amounts to the same thing, that we can find two points on the sphere such that any curve on the sphere which joins them must intersect C. On the other hand, if the torus is cut along the closed curve
593
Topology
FIGURE 13-Cuts on sphere and torus
C', the resulting surface still hangs together: any point of the surface can be joined to any other point by a curve that does not intersect C'. This difference between the sphere and the torus marks the two types of surfaces as topologically distinct, and shows that it is impossible to deform one into the other in a continuous way. Next let us consider the surface with two holes shown in Figure 14. On this surface we can draw two non-intersecting closed curves A and B which do not separate the surface. The torus is always separated into two parts by any two such curves. On the other hand, three closed nonintersecting curves always separate the surface with two holes.
FIGURE 14-A surface of genus 2
These facts suggest that we define the genus of a surface as the largest number of non-intersecting simple closed curves that can be drawn on the surface without separating it. The genus of the sphere is 0, that of the torus is I, while that of the surface in Figure 14 is 2. A simi! ar surface with p holes has the genus p. The genus is a topological property of a surface and remains the same if the surface is deformed. Conversely, it may be shown (we omit the proof) that if two closed surfaces have the same genus, then one may be deformed into the other, so that the genus p = 0, I, 2, . . . of a closed surface characterizes it completely from the topological point of view. (We are assuming that the surfaces considered are ordinary "two-sided" closed surfaces. Later in this section we shall consider "one-sided" surfaces.) For example, the two-holed doughnut and the sphere with two "handles" of Figure 15 are both closed surfaces of genus 2, and it is clear that either of these surfaces may be
Rlc1uU'd CO",.41I' 41Id Herblert Robbiru
S94
FIGUR.E l.s-8urfa may be represented as a mixture of three primary colours, taken in definite quantities. The particular mixtures can be actually made with the colour-top. In the same way we may consider the system of simple tones 6 as an aggregate of two dimensions, if we distinguish only pitch and intensity, and leave out of account differences of timbre. This generalisation of the idea is well suited to bring out the distinction between space of three dimensions and other aggregates. We can, as we know from daily experience, compare the vertical distance of two points with the horizontal distance of two others, because we can apply a measure first to the one pair and then to the other. But we cannot compare the difference between two tones of equal pitch and different intensity, with that between two tones of equal intensity and different pitch. Riemann showed, by considerations of this kind, that the essential foundation of any system of geometry, is the expression that it gives for the distance between two points lying in any direction towards one another, beginning with the infinitesimal interval. He took from analytical geometry the most general form for this expression, that, namely, which leaves altogether open the kind of measurements by which the position of any point is given. 7 Then he showed that the kind of free mobiHty without change of form which belongs to bodies in our space can only exist when certain quantities yielded by the calculation L-quantities that coincide wtih Gauss's measure of surface-curvature when they are expressed for surfaces-have everywhere an equal value. For this reason Riemann calls these quantities, when they have the same value in all directions for a particular spot, the measure of curvature of the space at this spot. To prevent misunderstanding, I will once more observe that this so-called measure of space-curvature is a quantity obtained by purely analytical calculation, and that its introduction involves no suggestion of relations that would have a meaning only for senseperception. The name is merely taken, as a short expression for a complex relation, from the one case in which the quantity designated admits of sensible representation. Now whenever the value of this measure of curvature in any space is everywhere zero, that space everywhere conforms to the axioms of Euclid; and it may be called a flat (homaloid) space in contradistinction to other spaces, analytically constructible, that may be called curved, because their measure of curvature has a value other than zero. Analytical geometry Helmholtz's PoplIlar Lectures, Series I. p. 243. Ibid., p. 86. ., For the square of the distance of two infinitely near points the expression is a homogeneous quadric function of the differentials of their co-ordinates. 8 They are algebraica) expressions compounded from the coefficients of the various terms jn the expression for the square of the distance of two contiguous points and from their differential quotients. :l
6
On the Origin and Significance 01 Geometrical Axioml
657
may be as completely and consistently worked out for such spaces as ordinary geometry can for our actually existing homaloid space. If the measure of curvature is positive we have spherical space, in which straightest lines return upon themselves and there are no parallels. Such a space would. like the surface of a sphere, be unlimited but not infinitely great. A constant negative measure of curvature on the other hand gives pseudospherical space, in which straightest lines run out to infinity, and a pencil of straightest lines may be drawn, in any flattest surface, through any point which does not intersect another given straightest line in that surface. Beltrami fl has rendered these last relations imaginable by showing that the points, lines, and surfaces of a pseudospherical space of three dimensions, can be so portrayed in the interior of a sphere in Euclid's homaloid space, that every straightest line or flattest surface of the pseudospherical space is represented by a straight line or a plane, respectively, in the sphere. The surface itself of the sphere corresponds to the infinitely distant points of the pseudospherical space; and the different parts of this space, as represented in the sphere, become smaller, the nearer they lie to the spherical surface, diminishing more rapidly in the direction of the radii than in that perpendicular to them. Straight Hnes in the sphere, which only intersect beyond its surface, correspond to straightest tines of the pseudospherical space which never intersect. Thus it appeared that space, considered as a region of measurable quantities, does not at all correspond with the most general conception of an aggregate of three dimensions, but involves also special conditions, depending on the perfectly free mobility of solid bodies without change of form to all parts of it and with all possible changes of direction; and, further, on the special value of the measure of curvature which for our actual space equals, or at least is not distinguishable from, zero. This latter definition is given in the axioms of straight lines and parallels. Whilst Riemann entered upon this new field from the side of the most general and fundamental questions of analytical geometry, I myself arrived at similar conclusions,lo partly from seeking to represent in space the system of colours. involving the comparison of one threefold extended aggregate with another, and partly from inquiries on the origin of our ocular measure for distances in the field of vision. Riemann starts by assuming the above-mentioned algebraical expression which represents in the most general form the distance between two infinitely near points, and deduces therefrom, the conditions of mobility of rigid figures. I, on the 9
Teoria jondamenlale, &c., ut sup.
Ueber die Thatsachen die der Geometrie zum Grunde liegen (Nachrichlen von der kon;gl. Ges. d. Wiss. Zit Gollingen, Juni 3, 1868). 10
other hand, starting from the observed fact that the movement of rigid figures is possible in our space, with the degree of freedom that we know, deduce the necessity of the algebraic expression taken by Riemann as an axiom. The assump~ions that I had to make as the basis of the calculation were the following. First, to make algebraical treatment at all possible, it must be assumed that the position of any point A can be determined. in relation to certain given figures taken as fixed bases, by measurement of some kind of magnitudes. as lines, angles between lines, angles between surfaces, and so forth. The measurements necessary for determining the position of A are known as its co-ordinates. In general, the number of co-ordinates necessary for the complete determination of the position of a point, marks the number of the dimensions of the space in question. It is further assumed that with the movement of the point A, the magnitudes used as co-ordinates vary continuously. Secondly, the definition of a solid body, or rigid system of points, must be made in such a way as to admit of magnitudes being compared by congruence. As we must not, at this stage, assume any special methods for the measurement of magnitudes, our definition can, in the first instance, run only as fol1ows: Between the co-ordinates of any two points belonging to a solid body, there must be an equation which, however the body is moved, expresses a constant spatial relation (proving at last to be the distance) between the two points, and which is the same for congruent pairs of points, that is to say, such pairs as can be made successively to coincide in space with the same fixed pair of points. However indeterminate in appearance, this definition involves most important consequences, because with increase in the number of points, the number of equations increases much more quickly than the number of coordinates which they determine. Five points. A, B. C, 0, E, give ten different pairs of points AB, AC, AD, AE, BC, BD. BE, CD, CE. DE, and therefore ten equations, involving in space of three dimensions fifteen variable co-ordinates. But of these fifteen, six must remain arbitrary, if the system of five points is to admit of free movement and rotation, and thus the ten equations can determine only nine co-ordinates as functions of the six variables. With six points we obtain fifteen equations for twelve quantities, with seven points twenty-one equations for fifteen, and so on. Now from n independent equations we can determine n contained quantities, and if we have more than n equations, the superfluous ones must be
deducible from the first n. Hence it follows that the equations which subsist between the co-ordinates of each pair of points of a solid body must have a special character, seeing that, when in space of three dimensions they are satisfied for nine pairs of points as formed out of any five points, the equation for the tenth pair follows by logical consequence. Thus our assumption for the definition of solidity, becomes quite sufficient to determine the kind of equations holding between the co-ordinates of two points rigidly connected. Thirdly, the calculation must further be based on the fact of a peculiar circumstance in the movement of solid bodies, a fact so famil iar to us that but for this inquiry it might never have been thought of as something that need not be. When in our space of three dimensions two points of a solid body are kept fixed, its movements are limited to rotations round the straight line connecting them. If we tum it completely round once, it again occupies exactly the position it had at first. This fact. that rotation in one direction always brings a solid body back into its original position, needs special mention. A system of geometry is possible without it. This is most easily seen in the geometry of a plane. Suppose that with every rotation of a plane figure its linear dimensions increased in proportion to the angle of rotation, the figure after one whole rotation through 360 degrees would no longer coincide with itself as it was original1y. But any second figure that was congruent with the first in its original position might be made to coincide with it in its second position by being also turned through 360 degrees. A consistent system of geometry would be possible upon this supposition, which does not come under Riemann's formula. On the other hand I have shown that the three assumptions taken together form a sufficient basis for the starting-point of Riemann's investigation, and thence for all his further results relating to the distinction of different spaces according to their measure of curvature. It still remained to be seen whether the laws of motion, as dependent on moving forces, could also be consistently transferred to spherical or pseudospherical space. This investigation has been carried out by Professor Lipschitz of Bonn. t 1 It is found that the comprehensive expression for all the laws of dynamics, Hamilton's principle. may be directly transferred to spaces of which the measure of curvature is other than zero. Accordingly, in this respect also, the disparate systems of geometry lead to no contradiction. We have now to seek an explanation of the special characteristics of our own flat space, since it appears that they are not implied in the general motion of an extended quantity of three dimensions and of the free mot t 'Untersuchungen tiber die ganzen homogenen Functionen von n Differentialen' (Borchardt's Journal Iii, Mathematik, Bd. Ixx. 3, 71; Ixxiii. 3, I): 'Untersuchung eines Problems der Varialionsrechnung' (Ibid. Bd. Ixxiv.).
660
Re",,,um Von Relmhol"
bility of bounded figures therein. Necessities of thought, such as are involved in the conception of such a variety, and its measurability, or from the most general of all ideas of a solid figure contained in it, and of its free mobility, they undoubtedly are not. Let us then examine the opposite assumption as to their origin being empirical, and see if they can be inferred from facts of experience and so established, or if, when tested by experience, they are perhaps to be rejected. If they are of empirical origin, we must be able to represent to ourselves connected series of facts, indicating a different value for the measure of curvature from that of Euclid's flat space. But if we can imagine such spaces of other sorts, it cannot be maintained that the axioms of geometry are necessary consequences of an apriori transcendental form of intuition, as Kant thought. The distinction between spherical, pseudospherical, and Euclid's geometry depends, as was above observed, on the value of a certain constant called, by Riemann, the measure of curvature of the space in question. The value must be zero for Euclid's axioms to hold good. If it were not zero, the sum of the angles of a large triangle would differ from that of the angles of a small one, being larger in spherical. smaller in pseudospherical, space. Again, the geometrical similarity of large and small solids or figures is possible only in Euclid's space. All systems of practical mensuration that have been used for the angles of large rectilinear triangles, and especially all systems of astronomical measurement which make the parallax of the immeasurably distant fixed stars equal to zero (in pseudospherical space the parallax even of infinitely distant points would be positive), confirm empirically the axiom of parallels, and show the measure of curvature of our space thus far to be indistinguiShable from zero. It remains, however, a question, as Riemann observed, whether the result might not be different if we could use other than our limited base-lines, the greatest of which is the major axis of the earth's orbit. Meanwhile, we must not forget that all geometrical measurements rest ultimately upon the principle of congruence. We measure the distance between points by applying to them the compass, rule, or chain. We measure angles by bringing the divided circle or theodolite to the vertex of the angle. We also determine straight lines by the path of rays of light which in our experience is rectilinear; but that light travels in shortest lines as long as it continues in a medium of constant refraction would be equally true in space of a different measure of curvature. Thus aU our geometrical measurements depend on our instruments being really, as we consider them, invariable in form, or at least on their undergoing no other than the small changes we know of, as arising from variation of temperature, or from gravity acting differently at different places. In measuring, we only employ the best and surest means we know of to determine, what we otherwise are in the habit of making out by sight
On Ih~ Origin and Significance 0/ Geonutlrical Axiom"
661
and touch or by pacing. Here our own body with its organs is the instrument we carry about in space. Now it is the hand, now the leg, that serves for a compass, or the eye turning in all directions is our theodolite for measuring arcs and angles in the visual field. Every comparative estimate of magnitudes or measurement of their spatial relations proceeds therefore upon a supposition as to the behaviour of certain physical things, either the human body or other instruments employed. The supposition may be in the highest degree probable and in closest harmony with all other physical relations known to us, but yet it passes beyond the scope of pure space-intuition. It is in fact possible to imagine conditions for bodies apparently solid such that the measurements in Euclid's space become what they would be in spherical or pseudospherical space. Let me first remind the reader that if all the linear dimensions of other bodies, and our own, at the same time were diminished or increased in like proportion, as for instance to half or double their size, we should with our means of space-perception be utterly unaware of the change. This would also be the case if the distension or contraction were different in different directions, provided that our own body changed in the same manner, and further that a body in rotating assumed at every moment, without suffering or exerting mechanical resistance, the amount of dilatation in its different dimensions corresponding to its position at the time. Think of the image of the world in a convex mirror. The common silvered globes set up in gardens give the essential features, only distorted by some optical irregularities. A well-made convex mirror of moderate aperture represents the objects in front of it as apparently solid and in fixed positions behind its surface. But the images of the distant horizon and of the sun in the sky lie behind the mirror at a limited distance, equal to its focal length. Between these and the surface of the mirror are found the images of all the other objects before it, but the images are diminished and flattened in proportion to the distance of their objects from the mirror. The flattening, or decrease in the third dimension, is relatively greater than the decrease of the surface-dimensions. Yet every straight line or every plane in the outer world is represented by a straight line or a plane in the image. The image of a man measuring with a rule a straight line from the mirror would contract more and more the farther he went, but with his shrunken rule the man in the image would count out exactly the same number of centimetres as the real man. And, in general, all geometrical measurements of lines or angles made with regularly varying images of real instruments would yield exactly the same results as in the outer world, an congruent bodies would coincide on being applied to one another in the mirror as in the outer worJd, all lines of sight in the outer world would be represented by straight lines of sight in the mirror. In short I do not see how men in the mirror are to discover that their
He,rruutn Von Relm11oll&
662
bodies are not rigid solids and their experiences good examples of the correctness of Euclid's axioms. But if they could look out upon our world as we can look into theirs, without overstepping the boundary, they must declare it to be a picture in a spherical mirror, and would speak of us just as we speak of them; and if two inhabitants of the different worlds could communicate with one another, neither, so far as I can see, would be able to convince the other that he had the true, the other the distorted, relations. Indeed I cannot see that such a question would have any meaning at all, so long as mechanical considerations are not mixed up with it. Now Beltrami's representation of pseudospherical space in a sphere of Euclid's space, is quite simiJar, except that the background is not a plane as in the convex mirror, but the surface of a sphere, and that the proportion in which the images as they approach the spherical surface contract, has a different mathematical expression. 12 If we imagine then, conversely, that in the sphere, for the interior of which Euclid's axioms hold good, moving bodies contract as they depart from the centre like the images in a convex mirror, and in such a way that their representatives in pseudospherical space retain their dimensions unchanged,----observers whose bodies were .regularly subjected to the same change would obtain the same results from the geometrical measurements they could make as if they lived in pseudospherical space. We can even go a srep further, and infer how the objects in a pseudospherical world, were it possible to enter one, would appear to an observer, whose eye-measure and experiences of space had been gained like ours in Euclid's space. Such an observer would continue to look upon rays of light or the lines of vision as straight lines, such as are met with in flat space, and as they really are in the spherical representation of pseudospherical space. The visual image of the objects in pseudospherical space would thus make the same impression upon him as if he were at the centre of Beltrami's sphere. He would think he saw the most remote objects round about him at a finite distance,13 let us suppose a hundred feet off. But as he approached these distant objects, they would dilate before him, though more in the third dimension than superficially, While behind him they would contract. He would know that his eye judged wrongly. If he saw two straight lines which in his estimate ran parallel for the hundred feet to his world's end, he would find on fol1owing them that the farther he advanced the more they diverged, because of the dilatation of all the objects to which he approached. On the other hand, behind him, their distance would seem to diminish, so that as he advanced they would appear always to diverge more and more. But two straight lines which from Compare the Appendix at the end of this Lecture. The reciprocal of the square of this distance, expressed in negative quantity, would be the measure of curvature of the pseudospherical space. 12
13
On tlte Orlgln and Significance 0/ Geometrical Axioms
663
his first position seemed to converge to one and the same point of the background a hundred feet distant, would continue to do this however far he went, and he would never reach their point of intersection. Now we can obtain exactly similar images of our real world, if we look through a large convex lens of corresponding negative focal length. or even through a pair of convex spectacles if ground somewhat prismatically to resemble pieces of one continuous larger lens. With these. like the convex mirror, we see remote objects as if near to us. the most remote appearing no farther distant than the focus of the lens. In going about with this lens before the eyes, we find that the objects we approach dilate exactly in the manner I have described for pseudospherical space. Now anyone using a lens, were it even so strong as to have a focal length of only sixty inches, to say nothing of a hundred feet. would perhaps observe for the first moment that he saw objects brought nearer. But after going about a little the illusion would vanish, and in spite of the false images he would judge of the distances rightly. We have every reason to suppose that what happens in a few hours to anyone beginning to wear spectacles would soon enough be experienced in pseudosphericaJ space. In short, pseudospherical space would not seem to us very strange, comparatively speaking; we should only at first be subject to illusions in measuring by eye the size and distance of the more remote objects. There would be illusions of an opposite description, if, with eyes practised to measure in Euclid's space, we entered a spherical space of three dimensions. We should suppose the more distant objects to be more remote and larger than they are, and should find on approaching them that we reached them more quickly than we expected from their appearance. But we should also see before us objects that we can fixate only with diverging lines of sight, namely, all those at a greater distance from us than the quadrant of a great circle. Such an aspect of things would hardly strike us as very extraordinary, for we can have it even as things are if we place before the eye a slightly prismatic glass with the thicker side towards the nose: the eyes must then become divergent to take in distant objects. This excites a certain feel ing of unwonted strain in the eyes, but does not perceptibly change the appearance of the objects thus seen. The strangest sight, however, in the spherical world would be the back of our own head, in which all visual lines not stopped by other objects would meet again, and which must fiU the extreme background of the whole perspective picture. At the same time it must be noted that as a small elastic flat disk, say of india-rubber, can only be fitted to a slightly curved spherical surface with relative contraction of its border and distension of its centre, so our bodies, developed in EucHd's fiat space, could not pass into curved space without undergoing similar distensions and contractions of their parts,
R erllUllfll Y on Relmholt~
their coherence being of course maintained only in as far as their elasticity permitted their bending without breaking. The kind of distension must be the same as in passing from a small body imagined at the center of Beltrami's sphere to its pseudospherical or spherical representation. For such passage to appear possible, it will always have to be assumed that the body is sufficiently elastic and small in comparison with the real or imaginary radius of curvature of the curved space into which it is to pass. These remarks will suffice to show the way in which we can infer from the known laws of our sensible perceptions the series of sensible impressions which a spherical or pseudospberical world would give us, if it existed. In doing so, we nowhere meet with inconsistency or impossibility any more than in the calculation of its metrical proportions. We can represent to ourselves the look of a pseudospherical world in all directions just as we can develop the conception of it. Therefore it cannot be allOWed that the axioms of our geometry depend on the native form of our perceptive faculty, or are in any way connected with it. It is different with the three dimensions of space. As all our means of sense-perception extend only to space of three dimensions, and a fourth is not merely a modification of what we have, but something perfectly new, we find ourselves by reason of our bodily organisation quite unable to represent a fourth dimension. In conclusion, I would .again urge that the axioms of geometry are not ptopositions pertaining only to the pure doctrine of space. As I said before, they are concerned with quantity. We can speak of quantities only when we know of some way by which we can compare, divide, and measure them. An space-measurements, and therefore in general all ideas of quantities applied to space, assume the possibility of figures moving without change of form or size. It is true we are accustomed in geometry to call such figures purely geometrical solids, surfaces, angles, and lines, because we abstract from aU the other distinctions, physical and chemical, of natural bodies; but yet one physical quality, rigidity~ is retained. Now we have no other mark of rigidity of bodies or figures but congruence, when~ ever they are applied to one another at any time or place, and after any revolution. We cannot, however, decide by pure geometry. and without mechanical considerations, whether the coinciding bodies may not both have varied in the same sense. If it were useful for any purpose, we might with perfect consistency look upon the space in which we live as the apparent space behind a convex mirror with its shortened and contracted background; or we might consider a bounded sphere of our space, beyond the limits of which we perceive nothing further, as infinite pseudospherical space. Only then we should have to ascribe to the bodies which appear to us to be solid, and to our own body at the same time. corresponding distensions and con-
tractions, and we should have to change our system of mechanical principles entirely; for even the proposition that every point in motion, if acted upon by no force, continues to move with unchanged velocity in a straight line, is not adapted to the image of the world in the convex-mirror. The path would indeed be straight, but the velocity would depend upon the place. Thus the axioms of geometry are not concerned with space-relations only but also at the same time with the mechanical deportment of soUdelt bodies in motion. The notion of rigid geometrical figure might indeed be conceived as transcendental in Kantts sense, namely, as formed independently of actual experience, which need not exactly correspond therewith, any more than natural bodies do ever in fact correspond exactly to the abstract notion we have obtained of them by induction. Taking the notion of rigidity thus as a mere ideal, a strict Kandan might certainly look upon the geometrical axioms as propositions given, a priori, by transcendental intuition, which no experience could either confirm or refute, because it must first be decided by them whether any natural bodies can be considered as rigid. But then we should have to maintain that the axioms of geometry are not synthetic propositions, as Kant held them; they would merely define what qualities and deportment a body must have to be recognised as rigid. But if to the geometrical axioms we add propositions relating to the mechanical properties of natural bodies, were it only the axiom of inertia, or the single proposition, that the mechanical and physical properties of bodies and their mutual reactions are, other circumstances remaining the same, independent of p]ace, such a system of propositions has a real import which can be confirmed or refuted by experience, but just for the same reason can also be gained by experience. The mechanical axiom, just cited, is in fact of the utmost importance for the whole system of our mechanical and physical conceptions. That rigid soHds, as we call them, which are really nothing else than elastic solids of great resistance, retain the same form in every part of space if no external force affects them, is a single case falling under the general principle. In conclusion, I do not, of course, maintain that mankind first arrived at space-intuitions, in agreement with the axioms of Euclid, by any carefuny executed systems of exact measurement. It was rather a succession of everyday experiences, especially the perception of the geometrical similarity of great and small bodies, only possible in flat space, that led to the rejection, as impossible, of every geometrical representation at variance with this fact. For this no knowledge of the necessary logical connection between the observed fact of geometrical similarity and the axioms was needed; but only an intuitive apprehension of the typicaJ relations between Jines, planes, angles, &c., obtained by numerous and attentive observations
Bemum,. YD,. Bel_Dt"
-an intuition of the kind the artist possesses of the objects he is to represent, and by means of which he decides with certainty and accuracy whether a new combination, which he tries, will correspond or not with their nature. It is true that we have no word but intuition to mark this; but it is knowledge empirically gained by the aggregation and reinforcement of similar recurrent impressions in memory, and not a transcendental form given before experience. That other such empirical intuitions of fixed typical relations, When not clearly comprehended, have frequently enough been taken by metaphysicians for Q priori principles, is a point on which I need not insist. To sum up, the final outcome of the whole inquiry may be thus expressed:( 1) The axioms of geometry. taken by themselves out of all connection with mechanical propositions, represent no relations of real things. When thus isolated, if we regard them with Kant as forms of intuition transcendentally given, they constitute a form into which any empirical content whatever will fit, and which therefore does not in any way limit or determine beforehand the nature of the content. This is true, however, not only of Euclid's axioms, but also of the axioms of spherical and pseudospherical geometry. (2) As soon as certain principles of mechanics are conjoined with the axioms of geometry, we obtain a system of propositions which has real import, and which can be verified or overturned by empirical observations, just as it can be inferred from experience. If such a system were to be taken as a transcendental form of intuition and thought, there must be assumed a pre-established harmony between form and reality. APPENDIX The elements of the geometry of spherical space are most easily obtained by putting for space of four dimensions the equation for the sphere x2 +
y2
+ Z2 + 12 = R2
• • • • • • ( 1)
and for the distance ds between the points (x, y, (y + dy) (z + dz) (I + dl)] the value ds2
=
dx 2
+ dy2 + dz 2 + dt 2
z.
t) and [(x
••••
+ dx)
(2)
It is easily found by means of the methods used for three dimensions that the shortest lines are given by equations of the form
ax + by + cz + It Q + yz + .,..t .I.. ax + /JY in which a, b, c,
It
as well as a, {J. y,
= O} =0 · · · · · ·
.p, are constants.
(3)
011 tire Orilill alld SilllilicQlIU
01 Geometrical Axioms
The length of the shortest are, s, between the points (x, y, (t, "I, " T) follows, as in the sphere, from the equation
s
cos-= R
xt + Y"I +
Z' +
tT
Z,
t), and
. . . . . (4)
R2
One of the co-ordinates may be eliminated from the values given in 2 to 4, by means of equation 1, and the expressions then apply to space of three dimensions. If we take the distances from the points
from which equation 1 gives
= R, then,
T
sin ( in which or,
= VX2
CT
So
= R . arc sin (
ji)= i
~)
+ y2 +Z2
= R . arc tang ( : ) . . . . . . (5)
In this, So is the distance of the point, x, y, z, measured from the centre of the co-ordinates. If now we suppose the point x, y, z, of spherical space, to be projected in a point of plane space whose co-ordinates are respectively Rx
Ry
Rz
t
t
t
t=-,J!=-,f=-
t:!
+ l!2 + ,2 =
R2q2 [2
= --
t2 then in the plane space the equations 3, which belong to the straightest lines of spherical space, are equations of the straight Hne. Hence the shortest lines of spherical space are represented in the system of t, l!, " by straight lines. For very small values of x, y, Z, t ::;;: R, and
t = x, l! = y, , = Z Immediately about the centre of the co-ordinates, the measurements of both spaces coincide. On the other hand, we have for the distances from the centre So = R. arc tang ( + : ) . . . . . . (6)
In this, [ may be infinite; but every point of plane space must be the pro-
jection of two points of the sphere, one for which So < 1h R11', and one for which So > 1h R11'. The extension in the direction of t is then
dso
R2
----dt R2 + t 2
In order to obtain corresponding expressions for pseudospherical space, let Rand t be imaginary; that is, R Itt, and t tt. Equation 6 gives then
=
=
So t tang-=±til tit
from which. eliminating the imaginary form, we get 80
It+t It-t
= ¥.a Illog. nat. - - -
=
=
Here So has real values only as long as t R; for t Il the distance So in pseudospherical space is infinite. The image in plane space is, on the contrary, contained in the sphere of radius R, and every point of this sphere forms only one point of the infinite pseudospherical space. The extension in the direction of t is tho
Il'
----dt 11 t 2 -
'
For linear elements, on the contrary, whose direction is at right angles to t. and for which t is unchanged, we have in both cases
Vdx2 + dy2 + dz'l
t
t
u
-------=-=-=Il t
COMMENTARY ON
Synlmetry N the everyday sense symmetry carries the meaning of balance. pro. portion. harmony. regularity of form. Beauty is sometimes linked with symmetry. but the relationship is not very illuminating since beauty is an even vaguer quaUty than symmetry. The protean character of the concept of symmetry is indicated by its widely different uses. A statue, a musical composition, a gaseous nebula, an ethical standard of action: each may be described as displaying symmetry. The ethical standard is that of moderation-a mean between extremes of action. The concept first acquires precision in geometry. Vague notions are replaced by the idea of bilateral symmetry, the symmetry of left and right, exhibited in the higher animals. "A body, a spatial configuration. is symmetric with respect to a given place E if it is carried into itself by reflection in E. tt The mathematical concept lends itself to remarkable elaboration. We are apt to think of symmetry as a static property; the analytic approach which deals with the genesis of symmetric forms is far more fruitful. Our understanding of these forms is deepened by regarding them as the product of various transformations or motions by means of which one pattern is converted, element by element. into another. Simple forms, when subjected to translatory and rotational motions, are transformed into marvelously intricate designs; these designs in two and three dimensions are encountered in art, biology, chemistry, astronomy. physics and crystallography. The ruling principle of symmetry is applied to forms and processes of every conceivable kind; that is to say, the forms and processes are defined in terms of the combination of motions which gives birth to them. This method of extending our insight is typical, as Weyl says, for all theoretic knowledge. "We begin with some general but vague principle (symmetry in the first sense), then find an important case where we can give that notion a concrete precise meaning (bilateral symmetry), and from that case we gradually rise again to generality, guided more by mathematical construction and abstraction than by the mirages of philosophy; and if we are lucky we end up with an idea no less universal than the one from which we started." On the eve of his retirement from the Institute for Advanced Study in Princeton, Hermann Weyl, a commanding figure among the mathematicians of this century, delivered a series of lectures on symmetry) They present a masterful survey of the applications of the principle of symmetry
I
I
For a biographica] note on Wey], see p. 1830. 669
in sculpture, painting, architecture, ornament and design; its manifestations in organic and inorganic nature; its philosophical and mathematical significance. Symmetry establishes a ridiculous and wonderful cousinship between objects, phenomena and theories outwardly unrelated: terrestrial magnetism, women's veils, polarized light, natural selection, the theory of groups, invariants and transformations, the work habits of bees in the hive, the structure of space, vase designs, quantum physics, scarabs, Hower petals, X-ray interference patterns, cell division in sea urchins, equilibrium positions of crystals, Romanesque cathedrals, snowflakes, music, the theory of relativity. The structure of these relationships is depicted by Weyl in a remarkable sweep. The style is not always easy; neither is the subject. Nevertheless the book affords an entry into a profound and fascinating subject which demonstrates, perhaps uniquely, the working of the mathematical intellect, the evolution of intuitive concepts into grand systems of abstract ideas. I have selected the first two of the lectures--on bilateral and related symmetries; I was tempted to give the entire series. You will discover within a few pages why it was so hard to resist the inclination.
• • . What immortal hand or eye. Dare frame thy fearful symmetry?
9
-WILLIAM BLAn
Symmetry By HERMANN WEYL BILATERAL SYMMBTRY
IF I am not mistaken the word symmetry is used in our everyday language in two meanings. In the one sense symmetric means something like wellproportioned, well-balanced, and symmetry denotes that sort of concordance of several parts by which they integrate into a whole. Beauty is bound up with symmetry. Thus Polykleitos, who wrote a book on proportion and whom the ancients praised for the harmonious perfection of his sculptures, uses the word, and DUrer follows him in setting down a canon of proportions for the human figure. 1 In this sense the idea is by no means restricted to spatial objects; the synonym Uharmony" points more toward its acoustical and musical than its geometric applications. Ebenmass is a good German equivalent for the Greek symmetry; for like this it carries also the connotation of "middle measure:' the mean toward which the virtuous should strive in their actions according to Aristotle's Nicomachean Ethics, and which Galen in De temperamentis describes as that state of mind which is equally removed from both extremes: crop.pA"('POJl hIp
ilCQ."('EpoV "(',sal I.lCptDJI d.1t~E&.
The image of the balance provides a natural link to the second sense in which the word symmetry is used in modem times: bilateral symmetry. the symmetry of left and right, which is so conspicuous in the structure of the higher animals, especially the human body. Now this bilateral symmetry is a strictly geometric and, in contrast to the vague notion of symmetry discussed before, an absolutely precise concept. A body, a spatial configuration, is symmetric with respect to a given plane E if it is 1 Durer, Vier Bucher von menschlicher Proportion, 1S28. To be exact, Durer him· self does not use the word symmetry, but the "authorized" Latin translation by his friend Joachim Camerarius (lS32) bears the title De symmetria partium. To Polykleitos the statement is ascribed hrEp~ fjc"07l"O"KWlI, IV, 2) that "the employment of a great many numbers would almost engender correctness in sculpture." See also Herbert Senk, Au sujet de l'expression (f1Jf.LfJATPla, dans Diodore [, 98, S-9, in Chronique d'Egypte 26 (1951), pp. 63-66. Vitruvius defines: "Symmetry results from proportion • . • Proportion is the commensuration of the various constituent parts with the whole:' For a more elaborate modern attempt in the same direction see George David Birkhoff. Aesthetic measure, Cambridge, Mass., Harvard University Press, 1933, and the lectures by the same author on "A mathematical theory of aesthetics and its applications to poetry and music," Rice institute PlII1Iph'et. Vol. 19 (July, 1932), pp. 189-342.
671
672
carried into itself by reflection in E. Take any line I perpendicular to E and any point p on I: there exists one and only one point pi on 1 which has the same distance from E but lies on the other side. The point p' coincides with p only if p is on E. Reflection in E is that mapping of space
p
E FIGURE I-Reftection in E.
upon itself, S: p -4 p', that carries the arbitrary point p into this its mirror image p' with respect to E. A mapping is defined whenever a rule is established by which every point p is associated with an image p'. Another example: a rotation around a perpendicular axis, say by 300 , carries each point p of space into a point p' and thus defines a mapping. A figure has rotational symmetry around an axis 1 if it is carried into itself by all rotations around I. Bilateral symmetry appears thus as the first case of a geometric concept of symmetry that refers to such operations as reflections or rotations. Because of their complete rotational symmetry, the circle in the plane, the sphere in space were considered by the Pythagoreans the most perfect geometric figures, and Aristotle ascribed spherical shape to the celestial bodies because any other would detract from their heavenly perfection. It is in this tradition that a modern poet 2 addresses the Divine Being as "Thou great symmetryu: God, Thou great symmetry, Who put a biting lust in me From whence my sorrows spring. For all the frittered days That I have spent in shapeless ways Give me one perfect thing. Symmetry, as wide or a8 narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection. The course these lectures will take is as follows. 3 First I will discuss 2 Anna Wickham, "Envoi," from The Contemplative Quarry, Harcourt, Brace and Co., 1921. 3 [The first two 1ectures are given here. Lecture 3 deals with ornamental symmetry, Lecture 4 with crystals and the general mathematical idea of symmetry. ED.]
m
bilateral symmetry in some detail and its role in art as well as oraanic and inorpnic nature. Then we shall amer· alize this concept gradually, in the direction indicated by our example of rotational sym· metry, first staying within the confines of geometry. but then going beyond these limits through the process of mathematical abstraction aloog a road that will finally }ead w to a mathematical idea of great generality, the Platonic idea as it were behind all the special appearances and ap-
plications of symmetry. To a certain degree this scheme is typical for all theoretic knowl· edge: We begin with some gmeral but vague principle (symmetry in the first sense). then find an important case where we can give that n0tion a concrete precise mean· ing (bilateral symmetry), and from that Coase we IIradually rise again to generality, guided more by mathematical construction and abstraction than by the mirages of philosophy; and if we are lucky we end up with an idea no Jess univenal than the one from which we started. Gone may be much of its emotional appeal, but it has the same or even greater unifying power in. the realm of thought and is exact in. stead of vague. J open the discussion on bilatera] symmetry by using this
no.,..,
."noble Greek sculpture from tbe fourth century
B.C.,
tbe statue of 8 praying
boy (Figure 2). to let you feel as in B symbol the areal significance of this type of symmetry both for life and art. One may ask whether the aesthetic value of symmetry depends on its vital value: Did tbe artist discover the symmetry with which nature according to some inherent law has endowed its creatures, and then copied and perfected what nature presented but in
imperfect realizations; or has tbe aesthetic value of symmetry an independent source? I am inclined to think with Plato that the mathematical idea is the eommon origin of both: the mathematical Jaws governing nature are the origin of symmetry in nature, the intuitive realization of the idea in the ereati~ artist's mind its origin in art; although I am ready to admit tbat in the arts the fact of the bilateral symmetry of the human body in its outward appearance has acted as an additional stimulw. Of aU ancient peoples the Sumerians seem to have been particularly fond of strict bilateral (If heraldic symmetry. A typical design on the famous silver vase of King Entemena, who ruled in the city of Lagash around 2700 B.C., shows a lion·headed eagle wilh spread wings en face. each of whose claws grips a stag in side view. which in its turn is frontaUy
FIOURE! )
FJOVlU! •
attacked by a lion (the stags in the upper design are replaced by goals in the lower) (Figure 3). Extension of the exact symmetry of the eagle to the other beasts obviously enforces their duplication. Not much later tbe eagle is given two heads facing in either direction, tbe fonnal principle of symmetry thus completely overwhelming tbe imitative principle of truth to nature. This heraldic desip can then be followed to Persia, Syria, later to Byzantium. and anyone who lived before the First World War will remember the double-beaded eagle in the coats-of-anns of Czarist Russia and the Austro-Hungarian monarchy. Look now at this Sumerian picture (Figure 4). lbe two eagle.headed men are nearly but not quite symmetric; wby not? ]n plane geometry reftection in a vertical line I can also be brougbt about by rotating the plane in space aroWld the axis I by ISO-. If you look at their arms you wouJd say these two monsters arise from each other by such rotation; the overlappings depicting their position in space prevent the plane pidure from having bilaterial symmetry. Yet the artist aimed at tbat symmetry by giving both figum: a balf tum toward 1M observer and also by the alTllJl,ement of feet and wings: the drooping wing is the right one in the left figure. the left one in the right figure. The designs on the cylindrical Babylonian seal "t~ are frequently ruled by heraldic symmetry. ] remember seeing in the collection of my former colleague, the late Ernst Herzfeld, samples where for symmetry's sake not the bead. but the lower bull·shaped part of a god's bodY. rendered in profile. was doubled and given four instead of two hind legs. In QJristian times one may see an analogy in certain representations of the Eucharist as on this Byzantine platen (Figure S). where two symmetric
.,. Cbrists are faeicg the disciples. But here symmetry is 001 cornplele: aDd bas clearly more than formal aianificance. for Christ on one side breaks the bread, on the other pours the wine. Between Sumeria and Byzantium Jet me insert Persia: "lbese enameled sphinxes (Figure 6) are from Darius' palace in Susa built in the days of Marathon. Crossing the Aegean we flod these floor patterns (flJUre 7) at the Megaron in Tiryns. latc helladic about 1200 B.C. Who believes strongly in historic: continuity and dependence will trace the graceful designs of marine life, dolphin and octopus. back to tbe Minoan culture of Crete. the heraldic symmetry to oriental. in tbe last instance Sumerian, influence. Skipping thousands of years we still see the same influences at work in tbis plaque (Figure 8) from the altar enclo$ure in the dome of Torcello, Italy, eleventh century A.ll. The peacocks drinking from a pine well among vine leaves are an ancient Christian symbol of immortality. the structural heraldic symmetry is oriental For ie contrast to the orient, occidental art. like life itself, is inclined to mitigate. to loosen. to modify, even to break strict symmetry. But seldom is asymmetry merely the absence of symmetry. Even in asym. metric designs one feels symmetry as tbe norm from which one deviates under the inftuence of forces of noa·formal character. I think the riders
.n
FlO"" •
FIOUJlE 7
from the famous Etruscan Tomb of Ihc Triclinium al Cometo (Figure 9) provide a good example, I have already mentioned representations of the Eucharist with Christ duplicated handing out bread and wine. The central group, Mary flanked by two angels. in this mosaic of the Lord's Ascension (Figure 12) in the caU.edral at Monreale. Sk:ily (twelfth century). has atmost perfect symmelry. {The band ornaments above and below the mosaic will demand our attention in the second lecture.] The principle of symmetry is somewhat less strictly observed in an earlier mosaic from San Apollinare in Ravenna (Figure 10). showing Christ surrounded by an angelic guard of honor. For instance Mary in the MORreale mosaic raises both hands symmetrically, in the orans gesture: hert only the right hands
".
....,.. . are raised. Asymmetry has made further inroads in the next picture (Figure 11). a Byzantine rdief ikon from San Marco, Venice. It is a Deesis. and. of course, the two figures praying fOr men:y as the Lord is about 10 pronounce the Jast judgment cannot be minor images of each other; for to the right stands his Virgin Mother, to the left lohn the Baptist. You may also think of Mary and John the Evangelist on both .sides of the cross in crucifixions as examples of broken symmetry. Clearly we touch ground here where the precise geometric notion of bilateral symmetry begins to dissolve into the vague notion of Aw~. wogenheit. balanced design with which we started. "Symmetry," says Dagobert Frey in an artk:le On the Problem of Symmetry in A,t,t "signifies rest and binding. asymmetry motion and loosening. the one orda' and law, the other arbitrariness and accident. the one formal rigidity and constraint. the other life, play and freedom. Wherever God or Christ are represenled as symbols for everlasling trulh or justice they are given in the symmetric frontal view. not in profile. Probably for similar reasons public buildings and houses of worship, whether Ihey are Greek temples or Christian basilicas and cathedrals, are bilaterally symmetric. It is, how· H
• Studiwn GmeraJe. p. 276.
".
PJou.e, ever, true that not infrequently the two towers of Gothic cathedrals are different, as for instance in Olartres. But in practically every case Ihis seems to be due to the history of the cathedral:namely to the fact that the towen were bWh in different periods. It is understandable that • later time was no longer satisfied with the design of an earlier period; hence one may speak here of historic asymmetry. MirTOr images occur where then~ is a mirror. be it a lake reflecting. landscape or a alass mirror into which a woman looks. Nature as wen as painters make use of this motif. I trust, examples will easily come to your mind. The one most familiar to me, because I look at it in my study every day. is Hodler's LAke of Silver plana. While we are about to tum from art to nature, let us tarry a few minutes and first consider what ont: may call the mathematical philosophy o f left nnd riRh,. To the scientific mind there is no inner difference, no polarity between left and right. as there is for instance in the contrast of male and female, or of the anterior and posterior ends of an animal. II requires an arbitrary act of choice to determine what is left and what is right. But afler it is made for one body it is determined for every body. I must try to make this a little clearer. In space the distinction of left and right con· cems the orientation of a screw. If you weak of turning left you mean that Ihe sense in which you tum combined with the upward direction from foot 10 bead of your body forms a left screw. TIte daily rotation of the earth together with the direction of its axis from South 10 North Pole is
...
FIOUD 10
FIOun II
a left screw, it is a right screw if you give the axis the opposite direction. There are certain crystalline substances called oplically active which betray the inner asymmetry of their constitution by turning the polarization
... plane of polarized light seal through them either 10 the left or to the right: by this, of course, we mean thai lhe sense in which the plane rotates while the light travels in a definite direction. combined with that direction. forms a left screw (or a riShi one, as the case may be). Hence when we said above and DOW repeat in a terminology due 10 Leibniz, ahat left and riShl are inJisurn1bJ~. we wanl to express that the inner structure of space does not permit us, except by amirrary choice. 10 distinguisb a left from a right screw. [ wish to make this fundamental notion stin more precise. for OIl it depends the entire theory of relativity, which is but another aspect of symmetry. According to Euclid one can describe the structure of space by a number of basic: relations between poinls, fiuch 85 ABC lie (In a saraiahl line. ABeD lie in a plane. AB is congruent CD. Perhaps the best way of describing the structure of space is the one Helmholtz adopted: by the
HerlMlIlI Weyl
682
single notion of congruence of figures. A mapping S of space associates with every point p a point p' : p ~ p'. A pair of mappings S, S' : p ~ p', p' ~ p, of which the one is the inverse of the other, so that if S carries p into p' then S' carries p' back into p and vice versa, is spoken of as a pair
01 Do Co
FIGURE 13
of one-to-one mappings or transformations. A transformation which preserves the structure of space--and if we define this structure in the Helmholtz way, that would mean that it carries any two congruent figures into two congruent ones-is called an automorphism by the mathematicians. Leibniz recognized that this is the idea underlying the geometric concept of similarity. An automorphism carries a figure into one that in Leibniz' words is "indiscernible from it if each of the two figures is considered by itself." What we mean then by stating that left and right are of the same essence is the fact that reflection in a plane is an automorphism. Space as such is studied by geometry. But space is also the medium of all physical occurrences. The structure of the physical world is revealed by the general laws of nature. They are formulated in terms of certain basic quantities which are functions in space and time. We would conclude that the physical structure of space "contains a screw," to use a
FIGURE 14
683
suggestive figure of speech, if these laws were not invariant throughout with respect to reflection. Ernst Mach tells of the intellectual shock he received when he learned as a boy that a magnetic needle is deflected in a certain sense, to the left or to the right. if suspended paranel to a wire through which an electric current is sent in a definite direction (Figure 14). Since the whole geometric and physical configuration, including the electric current and the south and north poles of the magnetic needle, to all appearances, are symmetric with respect to the plane E laid through the wire and the needle, the needle should react like Buridan's ass between equal bundles of hay and refuse to decide between left and right, just as scales of equal arms with equal weights neither go down on their left nor on their right side but stay horizontal. But appearances are sometimes deceptive. Young Mach's di1emma was the result of a too hasty assumption concerning the effect of reflection in E on the electric current and the positive and negative magnetic poles of the needle: whiJe we know a priori how geometric entities fare under reflection, we have to learn from nature how the physical quantities behave. And this is what we find: under reflection in the plane E the electric current preserves its direction. but the magnetic south and north poles are interchanged. Of course this way out, which re-establishes the equivalence of left and right, is possible only because of the essential equality of positive and negative magnetism. An doubts were dispelled when one found that the magnetism of the needle has its origin in molecular electric currents circulating around the needle's direction; it is dear that under reflection in the plane E such currents change the sense in which they flow. The net result is that in all physics nothing has shown up indicating an intrinsic difference of left and right. Just as all points and all directions in space are equivalent, so are left and right. Position. direction, left and right are relative concepts. In language tinged with theology this issue of relativity was discussed at great length in a famous controversy between Leibniz and Clarke, the Jatter a clergyman acting as the spokesman for Newton,a Newton with his belief in absolute space and time considers motion a proof of the creation of the world out of God's arbitrary will, for otherwise it would be inexplicab1e why matter moves in this rather than in any other direction. Leibniz is loath to burden God with such decisions as lack "sufficient reason." Says he, "Under the assumption that space be something in itself it is impossible to give a reason why God should have put the bodies (without tampering with their mutual distances and relative positions) just at this particular place and not somewhere else; for instance. why He should not have arranged everything in the opposite order by turning East and West about. If. on the other hand, 5 See G. W. Leibniz, Philosophische Schri/len, ed. Gerhardt (Berlin 1875 seq.), VB, pp. 352-440. in particular Leibniz' third letter. § 5.
space is nothing more than the spatial order and relation of things then the two states supposed above, the actual one and its transposition, are in no way different from each other . . . and therefore it is a quite inadmissible question to ask why one state was preferred to the other." By pondering the probJem of left and right Kant was first led to his conception of space and time as forms of intuition. 6 Kant's opinion seems to have been this: If the first creative act of God had been the forming of a left hand then this hand, even at the time when it could be compared to nothing else, had the distinctive character of left, which can only intuitively but never conceptually be apprehended. Leibniz contradicts: According to him it would have made no difference if God had created a "right" hand first rather than a "Jeft" one. One must follow the world's creation a step further before a difference can appear. Had God, rather than making first a left and then a right hand, started with a right hand and then formed another right hand, He would have changed the plan of the universe not in the first but in the second act, by bringing forth a hand which was equally rather than oppositely oriented to the first-created specimen. Scientific thinking sides with Leibniz. Mythical thinking has always taken the contrary view as is evinced by its usage of right and left as symbols for such polar opposites as good and evil. You need only think of the double meaning of the word right itself. In this detail from Michelangelo's famous Creation of Adam from the Sistine Ceiling (Figure 15) God's right hand, on the right, touches life into Adam's left. People shake right hands. Sinister is the Latin word for left, and heraldry still speaks of the left side of the shield as its sinister side. But sinistrum is at the same time that which is evil, and in common English only this figurative meaning of the Latin word survives. j Of the two malefactors who were crucified with Christ, the one who goes with Him to paradise is on His right. St. Matthew, Chapter 25, describes the last judgment as follows: "And he shall set the sheep on his right hand but the goats on the left. Then shall the King say unto them on his right hand, Come ye, blessed of my Father, inherit the Kingdom prepared for you from the foundation of the world. . . . Then he shall say also unto them on the left hand, Depart from me, ye cursed, into everlasting fire, prepared for the devil and his angels!' I remember a lecture Heinrich Wolfflin once delivered in Zurich on "Right and Jeft in paintings"; together with an article on "The problem of inversion (Umkehrung) in Raphael's tapestry cartoons," you now find it printed in abbreviated form in his Gedanken lour Kunstgeschichte, 1941. 6 Besides his UKritik der reinen Vernunft" see especially § 13 of the Prolegomena zu einer jeden kiinftigen Metaphysik . •.. . ., I am not unaware of the strange fact that as a terminus techniclls in the language of the Roman augurs sinislrum had just the opposite meaning of propitious.
..
....,.." By a number of exampies. as Raphael's Sinine MtJdonna and Rembrandt's etching Ltmdscope with the ,Ivu trus, WOlfftin tries to show that right in painting has another Stimnnmgswl!rt than.lefL Practically all methods of reproduction mterdlange lefl and riaht. and it seems that former limes ~ much iess sensitive than we are toward such inversion. (Even Rembnndt dad not hesitate 10 brins his Descent from the Cross as a convene etching upon the market.) Considering that we do a 101 more reading than the peop&e, say, of the sixteenth century, this suggests the hypothesis that the difference pointed out by Wijfflin is connected with OUl" habit of reading from left 10 rigbL AI. far as ] remember. be himself rejected this as well as a number of other psycboklrlical explanations put forward in the discussion after his lecture. The printed text concludes with the remark that the problem Mobviously bas deep roots,. roots which reach down to the VU'J fouodalions of our sensuous nature." ] for my part am disinclined to take the matter thai seriously.' In sc:ieoce the belief in tbe equivalence of kft and riabt btu been upbeld even in the face of certain bioIogjca1 racts presently to be mentioned which seem to suggest their inequivalmce even mor-c llronaly than does the de· viation of the magnetic needle which shocked youoS Mach. 'The ..me .Cf. abo A f'Utauer."Unb und rec:brs im 8iack," Amicis, lillubuell tkr 611nrridWdtnr Gahrit, 1926, p . n ; JulillS Y. ScNoun, "Intomo .u.. ietbl.ra dei quadri." Critka 28,1930, p. 12; Pali ~ "1UabI and left in ~ canoons." h/umtll til tilt W~ -.I C04UtffuIli IlUIiIIlln 7, 19«, p. 12.
686
Hernuanrl W.,I
problem of equivalence arises with respect to past and future, which are interchanged by inverting the direction of time, and with respect to positive and negative electricity. In these cases, especially in the second, it is perhaps clearer than for the pair left-right that a priori evidence is not sufficient to settle the question; the empirical facts have to be consulted. To be sure, the role which past and future play in our consciousness would indicate their intrinsic difference-the past knowable and unchangeable, the future unknown and still alterable by decisions taken now-and one would expect that this difference has its basis in the physical laws of nature. But those laws of which we can boast a reasonably certain knowledge are invariant with respect to the inversion of time as they are with respect to the interchange of left and right. Leibniz made it clear that the temporal modi past and future refer to the causal structure of the world. Even if it is true that the exact "wave laws" formulated by quantum physics are not altered by letting time flow backward, the metaphysical idea of causation, and with it the one way character of time, may enter physics through the statistical interpretation of those laws in terms of probability and particles. Our present physical knowledge leaves us even more uncertain about the equivalence or non-equivalence of positive and negative electricity. It seems difficult to devise physical laws in which they are not intrinsically alike; but the negative counterpart of the positively charged proton still remains to be discovered. This half-philosophical excursion was needed as a background for the discussion of the left-right symmetry in nature; we had to understand that the general organization of nature possesses that symmetry. But one will not expect that any special object of nature shows it to perfection. Even so, it is surprising to what extent it prevails. There must be a reason for this, and it is not far to seek: a state of equilibrium is likely to be symmetric. More precisely, under conditions which determine a unique state of equilibrium the symmetry of the conditions must carry over to the state of equilibrium. Therefore tennis balls and stars are spheres; the earth would be a sphere too if it did not rotate around an axis. The rotation flattens it at the poles but the rotational or cylindrical symmetry around its axis is preserved. The feature that needs explanation is, therefore, not the rotational symmetry of its shape but the deviations from this symmetry as exhibited by the irregular distribution of land and water and by the minute crinkles of mountains on its surface. It is for such reasons that in his monograph on the left-right problem in zoology Wilhelm Ludwig says hardly a word about the origin of the bilateral symmetry prevailing in the animal kingdom from the echinoderms upward, but in great detail discusses all sorts of secondary asymmetries superimposed upon the symmetrical ground plan. 9 I quote: "The human body like that of the 9
W. Ludwig, Rechts-links-Problem ;m Tie"e;ch und beim Menschen, Berlin 1932.
681
other vertebrates is basically built bilateral-symmetrically. All asymmetries occurring are of secondary character. and the more important ones affecting the inner organs are chiefly conditioned by the necessity for the intestinal tube to increase its surface out of proportion to the growth of the body, which lengthening led to an asymmetric folding and rolling-up. And in the course of phylogenetic evolution these first asymmetries concerning the intestinal system with its appendant organs brought about asymmetries in other organ systems." It is well known that the heart of mammals is an asymmetric screw, as shown by the schematic drawing of Figure 16.
FIGURE 16
If nature were all lawfulness then every phenomenon would share the full symmetry of the universal laws of nature as formulated by the theory of relativity. The mere fact that this is not so proves that contingency is an essential feature of the world. Clarke in his controversy with Leibniz admitted the latter's principle of sufficient reason but added that the sufficient reason often lies in the mere will of God. I think. here Leibniz the rationalist is definitely wrong and Clarke on the right track. But it would have been more sincere to deny the principle of sufficient reason altogether instead of making God responsible for all that is unreason in the world. On the other hand Leibniz was right against Newton and Clarke with his insight into the principle of relativity. The truth as we see it today is this: The laws of nature do not determine uniquely the one world that actually exists, not even if one concedes that two worlds arising from each other by an automorphic transformation, i.e.. by a transformation which preserves the universal laws of nature, are to be considered the same world. If for a lump of matter the overall symmetry inherent in the laws of nature is limited by nothing but the accident of its position P then it will assume the form of a sphere around the center P. Thus the lowest forms of animals. sman creatures suspended in water, are more or less spherical. For forms fixed to the bottom of the ocean the direction of gravity is an important factor, narrowing the set of symmetry operations from all rota-
Hermt#llt We)'1
tions around the center P to all rotations about an axis. But for animals capable of self-motion in water, air, or on land both the postero--anterior direction in which their body moves and the direction of gravity are of decisive influence. After determination of the antero--posterior, the dorso-ventral, and thereby of the 1eft-right axes, only the distinction between left and right remains arbitrary, and at this stage no higher symmetry than the bilateral type can be expected. Factors in the phylogenetic evolution that tend to introduce inheritable differences between left and right are likely to be held in check by the advantage an animal derives from the bilateral formation of its organs of motion, cilia or muscles and limbs: in case of their asymmetric development a screw-wise instead of a straightforward motion would naturally result. This may help to explain why our limbs obey the law of symmetry more strictly than our inner organs. Aristophanes in Plato's Symposium tells a different story of how the transition from spherica1 to bilateral symmetry came about. Originally, he says, man was round, his back and sides forming a circle. To humble their pride and might Zeus cut them into two and had Apollo turn their faces and genitals around; and Zeus had threatened, "If they continue insolent I will split them again and they shall hop around on a single leg." The most striking examp1es of symmetry in the inorganic world are the crystals. The gaseous and the crystalline are two clear-cut states of matter which physics finds relatively easy to exp1ain; the states in between these two extremes, like the fluid and the p1astic states, are somewhat less amenable to theory. In the gaseous state molecules move freely around in space with mutually independent random positions and velocities. In the crystalline state atoms oscillate about positions of equilibrium as if they were tied to them by elastic strings. These positions of equilibrium form a fixed regular configuration in space. to • • • While most of the thirty-two geometrically possible systems of crystal symmetry involve bilateral symmetry, not aU of them do. Where it is not involved we have the possibility of so-caned enantiomorph crystals which exist in a laevo- and dextra-form, each form being a mirror image of the other, like left and right hands. A substance which is optically active, i.e., turns the plane of polarized light either left or right, can be expected to crystaUize in such asymmetric forms. If the 1aeva-form exists in nature one would assume that the dextra-form exists likewise, and that in the average both occur with equal frequencies. In 1848 Pasteur made the discovery that when the sodium ammonium salt of optically inactive racemic acid was recrystallized from an aqueous solution at a lower temperature the deposit consisted of two kinds of tiny crystals which were mirror images of each other. They were carefully separated, and the acids set free from the one and the other proved to 10 [In a later lecture Weyl explains how the visible symmetry of crystals derives from their regular atomic arrangement. ED. j
Symmetry
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have the same chemical composition as the racemic acid, but one was optically laevo-active, the other dextra-active. The latter was found to be identical with the tartaric acid present in fermenting grapes, the other had never before been observed in nature. "Seldom," says F. M. Jaeger in his lectures On the principle of symmetry and its applications in natural science, "has a scientific discovery had such far-reaching consequences as this one had." Quite obviously some accidents hard to control decide whether at a spot of the solution a laevo- or dextro-crystal comes into being; and thus in agreement with the symmetric and optically inactive character of the solution as a whole and with the law of chance the amounts of substance deposited in the one and the other form at any moment of the process of crystallization are equal or very nearly equal. On the other hand nature, in giving us the wonderful gift of grapes so much enjoyed by Noah, produced only one of the forms, and it remained for Pasteur to produce the other! This is strange indeed. It is a fact that most of the numerous carbonic compounds occur in nature in one, either the laevo- or the dextroform only. The sense in which a snail's shell winds is an inheritable character founded in its genetic constitution, as is the "left heart" and the winding of the intestinal duct in the species Homo sapiens. This does not exclude that inversions occur, e.g. situs inversus of the intestines of man occurs with a frequency of about 0.02 per cent; we shall come back to that later! Also the deeper chemical constitution of our human body shows that we have a screw, a screw that is turning the same way in every one of us. Thus our body contains the dextro-rotatory form of glucose and laevo-rotatory form of fructose. A horrid manifestation of this genotypical asymmetry is a metabolic disease caned phenylketonuria, leading to insanity, that man contracts when a small quantity of laevo-phenylalanine is added to his food. while the dextra-form has no such disastrous effects. To the asymmetric chemical constitution of living organisms one must attribute the success of Pasteur's method of isolating the laeva- and dextroforms of substances by means of the enzymatic action of bacteria, moulds, yeasts, and the like. Thus he found that an original1y inactive solution of some racemate became gradually laevo-rotatory if Penicillium glaucum was grown in it. Clearly the organism selected for its nutriment that form of the tartaric acid molecule which best suited its own asymmetric chemical constitution. The image of lock and key has been used to i11ustrate this specificity of the action of organisms. In view of the facts mentioned and in view of the faHure of all attempts to "activate" by mere chemical means optical1y inactive material,ll it is understandable that Pasteur clung to the opinion that the production 11 There is known today one dear instance, the reaction of nitrocinnaminacid with bromine where circular-polarized light generates an optical1y active substance.
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of single optically active compounds was the very prerogative of Ufe. In 1860 he wrote, "This is perhaps the only wen-marked line of demarkation that can at present be drawn between the chemistry of dead and living matter." Pasteur tried to explain his very first experiment where racemic acid was transformed by recrystallization into a mixture of laevo- and dextra-tartaric acid by the action of bacteria in the atmosphere on his neutral solution. It is quite certain today that he was wrong; the sober physical explanation lies in the fact that at lower temperature a mixture of the two oppositely active tartaric forms is more stable than the inactive racemic form. If there is a difference in principle between life and death it does not lie in the chemistry of the material substratum; this has been fairly certain ever since Wohler in 1828 synthesized urea from purely mineral material. But even as late as 1898 F. R. Japp in a famous lecture on "Stereochemistry and Vitalism" before the British Association upheld Pasteur's view in the modified form: "Only the living organisms. or the living intelligence with its conception of symmetry can produce this result (i.e. asymmetric compounds)." Does he really mean that it is Pasteur's inteHigence that, by devising the experiment but to its own great surprise, creates the dual tartaric crystals? Japp continues. "Only asymmetry can beget asymmetry." The truth of that statement I am willing to admit; but it is of little help since there is no symmetry in the accidental past and present set-up of the actual world which begets the future. There is however a real difficulty: Wby should nature produce only one of the doublets of so many enantiomorphic forms the origin of which most certainly lies in living organisms? Pascual Jordan points to this fact as a support for his opinion that the beginnings of life are not due to chance events which, once a certain stage of evolution is reached, are apt to occur continuously now here now there, but rather to an event of quite singular and improbable character, occurring once by accident and then starting an avalanche by autocatalytic multiplication. Indeed had the asymmetric protein molecules found in plants and animals an independent origin in many places at many times, then their laevo- and dextro-varieties should show nearly the same abundance. Thus it looks as if there is some truth in the story of Adam and Eve, if not for the origin of mankind then for that of the primordial forms of life. It was in reference to these biological facts when I said before that if taken at their face value they suggest an intrinsic difference between left and right, at least as far as the constitution of the organic world is concerned. But we may be sure the answer to our riddle does not lie in any universal biological laws but in the accidents of the genesis of the organismic world. Pascual Jordan shows one way out; one would like to find a less radical one, for instance by reducing the asymmetry of the inhabitants on earth to some inherent, though accidental. asymmetry of the earth itself, or of the light received
691
on earth from the sun. But neither the earth's rotation nor the combined magnetic fields of earth and sun are of immediate help in this regard. Another possibility would be to assume that development actually started from an equal distribution of the enantiomorph forms, but that this is an unstable equilibrium which under a sUght chance disturbance tumbled over. From the phylogenetic problems of left and right let us finally turn to their ontogenesis. Two questions arise: Does the first division of the fertilized egg of an animal into two cells fix the median plane, so that one of the cells contains the potencies for its left, the other for its right half? Secondly what determines the plane of the first division? I begin with the second question. The egg of any animal above the protozoa possesses from the beginning a polar axis connecting what develops into the animal and the vegetative poles of the blastula. This axis together with the point where the fertilizing spermatozoon enters the egg determines a plane, and it would be quite natural to assume that this is the plane of the first division. And indeed there is evidence that it is so in many cases. Present opinion seems to incline toward the assumption that the primary polarity as wen as the subsequent bHateral symmetry come about by external factors actualizing potentialities inherent in the genetic constitution. In many instances the direction of the polar axis is obviously determined by the attachment of the oozyte to the wall of the ovary, and the point of entrance of the fertiHzing sperm is, as we said, at least one, and often the most decisive, of the determining factors for the median plane. But other agencies may also be responsible for the fixation of the one and the other. In the sea-weed Fucus light or electric fields or chemical gradients determine the polar axis, and in some insects and cephalopods the median plane appears to be fixed by ovarian influences before fertilization. 12 The underlying constitution on which these agencies work is sought by some biologists in an intimate preformed structure, of which we do not yet have a clear picture. Thus Conklin has spoken of a spongioplasmic framework, others of a cytoskeleton, and as there is now a strong tendency among biochemists to reduce structural properties to fibers. so much so that Joseph Needham in his Terry Lectures on Order and life (1936) dares the aphorism that biology is largely the study of fibers. one may expect them to find that that intimate structure of the egg consists of a framework of elongated protein molecules or fluid crystals. 12 Julian S. Huxley and G. R. de Beer in Iheir classical Elements 01 emhryologv (Cambridge University Press.. 1934) give Ihis formulation (Chapter XIV, Summary, p. 438): urn the earliest stnges. Ihe egg acquires a unitary organization of the gradientfield Iype in which quantitative differentials of one or more kinds extend across the substance of the egg in one or more directions. The constitution of the egg predetermines it to be able to produce a gradient-field of a particular type; however. the localization of the gradients is not predetermined. but is brought about by agencies external to the egg."
H.rmflllll
692
W.,.,
We know a little more about our first question whether the first mitosis of the cell divides it into left and right. Because of the fundamental character of bilateral symmetry the hypothesis that this is so seems plausible enough. However, the answer cannot be an unqualified affirmation. Even if the hypothesis should be true for the normal development we know from experiments first performed by Hans Driesch on the sea urchin that a single blastomere isolated from its partner in the two-cell stage develops into a whole gastrula differing from the normal one only by its smaller size. Here are Driesch's famous pictures. It must be admitted that this is not so for all species. Driesch's discovery led to the distinction between the actual and the potential destiny of the several parts of an egg. Driesch himse1f speaks of prospective significance (prospektive Bedeutung), as against prospective potency (prospektive Potenz); the latter is wider than the former, but shrinks in the course of development. Let me illustrate this basic point by another example taken from the determination of limb-buds of amphibia. According to experiments performed by R. G. Harrison, who transplanted discs of the outer wal1 of the body representing the buds of
a,
FIGURE 17
Experiments on pluripotence in "Echinus." al and b l • Normal gastrula and normal pluteus. a2 and b z• Half-gastrula and half-pluteus, expected by Driesch. a3 and ba. The small but whole gastrula and pluteus, which he actually obtained.
future limbs, the antero-posterior axis is determined at a time when transplantation may still invert the dorso-ventral and the medio-lateral axes; thus at this stage the opposites of left and right still belong to the prospec-
693
p b
(J
dorsol
anterior.
.~
l:
t
.~
~
.&.;;
& posterior
c
d FIGURE 18
tive potencies of the discs, and it depends on the influence of the surrounding tissues in which way this potency will be actualized. Driesch's violent encroachment on the normal development proves that ,the first cell division may not fix left and right of the growing organism for good. But even in normal development the plane of the first division may not be the median. The first stages of cell division have been closely studied for the worm Ascaris megalocephaJa, parts of whose nervous system are asymmetric. First the fertilized egg splits into a cell I and a smaller P of obviously different nature (Figure 18). In the next stage they divide along two perpendicular planes into I' + /" and PI + P,! respectively. Thereafter the handle P t + P"!. turns about so that P,! comes into contact with either I' or /"; call the one it contacts B. the other A. We now have a sort of rhomboid and roughly AP2 is the antero-posterior axis and BP t the dorsal-ventral one. Only the next division which along a plane perpendicular to the one separating A and B splits A as well as B into symmetric halves A == a + a, B == b + {J, is that which determines left and right. A further slight shift of the configuration destroys this bilateral symmetry. The question arises whether the direction of the two consecutive shifts is a chance event which decides first between anterior and posterior and then between left and right, or whether the constitution of
694
the egg in its one-cell stage contains specific agents which determine the direction of these shifts. The hypothesis of the mosaic egg favoring the second hypothesis seems more likely for the species Ascaris. There are known a number of cases of genotypical inversion where the genetic constitutions of two species are in the same relation as the atomic constitutions of two enantiomorph crystals. More frequent, however, is phenotypical inversion. Left-handedness in man is an example. I give another more interesting one. Several crustacea of the lobster type have two morphologically and functionally different claws, a bigger A and a smaller a. Assume that in normally developed individuals of our species, A is the right claw. If in a young animal you cut off the right claw. inversive regeneration takes place: the left claw develops into the bigger form A while at the place of the right claw a small one of type a is regenerated. One has to infer from such and similar experiences the bipotentiality of plasma, namely that all generative tissues which contain the potency of an asymmetric character have the potency of bringing forth both forms. so however that in normal development always one form develops, the left or the right. Which one is genetically determined, but abnormal external circumstances may cause inversion. On the basis of the strange phenomenon of inversive regeneration Wilhelm Ludwig developed the hypothesis that the decisive factors in asymmetry may not be such specific potencies as, say, the development of a "right claw of type A ," but two Rand L (right and left) agents which are distributed in the organism with a certain gradient, the concentration of one faJling off from right to left, the other in the opposite direction. The essential point is that there is not one but that there are two opposite gradient fields Rand L. Which is produced in greater strength is determined by the genetic constitution. If, however, by some damage to the prevalent agent the other previously suppressed one becomes prevalent, then inversion takes place. Being a mathematician and not a biologist J report with the utmost caution on these matters, which seem to me of highly hypothetical nature. But it is clear that the contrast of left and right is connected with the deepest problems concerning the phylogenesis as well as the ontogenesis of organisms. TRANSLATORY, ROTATIONAL, AND RELATED SYMMETRIES
From bilateral, we shan now turn to other kinds of geometric symmetry. Even in discussing the bilateral type J could not help drawing in now and then such other symmetries as the cylindrical or the spherical ones. It seems best to fix the underlying general concept with some precision heforehand, and to that end a little mathematics is needed, for which J ask your patience. J have spoken of transformations. A mapping S of space associates with every space point p a point p' as its image. A special such mapping is the identity 1 carrying every point p into itself. Given two
695
S,.mmttry
mappings S, T, one can perform one after the other: if S carries pinto p' and T carries p' into p" then the resulting mapping, which we denote by ST, carries pinto p". A mapping may have an inverse S' such that SS' = 1 and S'S = I; in other words, if S carries the arbitrary point pinto p' then S' carries p' back into p, and a similar condition prevails with S' performed in the first and S in the second place. For such a one-to-one mapping S the word transformation was used in the first lecture; let the inverse be denoted by S-I, Of course, the identity 1 is a transformation, and 1 itself is its inverse. Reflection in a plane, the basic operation of bilateral symmetry, is such that its iteration SS results in the identity; in other words, it is its own inverse. In general composition of mappings is not commutative; ST need not be the same as TS. Take for instance a point 0 in a plane and let S be a horizontal translation carrying 0 into 0 1 and T a rotation around 0 by 90", Then ST carries 0 into the point o;! (Figure 19), but TS carries 0 into 0 1 , If S is a transformation with the inverse S-l, then S-1 is also a transformation and its inverse is S. The composite of two transformations ST is a transformation again. and (Sn- 1 equals T-1S-1 (in this order!), With this rule. although perhaps not with its mathematical expression, you are all familiar. When you dress. it is not immaterial in which order you perform the operations; and when in dressing you start with the shirt and end up with the coat, then in undressing you observe the opposite order; first take off the coat and the shirt comes last.
FIGURE 19
I have further spoken of a special kind of transformations of space called similarity by the geometers. But I preferred the name of automorphisms for them, defining them with Leibniz as those transformations which leave the structure of space unchanged. For the moment it is immaterial wherein that structure consists. From the very definition it is clear
696
Herman,. Wey'
that the identity I is an automorphism, and if S is, so is the inverse S-1. Moreover the composite ST of two automorphisms S, T is again an automorphism. This is only another way of saying that (1) every figure is similar to itself, (2) if figure F' is similar to F then F is similar to F', and (3) if F is similar to F' and F' to Fit then F is similar to F". The mathematicians have adopted the word group to describe this situation and therefore say that the aUlomorphisms form a group. Any totality, any set r of transformations form a group provided the following conditions are satisfied: (I) the identity I belongs to r~ (2) if S belongs to r then its inverse S-1 does; (3) if Sand T belong to r then the composite ST does. One way of describing the structure of space, preferred by both Newton and Helmholtz, is through the notion of congruence. Congruent parts of space V, V' are such as can be occupied by the same rigid body in two of its positions. If you move the body from the one into the other position the particle of the body covering a point p of V will afterwards cover a certain point p' of V'. and thus the result of the motion is a mapping p ~ p' of V upon V'. We can extend the rigid body either actually or in imagination so as to cover an arbitrarily given point p of space, and hence the congruent mapping p -+ p' can be extended to the entire space. Any such congruent transformation-I call it by that name because it evidently has an inverse p' ~ p-is a similarity or an automorphism; you can easily convince yourselves that this fo]Jows from the very concepts. It is evident moreover that the congruent transformations form a group, a subgroup of the group of automorphisms. In more detail the situation is this. Among the similarities there are those which do not change the dimensions of a body; we shall now call them congruences. A congruence is either proper, carrying a 1eft screw into a left and a right one into a right, or it is improper or reflexive, changing a left screw into a right one and vice versa. The proper congruences are those transformations which a moment ago we called congruent transformations, connecting the positions of points of a rigid body before and after a motion. We shall now call them simply motions (in a non kinematic geometric sense) and call the improper congruences reflections, after the most important example: reflection in a plane, by which a body goes over into its mirror image. Thus we have this step-wise arrangement: similarities ~ congruences - similarities without change of scale -+ motions - proper congruences. The congruences form a subgroup of the similarities, the motions form a subgroup of the group of congruences, of index 2. The latter addition means that if B is any given improper congruence. we obtain all improper congruences in the form BS by composing B with all possible proper congruences S. Hence the proper congruences form one half, and the improper ones another half, of the group of all congruences. But only the first half is a
697
Symmt!lry
group; for the composite A B of two improper congruences A, B IS a proper congruence. A congruence leaving the point 0 fixed may be called rotation around 0; thus there are proper and improper rotations. The rotations around a
A'
B' A
B FIGURE 20
given center 0 form a group. The simplest type of congruences are the ~
translations. A translation may be represented by a vector A A'; for if a translation carries a point A into A' and the point B into B' then BB' has the same direction and length as A A', in other words the vector BB' = A A'.13 The translations form a group; indeed the succession of the two ~~
~
translations A B. Be results in the translation A C. What has all this to do with symmetry? It provides the adequate mathematicallanguage to define it. Given a spatial configuration jf, those automorphisms of space which leave jf unchanged form a group r, and this group describes exactly the symmetry possessed by jf. Space itself has the full symmetry corresponding to the group of all automorphisms, of all similarities. The symmetry of any figure in space is described by a subgroup of that group. Take for instance the famous pentagram (Figure 21) by which Dr. Faust banned Mephistopheles the devil. It is carried into itself by the five proper rotations around its center 0, the angles of which are multiples of 360 0 /5 (including the identity), and then by the five reflections in the lines joining 0 with the five vertices. These ten operations form a group. and that group tells us what sort of symmetry the pentagram possesses. Hence the natural generaHzation which leads from bilateral symmetry to symmetry in this wider geometric sense consists in replacing reflection in a plane by any group of automorphisms. The circle in a plane with center 13 While a segment has only length, a vector has length and direction. A vector is really the same thing as a translation. although one uses different phraseologies for vectors and translations. Instead of speaking of the translation ft which carries
the point A into A' one speaks of the vector ft = AA'; and instead of the phrase: the translation ft carries A into A' one says that A' is the end point of the vector laid off from A. The same vector laid off from B ends in B' if the translation carrying A into A' carries B into B'.
a
698
FIGURE 21
o
and the sphere in space around 0 have the symmetry described by the group of aU plane or spatial rotations respectively. If a figure J does not extend to infinity then an automorphism leaving the figure invariant must be scale-preserving and hence a congruence t unless the figure consists of one point only. Here is the simple proof. Had we an automorphism leaving J unchanged, but changing the scale, then either this automorphism or its inverse would increase (and not decrease) all linear dimensions in a certain proportion a : ) where a is a number greater than 1. Call that automorphism S, and let a, fJ be two different points of our figure J. They have a positive distance d. Iterate the transformation St S
=
SI, SS
=
S2t SSS _ S3, • . • .
The n-times iterated transformation sn carries a and fJ into two points an. fJn of our figure whose distance is d· al'. With increasing exponent n this distance tends to infinity. But if our figure J is bounded, there is a number c such that no two points of J have a distance greater than c. Hence a contradiction arises as soon as n becomes so large that d'a'" > c. The argument shows another thing: Any finite group of automorphisms consists exclusively of congruences. For if it contains an S that enlarges linear dimensions at the ratio a : I, a > 1, then all the infinitely many iterations Sl, S.!, S3, . . . contained in the group would be different because they enlarge at different scales aI, a 2, aX, . . '. For such reasons as these we shall almost exclusively consider groups of congruenceseven if we have to do with actually or potentially infinite configurations such as band ornaments and the like. After these general mathematical considerations let us now take up some special groups of symmetry which are important in art or nature. The operation which defines bilateral symmetry, mirror reflection, is essen-
699
Symmt!lry
tial1y a one·dimensional operation. A straight line can be reflected in any of its points 0; this reflection carries a point P into that point P' that has the same distance from 0 but Jies on the other side. Such reflections are the only improper congruences of the one·dimensional line, whereas its only proper congruences are the translations. Reflection in 0 followed by the translation OA yields reflection in that point A 1 which halves the dis· tance OA. A figure which is invariant under a translation I shows what in tbe art of ornament is called "infinite rapport," i.e. repetition in a regu· lar spatial rhythm. A pattern invariant under the translation I is also invariant under its iterations 11 , ,2, ,a, . . " moreover under the identity to = I, and under the inverse 1-1 of t and its iterations ,-1, ,-2, t- 3 , If I shifts the line by the amount a then t''' shifts it by the amount (n = 0, ± 1, ±2, . . .) .
na
Hence if we characterize a translation t by the shift a it effects then the iteration or power t'" is characterized by the multiple na. AU translations carrying into itself a given pattern of infinite rapport on a straight line are in this sense multiples na of one basic translation a. This rhythmic may be combined with reflexive symmetry. If so the centers of reflections follow each other at half the distance lha. Only these two types of sym~ metry, as iJIustrated by Figure 22, are possible for a one-dimensional pattern or ··ornament." (The crosses X mark the centers of reflection.)
I
II
II
•
••
It
a
•
•
I
K
J(
•
•
•
•
FIGURE 22
Of course the real band ornaments are not strictly one-dimensional, but their symmetry as far as we have described it now makes use of their longitudinal dimension only. Here are some simple examples from Greek art. The first (Figure 23) which shows a very frequent motif, the palmette, is of type I (translation + reflection). The next (Figure 24) are without reflections (type II). This frieze of Persian bowmen from Darius' palace in Susa (Figure 25) is pure translation; but you should notice that the basic translation covers twice the distance from man to man because the costumes of the bowmen alternate. Once more I shall point out the Monreale mosaic of the Lord's Ascension (Figure 10). but this time drawing your attention to the band ornaments framing it. The widest, carried out in a peculiar technique, later taken up by the Cosmati, displays the translatory symmetry only by repetition of the outer contour of the basic tree-like motif, while each copy is filled by a different highly
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:u
FKiURE
1-(
symmetric two-dimensional mosaic. The palace of the doges in Venice (Figure 26) may stand for translatory symmetry in architecture. Innumerable examples could be added. As I said before, hand om::aments rcally consist of a two-dimensional strip around a central line and thus have a second lransvenal dimension. As such they can have further symmetries. The pattern may be carried inlo itself by reflection in the central line I; leI liS clislinguish this as longitudinal reflection from the transversal reneetion in a line perpendicular 10 I. Or the patlem may be carried into ilSelf by longitudinal reflection combined with the translation by 'ha (longitudinal slip reflection) , A freo
f'IOVltE
:u
FJOVU 16
702
quent motif in band ornaments are cords, strings, or pJaits of some sort, tbe design of which suggests that one strand crosses the other in space (and thus makes part of it invisible). If this interpretation is accepted, further operations become possible; for example, reflection in the plane of the ornament would change a strand slightly above the plane into one below. All this can be thoroughly analyzed in tenns of group theory as is for instance done in a section of Andreas Speiser's book, Theorie der Gruppen von endlicher Ordnung. In the organic world the translatory symmetry, which the zoologists call metamerism, is seldom as regular as bilateral symmetry frequently is. A maple shoot and a shoot of Angraecum distichum (Figure 27) may serve as examples. 14 In the latter Ii case translation is accompanied by longitudinal slip reflection. Of course the pattern does not go on into infinity (nor does a band ornament). but one may say that it is potential1y infinite at least in one direction, as in the course of time ever new If If segments separated from each other by a bud come into being. Goethe said of the tails of vertebrates that they allude as it were to the potential infinity of organic existence. The central part of the animal shown in this picture, a scolopendrid (Figure 28), possesses fairly reguJar translational, combined with bilateral, symmetry, the basic operations of which are translation by one segment and longitudinal reflection. In one-dimensional time repetition at equal intervals is the musical principle of rhythm. As a shoot grows it translates, one might say. a slow temporal into a spatial rhythm. Reflection, inversion in time, plays a far less important part in music than rhythm does. A melody changes its character to a considerable degree if played backward, and I, who am a poor musician, find it hard to recognize reflection when it is used in the construction of a fugue; it certainly has no such spon14 This and the next picture are taken from Silldiwn Generale. p. 249 and p. 241 (article by W. Troll, "Symmetriebetrachtung in der Biologie").
FIGURE 27
FIGURE
28
703
taneous effect as rhythm. All musicians agree that underlying the emotional element of music is a strong formal element. It may be that it is capable of some such mathematical treatment as has proved successful for the art of ornaments. If so, we have probably not yet discovered the appropriate mathematical tools. This would not be so surprising. For after all, the Egyptians excelled in the ornamental art four thousand years before the mathematicians discovered in the group concept the proper mathematical instrument for the treatment of ornaments and for the derivation of their possible symmetry classes. Andreas Speiser, who has taken a special interest in the group-theoretic aspect of ornaments, tried to apply combinatorial principles of a mathematical nature also to the formal problems of music. There is a chapter with this title in his book. "Die mathematische Denkweise," (Zurich, 1932). As an example, he analyzes Beethoven's pastoral sonata for piano, opus 28, and he also points to Alfred Lorenz's -investigations on the fonnal structure of Richard Wagner's chief works. Metrics in poetry is closely related, and here, so Speiser maintains, science has penetrated much deeper. A common principle in music and prosody seems to be the configuration a a b which is often called a bar: a theme a that is repeated and then foI1owed by the "envoy" b; strophe, antistrophe, and epode in Greek choric lyrics. But such schemes fall hardly under the heading of symmetry yi We return to symmetry in space. Take a band ornament where the individual section repeated again and again is of length a and sling it around a circular cylinder, the circumference of which is an integral multiple of a, for instance 25a. You then obtain a pattern which is carried over into itself through the rotation around the cylinder axis by a. = 360 0 /25 and its repetitions. The twenty-fifth iteration is the rotation by 360 0 , or the identity. We thus get a finite group of rotations of order 25, i.e. one consisting of 25 operations. The cylinder may be replaced by any surface of cylindrical symmetry, namely by one that is carried into itself by all rotations around a certain axis, for instance by a vase. Figure 29 shows an attic vase of the geometric period which displays quite a number of simple ornaments of this type. The principle of symmetry is the same, although the style is no longer "geometric," in this Rhodian pitcher (Figure 30), Ionian school of the seventh century B.C. Other illustrations are such capitals as these from early Egypt (Figure 31). Any finite group of proper rotations around a point 0 in a plane, or around a given axis in space, contains a primitive rotation t whose angle is an aliquot part 360 In of the full rotation by 3600, and consists of its iterations It, (1., . . . ,t" -1, til = identity. The order n completely characterizes this group. The result follows from the analogous fact that any group of translations 0
15 The reader should compare what G D Birkhoff has to say on the mathematics of poetry and music in the two publications quoted in note 1.
FIGURE lO
FIGURE 19
FIGURE 11
'"
of a line, provided it contains no operations arbitrarily Rcar to the idenlity except the identity itself, consists of the iterations va of II single translatWm a(v=O. ±1. ±2.· . ' ). The wooden dome in the Bardo of Tunis, once the palace of the Beys of Tunls (Figure 32). may serve as an example from interior architecture.
FIOUAE 12
The next picture (Figure 33) takes you to Pisa; the BaptiSCerium with the tiny-looking statue of John the BaptiSi on lop is a cenlral building in whose exterior you can distinguish six horizontal layers each of rotary symmetry of a different order fl . One could make the picture still more impressive by adding the leaning lower with its six ganeries of arcades all having rotary symmetry of the same higb order and the dome itself. the exterior of whose nave displays in columns and friezes paUems of the lineal translatory type of symmetry while the cupola is surrounded by a colonnade of rugh order rotary symmetry. An entirely different spirit speaks to us from the view, seen from the rear or the choir, of the Rom,lne5quc cathedral in Mainz, Germany
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(Figure 34). Yet again repetition in the round arcs of the friezes. octagonal central symmetry (n 8, a low value compared to Ibose embodied in tbe several layers of the Pisa Baptisterium) in the small rosette and the three lowers, while bilateral symmelI}' rules the structure as a whole as well as almost every detail. Cyclic symmetry appears in its simplest form if the surface of fully cylindrical symmetry is a plane perpendicular 10 the axis. We then can limit ourselves to the two-dimensional plane with a eenter O. Magnificent examples of such central plane syrnmelCy are provided by the rose WiDdows of Gotbic cathedrals with their brilliant-colored glasswork. The richest I remember is the rosette of St. Pierre in Troyes. France. which is based on the number 3 throughout.
=
FlOURE 14
F1owcrs. nature's gentlest children, are also conspicuous for thcir colors and their cyclic symmetry. Here (Figure 35) is a picture of an iris with its triple pole. 1be symmetry of 5 is most frequent among flowers. A page like the following (Figure 36) from Emst Haeckel's Kllnsljo,men de, Nalu, seems to indicate that it also occurs not infrequently among the lower animals. But the biologists warn me that the outward appearance of lhese echinoderms of the class of OphiQdea is to a certain degree deceptive: their larvae are organized according to the principle of bilateral symmetry. No such objection attaches to the next picture from the same source (Figure 31), a Discomedusa of octagonal symmetry. For the c0elentera occupy a place in the phylogenetic evolution where cyclic has not yet given way to bilateral symmetry. Haeckers extraordinary work, in which his interest in the concrete forms of organisms finds expression in
..
FlOOIU! U
countless drawings executed in minutest detail, is a true nature's codex of symmtlry. Equally revealing for Haeckel. the biologist, are the thousands and thousands of figures in his Challenger Monograph. in which be describes for the first time 3,508 new species of radiolarians discovered by him on the Challenger Expedition, 1881. One should not forget these accomplishments over the often all-loa-speculative phylogenetic constructions in which this enthusiastic apostle of Darwinism indulged, and over his rather shallow materialistic philosophy of monism. which made quite a splash in Germany around the turn of the century. Speaking of Metiusoe I cannot resist the temptation of quoting a few lin" from D'Arcy Thompson's classic work on Growth Qnd Form, a masterpiece of English literature. which combines profound knowledge in geometry. physics, and biology with humanistic erudition and scientific insight or unusual originality. Thompson repons on physical experiments
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FlOVRB l6
710
with hanging drops which serve to illustrate by analogy the formation of medusae. "The living medusa", he says, "has geometrical symmetry so marked and regular as to suggest a physical or mechanical erement in the little creatures~ growth and construction. It has, to begin with, its vortexlike bell or umbrella, with its symmetrical handle or manubrium. The ben is traversed by radial canals, four or in multiples of four; its edge is beset with tentacles, smooth or often beaded, at regular intervals or of graded sizes; and certain sensory structures, including solid concretions or 'otoliths,' are also symmetrically interspersed. No sooner made, then it begins to pulsate; the bell begins to 'ring.' Buds, miniature replicas of the parent-organism, are very apt to appear on the tentacles, or on the manubrium or sometimes on the edge of the bell; we seem to see one vortex producing others before our eyes. The development of a medusoid deserves to be studied without prejudice from this point of view. Certain it is that the tiny medusoids of Obelia, for instance, are budded off with a rapidity and a complete perfection which suggests an automatic and all but instantaneous act of conformation, rather than a gradual process of growth." WhUe pentagonal symmetry is frequent in the organic world, one does not find it among the most perfectly symmetrical creations of inorganic nature, among the crystals. There no other rotational symmetries are possible than those of order 2, 3, 4, and 6. Snow crystals provide the best known specimens of hexagonal symmetry. Figure 38 shows some of these little marvels of frozen water. In my youth, when they came down from heaven around Christmastime blanketing the landscape, they were the delight of old and young. Now only the skiers like them, while they have become the abomination of motorists. Those versed in English literature will remember Sir Thomas Browne's quaint account in his Garden 0/ Cyrus (1658) of hexagonal and "quincuncial" symmetry which "doth neatly declare how nature Geometrizeth and observeth order in all things." One versed in German literature will remember how Thomas Mann in his Magic Mountain 16 describes the Uhexagonale Unwesen" of the snow storm in which his hero, Hans Castorp, nearly perishes when he fans asleep with exhaustion and leaning against a barn dreams his deep dream of death and love. An hour before when Hans sets out on his unwarranted expedition on skis he enjoys the play of the flakes "and among these myriads of enchanting little stars." so he philosophizes, "in their hidden splendor, too small for man's naked eye to see, there was not one like unto another; an endless inventiveness governed the development and unthinkable differentiation of one and the same basic scheme, the equilateral, equiangled hexagon. Yet each in itself-this was the uncanny, the antiorganic, the 16
I quote Helen Lowe-Porter's translation, Knopf, New York, 1927 and 1939.
711
Re",umn We)"
life-denying character of them all--each of them was absolutely symmetrical, icily regular in form. They were too regular, as substance adapted to life never was to this degree-the living principle shuddered at this perfect precision, found it deathly, the very marrow of death-Hans Castorp felt he understood now the reason why the builders of antiquity purposely and secretly introduced minute variation from absolute symmetry in their columnar structures." 17 Up to now we have paid attention to proper rotations only. If improper rotations are taken into consideration, we have the two following possibilities for finite groups of rotations around a center 0 in plane geometry, which correspond to the two possibilities we encountered for ornamental symmetry on a line: (1) the group consisting of the repetitions of a single proper rotation by an aliquot part a = 360 In of 360 (2) the group of these rotations combined with the reflections in n axes forming angles of %a. The first group is ca:lled the cyclic group C n and the second the dihedral group Dfi' Thus these are the only possible central symmetries in two-dimensions: 0
0
;
(1)
C 1 means no symmetry at all, D] bilateral symmetry and nothing else. In architecture the symmetry of 4 prevails. Towers often have hexagonal symmetry. Central buildings with the symmetry of 6 are much less frequent. The first pure central building after antiquity, S. Maria degJi Angeli in Florence (begun 1434), is an octagon. Pentagons are very rare. When once before I lectured on symmetry in Vienna in 1937 I said I knew of only one example and that a very inconspicuous one, forming the passageway from San Michele di Murano in Venice to the hexagonal Capella EmHiana. Now, of course, we have the Pentagon building in Washington. By its size and distinctive shape, it provides an attractive landmark f01 bombers. Leonardo da Vinci engaged in systematically determining the possible symmetries of a central building and how to attach chapels and niches without destroying the symmetry of the nudeus. In abstract modern terminology, his result is essentially our above table of the possible finite groups of rotations (proper and improper) in two dimensions. So far the rotational symmetry in a plane had always been accompanied by reflective symmetry; I have shown you quite a number of examples for the dihedral group D" and none for the simpler cyclic group e". But this is more or less accidental. Here (Figure 39) are two flowers, a gera17 Durer considered his canon of the human figure more a; 388, 394; Arilhmetik, Grundgesetze der, 389, 537/n.: Arilh· mellk, Grundla&en -dn, 537; vocabulary for mathematics, 390 French Revolution, 59111., 300, 330 Frey, Dagobert, Symmetry in Art, On the Problem 01, 678 Frisch, Karl von, 488 Fuller, Tom. idiot Savant. 465, 470 function: analysis of, 412; conception, 19. 20, 39; dimensions, 51; generaJization, 411; propositional. 70; signs for, 32 In.; theory, 125, 160; variability, 22. 29. 32, 38, 70, 411 functional dependence,29
Batspace,6S6-657,6S~
G Galen, De temperamenlis, 671 Galileo (Galilei); astronomical discoveries. 218; book of the Universe. 153; cycloid, 135; dynamics, 120; falling bodies, 44-49, 51, 52. 57. 58, 61, 549; motions, composition of, 52; pendulum, behaVior, 412; pentagon, 615; telescope, 218 Galois. Evariste, group theory, 509 Galton. Francis, Explorer in Tropical South Alrica, Narrative 01. 433 gases. kinetic theory. 515 Gauricus. Pomponius. 601 Gauss, Carl Friedrich: algebra, 308, 313; analysis, 165. 299. 334, 338, 339; Anthmel;cae, DisqulsltioIWs, 305, 306, 307, 309-312, 321, 325, 327, 333, 498; astronomical calculations, 314-316, 320, 324, 334, 498; binary theory, 310; binomial theorem,. 298-299, 310, 311, 325; biquadratic rec;iprocity, 325; Ceres. discovery of, 315-316; commentary on, 294; complex numbersr analytical functions. 322, 332; congruences, theory of, 310, 498; di1ferential geometry, 334-335; discoveries, 317, 326; electromagnetic research, 311. 327, 338, 339, 498; elliptic functions.. 304, 305, 332; errors, theory of. 303; Fermat's Last Theorem. 312, 313, 326; geodetic research, 301, 334. 337; geometry, contributions to, 498; heliotrope.
Floating Bodies (Archimedes), 199 Bowers, cyclic symmetry, 707 Bowing quantities, 289 Buid: motion. 267, 323, 116; solid immersed in. 185-186, 188, 199-20Cl Bullions, method of, 48. 52, 58, 61, 142, 152,258,286,288-293 FluxiolU, Sir Istldc Newton's Method 01 (Robins),2I1In. Fluxions, A Treatise 01 (Maclaurin), 287 In. Fontana, Niccolo (Tartaglia). 78, 118119, 615 Fontenelle, Bernard de, 256 Foppa, Vincenzo. 606 Forsyth. Andrew Russen, 267; quoted, 362 four: early numeral, 445. 453; Roman numeral, 448 four-color problem, topology, 590-591 four-dimensional geometry, 359-360 Fourier, Jean B. J., 59, 1S5 Fowler, R. H., approximative calculation, 508 fractional indices. 131 fractional numbers. 25 r 26. 27, 31, 34,
162
fractions: continued. 104, 105, 374; Egyptians', 171; Greek mathematicians, 172; multiplication, 17, 25 In., 33; reduction, 12, 79; unit, 171, 172 France: geometrical work, decline, 133:
",..
iD.ftntion, 327; hyperboUc geometry, IS8-159; hypergeometric series, 324; infinitesimal calculus, 60; interpolation, 140; invariants, 350; least squares, mathod of, 303, 331, 334; mathematieal physics, 334; non-Euclidean geometry, 314, 332: numbers, theory of. 68, 338. 498, 499, 511: polygon, Euclidean construction, SOl; prime number theorem, 157-158; quadratic reciprocity, law of. 301, 302, 310, 325; quater· nions, 332; space geometry, 546: surfaces, theory of, 160, 335-337, 652656; Thl!oria motlU. 316, 331: writinp, 157; quoted, 314. 326, 333. 334, SOl; 8ft also Ben, "Prince of Madllmaticians" Garus. C. F.• und dil! ~inl!n (Mack), 294/n. Gaussian complex integers, 31S, 326, 513 Gdfond, A., 514 Gelon (king of Syracuse). Archimedes" Sand Rl!ckonl!~ 420 Geminus. perspective, 605 generalizations of number, 25, 26, 30 genotypical inversion, 695 geodesic line. 563-564 geodesic surface, 564, 565 geodesy, heliotrope, invention, 327 geodetic surveying, least squares. method of, 303, 331, 334 geometric mean, 92-93 geometric symmetry, 694 "Geometrical Axioms, On the Origin and Significance of' (von Helmholtz). 647~8
geometrical conditions, 404 geometrical measurements, 6~1 GtWml!tricai Ml!thods. A History 01 (Coolidge). 169 In.. 236In.; quotH, 237 In. geometrieal progression, 101. 137-138 Gloml!tril!, ~/lmmts dl! (Legendre), 158 geometry: abstract mathematics, 62; algebra, application of, 61, 130. 162; analytical, 17, 31-43, 56, 112. 116, 129, 133. 155, 236-237, 359. 41t. 631; areas and solids, 82; arithmetic applied to. 25. 189; assumptions analyzed, 546. SSO, 565; bee. 110, 188, 207/n •• 208209; characteristics, 22; circle, 147; conformal mapping, 337; cross-classification, 77; deductive. 13-14; descriptive, 601, 631; differential, 104. 334; Egyptian, 13-14, 68: elliptic, 159: Euclidean, 570; Euclidean and non.Euclidean unified, 359; four-dimensional, 359-360: Greek, 10, 61. 68; higher, lSI; hyperbolic. 158-159; infinny, upon th: plane at, 554; magnitudes, 585; many-dimensional, 134; metrical, 103. 359, 570; non-Euclidean, 102, 158, 165, 306, 332, 342, 396, 546; number allocation, 85; of position, 339,
573; origin. 10, 11. 79; parabolic, 1S9; plrme, 130, 568, 719; Plato's views, 96; practical and theoretical, 80; problems, 239, 649; projective, SH projective geometry; pure and physical, 152; revival, 207; rhetorical. 130; Rhind papyrus, 174; scientific method, 647; solid, 14, 568; space, 67. 86. 110. 342; Thales', 82; topology. S70; trigonometry separated from, 40 GtWmetry, Thl! (Descartes), 26-27, 31. 32. 237. 520 In.; quoted, 239-253 Gl!oml!try, ThI! Axioms 01 (Helmholtz). 550 In. Gl!Oml!try, &wry on the Foundations 01 (Russell). 377 Gl!Ometry, On thl! Hypothl!.s That UI! lit thl! Bases 01 (Riemann), 546, 645 Germain, Sophie, 325, 332, 333-334 Germany: mathematics, rise of, 118, 120; number designatioB by knots, 463 Ghiberti. Lorenzo, 616 Gibbon, Edward, qUDtfJd, 381 Gilman, Daniel Coit, 355 Ginsburg, JetuthieI. NUIftbn80 tlIfIl Nilml!rall, 430; "From Numbers to Numerals," 442-464 Girard. Albert, 23 G1anvill, Joseph, quotl!d, 552 gnomon, L--shaped bonier, 84-8S Goethe. Johann Wolfgang von: Faust, 271-272; spiral tendf.:nqr in nature, 718; tails of ftrtebrates, 702 Goldbach. Christian, 365, 367, 499, 507 golden section: Great Pyramid, 80; ladder of numben. 98; star pentagram, 88 Golenischev papyrus, 169 Goths, alphabetic numerals, 451 Gottschalk, E., 510 GrMe. A. von, 643/n. grammar, arithmetic needful for, 214 grapes, fermentin&' (optically dextroactive), 689 graph, configuration, 572 graphic representation, 605 Grassmann, Hermann, Awdl!hnungsiehrl!. 396 ~ Grassmann, Robert. 293 In. gravitation, Einstein's theory, 146, 328, 409 gravity, law of, 62, Ill, 142. 144, 145, 146, 256. 257-258, 266. 268, 315, 548 "Great CompOation, n.e" (Ptolemy).
lOO/n.
GrI!ater Logic (Hegel), 387 greatest common measure, 16-17. 101 GrI!l!k Mathl!malks. A Manual 0/ (Heath). 76, 191 In. Greeks: algebra, n, 16; analytical geometry, 236; astronomy, 14; curvilinear figures, 34, 40, 41, 42, 53; ellipse. 229; Euclidean construction. 502; fractions, reducing, 12; geometry. 10, 11, 13, 61,
lrula 6~9; incommensurable lines, 26, 528; irrational numbers, 89; Iogistica, 498; mathematics, 18, 89, 116, 188; motion, conception of, 19; notation, 92, 99, 114-115; number symbols, 418, 443, 446; numbers and maptitudes, distinction, 17, 61-63; polyhedra, tabulation. 723; problems, classification, 91; projective geometry, 632; prolixity, 38; "symmetric," use of word, 720; tangent to a curve, 36; triquetrum, 714 Greene, Robert, quoted, 489 Greaory, James, 78, 137, 138, 139, 140, 143 GrimalcU, Francesco Maria. on nature of light, 260, 262 Gross, Mason, 399 In. Grosseteste, Robert, 60S Ground 0/ Artes, The (Recorde), 210211 groups: automorphisms, 696; relationships, numbers, 402; theory, 167; units, comparing, 492 Growth and Form (Thompson), 708 G".p~n yon endlichn Ordnung, Theorie der (Speiser), 702 Guldin, Paul, 111 Gulliyer's Travels (Swift), 26
H Hadamard, Jacques S., prime number theory, 1S8, S07 Haddon, A. C., 440 In. Haeckel, Ernst: Challenger Monograph, 708, 720; Kurut/ormen der NatUl, 707 Haldane, Elizabeth S., 235 In. Halley, Edmund: Newton's Principia, 286; planetary motion, 144, 264. 278 Halley's comet, 144 Halphen. Georps, 363 Hambidge, J., Dynamic Symmetry, 718/n. Hamilton, WilHam Rowan: algebraic discoveries, 76; biography, 161; Characteristic Function, 161, 162, 164; dynamics, laws of, 659; least action., principle, 110, 132; quaternions, 76, 162, 163, 164, 332; Rays, Account 0/ a Theory 0/, quoted, 161 Hardy, Godfrey Harold: analytic theory of numbeR, 158; definite integrals, 371; partitions of n, 167; prime number theorem, 508; Ramanujan, 366/n., 367, 370-373, 375; Waring problem, 515 harmonic progressions, civil life, 127 Harmonices Mundi (Kepler), 126, 127 Harmonics (Ptolemy), 232 hannony,appHcations, 671 Harmony, Elements 0/ (Aristoxenus). 192 Harriot, Thomas, 23, 29 Harrison. R. G., 692 Hartshorne, Charles, 395 In.
heap, symbol, 12 heart, mammals, an asymmetric screw, 687 heart-beats, atom, 164 Heath. Thomas: Archimedes, The Works 0/, 181/n., 197/n., 198 In.; Euclid's Ekments, The Thirteen Books 0/, 191/n.; Greek Mathematics, Manual 0/, 76, 191 In. Heavenly Bodies, The Circuit" Motion 0/ the (Cleomedes), 20S Heawood, four-color problem, S91 Hebrews, numerals, 451 Hegel, Geol'l Wilhelm Friedrich, 314, 316, 387 Heibel'l, Johan Ludvig, 94 Heisenberg, Werner: quantum mechanics, 362, 396; wave mechanics, 164 heliocentric theory, Copernicus, 110, 120, 218, 226, 227 Heliodorus of Larissa, perspective, 605 heliotrope, invention, 327 Helmholtz, Hermann von: Axioms 0/ Geometry, The, S50/n.; commentary on, 642-646; energy, conservation, 643; geometry, assumptions, S50; metaphysics, 645 /n.; "On the Origin and Significance of Geometriea! Axioms," 644, 647-668; metaphysics, 64S /n.; ophthalmoscope, invention, 643; Physiological Optics, 643. 644; seDl8tions of tone, 643; space geometry, S46; space structure, 645, 647, 681~2, 696; Theory 0/ Animal Heat, 643 hemisphere, area of, 175 heptagon, anaie, determining, 80 Heracleides, 179 heralcUc symmetry, 674, 675 Hennite. Charles, 514 Hero (Heron) of Alexandria: fractions. 172; inventions, 110; plane mirrors, 110, 132, 154; relativity, 110 Herodotus, 10, 80; quoted, 79 Hertz, Heinrich: wireless telegraphy, 9, 644; quoted, 521 Herzfeld, Ernst, 675 hexagon: Egyptian knowledge of, 80; inscribed in a conic, 134 hexagonal cells, bees', 80, 111, 208 hexagonal symmetry, 712 hexagram. mystic, 134, 151 Hiero II (king of Sicily), 103, 181, 185, 188, 200 /no higher algebra, 136 higher geometry, 151 higher plane curves, theory of, 113 higher space, geometry of, 342 Hilbert, David: cUophantine equations, 50:); transcendental numbers, 514; Waring's conjecture, 514; quoted, 510 Hill, G. F., Arabic Numerals in Europe. 454/n. Hindu-Arabic numerals, 453
,..-.
1111
Hindu·Arabic Num~rals. Th~ (Smith and Karpinski), 4S4 fn. Hindus: diophantine equations, S09; rotation system, 13; trigonometry, 18; see also India Hipparchus: planetary motion, 110, 226; trigonometry. 18, 109 Hippasus, 83 Hippocrates of Chios: area of a circle, 97; method of exhaustion, 92, 101. 193 tn.; squaring the circle, 90 Hire, Phillippe de la, 631 historic asymmetry, 679 historic continuity: heraldic Symmetry. 676 Histories, Book ot (Tzetzes), quoted, 187 Hobbes. Thomas, quoted, 622 Holland, freedom of thought, 235 Holmes, Oliver Wendell, quoted, 467 holy teractys. 84 homaloid space, 6S6-651 Homer, 99 honey cell, hexagon, 80, 110, 208 Hooke, Robert: inverse square law, 264, :l6S; Mlcrographia, 261; planetary motion, 144,264 Houdin, Jean Eug~ne Robert, 477 Huaman Poma de Ayala, Don Felipe, 463 Hughes, Richard, quoted, S37 human bodies, in curved space, 663664 human body, bilateral symmetry, 671, 674, 686-687 Humboldt, Baron Alexander von, 316, 318, 330 hllDdrM, Roman numeral, 448 Huxley, Aldous, quoted, 277 Huxley, Julian S., Embryology, Elements ot, 691 In. Huxley, Thomas Henry, 166,239 Huygens, Christian, light; theory' of. 146, 261. 262, 412 hydrodynamics: conformal mapping. 337; text·book, 267 hydrostatics: Archimedes' work on, 107; Stevinus' work on, 120 hyperbola. 34, 87, 91, 93, 97. 2S8 hyperbolic geometry, IS8-1S9 hyperbolic table. Dase's. 477 hypergeometric series, 324. 369 I I (defined), complex unity. 30
Iceland spar, 263 icosahedron, 86,96,719-720 ideal numbers, 499, S13 IdeM, Adventure. in (Whitehead). 399 idiot savants, 46s-466 imaginary numbers, 29-30, 34, 119, 148. 309. 8~e also complex numbers immediate occasion, 407 implication, mathematics. 70 inanimate objects, motions and rests. 44
Inaudi, Jacques. mental calculations, 475, 478-480, 481, 482 incommensurables, 16. 26, 62, 99, 191 In., S25 increasing arithmetically, logarithms, 124 increment of, symbol, 47-48 indeterminaie analysis, 197, 207 In. indeterminate equations, lOS, 116 India: decimal notation, 23, 114, 117; four, forms for, 453: mathematicians, 166i numerals, 443; se~ also Hindus indices: fractional; 131, 140i negative, 131; notatia, 122: theory, 89, 106, 125 indivisibles. method of, 31-43, 53, 13S. 137 induction: process. 68; theory of. 40S inertia, law of, 52, 665 infinite descent, method, 311, 50S infinite geometrical progression, 94 infinite plane dwellers, geometry. 651 infinite rapport, 699 infinite series, S21-S24 infinitesimal, meaning, 152 infinitesimal calculus, 43, 48, S2, 53-62,. 95, 127, 142, 146, 286-293, 411 infinitesimals, 40, 41, 51 in1lected language, 406 Inkeri, K., S13 integers. 13, 28, 60, 65, 66 integral calculus, 42, 47, 90, 103, 104, n5, 139, 142, 143, 2S8 integration: differential, 55-56, 61~ Le1~ niz's problem, S8 inteUectual needs, satisfaction of. 8-9, 10 interpolation, mathematics of. 140, 14S invariancc: illustrations, 352-3S3; projec. tive .eometry, 640 "Invariant Twins, Cayley and Sylvester" (Bell). 341-365 invariants: differential, 363; finite. 164: theory of, 163. 341-342, 344. 348, 350. 361 invention, primitive.,. 8 inve:rse ftuxicms. 258 inverse square law, 14S, 258, 264, 265, 266 inversion, art. 684-68S involution of six points, 112-113 Iran (Persia), numerals, 454 irrational numbers, 26,27. 34, 62-63.64, 87-88, 89, 92, 9S-96, 97. 101, 162, 168. 52S-536 "Irrational Numbers" (Dcdetind), 528536 UTatio~,unord~,l08
irreducible equation of degree n, S12 isoperimetric figures, 188, 207-209 isoperimetrical problem, lS3 isosceles triangle. IS, 81. 101. 175 Italy: algebra introduced, 23; mathematical treatises. 210; mathematics, 18. 118, 119 Ivory, James. lSI
"III
J Jacobi, K.arl Gustav, 306, 331, 370 Jaeger, F. M., "Symmetry and Its Applications in Natural Science," 689 Japan: abacus, 457; numerals, 445; symbol for "three," 444; symbol for "two," 443 Japp, F. R., "Stereochemistry and Vita]ism" 690 Johns' Hopkins University, founding, 354355 Johnson, A. H.• 395 /n .• 398 In. Johnson, Samuel, 287; quoted, 528 Jordan, Camille, 589; curve theorem, 588-590; quantum mechanics, 396 Jordan, Pascual, life, beginnings, 690 Joseph, Abraham, 346 Joule, James Prescott, conservation of energy. 643/n. Jourdain, Philip E. B.: commentary on, 2-3; "Mathematics, The Nature of,"
4-72
Cuiture, 602; "Projective Geometry," 622-641 knots: number designation, 436, 463; topology, 592 knowledge: communication of, 9; theory of, 391 known quantities. denoting, 28-29 K.oehler, 0., "Ability of Birds to Count," 489-496 Koenigsberger, Leo: Hermann 'Von Htlmholtl., 642 /n.. 643 In.; quoted. 645 In. Kohts, N. N. Ladygina, 492.496 UKonigsberg, The Seven Bridges or' (Euler),571-580 Kronecker, Leopold: algebraic numbers, 313,499; quoted, 168,295 Krumbieget, B., 197 In. Kummer, E. E .• algebraic numbers, theory of,313,499,512-513 Kunstlormen del' Natur (Haec:ke1), 707 Kusmin, R., 514
Jurin, James, 291/n. Juvenal. quOltd, 189,489
L
K Kaestner, Abraham Gotthe]f. 156, 158 Kant, Immanuel: axioms of geometry, 647, 660, 665, 666; properties of space, 645; space and time, 684; quoted, 274 Kapferer, H., 510 Kasner, Edward, quoted. 420 Kelvin, Lord (W. Thomson), energy, conservation, 643 In. Kempe, A. B., four-4;O]or problem, 591 Kepler, Johann (Johannes): astronomical speculations, 120; Brahe, association with, 125-126, 223-225; commentary on. 218-219; DUrer's analytical geometry, 613; Ephemtridts, 124; Harmoniees Mundi, 126, 127; indivisibles. 42; infinitesimal calculus, 127. 152; Mars, motion of, 226-228, 230; methods, 233234; pentagon, 615; planetary laws, 126-127, 132, 218, 220-234, 257, 266, 412; planetary system, reducing distances, 720; planetary tunes, 127-128; projective geometry, 133; prophesying almanack, 231; wine-cask geometry, 127, 137; quoted, 220 Keser, Jacob,609 Keynes, Geoffrey, 254 Keynes. John Maynard: "Newton, the Man," 277-285; quoted, 254 Keyser, Cassius J., Mathematical Philosophy, 570 In. kinematical thought, 142 King Leal' (Shakespeare), 306/n. Klein, Felix: analysis, 151; geometry discoveries, 342, 359. 639; groups, theory of,167 Klein bottle, 597 Khne, Morris: Mathematics in Western
L-Shaped border, 84 La Condamine, Charles M. de, 268 ladder arithmetic, 97-98, 104, 138 Lagrange, Joseph Louis: analytical geometry, 133, 360; biography. 153; differential calculus, 152; infinitesimal calculus, 59, 292/n.; invariants, 3S0; mathematical analysis, 299; Mecanique AnaIyliqut, 153, 154, 155; music aid to thought, 155; sound, mathematics of, 154; universe, picture of, 153 Lambert, J. H., calculus of logic, 20 land surveyors: Egyptians, 80; measuring problem, 14 Landau, E., 158, 166,371, S08 language: development, 406; human. origin of, 490-491 languages, scientific workers, 300 Laplace. Pierre Simon de: formula, 370; infinitesimal calculus, 59; least squares, method of, 331; mathematical analysis, 299; mathematical astronomy. 313, 314, 315, 316; Meeanique CIleste, ISS. 321; quoted,27S Last Supper (Bouts), 604, 60S lateral vanishing points, 604 Latin, infiected language, 406 latitudes and longitudes, Oresme's system, 236 law, arithmetic needful for, 214 laws of nature, 39 Jeast action, principle, 110, 132. 149, 153, 166 least common measure, 101 least resistance, solid of revolution of, 267 least squares, method of, 303, 331, 334 Leetiones Geo~trical (Barrow), 139 Leetiones Opticae (Barrow), 256 Leetiones Optieae (Newton), 259
,--
Lefscbetz, S ... 339 left. meaninp. 684 left and riaht: equivalence. 685; mathematicalphDosophY.679 Legendre, Adrien-Marie: Glometrie. £11· men/s tk. 158; infinitesimal calcu1UL 59; least squan::s. method of. 303. 331~ auadratic reciorocity, 301, 302; series, 371; quoted, 312 Leggett, H. W., 378 tn. Lehman. Harvey C.. Age and Achievement. 467 In. Lehmer, D. H., 503, 504 Lehmer, D. N •• 503-504 Leibniz. Gottfried Wilhelm von: binary scale. 516. 517; calculating machiae, 516; differential calculus, 152, 275, 286; 8uxions. 143; aeometry of position, 573; infinitesimal calculus. 48. S7-59, 61. 292; left and riabt indiscernible, 680; logarithm, 137, 138; mathematical analysis, 299; motion, 20; notation, 32, 54, S7-S8, 59; past and future, 686; PhU080phUche Schrilten, 683 In.; plane. 564; relative concepts, 683-684, 687; space structure, 682. 69S; straight line, 564.567 Leibnl1., Crltkal Expotilion 0/ the Plai/osophy 01 (Russell), 378, 388 LeibDiz-Clark.e. controversy, 683, 687 leagtb. measurements, 17. 35 lens: shape improvement. 132; telescope, imperfection, 142 Leoaardo of Pi.sa (Fibonacci), 23. 78. 98,
lt7 Lessiq, G. E .• cattle problem. 197 In. utler Written to a P,(Winciol, A (Pascal),13S leUers: Gotbie. construction. 617; meaning of geometrical and mechanical equations. So-SI; Roman, geometrical construction. 616; symbols. 24-25 lever: invention, 8, 68; motions and rest, 44; principle, 188. 195-197 Levi-Civita. Tullio, 328, 3S2 Lie. Marius Sophus: group theory, 167; invariants. 352 life, beginrtinp of. 690-691 light: corpuscular theory, 146; rays. 161: transverse waves, 263; wave theory, 146, 154. 164,262.263.267,412; white, 141-142. 146.2S6,260-263 limit. conception, 37, 48. S9, 60, 62. 95.
291/n. Limil. and Fluxio,.. in Great Britain, Hislory 01 1M Conception 01 (Cajori). 286/n. limits and numbers, modem views, 62-67 ~,Ferdinand, 166,168,5)0.514 line: at infinity. 629; boundaries of a part of, SSS-SS6; composed of points, 42; concurrence. 638; aeodesic. 563-564; Greek p:ometriciaDs, 11; group of translations, 703, 705; measuring one
against another, 81-88; point auregate, 556: point.event, 352; straiabtness and direction, 159; He auo straiabt lines line circle, tangents to, 636-637 line integral, curve, 324 linear problems, 91
Linear TrtllUlormotioru, On the TMory 01 (Cayley). 351 Linus, Franciscus. 260. 270 Liouville. Joseph, ttanscendentals, 514 Lipschitz. Professor, 659 Lipslorpius, Daniel7 237 Listing, J. B., Topologie, Vorstudien zur. 57l Littlewood. J. E., IS8. 37S, 508; quoted, t67 liviDa: organisms, asymmetric chemical -:onstilution, 689 Livy, 181/11. loaves and men problem, Rhind papyrus, 172-174 Lobachevski, Nikolai I.: Euclidean postulates, 56S; geometry, foundations critiCIZed, 550; non-Euclidean geometry, 639; parallel axiom. 654; revolution in scientific ideas. S53; space geometry, '46, 547, 554 loci: curve, 17, 19, 31, 82; on surfaces, 91; order of the locus, 113 locus ad Ires et qUiltlUDl' lineas, J08, 112, 130 Lodge, Sir Oliver: commentary on, 218219; "Johann Kepler," 220-234 logarithmic series, 138 loprithmic spiral, 717 logarithmic tables, publication of, 123 logarithms: algebraic notation. 23: coordinates, 137; definition. 124; equiangular spiral, 147; harmonical progression, 137; practical beneftt, 121. 123 Logarithms, Table 01 (Schulze). 158 logic: abstract, 407; Aristotelian, 409; laws of. 68-69; mathematical analysis. 19, 27, 70; proposition, 70; satisfaction of intellectual needs, 10; science of, 6; use of word, 67 logical allebra, 163, 397 10Bical multiplication, 25 logical relations, analolous types, 403 logistica, 498 Lomazzo. Giovanni Paolo, 606 London, Royal Society of, founded, 143 long division, 461 longitudinal reftection. 700. 702 Lorenz, Alfred. music, mathematical treatment, 703 Lorenz, Konrad, 488 Lowe, Victor, 395 111. Lubbock. Sir John. 434; quoted, 433 Lucas, E., 260, 504 Ludwig, Wilhelm: asymmetry. decisive factors. 694; left-right problem, in zoology, 686. 686/11.
Ilflla
M Macdonald. James. 439 /,,_ Macfarlane. Alexander. 340 ",. Mach. Ernst, magnetic needle deviation. 683.685 Mack. Heinrich. 294/". Maclaurin, Colin: analysis. 165; biography, IS 1; ftuxions, 1S2. 287; Fluxions, A Tr~atis~ 0/. 287/n.; lines and figures. 1S3; planets, equilibrium, 152; projective geometry. 102; spheroids, elliptical,149 MacMahon, P. A .• quot~d, 341 McTaggart, J. M. E .• 377. 385. 386, 388 magnetic needle deviation, 683 magnifying glass. invention, 116-117 Mahaffy, J. P., D~sc(Utes, 235 In. Maine. Henry James Sumner, quot~d. 189 Malus, aienne Louis, 161 man: primitive, inventions, 8; unnamed numbers. grasping, 495 Mangiamele, Vito, 476 manifold, three-dimensional closed, 598599 Mann, Thomas, snow crystals, symmetry, 712-713 Mansfield, Lord, quoted, 288 Mantegna. Andrea. 616 many-dimensional geometry, 134 mapping: conformal, 337; conaruent, 696; defined, 672; t~fonnation, 694695 maps, geographical: coloring, 590-591; measuring angles, 17 MarceUus (Plutarch). quoted, 180-185 MarJowe, Christopher. quoted. 220 Mars (planet), motion of, 226-228. 230 Martini, Francesco di Oiorgio, 616 Martini, Simone, "The AnDunciation," 622~ 623 mass, gravitational and inertial, 268 mass point attraction, 266 Mathematica, Philosophiae Naturalis Principia, 290 In. Mathematical Discoveries of Newton (Turnbull), 520 In. mathematical entities, status, 408-409 "MathematiCal Laws, The Exactness of" (Clifford),548-551 "Mathematical Machines" (Davis), 518 Mathematical Papers (Sylvester), 348 Mathematical Papers, Collected (Cayley),362 Mathematical Philosophy (Keyser), 570
In. Mathematical Philosophy, Introduction to (Russell), 378 mathematical philosophy of left and right. 679 mathematical physics, first text-book. 267
mathematical problems. Rhind papyrus, 171-178 Math~matical RecHaliDns and E8MlYs (BaU),466 mathematical .. science, growth in ancient times, 8-18 Mathematical Society, American. 503 mathematical terms, Oreek dcrjvation, 152 Mathematical TTtlCts (Robins), 287 In. mathematicians, date list. 18th century B.c.-20th century A.D•• 78 "Mathematicians, The Oreat" (TumbuU), 75-168 "Mathematicians, The Prince of" (BeD), 295-339 mathematics: advances before 17th oeD.tury. 410; autonomous science, 315; classification, 85-86; Descartes' contribution, 236; discovery, 66, 407; foundations, 413; functionality. idea of, 411; indirect proof, 89; interpolation, 140, 145; logicism. 6. 7, 44; modem, generalization, 511; modem, rise and progress, 19-43, 53-62; monomania, 403; natural science, application to, 43-52, 69-70; originality, 402: perspective. 611; philosophy. 156. 188, 193~ 410; pure, Me pure mathematics; Pythagoreaus' progress. 83, 408; queen of. 498-518 Mathematics, American /oul1lll1 01. 357 Mathematics, Bml History 0/ (Fink). 71
Mathematics, Concise History 01 (Struik). 74 Mathematics, Development 01 (BeD), 294 Mathematics, History 0/ (Ball). 71. 74, 466
Mathematics, History 0/ (Cajori), 286 In. Mathematics. History 01 (Smith), 210
In., 454 In.
Mathematics, Introduction to (Whitehead). 3, 291
In., 398, 412 In.
Mathematics, Principws 01 (RusseD), 378 "Mathematics. Queen of" (Bell), 498518 Mathematics-Queens and Servant of Science (Bell), 497 Mathematics, Sou,ce Book in (Smith). 291 In., 509 MathematicS, Synopsis 01 Pure (Carr), 366.368 Mathematics, What Is? (Courant and Robbins), 571 "Mathematics as an Element in the History of Thought" (Whitehead), 402416
Mathematics lor the Gene,aI R.eatIe, (Titchmarsh), 3
Mathematics in Europe, The History 01 (Sullivan). 74
Math~mat;cal
Mathematics in Wesle". CuilUn (ICJinc).
mathematical probability. birth of idea, 135
602 "Mathematics of Communication, The" (Weaver), 518
Principles 0/ Natural Philosophy (Newton), 8U P,incipia
7:111 Ma/h~ma/ics
T_
01 Gr~a/ AmG/~urs. (Coolidge),601In. Math~ma/ik. Yersuch ~in~s vollk.omm~n cOIU~qu~nlm Sys/~ms d~r (Ohm), 293 Malh~ma/ick, Yor/~sungen tiber Geschicht~ der (Cantor), '72/11. Malh~ma/Uc_r Papyrus (Struve), 169/11.
matrices, theory of, 342, 3'9, 361. 396 matter, motion of. '68 Maxwell, James Clerk; colors, aggregate, 6'6; electro-magnetism 7 644 Mayer, Julius Robert, conservation of energy, 643 In.
M t!II1Iing and Truth, An Inquiry into (Russell), 389 measurement: curvature, 6'2, 6'3, 6'66",6'9,660; leometricaJ, 660-661
Mtta,ur~ment with Compass and Rulttr, Cours~ in th~ Art 01 (DUrer), 601,
609,610,611,612-619 Ana1yt;qu~ (Lagrange), 1S3, 1'4, I" Mecanlqw Cllttsttt (Laplace), 155, 321 mechanics: art of, 103, 180; foundations, 219; military art, 181, 182, 187; model of reality, 44; MechanicS, Th~ Selenctt 01 (Mach), 71 medicine. Pythagoreans' study of, 83 MttdltatiolU (Descartes), 129 medusa, geometricaJ symmetry, 712 Meinong. Alexius, 389 memory: caJculating machines, 518; experience, 10 Mttn 01 Mathttmatics (Bell), 294,340 Menaechmus: conic sections, 103. 236; cube duplication, 93; mechanical instruments. use of. 15; parabola, 91 Menelaus. 78, 109 Mengoli, Pietro, 137 mental calculation, 465, 467 mental work, Jaw for rapidity of, 486-487 mentality, mathematical, submergence of, 414 Mercator, Nicolaus. 137,2'8. 337 Mercury (planet): motions, 229; orbit, 128 meridian circles, 205 Mersenne, Marin. 134, 247 /n.-249 In.: logarithm, 137; numbers, '00, '02-50~ 506; vibration, theory of. 412 mesolabes. geometric instrument. 181 In. Mesopotamia: cuneiform writing, 443; Sumerians' numbers and numerals, 430, 442 metals, transmutation, 272 metamerism, 702 meteorology. 234 Mttthod (Archimedes), 94, 104 Mttthod, DircOurstt on (Descartes), 129 methods, infinitesimal, 40 metric geometry, 570 metric system adopted in France, ISS metricalleometry. 103, 359 Metrodorus, 207 In.
Mlcaniqw
11I4G
Michelanlelo. Cr~alion 01 Adam. 684 Michelson. Albert A., quoted, 262 Michelson-Morley experiment, 640 Micrograph", (Hooke), 261 military machines, Archimedes', 103, 179, 180, 181-182, 187 military problems, application of mathematics,119 Mill, John Stuart, 381,433 million, use of word, 4SS Milne, E. A., Sir lames I~QIU, 645 Mind, Thtt Analysis 01 (Russell), 389 mind and nature, bifurcation, 399 minimum vocabularies, 390-391 mirrors: Archimedes' concave, 179; images, 679, 696, 698; metaJ, 260, 263; reflecting, 110, 132, 154 Mitchell, F. D., 467 !n., 471 /n., 478, 483 mnemonic equations, 50-51 .model, working of the wUverse. 43-44 modem numerals, 453 modular equations, 374 Moebius, August Ferdinand: four-color problem, 591; one-sided surface, 595; strip, 595-597 Moeris, geometry discovery, 189 Moile, Damiano da, 616 molecular theory, 96 moments, 289 Mondeux, Henri, 476, 480 Monge. (laspard, 155,334,601,631 monochord, musical intervals, 189 monochromatic light, 263 monogenic functions, 322 Monrcale mosaic, symmetry, 617, 699 moon: celestial motions, 145, 266, 268; diameter, 421; gravitational pull, 257, 258 Moore, (I. E., 377. 386, 387 More,lIenry.261,268,281 Morice, A., 439 In. Moscow Papyrus, 89, 169, 175 Mother (loose rhyme, problem, 178 motion: chanae of position, 64 /n.; circular, 227; composition, 52; conception, 19; fluid. 716; law of, 44-48; matter, 568; parabolic, 49; resisting medium, 267; rigid, 717; transferred to space, 659; transfonnation, 570; variability, 70 Motte, Andrew. 266 In. Motu Corporum, D~ (Newton), 264-265 MiiJIer, G. E., 467 In. mUltiplication: Ahmes' treatment, 12; calculating prodigies, 469, 472, 481-482; complex. 371; Egyptians, 172; fractions, 17, 25 In.; generalization of number, 30-31; grating method, 461; (Jreek guiding process, 117; logarithms, 123; logical, 25; matrices, 361-362; negative number, 29-30; repetitions, 24; Roman methods, 451; symbols, 27; tables, 460; velocities and times, 50 music: numericaJ ratios, 99-100; periodicities of notes, 416; rhythm, 702-703
Index
musical intervals, planetary relations, 127 musical scale, harmonic progressions, 85, 189 myriad,425 Mysterium (Kepler), 125 Mysticism and Logic (Russell), 379 N n dimensions, geometry of, 342, 344, 359, 640
n partitions of a number (theorem), 167 n roots of algebraic equation, SIt named numbers, 491, 494 Napier, John: arithmetic triangle, 135; biography, 121; inventions, 122; logarithms, 23, 123-125, 137, 140, 142, 148; spherical trigonometry, 125; symbolic algebra, 115; quoud, 123 Napier's Bones, 122 Natura Acidorum, De (Newton), 272 natural bodies, mechanical properties, 665 Natural Knowledge, An Enquiry concerning the Principles 01 (Whitehead), 398 natural numbers, 5()() natural science: ancient, 15; application of mathematics, 43-52, 69-70; prevention of waste, 5-6; satisfaction of intellectual needs, 10 nature: economical tendencies, 154; laws of, mathematically expressed, 411; leftright symmetry, 686; observation by scientists, 411; uniformity, 549 Nature, The Concept 01, 398 Nautical Almanac, 5-6,43,44,233 Needham, Joseph, Order and Lile, 691 Nees von Essenbeck, Christian, 314 negative indices, 131 negative numbers, 6, 27-28, 30, 33, 34, 65, 96, 115. 119, 147-148, 162 Neugebauer, 0., 89, 169 In., 449 In. Newman, James R.: "Rhind Papyrus, The," 169-178; "Srinivasa Ramanujan," 368-376 Newton, Humphrey, 259, 280, 282; quoted,272 Newton, Isaac: achievement, period of, 25.6-257; analytical geometry, 133, 155, 360; apple story, 145, 257, 328; Arithmetica Un;versalis. 145; binomial theorem, 139, 140,256,258,519-524; brachistrochrone problem, 267, 274; calculus, 256, 258, 267, 279; chemical operations, 272-273; chromatic aberration, 260; Chronology 01 the Ancient Kingdoms Amended. 273, 274; commentary on, 254; curves, moving points, 58; earth measurements, 267-268; equations, imaginary roots, 273, 364; experiments, 268, 279; finite differences, calculus, 267; fits, theory of, 262; ftuxions, method of, 48, 58, 61, 142-143, 145, 258; gravitation, theory of, 45, 62, 142. 144, 256-258, 266-268; honors, 145146, 254, 268, 269-270; Hooke misun-
~II
derstandings, 261, 271, 278; hypotheses, 260, 266; infinite series, 521-524; inverse square law, 145, 258, 264, 265, 266; least resistance, problem of, 267; Leibniz controversy, 58-59, 143, 152, 165, 278; letter to Royal Society, 259; light waves, 262, 263, 267; limits, 165, 291 In .•' logarithm, 137; mathematical analysis, 299; mathematical astronomy, 313, 315; mathematical discoveries, 258, 295; metallic mirrors, 260, 263; motion, theory of, 45; Motu Corporum. De, 264-265; Natura Acidorum. De, 272; Opticks, 259, 260, 261, 263, 264, 269, 272, quoud. 266, 273, 286; optics, lectures, 258-259, 271; planetary motions, 52, 132, 140, 144-145, 257. 264, 267-268; Portsmouth Papers, 274, 284: Principia. 62, 144, 145, 257259, 264, 268, 269, 272-275, 279, 280, 282. 284, 286, quoted. 265-266, 267; projective geometry, 102; Prophecies oj Daniel and the Apocalypse 01 St. John. Observations upon, 274; relative concepts, 683; sound, theory of, 154, 412; space structure, 696; spectrum, 260; speculative theory, attitude to, 262; sphere. gravitational pull. 268, 279: telescope, reflecting, 142, 259, 260, 264: universe, system of the, 153; unpublished papers, 280-282, 284; variations, calculus of. 267; waterclock, 255-256; white light discoveries, 141142, 146, 256, 260-263; wind force, ascertaining, 140; quoted, 141, 326 "Newton, Isaac" (Andrade), 255-276 "Newton: The Algebraist and Geometer" (Turnbu)]), 519 In. "Newton, the Man" (Keynes), 277-285 Newtonian viscosity, 267 "Newton's Principles and Modem Atomic Mechanics" (Bohr), 254 nine, Roman numerals, 449 nineteen, Roman numerals, 449 Niven, I. M., Waring problem, 515 Noether. Emmy, 333 Nombres. Thlorie des (Lucas), 504 non-Euclidean geometry, 102, 158, 165, 306,332,342,396,546 normal to a surface. 335 normals, 108 Northrop, F. S. C., 399 In. notation: algebraic, 12, 22-24, 26-27, 32/n., 92, 120, 130; Arabic, 117; decimal, 23, 114, 116, 117; development, 13. 14; fluxions, 58; integral calculus, 54; Leibniz's, 57-58; numerical, 99; symbolic, see symbolic notation; Viete's improvements, 236 Notizen-/ournal (Gauss), 304 Nouvelles Experiences sur Ie vide (Pascal), 134 number: concept, 525-527; definition, 537-543, djgitaJ, 483; group relation-
Indn
xvIII
ships, 402; origin, 432, 433, 434; properties, 156 "Number, Definition of' (Russell), 537543 number intelligence, birds, 494 number knowledge, poverty of, 432 numbers: abstract, 24, 511; algebraic, 26, 313, 326, 499, 511; analytic theory of, 158, 374, 499, 508; binary scale, 516; cardinal, 389; characteristic designs, 84; classes, 65-66; complex, 31. 60, 71, 119. 162, 308, 309, 321-322, 529; compo. for ''The ROnt· gen Rays," from The UltiveTst! of Lipt. by Sir WHiiam Braa,. McGraw-Hili Book Company for "Kinetic Theory of Gases." by Daniel Bernoulli. from SouTce Book lit Pllyslcs, by W. F. Marie. @ 1935 by McGraw-Hili Book Company. Inc. Penguin Books Ltd. for "Theory of Games." by S. Valda. from PeltPilt ScleltCt! News Philosophical Library for "Mathematics of War and Foreian Politics" and "Statistics of Deadly Quarrels," from PsychoioBical FIJCIOTS of PeIJCe and War. by Lewis Fry Richardson and P Pear. if) 1946 by P. Pear. Oxford University Press for "Plateau's Problem," from What Is Mathemflllcs? by Richard COMMltt altd HeTbeTl Robbins. ® 1941 by Richard Courant. 1be Science Press, Lancaster. Pennsylvania. for "Chance," from FOMltdatiolts of Science. by Henri Poincare. translated by George Bruce Halsted. Scitmtific AmeTictJIt for "Crystals and the Fu· ture of Physt"," by Philippe Le ComeiJIer. ® 1953 by Scientific American. Inc. Simon and Schuster. Inc.• for "Mendel~ff'" from Crru:ibles. The StOTY 01 Chemistry. © 1930, 1942. 1948 by Bernard Jaffe and the Forum Printin, Corporation. Society for Promoting Christian Knowledge for selection from Soap BMbbles. by C. V. Boys. Messrs Taylor and Francis. Ltd. and The Philosophical Magazilte for "Atomic Numbers," bY H. G. J. Moseley. University of Chicago Press for ''The Uncertainty Principle," from The PII,slcal PTlnclples 01 tile OMaltt"m Theory, by Werner Heisenberg. John Wiley Ie Sons for "Concerning Probability," 'rom Philosophical Essay Olt Probabllitie!. by Pierre Simon de Laplace.
2'.
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Table of Contents C>: • -2
VOLUME
TWO
::1-
PART
v: Mathematics and the Physical World
Galileo GaWel: Commentary
1. Mathematics of Motion by
726
734
GALILEO GALILEI
The Bernoullis: Commentary
2. Kinetic Theory of Gases by
771
DANIEL BERNOULLI
774
A Great Prize, a Long-SuDering Inventor and the First Accurate Clock: Commentary 778
3. The Longitude by
780
LLOYD A. BROWN
John Couch A dams: Commentary
820
4. John Couch Adams and the Discovery of Neptune by
SIR HAROLD SPENCER JONES
H. G. J. Moseley: Commentary
5. Atomic Numbers by
840
842
H. G. J. MOSELEY
The Small Furniture of Earth: Commentary
6. The Rontgen Rays by SIR WILLIAM 7. Crystals and the Future of Physics by
822
851
BRAGG
854 871
PHILIPPE LE CORBEILLER
Queen Dido, Soap Bubbles, and a Blind Mathematician: Commentary 882
8. What Is Calculus of Variations and What Are Its Applications? by KARL MENGER 9. The Soap-bubble by C. VERNON BOYS 10. Plateau's Problem by
RICHARD COURANT
and
HERBERT ROBBINS y
886 891 901
vi
The Periodic Law and MendeleeD: Commentary
910
11. Periodic Law of the Chemical Elements by DMITRI MENDELEEFF 12. Mendeleeff by BERNARD JAFFE Gregor Mendel: Commentary
13. Mathematics of Heredity by
913 919
932
GREGOR MENDEL
I. B. S. Haldane: Commentary
950
14. On Being the Right Size by J. B. S. HALDANE 15. Mathematics of Natural Selection by J. B. S. HALDANE Erwin Schrodinger: Commentary
975
D'A.rcy Wentworth Thompson: Commentary
996
D'ARCY WENTWORTH THOMPSON
Uncertainty: Commentary
Sir Arthur Stanley Eddington: Commentary
1051 1056
1069
20. The Constants of Nature by SIR ARTHUR STANLEY EDDINGTON 21. The New Law of Gravitation and the Old Law by SIR ARTHUR STANLEY EDDINGTON
1074 1094
1105
22. The Theory of Relativity by
PART VI:
1001
1047
18. The Uncertainty Principle by WERNER HEISENBERG 19. Causa1ity and Wave Mechanics by ERWIN SCHRODINGER
Commentary
952 958
973
16. Heredity and the Quantum Theory by ERWIN SCHRODINGER 17. On Magnitude by
937
CLEMENT V. DURELL
1107
Mathematics and Social Science
The Founder 01 Psychophysics: Commentary
1. Gustav Theodor Fechner by
1146
EDWIN G. BORING
Sir Francis Galton: Commentary
1148
1167
2. Qassification of Men According to Their Natural Gifts by SIR FRANCIS GALTON Thomas Robert Malthlls: Commentary
1173
1189
3. Mathematics of Population and Food by THOMAS ROBERT MAL THUS
1192
COllrnot, Jevons, and the Mathematicl 0/ Money: Commentary 1200
4. Mathematics of Value and Demand by AUGUSTIN COURNOT 5. Theory of Political Economy by WILLIAM STANLEY JEVONS
1203 1217
Contents
vii
A Distinguished Quaker and War: Commentary
1238
6. Mathematics of War and Foreign Politics
1240
by LEWIS FRY RICHARDSON
7. Statistics of Deadly Quarrels by
LEWIS FRY RICHARDSON
The Social Application of Mathematics: Commentary
1264
8. The Theory of Economic Behavior by LEONID HURWICZ 9. Theory of Games by S. VAJDA 10. Sociology Learns the Language of Mathematics by ABRAHAM KAPLAN
PART VII:
1254 1267 1285 1294
The Laws of Chance
Pierre Simon de Lap/ace: Commentary
1316
1. Concerning Probability by PIERRE SIMON DE LAPLACE 2. The Red and the Black by CHARLES SANDERS PEIRCE 3. The Probability of Induction by CHARLES SANDERS PEIRCE Lord Keynes: Commentary
1325 1334 1341
1355
4. The Application of Probability to Conduct
1360
by JOHN MAYNARD KEYNES An Absent-minded Genius and the Laws of Chance: Commentary 1374
5. Chance by
1380
HENRI POINCARE
Ernest Nage/ and the Laws of Probability: Commentary
6. The Meaning of Probability by INDEX
ERNEST NAGEL
1395
1398
follows page 1414
PART V
Mathematics and the Physical World 1. 2. 3. 4. S. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
Mathematics of Motion by GALILEO GALILEI Kinetic Theory of Gases by DANIEL BERNOULLI The Longitude by LLOYD A. BROWN John Couch Adams and the Discovery of Neptune by SIR HAROLD SPENCER JONES Atomic Numbers by H. G. J. MOSELEY The Rontgen Rays by SIR WILLIAM BRAGG Crystals and the Future of Physics by PHILIPPE LE CORBEILLER What Is Calculus of Variations and What Are Its Applications? by KARL MENGER The Soap-bubble by C. VERNON BOYS Plateau's Problem by RICHARD COURANT and HERBERT ROBBINS Periodic Law of the Chemical Elements by DMITRI MENDELEEFF MendeJ.eefl by BERNARD JAFFE Mathematics of Heredity by GREGOR MENDEL On Being the Right Size by J. B. S. HALDANE Mathematics of Natural Selection by J. B. S. HALDANE Heredity and the Quantum Theory by ERWIN SCHRODINGER On Magnitude by D'ARCY WENTWORTH THOMPSON The Uncertainty Princip1e by WERNER HEISENBERG Causality and Wave Mechanics by ERWIN SCHRODINGER The Constants of Nature by SIR ARTHUR STANLEY EDDINGTON The New Law of Gravitation and the Old Law by SIR ARTHUR STANLEY EDDINGTON The Theory of Relativity by CLEMENT V. DURELL
COMMENTARY ON
GALILEO GALILEI ODERN science was founded by men who asked more searching questions than their predecessors. The essence of the scientific revolution of the sixteenth and seventeenth centuries is a change in mental outlook rather than a flowering of invention and Galileo, more than any other single thinker, was responsible for that change. Galileo has been called the first of the moderns. "As we read his writings we instinctively feel at home~ we know that we have reached the method of physical science which is still in use," 1 Galileo's primary interest was to discover how rather than why things work. He did not depreciate the role of theory and was himself unrivaled in framing bold hypotheses. But he recognized that theory must conform to the results of observation, that the schemes of Nature are not drawn up for our easy comprehension. "Nature nothing careth," he says, "whether her abstruse reasons and methods of operating be or be not exposed to the capacity of men," He insisted on the supremacy of the "irreducible and stubborn facts" however "unreasonable" they might seem. 2 '" know very well," says Salviati, a character representing Galileo himself in the Dialogues Concerning the Two Principal Systems 01 the World, "that one sole experiment, or concludent demonstration, produced on the contrary part, sufftceth to batter to the ground . . . a thousand . . . probable Arguments." The origins of modern science can of course be traced much further back-at least to the thirteenth- and fourteenth-century phi1osophers. Robert Grosseteste. Adam Marsh. Nicole Oresme. Albertus Magnus. WiJ1iam of Occam.a Recent historical researches have broadened our understanding of the evolution of scientific thought. proved its continuity (history. like nature, evidently abhors making jumps) and helped to kill the already tottering myth that the science of the Middle Ages was little more than commentary and sterile exegesis. 4 The enlarged perspective
M
'Sir William Dampier, A History 01 Science, a"d Its Relations with Philosophy and Religion, Fourth Edition. Cambridge (England), 1949. p. 129. t It is a great mistake, as Whitehead points out. "to conceive ,his historical revolt as an appeal to reason. On the contrary it was through and through an antiintellectualist movement. It was the return to the contemplation of brute fact; and it was based on a recoil from the inftexible rationality of medieval thought." Science and the Modern World. Chapter f. a See for example Herbert Butterfield. The Origins 01 Modern Science, London, 1949; A. C. Crombie. Rober, Grosseteste and the Origins 01 Experimental Science, Oxford, 1953; A. C. Crombie, Augustine 10 Galileo. the Hislory 01 Science, A.D. 4001650, London, 1952; A. R. Hall, The Scientipc Re\'olulion, 1500-1800. London, ]954. .. In this discussion I have drawn on material of mine published in the pages of Scientipc American; in particular on a review of the Butterfield book cited in the preceding note (Scientipc Americall, July ]950, p. 56 et seq.) I am much indebted to The Origins 0/ Modern Science for its masterly presentation of the period treated above. 726
727
does not, however, diminish one's admiration for the stupendous achievements of Galileo. It is in his approach to the problems of motion that his imaginative powers are most wonderfully exhibited. Let us examine briefly the ideas he had to overthrow and the system he had to create in order to found a rational science of mechanics. According to Aristotle, all heavy bodies had a "natural" motion toward the center of the universe, which, for medieval thinkers was the center of the earth. All other motion was "violent" motion, because it required a constant motive force, and because it contravened the tendency of bodies to sink to their natural place. The acceleration of falling bodies was explained on the ground that they moved more "jubilantly"-somewhat like a horse-as they got nearer home. The planetary spheres, seen to be exempt (rom the "natural" tendency, were kept wheeJing in their great arcs by the labors of a sublime Intelligence or Prime Mover. Except for falling bodies things moved only when and as long as effort was expended to keep them moving. They moved fast when the mover worked hard~ their motion was impeded by friction; they stopped when the mover stopped. For the motion of terrestrial objects Aristotle had in mind the example of the horse and cart; in the celestial regions his mechanics left "the door halfway open for spirits already." On the whole. Aristotle's theory of motion squared well enough with common experience, and his teachings prevailed for more than fifteen centuries. Eventually, however, men began to discover small but disturb· ing inconsistencies between experimental data and the Aristotelian dictates. There was the anomaly of the misbehaving arrow which, according to the horse and cart concept of motion, should have fanen to earth the instant it lost contact with the bowstring. Nor was the traditional explana· tion of the acceleration of falUng bodies swa]Jowed forever without pro. test. In each case the paradox was met by an ingenious modification of the accepted system (this was known, from Plato's celebrated phrase, as "saving the phenomena"); yet every such tailoring, however skillfuJ, was a source of controversy and raised new suspicions as to the validity of all Aristotle's teachings. In the fourteenth century Jean Buridan and others at the University of Paris developed a "theory of impetus" which proved to be a major factor in dethroning Aristotelian mechanics. This theory, later picked up by Leonardo da Vinci, held that a projectile kept moving by virtue of a something "inside the body itself" which it had acquired in the course of getting under way. Falling bodies accelerated because "impetus" was continually being added to the constant fan produced by the original weight. The importance of the theory lay in the fact that men for the first time were presented with the idea of motion as a lingering aftereffect derived from an initial impUlse. This was midway to the modern view,
728
EilItOl" COIfUfIeI'I'
pretty clearly expressed in Galileo, that a body "continues its motion in a straight line until something intervenes to halt or slacken or defiect it." What was needed to complete the journey was an extraordinary transposition of ideas from the real to an imaginary world. The ghosts of Plato and Pythagoras returned triumphantly to point the way. Modem mechanics describes quite well how real bodies behave in the real world; its principles and laws are derived, however, from a nonexistent con· ceptual world of pure, clean, empty, boundless Euclidean space, in which perfect geometric bodies execute perfect geometric figures. Until the great thinkers, operating, in Butterfield's words, "on the margin of contem~ porary thought," were able to establish the mathematical hypothesis of this ideal Platonic world, and to draw their mathematical consequences, it was impossible for them to construct a rational science of mechanics applicable to the physical world of experience. This was the forward leap of imagination required-the new look at familiar things in an unfamiliar way. to see what in fact there was to be seen, rather than what some classical or medieval writer had said ought to be seen. Buridan, Nicole Oresme and Albert of Saxony with their theory of impetus; Galileo with his beautiful systematization of everyday mechanical occurrences and his ability to picture such situations as the behavior of perfectly spherical bans moving on perfectly horizontal planes; Tycho Brahe with his immense and valuable observational labors in astronomy; Copernicus with the De Revolutionibus Orbium and heliocentric hypothesis; Kepler with his laws of planetary motion and his passionate search for harmony and "sphericity"; Descartes with his discourse on method, his determination to have an science as closely knit as mathematics, his wedding of algebra to geometry; Huygens with his mathematical analysis of circular motion and centrifugal force; Gilbert with his terella, his theory of magnetism and gravitation; Viete, Stevin and Napier with their aids to simplicity of mathematical notation and operations: each took a part in the grand renovation not only of the physical sciences but of the whole manner of thinking about the furniture of the outside world. They made possible the culminating intellectual creation of the seventeenth century, the clockwork universe of Newton in which marbles and planets rolled about as a result of the orderly interplay of gravitational forces, in which motion was as "natural" as rest, and in which God, once having wound the clock. had no further duties. Galileo was the principal figure in this drama of changing ideas.' He 5 "Taking his achievements in mechanics as a whole, we must admit that in the progress from the pre-Galileon to the post-Newtonian period, Galileo's contribution extends more than halfway. And it must be remembered that he was the pioneer. Newton s&id no more than the truth when he declared that if he saw further than other men it was because he stood On the shoulders of giants. Galileo in these matters had no giants on whom to mount; the only giants he encountered were those who had first to be destroyed before vision of any kind became possible," Herbert Dingle, The Scientific Ad\'ent"re ("Galileo Galilei (lS64 1642)"), London, 1952, p. 106.
Gall1eo Ga1l1el
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was the first to grasp the importance of the concept of acceleration in dynamics. Acceleration means change of velocity, in magnitude or direction. As against Aristotle's view that motion required a force to maintain it, Galileo held that it is not motion but rather ""the creation Or destruction" of motion or a change in its direction-i.e. acceleration, which requires the application of external force. He discovered the law of falling bodies. This law, as Bertrand Russell remarks, given the concept of "acceleration," is of Uthe utmost simplicity." 8 A falling body moves with constant acceleration except for the resistance of the air. At first Ga1i1eo supposed that the speed of a fa11ing body was proportional to the distance fallen through. When this hypothesis proved unsatisfactory, he modified it to read that speed was proportional to the time of fall. The mathematical consequences of this assumption he was able partially to verify by experiment. Because free-falling bodies attained a velocity beyond the capacity of the measuring instruments then available, Galileo approached the problem of verification by experiments in which the effect of gravity was "diluted." He proved that a body moving down an inclined plane of given height attains a velocity independent of the angle of slope, and that its terminal speed is the same as if it had fallen through the same vertical height. The trials on the inclined plane thus confirmed his law. The famous story about his dropping different weights from the Tower of Pisa to disprove Aristotle's contention that heavy objects fall faster than light ones is probably untrue; but it follows from Ga1ileo's law that an bodies, heavy or light, are subject to the same acceleration. He himself had no doubt on this score-whether or not he actually corroborated the principle by experiment-but it was not until after his death, when the air pump had been invented, that a complete proof was given by causing bodies to fall in a vacuum. Experimenting with a pendulum, Galileo obtained further evidence to sustain the principle of "persistence of motion." The swinging bob of a pendulum is analogous to a body falling down an inclined plane. A bal] rolling down a plane, assuming negligible friction, will climb another plane to a height equal to that of its starting point. And so, as Galileo found, with the bob. If released at one horizontal level C (as in the diagram), it will ascend to the same height DC, whether it moves by the arc BD or, when the string is caught by nails at E or F, by the steeper arcs BG or BI." It was a short step from his work on the problem of persistence of motion to Newton's first law of motion, also known as the law of inertia. Another of Ga1i1eo's important discoveries in dynamics resulted from his study of the path of projectiles. That a cannon ball moves forward and 6 '7
Bertrand Russell, A History oj Western Philosophy. New York, 1945, p. 532. See Crombie (Augustine 10 Gali/eo) , op. cit., pp. 299-300.
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also falls was obvious, but how these motions were combined was not understood. Galileo showed that the trajectory of the projectile could be resolved into two simultaneous motions: one horizontal, with the velocity (disregarding the small air resistance) remaining unchanged, the other vertical, conforming to the law of falling bodies, i.e., 16 feet in the first second, 48 feet in the second second, 80 feet in the third, and so on. Combining the two motions makes the path a parabola. Galileo's principle of the persistence of motion, and his method of dissecting compound motions solved the apparent anomalies gleefully urged by those opposed to the Copernican system. It could now be explained why an object dropped from the mast of a ship feU to the foot of the mast and was not left behind by the ship's motion; why a stone dropped from a tower landed at its base and not to the west of it, even though the rotating earth had moved towards the east while the stone fell. The stone, as Ga1ileo realized, shares the velocity of rotation of the earth. and retains it on the way down. 8 It was Galileo's way to turn back and forth from hypothesis and deduc8 "In fact if the tower were high enough. there would be the opposite effect to that expected by the opponents of Copernicus. The top of the tower, being further from the center of the earth than the bottom, is moving faster, and therefore the stone should fall slightly to the east of the foot of the tower. This effect, however, would be too s1ight to be measurable." Russell, op. cit., p. 534.
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tion to experiment: no one before him attained a comparable skill in blending experiments with mathematical abstractions. In all his investigations he followed the procedure epitomized in a famous passage of Francis Bacon: "to educe and form axioms from experience ..• to deduce and derive new experiments from axioms. . . . For our road does not lie on a level, but ascends and descends; first ascending to axiom~ then descending to works." In the true Platonic tradition he was convinced that the mathematical models which led him to observations were the "enduring reaJity. the substance. underlying phenomena." 0 "Philosophy," he wrote in his polemic treatise, II Saggiatore, "is written in that vast book which stands forever open before our eyes, I mean the universe; but it cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles. circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word."
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The facts of Galileo's life are well known and I shall restrict myself to a bare summary. He was born in 1564, the year Michelangelo died, and he died in 1642, the year in which Newton was born. ("I commend these facts," says Bertrand Russell, Uto those (if any) who still believe in metempsychosis. tt) His father was a noble of Florence and Ga1ileo was well educated. first in medicine. at the University of Pisa. then in mathematics and physics~ One of his first discoveries was that of the isochronism of the pendulum; he was seventeen when the sight of a lamp set swinging in the cathedral of Pisa-which he measured by his pulse beats-inspired this conjecture. Another of his early works was the invention of a hydrostatic balance. For eighteen years (1592-1610) he held the chair of mathematics at Padua. He was an enormously popular lecturer, and "such was the charm of his demonstrations that a hall capable of containing 2000 people had eventually to be assigned for the accommodaI) "Galileo's Platonism was of the same kind as that which had led to Archimedes being known in the sixteenth century as the 'Platonic philosopher,' and with Galiteo mathematical abstractions got their validity as statements about Nature by being solutions of particular physical problems. By using this method of abstracting from immediate and direct experience, and by correlating observed events by means of mathematical relations which could not themselves be observed, he was led to experiments of which he could not have thought in terms of the old commonsense empiricism. A good example of this is his work on the pendulum. "By abstracting from the inessentials of the situation. 'the opposition of the air, and line, or other accidents,' he was able to demonstrate the law of the pendulum, that the period of oscillation is independent of the arc of swing and simply proportional to the square-root of the length. This having been proved, he could then reintroduce the previously excluded factors. He showed. for instance, tbat the reason why a real pendulum. of which the thread was not weightless. came to rest, was not simply because of air resistance, but because each particle of the tbread acted u a small pendulum. Since they were at different distances from the point of suspension, they had different frequencies and tberefore inbibited each other." Crombie. op. cit., (Augustine to Ga/ileo). pp. 295-296.
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tion of the overftowing audiences which they attracted." 10 Galileo made first-class contributions to hydrostatics, to the mechanics of fluids and to acoustics. He designed the first pendulum clock, invented the first thermometer (a glass bulb and tube, filled with air and water, with the end of the open tube dipping in a vessel of water), and constructed, from his own knowledge of refraction, one of the first telescopes and a compound microscope.ll Although he was early drawn to the Copernican system, it was not until 1604, on the appearance of a new star, that he publicly renounced the Aristotelian axiom of the "incorruptibility of the heavens"; a short time later he entirely abandoned the Ptolemaic principles. Having greatly improved on his flrst te1escope, GaHleo made a series of discoveries which opened a new era in the history of astronomy. He observed the mountains in the moon and roughly measured their height; the visibility of 4'the old moen in the new moon's arms" he explained by earth-shine; 12 he discovered four of the eleven satellites of Jupiter,18 innumerable stars and nebulae, sun spots, the phases of Venus predicted by Copernicus, the librations of the moon. In 161S, after Ga1i1eo had removed from Padua to Florence, his advocacy of the Copernican doctrines began to bring him into conflict with the Church. At first the warnings were mild. UWrite freely," he was told by a high ecclesiastic, Monsignor Dini, "but keep outside the sacristy:' He made two visits to Rome to exp1ain his position; the second on the accession of Pope Urban VIII, who received him warmly. But the publication in 1632 of his powerfully argued, beautifully written masterpiece, Dialogue on the Great World Systems, an evaluation, "sparkling with malice," of the comparative merits of the old and new theories of celestial motion, brought a head-on collision with the Inquisition.14 Ga1i1eo was Encyclopaedia Britannica, Eleventh Edition. article on Galileo. It is the Dutchman, Jan Lippershey, to whom priOl ity in the invention of these . truments (1600) is usually attributed. See H. C. King, The History 0/ ,_ Telesc pe, Cambridge, 1955. small excursion on Galileo's work with the microscope will perhaps be pennitted. .. ben tbe Frenchman, Jean Tarde, called on Galileo in 1614. he said 'Galileo told that the tube of a telescope for observing the stars is no more than two feet in length; but to see objects weU, which are very near, and which on account of their smaller size are hardly visible to tbe naked eye, the tube must be two or three times longer. He tells me that with this long tube be has seen fties which look as big as a Jamb, are covered all over with hair.. and have very pointed nails. by means of which they keep themselves up and walk on glass although hanging feet upwards.'" Galileo. Opera, Ed. Nar.. Vol. 19, p. 589, as quoted in Crombie, op. cit., p. 352. 111 Sir Oliver Lodge, Pioneers 0/ Sciellce. London. 1928. p. 100. 18 It is not always possible to prove to a philosopher the existence of a thing by bringing it into plain view. The professor of philosophy at Padua "refused to look throu(th Galileo's telescope. and his colleague at Pisa labored before the Grand Duke with 10gicaJ arguments, 'as if with magical incantations to charm the new planets out of the sky.''' Dampier. op. cit., p. 130. 14 In 1953 appeared two new excellent editions of the Dialogue, last translated into English by Sir Thomas Salusbury in 1661: (1) Galil"o's Dialogue on the Grea, World S}'stems. edited by Giorgio de Santillana. Chicago. 1953 (based on the Salusbury translation); (2) Galileo Galilel-Dialogue Concerning the Two Chie/ World 10
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summoned to Rome and tried for heresy. Before his spirit was broken he observed: "In these and other positions certainly no man doubts but His Holiness the Pope hath always an absolute power of admitting or condemning them; but it is not in the power of any creature to make them to be true or false or otherwise than of their own nature and in fact they are." Long questioning-though it is unlikely physical torture was applied-brought him to his knees. He was forced to recant. to recite penitential psalms. and was sentenced to house imprisonment for the rest of his life. He retired to a villa at Arcetri, near Florence. where he continued, though much enfeebled and isolated, to write and meditate. In 1637 he became bHnd and thereafter the rigor of his confinement was relaxed so as to permit him to have visitors. Among those who came, it is said. was John Milton. He died aged seventy-eight. I have taken a substantial excerpt from the Dialogues Concerning Two New Sciences (Discorsi e Dimostriazioni Matematiche Intorno a due nuove scienze),l1 a work completed in 1636. It presents his mature and final reflections on the science of mechanics and is a monument of literature and science. On the margin of Galileo's own copy of the Dialogue on the Great World Systems appears a note in his handwriting which sums up his lifelong, passionate and courageous dedication to the unending struggle of reason against authority ~ "In the matter of introducing novelties. And who can doubt that it will lead to the worst disorders when minds created free by God are compelled to submit slavishly to an outside will? When we are told to deny our senses and subject them to the whim of others? When people devoid of whatsoever competence are made judges over experts and are granted authority to treat them as they please? These are the novelties which are apt to bring about the ruin of commonwealths and the subversion of the state." Syslems-Proiemaic and Copernican, translated by Stillman Drake, Foreword by Albert Einstein, Berkeley and Los Angeles, 1953. For the most detailed and searching modern account of Galileo's clash with the Church, and trial for heresy, see Giorgio de Santillana, The Crime 01 Galileo, Chicago. 1955. 15 The standard English translation is by Henry Crew and Alfonso de Salvio, The Macmillan Company, New York, 1914. A reprint has recently (952) been issued by Dover Publications, Inc., New York.
[My uncle Toby] proceeded next to Galileo and Torricellius. wherein. by certain Geometrical rules, inlallibly laid down. he lound the precise part to be a "Parahola"--or else an "Hyperhola,"--tmd that the para mete,.. or "latus rectum," 01 the conic section 01 the said path. was to the quantity and amplitude in a direct ratio, as the whole line to the sine 01 double the angle of incidence, lormed by the breech upon an horizontal line;--and that the semiparameter.-stopl my dear uncle T oby-stop! -LAWRENCE STERNE
In questions 01 science the authority 01 a thousand is not worth the humble -GALILEO GALJLEI reasoning 01 a single individual.
1
Mathematics of Motion By GALILEO GALILEI
CHANGE OF
THIRD DAY POSITION [De Motu Locall1
MY purpose is to set forth a very new science dealing with a very ancient subject. There is, in nature, perhaps nothing older than motion, concerning which the books written by philosophers are neither few nor small; nevertheless I have discovered by experiment some properties of it which are worth knowing and which have not hitherto been either observed or demonstrated. Some superficial observations have been made, as, for instance, that the free motion [naturalem mOlum] of a heavy falling body is continuously accelerated; 1 but to just what extent this acceleration occurs has not yet been announced; for so far as I know, no one has yet pointed out that the distances traversed, during equal intervals of time, by a body falling from rest, stand to one another in the same ratio as the odd numbers beginning with unity.2 It has been observed that missiles and prOjectiles describe a curved path of some sort; however no one has pointed out the fact that this path is a parabola. But this and other facts, not few in number or less worth knowing, I have succeeded in proving; and what I consider mOre impor4 tant, there have been opened up to this vast and most excellent science. of which my work is merely the beginning, ways and means by which other minds more acute than mine will explore its remote comers. NATURALLY ACCELERATED MOTION
The properties belonging to uniform motion have been discussed in the preceding section; but accelerated motion remains to be considered. 1 "Natural motion" of the author has here been translated into "free motion"since this is the term used to-day to distinguish the "natural" from the "violent" motions of the Renaissance. f Trans.] 2 A theorem demonstrated in Corollary I, p. 746. [Trans. I
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And first of all it seems desirable to find and explain a definition best fitting natural phenomena. For anyone may invent an arbitrary type of motion and discuss its properties; thus. for instance. some have imagined helices and conchoids as described by certain motions which are not met with in nature. and have very commendably established the properties which these curves possess in virtue of their definitions; but we have decided to consider the phenomena of bodies falling with an acceleration such as actually occurs in nature and to make this definition of accelerated motion exhibit the essential features of observed accelerated motions. And this, at last, after repeated efforts we trust we have succeeded in doing. In this belief we are confirmed mainly by the consideration that experimental results are seen to agree with and exacdy correspond with those properties which have been. one after another. demonstrated by us. Finally, in the investigation of naturally accelerated motion we were led, by hand as it were, in fol1owing the habit and custom of nature herself. in all her various other processes, to employ only those means which are most common, simple and easy. For I think no one believes that swimming or fiying can be accomplished in a manner simpler or easier than that instinctively employed by fishes and birds. When, therefore. I observe a stone initially at rest faUing from an elevated position and continually acquiring new increments of speed. why should I not believe that such increases take place in a manner which is exceedingly simple and rather obvious to everybody? If now we examine the matter carefully we find no addition or increment more simple than that which repeats itself always in the same manner. This we readily understand when we consider the intimate relationship between time and motion; for just as uniformity of motion is defined by and conceived through equal times and equal spaces (thus we call a motion uniform when equal distances are traversed during equal time-intervals), so also we may. in a similar manner. through equal time-intervals, conceive additions of speed as taking place without complication; thus we may picture to our mind a motion as uniformly and continuously accelerated when, during any equal intervals of time whatever. equal increments of speed are given to it. Thus if any equal intervals of time whatever have elapsed. counting from the time at which the moving body left its position of rest and began to descend. the amount of speed acquired during the first two time-intervals will be double that acquired during the first timeinterval alone; so the amount added during three of these time-intervals will be treble; and that in four. quadruple that of the first time-interval. To put the matter more dearly, if a body were to continue its motion with the same speed which it had acquired during the first time-interval and were to retain this same uniform speed. then its motion would be twice
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as slow as that which it would have if its velocity had been acquired during two time-intervals. And thus, it seems, we shall not be far wrong if we put the increment of speed as proportional to the increment of time; hence the definition of motion which we are about to discuss may be stated as follows: A motion is said to be uniformly accelerated, when starting from rest, it acquires, during equal time-intervals, equal increments of speed. SAGREDO. Ahhough I can offer no rational objection to this or indeed to any other definition, devised by any author whomsoever, since all definitions are arbitrary, I may nevertheless without offense be allowed to doubt whether such a definition as the above, established in an abstract manner, corresponds to and describes that kind of accelerated motion which we meet in nature in the case of freely falling bodies. And since the Author apparently maintains that the motion described in his definition is that of freely falling bodies, I would Uke to clear my mind of certain difficulties in order that I may later apply myself more earnest1y to the propositions and their demonstrations. SALVlATI. It is welJ that you and Simplicio raise these difficulties. They are, I imagine. the same which occurred to me when I first saw this treatise. and which were removed either by discussion with the Author himself, or by turning the matter over in my own mind. SAGR. When I think of a heavy body faning from rest. that is, starting with zero speed and gaining speed in proportion to the time from the beginning of the motion; such a motion as would, for instance, in eight beats of the pulse acquire eight degrees of speed; having at the end of the fourth beat acquired four degrees; at the end of the second, two; at the end of the first. one: and since time is divisible without limit, it follows from al1 these considerations that if the earlier speed of a body is less than its present speed in a constant ratio, then there is no degree of speed however small (or, one may say, no degree of slowness however great) with which we may not find this body travelling after starting from infinite slowness, i.e., from rest. So that if that speed which it had at the end of the fourth beat was such that, if kept uniform, the body would traverse two mites in an hour, and if keeping the speed which it had at the end of the second beat, it would traverse one mile an hour. we must infer that, as the instant of starting is more and more nearly approached, the body moves so slowly that, if it kept on moving at this rate, it wouJd not traverse a mile in an hour, or in a day, or in a year or in a thousand years; indeed, it would not traverse a span in an even greater time; a phenomenon which baffles the imagination. while our senses show us that a heavy falling body suddenly acquires great speed. SALvo This is one of the difficulties which I also at the beginning, experienced, but which I shortly afterwards removed; and the removal
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was effected by the very experiment which creates the difficulty for you. You say the experiment appears to show that immediately after a heavy body starts from rest it acquires a very considerable speed: and I say that the same experiment makes clear the fact that the initial motions of a falling body, no matter how heavy, are very slow and gentle. Place a heavy body upon a yielding material, and leave it there without any pressure except that owing to its own weight; it is clear that jf one lifts this body a cubit or two and allows it to fall upon the same material, it will, with this impulse, exert a new and greater pressure than that caused by its mere weight; and this effect is brought about by the [weight of the] falling body together with the velocity acquired during the fall, an effect which will be greater and greater according to the height of the fall, that is according as the velocity of the falling body becomes greater. From the quality and intensity of the blow we are thus enabled to accurately estimate the speed of a falling body. But teU me, gentlemen, is it not true that if a block be allowed to fan upon a stake from a height of four cubits and drives it into the earth, say, four finger-breadths, that coming from a height of two cubits it win drive the stake a much Jess distance, and from the height of one cubit a still less distance; and finally if the block be lifted only one finger-breadth how much more will it accomplish than if merely laid on top of the stake without percussion? Certainly very little. If it be lifted only the thickness of a leaf, the effect wiIJ be altogether imperceptible. And since the effect of the blow depends upon the velocity of this striking body, can anyone doubt the motion is very slow and the speed more than small whenever the effect [of the blow] is imperceptible? See now the power of truth; the same experiment which at first glance seemed to show one thing. when more carefuny examined. assures us of the contrary. But without depending upon the above experiment, which is doubtless very conclusive, it seems to me that it ought not to be difficu1t to establish such a fact by reasoning alone. Imagine a heavy stone held in the air at rest; the support is removed and the stone set free; then since it is heavier than the air it begins to fall, and not with uniform motion but slowly at the beginning and with a continuously accelerated motion. Now since velocity can be increased and diminished without limit, what reason is there to believe that such a moving body starting with infinite slowness. that is, from rest, immediately acquires a speed of ten degrees rather than one of four, or of two, or of one, or of a half, or of a hundredth; or, indeed, of any of the infinite number of small values [of speed]? Pray listen. I hardly think you win refuse to grant that the gain of speed of the stone falling from rest follows the same sequence as the diminution and Joss of this same speed when, by some impelling force. the stone is thrown to its former elevation: but even if you do not grant this, I do not see
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how you can doubt that the ascending stone, diminishing in speed, must before coming to rest pass through every possible degree of slowness. SIMPLICIO. But if the number of degrees of greater and greater slowness is limitless, they wiH never be al1 exhausted, therefore such an ascending heavy body wi1l never reach rest, but will continue to move without limit always at a slower rate; but this is not the observed fact. SALvo This would happen, Simplicio, if the moving body were to maintain its speed for any length of time at each degree of velocity; but it merely passes each point without delaying more than an instant: and since each time-interval however small may be divided into an infinite number of instants, these wi1l always be sufficient [in number] to correspond to the infinite degrees of diminished velocity. That such a heavy rising body does not remain for any length of time at any given degree of velocity is evident from the following: because if, some time-interval having been assigned, the body moves with the same speed in the last as in the first instant of that time-interval, it could from this second degree of elevation be in like manner raised through an equal height, just as it was transferred from the first elevation to the second, and by the same reasoning would pass from the second to the third and would finally continue in uniform motion forever . . . . SALvo The present does not seem to be the proper time to investigate the cause of the acceleration of natural motion concerning which various opinions have been expressed by various philosophers, some explaining it by attraction to the center, others to repUlsion between the very small parts of the body, While still others attribute it to a certain stress in the surrounding medium which closes in behind the falling body and drives it from one of its positions to another. Now, al1 these fantasies, and others too, ought to be examined; but it is not really worth while" At present it is the purpose of our Author merely to investigate and to demonstrate some of the properties of accelerated motion (whatever the cause of this acceleration may bel-meaning thereby a motion, such that the momentum of its velocity [i moment; della sua velocila] goes on increasing after departure from rest, in simple proportionality to the time, which is the same as saying that in equal time-intervals the body receives equal increments of velocity; and if we find the properties [of accelerated motion] Which will be demonstrated later are realized in freely falling and accelerated bodies, we may conclude that the assumed definition includes such a motion of fa11ing bodies and that their speed [accelerazione] goes on increasing as the time and the duration of the motion. SAGR. So far as I see at present, the definition might have been put a little more clearly perhaps without changing the fundamental idea, namely, uniformly accelerated motion is such that its speed increases in proportion to the space traversed; so that, for example, the speed acquired
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by a body in falling four cubits would be double that acquired in falling two cubits and this latter speed would be double that acquired in the first cubit. Because there is no doubt but that a heavy body falling from the height of six cubits has, and strikes with~ a momentum {impeto] double that it had at the end of three cubits, triple that which it had at the end of one. SALVe It is very comforting to me to have had such a companion in error; and moreover let me ten you that your proposition seems so highly probable that our Author himself admitted, when I advanced this opinion to him, that he had for some time shared the same fallacy. But what most surprised me was to see two propositions so inherently probable that they commanded the assent of everyone to whom they were presented, proven in a few simple words to be not only false, but impossible. SIMP. I am one of those who accept the proposition, and believe that a falling body acquires force [vires] in its descent, its velocity increasing in proportion to the space, and that the momentum [momento] of the falling body is doubled when it fans from a doubled height; these proposi. tions, it appears to me, ought to be conceded without hesitation or controversy. SALv. And yet they are as false and impossible as that motion should be completed instantaneously; and here is a very clear demonstration of it. If the velocities are in proportion to the spaces traversed, or to be traversed, then these spaces are traversed in equal intervals of time; if, therefore, the velocity with which the falling body traverses a space of eight feet were double that with which it covered the first four feet (just as the one distance is double the other) then the time-intervals required for these passages would be equal. But for one and the same body to fall eight feet and four feet in the same time is possible only in the case of instantaneous [discontinuous] motion; but observation shows us that the motion of a falling body occupies time, and less of it in covering a distance of four feet than of eight feet; therefore it is not true that its velocity increases in proportion to the space. The falsity of the other proposition may be shown with equal clearness. For if we consider a single striking body the difference of momentum in its blows can depend only upon difference of velocity; for if the striking body falling from a double height were to deliver a blow of double momentum, it would be necessary for this body to strike with a doubled velocity~ but with this doubled speed it would traverse a doubled space in the same time-interval; observation however shows that the time required for fan from the greater height is longer. SAGR. You present these recondite matters with too much evidence and ease; this great facility makes them less appreciated than they would
740
be had they been presented in a more abstruse manner. For, in myopinion, people esteem more lightly that knowledge which they acquire with so little labor than that acquired through long and obscure discussion. SALVo If those who demonstrate with brevity and clearness the fallacy of many popular beliefs were treated with contempt instead of gratitude the injury would be quite bearable; but on the other hand it is very unpleasant and annoying to see men, who claim to be peers of anyone in a certain field of study, take for granted certain conclusions which later are quickly and easily shown by another to be false. I do not describe such a feeling as one of envy, which usually degenerates into hatred and anger against those who discover such fal1acies; I would call it a strong desire to maintain old errors, rather than accept newly discovered truths. This desire at times induces them to unite against these truths. although at heart believing in them. merely for the purpose of lowering the esteem in which certain others are held by the unthinking crowd. Indeed. I have heard from our Academician many such fallacies held as true but easily refutable; some of these I have in mind. SAGR. You must not withhold them from us, but. at the proper time. tell us about them even though an extra session be necessary. But now. continuing the thread of our talk, it would seem that up to the present we have established the definition of uniformly accelerated motion which is expressed as f01l0ws: A motion is said to be equally or uniformly accelerated when, starting from rest, its momentum (eeleritatis momenta) receives equal increments in equal times. SALvo This definition established, the Author makes a single assumption, namely, The speeds acquired by one and the same body moving down planes of different inclinations are equal when the heights of these planes are equal. By the height of an inclined plane we mean the perpendicular let faU from the upper end of the plane upon the horizontal line drawn through the lower end of the same plane. Thus, to illustrate, let the line AB be horizontal. and let the planes CA and CO be inclined to it; then the Author calls the perpendicular CB the "height" of the planes CA and CD; he supposes that the speeds acquired by one and the same body. descending along the planes CA and CD to the terminal points A and 0 are equal since the heights of these planes are the same, CB; and also it must be understood that this speed is that which would be acquired by the same body falling from C to B. SAGR. Your assumption appears to me so reasonable that it ought to be conceded without question. provided of course there are no chance or outside resistances, and that the planes are hard and smooth, and that
741
MtI'''emtltu:s of 'Motion
c
FIGURE 1
the figure of the moving body is perfectly round, so that neither plane nor moving body is rough. All resistance and opposition having been removed, my reason tells me at once that a heavy and perfectly round baH descending along the lines CA, CD, CB would reach the terminal points A, 0, B, with equal momenta [impel; eguali]. SALVo Your words are very plausible; but I hope by experiment to increase the probability to an extent which shall be little short of a rigid demonstration. Imagine this page to represent a vertical wall, with a nai] driven into it; and from the nail let there be suspended a lead bullet of one or two ounces by means of a fine vertical thread, AB, say from four to six feet long, on this wan dr.aw a horizontal Hne DC, at right angles to the vertical thread AB, which hangs about two finger-breadths in front of the wan. Now bring the thread AB with the attached ban into the posi. tion AC and set it free; first it will be observed to descend along the arc CBD. to pass the point B. and to travel along the arc BD, tm it almost reaches the horizontal CD, a slight shortage being caused by the resistance of the air and the string; from this we may rightly infer that the ban in its descent through the arc CB acquired a momentum [impelo] on reaching B, which was just sufficient to carry it through a similar arc BD to the same height. Having repeated this experiment many times, let us noW drive a nail into the waH close to the perpendicular AB t say at E or F, so that it projects out some five or six finger-breadths in order that the thread, again carrying the bullet through the arc CB, may strike upon the nail E when the bullet reaches B, and thus compel it to traverse the arc BO. described about E as center. From this we can see what can be done by the same momentum [impelo] which previously starting at the same point B carried the same body through the arc BD to the horizontal CD. Now, gentlemen, you will observe with pleasure that the ball swings to the point a in the horizontal, and you would see the same thing happen if the obstacle were placed at some lower point, say at F, about which the ban would describe the arc BIt the rise of the ball always terminating exactly on the line CD. But when the nail is placed so low that the remainder of the thread below it will not reach to the height CD
Gam'D G"",.,
742
c
FIGURE 2
(which would happen if the nail were placed nearer B than to the intersection of AB with the horizontal CD) then the thread leaps over the nail and twists itself about it. This experiment leaves no room for doubt as to the truth of our supposition; for since the two arcs CB and DB are equal and similarly placed, the momentum [momento] acquired by the fall through the arc CB is the same as that gained by fall through the arc DB; but the momentum [momentoJ acquired at B, owing to faU through CB, is able to lift the same body [mobile] through the arc BD; therefore, the momentum acquired in the fall BD is equal to that which lifts the same body through the same arc from B to 0; so, in general, every momentum acquired by fall through an arc is equal to that which can lift the same body through the same arc. But all these momenta [momenti] which cause a rise through the arcs BD, BG, and BI are equal, since they are produced by the same momentum, gained by fall through CB. as experiment shows. Therefore all the momenta gained by fall through the arcs DB, GB, IB are equal. SAGR. The argument seems to me so conclusive and the experiment so well adapted to establish the hypothesis that we may, indeed, consider it as demonstrated. SALVo I do not wish, Sagredo, that we trouble ourselves too much about this matter, since we are going to apply this principle mainly in motions which occur on plane surfaces, and not upon curved, along which acceleration varies in a manner greatly different from that which we have assumed for planes. So that, although the above experiment shows us that the descent of the moving body through the arc CB confers upon it momentum [momento] just sufficient to carry it to the same height through any of
Math~mQtlcs
743
01 Mollo"
the arcs BD. BG, BI. we are not able, by similar means, to show that the event would be identical in the case of Ii perfectly round ball descending along planes whose incHnations are respectively the same as the chords of these arcs. It seems likely, on the other hand, that, since these planes form angles at the point B, they will present an obstacle to the ba11 which has descended along the chord CB, and starts to rise a10ng the chord BD, BG, BI. In striking these planes some of its momentum [impeto1 will be lost and it will not be able to rise to the height of the line CD; but this obstacle, which interferes with the experiment, once removed, it is clear that the momentum [impeto] (which gains in strength with descent) will be able to carry the body to the same height. Let us then, for the present, take this as a postulate, the absolute truth of which will be established when we find that the inferences from it correspond to and agree perfectly with experiment. The Author having assumed this single principle passes next to the propositions which he clearly demonstrates; the first of these is as follows: THEOREM I, PROPOSITION I
The time in which any space is traversed by a body starting from rest and uniformly accelerated is equal to the time in which that same space would be traversed by the same body moving at a uniform speed whose value is the mean of the highest speed and the speed just before acceleration began. Let us represent by the line AB the time in which the space CD is traversed by a body which starts from rest at C and is uniformly accelerated; let the final and highest value of the speed gained during the interval AB be represented by the line EB drawn at right angles to AB; draw the line AE. then an lines drawn from equidistant points on AB and parallel to BE will represent the increasing values of the speed, beginning with the instant A. Let the point F bisect the line EB; draw FG parallel to BA, and GA parallel to FB, thus forming a parallelogram AGFB which will be equal in area to the triangle AEB. since the side GF bisects the side AE at the point I; for if the parallel lines in the triangle AEB are extended to GI, then the sum of all the parallels contained in the quadrilateral is equal to the sum of those contained in the triangle AEB; for those in the triangle IEF are equal to those contained in the triangle G lA, while those included in the trapezium AIFB are common. Since each and every instant of time in the time-interval AB has i1s corresponding point on the line AB, from which points parallels drawn in and 1imited by the triangle AEB represent the increasing values of the growing velocity. and since parallels contained within the rectangle represent the values of a speed which is not increasing, but constant, it appears, in like manner. that the momenta [momenti] assumed by the moving body may also be
744
c G .A
£ F
B
p FIGURE 3
represented. in the case of the accelerated motion. by the increasing parallels of the triangle AEB. and, in the case of the uniform motion. by the paranels of the rectangle GB. For, what the momenta may lack in the first part of the accelerated motion (the deficiency of the momenta being represented by the parallels of the triang.e AGI) is made up by the momenta represented by the parallels of the triangle 1EF. Hence it is clear that equal spaces will be traversed in equal times by two bodies, one of Which, starting from rest, moves with a uniform acceleration, while the momentum of the other, moving with uniform speed, is one-half its maximum momentum under accelerated motion. Q. E. D. THEOREM II, PROPOSITION II
The spaces described by a body falling from rest with a uniformly accelerated motion are to each other as the squares of the time-intervals employed in traversing these distances. Let the time beginning with any instant A be represented by the straight line AB in which are taken any two time-intervals AD and AE. Let HI represent the distance through which the body, starting from rest at H, falls with uniform acceleration. If HL represents the space traversed during the time-interval AD. and HM that covered during the interval AB. then the space MH stands to the space LH in a ratio which is the square of the ratio of the time AE to the time AD; or we may say simply that the distances HM and HL are related as the squares of AE and AD. Draw the line AC making any angle whatever with the line AB; and
745
Mathematics of Motlo,.
AH o
0
L
P £
c
FIGUllE 4
from the points D and E, draw the paralle11ines DO and EP; of these two lines, DO represents the greatest velocity attained during the interval AD, whi1e EP represents the maximum velocity acquired during the interval AE. But it has just been proved that so far as distances traversed are concerned it is precisely the same whether a body falls from rest with a uniform acceleration or whether it falls during an equal time-interval with a constant speed which is one-half the maximum speed attained during the accelerated motion. It follows therefore that the distances HM and HL are the same as would be traversed, during the time-intervals AE and AD, by uniform velocities equal to one-half those represented by DO and EP respectively. If, therefore, one can show that the distances HM and HL are in the same ratio as the squares of the time-intervals AE and AD, our proposition will be proven. But in the fourth proposition of the first book it has been shown that the spaces traversed by two particles in uniform motion bear to one another a ratio which is equal to the product of the ratio of the velocities by the ratio of the times. But in this case the ratio of the velocities is the same as the ratio of the time-intervals (for the ratio of AE to AD is the same as that of tAl EP to % DO or of EP to DO). Hence the ratio of the spaces traversed is the same as the squared ratio of the time-intervals. Q. E. D.
Gal,'-o Gal''-'
746
Evidently then the ratio of the distances is the square of the ratio of the final velocities, that is, of the lines EP and DO, since these are to each other as AE to AD. COROLLARY I
Hence it is dear that if we take any equal intervals of time whatever, counting from the beginning of the motion, such as AD, DE, EF, FG, in which the spaces HL, LM, MN, NI are traversed, these spaces wil1 bear to one another the same ratio as the series of odd numbers, 1, 3, S, 7; for this is the ratio of the differences of the squares of the Hnes [which represent time], differences which exceed one another by equal amounts, this excess being equal to the smallest line [viz. the one representing a single time-interval]: or we may say [that this is the ratio] of the differences of the squares of the natura] numbers beginning with unity. While, therefore, during equal intervals of time the velocities increase as the natural numbers, the increments in the distances traversed during these equal time-intervals are to one another as the odd numbers beginning with unity. . . . SIMP. I am convinced that matters are as described, once having accepted the definition of uniformly accelerated motion. But as to whether this acceleration is that which one meets in nature in the case of falling bodies, I am still doubtful; and it seems to me, not only for my own sake but also for aU those who think as I do, that this would be the proper moment to introduce one of those experiments-and there are many of them, I understand-which illustrate in several ways the conclusions reached. SALVo The request which you, as a man of science, make, is a very reasonable one; for this is the custom-and properly so--in those sciences where mathematical demonstrations are applied to natural phenomena, as is seen in the case of perspective, astronomy, mechanics, music, and others where the principles, once established by wen-chosen experiments, become the foundations of the entire superstructure. I hope therefore it will not appear to be a waste of time if we discuss at considerable length this first and most fundamental question upon which hinge numerous consequences of which we have in this book only a small number, placed there by the Author, who has done so much to open a pathway hitherto closed to minds of speCUlative turn. So far as experiments go they have not been neglected by the Author; and often, in his company, I have attempted in the following manner to assure myself that the acceleration actually experienced by falling bodies is that above described. A piece of wooden moulding or scantling, about 12 cubits long, half a cubit wide, and three finger-breadths thick, was taken; on its edge was cut a channel a little more than one finger in breadth; having made this
747
groove very straight, smooth, and polished, and having lined it with parchment, also as smooth and polished as possible, we ro1Jed along it a hard. smooth, and very round bronze ball. Having placed this board in a s10ping position, by lifting one end some one or two cubits above the other, we rolled the ball, as J was just saying, along the channel, noting, in a manner presently to be described, the time required to make the descent. We repeated this experiment more than once in order to measure the time with an accuracy such that the deviation between two observations never exceeded one-tenth of a pulse-beat. Having performed this operation and having assured ourselves of its reHabiHty, we now rolled the ball only one-quarter the length of the channel; and having measured the time of its descent, we found it precisely one-half of the former. Next we tried other distances, comparing the time for the whole length with that for the half, or with that for two-thirds, or three-fourths, or indeed for any fraction; in such experiments, repeated a full hundred times, we always found that the spaces traver~ed were to each other as the squares of the times, and this was true for all inclinations of the plane, i.e., of the channel, along which we rolled the bal1. We a1so observed that the times of descent, for various inclinations of the plane, bore to one another precisely that ratio which, as we shan see later, the Author had predicted and demonstrated for them. For the measurement of time, we employed a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected Was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results. SIMP. I would like to have been present at these experiments; but feeling confidence in the care with which you performed them, and in the fidelity with which you relate them, I am satisfied and accept them as true and valid. SALVo Then we can proceed without discussion. COROLLARY n Secondly, it follows that, starting from any initial point, if we take any two distances, traversed in any time-intervals whatsoever, these timeintervals bear to one another the same ratio as one of the distances to the mean proportional of the two distances. For if we take two distances ST and SY measured from the initial
748
s
r x y FIGURE S
point S, the mean proportional of which is SX, the time of faU through ST is to the time of fall through SY as ST is to SX; or one may say the time of fall through SY is to the time of faU through ST as SY is to SX. Now since it has been shown that the spaces travened are in the same ratio as the squares of the times~ and since, moreover, the ratio of the space SY to the space ST is the square of the ratio SY to SX, it follows that the ratio of the times of fall through SY and ST is the ratio of the respective distances SY and SX. SCHOLIUM
The above corollary has been proven for the case of vertical fall; but it holds also for planes inclined at any angle; for it is to be assumed that along these planes the velocity increases in the same ratio, that is, in proportion to the time, or, if you prefer, as the series of natural numbers. THEOREM III, PROPOSITION III
If one and the same body, starting from rest, falls along an inclined plane and also along a vertical, each having the same height, the times of descent will be to each other as the lengths of the inclined plane and the vertical.
Let AC be the inclined plane and AB the perpendicular, each having the same vertical height above the horizontal, namely, BA; then I say, the time of descent of one and the same body along the plane AC bears a ratio to the time of fall along the perpendicular AB, which is the same as the ratio of the length AC to the length AB. Let DO, EI and LF be any lines paranel to the horizontal CB; then it follows from what has preceded that a body starting from A will acquire the same speed at the point 0 as at D, since in each case the vertical fan is the same; in like manner the speeds at I and E will be the same; so also those at Land F. And in general the speeds at the two extremities of any parallel drawn from any point on AB to the corresponding point on AC wilt be equal.
749
A
FIGURE t.
Thus the two distances AC and AB are traversed at the same speed. But it has already been proved that if two distances are traversed by a body moving with equal speeds. then the ratio of the times of descent will be the ratio of the distances themselves; therefore, the time of descent along AC is to that along AB as the length of the plane AC is to the vertical distance AB. Q. E. D. SAGR. It seems to me that the above could have been proved clearly and briefly on the basis of a proposition already demonstrated, name1y, that the distance traversed in the case of accelerated motion along AC or AB is the same as that covered by a uniform speed whose value is onehalf the maximum speed. CB; the two distances AC and AB having been traversed at the same uniform speed it is evident, from Proposition I, that the times of descent will be to each other as the distances. COROLLARY
Hence we may infer that the times of descent along planes having different inclinations. but the same vertical height stand to one another in the same ratio as the-lengths of the planes. For consider any plane AM extending from A to the horizontal CB; then it may be demonstrated in the same manner that the time of descent along AM is to the time along AB as the distance AM is to AB; but since tbe time along AB is to that along AC as the length AB is to the length AC, it follows, ex requali, that as AM is to AC so is the time along AM to the time along AC. THEOREM IV, PROPOSITION IV
The times of descent along planes of the same length but of different inclinations are to each other in the inverse ratio of the square roots of their heights. From a single point B draw the planes BA and BC, having the same length but different inclinations; let AE and CD be horizontal lines drawn to meet the perpendicular BD; and let BE represent the height of the
GameD GIIII,.,
750
plane AB, and BD the height of BC; also let BI be a mean proportional to DD and BE; then the ratio of BD to BI is equal to the square root of the ratio of BD to BE. Now, I say, the ratio of the times of descent along BA and BC is the ratio of BD to BI; so that the time of descent along BA is related to the height of the other plane BC, namely BD as the time
FIGURE 7
along BC is related to the height BI. Now it must be proved that the time of descent along BA is to that along BC as the length BD is to the length BI. Draw IS parallel to DC; and since it has been shown that the time of fall along BA is to that along the vertical BE as BA is to BE; and also that the time along BE is to that along BD as BE is to BI; and likewise (hat the time along BO is to that along BC as BD is to BC, or as BI to BS; it foHows, ex tequali, that the time along BA is to that along BC as BA to BS, or BC to BS. However, BC is to BS as BD is to Bf; hence follows our proposition. THEOREM V, PROPOSITION V
The times of descent along planes of different length, slope and height bear to one another a ratio which is equal to the product of the ratio of the lengths by the square root of the inverse ratio of their heights. Draw the planes AB and AC, having different inclinations, lengths, and heights. My theorem then is that the ratio of the time of descent along AC to that along AB is equal to the product of the ratio of AC to AB by the square root of the inverse ratio of their heights. For Jet AD be a perpendicular to which are drawn the horizontal Hnes 80 and CD; also let AL be a mean proportional to the heights AO and AD; from the point L draw a horizontal line meeting AC in F; accordihg]y A F will be a mean proportional between AC and AE. Now since the time of de~cent along AC is to that along AE as the length AF is to AE;
751
A
jIooooo---...
L
C-_ _---'D FIGURE 8
and since the time along AE is to that along AB as AE is to AB, it is clear that the time along AC is to that along AB as AF is to AB. Thus it remains to be shown that the ratio of AF to AB is equal to the product of the ratio of AC to AB by the ratio of AG to AL, which is the inverse ratio of the square roots of the heights DA and GA. Now it is evident that, if we consider the line AC in connection with AF and AB, the ratio of AF to AC is the same as that of AL to AD, or AG to AL which is the square root of the ratio of the heights AG and AD; but the ratio of AC to AB is the ratio of the lengths themselves. Hence follows the theorem. THEOREM VI, PRoposmON VI
If from the highest or lowest point in a vertical circle there be drawn any inclined planes meeting the circumference the times of descent along these chords are each equal to the other.
On the horizontal line GH construct a vertical circle. From its lowest point-the point of tangency with the horizontal--draw the diameter FA and from the highest point, A, draw inclined planes to Band C, any points whatever on the circumference; then the times of descent along these are equal. Draw BD and CE perpendicular to the diameter; make AI a mean proportional between the heights of the planes, AE and AD; and since the rectangles FA.AE and FA.AD are respectively equal to the squares of AC and AB, while the rectangle F A.AE is to the rectangle F A.AD as AE is to AD, it follows that the square of AC is to the square of AB as the length AE is to the length AD. But since the length AE is to AD as the square of AI is to the square of AD, it follows that the squares on the lines AC and AB are to each other as the squares on the lines AI and AD, and hence also the length AC is to the length AB as AI
752
A
C~
_ _ _---1
G
H FIGURE 9
is to AD. But it has previously been demonstrated that the ratio of the time of descent along AC to that along AD is equal to the product of the two ratios AC to AD and AD to AI; but this last ratio is the same as that of AD to AC. Therefore the ratio of the time of descent along AC to that along AD is the product of the two ratios, AC to AB and AD to AC. The ratio of these times is therefore unity. Hence follows our proposition. By use of the principles of mechanics [ex mechanicisJ one may obtain the same result. . . . SCHOLIUM
We may remark that any velocity once imparted to a moving body will be rigidly maintained as long as the external causes of acceleration or retardation are removed, a condition which is found only on horizontal planes; for in the case of planes which slope downwards there is already present a cause of acceleration, while on planes sloping upward there is retardation; from this it follows that motion along a horizontal plane is perpetual; for, if the velocity be uniform, it cannot be diminished or slackened, much less destroyed. Further, although any velocity which a body may have acquired through natural fall is permanently maintained so far as its own nature [suapte natura] is concerned, yet it must be remembered that if, after descent along a plane inclined downwards, the body is deflected to a plane inclined upward, there is already existing in this latter plane a cause of retardation; for in any such plane this same body is subject to a natural acceleration downwards. Accordingly we have here the superposition of two different states, namely, the velocity acquired during the preceding fan which if acting alone would carry the body at a uniform rate to infinity. and the velocity which results from a natural acceleration downwards common to al1 bodies. It seems altogether
753
reasonable, therefore, if we wish to trace the future history of a body which has descended along some inclined plane and has been deflected along some plane inclined upwards, for us to assume that the maximum speed acquired during descent is permanently maintained during the ascent. In the ascent, however, there supervenes a natural inclination downwards, namely, a motion which, starting from rest, is accelerated at the usual rate. If perhaps this discussion is a little obscure, the following figure wi11 help to make it clearer. Let us suppose that the descent has been made along the downward sloping plane AB, from which the body is deflected so as to continue its motion along the upward sloping plane BC~ and first let these planes be of equal length and placed so as to make equal angles with the horizontal line GH. Now it is wen known that a body, starting from rest at A. and descending along AB, acquires a speed which is proportional to the time,
c
,.
A ... ...
... ...
... ...
.... ...
G ---------~--------FIGURE 10
which is a maximum at B, and which is maintained by the body so long as al1 causes of fresh acceleration or retardation are removed; the acceleration to which I refer is that to which the body would be subject if its motion were continued along the plane AB extended, while the retardation is that which the body would encounter if its motion were deflected along the plane BC inclined upwards; but, upon the horizontal plane GR, the body would maintain a uniform velocity equal to that which it had acquired at B after fall from A; moreover this velocity is such that, during an interval of time equal to the time of descent through AB, the body will traverse a horizontal distance equal to twice AB. Now let us imagine this same body to move with the same uniform speed along the plane BC so that here also during a time-interval equal to that of descent along AB, it will traverse along BC extended a distance twice AB; but let us suppose that, at the very instant the body begins its ascent it is subjected, by its very nature, to the same influences which surrounded it during its descent from A along AB, namely, it descends from rest under the same acceleration as that which was effective in AB, and it traverses, during an equal interval of time, the same distance along this second plane as it did along AB; it is clear that, by thus superposing upon the body a uniform motion of ascent and an accelerated motion of descent, it will be carried along
7!4
the pJane BC as far as the point C where these two velocities become equal. If now we assume any two points D and E, equally distant from the vertex B, we may then infer that the descent along BD takes place in the same time as the ascent along BE. Draw DF parallel to BC; we know that, after descent along AD, the body will ascend along DF; or, if, on reaching D, the body is carried along the horizontal DE, it will reach E with the same momentum [impetus] with which it left D; hence from E the body wiJI ascend as far as C, proving that the velocity at E is the same as that at D. From this we may logically infer that a body which descends along any inclined plane and continues its motion along a plane inclined upwards will, on account of the momentum acquired, ascend to an equal height above the horizontal; so that if the descent is along AB the body will be carried up the plane BC as far as the horizontal line ACD: and this is true whether the inclinations of the planes are the same or different, as in the case of the planes AB and BD. But by a previous postulate the speeds acquired by fall along variously inclined planes having the same vertical
D
FIGURE II
height are the same. If therefore the planes EB and BD have the same slope, the descent along EB will be able to drive the body along BD as far as D; and since this propulsion comes from the speed acquired on reaching the point B, it follows that this speed at B is the same whether the body has made its descent along AB or EB Evidently then the body will be carried up DD whether the descent has been made along AB or along ED. The time of ascent along BD is however greater than that along BC, just as the descent along EB occupies more time than that along AD; moreover it has been demonstrated that the ratio between the lengths of these times is the same as that between the lengths of the planes. . . . FOURTH DAY SALVo Once more, Simplicio is here on time; so let us without delay take up the question of motion. The text of our Author is as follows:
755
THE MOTION OF PROJECTILES
In the preceding pages we have discussed the properties of uniform motion and of motion naturally accelerated along planes of all inclinations. I now propose to set forth those properties which belong to a body whose motion is compounded of two other motions, namely, one uniform and one natura)]y accelerated; these properties, well worth knowing, I propose to demonstrate in a rigid manner. This is the kind of motion seen in a moving projectile; its origin I conceive to be as follows: Imagine any particle projected along a horizontal plane without friction; then we know, from what has been more fully explained in the preceding pages, that this particle will move along this same plane with a motion which is uniform and perpetual, provided the plane has no Hmits. But if the plane is limited and elevated, then the moving particle, which we imagine to be a heavy one, will on passing over the edge of the plane acquire, in addition to its previous uniform and perpetual motion, a downward propensity due to its own weight; so that the resulting motion which I call projection [pTojectio], is compounded of one which is uniform and horizontal and of another which is vertical and naturally accelerated. We now proceed to demonstrate some of its properties, the first of which is as follows: THEOREM I, PROPOSITION I
A projectile which is carried by a uniform horizontal motion compounded with a naturally accelerated vertical motion describes a path which is a semi-parabola. SAGR. Here, Salviati, it will be necessary to stop a little while for my sake and, J believe, also for the benefit of Simplicio; for it so happens that I have not gone very far in my study of Apollonius and am merely aware of the fact that he treats of the parabola and other conic sections, without an understanding of which I hardly think one will be able to follow the proof of other propositions depending upon them. Since even in this first beautiful theorem the author finds it necessary to prove that the path of a projectile is a parabola, and since, as I imagine, we shall have to deal with only this kind of curves, it will be absolutely necessary to have a thorough acquaintance, if not with all the properties which Apollonius has demonstrated for these figures, at least with those which are needed for the present treatment. SALVo You are quite too modest, pretending ignorance of facts which not long ago you acknowledged as well known-I mean at the time when we were discussing the strength of materials and needed to use a certain theorem of Apollonius which gave you no trouble.
I may have chanced to know it or may possibly have assumed it, so long as needed, for that discussion; but now when we have to follow all these demonstrations about such curves we ought not, as they say, to swallow it whole, and thus waste time and energy. SIMP. Now even though Sagredo is, as I believe, wen equipped for atl his needs, I do not understand even the elementary terms; for although our philosophers have treated the motion of projectiles, I do not recall their having described the path of a projectile except to state in a general way that it is always a curved line, un1ess the projection be vertically upwards. But if the little Euclid which I have learned since our previous discussion does not enable me to understand the demonstrations which are to follow, then I shall be obliged to accept the theorems on faith without fully comprehending them. SALV • On the contrary, I desire that you should understand them from the Author himse1f. who. when he allowed me to see this work of his, was good enough to prove for me two of the principal properties of the parabola because I did not happen to have at hand the books of Apollonius. These properties, which are the on1y ones we shall need in the present discussion, he proved in such a way that no prerequisite knowledge Was required. These theorems are, indeed, given by Apol1onius, but after many preceding ones, to follow which would take a long while. J wish to shorten our task by deriving the first property purely and simply from the mode of generation of the parabola and proving the second immediately from the first. Beginning now with the first, imagine a right cone, erected upon the circular base ibkc with apex at I. The section of this cone made by a plane drawn parallel to the side Ik is the curve which is called a parabola. The base of this parabola be cuts at right angles the diameter ik of the circle ibkc, and the axis ad is parallel to the side Ik; now having taken any SAGa.
t "............-~~_.....
FIGUltE 12
751
MtIIINmtlllcl of MottoN
point I in the curve bla draw the straight line Ie parallel to bd; then, I say, the square of bd is to the square of Ie in the same ratio as the axis ad is to the portion ae. Through the point e pass a plane parallel to the circle ibke, producing in the cone a circular section whose diameter is the line geh. Since bd is at right angles to ik in the circle ibk, the square of bd is equal to the rectangle formed by id and dk; so also in the upper circle which passes through the points glh the square of Ie is equal to the rectangle formed by ge and eh; hence the square of bd is to the square of Ie as the rectangle id.dk is to the rectangle ge.eh. And since the line ed is parallel to hk, the line eh, being parallel to dk, is equal to it; therefore the rectangle id.dk is to the rectangle ge.eh. And since the line ed is parallel to hk, the line eh, being parallel to dk, is equal to it; therefore the rectangle id.dk is to the rectangle ge.eh as id is to ge, that is, as da is to ae; whence also the rectangle id.dk is to the rectangle ge.eh, that is. the square Q. E. D. of bd is to the square of Ie, as the axis da is to the portion ae. The other proposition necessary for this discussion we demonstrate as follows. Let us draw a parabola whose axis ea is prolonged upwards to a point d; from any point b draw the line be paral1el to the base of the parabola; if now the point d is chosen so that da = ea, then, I say. the straight line drawn through the points band d will be tangent to the parabola at b. For imagine, if possible, that this line cuts the parabola above or that its prolongation cuts it below, and through any point g in it draw the straight
J
FIGURE 13
758
GameD Gamel
line Ige. And since the square of Ie is greater than the square of get the square of Ie will bear a greater ratio to the square of be than the square of ge to that of be; and since, by the preceding proposition, the square of Ie is to that of be as the line ea is to ea, it follows that the line ea will bear to the line ea a greater ratio than the square of ge to that of be, or, than the square of ed to that of cd (the sides of the triangles deg and deb being proportional). But the line ea is to ea, or da, in the same ratio as four times the rectangle ea.ad is to four times the square of ad, or, what is the same, the square of cd, since this is four times the square of ad; hence four times the rectangle ea.ad bears to the square of cd a greater ratio than the square of ed to the square of cd; but that would make four times the rectangle ea.ad greater than the square of ed; which is false, the fact being just the opposite, because the two portions ea and ad of the line ed are not equal. Therefore the line db touches the parabola without cutting it. Q. E. D. SIMP. Your demonstration proceeds too rapidly and, it seems to me, you keep on assuming that all of Euclid's theorems are as familiar and available to me as his first axioms, which is far from true. And now this fact which you spring upon us, that four times the rectangle ea.ad is less than the square of de because the two portions ea and ad of the line de are not equal brings me little composure of mind, but rather leaves me in suspense. SALvo Indeed. all real mathematicians assume on the part of the reader perfect famiHarity with at least the elements of Euclid; and here it is necessary in your case only to recall a proposition of the Second Book in which he proves that when a line is cut into equal and also into two unequal parts. the rectangle formed on the unequal parts is less than that formed on the equal (i.e., less than the square on half the line), by an amount which is the square of the difference between the equal and unequal segments. From this it is clear that the square of the whole Hne which is equal to four times the square of the half is greater than four times the rectangle of the unequal parts. In order to understand the following portions of this treatise it will be necessary to keep in mind the two elemental theorems from conic sections which we have just demonstrated; and these two theorems are indeed the only ones which the Author uses. We can now resume the text and see how he demonstrates his first proposition in which he shows that a body falling with a motion compounded of a uniform horizontal and a naturally accelerated [naturale de.rcendente] one describes a semi-parabola. Let us imagine an elevated horizontal line or plane ab along which a body moves with uniform speed from a to b. Suppose this plane to end abruptly at h; then at this point the body will, on account of its weight, acquire also a natural motion downwards along the perpendicular bn.
759
Mathematks of Motion
Draw the Hne be along the plane ba to represent the flow, or measure. of time; divide this line into a number of segments, be, cd, de, representing equal intervals of time; from the points b, e, d, e, let faU lines which are parallel to the perpendicular bn. On the first of these layoff any distance ci, on the second a distance four times as long, dl; on the third, one nine times as long, eh; and so on, in proportion to the squares of eb, db. eb,
o
1
;;;r---+--t-----I l
FIGURE ]4
or, we may say, in the squared ratio of these same lines. Accordingly we see that while the body moves from b to e with uniform speed, it also faUs perpendicularly through the distance ci, and at the end of the time-interval be finds itself at the point i. In like manner at the end of the time-interval bd, which is the double of be, the vertical fall will be four times the first distance ci; for it has been shown in a previous discussion that the distance traversed by a freely falling body varies as the square of the time; in 'like manner the space eh traversed during the time be will be nine times ci; thus it is evident that the distances eh, dl, ci will be to one another as the squares of the lines be, bd, be. Now from the points i, I, h draw the straight lines io, fg, hi parallel to be; these lines hi, Ig, io are equal to eb, db and eb, respectively; so also are the lines bo, bg, bl respectively equal to ci, df, and eh. The square of hi is to that of Ig as the line lb is to bg; and the square of Ig is to that of io as gb is to bo; therefore the points i, I, h, lie on one and the same parabola. In like manner it may be shown that, if we take equal time-intervals of any size whatever, and if we imagine the particle to be carried by a similar compound motion, the positions of this particle, at the ends of these time-intervals, will lie on one and the same parabola. Q. E. D. SALV, This conclusion follows from the converse of the first of the two propositions given above. For, having drawn a parabola through the points band h, any other two points, f and i, not falling on the parabola must lie either within or without; consequently the line fg is either longer or shorter than the line which terminates on the parabola. Therefore the
GlIlt,./) Gall,.,
760
square of hI will not bear to tbe square of fg the same ratio as the line Ib to bg, but a greater or smaller; the fact is, however, that the square of hi does bear this same ratio to the square of fg. Hence the point f does lie on the parabola, and so do all the others. SAGR. One cannot deny that the argument is new, subtle and conclusive, resting as it does upon this hypothesis, namely, that the horizontal motion remains uniform, that the vertical motion continues to be accelerated downwards in proportion to the square of the time, and that such motions and velocities as these combine without altering, disturbing, or hindering each other,8 so that as the motion proceeds the path of the projectile does not change into a different curve: but this, in my opinion, is impossible. For the axis of the parabola along which we imagine the natural motion of a falling body to take place stands perpendicular to a horizontal surface and ends at the center of the earth; and since the parabola deviates more and more from its axis no projectile can ever reach the center of the earth or, if it does, as seems necessary, then the path of the projectile must transform itself into some other curve very different from the parabola. SIMP. To these difficulties, I may add others. One of these is that we suppose the horizontal plane, which slopes neither up nor down, to be represented by a straight line as if each point on this line were equally distant from the center, which is not the case; for as one starts from the middle [of the line] and goes toward either end, he departs farther and farther from the center [of the earth] and is therefore constantly going uphill. Whence it follows that the motion cannot remain uniform through any distance whatever, but must continually diminish. Besides, I do not see how it is possible to avoid the resistance of the medium which must destroy the uniformity of the horizontal motion and change the law of acceleration of falling bodies. These various difficulties render it highly improbable that a result derived from such unreliable hypotheses should hold true in practice. SALVo All these difficulties and objections which you urge are so well founded that it is impossible to remove them; and, as for me, I am ready to admit them a11, which indeed I think our Author would also do. I grant that these concl usions proved in the abstract will be different when applied in the concrete and will be fal1acious to this extent, that neither will the horizontal motion be uniform nor the natural acceleration be in the ratio assumed, nor the path of the projectile a parabola, etc. But, on the other hand, I ask you not to begrudge our Author that which other eminent men have assumed even if not strictly true. The authority of Archimedes alone will satisfy everybody. In his Mechanics and in his first quadrature of the parabola he takes for granted that the beam .ok-ba1ance 3
A very near approach to Newton's Second Law of Motion. [Trans.]
MQtlremQtlcl 01 Motion
7151
or steelyard is a straight 1ine, every point of which is equidistant from the common center of aU heavy bodies. and that the cords by which heavy bodies are suspended are paranel to each other. Some consider this assumption permissible because, in practice. our instruments and the distances involved are so small in comparison with the enormous distance from the center of the earth that we may consider a minute of arc on a great circle as a straight line, and may regard the perpendiculars let fall from its two extremities as parallel. For if in actual practice one had to consider such small quantities, it would be necessary first of all to criticise the architects who presume, by use of a plumbJine, to erect high towers with paraUel sides. I may add that, in all their discussions. Archimedes and the others considered themselves as located at an infinite distance from the center of the earth, in which case their assumptions were not false, and therefore their conclusions were absolutely correct. When we wish to apply our proven conclusions to distances which, though finite. are very large. it is necessary for us to infer, on the basis of demonstrated truth, what correction is to be made for the fact that our distance from the center of the earth is not real1y infinite, but merely very great in comparison with the small dimensions of our apparatus. The largest of these wiJ1 be the range of our projectiles--and even here we need consider only the artillery-which, however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth; and since these paths terminate upon the surface of the earth only very slight changes can take place in their parabolic figure which. it is conceded. would be greatly altered if they terminated at the center of the earth. As to the perturbation arising from the resistance of the medium this is more considerable and does not, on account of its manifold forms. submit to fixed laws and exact description. Thus if we consider only the resistance which the air offers to the monons studied by us, we shall see that it disturbs them all and disturbs them in an infinite variety of ways corresponding to the infinite variety in the form, weight. and velocity of the projectiJes. For as to velocity. the greater this is, the greater will be the resistance offered by the air; a resistance which will be greater as the moving bodies become less dense [men gravJ1. So that although the falJing body ought, to be displaced [andare accelerandosll in proportion to the square of the duration of its motion, yet no matter how heavy the body, jf it faUs from a very considerable height, the resistance of the air will be such as to prevent any increase in speed and wiJI render the motion uniform; and in proportion as the moving body is less dense [men grave] this uniformity will be so much the more quickly attained and after a shorter fall. Even horizontal motion which, if no impediment were offered, would be uniform and constant is altered by the resistance of the air and
762
Oal,"o Oallk,
finany ceases; and here again the less dense [piu leggiero] the body the quicker the process. Of these properties [accidenti] of weight, of velocity, and also of form (figura], infinite in number, it is not possible to give any exact description; hence, in order to handle this maUer in a scientific way, it is necessary to cut loose from these difficulties; and having discovered and demonstrated the theorems, in the case of no resistance, to use them and apply them with such limitations as experience will teach. And the advantage of this method will not be small; for the material and shape of the projectile may be chosen, as dense and round as possible, so that it will encounter the least resistance in the medium. Nor will the spaces and velocities in general be so great but that we shall be easily able to correct them with precision. In the case of those projectiles which we use, made of dense [grave] material and round in shape, or of lighter material and cylindrical in shape, such as arrows, thrown from a sling or crossbow, the deviation from an exact parabolic path is quite insensible. Indeed, if you will allow me a little greater liberty, I can show you, by two experiments. that the dimensions of our apparatus are so sma11 that these external and incidental resistances, among which that of the medium is the most considerable, are scarcely observable. I now proceed to the consideration of motions through the air, since it is with these that we are now especially concerned; the resistance of the air exhibits itself in two ways: first by offering greater impedance to less dense than to very dense bodies, and secondly by offering greater resistance to a body in rapid motion than to the same body in slow motion. Regarding the first of these, consider the case of two baJJs having the same dimensions, but one weighing ten or twelve times as much as the other; one, say, of lead, the other of oak. both allowed to fall from an elevation of I SO or 200 cubits. Experiment shows that they will reach the earth with slight difference in speed, showing us that in both cases the retardation caused by the air is smatl; for if both balls start at the same moment and at the same elevation, and jf the leaden one be slightly retarded and the wooden one greatly retarded, then the former ought to reach the earth a considerable distance in advance of the latter, since it is ten times as heavy. But this does not happen; indeed, the gain in distance of one over the other does not amount to the hundredth part of the entire fall. And in the case of a baH of stone weighing only a third or half as much as one of lead. the difference in their times of reaching the earth win be scarcely noticeable. Now since the speed [impeto] acquired by a leaden ball in fal1ing from a height of 200 cubits is so great that if the motion remained uniform the ball WOUld, in an interval of time equal to that of the fall, traverse 400 cubits, and since this speed is so considerable in comparison with those which, by use of
163
bows or other machines except fire arms, we are able to give to our projectiles, it fol1ows that we may, without sensible error, regard as absolutely true those propositions which we are about to prove without considering the resistance of the medium. Passing now to the second case, where we have to show that the resistance of the air for a rapidly moving body is not very much greater than for one moving slowly, ample proof is given by the following experiment. Attach to two threads of equal length-say four of five yards--two equal leaden balls and suspend them from the ceiling; now pull them aside from the perpendicular, the one through 80 or more degrees, the other through not more than four or five degrees; so that, when set free, the one faUs, passes through the perpendicular, and describes large but slowly decreasing arcs of 160, ISO, 140 clegrees, etc.; the other swinging through small and also slowly diminishing arcs of 10, 8, 6 degrees, etc. In the first place it must be remarked that one pendulum passes through its arcs of J80°, 160°, etc., in the same time that the other swings through its 10°, 8°, etc., from which it follows that the speed of the first ball is 16 and 18 times greater than that of the second. Accordingly, if the air offen more resistance to the high speed than to the low, the frequency of vibration in the large arcs of 180° or J60°, etc., ought to be less than in the small arcs of 10°, 8°, 4°, etc., and even less than in arcs of 2°, or 1°; but this prediction is not verified by experiment; because if two persons start to count the vibrations, the one the large, the other the small, they wi11 discover that after counting tens and even hundreds they will not differ by a single vibration, not even by a fraction of one. This observation justifies the two following propositions, namely, that vibrations of very large and very small amplitude all occupy the same time and that the resistance of the air does not affect motions of high speed more than those of low speed, contrary to the opinion hitherto generally entertained. SAGR. On the contrary, since we cannot deny that the air hinders both of these motions, both becoming slower and finally vanishing, we have to admit that the retardation occurs in the same proportion in each case. But how? How, indeed, could the resistance offered to the one body be greater than that offered to the other except by the impartation 'of more momentum and speed [impeto e velocita] to the fast body than to the slow? And if this is so the speed with which a body moves is at once the cause and measure [cagione e misura] of the resistance which it meets. Therefore, all motions, fast or slow, are hindered and diminished in the same proportion; a result, it seems to me, of no small importance. SALVo We are able, therefore, in this second case to say that the errors, neglecting those which are accidental, in the results which we are about to demonstrate are small in the case of our machines where the velocities
764
Gtdlleo GalIlei
employed are mostly very great and the distances negligible in comparison with the semi-diameter of the earth or one of its great circles. SIMP. I would like to hear your reason for putting the projectiles of fire arms, i.e., those using powder, in a different class from the projectiles employed in bows, slings, and crossbows, on the ground of their not being equally subject to change and resistance from the air. SALVo I am led to this view by the excessive and, so to speak, supernatural violence with which such projectiles are launched; for, indeed, it appears to me that without exaggeration one might say that the speed of a ball fired either from a musket or from a piece of ordnance is supernatural. For if such a ball be allowed to fall from some great elevation its speed will, owing to the resistance of the air, not go on increasing indefinitely; that which happens to bodies of small density in falling through short distances-I mean the reduction of their motion to uniformity-will also happen to a ball of iron or lead after it has fallen a few thousand cubits; this terminal or final speed [terminata veiocita] is the maximum which such a heavy body can naturally acquire in falling through the air. This speed J estimate to be much smal1er than that impressed upon the ball by the burning powder. An appropriate experiment will serve to demonstrate this fact. From a height of one hundred or more cubits fire a gun [archibuso] loaded with a lead bullet, vertical1y downwards upon a stone pavement; with the same gun shoot against a similar stone from a distance of one or two cubits, and observe which of the two balls is the more flattened. Now if the ball which has come from the greater elevation is found to be the less flattened of the two, this will show that the air has hindered and diminished the speed initially imparted to the bullet by the powder, and that the air will not permit a bullet to acquire so great a speed, no matter from what height it falls; for if the speed impressed upon the ball by the fire does not exceed that acquired by it in falling freely [natura/mente] then its downward blow ought to be greater rather than less. This experiment I have not performed, but 1 am of the opinion that a musket-ball or cannon-shot, falling from a height as great as you please, will not deliver so strong a blow as it would if fired into a wall only a few cubits distant, i.e., at such a short range that the splitting or rending of the air will not be sufficient to rob the shot of that excess of supernatural violence given it by the powder. The enormous momentum [impeto] of these violent shots may cause some deformation of the trajectory, making the beginning of the parabola flatter and less curved than the end; but, so far as our Author is concerned, this is a matter of smal1 consequence in practical operations, the main one of which is the preparation of a table of ranges for shots of high ele.ation, giving the distance attained by the bal1 as a function of the angle of eleva-
165
M «h. mallcs of M olio"
tioo; and since shots of this kind are fired from mortars [mortarll using smal1 charges and imparting no supernatural momentum [impeto sopranaturale] they follow their prescribed paths very exactly. But now let us proceed with the discussion in which the Author invites us to the study and investigation of the motion of a body [impeto del mobile] when that motion is compounded of two others; and first the case in which the two are uniform, the one horizontal, the other vertical. THEOREM n, PROPOSITION
II
When the motion of a body is the resu1tant of two uniform motions, one horizontal, the other perpendicular. the square of the resultant momentum is equal to the sum of the squares of the two component momenta. Let us imagine any body urged by two uniform motions and let ab represent the vertical displacement, while be represents the displacement which, in the same interval of time. takes place in a horizontal direction.
FIGURE l5
If then the distances ab and be are traversed, during the same timeinterval, with uniform motions the corresponding momenta will be to each other as the distances ab and be are to each other; but the body which is urged by these two motions describes the diagonal ae; its momentum is proportional to ac. Also the square of ae is equal to the sum of the squares of ab and be. Hence the square of the resultant momentum is equal to the sum of the squares of the two momenta ab and be. Q. E. D. SIMP. At this. point there is just one slight difficulty which needs to be cleared up; for it seems to me that the conclusion just reached contradicts a previous proposition in which it is claimed that the speed [impeto] of a body coming from a to b is equal to that in coming from a to e; while now you conclude that the speed [impeto] at e is greater than that at b. SALvo Both propositions, Simplicio, are true, yet there is a great difference between them. Here we are speaking of a body urged by a single motion which is the resultant of two uniform motions, while there we were speaking of two bodies each urged with naturally accelerated motions, one along the vertical ab the other along the inclined plane ae. Besides the time-intervals were there not supposed to be equal, that along the incline ae being greater than that along the vertical ab; but the motions of which we now speak. those along ab, be, ae, are uniform and simultaneous.
166
GtIllleo GtIlIld
Pardon me; I am satisfied; pray go on. SALVo Our Author next undertakes to explain what happens when a body is urged by a motion compounded of one which is horizontal and uniform and of another which is vertical but naturally accelerated; from these two components results the path of a projectile. which is a parabola. The problem is to determine the speed [impeto] of the projectiJe at each point. With this purpose in view our Author sets forth as follows the manner. or rather the method, of measuring such speed [impeto] along the path which is taken by a heavy body starting from rest and falling with a naturally accelerated motion. SIMP.
THEOREM In, PROPOSITION III
Let the motion take place along the line ab, starting from rest at a, and in this line choose any point c. Let ac represent the time, or the measure of the time, required for the body to fan through the space ac; let ac also represent the velocity [impetus seu momentum] at c acquired by a fall through the distance ac. In the Hne ab select any other point b. The problem now is to determine the velocity at b acquired by a body in famng through the distance ab and to express this in terms of the velocity at c, the measure of which is the length ac. Take as a mean proportional between ac and abo We shall prove that the velocity at b is to that at c as the length as is to the length ac. Draw the horizontal line cd, having twice the length of ac. and be. having twice the length of ba. It then fol-
c
•
I
~----+-~----~
,
FIGURE U5
lows, from the preceding theorems, that a body falling through the distance ac, and turned so as to move along the horizontal cd with a uniform speed equal to that acquired on reaching c will traverse the distance cd in the same interval of time as that required to fall with accelerated motion from a to C. Likewise be wi11 be traversed in the same time as ba. But the time of descent through ab is as; hence the horizontal distance be is also traversed in the time as. Take a point I such that the time as is to the time ac as be is to bl; since the motion along be is uniform, the distance bl, if tra.versed with the speed [momentum celerilalis] acquired at h, wi11 occupy the time ac; but in this same time-interval, ac, the dista.nce cd
167
Mathematics (II M(ltiM
is traversed with the speed acquired in c. Now two speeds are to each other as the distances traversed in equal intervals of time. Hence the speed at c is to the speed at b as cd is to bl. But since dc is to be as their halves, namely, as ca is to ba, and since be is to bl as ba is to sa; it follows that dc is to bl as ca is to sa. In other words, the speed at c is to that at b as ca is to sa, that is, as the time of fa11 through abo The method of measuring the speed of a body along the direction of its fall is thus clear; the speed is assumed to increase directly as the time. PROBLEM 1, PROPOSITION IV
SALvo Concerning motions and their velocities or momenta [movimenli e lor velocita 0 impeti] whether uniform or naturally accelerated, one cannot speak definitely until he has established a measure for such velocities and also for time. As for time we have the already widely adopted hours, first minutes and second minutes. So for velocities, just as for intervals of time, there is need of a common standard which shall be understood and accepted by everyone, and which shall be the same for all. As has already been stated, the Author considers the velocity of a freely famng body adapted to this purpose, since this velocity increases according to the same law in an parts of the world; thus for instance the speed acquired by a leaden ball of a pound weight starting from rest and falling vertically through the height of, say, a spear's length is the same in al1 places; it is therefore excellently adapted for representing the momentum [impeto) acquired in the case of natural fall. It stiH remains for us to discover a method of measuring momentum in the case of uniform motion in such a way that all who discuss the subject will form the same conception of its size and velocity [grandeua e velocita). This will prevent one person from imagining it larger, another smal1er, than it really is; so that in the composition of a given uniform motion with one which is accelerated different men may not obtain different values for the resultant. In order to determine and represent such a momentum and particular speed [impelo e veloeita particolare] our Author has found no better method than to use the momentum acquired by a body in naturally accelerated motion. The speed of a body which has in this manner acquired any momentum whatever will, when converted into uniform motion, retain precisely such a speed as, during a timeinterval equal to that of the fall, will carry the body through a distance equal to twice that of the fall. But since this matter is one which is fundamental in our discussion it is well that we make it perfectly clear by means of some particular example. Let us consider the speed and momentum acquired by a body falling through the height, say, of a spear [picca] as a standard which we may use in the measurement of other speeds and momenta as occasion de-
761
mands; assume for instance that the time of such a fall is four seconds [minuti secondi €fora]; now in order to measure the speed acquired from a fall through any other height, whether greater or less, one must not conclude that these speeds bear to one another the same ratio as the heights of fall; for instance, it is not true that a fa'll through four times a given height confers a speed four times as great as that acquired by descent through the given height; because the speed of a naturally accelerated motion does not vary in proportion to the time. As has been shown above, the ratio of the spaces is equal to the square of the ratio of the times. If, then, as is often done for the sake of brevity, we take the same limited straight line as the measure of the speed, and of the time, and also of the space traversed during that time, it follows that the duration of fall and the speed acquired by the same body in passing over any other distance, is not represented by this second distance, but by a mean proportional between the two distances. This I can better ill ustrate by an example.
, c FIGURE 17
In the vertical line ac, layoff the portion ab to represent the distance traversed by a body falling freely with accelerated motion: the time of fall may be represented by any limited straight line, but for the sake of brevity, we shan represent it by the same length ab; this length may also be employed as a measure of the momentum and speed acquired during the motion; in short, let ab be a measure of the various physical quantities which enter this discussion. Having agreed arbitrarily upon ab as a measure of these three different quantities, namely, space, time, and momentum, our next task is to find the time required for fall through a given vertical distance ac, also the momentum acquired at the terminal point c, both of which are to be expressed in terms of the time and momentum represented by abo These two required quantities are obtained by laying off ad, a mean proportional between ab and ac; in other words, the time of fall from a to c is represented by ad on the same scale on which we agreed that the time of fall from a to b should be represented byab. In like manner we may say that
761
the momentum [impeto 0 grado di velocita] acquired at e is related to that acquired at b, in the same manner that the line ad is related to ab, since the velocity varies directly as the time, a conclusion, which although employed as a postulate in Proposition JII, is here amplified by the Author. This point being clear and well-established we pass to the consideration of the momentum [impeto] in the case of two compound motions, one of which is compounded of a uniform horizontal and a uniform vertical motion, while the other is compounded of a uniform horizontal and a naturally accelerated vertical motion. If both components are uniform, and one at right angles to the other, we have already seen that the square of the resultant is obtained by adding the squares of the components [po 765] as will be clear from the following illustration. Let us imagine a body to move a10ng the vertical ab with a uniform momentum [impeto] of 3, and on reaching b to move toward e with a momentum [velocittl ed impeto] of 4, so that during the same time-interval it will traverse 3 cubits along the vertical and 4 along the horizontal. But
c"'-____
.....I
PlGU'ltB 18
a particle which moves with the resultant velocity [veiocita] will, in the same time, traverse the diagonal ae, whose length is not 7 cubits-the sum of ab (3) and be (4)-but 5, which is in potenza equal to the sum of 3 and 4. that is, the squares of 3 and 4 when added make 25, which is the square of ac, and is equal to the sum of the squares of ab and be. Hence ac is represented by the side-or we may say the root--of a square whose area is 25, namely 5. As a fixed and certain rule for obtaining the momentum which results from two uniform momenta, one vertical. the other horizontal, we have therefore the following: take the square of each, add these together, and extract the square root of the sum, which will be the momentum resulting from the two. Thus, in the above example, the body which in virtue of its vertical motion would strike the horizontal plane with a momentum r/orza] of 3, would owing to its horizontal motion alone strike at e with a momentum of 4; but if the body strikes with a momentum which is the resultant of these two, its blow will be that of a body moving with a momentum [velocittl e !orza] of 5; and such a blow will be the same at all points of the diagonal ae, since its components are always the same and never increase or diminish.
110
Let us now pass to the consideration of a uniform horizontal motion compounded with the vertical motion of a freely falling body starting from rest. It is at once clear that the diagonal which represents the motion compounded of these two is not a straight line, but, as has been demonstrated, a semi-parabola, in which the momentum [impeto] is always increasing because the speed [velocita] of the vertical component is always increasing. Wherefore, to determine the momentum [impeto] at any given point in the parabolic diagonal, it is necessary first to fix upon the uniform horizontal momentum [impeto) and then, treating the body as one faUing freely, to find the vertical momentum at the given point; this latter can be determined only by taking into account the duration of fall, a consideration which does not enter into the composition of two uniform motions where the velocities and momenta are always the same; but here where one of the component motions has an initial value of zero and increases its speed [velocita] in direct proportion to the time, it fol1ows that the time must determine the speed [velocita] at the assigned point. It only remains to obtain the momentum resulting from these two components (as in the case of uniform motions) by placing the square of the resultant equal to the sum of the squares of the two components. . . .
COMMENTARY ON
THE BERNOULLIS N EIGHT generations the Bach family produced at least two dozen eminent musicians and several dozen more of sufficient repute to find their way into musical dictionaries. So numerous and so eminent were tbey that, according to the Britannica, musicians were known as "Bacbs" in Erfurt even when tbere were no longer any members of the family in the town. What the Bachs were to music, the Bernoulli clan was to science. In the course of a century eight of its members pursued matbematical studies, several attaining the foremost rank in various branches of this science as well as in related disciplines. From this group came a "swarm of descendants about half of whom were gifted above the average and nearly all of whom, down to tbe present day, bave been superior human beings." 1 The Bernoullis were a Protestant famHy driven from Antwerp in the last quarter of the sixteenth century by religious persecution. In 1583 they found asylum in Frankfurt; after a few years they moved to Basel in Switzerland. Nicolaus Bernoulli (1623-1708) was a wealthy merchant and a town councilor. This in itself is not a noteworthy achievement; he deserves rather to be remembered for his three sons Jacob, Nicolaus and John, and their descendants. 2 It is peculiar that nothing is ever said about the women the Bernoullis married; they must have made at least a genetic contribution to this illustrious spawn. Jacob (I), for eighteen years professor of mathematics at Basel, had started out at his father?s insistence as a theologian, but soon succumbed to his passion for science. He became a master of the calculus, developing
I
• E. T. Bell, M~n of MQth~mQt;cs. N. Y., 1937, p. 131. "No fewer than 120 of the descendants of the mathematical Bernoullis have been traced genealogically, and of this considerable posterity the majority achieved distinction-sometimes amounting to eminence--in the law, scholarship, science. literature, the learned professions. administration and the arts. None were fanures." • There is a confusion, understandabJe, about the Bernoum genealogical lines, and another, less understandable, about their names. Jacob, for example. is also known as Jakob, Jacques and James; Johannes. as Johann, John and Jean. I shall use the famUiar fonns in the following table: Nicolaus Senior (162~1708)
r
Jacob (I) (16S4-170S)
r Nicolaus (I) (1662-1716)
• (I) John (1667-1748)
Nicollus (II)
Nicolaus (m)
(1687-17S9)
(169S-1726)
f
r
Daniel (1700-1782)
I
John (n) (1710-1790)
I JotJ. (III) (1746-1807) 771
Tn
and applying it successfully to a considerable number of problems. Among his more celebrated investigations were those into the properties of the curve known as the catenary (it is formed by a heavy chain hanging freely from its two extremities), into isoperimetrical figures (those enclosing, for any given perimeter, the greatest area) and into various spiral curves. His other works include a great treatise on probability, the Ars Conjectandi (a selection from it appean elsewhere in these pages: see pp. 1452-1455), A Method of Teaching Mathematics to the Blind, based on his experience teaching the elements of science to a blind girl at Geneva, and many verses in Latin, German and French, regarded as "elegant" in their time but now forgotten. Jacob, according to Francis Galton, suffered from "a bilious and melancholic temperament 8 his brother John did nothing to soothe it. John also was an exceptional mathematician. He was more prolific than Jacob, made many independent and important mathematical discoveries, and enlarged scientific knowledge in chemistry, physiCs and astronomy. The BemouUis. Galton says, "were mostly quarrelsome and unamiable"; John was a prime example. He was violent, abusive, jealous, and, when necessary. dishonest. He claimed a reward Jacob had offered for a solution of the isoperimetrical problem. The solution he presented was wrong; he waited until Jacob died and then pubHshed another solution which he knew to be wrong-a fact he admitted seventeen yean later. His son Daniell again a brilliant mathematician, had the temerity to win a French Academy of Sciences prize which his father had sought. John gave him a special reward by throwing him out of the house." These agreeable traits were "lived out," as psychoanalysts might observe, and thus did nothing to shorten John's life. He died at the age of eighty, retaining his powen and his meanness to the end. It is with Daniel Bernoulli that we are here mainly concerned. He was a second son, born at Groningen-where his father was then professor of mathematics-in 1700. His father did everything possible to tum him from mathematical punuits. The program consisted of cruel mistreatment, when Daniel was a child, to destroy his self-confidence, and of later attempts to force him into business. John should have known this wouldn't work; the Bernoullis were tough as well as dedicated. When he was eleven, Daniel got instruction in geometry from his brother Nicolaus. s He studied U
;
Francis Galton. Hereditary Genius; London, reprint of 1950, p. 195. E. T. Bell, op. cit., p. 134. S Nicolaus Bernoulli (1695-1726) was no exception to the Bernoulli rule of extraordinariness. At the age of eight he could speak Gennan. Dutch. French and Latin; at sixteen he became a Doctor of Philosophy at the University of Basel; he was appointed professor of mathematics at St. Petersburg at the same time as Daniel. His early death (of a "lingering fever") prevented him from developing his evident powers. As the eldest son he was better treated by his father than Daniel; at any rate he was permitted, even encouraged, to study mathematics. When he was twenty-one his father pronounced him "worthy of receiving the torch of science from his own hands." 3
4
medicine, became a physician and finaJ1y, at the age of twenty-five, accepted an appointment as professor of mathematics at St. Petersburg. In 1733 he returned to Basel to become professor of anatomy, botany, and later, "experimental and speculative philosophy," i.e., physics. He remained at this post until be was almost eighty, pub1ishing a large number of important memoirs on physical problems and doing first-rate work in probability theory, calculus, differential equations and related fields. He won, or divided equally, no less than ten prizes put up by the French Academy of Sciences, inc1uding the one which so infuriated his father. Late in life, he particularly enjoyed reca1ling that. in his youth, a stranger once answered his self-introduction, "I am Daniel Bernoulli," with an "incredulous and mocking" ·'and I am Isaac Newton." Bernoulli's most famous book is the Hydrodynamica, in which he Jaid the foundations, theoretical and practical, for the Uequilibrium, pressure, reaction and varied velocities" of ftuids. The Hydrodynamica is notable also for presenting the first formulation of the kinetic theory of gases, a keystone of modern physics. Bernoulli showed that, if a gas be imagined to consist of "very minute corpuscles," "practically infinite in number," "driven hither and thither with a very rapid motion," their myriad comsions with one another and impact on the wans of the containing vessel would explain the phenomenon of pressure. Moreover, if the volume of the container were slowly decreased by sliding in one end like a piston, the gas would be compressed, the number of collisions of the corpusc]es would be increased per unit of time, and the pressure would rise. The same effect would follow from heating the gas; heat, as Bernoulli perceived, being nothing more than "an increasing internal motion of the particles." This "astonishing prevision of a state of physics which was not actuaJIy reached for 110 years" (notably by Joule, who calculated the statistical averages of the enormous number of molecular collisions and thus derived Boyle's )aw--pressure X volume constant-from the laws of impact) was fully sustained by Bernou11i's remarkable experimental and theoretical labors. 8 He provided an algebraic formulation of the relation between impacts and pressure; he even calculated the magnitude of the pressure increase resulting from decreased volume and found it corresponded to the hypothesis Boyle had confirmed by experiment, that ''the pressures and expansions are in reciprocal proportions.ft .,. The following selection covers these topics; it is from the tenth section of the Hydrodynamica, (1738).
=
G Lloyd W. Taylor, Physics-The Pioneer Science, Boston, 1941, p. 109. ., "Bernoulli in effect bad made in his thinking two enormous jumps, for wbich the temper of his time was not ready for over three generations: first the equivalence between beat and energy, and, secondly, the idea that a weJJ~defined relationship. such as Boyle'S simple law, could be deduced from the chaotic picture of randomly moving particles." Gerald Holton, Introduction 10 Concepts and Theories In Physical Science, Cambridge (Mass.), 1952, p. 376.
DanM
B~rnoulli
has
b~~n call~d th~ lath~r
01
math~matical physics. -ERIC TEMPLE BELL
So many 01 th~ properties oj matter, esp~cially wh~n in th~ gQS~ous form. can ~ d~tiuced jrom tM hypoth~sis that lh~ir minut~ paris ar~ in rapid motion, the velocity incr~Q8ing with 1M t~mp~,atur~, that th~ pr~cis~ nalUr~ of this motion becomes a subi~ct oj rational curios;IY. Dani~1 B~rnoulli, H~rapath, Joul~, KriJnig, Clausius, «c., have shewn that the relations be· tween pr~ssure, temperalur~ and density in a perject gas can be explained by supposing the partic/~s to mov~ with unijorm v~/ocity in straight lin~s, striking against th~ sid~s oj 1M containing vessel and thus producing pressure. -JAMES CLERK MAXWELL (Illustrations oj tM Dynamical Theory oj Gases)
2
Kinetic Theory of Gases By DANIEL BERNOULLI
1. IN the consideration of elastic fluids we may assign to them such a constitution as will be consistent with all their known properties, that so we may approach the study of their other properties, which have not yet been sufficiently investigated. The particular properties of elastic fluids are as follows: 1. They are heavy; 2. they expand in all directions unless they are restrained; and 3. they are continually more and more compressed when the force of compression increases. Air is a body of this sort, to which especially the present investigation pertains. 2.. Consider a cylindrical vessel ACDB (Figure 44) set vertically, and a movable piston EF in it, on which is pJaced a weight P: let the cavity ECDF contain very minute corpuscles, which are driven hither and thither with a very rapid motion; so that these corpuscles, when they strike against the piston EF and sustain it by their repeated impacts, form an elastic fluid which will expand of itself if the weight P is removed or diminished, which wi11 be condensed jf the weight is increased, and which gravitates toward the horizontal bottom CD just as if it were endowed with no elastic powers: for whether the corpuscles are at rest br are agitated they do not lose their weight, so that the bottom sustains not only the weight but the elasticity of the fluid. Such therefore is the fluid which we shall substitute for air. Its properties agree with those which we have already assumed for elastic fluids, and by them we shall explain other properties which have been found for air and shan point out others which have not yet been sufficiently considered. 3. We consider the corpuscles which are contained in the cyJindrical cavity as practically infinite in number. and when they occupy the space ECDF we assume that they constitute ordinary air, to which as a standard 774
775
Kllft!tic TMDry Df GMeS
all our measurements are to be referred: and so the weight P holding the piston in the position EF does not differ from the pressure of the superincumbent atmosphere, which therefore we shalt designate by P in what follows. It should be noticed that this pressure is not exactly equal to the absolute weight of a vertical cylinder of air resting on the piston EFt as hitherto most authors have asserted without sufficient consideration; rather it is equal to the fourth proportional to the surface of the earth, to the size of the piston EF, and to the weight of all the atmosphere on the surface of the earth. 4. We shall now investigate the weight 1[, which is sufficient to condense the air ECDF into the space eCDf, on the assumption that the ve]ocity of the particles is the same in both conditions of the air, the natural condition as wen as the condensed. Let EC 1 and eC s. When the piston EF is moved to eft it appears that a greater effort is made by the fluid for two reasons: first, because the number of particles is now greater in the ratio of the space in which they are contained, and secondly, because each particle repeats its impacts more often. That we may proper]y calculate the increment which depends on the first cause we may consider the particles as if they were at rest. We shall set the number of them which are contiguous to the piston in the position EF n; then the like number
=
=
=
when the piston is in the position e' will be
= n : (:~}fsor = n : slI.
It should be noticed that the fluid is no more condensed in the lower part than in the upper part. because the weight P is infinitely greater than
776
Daniel 1J'l'IIDlIlli
the weight of the fluid itself: hence it is plain that for this reason the force of the fluid is in the ratio of the numbers nand n : s% that is, as s% is to 1. Now in reference to the other increment arising from the second cause, this is found by considering the motion of the particles, and it appears that their impacts are made more often by as much as the particles are c10ser together: therefore the numbers of the impacts will be recipro-cally as the mean distances between the surfaces of the particles, and these mean distances will be thus determined. We assume that the particles are spheres. We represent by D the mean distance between the centers of the spheres when the piston is in the position EF. and by d the diameter of a sphere. Then the mean distance between the surfaces of the Spheres will be D - d. But it is evident that when the piston is in the position ef. the mean distance between the centers of the spheres DV's and therefore the mean distance between the surfaces of the spheres DV's ~ d. Therefore, with respect to the second cause, the force of the natura' air in ECDF will be to the force of
=
=
1
the compressed air in eCDf as
1 to - - - - , or as DV's - d to D-d DV's-d
D - d. When both causes are joined the predicted forces will be as s% X (DV's - d) to D - d. For the ratio of D to d we may substitute one which is easier to understand: for if we thin k of the piston EF as depressed by an infinite weight, so that it descends to the position mn, in which all the particles are in contact, and if we represent the line mC by m, we shall have D is to d as 1 is to V'm. If we substitute this in the ratio above, we shall find that the force of the natural air in ECDF is to the force of the compressed air in
eCDf as s%
X
(V's - V'm) is to 1 - V'm. or as s - V'mss is to I -
V'm. Therefore 1£
=
1-
V'm X
P.
., - V'ms.,
5. From all the facts known we may conclude that natural air can be very much condensed and compressed into a practically infinitely small space; so that we may set m 0, and hence 1t Pis; so that the compressing weights are almost in the inverse ratio of the spaces which air occupies when compressed by different amounts. This law has been proved by many experiments. It certainly may be safely adopted' for air that is less dense than natural air; whether it holds for considerably denser air I have not sufficiently investigated: nor have there yet been' experiments instituted with the accuracy which is necessary in this case. There is special need of an experiment to find the value of m, but this experiment must be most accurately carried out and with air under very high pressure;
=
=
KI"etlc TIIeory 01
777
Gas"
and the temperature of the air while it is being compressed must be carefully kept constant. 6. The elasticity of air is not only increased by condensation but by heat supplied to it, and since it is admitted that heat may be considered as an increasing internal motion of the particles. it fol1ows that if the e]asticity of air of which the volume does not change is increased. this indicates a more intense motion in the particles of air~ which fits in wen with our hypothesis; for it is plain that so much the greater weight P is needed to keep the air in the condition ECDF, as the aerial particles are agitated by the greater velocity. It is not difficult to see that the weight P should be in the duplicate ratio of this velocity because, when the velocity increases, not only the number of impacts but also the intensity of each of them increases equally, and each of them is proportional to the weight P. Therefore, if the velocity of the particles is caUed v, the weight which is able to sustain the piston in the position EF = vvP and in the position e/
=
l-~m
s-~mss
vvP
X vvP, or approximately
= -, because as we have seen s
the number m is very small in comparison with unity or with the number s. 7. This theorem, as I have presented it in the preceding paragraph, in which it is shown that in air of any density but at a fixed temperature, the elasticities are proportional to the densities, and further that the increments of elasticity which are produced by equal changes of temperature are proportional to the densities, this theorem, I say, D. Amontons discovered by experiment and presented it in the Memoirs of the Royal Academy of Sciences of Paris in 1702.
COMMENTARY ON
A Great Prize, a Long-Suffering Inventor and the First Accurate Clock
T
HE reckoning of latitude-the distance north or south from the
equator of a point on the earth's surface-was well understood by the ancients. Since the Pole Star holds approximately the same position in the heavens throughout every night, and since the earliest sailors observed that it dipped toward the horizon as they sailed south, they measured its angle with the horizon (its altitude), using an astrolabe, cross-staff or other angle~measuring device. and thus fixed their position with reference to the equator. Over the centuries the ancient methods were modified and improved, but their basic features are retained by sea and air navigators to the present day.l Longitude, the measure of distance east or west from an arbitrary line, presented much greater difficulties and defied exact calculation until the eighteenth century. The combined efforts of astronomers, physicists, mathematicians and clock makers were required to solve this important problem upon which scientific cartography, sound navigation and systematic exploration and discovery depended. Curiously enough, it was not until almost half a century after Columbus had made his "long voyage" that anyone connected the fixing of longitui::le with the construction of reliable and portable timekeepers. The relationship between clocks and longitude is doubtless obvious to many readers. but I had better be safe and explain what is involved. Longitude is determined, in effect. by translating space into time. Take as a baseline from which east-west distances are to be measured a meridian (half a great circle included between the poles) passing through a convenient place~ the modem convention fixes on Greenwich. Designate this Hne as the zero or prime meridian and imagine other meridians marking off the globe at intervals of 15°. Since it takes the earth twenty-four hours to complete a rotation of 360 0 • each meridian may be regarded as separated from its immediate neighbors to the east and west by one hour. Finding your longitude then, is "merely a matter 1 "The quadrants. sextants and octanls, developed throughout the centuries. were little more than segments of the ancient astrolabe, refined and adapted to meet the special requirements of surveyors and navigators. The modern Nautical Almanac, with its complex and multifarious tables that make it possible to find the latitude at any hour of the day or night. is nothing more than the sum total of ancient astrology, streamlined and perfected by astronomical instruments, including telescopes." Lloyd A. Brown, The Story of Maps, Boston, 1949, p. 180.
778
A O,.,al Prlu. a LlJPI,·SllfJtrI1l, l11vr1llor lind ,1ft First ACCllratt Clock
719
of comparing noons with Greenwich. You are just as long a distance from Greenwich as your noon is long a time from the Greenwich noon." 2 Nowadays Greenwich noon is ascertained by radio; but a clock set to run on Greenwich time wilt do the job almost as well. Starting out on a sea voyage, you take along a chronometer set to Greenwich time; after sailing west for a few days you observe, let us suppose, that when the sun is directly overhead (12 noon), your Greenwich chronometer says 3 P.M. This means that the sun has required three hours to "move" from directly above Greenwich to directly above the spot where you find yourself. To be accurate, it means that the earth has turned for three hours. Thus you have reached a point three times 15° west longitude. To give another illustration, if it is noon where you are when it is midnight in Greenwich, you are halfway around the globe, at longitude 180 It is easy to see, therefore, that precise clocks were needed to calculate longitude. The search for a reliable method of keeping time at sea produced its share of fantastic as well as sensible suggestions. Sir Kenelm Digby, for example, invented a "powder of sympathy" which. by a method I shan not attempt to repeat, caused a dog on shipboard to "yelp the hour on the dot." This was not the final answer to the problem. John Harrison, a Yorkshire carpenter, did better with his famous No. 4 chronometer, which took fifty years to make but lost only one second per month in trials at sea. Parliament had offered a prize of £20.000 for a dependable chronometer and Harrison-this "very ingenious and sober man, to a contemporary cal1ed him--quite properly claimed the money. He thereupon became the victim of a series of unsurpassed chicaneries perpetrated by scientists and politicians in concert. He got his reward but only after a parliamentary crisis and direct intervention by the King. The story of "The Longitude," a chronicle of science, politics, mathematics, human determination and brilliant craftsmanship, is admirably told by Lloyd A. Brown, a leading cartographer, in The Story of Maps. The following material is selected from his book. 0
•
2 David Greenhood, Down to Earth: Mapping lor Everybody. New York, 1951, p.15. .
The A.rt 0/ Navigation Is to bl! pt!rfected by the Solutwn 0/ this Pl'OjJlt!m. To find, at any Time, the Lon,Uude 0/ a Plact! at Sea. A. Public Reward Is promised lor the Discovery. Let him obtain it who is able. -BERNHARD VARENIUS (Geographia Generalis, 1650)
3
The Longitude By LLOYD A. BROWN
SCIENTIFIC cartography was born in France in the reign of Louis XIV (1638-1715), the offspring of astronomy and mathematics. The principles and methods which had been used and talked about for over two thousand years were unchanged; the ideal of Hipparchus and Ptolemy, to locate each place on earth scientifically, according to its latitude and longitude. was still current. But something new had been introduced into the picture in the form of two pieces of apparatus--a telescope and a timekeeper. The result was a revolution in map making and a start towards an accurate picture of the earth. With the aid of these two mechanical contrivances it was possible. for the first time, to solve the problem of how to determine longitude, both on land and at sea. The importance of longitude, the distance of a place east or west from any other given place, was fully appreciated by the more literate navigators and cartographers of history, but reactions to the question of how to find it varied from total indifference to complete despondency. Pigafetta, who sailed with Magellan, said that tbe great explorer spent many hours studying the problem of longitude, "but," he wrote, "the pilots content themselves with knowledge of the latitude, and are so proud [of themselves], they will not hear speak of the longitude." Many explorers of the time felt the same way, and rather than add to their mathematical burdens and observations. they were content to let welt enough alone. However, "there be some," says an early writer, "that are very inquisitive to have a way to get the longitude, but that is too tedious for seamen, since it requireth the deep knowledge of astronomy, wherefor I would not have any man think that the longitude is to be found at sea by any instrument; so let no seamen trouble themselves with any such rule, but (according to their accustomed manner) let them keep a perfect account and reckoning of the way of their ship." What he meant was, let them keep their dead reckoning with a traverse board, setting down the ship's estimated dai1y speed and her course, 1 Like the elixir of life and the pot of gold, longitude was a wil1-o'-thewisp which most men refused to pursue and others talked about with awe. t
W. R. Martin, article "Navigation" in Encyclopaedia Britannica, 11th Edition. 18G
Tilt Lo,.,ltlldt
18J
"Some doo understand," wrote Richard Eden, "that the Knowledge of the Longitude myght be founde, a thynge doubtlesse greatly to be desyred. and hytherto not certaynly knowen, although Sebastian Cabot, on his death-bed told me that he had the knowledge thereof by divine revelation, yet so. that he myght not teache any man. But," adds Eden, with a certain amount of scorn. ". thinke that the good olde man, in that extreme age, somewhat doted, and had not yet even in the article of death. vtterly shaken off all wordly vayne glorie." 2 Regardless of pessimism and indifference, the need for a method of finding the longitude was fast becoming urgent. The real trouble began in 1493, less than two months after Columbus returned to Spain from his first voyage to the west. On May 4 of that year, Pope Alexander VI issued the Bull of Demarcation to settle the dispute between Spain and Portugal, the two foremost maritime rivals in Europe. With perfect equanimity His Holiness drew a meridian Jine from pole to pole on a chart of the Western Ocean one hundred leagues from the Azores. To Spain he assigned all lands not already belonging to any other Christian prince which had been or would be discovered west of the line, and to Portugal all discoveries to the east of it; a masterful stroke of diplomacy, except for the fact 'that no one knew where the line feU. Naturally both countries suspected the worst, and in later negotiations each accused the other of pushinlJ the line a little in the wrong direction. For all practical purposes, the term "100 leagues west of the Azores" was meaningless, as was the Line of Demarcation and all other meridians in the New World laid down from a line of reference in the Old. Meanwhile armed convoys heavily laden with the wealth of the Indies ploughed the seas in total darkness so far as their longitude was concerned. Every cargo was worth a fortune and an the risk involved, but too many ships were lost. There were endless delays because a navigator was never sure whether he had overreached an island or was in imminent danger of arriving in the middle of the night without adequate preparations made for landing. The terrible uncertainty was wearing. In 1598 Philip III of Spain offered a perpetual pension of 6000 ducats, together with a life pension of 2000 ducats and an additional gratuity of 1000 more to the f'discoverer of longitude." Moreover, there would be smaller sums available in advance for sound ideas that might lead to the discovery and for partially completed inventions that promised tangible results. and no questions asked. It was the clarion can for every crank. lunatic and under· nourished inventor in the land to begin research on "the fixed point" or the UEast and West navigation" as it was caUed. In a short time the $I See Richard Eden's "Epistle Dedicatory" in his translation of Iohn Taisnier's A. very necessarie and profitable book conceming nauigatwn . . . london, 1579 (1). (Quoted from Blbliotheca Americana. A catalogue oj books . . . in the library 0/ the late lohn Carter Brown. Providence, 1815, Part I, No. 310.)
182
Lloyd A. BroWft
Spanish government was so deluged with wild, impractical schemes and Philip was so bored with the whole thing that when an ItaHan named GaUJeo wrote the court in 1616 about another idea, the king was unimpressed. After a long. sporadic correspondence covering sixteen years, Galileo reluctantly gave up the idea of selling his scheme to the court of Spain.! Portugal and Venice posted rewards. and drew the same motley array of talent and the same results as Spain. Holland offered a prize of 30,000 scudi to the inventor of a reliable method of finding the longitude at sea, and Willem Blaeu, map publisher, was one of the experts chosen by the States Genera) to pass on an such inventions. In August, 1636, Galileo came forward again and offered his plan to HoUand. this time through his Paris friend Diodati, as he did not care to have his correspondence investigated by the Inquisition. He told the Dutch authorities that some years before, with the aid of his telescope, he had discovered what might be a remarkable celestial timekeeper-Jupiter. He, Galileo, had first seen the four satellites, the "Cosmian Stars" (Sidera Medicea, as he called them), and had studied their movements. Around and around they went, first on one side of the planet, then on the other, now disappearing. then reappearing. In 1612, two years after he first saw them, he had drawn up tables, plotting the positions of the sate11ites at various hours of the night. These, he found. could be drawn up several months in advance and used to determine mean time at two different places at once. Since then, he had spent twenty-four years perfecting his tables of the satellites, and now he was ready to offer them to Honand, together with minute instructions for the use of any who wished to find the longitude at sea or on land .• The States General and the four commissioners appointed to investigate the merits of Galileo's proposition were impressed. and requested further details. They awarded him a golden chain as a mark of respect and Hortensius, one of the commissioners, was elected to make the journey to Italy where he could discuss the matter with Ga1i1eo in person. But the Holy Office got wind of things and the trip was abandoned. In 1641 after a lapse of nearly three years. negotiations were renewc!d by the Dutch scientist Christian Huygens. but Galileo died a short time after and the idea of using the sateUites of Jupiter was set aside.1S Tn the two thousand year search for a solution of the longitude problem 'J J. J. Fahie, Galileo. Hif life and work, New York. 1903. pp. 172, 372 ff.: Rupert T. Gould. The marine chronometer' Its histor)' and del'elopme"t, London. 1923. pp. 11. 12. 4 J. J. Fahie. op. cit., pp. 372 fr. Galileo named the satellites of Jupiter the "Cosmian Stars" after Cosmo Medici (Cosmo n, grandduke of Tuscany). See Ga1ileo's Opere edited by Eugenio Alberi, 16 vols., Firenze. 1842 S6. Tome III contains his "Sydereus Nuncius," pp. S9-99, describing his observations of Jupiter's satellites. and his suggestion that they be used in the determination of longitude. 1$ J. J. Fahie. op. cit., pp. 373-7S.
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it was never a foregone conclusion that the key lay in the transportation of timekeepers. But among the optimistic who believed that a solution could somehow be found, it was agreed that it would have to come from the stars, especially for longitude at sea, where there was nothing else to observe. It might be found in the stars alone or the stars in combination with some terrestrial phenomenon. However, certain fundamental principles were apparent to all who concerned themselves with the problem. Assuming that the earth was a perfect sphere divided for convenience into 360 degrees, a mean solar day of 24 hours was equivalent to 360 degrees of arc, and t hour of the solar day was equivalent to 15 degrees of arc or 15 degrees of longitude. Likewise, 1 degree of longitude was the equivalent of 4 minutes of time. Finer measurements of time and longitude (minutes and seconds of time, minutes and seconds of arc) had been for centuries the stuff that dreams were made of. Surveys of the earth in an east-west direction. expressed in leagues, miles or some other unit of linear measure. would have no significance unless they could be translated into degrees and minutes of arc, fractional parts of the circumference of the earth. And how big was the earth? The circumference of the earth and the length of a degree (1 /360th part of it) had been calculated by Eratosthenes and others but the values obtained were questionable. Hipparchus had worked out the difference between a solar day and a sidereal day (the interval between two successive returns of a fixed star to the meridian), and had plotted a list of 44 stars scattered across the sky at intervals of right ascension equal to exactly one hour, so that one or more of them would be on the meridian at the beginning of every sidereal hour. He had gone a step further and adopted a meridian line through Rhodes. suggesting that longitudes of other places could be determined with reference to his prime meridian by the simultaneous observation of the moon's eclipses. This proposal assumed the existence of a reliable timekeeper which was doubtless nonexistent. The most popular theoretical method of finding longitude. suggested by the voyages of Columbus. Cabot, Magellan, Tasman and other explorers, was to plot the variation or declination of the compass needle from the true north. This variation could be found by taking a bearing on the polestar and noting on the graduated compass card the number of points, half and quarter points (degree and minutes of arc) the needle pointed east or west of the pole. Columbus had noted this change of compass variation on his first voyage, and later navigators had confirmed the existence of a "line of no variation" passing through both poles and the fact that variation changed direction on either side of it. This being so, and assuming that the variation changed at a uniform rate with a change of 10ngitude, it was logical to assume that here at last was a solution to the whole problem. All you had to do was compare the amount of variation at your place of
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observation with tbe tabulated variation at places whose longitude had already been determined. It was tbis fond hope that induced Edmund Halley and others to compile elaborate charts showing tbe supposed lines of equal valiation throughout the world. However, it was by no means that simple, as Gellibrand and others discovered. Variation does not change uniformly with a change of longitude; likewise, changes in variation occur very slowly; so slowly, in fact, tbat precise east-west measurements are impractical, especially at sea. And, too, it was found that lines of equal variation do not always run north and soutb; some run nearly east and west. However, in spite of the flaws that cropped up, one by one, the method had strong supporters for many years, but finally died a painful, lingering death.8 In addition to discovering Jupiter's satellites, Galileo made a second important contribution to the solution of longitude by his studies of the pendulum and its behavior, for the application of the swinging weight to the mechanism of a clock was the first step towards the development of an accurate timekeeper.1 The passage of time was noted by the ancients and their astronomical observations were "timed" with sundials, sandglasses and water clocks but little is known about how the latter were controlled. Bernard Walther, a pupil of Regiomontanus, seems to have been the first to time his observations with a clock driven by weights. He stated that on the 16th of January, 1484, he observed tbe rising of the planet Mercury, and immediately attached the weight to a clock having an bour-wbeel with fifty-six teeth. By sunrise one hour and thirty-five teeth had passed, so that the elapsed time was an hour and thirty-seven minutes, according to his calculations. The next important phase in the development of a timekeeper was the attachment of a pendulum as a driving force. This clock was developed by Christian Huygens, Dutch physicist and astronomer, the son of Constantine Huygens. He built the first one in 1656 in order to increase the accuracy of his astronomical observations, and later presented it to the States General of Honand on the 16th of June, 1657. The fol1owing year he published a full description of the principles involved in the mechanism of his timekeeper and the physical laws governing the pendulum. It was a classic piece of writing, and established Huygens as one of the leading European scientists of the day. 8 By 1666 there were many able scientists scattered throughout Europe. Their activities covered the entire fields of physics, chemistry. astronomy, mathematics, and natural history, experimental and applied. For the most 6 R. T. Gould, op. cit .. pp. 4 and 4 n. ,. See Oalileo's Dialogues cOllcernillg two new sciences, by Galileo Galilei. translated by Henry Crew and Alfonso de Salvio, introduction by Antonio Favaro, New York, ]9]4, pp. 84, 95, t 70, 254. 8 See John L. E. Dreyer's article "Time. Measurement of' (in the Encyclopaedia Britannica, 1Jth edition, pp. 983d. 984a).
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part they worked independently and their interests were widely diversified. Occasionally the various learned societies bestowed honorary memberships on worthy colleagues in foreign countries, and papers read in the various societies were exchanged with fellow scientists in foreign lands. The stage was set for the transition of cartography from an art to a science. The apparatus was at hand and the men to use it. Pleading for the improvement of maps and surveys, Thomas Burnet made a useful distinction between the popular commercial map publications of the day and what he considered should be the goal of future map makers. "I do not doubt," he wrote, "but that it would be of very good use to have natural Maps of the Earth . . . as well as civil . . . . Our common Maps I call Civil, which note the distinction of Countries and of Cities, and represent the Artificial Earth as inhabited and cultivated: But natural Maps leave out an that, and represent the Earth as it would be if there were not an Inhabitant upon it, nor ever had been; the Skeleton of the Earth, as I may so say, with the site of all its parts. Methinks also every Prince should have such a Draught of his Country and Dominions, to see how the ground lies in the several parts of them, which highest, which lowest; what respect they have to one another, and to the Sea; how the Rivers flow, and why; how the Mountains lie, how Heaths, and how the Marches. Such a Map or Survey would be useful both in time of War and Peace, and many good observations might be made by it, not only as to Natural History and Philosophy, but also in order to the perfect improvement of a Country." 8 These sentiments regarding "natural n maps were fully appreciated and shared by the powers in France, who proceeded to..do something about it. All that was needed was an agency to acquire the services and direct the work of the available scientific talent, and someone to foot the bilts. The agency was taken care of by the creation of the Academie Royale des Sciences, and the man who stood prepared to foot the bills for better maps was His Majesty Louis XIV, king of France. Louis XIV ascended the throne when he was five years old, but had to wait sixteen years before he could take the reins of government. He had to sit back and watch the affairs of state being handled by the queen~ mother and his minister, Cardinal Mazarin. He saw the royal authority weakened by domestic troubles and the last stages of the Thirty Years' War. Having suffered through one humiliation after another without being able to do anything about it, Louis resolved, when he reached the age of twenty-one, to rule as well as reign in France. He would be his own first minister. Foremost among his few trusted advisors was Jean Baptiste Colbert, minister for home affairs, who became, in a short time, the chief power behind the throne. Colbert, an ambitious and industrious man with I»
Thomas Burnet: TM theory O/Ihe eQrth ••• London, 1684, p. 144.
Lloyd. A • .BroWIJ
expensive tastes, contrived to indulge himself in literary and artistic extravagances while adding to the stature and glory of his monarch. As for the affairs of state over which he exercised control, there were two enterprises in particular which entitle Colbert to an important place in the history of France. The first was the establishment of the French Marine under a monarch who cared little for naval exploits or the importance of sea power in the growth and defense of his realm; the second was the founding, in 1666, of the Academie Royale des Sciences, now the Institut de France. 1o The Academie Royale was Colbert's favorite project. An amateur scientist, he realized the potential value of a distinguished scientific body close to the throne, and with his unusual skill and seemingly unlimited funds he set out to make France pre-eminent in science..as it was in the arts and the art of war. He scoured Europe in search of the top men in every branch of science. He addressed personal invitations to such figures as Gottfried Wilhelm von Leibnitz, German philosopher and mathematician; Niklaas Hartsoeker, Dutch naturalist and optician; Ehrenfried von Tschirnhausen, German mathematician and manufacturer of optical lenses and mirrors; Joannes Hevelius, one of Europe's foremost astronomers; Vincenzo Viviani, Italian mathematician and engineer; Isaac Newton, England's budding mathematical genius. The pensions that went with the invitations were without precedent, surpassing in generosity those established by Cardinal Richelieu for the members of the Academie Fran~aise, and those granted by Charles II for the Royal Society of London. Additional funds were available for research, and security and comfort were assured to those scientists who would agree to work in Paris, surrounded by the most brilliant court in Europe. Colbert's ambition to make France foremost in science was realized, though some of the invitations were declined with thanks. Christian Huygens joined the Academie in 1666 and received his pension of 6000 livres a year until 1681, when he returned to Holland. Olaus Romer, Dutch astronomer, also accepted. These celebrities were followed by Marin de la Chambre who became physician to Louis XIV; Samuel Duclos and Claude Bourdelin in chemistry; Jean Pecquet and Louis Gayant in anatomy; Nicholas Marchant in botany.ll In spite of the broad scope of its activities, the avowed purpose of founding the Academie Royale, according to His Majesty, was to correct and improve maps and sailing charts. And the solution of the major problems of chronology, geography and navigation, whose practical importance was incontestable, Jay in the further study and application of astronomy.12 10 See Charles J. E. Wolf, Histoire de l'obseT\'atolre de Paris de sa /ondation a 179J, Paris, 1902; also L'lnslitlil de France by Gaston Darboux, Henry Roujon and George Picot, Paris, 1907 ("Les Grandes Institutions de France"). 11 C. J. E. Wolf, op. cit., pp. 5 ft. 12 Memoires de rAcademie Royale des Sciences, Vol. VIII, Paris, 1730.
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To this end, astronomical observations and conferences were begun in January, 1667. The Abbe Jean Picard, Adrian Auzout, Jacques Duot and Christian Huygens were temporarily insta11ed in a house near the CordeHers, the garden of which was taken over for astronomical observations. There the scientists set up a great quadrant, a mammoth sextant and a highly refined version of the sundial. They also constructed a meridian line. Sometimes observations were made in the garden of the Louvre. On the whole, faciHties for astronomical research were far from good, and there was considerable grumbling among the academicians. As early as 1665, before the Academie was founded, Auzout had written Colbert an impassioned memorandum asking for an observatory, reminding him that the progress of astronomy in France would be as nothing without one. When Colbert fina11y made up his mind, in 1667, and the king approved the money, events moved rapidly. The site chosen for the observatory was at Faubourg St. Jacques, well out in the country, away from the lights and disconcerting noises of Paris. Colbert decided that the Observatory of Paris should surpass in beauty and utility any that had been built to date, even those in Denmark, England and China, one which would reflect the magnificence of a king who did things on a grand scale. He called in Claude Perrault, who had designed the palace at Versai11es with accommodations for 6000 guests, and told him what he and his Academie wanted. The building should be spacious; it should have ample laboratory space and comfortable living quarters for the resident astronomers and their families. 18 On the 21 st of June, 1667, the day of the summer solstice, the members of the Academie assembled at Faubourg St. Jacques, and with great pomp and circumstance made observations for the purpose of "locating" the new observatory and establishing a meridian line through its center, a line which was to become the official meridian of Paris. The building was to have two octagonal towers flanking the southern fa~ade, and eight azimuths were carefully computed so that the towers would have astronomical as well as architectural significance. Then, without waiting for their new quarters, the resident members of the Academie went back to work, attacking the many unsolved riddles of physics and natural history, as well as astronomy and mathematics. They designed and built much of the apparatus for the new observatory. They made vast improvements in the telescope as an astronomical tool; they solved mechanical and physical problems connected with the pendulum and what gravity does to it, helping Huygens get the few remaining "bugs" out of his pendulum timekeeper. They concentrated on the study of the earth, its size and shape and place in the universe; they investigated the nature and behavior of the moon and other celestial bodies; they worked towards the establishment of a 13
C. J. E. Wolf, op. cit., p. 4.
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standard meridian of longitude for all nations. the meridian of Paris running through the middle of their observatory. They worked on the problem of establishing the linear value of a degree of longitude which would be a universally acceptable constant. In all these matters the Academie Royale des Sciences was fortunate in having at its disposal the vast resources of the court of France as well as the personal patronage of Louis XIV. An accurate method of determining longitude was first on the agenda of the Academie Royale. for obviously no great improvement could be made in maps and charts until such a method was found. Like Spain and the Netherlands, France stood ready to honor and reward the man who could solve the problem. In 1667. an unnamed German inventor addressed himself to Louis XIV. stating that he had solved the problem'of determining longitude at sea. The king promptly granted him a patent (brevet) on his invention. sight unseen, and paid him 60.000 Jivres in cash. More than this, His Majesty contracted to pay the inventor 8000 livres a year (Huygens was getting 6ooo!) for the rest of his life. and to pay him four sous on every ton of cargo moved in a ship using the new device, reserving for himself only the right to withdraw from the contract in consideration of 100,000 livres. An this His Majesty would grant, but on one condition: the inventor must demonstrate his invention before Colbert. Abraham Duquesne, Lieutenant-General of His Majesty's navaJ forces, and Messrs. Huygens, Carcavi. Roberval, Picard and Auzout of the Academie Royale des Sciences. 14 The invention proved to be nothing more than a variation on an old theme. an ingenious combination of water wheel and odometer to be inserted in a hole drilled in the keel of a ship. The passage of water under the keel would turn the water wheel. and the distance traversed by the ship in a given period would be recorded on the odometer. The inventor also claimed that by some strange device best known to himself his machine would make any necessary compensations for tides and crosscurrents of one kind and another; it was, in fact, an ideal and perfect solution to the longitude problem. The royal examiners studied the apparatus, praised its ingenuity and then submitted their report to the king in writing. They calmly pointed out, among other things, that if a ship were moving with a current, it might be most stationary with respect to the water under the keel and yet be carried along over an appreciable amount of longitude while the water wheel remained motionless. If, on the other hand, the ship were breasting a current, the odometer would register considerable progress when actuaUy the ship might be getting nowhere. The German 14
Histoire de r Acadlmie Royale des Sciences. Vol. 1, pp. 45--46.
inventor departed from Paris richer by 60,000 livres and the members of the Academie went back to work. 11 In 1669, after three years of intensive study~ the scientists of the Academie Royale had gathered together considerable data on the celestial bodies, and had studied every method that had been suggested for the determination of longitude. The measurement of lunar distances from the stars and the sun was considered impractical because of the complicated mathematics involved. Lunar eclipses might be all right except for the infrequency of the phenomenon and the slowness of eclipses, which increase the chance of error in the observer. Moreover, lunar eclipses were utterly impractical at sea. Meridional transits of the moon were also tried with indifferent success. What the astronomen were looking for was a celestial body whose distance from the earth was so great that it would present the same appearance from any point of observation. Also wanted was a celestial body which would move in a constantly predictable fashion, exhibiting at the same time a changing picture that could be observed and timed simultaneously from different places on the earth. Such a body was Jupiter. whose four sateJlites. discovered by Galileo. they had observed and studied. The serious consideration of Jupiter as a possible solution of the longitude problem brought to mind a publication that had come out in 1668 written by an Italian named Cassini. Wbi1e the members of the Academie continued their study of Jupiter's sateUites with an eye to utilizing their frequent eclipses as a method of determining longitude, Colbert investigated the possibilities of luring Cassini to Paris. Giovanni Domenico Cassini was born in Perinaldo, a village in the Comte of Nice, June 8, 1625, the son of an Italian gentleman. After completing his elementary schooling under a preceptor he studied theology and law under the Jesuits at Genoa and was graduated with honon. He developed a decided love of books. and while browsing in a library one day he came across a book on astrology. The work amused him, and after studying it he began to entertain his friends by predicting coming events. His phenomenal success.. as an astrologer plus his intellectual honesty made him very suspicious of his new-found talent. and he promptly abandoned the hocus-pocus of astrology for the less dramatic study of astron1$ Justin Winsor, Narrative and Critical History oj America, Boston. 1889. Vol. n, pp. 98-99 has an interesting note on the uJog." In Pigafetta's journal (January, 1521). be mentions the use of a chain dragged astern on Magellan's ships to measure their speed, The "Jog" as we know it was described in Bourne's Regiment oj the Sea, 1573, and Humphrey Cole is said to have invented it. In Eden's translation of Taisnier he speaks of an artifice "not yet divuJgate, which, placed in the pompe of a shyp, whyther the water hath recourse, and moved by the motion of the shypp, with wheels and weyghts, doth exactly shewe what space the shyp hath gone." See the article "Navigation" in the Encyclopaedia Britannica, 9th edition. For further comments on the history of the Jag, see L. C. Wroth, The Way oj Q Ship. Portland, Maine. 1937, pp.72-74.
omy. He made such rapid progress and displayed such remarkable aptitude, that in 1650, when he was only twenty-five years old, he was chosen by the Senate of Bologna to fill the first chair of astronomy at the university, vacant since the death of the celebrated mathematician Bonaventura Cavalieri. The Senate never regretted their choice.10 One of Cassini's first duties was to serve as scientific consultant to the Church for the precise determination of Holy Days, an important application of chronology and longitude. He retraced the meridian line at the Cathedral of Saint Petronius constructed in 1575 by Ignazio Dante, and added a great mural quadrant which took him two years to build. In 1655 when it was completed. he invited aU the astronomers in Italy to observe the winter solstice and examine the new tables of the sun by which the equinoxes, the solstices and numerous Holy Days could now be accurately determined. Cassini was next appointed by the Senate of Bologna and Pope Alexander VII to determine the difference in level between Bologna and Ferrara, relative to the navigation of the Po and Reno rivers. He not only did a thorough job of surveying, but wrote a detailed report on the two rivers and their peculiarities as wen. The Pope next engaged Cassini, in the capacity of a hydraulic engineer, to straighten out an old dispute between himself and the Duke of Tuscany relative to the diversion of the precious water of the Chiana River, alternate affluent of the Arno and the Tiber. Having settled the dispute to the satisfaction of the parties concerned, he was appointed surveyor of fortifications at Perugia, Pont Felix and Fort Urbino, and was made superintendent of the waters of the Po, a river vital to the conservation and prosperity of the country. In his spare time Cassini busied himself with the study of insects and to satisfy his curiosity repeated several experiments on the transfusion of blood from one animal to another, a daring procedure that was causing a flurry of excitement in the scientific world. But his major hobby was astronomy and his favorite planet was Jupiter. Whi1e he worked on the Chi ana he spent many evenings at Citta dena Pieve observing Jupiter's satellites. His telescope was better than Galileo's and with it he was able to make some additional discoveries. He noted that the plane of the revolving sate11ites was such that the satellites passed across Jupiter's disc close to the equator; he noted the size of the orbit of each satellite. He was certain he could see a number of fixed spots on Jupiter's orb, and on the strength of his findings he began to time the rotation of the planet as well as the satellites, using a fairly reliable pendulum clock.17 to OeU\',es de Fonttnelle. Eloges. Paris, 182S, Vol. I, p. 2S4.
Joseph Fran~ois Michaud: Biog,aphit Unil't,selle. Paris, 18S4-6S. Cassini gave Jupiter's rotation as 9" S6m • The COTrect time is not yet known with certainty. Slightly different results are obtained by using different markings. The value 9" Ssm is frequently used in modem texIS. 17
The Lon,ltu.
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After sixteen years of patient toil and constant observations, Cassini published his tables (Ephemerides) of the eclipses of Jupiter's satellites for the year 1668, giving on one page the appearance of the planet in a diagram with the sate11ites grouped around it and on the opposite page the time of the ecHpse (immersion) of each sate1lite in hours, minutes and seconds, and the time of each emersion. 18 Cassini, then forty-three years old, had become widely known as a scholar and skilled astronomer, and when a copy of his Ephemerides reached Paris Colbert decided he must get him for the Observatory and the Academie Royale. In this instance, however, it took considerable diplomacy as well as gold to get the man he wanted, for Cassini was then in the employ of Pope Clement IX, and neither Louis XIV nor Colbert cared to offend or displease His Holiness. Three distinguished scholars, Vaillant, Auzout and Count Graziani, were selected to negotiate with the Pope and the Senate of Bologna for the temporary loan of Cassini, who was to receive 9000 livres a year as long as he remained in France. The arrangements were final1y completed, and Cassini arrived in Paris on the 4th of April, t 669. Two days later he was presented to the king. Although Cassini had no intention of staying indefinitely, Colbert was insistent, and in spite of the remonstrances of the Pope and the Senate of Bologna, Cassini became a naturalized citizen of France in 1673, and was thereafter known as Jean Domenique Cassini. 1D Observations were in full swing when Cassini took his place among the savants of the Academie Royale who were expert mechanics as well as physicists and mathematicians. Huygens and Auzout had ground new lenses and mirrors, and had built vastly improved telescopes for the observatory. With the new instruments Huygens had already made some phenomenal discoveries. He had observed the rotation period of Saturn, discovered Saturn's rings and the first of the sateHites. Auzout had built other instruments and applied to them an improved filar micrometer, a measuring device all but forgotten since its invention by Gascoyne (Gascoigne) about 1639. After Cassini's arrival, more apparatus was order~, including the best telescopes available in Europe, made by Campani in Italy.2o One of the first important steps toward the correction of maps and charts was the remeasurement of the circumference of the earth and the establishment of a new value for a degree of arc in terms of linear measure. There was still a great deal of uncertainty as to the size of the earth, 18 The first edition of Cassini's work was published under the title Ephemerides Bononiellses Mediceorvm sydervm ex hypothesibvs, et tabvJis 10: Dominici Cassin; . . . Bononiae [Bologne], 1668. 19 C. J. E. Wolf, op. cit., p. 6. 20 For a detailed inventory of the equipment built and purchased by the Academie Royale, see C. 1. E. Wolf, op. cit.
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and the astronomers were reluctant to base their new data on a fundamental value which might negate all observations made with reference to it. After poring over the writings of Hipparchus, Poseidonius, Ptolemy and later authorities such as Snell, and after studying the methods these men had used, the Academie worked out a detailed plan for measuring the earth, and in 1669 assigned Jean Picard to do the job. The measurement of the earth at the equator, from east to west, was out of the question; no satisfactory method of doing it was known. Therefore the method used by Eratosthenes was selected, but with several important modifications and with apparatus that the ancients could only have dreamed of. Picard was to survey a line by triangulation running approximately north and south between two terminal points; he would then measure the arc between the two points (that is, the difference in latitude) by astronomical observations. After looking over the country around Paris, Picard decided he could run his line nearly northward to the environs of Picardy without encountering serious obstructions such as heavy woods and high hills.21 Picard selected as his first terminal point the "Pavillon" at Malvoisine near Paris, and for his second-.point the clock tower in Sourdon near Amiens, a distance of about thirty-two French leagues. Thirteen great triangles were surveyed between the two points, and for the purpose Picard used a stoutly reinforced iron quadrant with a thirty-eight inch radius fixed on a heavy standard. The usual pinhole alidades used for sighting were replaced by two telescopes with oculars fitted with cross hairs, an improved design of the instrument used by Tycho Brahe in Denmark. The limb of the quadrant was graduated into minutes and seconds by transversals. For measuring star altitudes involving relatively acute angles. Picard used a ta11 zenith sector made of copper and iron with an amplitude of about 18°. Attached to one radius of the sector was a telescope ten feet long. Also part of his equipment were two pendulum clocks, one regulated to beat seconds, the other half-seconds. For genera) observations and for observing the satellites of Jupiter. he carried three telescopes: a sma1l one about five feet long and two larger ones. fourteen and eighteen feet long. Picard was well satisfied with his equipment. In describing his specially fitted quadrant, he said it did the work so accurately that during the two years it took to measure the arc of the meridian, there was never an error of more than a minute of arc in any of the angles measured on the entire circumference of the horizon, and that in many cases, on checking the instrument for accuracy, it was found to be absolutely true. And as for the 21 For a complete account of Picard's measurement of the earth. including tables of data and a historical summary, see the Memoires de I'Academie Royale des Sciences, Vol. VII, Pt. I. Paris, 1729. See also the article "Earth, Figure of the" by Alexander R.oss Clarke and Frederick R.obert Helmert in the Encyclopaedia Britannica. 11th edition, p. 801.
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pendulum clocks he carried, Picard was pleased to report that they "marked the seconds with greater accuracy than most clocks mark the half hours." 22 When the results of Picard's survey were tabulated, the distance between his two terminal points was found to be 68,430 toises 3 pieds. The difference in latitude between them was measured, not by taking the altitude of the sun at the two terminal points, but by measuring the angle between the zenith and a star in the kneecap of Cassiopeia, first at Malvoisine and then at Sourdon. The difference was 10 11' 57". From these figures the value of a degree of longitude was calculated as 57,064 toises 3 pieds. But on checking from a second base line of verification which was surveyed in the same general direction as the first, this value was revised to 57,060 toises, and the diameter of the earth was announced as 6,538,594 toises. An measurements of longitude made by the Academie Royale were based on this value, equivalent to about 7801 miles, a remarkably close result. 28 In 1676, after the astronomers had revised and enlarged his Ephemerides of 1668, Cassini suggested that the corrected data might now be used for the determination of longitude, and Jupiter might be given a trial as a celestial clock. The idea was approved by his colleagues and experimental observations were begun, based on a technique developed at the Observatory and on experience acquired by a recent expedition to Cayenne for the observation of the planet Mars. The scientists were unusually optimistic as the work began, and in a rare burst of enthusiasm one of them wrote, us; ce n'est pas -/d Ie veritable secret des Longitudes, au-moins en approche-t-il de bien pres." Because of his tremendous energy, skill and patience, Cassini had by this time assumed the leadership of the scientists working at the observatory, even though he did not have the title of Director. He carried on an ext~sive correspondence with astronomers in other countries, particularly in Italy where the best instruments were available and where he and his work were wen known. Astronomers in foreign parts responded with enthusiasm when they learned of the work that was being done at the Paris Observatory. New data began to pour in faster than the resident astronomers could appraise and tabulate it. Using telescopes and the satellites of Jupiter, hundreds of cities and towns were now being located for the first time with reference to a prime meridian and to each other. AU of the standard maps of Europe, it seemed, would have to be scrapped.24 22 Mlmoires de l'Acadlmie Royale des Sciences, Vol. VII. Pt. I. Pagination varies widely in different editions of this series. 23 Ibid., pp. 306-07, gives a table of the linear measures used by Picard in making his computations. Measurements were made between the zenith and a star in the kneecap of Cassiopeia, probably 8 (AI Rukbab). See the Mlmoires de l'Acad~mie Royale des Sciences, Vol. VII, Pt. II, Paris, 1730, p. 305. 24 For Cassini's status in the Academie Royale, see C. J. E. Wolf. op. cit.
794
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With so much new information available, Cassini conceived the idea of compiling a large-scale map of the world (planisphere) on which revised geographical information could be laid down as it came in from various parts of the world, especially the longitudes of different places, hitherto unknown or hopelessly incorrect. For this purpose the third floor of the west tower of the Observatory was selected. There was plenty of space, and the octagonal walls of the room had been oriented by compass and quadrant when the foundation of the building was laid. The planisphere, on an azimuthal projection with the North Pole at the center, was executed in ink by sedileau and de Chazelles on the floor of the tower under the watchful eye of Cassini. The circular map was twenty-four feet in diameter, with meridians radiating from the center to the periphery, like the spokes of a wheel, at intervals of 10°. The prime meridian of longitude (through the island of Ferro) was drawn from the center at any angle "half way between the two south windows of the tower" to the point where it bisected the circumference of the map. The map was graduated into degrees from zero to 360 in a counterclockwise direction around the circle. The parallels of latitude were laid down in concentric circles at intervals of 10°, starting with zero at the equator and numbering both ways. For convenient and rapid "spotting" of places, a cord was attached to a pin fastened to the center of the map with a small rider on it, so that by swinging the cord around to the proper longitude and the rider up or down to the proper latitude, a place could be spotted very quickly. On this great planisphere the land masses were of course badly distorted, but it did not matter. What interested the Academie was the precise location, according to latitude and longitude, of the important places on the earth's surface, places that could be utilized in the future for bases of surveying operations. For this reason it was much more important to have the names of a few places strategically located and widely distributed. according to longitude, than it was to include a great many places that were scientifically unimportant. For the same reason, most of the cities and towns that boasted an astronomical observatory, regardless of how sman. were spotted on the map. The planisphere was highly praised by al1 who saw it. The king came to see it with Colbert and all the court. His Majesty graciously allowed Cassini, Picard and de la Hire to demonstrate the various astronomical instruments used by the members of the Academie to study the heavens and to determine longitude by remote control, as it were. They showed him their great planisphere and explained how the locations of different places were being corrected on the basis of data sent in from the outside world. It was enough to make even Louis pause. 23 2~ Ibid., pp. 62-65. For an account of the great planisphere. see the His/oire de /'Acadhnie Royale des Science.f, Vol. T. pp. 225-26: C. J. E. Wolf. op. cit A facsimile
T"~
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195
What effect the king's visit had on future events it is difficult to estimate, but in the next few years a great deal of surveying was done. Many surveying expeditions were sent out from the Observatory, and the astronomen went progressively further afield. Jean Richer led an expedition to Cayenne and Jean Mathieu de Chazel1es went to Egypt. Jesuit missionaries observed at Madagascar and in Siam. Edmund Haney, who was in close touch with the work going on in France, made a series of observations at the Cape of Good Hope. Thevenot, the historian and explorer, communicated data on several lunar ~clipses observed at Goa. About this time, Louis-Abel Fontenay, a Jesuit professor of mathematics at the Col1ege of Louis Ie Grand. was preparing to leave for China. Hearing of the work being done by Cassini and his colleagues, Fontenay volunteered to make as many observations as he could without interfering with his missionary duties. Cassini trained him and sent him on his way prepared to contribute data on the longitudes to the Orient. Thus the importance of remapping the world and the feasibiUty of the method devised by the Academie Royale began to dawn on the scholars of Europe, and many foreigners volunteered to contribute data. Meanwhile Colbert raised more money and Cassini sent more men into the field. One of the longest and most difficult expeditions organized by the Academie Royale was led by Messrs. Varin and des Hayes, two of His Majesty's engineers for hydrography. to the island of Goree and the West Indies. It was also one of the most important for the determination of longitudes in the Western Hemisphere, involving as it did the long jump across the Atlantic Ocean, a span where some of the most egregious errOR in longitude had been made. Cassini's original plan, approved by the king. was to launch the expedition from Ferro. on the extreme southwest of the Canary Islands, an island frequently used by cartographers as a prime meridian of longitude. But as there was some difficulty about procuring passage for the expedition, it was decided to take a departure from Gor~, a small island off Cape Verde on the west coast of Africa, where a French colony had recently been established by the Royal Company of Africa. 28 Before their departure, Varin and des Hayes spent considerable time at the Observatory, where they were thoroughly trained by Cassini and where they could make trial observations to perfect their technique. They received their final instructions in the latter part of 1681, and set out for of one of the printed versions of the map, reduced. was issued with Christian Sandler's Die Re/ormalion der Kartographie um 1700, Munich, 19O5; a second one, colored. was published in 1941 by the University of Michigan from the original in the William L. Clements Library. For bibliographical notes regarding the publication of the map. see L. A. Brown, Jean Domeniqlle Cassini . . . pp. 62-73. 28 The island of Ferro. the most southwest of the Canary Islands. was a common prime meridian among cartographers as late as 1880. It was considered the dividing line between the Eastern and Western hemispheres! (See Lippincott's A complete prollollncing galelleer 0' geographical dictlolrary 0/ the world. Philadelphia, 1883.)
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Rouen equipped with a two and a half foot quadrant, a pendulum clock, and a nineteen foot telescope. Among the smaJler pieces of apparatus they carried were a thermometer. a barometer and a compass. From Rouen they moved to Dieppe, where they were held up more than a month by storm weather and contrary winds. With time on their hands, they made a series of observations to determine the latitude and longitude of the city. The two men finally arrived at Goree in March, 1682, and there they were joined by M. de Glos, a young man trained and recommended by Cassini. De Glos brought along a six foot sextant, an eighteen foot telescope, a small zenith sector, an astronomical ring and another pendulum clock. Although the primary object of the expedition was to determine longitudes by observing the ecJipses of the satellites of IUpiter, the three men had orders to observe the variation of the compass at every point in their travels, especially during the ocean voyage, and to make thermometrical and barometric observations whenever possible; in short, they were to gather all possible scientific data that came their way. From Goree the expedition sailed for Guadeloupe and Martinique, and for the next year extensive observations were made. The three men returned to Paris in March, 1683.21 Cassini's instructions to the party were given in writing. They furnish a clear picture of the best seventeenth century research methods and at the same time explain just how terrestrial longitudes were determined by timing the eclipses of the satellites of Iupiter. The object was simple enough: to find the difference in mean or local time between a prime meridian such as Ferro or Paris and a second place such as Guadeloupe, the difference in time being equivalent to the difference in longitude. Two pendulum clocks were carried on the expedition, and before leaving they were carefully regulated at the Observatory. The pendulum of one was adjusted so that it would keep mean time, that is twenty-four hours a day. The second clock was set to keep sidereal or star time (23 b S61ll 4"') .2R The rate of going for the two timekeepers was carefu1ly tabulated over a long period so that the observers might know in advance what to expect when the temperature, let us say. went up or down ten degrees in a twenty-four hour period. These adjustments were made by raising or lowering the pendulum bob to speed up or slow down the clock. After the necessary adjustments were made, the position of the pendulum bobs on the rods was marked and the clocks were taken apart for shipping. Having arrived at the place where observations were to be made, the L. A. Brown: Jean Dome,,;qlle Cossin; . . . pp. 42-44. Cassini suggested two methods of adjusting a clock to meall time. The first was to make a series or observations of the sun (equal altitudes) and afterwards correct them with tables of the equation of time. and second, to regulate one clock to keep sidereal time by observing two successive transits of a star and correct the second dock from it. 27
III
astronomers selected a convenient, unobstructed space and set up their instruments. They fixed their pendulum bobs in position and started their clocks, setting them at the approximate hour of the day. The next operation was to establish a meridian line, running true north and south, at the place of observation. This was done in several ways, each method being used as a check against the accuracy of the others. The first was to take a series of equal altitude observations of the sun, a process which would also give a check on the accuracy of the clock which was to keep mean time. To do this, the altitude of the sun was taken with a quadrant or sextant approximately three (or four) hours before apparent noon. At the moment the sight was taken the hour, minute and second were recorded in the log. An afternoon sight was taken when the sun had descended to precisely the same angle recorded in the morning observation. Again the time was taken at the instant of observation. The difference in time between the two observations divided by two and added to the morning time gave the hour, minute and second indicated at apparent noon. This observation was repeated two days in succession, and the difference in minutes and seconds recorded by the clock on the two days (always different because of the declination of the sun) divided by two and added to the first gave the observers what the clock did in twenty-four hours; in other words, it gave them mean time. A very simple check on the arrival of apparent noon, when the sun reaches the meridian, was to drop a plumb line from the fixed quadrant and note the shadow on the ground as each observation was taken. These observations were repeated daily so that the observers always knew their local time. The second pendulum clock was much simpler to adjust. All they bad to do was set up a telescope in the plane of the meridian, sight it on a fixed star and time two successive transits of the star. When the pendulum was finally adjusted so that 23 b 56m 4' elapsed between two successive transits, the job was done. The latitude of the place of observation was equany simple to determine. The altitude of the sun at apparent noon was taken with a quadrant and the angle, referred to the tables of declination, gave the observers their latitude. A check on the latitude was made at night, by observing the height of the polestar. With the meridian line established, and a clock regulated to keep mean time, the next thing was to observe and time the eclipses of the satellites of Jupiter, at least two of which are eclipsed every two days. As Cassini pointed out, this was not always a simple matter, because not all eclipses are visible from the same place and because bad weather often vitiates the observations. Observations called for a very fussy technique. The most satisfactory time observations of Jupiter, in Cassini's opinion, could be made of the immersions and emersions of the first satellite. Six phases of the eclipse should be timed: during the immersion of the satellite
Lloyd A. B1'OWJI
'"( 1) when the satellite is at a distance from the limb of Jupiter equal to its own diameter; (2) when the satellite just touches Jupiter; (3) when it first becomes entirely hidden by Jupiter's disc. During the emersion of the satellite (4) the instant the satelUte begans to reappear; (5) when it becomes detached from Jupiter's disc; (6) when the satellite has moved away from Jupiter a distance equal to its own diameter. To observe and time these phases was a two-man job: one to observe and one to keep a record of the time in minutes and seconds. If an observer had to work alone, Cassini recommended the "eye and ear" method of timing observations, which is still good observational practice. The observer begins to count out loud "one-five-hundred, two-five-hundred, three-five-hundred" and so on, the instant the eclipse begins, and he continues to count until he can get to his clock and note the time. Then by subtracting his count from the clock reading, he has the time at which the observation was made. The emersion of the satellite, Cassini warned, always requires very careful observation, because you see nothing while you are waiting for it. At the instant you see a faint light in the region where the satellite should reappear, you should begin counting without leaving the telescope until you are sure you are seeing the actual emersion. You may make several false starts before you actually see and can time the actual emersion. Other observations worth using, according to Cassini, were the conjunctions of two satellites going in opposite directions. A conjunction was said to occur when the centers of the two satellites were in a straight perpendicular line. In all important observations requiring great accuracy, Cassini recommended a dress rehearsal the day before and at the same hour, so that if the instruments did not behave or the star was found to be in a difficult position. all necessary adjustments would have been made in advance.29 In addition to the observations for the determination of longitude, all expeditions sent out from the Paris Observatory were cautioned to note any variation in the functioning of their pendulum clocks. This did not mean normal variations caused by changes in temperature. Such variations could be predicted in advance by testing the metal pendulum rods: determining the coefficient of expansion at various temperatures. What they were watching for was a change caused by a variation in gravity. There were two reasons involved, one practical and one theoretical. The pendulum was an extremely important engine, since it was the driving force of the best clocks then in use. And too, the whole subject of gravitation, whose leading exponents were Christian Huygens, Isaac Newton and 29 Cassini's "Instructions" were printed in full in the Mlmoires de J'AcatUmie Royale des Sciences, Vol. VllI, Paris, 1730. For a translation into English see L. A. Brown, lean Domenique Cassini. . . pp. 48-60.
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Robert Hooke, was causing a stir in the scientific world. The idea of IlsinB the pendulum experimentally for studying gravitation came from Hooke, and the theories of Newton and Huygens might well be proved or disproved by a series of experiments in the field. What no one knew was that UlCSC field trials would result in the discovery that tbe earth is not 8 perfect sphere but an oblate spheroid, a sphere flattened at the poles. What effect, if any, did a change of latitude produce in the oscillations of a pendulum if the temperature remained unchanged? Many scientists said none. and experimenfs seemed to prove it. Members of the Academie bad transponed timekeepers to Copenhagen and The Hague to try them at different latitudes., and a series of experiments had been conducted in London. The results were all negative; al every place a pendulum of a given length (39.1 inches) beat seconds or made 3600 oscillations an bour. However. there was one exception. In 1672, Jean Richer had made an expednion to Cayenne (4" 56' 5" N.) to observe the opposition of Mars. On the whole the expedilion was a success, but Rkher had had trouble with his timekeeper. Although the ~nglh of the pendulum had been carefully adjusted at the Observatory before he sailed, Richer found that in Cayenne his clock lost about two minutes and a half a day, and that in order to get it to keep mean time he had to shorten the pendulum (raise the bob) by more Ihan a "lilM" (about Yl2 of an incb). All this was very trying to Cassini, who was a meliculous observer. Mit is suspected," he wrote, "thai this resulted from some error in the observation."
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Had he not been a gentleman as well as a scholar, he would have said that Richer was just plain care1ess. 80 The following year, 1673, Huygens published his masterpiece on the oscillation of the pendu1um, in which he set down for the first time a sound theory on the subject of centrifugal force, principles which Newton later applied to his theoretical investigation of the earth.St The first opportunity to confirm the fallacy of Richer's observations on the behavior of his timekeeper came when Varin and des Hayes sailed for Martinique (14° 48' N.) and Guadeloupe (between 15° 47' and 16° 30' N.). Cassini cautioned them to check their pendulums with the greatest possible care and they did. But, unfortunately, their clocks behaved badly, and they, too, had to shorten the pendulums in order to make them beat mean time. Cassini was still dubious, but not Isaac Newton. In the third book of his Principia he concluded that this variation of the pendulum in the vicinity of the equator must be caused either by a diminution of gravity resu1ting from a bUlging of the earth at the equator, or from the strong, counteracting effect of centrifugal force in that region. S2 The discoveries made by the Academie Royale des Sciences set a fast pace in the scientific world and pointed the way towards many others. The method of finding longitude by means of the eclipses of Jupiter's satellites had proved to be feasible and accurate, but it was not accepted by foreign countries without a struggle. Tables of Jupiter's satellites were finally included in the English Nautical Almanac and remained there in good standing for many years, along with tables of lunar distances and other star data associated with rival methods of finding longitude. It was generally conceded, however, that Jupiter could not be used for finding longitude at sea, in spite of Galileo's assertions to the contrary. Many inventors besides the great ltaJian had come up with ingenious and wholly impractical devices to provide a steady platform on shipboard from which astronomical observations could be made. But the fact remained that the sea was too boisterous and unpredictable for astronomers and their apparatus. England made her official entry in the race for the longitude when Charles II ordered the construction of a Royal Observatory, for the advancement of navigation and nautical astronomy, in Greenwich Park, overlooking the Thames and the plain of Essex.s3 In England things moved slowly at first, but they moved. The king was determined to have the tables of the heavenly bodies corrected for the use of his seamen and L. A. Brown: Jean Domeniqlle Cassin; . . . p. 57. Christian Huygens: Horologivm oscillaIOril'm; sil'e de molll pendu/orvm ad hor%gia aptalo demonslrat;ones geomelr;cae, Paris, 1673. 32 Isaac Newton: Philosoph;ae Naturalis Principia Mathemal;ca, 3 Yols., London, 1687. 83 R. T. Gou]d, op. cit., p. 9; a]so Henry S. Richardson's Greenwich: its history, antiquities, improvements and public buildings, London, 1834. 30 31
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so appointed John FJamsteed "astronomical observator" by a royal warrant dated March 4, 1675. at the handsome salary of £100 a year, out of which he paid £10 in taxes. He had to provide his own instruments, and as an additional check to any delusions of grandeur he might have, he was ordered to give instruction to two boys from Christ's Hospital. Stark necessity made him take several private pupils as well. Dogged by ill health and the irritations common to the life of a public servant, Flamsteed was nevertheless buoyed up by the society of Newton, Halley, Hooke and the scientists of the Academie Royale, with whom he corresponded. A perfectionist of the first magnitude, Flamsteed was doomed to a life of unhappiness by his unwillingness to publish his findings before he had had a chance to check them for accuracy. To Flamsteed, no demand was sufficiently urgent to justify such scientific transgression. Aamsteed worked under constant pressure. Everybody, it seemed, wanted data of one sort of another, and wanted it in a hurry. Newton needed full information on "places of the moon" in order to perfect his lunar theory. British scientists, as a group, had set aside the French method of finding longitude and all other methods requiring the use of sustained observations at sea. They were approaching the problem from another angle. and demanded complete tables of lunar distances and a complete catalogue of star places. Flamsteed did as he was told, and for fifteen years (1689-1704) spent most of his time at the pedestrian task of compiling the first Greenwich star catalogue and tables of the moon, meanwhile reluctantly doling out to his impatient peers small doses of what he considered incomplete if not inaccurate data.84 The loudest clamors for information came from the Admiralty and from the waterfront. In 1689 war broke out with France. In 1690 (June 30) the English fleet was defeated by the French at the battle of Beachy Head. Lord Torrington, the English admiral. was tried by court-martial and acquitted, but nevertheless dismissed from the service. In 1691 several ships of war were lost off Plymouth because the navigators mistook the Deadman for Berry Head. In 1707 Sir Cloudesley Shovel, returning with his fleet from Gibraltar, ran into dirty weather. After twelve days of groping in a heavy overcast, all hands were in dOUbt as to the fleet's position. The Admiral called for the opinion of his navigators, and with one exception they agreed that the fleet was well to the west of Ushant, off the Brittany peninsula. The fleet stood on, but that night, in a heavy fog, they ran into the Scilly Islands off the southwest coast of England. Four ships and two thousand men were lost, including the Admiral. There was a story current, long after, that a seaman on the fiagship had estimated from his own dead reckoning that the fleet was in a dangerous position. He had the temerity to point this out to his superiors, who 8.
See Francis Baily's Account 0/ the Rev. Joh" Flam steed, London. 1835.
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sentenced him forthwith to be hanged at the yardann for mutiny. The longitude had to be found! 85 There was never a shortage of inventive genius in England. and many fertile minds were directed towards the problem of finding longitude at sea. In 1687 two proposals were made by an unknown inventor which were novel, to say the least. He had discovered that a glass filled to the brim with water would run over at the instant of new and full moon, so that the longitude could be detennined with precision at least twice a month. His second method was far superior to the first, he thought, and involved the use of a popular nostrum concocted by Sir Kenelm Digby caned the "powder of sympathy!' This miraculous healer cured open wounds of all kinds, but unlike ordinary and inferior brands of medicine, the powder of sympathy was applied. not to the wound but to the weapon that inflicted it. Digby used to describe how he made one of his patients jump sympathetically merely by putting a dressing he had taken from the patient's wound into a basin containing some of his curative powder. The inventor who suggested using Digby's powder as an aid to navigation proposed that before sailing every ship should be furnished with a wounded dog. A reliable observer on shore. equipped with a slandard clock and a bandage from the dog's wound, would do the rest. Every hour, on the dot, he would immerse the dog's bandage in a solution of the powder of sympathy and the dog on shipboard would yelp the hour. 8ft Another serious proposal was made in 1714 by Wi1liam Whiston, a clergyman, and Humphrey Ditton, a mathematician. These men suggested that a number of lightships be anchored in the principal shipping lanes at regular intervals across the Atlantic ocean. The lightships would fire at regular intervals a star shell timed to explode at 6440 feet. Sea captains could easily calculate their distance from the nearest lightship merely by timing the interval between the flash and the report. This system would be especially convenient in the North Atlantic, they pointed out, where the depth never exceeded 300 fathoms! For obvious reasons, the proposal of Whiston and Ditton was not carried out, but they starled something. Their plan was published, and thanks to the publicity it received in various periodicals. a petition was submitted to Parliament on March 25, 1714. by "several Captains of Her Majesty's Ships, Merchants of London, and Commanders of Merchantmen," setting forth the great importance of finding the longitude and praying that a public reward be offered for some practicable method of doing it. 37 Not only the petition but the R. T. Gould. op. cit., p. 2. See Curious Enquiries, London. 1687; R. T. Gould. op. cit., p. 11. The title of Sir Kenelm Digby's famous work. which appeared in French as well as English, was A lale discourse . . . tOllching the ClIrf! 0/ wounds by the powder 01 sympath)': with inSlruclions how to make the said powder . . . Second edition. augmented, London, 1658. 37 See Whiston and Ditton's A new method lor disco,'eriug the 10l1gitude. London. 35
36
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proposaJ of Whiston and Ditton were referred to a committee, who in turn consulted a number of eminent scientists induding Newton and Haney. That same year Newton prepared a statement which he read to the committee. He said, "That, for determining the Longitude at Sea, there have been several Projects, true in the Theory. but difficult to execute." Newton did not favor the use of the eclipses of the satellites of Jupiter, and as for the scheme proposed by Whiston and Ditton, he pointed out that it was rather a method of "keeping an Account of the Longitude at Sea, than for finding it, if at any time it should be lost." Among the methods that are difficult to execute, he went on. "One is, by a Watch to keep time exactly: But, by reason of the Motion of a Ship, the Variation of Heat and Cold. Wet and Dry, and the Difference of Gravity in Different Latitudes, such a Watch hath not yet been made!' That was the trouble: such a watch had not been made. as The idea of transporting a timekeeper for the purpose of finding longitude was not new, and the futility of the scheme was just as 01d. To the ancients it was just a dream. When Gemma Frisius suggested it in 1530 there were mechanical clocks, but they were a fairly new invention, and crudely built, which made the idea improbable if not impossible.s9 The idea of transporting "some true Horologie or Watch. apt to be carried in journeying, which by an Astrolabe is to be rectified . . . was again stated by Blundevi11e in t 622, but still there was no watch which was Utrue" in the sense of being accurate enougb to use for determining longitude.4.0 If a timekeeper was the answer, it would have to be very accurate indeed. According to Picard's value, a degree of longitude was equal to about sixty-eight miles at the equator, or four minutes, by the dock. One minute of time meant seventeen miles-towards or away from danger. And jf on a six weeks' voyage a navigator wanted to get his 10ngitude within half a degree (tbirty-four miles) the rate of his timekeeper must not gain or lose more than two minutes in forty-two days, or three seconds a day. Fortified by these calculations, which spelled the impossible, and the report of the committee, Par1iament passed a bi1l (t 7 t 4) ufor providing a publick reward for such person or persons as shan discover the Longitude!' It was the largest reward ever offered. and stated that for any practical invention the following sum would be paid: 41 £10,000 for any device that would determine the Jongitude within 1 degree. It
1714. The petition appeared in various periodicals: The Guardian, luly 14; The Entrlishman. Dec. 19, 1713 (R. T. Gould, op. cit., p. 13). 38 R. T. GouJd, op. cit., p. 13. 38 Gemma Frisius: De principiis astronomiae e/ cosmographiae, Antwerp, 1530. 40 Thomas Blundevil1e. M. BlundeviUe his exercises . . .• Sixth Edition, London, 1622. p. 390. . oil 12 Anne, Cap. 15; R. T. Goold, op. cit., p. 13.
£IS,OOO for any device that would detennine the longitude within 40 minutes. £20,000 for any device that would determine the longitude within 30 minutes (2 minutes of time or 34 miles). As though aware of the absurdity of their terms, Parliament authorized the formation of a permanent commission-the Board of Longitude-and empowered it to pay one half of any of the above rewards as soon as a majority of its members were satisfied that any proposed method was practicable and useful, and that it would give security to ships within eighty miles of danger, meaning land. The other half of any reward would be paid as soon as a ship using the device should sail from Britain to a port in the West Indies without erring in her longitude more than the amounts specified. Moreover, the Board was authorized to grant a smaller reward for a less accurate method, provided it was practicable, and to spend a sum not to exceed £2000 on experiments which might lead to a useful invention. For fifty years this handsome reward stood untouched t a prize for the impossible, the butt of English humorists and satirists. Magazines and newspapers used it as a stock cliche. The Board of Longitude failed to see the joke. Day in and day out they were hounded by fools and charlatans, the perpetual motion lads and the geniuses who could quarter a circle and trisect an angle. To handle the flood of crackpots, they employed a secretary who handed out stereotyped replies to stereotyped proposals. The members of the Board met three times a year at the Admirality, contributing their services and their time to the Crown. They took their responsibilities seriously and frequently called in consultants to help them appraise a promising invention. They were generous with grants-in-aid to struggling inventors with sound ideas, but what they demanded was resultS.42 Neither the Board nor anyone else knew exactly what they were looking for, but what everyone knew was that the longitude problem had stopped the best minds in Europe, including Newton, Halley, Huygens, von Leibnitz and all the rest. It was solved, finally, by a ticking machine in a box, the invention of an uneducated Yorkshire carpenter named John Harrison. The device was the marine chronometer. Early clocks fell into two general classes: nonportable timekeepers driven by a falling weight, and portable timekeepers such as table clocks and crude watches, driven by a coiled spring. Gemma Frisius suggested the latter for use at sea, but with reservations. Knowing the unreliable temperament of spring-driven timekeepers, he admitted that sand and water clocks would have to be carried along to check the error of a 411 R. T. Gould, op. cit., p. 16. According to the Act of 1712 the Board was comprised of: "The Lord High Admiral or the First Lord of the Admiralty; The Speaker of the House of Commons; The First Commissioner of the Navy; The First Commissioner of Trade; The Admirals of the Red. White and Blue Squadrons; The Master of the Trinity House; The President of the Royal Society; The AstronomerRoyal; The Savilian, Lucasian, and Plumian Professors of Mathematics."
spring-driven machine. In Spain, during the reign of Philip II, clocks were solicited which would run exactly twenty-four hours a day, and many different kinds had been invented. According to Alonso de Santa Cruz there were "some with wheels, chains and weights of steel: some with chains of catgut and steel: others using sand, as in sandglasses: others with water in place of sand, and designed after many different fashions: others again with vases or large glasses filled with quicksilver: and, lastly, some, the most ingenious of aU, driven by the force of the wind, which moves a weight and thereby the chain of the clock, or which are moved by the flame of a wick saturated with oil: and all of them adjusted to measure twenty-four hours exactly." fa Robert Hooke became interested in the development of portable timekeepers for use at sea about the time Huygens perfected the pendulum clock. One of the most versatile scientists and inventors of all time, Hooke was one of those rare mechanical geniuses who was equally clever with a pen. After studying the faults of current timekeepers and the possibility of building a more accurate one, he slyly wrote a summary of his investigations, intimating that he was completely baffled and discouraged. '~Al1 I could obtain:' he said, '''was a Catalogue of Difficulties, first in the doing of it, secondly in the bringing of it into publick use, thirdly, in making advantage of it. Difficulties were proposed from the alteration of Climates, Airs. heats and colds, temperature of Springs, the nature of Vibrations, the wearing of Materials, the motion of the Ship, and divers others." Even if a reliable timekeeper were possible, he concluded, "it would be difticult to bring it to use, for Sea..men know their way already to any Port. . . ." As for the rewards: "the Praemium for the Longitude," there never was any such thing, he retorted scornfully. UNo King or State would pay a farthing for it.1t In spite of his pretended despondency, Hooke nevertheless lectured in 1664 on the subject of applying springs to the balance of a watch in order to render its vibrations more uniform, and demonstrated, with models, twenty different ways of doing it. At the same time he confessed that he had one or two other methods up his sleeve which he hoped to cash in on at some future date. Like many scientists of the time, Hooke expressed the principle of his balance spring in a Latin anagram; roughly: Ut tensio, sic vis. '~as the tension is, so is the force," or, "the force exerted by a spring is direct1y proportional to the extent to which it is tensioned!' 44 The first timekeeper designed specificany for use at sea was made by Christian Huygens in 1660. The escapement was controlled by a pendu43 Ibid., p. 20; this translation is from a paraphrase by Duro in his Disquisicionel Naullcas. 44 Ibid., p. 25. 'The anagram was a device commonly used in the best scientific circles of the time to establish priority of invention or discovery without actually disclosing anything that might be seized upon by a zealous coUeague.
£10)1(1 A. Brotm
lum instead of a spring balance, and like many of the clocks that followed, it proved useless except in a flat calm. Its rate was unpredictable; when tossed around by the sea it either ran in jerks or stopped altogether. The length of the pendulum varied with changes of temperature, and the rate of going changed in different latitudes, for some mysterious reason not yet determined. But by 1715 every physical principal and mechanical part that would have to be incorporated in an accurate timekeeper was understood by watchmakers. An that remained was to bridge the gap between a good clock and one that was nearly perfect. It was that half degree of longitude, that two minutes of time, which meant the difference between conquest and failure, the difference between £20,000 and just another timekeeper.~G
One of the biggest hurdles between watchmakers and the prize money was the weather: temperature and humidity. A few men included barometric pressure. Without a doubt, changes in the weather did things to clocks and watches, and many suggestions were forthcoming as to how this principal source of trouble could be overcome. Stephen Plank and William Palmer, watchmakers, proposed keeping a timekeeper close to a fire, thus obviating errors due to change in temperature. Plank suggested keeping a watch in a brass box over a stove which would always be hot. He claimed to have a secret process for keeping the temperature of the fire uniform, Jeremy Thacker, inventor and watchmaker, published a book on the subject of the longitude, in which he made some caustic remarks about the efforts of his contemporaries.~8 He suggested that one of his colleagues, who wanted to test his clock at sea, should first arrange to have two consecutive Junes equally hot at every hour of every day. Another colleague, referred to as Mr. Br . . . e, was dubbed the Corrector of the Moon's Motion. In a more serious vein, Thacker made several sage observations regarding the physical laws with which watchmakers were struggling. He verified experimentally that a coiled spring loses strength when heated and gains it when cooled. He kept his own clock under a kind of bell jar connected with an exhaust pump, so that it could be run in a partial vacuum. He also devised an auxiliary spring which kept the clock going while the mainspring was being wound. Both springs were wound outside the ben by means of rods passed through stuffing boxes, so that neither the vacuum nor the clock mechanism 4'1lbid.• pp. 27-30; Huygens described his pendulum clock in his Horologlum Oscillatorium. Paris, 1673. 46 Ibid .• pp. 32, 33; Jeremy Thacker wrote a clever piece entitled: The LOlfgitlldes Eramined. beginlfing with Q short epistle 10 the Lom:itudinarians and endinR with the description 01 a smart, pretty Machine 01 my Own which I am (almost) sllre will do lor the LonRitllde and procure me The Twenty Thousand Pounds. By Jeremy Thacker. of Beverly in Yorkshire. ", .. qftid nOli morlalia pectora cog;s Allr; sacra Fames • •." London. Printed for J. Roberts at the Oxford Arms in Warwick Lane. 1714. Price Sixpence.
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would have to be disturbed. In spite of these and other devices, watchmakers remained in the dark and their problems remained unsolved until John Harrison went to work on the physical laws behind them. After that they did not seem so difficult. 47 Harrison was born at Foulby in the parish of Wragby, Yorkshire, in May, 1693. He was the son of a carpenter and joiner in the service of Sir Rowland Winn of Nostell Priory. John was the oldest son in a large family. When he was six years old he contracted smallpox, and while convalescing spent hours watching the mechanism and listening to the ticking of a watch laid on his pillow. When his fami1y moved to Barrow in Lincolnshire, John was seven years old. There he learned his father's trade and worked with him for several years. Occasionally he earned a Httle extra by surveying and measuring land, but he was much more interested in mechanics, and spent his evenings studying Nicholas Saunderson's published lectures on mathematics and physics. These he copied out in longhand including all the diagrams. He also studied the mechanism of clocks and watches. how to repair them and how they might be improved. In 171 S, when he was twenty-two, he built his first grandfather clock or "regu1ator." The only remarkable feature of the machine was that all the wheels except the escape wheel were made of oak, with the teeth, carved separately, set into a groove in the rim.48 Many of the mechanical faults in the clocks and watches that Harrison saw around him were caused by the expansion and contraction of the metals used in their construction. Pendulums, for example. were usually made of an iron or steel rod with a lead bob fastened at the end. In winter the rod contracted and the clock went fast, and in summer the rod expanded, making the clock lose time. Harrison made his first important contribution to clockmaking by developing the "gridiron" pendulum, so named because of its appearance. Brass and steel, he knew, expand for a given increase in temperature in the ratio of about three to two (100 to 47 The invention of a "maintaining power" is erroneously attributed to Harrison. See R. T. Gould, op. cil., p. 34, who says that Thacker antedates Harrison by twenty years on the invention of an auxiliary spring to keep a machine going white it was being wound. In spite of the magnitude of John Harrison's achievement, the inventive genius of Pierre Le Roy of Paris produced the prototype of the modern chronometer. As R.upert Gould points out, Harrison's Number Four was a remarkable piece of mechanism, "a satisfactory marine timekeeper. one, too, which was of permanent usefullness, and which could be duplicated as often as necessary. But No.4. in spite of its fine performance and beautiful mechanism, cannot be compared, for efficiency and design, with Le Roy's wonderful machine. The Frenchman. who was but little indebted to his precessors, and not at an to his contemporaries, evolved, by sheer force of genius, a timekeeper which contains all the essential mechanism of the modern chronometer!' (See R. T. Gould, op. cit., p. 65.) 48 For a biographical sketch of Harrison see R. T. Gould, op. cit., pp. 40 If. Saunderson (1682-1739) was Lucasian professor of mathematics at Cambridge. Harrison's first "regulator" is now in the museum of the C]ockmakers' Co. of London. "The term 'regulator' is used to denote any high-class pendu]um clock designed for use solely as an accurate time-measurer. without any additions such as striking mechanism, calendar work, &c." (Gould. p. 42 n.)
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62}. He therefore built a pendulum with nine alternating steel and brass rods, so pinned together that expansion or contraction caused by variation in the temperature was eliminated, the unlike rods counteracting each other.49 The accuracy of a clock is no greater than the efficiency of its escapement_ the piece which releases for a second, more or less, the driving power, such as a suspended weight or a coiled mainspring. One day Harrison was ca1led out to repair a steeple clock that refused to run. After looking it over he discovered that all it needed was some oil on the pallets of the escapement. He oiled the mechanism and soon after went to work on a design for an escapement that would not need oiling. The result was an ingenious "grasshopper" escapement that was very nearly frictionless and also noiseless. However, it was extremely delicate_ unnecessarily so, and was easily upset by dust or unnecessary oil. These two improved parts alone were almost enough to revolutionize the clockmaking industry. One of the first two grandfather clocks he built that were equipped with his improved pendulum and grasshopper escapement did not gain or lose more than a second a month during a period of fourteen years. Harrison was twenty-one years old when Parliament posted the £20,000 reward for a reliable method of determining longitude at sea. He had not finished his first clock, and it is doubtful whether he seriously aspired to winning such a fortune, but certainly no young inventor ever had such a fabulous goal to shoot at, or such limited competition. Yet Harrison never hurried his work, even after it must have been apparent to him that the prize was almost within his reach. On the contrary, his real goal was the perfection of his marine timekeeper as a precision instrument and a thing of beauty. The monetary reward, therefore, was a foregone conclusion. His first two fine grandfather clocks were completed by 1726, when he was thirty-three years old, and in 1728 he went to London, carrying with him fun-scale models of his gridiron pendulum and grasshopper escapement, and working drawings of a marine clock he hoped to build if he could get some financial assistance from the Board of Longitude. He called on Edmund Haney, Astronomer Royal, who was also a member of the Board. Haney advised him not to depend on the Board of Longitude, but to talk things over with George Graham, Eng1and's leading horologist. GO Harrison called on Graham at ten o'clock one morning, and together they talked pendulums, escapements, remontoires and springs until eight o'clock in the evening, when Harrison departed a happy man. Graham had advised him to build his clock first and then apply to the 49Ibid., pp. 40-41. Graham had experimented with a gridiron pendulum and in 1725 had produced a pendulum with a small jar of mercury attached to the bob which was supposed to counteract the expansion of the rod caused by a rise in temperature. 1!o0 Ibid .• pp. 42, 43; Graham and Tompion are tbe only two horologists buried in Westminster Abbey.
Board of Longitude. He had also offered to loan Harrison the money to build it with, and would not listen to any talk about interest or security of any kind. Harrison went home to Barrow and spent the next seven years building his first marine timekeeper, his "Number One," as it was later called. In addition to heat and cold, the archenemies of all watchmakers, he concentrated on eliminating friction, or cutting it down to a bare minimum, on every moving part, and devised many ingenious ways of doing it; some of them radical departures from accepted watchmaking practice. Instead of using a pendulum, which would be impractical at sea, Harrison designed two huge balances weighing about five pounds each. that were connected by wires running over brass arcs so that their motions were always opposed. Thus any effect on one produced by the motion of the ship would be counteracted by the other. The "grasshopper" escapement was modified and simplified and two mainsprings on separate drums were instaUed. The clock was finished in 1735. There was nothing beautiful or graceful about Harrison's Number One. It weighed seventy-two pounds and looked like nothing but an awkward, UDwieldly piece of machinery. However, everyone who saw it and studied its mechanism declared it a masterpiece of ingenuity, and its perfonnance certainly belied its appearance. Harrison mounted its case in gimbals and for a while tested it unofficially on a barge in the Humber River. Then he took it to London where he enjoyed his first brief triumph. Five members of the Royal Society examined the clock, studied its mechanism and then presented Harrison with a certificate stating that the principles of this timekeeper promised a sufficient degree of accuracy to meet the requirements set forth in the Act of Queen Anne. This historic document, which opened for Harrison the door to the Board of Longitude, was signed by Haney, Smith. Bradley, Machin and Graham. On the strength of the certificate, Harrison applied to the Board of Longitude for a trial at sea, and in 1736 he was sent to Lisbon in H.M.S. Centurion, Captain Proctor. In his possession was a note from Sir Charles Wager. First Lord of the Admiralty, asking Proctor to see that every courtesy be given the bearer, who was said by those who knew him best to be "a very ingenious and sober man." Harrison was given the run of the ship, and his timekeeper was placed in the Captain's cabin where he could make observations and wind his clock without interruption. Proctor was courteous but skeptical. 'The difficulty of measuring Time truly," he wrote, "where so many unequal Shocks and Motions stand in Opposition to it, gives me concern for the honest Man, and makes me feel he has attempted Impossibilities." 11 No record of the clock's going on the outward voyage is known, but 51
Ibid .• pp. 45-46.
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after the return trip, made in H.M.S. Orford, Robert Man, Harrison was given a certificate signed by the master (that is, navigator) stating: "When we made the land, the said land, according to my reckoning (and others), ought to have been the Start; but before we knew what land it was, John Harrison declared to me and the rest of the ship's company, that according to his observations with his machine, it ought to be the Lizard-the which, indeed, it was found to be, his observation showing the ship to be more west than my reckoning, above one degree and twenty-six miles." It was an impressive report in spite of its simplicity, and yet the voyage to Lisbon and return was made in practically a north and south direction; one that would hardly demonstrate the best qualities of the clock in the most dramatic fashion. It should be noted, however. that even on this well-worn trade route it was not considered a scandal that the ship's navigator should make an error of 90 miles in his landfall. On June 30, 1737, Harrison made his first bow to the mighty Board of Longitude. According to the official minutes, "Mr. John Harrison produced a new invented machine, in the nature of clockwork, whereby he proposes to keep time at sea with more exactness than by any other instrument or method hitherto contrived . . . and proposes to make another machine of smal1er dimensions within the space of two years, whereby he will endeavour to correct some defects which he hath found in that already prepared, so as to render the same more perfect . . ." The Board voted him £500 to help defray expenses, one half to be paid at once and the other hal f when he completed the second clock and delivered same into the hands of one of His Majesty's ship's captains. 52 Harrison's Number Two contained several minor mechanical improvements and this time all the wheels were made of brass instead of wood. In some respects it was even more cumbersome than Number One, and it weighed one hundred and three pounds. Its case and gimbal suspension weighed another sixty-two pounds. Number Two was finished in 1739, but instead of turning it over to a sea captain appointed by the Board to receive it, Harrison tested it for nearly two years under conditions of "great heat and motion." Number Two was never sent to sea because by the time it was ready, England was at war with Spain and the Admiralty had no desire to give the Spaniards an opportunity to capture it. In Janu,ary, 1741, Harrison wrote the Board that he had begun work on a third clock which promised to be far superior to the first two. They voted him another £500. Harrison struggled with it for several months, but seems to have miscalculated the "moment of inertia" of its balances. He thought he could get it going by the first of August, 1741, and have it ready for a sea trial two years later. But after five years the Board learned "that it does not go wel1, at present, as he expected it would, yet he plainly ~2
Ibid., p. 47.
811
perceived the Cause of its present Imperfection to lye in a certain part [the balances1 which. being of a different form from the corresponding part in the other machines, had never been tried before." Harrison had made a few improvements in the parts of Number Three and had incorporated in it the same anti friction devices he had used on Number Two, but the clock was still bulky and its parts were far from delicate; the machine weighed sixty-six pounds and its case and gimbals another thirtyfive." Harrison was again feeling the pinch, even though the Board had given him several advances to keep him going, for in 1746, when he reported on Number Three, he laid before the Board an impressive testimonial signed by twelve members of the Royal Society inc1uding the President. Martin Folkes, Bradley, Graham, Haney and Cavendish, attesting the importance and practical value of his inventions in the solution of the longitude problem. Presumably this gesture was made to insure the financial support of the Board of Longitude. However, the Board needed no prodding. Three years later, acting on its own volition, the Royal Society awarded Harrison the Copley medal, the highest honor it could bestow. His modesty, perseverance and skill made them forget, at least for a time. the total lack of academic background which was so highly revered by that august body.M Convinced that Number Three wou1d never satisfy him, Harrison proposed to start work on two more timekeepers, even before Number Three was given a trial at sea. One would be pocketsize and the other slightly larger. The Board approved the project and Harrison went ahead. Abandoning the idea of a pocketsize chronometer, Harrison decided to concentrate his efforts on a s1ightly larger clock. which could be adapted to the intricate mechanism he had designed without sacrificing accuracy. In 1757 he began work on Number Four, a machine which Uby reason alike of its beauty, its accuracy, and its historical interest, must take pride of place as the most famous chronometer that ever has been or ever will be made." It was finished in 1759.155 Number Four resembled an enormous "pair-case" watch about five inches in diameter, complete with pendant, as though it were to be worn. The dial was white enamel with an ornamental design in black. The hour and minute hands were of blued steel and the second hand was polished. Instead of a gimbal suspension, which Harrison had come to distrust, he used only a soft cushion in a plain box to support the clock. An adjustable 53 Ibid., pp. 47-49. for details of the technical improvements made in No.2 and No.3. M Ibid., p. 49. Some years later he was oft'ered the honor of Fellow of the Royal Society, but he declined it in favor of his son William. S~ Ibid., pp. 50, 53; for a full description of No. 4 see H. M. Frodsham in the Hor%gical Jo"rnal for May. 1878-with drawings of the escapement. train and remontoire taken from the duplicate of No. 4 made by Larcum Kendall.
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outer box was fitted with a divided arc so that the timekeeper could be kept in the same position (with the pendant always slightly above the horizontal) regardless of the lie of the ship. When it was finished, Number Four was not adjusted for more than this one position, and on its first voyage it had to be carefully tended. The watch beat five to the second and ran for thirty hours without rewinding. The pivot holes were jeweled to the third wheel with rubies and the end stones were diamonds. Engraved in the top-plate were the words "lohn Harrison & Son, A.D. 1759." Cunningly concealed from prying eyes beneath the plate was a mechanism such as the world had never seen; every pinion and bearing, each spring and wheel was the end product of careful planning, precise measurement and exquisite craftsmanship. Into the mechanism had gone '1ifty years of self-denial, unremitting toil, and ceaseless concentration." To Harrison, whose singleness of purpose had made it possible for him to achieve the impossible, Number Four was a satisfactory climax to a lifetime of effort. He was proud of this timekeeper, and in a rare burst of eloquence he wrote, "I think I may make bold to say, that there is neither any other Mechanical or Mathematical thing in the World that is more beautiful or curious in texture than this my watch or Time-keeper for the Longitude . . . and I heartily thank Almighty God that I have liyed so long, as in some measure to complete it." 158 After checking and adjusting Number Four with his pendulum clock for nearly two years, Harrison reported to the Board of Longitude, in March 1761, that Number Four was as good as Number Three and that its performance greatly exceeded his expectations. He asked for a trial at sea. His request was granted, and in April, 1761, William Harrison, his son and right-hand man, took Number Three to Portsmouth. The father arrived a short time later with Number Four. There were numerous delays at Portsmouth, and it was October before passage was finally arranged for young Harrison aboard H.M.S. Deptford, Dudley Digges, bound for Jamaica. lohn Harrison, who was then sixty-eight years old, decided not to attempt the long sea voyage himself; and he also decided to stake everything on the performance of Number Four, instead of sending both Three and Four along. The Deptford finally sailed from Spithead with a convoy, November 18, 1761, after first touching at Portland and Plymouth. The sea trial was on. Number Four had been placed in a case with four locks. and the four keys were given to William Harrison, Governor Lyttleton of Jamaica, who was taking passage on the Deptford, Captain Digges, and his first lieutenant. All four had to be present in order to open the case, even for winding. The Board of Longitude had further arranged to have the longitude of Jamaica determined de novo before the trial, by a series of obser'I
Ibid., p. 63.
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vations of the sateUites of Jupiter, but because of the lateness of the season it was decided to accept the best previously established reckoning. Local time at Portsmouth and at Jamaica was to be determined by taking equal altitudes of the sun, and the difference compared with the time indicated by Harrison's timekeeper. As usual, the first scheduled port of call on the run to Jamaica was Madeira. On this particular voyage, all hands aboard the Deptford were anxious to make the island on the first approach. To Wi11iam Harrison it meant the first crucial test of Number Four; to Captain Digges it meant a test of his dead reckoning against a mechanical device in which he had no confidence; but the ship's company had more than a scientific interest in the proceedings. They were afraid of missing Madeira altogether, "the consequence of which, would have been Inconvenient." To the horror of al1 hands, it was found that the beer had spoiled, over a thousand gal10ns of it, and the people had already been reduced to drink· ing water. Nine days out from Plymouth the ship's longitude. by dead reckoning, was 13 0 50' west of Greenwich, but according to Number Four and WiUiam Harrison it was 1So 19' W. Captain Digges naturally favored his dead reckoning calculations, but Harrison stoutly maintained that Number Four was right and that if Madeira were properly marked on the chart they would sight it the next day. Although Digges offered to bet Harrison five to one that he was wrong, he held his course, and the follow· ing morning at 6 A.M. the lookout sighted Porto Santo, the northeastern island of the Madeira group, dead ahead. The Deptford'9 officers were greatly impressed by Harrison's uncanny predictions throughout the voyage. They were even more impressed when they arrived at Jamaica three days before H.M.S. Beaver, which had sailed for Jamaica ten days before them. Number Four was promptly taken ashore and checked. After aUowing for its rate of going (2~ seconds per day losing at Portsmouth), it was found to be 5 seconds slow t an error in longitude of 1lA' only, or 1% nautical miles. IST The official trial ended at Jamaica. Arrangements were made for William Harrison to make the return voyage in the Merlin, stoop, and in a burst of enthusiasm Captain Digges placed his order for the first Harrisonbuilt chronometer which should be offered for sale. The passage back to Eng1and was a severe test for Number Four. The weather was extremely rough and the timekeeper, still carefully tended by Harrison, had to be moved to the poop, the only dry place on the ship, where it was pounded unmercifully aad ureceived a number of violent shocks." However, when it was again checked at Portsmouth. its total error for the five months' voyage, through heat and cold, fair weather and foul (after allowing for its rate of going), was only 1m 531,4,11, or an error in longitude of 281,4,' 3'7
Ibid., pp. 55-56 and 56 n.
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(28% nautical miles). This was safely within the. limit of half a degree specified in the Act of Queen Anne. John Harrison and son had won the fabulous reward of £20,000. The sea trial had ended, but the trials of John Harrison had just begun. Now for the first time, at the age of sixty-nine, Harrison began to feel the lack of an academic background. He was a simple man; he did not know the language of diplomacy, the gentle art of innuendo and evasion. He had mastered the longitude but he did not know how to cope with the Royal Society or the Board of Longitude. He had won the reward and all he wanted now was his money. The money was not immediately forthcoming. Neither the Board of Longitude nor the scientists who served it as consultants were at any time gUilty of dishonesty in their dealings with Harrison; they were only human. £20,000 was a tremendous fortune, and it was one thing to dole out living expenses to a watchmaker in amounts not exceeding £500 so that he might contribute something or other to the general cause. But it was another thing to hand over £20,000 in a lump sum to one man,- and a man of humble birth at that. It was most extraordinary. Moreover, there were men on the Board and members of the Royal Society who had designs on the reward themselves or at least a cut of it. James Bradley and Johann Tobias Mayer had both worked long and hard on the compilation of accurate lunar tables. Mayer's widow was paid £3000 for his contribution to the cause of longitude, and in 1761 Bradley told Harrison that he and Mayer would have shared £10,000 of the prize money between them if it had not been for his blasted watch. Halley had struggled long and manfully on the solution of the longitude by compass variation, and was not in a position to ignore any part of £20,000. The Reverend Nevil Maskelyne, Astronomer Royal, and compiler of the Nautical Almanac. was an obstinate and uncompromising apostle of "lunar distances" or "lunars" for finding the longitude, and had closed his mind to any other method whatsoever. He loved neither Harrison nor his watch. In view of these and other unnamed aspirants, it was inevitable that the Board should decide that the amazing performance of Harrison's timekeeper was a fiuke. They had never been allowed to examine the mechanism, and they pointed out that if a gross of watches were carried to Jamaica under the same conditions. one out of the lot might perform equally wen-at least for one trip. They accordingly refused to give Harrison a certificate stating that he had met the requirements of the Act until his timekeeper was given a further trial, or trials. Meanwhile, they did agree to give him the sum of £2500 as an interim reward, since his machine had proved to be a rather useful contraption, though mysterious beyond words. An Act of Par1iament (February. 1763) enabling him to receive £5000 as soon as he disclosed the secret of his invention,
liS
was completely nullified by the absurdly rigid conditions set up by the Board. He was finally granted a new trial at sea. 1I8 The rules laid down for the new trial were elaborate and exacting. The difference in longitude between Portsmouth and Jamaica was to be determined de novo by observations of Jupiter's satellites. Number Four was to be rated at Greenwich before sailing, but Harrison balked, saying "that he did not chuse to part with it out of his hands till be shall have reaped some advantage from it." However, he agreed to send his own rating, sealed, to the Secretary of the Admiralty before tbe trial began. After endless delays the trial was arranged to take place between Portsmouth and Barbados, instead of Jamaica, and William Harrison embarked on February 14, 1164, in H.M.S. Tartar, Sir Jobn Lindsay, at tbe Nore. The Tartar proceeded to Portsmouth, where Harrison checked the rate of Number Four with a regulator installed there in a temporary observatory. On March 28, 1764, the Tartar sailed from Portsmouth and the second trial was on. It was the same story all over again. On April 18, twenty-one days out, Harrison took two altitudes of the sun and announced to Sir Jobn tbat tbey were forty-three miles east of Porto Santo. Sir John accordingly steered a direct course for it, and at one o'clock the next morning the island was sighted, "which exactly agreed with the Distance mentioned above,'· They arrived at Barbados May 13, "Mr. Harrison all along in the Voyage declaring how far he was distant from that Island, according to the best settled longitude thereof. The Day before they made it, he declared the Distance: and Sir John sailed in Consequence of tbis Declaration, till Eleven at Night, which proving dark he thought proper to lay by. Mr. Harrison then declaring tbey were no more tban eigbt or nine MiJes from tbe Land, wbicb accordingly at Day Break tbey saw from tbat Distance." III When Harrison went ashore with Number Four he discovered that none other than Maskelyoe and an assistant, Green, had been sent ahead to check the longitude of Barbados by observing Jupiter's satellites. More.. over, MaskeJyoe bad been orating loudly on the superiority of his own method of finding longitude, namely, by lunar distances. When Harrison heard what had been going on be objected strenuously, pointing out to Sir John that Maskelyne was not only an interested party but an active and avid competitor t and sbould not have anything to do with tbe trials. A compromise was arranged, but, as it turned out, Maske)yoe was suddenly indisposed and unable to make the observations. liB Ibid., pp. S7 ff. On August 17, 1762, tbe Board refused to give Harrison a certificate stating that be bad complied with tbe tenns set fortb in the Act of Queen Anne. 59 Ibid., p. S9; see also A. narrative 01 the proceedings relative to the disco 'Very oj the longitude QJ sea; by Mr. John Harrison's time-keeper . .. [By James Sbort] London: printed for tbe author, 176S, pp. 7, 8.
Uoyd A. BrtIWII
816
After comparing the data obtained by observation with Harrison·. chronometer, Number Four showed an error of 38.4 seconds over a period of seven weeks, or 9.6 miles of longitude (at the equator) between Portsmouth and Barbados. And when the clock was again checked at Portsmouth, after 156 days, elapsed time, it showed, after allowing for its rate of going, a total gain of only 54 seconds of time. If further allowance were made for changes of rate caused by variations in temperature, information posted beforehand by Harrison, the rate of Number Four would have been reduced to an error of 15 seconds of loss in 5 months, or less than of a second a day.eo The evidence in favor of Harrison's chronometer was overwhelming, and could no longer be ignored or set aside. But the Board of Longitude was not through. In a Resolution of February 9, 1765, they were unanimously of the opinion that "the said timekeeper has kept its time with sufficient correctness, without losing its longitude in the voyage from Portsmouth to Barbados beyond the nearest limit required by the Act 12th of Queen Anne, but even considerably within the same." Now, they said, a1l Harrison had to do was demonstrate the mechanism of his clock and explain the construction of it, "by Means whereof other such Timekeepers might be framed, of sufficient Correctness to find the Longitude at Sea. . . ." In order to get the first £10,000 Harrison had to submit, on oath, complete working drawings of Number Four; explain and demonstrate the operation of each part, including the process of tempering the springs; and finally, hand over to the Board his first three timekeepers as well as Number Four.ll Any foreigner would have acknowledged defeat at this juncture, but not Harrison, who was an Englishman and a Yorkshireman to boot. uI cannot help thinking," he wrote the Board, after hearing their harsh terms, "but I am extremely ill used by gentlemen who I might have expected different treatment from. . . . It must be owned that my case is very hard, but I hope I am the first, and for my country's sake, shall be the last that suffers by pinning my faith on an English Act of Parliament." The case of "Longitude Harrison" began to be aired publicly, and several of his friends launched an impromptu publicity campaign against the Board and against Parliament. The Board final1y softened their terms and Harrison reluctantly took his clock apart at his home for the edification of a committee of six, nominated by the Board; three of them, Thomas Mudge, William Matthews and Larcum Kendall, were watchmakers. Harrison then received a certificate from the Board (October 28, 1765) entitling him to £7500, or the balance due him on the first half of the reward. The second half did not come so easily.
*0
R. T. Gould, op. cit., pp. 59. 60. Ibid., pp. 60 ft.; Act of Parliament 5 George III, Cap. 20; see also James Short. op. cit., p. 15. 80
61
117
Number Four was now in the hands of the Board of Longitude, held in trust for the benefit of the people of England. As such. it was carefully guarded against prying eyes and tampering, even by members of the Board. However, that learned body did its humble best. First they set out to publicize its mechanism as widely as possible. Unable to take the thing apart themselves, they had to depend on Harrison's own drawings, and these were redrawn and carefully engraved. What was supposed to be a full textual description was written by the Reverend Nevil Maskelyne and printed in book form with illustrations appended: The Principles of Mr. Harrison's Time-Keeper, with Plates of the Same. London, 1767. Actually the book was harmless enough. because no human being could have even begun to reproduce the clock from Maskelyne's description. To Harrison it was just another bitter pm to swal1 ow. "They have since pubUshed all my Drawings," he wrote, "without giving me the last Moiety of the Reward, or even paying me and my Son for our Time at a rate as common Mechanicks; an Instance of such Cruelty and Injustice as I believe never existed in a learned and civilised Nation before." Other ga11ing experiences fol1owed. 61 With great pomp and ceremony Number Four was carried to the Royal Observatory at Greenwich. There it was scheduled to undergo a prolonged and exhaustive series of trials under the direction of the Astronomer Royal. the Reverend Nevil Maskelyne. It cannot be said that Maskelyne shirked his duty. although he was handicapped by the fact that the timekeeper was always kept locked in its case, and he could not even wind it except in the presence of an officer detailed by the Governor of Greenwich to witness the performance. Number Four, after all, was a £10,000 timekeeper. The tests went on for two months. Maskelyne tried the watch in various positions for which it was not adjusted, dial up and dial down. Then for ten months it was tested in a horizontal position, dial up. The Board published a fun account of the results with a preface written by Maskelyne, in which he gave it as his studied opinion "'lbat Mr. Harrison's Watch cannot be depended upon to keep the Longitude within a Degree, in a West-India Voyage of six weeks, nor to keep the Longitude within Half a Degree for more than a Fortnight, and then it must be kept in a Place where the Thermometer is always some Degrees above freezing." (There was still £10,000 prize money outstanding.)68 The Board of Longitude next commissioned Larcum Kendall, watchmaker, to make a duplicate of Number Four. They also advised Harrison that he must make Number Five and Number Six and have them tried at 611 R. T. Gould, op. cit., pp. 61-62; 62 n; the unauthorized publication of Harrison's drawinss with a preface by the Reverend Maskelyne, appeared under the title: The principles oj Mr. Harriso,,'s time-keeper, with plates 01 the same. London, 1767. liS R. T. Gould, op. cit., p. 63; see also Nevil Maskelyne's An account 01 the going of Mr. John Harrison's walch . .. London, 1767.
818
Lloyd A. BI'O"'"
sea, intimating that otherwise he would not be entitled to the other half of the reward. When Harrison asked if he might use Number Four for a short time to help him build two copies of it, he was told that Kendall needed it to work from and that it would be impossible. Harrison did the best he could, while the Board laid plans for an exhaustive series of tests for Number Five and Number Six. They spoke of sending them to Hudson's Bay and of letting them toss and pitch in the Downs for a month or two as well as sending them out to the West Indies. After three years (1767-1770) Number Five was finished. In 1771, just as the Harrisons were finishing the last adjustments on the clock, they heard that Captain Cook was preparing for a second exploring cruise, and that the Board was planning to send Kendall's duplicate of Number Four along with him. Harrison pleaded with them to send Number Four and Number Five instead, telling them he was willing to stake his claim to the balance of the reward on their performance. or to submit Uta any mode of trial, by men not already proved partial, which shall be definite in its nature." The man was now more than ever anxious to settle the business once and for all. But it was not so to be. He was told that the Board did not see fit to send Number Four out of the kingdom, nor did they see any reason for departing from the manner of trial already decided upon. John Harrison was now seventy-eight years old. His eyes were failing and his skilled hands were not as steady as they were, but his heart was strong and there was still a lot of fight left in him. Among his powerful friends and admirers was His Majesty King George the Third, who had granted Harrison and his son an audience after the historic voyage of the Tartar. Harrison now sought the protection of his king, and "Farmer George," after hearing the case from start to finish, lost his patience. "By God, Harrison, I'll see you righted:' he roared. And he did. Number Five was tried at His Majesty's private observatory at Kew. The king attended the daily checking of the clock's performance, and had the pleasure of watching the operation of a timekeeper whose total error over a ten week's period was 4* seconds." Harrison submitted a memorial to the Board of Longitude, November 28, 1772, describing in detail the circumstances and results of the trial at Kew. In return, the Board passed a resolution to the effect that they were not the slightest bit interested; that they saw no reason to alter the manner of trial they had already proposed and that no regard would be paid for a trial made under any other conditions. In desperation Harrison decided to play his last card-the king. Backed by His Majesty's personal interest in the proceedings, Harrison presented a petition to the House of Commons with weight behind it. It was heralded as follows: "The Lord North. by His Majesty's Command, acquainted the House that His MajOR. T. Gould. op. cit .• pp. 64-65.
819
esty, having been informed of the C,ontents of the said Petition, recommended it to the Consideration of the House." Fox was present to give the petition his full support, and the king was willing, if necessary, to appear at the Bar of the House under an inferior title and testify in Harrison's behalf. At the same time, Harrison circulated a broadside, The Case 01 Mr. John Harrison, stating his claims to the second half of the reward.81 The Board of Longitude began to squirm. Public indignation was mounting rapidly and the Speaker of the House informed the Board that consideration of the petition would be deferred until they had an opportunity to revise their proceedings in regard to Mr. Harrison. Seven Admiralty clerks were put to work copying out all the Board's resolutions concerning Harrison. While they worked day and night to finish the job, the Board made one last desperate effort. They summoned William Harrison to appear before them; but the hour was late. They put him through a catechism and tried to make him consent to new trials and new conditions. Harrison stood fast, refusing to consent to anything they might propose. MeanwhUe a money bill was drawn up by Parliament in record time; the king gave it the nod and it was passed. The Harrisons had won their fight. 8$
Ibid., p. 66; see also the Journal 01 the House 01 Commons, 6. S. 1772.
COMMENTARY ON
JOHN COUCH ADAMS
T
HE discovery in 1846 of the planet Neptune was a dramatic and spectacular achievement of mathematical astronomy. The very existence of this new member of the solar system, and its exact location, were demonstrated with pencil and paper; there was left to observers only the routine task of pointing their telescopes at the spot the mathematicians had marked. It is easy to understand the enthusiastic appraisal of Sir John Herschel, who, on the occasion of the presentation of gold medals of the Royal Astronomical Society to Leverrier and Adams, said that their codiscovery of Neptune "surpassed by intelligible and legitimate means, the wildest pretensions of clairvoyance." 1 As early as about 1820 it had been recognized that irregularities in the motion of Uranus, deviations of its observed orbit from its calculated positions, could be accounted for only by an outside disturbing force. The German astronomer Bessel at that time remarked to Humboldt that sooner or later the "mystery of Uranus" would "be solved by the discovery of a new planet. 2 The problem was solved by two astronomers working entirely independently of each other: Urban Jean Leverrier, director of the Paris Observatory, and John Couch Adams, a twenty-six-year-old Fellow of St. John's College in Cambridge. The conquest of this intrjcate and incredibly laborious problem of "inverse perturbations"-Le., given the perturbations, to find the planet-was an intellectual feat deservedly acclaimed as "sublime"; but the personal behavior of various participants in the event faUs short of sublimity. The affair was marked by episodes of confusion and fecklessness, of donnish hairpulling and Gallic backbiting, of stuffiness, jealousy and general academic blight. Sir Harold Spencer Jones gives a balanced account~ as the present Astronomer Royal he is understandably charitable to Sir George Airy, the Astronomer Royal in Adams' day, who almost succeeded-more or less innocently-in doing Adams out of proper recognition for his work. I add a few biographical details about Adams not covered in Jones' lecture. While the discovery of Neptune was his most sensational achievement, it came at the beginning of a long and distinguished service to astronomy and mathematics. Adams succeeded George Peacock, the noted algebraist, as professor of astronomy and geometry at Cambridge. This post, as well as a fellowship in mathematics at Pembroke College, he held until his death. His researches are represented by important memoirs on It
1 2
Nature, Vol. XXXIV, p. 565. Giorgio Abetti, The His/ory 0/ Aflronomy, 1952, New York, p. 214. 820
821
lunar motion, on the effect of planetary perturbations on the orbits and periods of certain meleors, on various problems of pure mathematics. Adams was a stylish craftsman in his mathematical work; even the examination questions he set in prize competitions were admired for their finish. Like Euler and Gauss. he found pleasure in undertaking immense numerical calculations, which he carried out with consummate accuracy. But he was not chained to his specialty. He read widely in history, biology, geology and general literature; he took a deep interest in political questions. During the Franco-Prussian war, he was so moved "that he could scarcely work or sleep." S Adams was a shy, gentle and unaffected man. He refused knighthood in 1847, just as he had refused to be drawn into the bitter controversy over the question of who was first to discover Neptune. The honor was tendered in a foolish attempt to settle a foolish question. The entire business was beneath Adams. "He uttered no complaint; he 1aid no claim to priority; Leverrier had no warmer admirer."" He died after a 10ng illness on January 21, 1892}' 8 J. W. L. Glaisher, biographical memoir in The Scientific Papers 0/ John Couch Adams, ed. by William Grylls Adams, Cambridge, 1896. P. XLIV. 4 The Dictionary 0/ National Biography; Vol. XXII, Supplement, article on Adams, p. 16. Ii For further information on the Adams·Airy affair see 1. E. Littlewood, A Mathematician's Miscellany, London. 1953, pp. ) 16-134.
There are seven windows in the head, two nostrils, two eyes, two ears, tutti a mouth; so in the heavens there are two favorable stars, two unpropitious, two luminaries, and Mercury alone undecided and indifferent. From which and many other similar phenomena of nature, such as the seven metals, etc., which it were tedious to enumerate, we gather that the number of planets is necessarily seven.-FRANCESCO SIZZI (argument against Gailleo's discovery of the satellites of Jupiter)
4
John Couch Adams and the Discovery 0 f Neptune By SIR HAROLD SPENCER JONES
ON the night of 13 March 1781 William Herschel, musician by profession but assiduous observer of the heavens in his leisure time, made a discovery that was to bring him fame. He had for some time been engaged upon a systematic and detailed survey of the whole heavens, using a 7 in. telescope of his own construction; he carefully noted everything that appeared in any way remarkable. On the night in question, in his own words: 'In examining the small stars in the neighbourhood of H Geminorum I perceived one that appeared visibly larger than the rest; being struck with its uncommon appearance I compared it to H Geminorum and the small star in the quartile between Auriga and Gemini, and finding it so much larger than either of them, I suspected it to be a comet.' Most observers would have passed the object by without noticing anything unusual about it, for the minute disk was only about 4 sec. in diameter. The discovery was made possible by the excellent quality of Herschel's telescope, and by the great care with which his observations were made. The discovery proved to be of greater importance than Herschel suspected, for the object he had found was not a comet, but a new planet, which revolved round the Sun in a nearly circular path at a mean distance almost exactly double that of Saturn; it was unique, because no planet had ever before been discovered; the known planets, easily visible to the naked eye, did not need to be discovered. After the discovery of Uranus, as the new planet was caned, it was ascertained that it had been observed as a star and its position recorded on a score of previous occasions. The earliest of these observations was made by Flamsteed at Greenwich in 1690. Lemonnier in 1769 had observed its transit six times in the course of 9 days and, had he compared 821
823
the observations with one another, he could not have failed to anticipate Herschel in the discovery. As Uranus takes 84 years to make a complete revolution round the Sun, these earlier observations were of special value for the investigation of its orbit. The positions of the planet computed from tables constructed by Delambre soon began to show discordances with observation, which became greater as time went on. As there might have been error or incompleteness in Delambre's theory and tables, the task of revision was undertaken by Bouvard, whose tables of the planet appeared in 1821. Bouvard found that, when every correction for the perturbations in the motion of Uranus by the other planets was taken into account, it was not possible to reconcile the old observations of F1amsteed, Lemonnier, Bradley, and Mayer with the observations made subsequently to the discovery of the planet in 1781. 'The construction of the tables, then,' said Bouvard, 'involves this alternative: if we combine the ancient observations with the modem, the former will be sufftciently wen represented, but the latter will not be so, with all the precision which their superior accuracy demands; on the other hand, if we reject the ancient observations altogether, and retain only the modem, the resulting tables will faithfully conform to the modem observations, but will very inadequately represent the more ancient. As it was necessary to decide between these two courses, I have adopted the latter. on the ground that it unites the greatest number of probabilities in favour of the truth, and I leave to the future the task of discovering whether the difficulty of reconciling the two systems is connected with the ancient observations, or whether it depends on some foreign and unperceived cause which may have been acting upon the planet.' Further observations of Uranus were for a time found to be pretty well represented by Bouvard's Tables, but systematic discordances between observations and the tables gradually began to show up. As time went on, observations continued to deviate more and more from the tables. It began to be suspected that there might exist an unknown distant planet, whose gravitational attraction was disturbing the motion of Uranus. An alternative suggestion was that the inverse square law of gravitation might not be exact at distances as great as the distance of Uranus from the Sun. The problem of computing the perturbations in the motion of one pJanet by another moving planet, when the undisturbed orbits and the masses of the planets are known is fairly straightforward, though of some mathematical complexity. The inverse problem, of analysing the perturbations in the motion of one planet in order to deduce the position, path and mass of the planet whjch is producing these perturbations, is of much greater complexity and difficulty. A little consideration will, I think, show that this must be so. If a planet were exposed solely to the attractive
824
SI, Rt#D14 SIMMn lou.
infiuence of the Sun, its orbit would be an ellipse. The attractions of the other planets perturb its motion and cause it to deviate now on the one side and now on the other side of this ellipse. To determine the elements of the elliptic orbit from the positions of the planet as assigned by observation, it is necessary first to compute the perturbations produced by the other planets and to subtract them from the observed positions. The position of the planet in this orbit at any time, arising from its undisturbed motion, can be calculated; if the perturbations of the other planets are then computed and added, the true position of the planet is obtained. The whole procedure is, in practice, reduced to a set of tables. But if Uranus is perturbed by a distant unknown planet, the observed positions when corrected by the subtraction of the perturbations caused by the known planets are not the positions in the true elliptic orbit; the perturbations of the unknown planet have not been allowed for. Hence when the corrected positions are analysed in order to determine the elements of the elliptic orbit, the derived elements will be falsified. The positions of Uranus computed from tables such as Bouvard's would be in error for two reasons; in the first place, because they are based upon incorrect elements of the elliptic orbit; in the second place, because the perturbations produced by the unknown planet have not been applied. The two causes of error have a common origin and are inextricably entangled in each other, so that neither can be investigated independently of the other. Thus though many astronomers thought it probable that Uranus was perturbed by an undiscovered planet, they could not prove it. No occasion had arisen for the solution of the extremely complicated problem of what is termed inverse perturbations, starting with the perturbed positions and deducing from them the position and motion of the perturbing body. The first solution of this intricate problem was made by a young Cambridge mathematician, John Couch Adams. As a boy at school Adams had shown conspicuous mathematical ability, an interest in astronomy, and skill and accuracy in numerical computation. At the age of 16 he had computed the circumstances of an annular eclipse of the Sun, as visible from Lidcot, near Launceston, where his brother Jived. He entered St John's College in October, 1839, at the age of 20, and in 1843 graduated Senior Wrangler, being reputed to have obtained more than double the marks awarded to the Second Wrangler. In the same year he became first Smith's Prizeman and was elected Fellow of his College. Whilst still an undergraduate his attention had been drawn to the irregularities in the motion of Uranus. After his death there was found among his papers this memorandum, written at the beginning of his second long vacation:
825
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l1841, July 3. Formed a design in the beginning of this week, of investigating, as soon as possibJe after taking my degree. the irregularities in the motion of Uranus, which are yet unaccounted for; in order to find whether they may be attributed to the action of an undiscovered planet beyond it; and if possible thence to determine the elements of its orbit, etc. approximately, which would probably lead to its discovery! As soon as Adams had taken his degree he attempted a first rough solution of the problem, with the simplifying assumptions that the unknown planet moved in a circular orbit, in the plane of the orbit of Uranus, and that its distance from the Sun was twice the mean distance of Uranus, this being the distance to be expected according to the empirical law of Bode. This preliminary solution gave a sufficient improvement in the agreement between the corrected theory of Uranus and observation to encourage him to pursue the investigation further. In order to make the observational data more complete application was made in February 1844 by Challis, the Plumian Professor of Astronomy, to Airy, the Astronomer Royal, for the errors of longitude of Uranus for the years 1818-26. Challis explained that he required them for a young friend, Mr Adams of St John's College, who was working at the theory of Uranus. By return of post, Airy sent the Greenwich data not merely for the years 1818-26 but for the years 1754-1830. Adams now undertook a new solution of the problem, still with the assumption that the mean distance of the unknown planet was twice that of Uranus but without assuming the orbit to be circular. During term·
826
Sir RfJrDld SIMnc" 10MI
time he bad little opportunity to pursue his investigations and most of the work was undertaken in the vacations. By September 1845, he bad completed the solution of the problem, and gave to Challis a paper with the elements of the orbit of the planet, as well as its mass and its position for 1 October 1845. The position indicated by Adams was actually within r of the position of Neptune at that time. A careful search in the vicinity of this position should have led to the discovery of Neptune. The comparison between observation and theory was satisfactory and Adams, confident in the validity of the law of gravitation and in his own mathematics, referred to the 'new planet'. Challis gave Adams a letter of introduction to Airy, in which he said that 'from his character as a mathematician, and his practice in calculation, I should consider the deductions from his premises to be made in a trustworthy manner'. But the Astronomer Royal was in France when Adams called at Greenwich. Airy, immediately on his return, wrote to Challis saying: 'would you mention to Mr Adams that I am very much interested with the subject of his investigations, and that I should be delighted to hear of them by letter from him?' Towards the end of October Adams called at Greenwich, on his way from Devonshire to Cambridge, on the chance of seeing the Astronomer Royal. At about that time Airy was occupied almost every day with meetings of the Railway Gauge Commission and he was in London when Adams called. Adams left his card and said that he would can again. The card was taken to Mrs Airy, but she was not told of the intention of Adams to can later. When Adams made his second call, he was informed that the Astronomer Royal was at dinner; there was no message for him and he went away feeling mortified. Airy, unfortunately, did not know of this second visit at the time. Adams left a paper summarizing the results which he had obtained and giving a list of the residual errors of the mean longitude of Uranus, after taking account of the disturbing action of the new planet. These errors were satisfactorily small, except for the first observation by Flamsteed in 1690. A few days later Airy wrote to Adams acknowledging the paper and enquiring whether the perturbations would explain the errors of the radius vector of Uranus as well as the errors of longitude; in the reduction of the Greenwich observations, Airy had shown that not only the longitude of Uranus but also its distance from the Sun (caned the radius vector) showed discordances from the tabular values. Airy said at a later date that he waited with much anxiety for the answer to this query, which he looked upon as an experimentum crucis, and that if Adams had replied in the affirmative, he would at once have exerted all his influence to procure the publication of Adams's theory. It should be emphasized that neither Challis nor Airy knew anything about the details of Adams's investigation. Adams had attacked this
827
difficult problem entirely unaided and without guidance. Confident in his own mathematical abiJity he sought no help and he needed no help. Adams never replied to the Astronomer Royal's query; but for this failure to reply, he would almost certainly have had the sole glory of the discovery of Neptune. Airy and Adams were looking at the same problem from different points of view; Adams was so convinced that the discordances between the theory of Uranus and observation were due to the perturbing action of an unknown planet that no alternative hypothesis was considered by him; Airy, on the other hand, did not exclude the possi-
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bility that the law of gravitation might not apply exactly at great distances. The purpose of his query, to which he attached great importance, was to decide between the two possibilities. As he later wrote to Chams (21 December 1846): 'There were two things to be explained, which might have existed each independently of the other, and of which one could be ascertained independently of the other: viz. error of longitude and error of radius vector, And there is no a priori reason for thinking that a hypothesis which will
829
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explain the error of longitude will also explain the error of radius vector. If, after Adams had satisfactorily explained the error of longitude he had (with the numerical values of the elements of the two planets so found) converted his formula for perturbation of radius vector into numbers. and if these numbers had been discordant with the observed numbers of discordances of radius vector, then the theory would have been false, NOT from any error of Adams's BUT from a failure in the law of gravitation. On this question therefore turned the continuance or fall of the law of gravitation.' What were the reasons for Adams's failure to reply? There were several; he gave them himself at a later date (18 November 1846) in a letter to Airy. He wrote as follows: 'I need scarcely say how deeply I regret the neglect of which I was guilty in delaying to reply to the question respecting the Radius Vector
of Uranus, in your note of November 5th, 1845. In paJliation, though not in excuse of thia neglect, I may say that I was not aware of the importance which you attached to my answer on this point and I had not the smallest notion that you felt any difficulty on it. . . . For several years past, the observed place of Uranus has been faJling more and more rapidly behind its tabular place. In other words, the real angular motion of Uranus is considerably slower than that given by the Tables. This appeared to me to show clearly that the Tabular Radius Vector would be considerably increased by any Theory which represented the motion in Longitude. . . . Accordingly, I found that if I simply corrected the elJiptic elements. so as to satisfy the modem observations as nearly as possible without taking into account any additional perturbations, the corresponding increase in the Radius Vector would not be very different from that given by my actual Theory. Hence it was that I waited to defer writing to you till I could find time to draw up an account of the method employed to obtain the results which I had communicated to you. More than once I commenced writing with this object, but unfortunately did not persevere. I was also much pained at not having been able to see you when I called at the RoyaJ Observatory the second time, as I felt that the whole matter might be better explained by half an hour's conversation than by several letters, in writing which I have always experienced a strange difficulty. I entertained from the first the strongest conviction that the observed anomalies were due to the action of an exterior planet; no other hypothesis appeared to me to possess the slightest claims to attention. Of the accuracy of my calculations I was quite sure, from the care with which they were made and the number of times I had examined them. The only point which appeared to admit of any doubt was the assumption as to the mean distance and this I soon proceeded to correct. The work however went on very slowly throughout, as I had scarcely any time to give to these investigations, except during the vacations. 'I could not expect, however, that practical astronomers, who were already fully occupied with important labours, would feel as much confidence in the results of my investigation, as I myself did; and I therefore had our instruments put in order, with the express purpose, if no one else took up the subject, of undertaking the search for the planet myself, with the small means afforded by our observatory at St John's.' Airy was a man with a precise and orderly mind, extremely methodical and prompt in answering letters. Another person might have followed the matter up, but not Airy. In a letter of later date to Challis, he said that 'Adams's silence . . . was so far unfortunate that it interposed an effectual barrier to all further communication. It was clearly impossible for me to write to him again.'
loll" Couell A."" a,,,, 'lie Disrovery 01
N~ptu"e
831
Meanwhile, another astronomer had turned his attention to the problem of accounting for the anomalies in the motion of Uranus. In the summer of 184S Arago, Director of the Paris Observatory, drew the attention of his friend and protege, Le Verrier, to the importance of investigating the theory of Uranus. Le Verrier was a young man, 8 years older than Adams, with an established reputation in the astronomical world, gained by a brilliant series of investigations in celestial mechanics. In contrast, Adams was unknown outside the circle of his Cambridge friends and he had not yet pUblished anything. Le Verrier decided to devote himself to the problem of Uranus and laid aside some researches on comets, on which he had been engaged. His investigations received full publicity, for the results were published, as the work proceeded, in a series of papers in the Comptes Rendus of the French Academy. In the first of these, communicated in November 1845 (a month after Adams had left his solution of the problem with the Astronomer Royal), Le Verrier recomputed the perturbations of Uranus by Jupiter and Saturn, derived new orbital elements for Uranus, and showed that these perturbations were not capable of explaining the observed irregularities of Uranus. In the next paper, presented in June 1846, Le Verrier discussed possible explanations of the irregularities and concluded that none was admissible, except that of a disturbing planet exterior to Uranus. Assuming, as Adams had done, that its distance was twice the distance of Uranus and that its orbit was in the plane of the ecliptic, he assigned its true longitude for the beginning of 1847; he did not obtain the elements of its orbit nor determine its mass. The position assigned by Le Verrier differed by only 10 from the position which Adams had given seven months previously. Airy now felt no doubt about the accuracy of both calculations; he still required to be satisfied about the error of the radius vector, however, and he accordingly addressed to Le Verrier the query that he had addressed to Adams, but this time in a more explicit form. He asked whether the errors of the tabular radius vector were the consequence of the disturbance produced by an exterior planet, and explained why, by analogy with the moon's variation, this did not seem to him necessarily to be so. Le Verrier replied a few days later giving an explanation which Airy found completely satisfactory. The errors of the tabular radius vector, said Le Verrier, were not produced actual1y by the disturbing planet; Bouvard's orbit required correction, because it had been based on positions which were not true elliptic positions, including, as they did, the perturbations by the outer planet; the correction of the orbit, which was needed on this account, removed the discordance between the observed and tabular radius vector. Airy was a man of quick and incisive action. He was now fuJ1y convinced that the true explanation of the irregularities in the motion of
832
SI, BIlrDI4 SPft"' 10M'
Uranus had been provided and he felt confident that the new planet would soon be found. He had already, a few days before receiving the reply from Le Verrier, informed the Board of Visitors of the Royal Observatory, at their meeting in June, of the extreme probability of discovering a new planet in a very short time. It was in consequence of this strongly expressed opinion of Airy that Sir John Herschel (a member of the Board) in his address on 10 September to the British Association, at its meeting at Southampton, said: 'We see it [a probable new planet] as Columbus saw America from the shores of Spain. Its movements have been felt, trembling along the far-reaching line of our analysis, with a certainty hardly inferior to that of ocular demonstration.' Airy considered that the most suitable telescope with which to make the search for the new planet was the Northumberland telescope of the Cambridge Observatory, which was larger than any telescope at Greenwich and more likely to detect a planet whose light might be feeble. Airy offered to lend Challis one of his assistants, if Challis was too busy to undertake the search himself. He pointed out that the most favourable time for tbe search (when the undiscovered planet would be at opposition) was near at hand. A few days later, Airy sent Challis detailed directions for carrying out the search and in a covering letter said that, in his opinion, the importance of the inquiry exceeded that of any current work, which was of such a nature as not to be totally lost by delay. Challis decided to prosecute the search himself and began observing on 29 July 1846, three weeks before opposition. The method adopted was to make three sweeps over the area to be searched, mapping the positions of all the stars observed, and completing each sweep before beginning the next. If the planet was observed it would be revealed, when the different sweeps were compared, by its motion relative to the stars. What followed was not very creditable to Chal1is. He started by observing in the region indicated by Adams: the first four nights on which observations were made were 29 July, 30 July, 4 August and 12 August. But no comparison was made, as the search proceeded, between the observations on different nights. He did indeed make a partial comparison between the nights of 30 July and 12 August, merely to assure himself that the method of observation was adequate. He stopped short at No. 39 of the stars observed on 12 August; as he found that all these had been observed on 30 July, he felt satisfied about the method of observation. If he had continued the comparison for another ten stars he would have found that a star of the 8th magnitude observed on 12 August was missing in the series of 30 July. This was the planet: it had wandered into the zone between the two dates. Its discovery was thus easily within his grasp. But 12 August was not the first time on which Chams had observed the planet; he had already observed it on 4 August and if he had com-
833
pared the observations of 4 August with the observations of either 30 July or of 12 August, the planet would have been detected. When we recall Airy's strong emphasis on carrying on the search in preference to any current work, Challis's subsequent excuses to justify his failure were pitiable. He had delayed the comparisons, he said, partly from being occupied with comet reductions (which could wen have waited), and partly from a fixed impression that a long search was required to ensure success. He confessed that, in the whole )f the undertaking, he had too little confidence in the indications of theory. Oh! man of little faith! If only he had shared Airy's conviction of the great importance of the search. But we have anticipated somewhat. While Challis was laboriously continuing his search, Adams wrote on 2 September an important letter to Airy who, unknown to Adams, was then in Germany. He referred to the assumption in his first calculations that the mean distance of the supposed disturbing planet was twice that of Uranus. The investigation, he said, could scarcely be considered satisfactory While based on anything arbitrary. He had therefore repeated his calculations, assuming a somewhat smaller mean distance. The result was very satisfactory in that the agreement between theory and observations was somewhat improved and, at the same time, the eccentricity of the orbit, which in the first solution had an improbably large value, was reduced. He gave the residuals for the two solutions, and remarked that the comparison with recent Greenwich observations suggested that a stm better agreement could be obtained by a further reduction in the mean distance. He asked for the results of the Greenwich observations for 1844 and 1845. He then gave the corrections to the tabular radius vector of Uranus and remarked that they were in close agreement with those required by the Greenwich observations. Two days earJier, on 31 August, Le Verrier had communicated a third paper to the French Academy which was published in a number of the Comptes Rendus that reached England near the end of September. Challis received it on 29 September. Le Verrier gave the orbital elements of the hypothetical planet, its mass, and its position. From the mass and distance Qf the planet he inferred, on the reasonable assumption that its mean density was equal to the mean density of Uranus, that it should show a disk with -an angular diameter of about 3'3 sec. Le Verrier went on to remark as follows: 'It should be possible to see the new planet in good telescopes and also to distinguish it by the size of its disk. This is a very important point. For if the .planet could not be distinguished by its appearance from the stars it would be necessary, in order to discover it, to examine all the sman stan in the region of the sky to be explored, and to detect a proper motion
834
SI, BMo'" S"'IIC~' 10."
of one of them. This work would be long and wearisome. But if, on the contrary, the planet has a sensible disk which prevents it from being confused with a star, if a simple study of its physical appearance can replace the rigorous determination of the positions of a II the stars, the search wiH proceed much more rapidly.' After reading this memoir on 29 September, Chal1is searched the same night in the region indicated by Le Verrier (which was almost identical with that indicated by Adams, in the first instance, a year earlier), looking out particularly for a visible disk. Of 300 stars observed he noted one and one only as seeming to have a disk. This was, in actual fact, the planet. Its motion might have been detected in the course of a few hours, but Challis waited for confirmation until the next night, when no observation was possible because the Moon was in the way. On 1 October he learnt that the planet had been discovered at Berlin on 23 September. His last chance of making an independent discovery had gone. For on 23 September Gane, Astronomer at the Berlin Observatory, had received a letter from Le Verrier suggesting that he should search for the unknown planet, which would probably be easily distinguished by a disk. D'Arrest, a keen young volunteer at the Observatory, asked to share in the search, and suggested to Galle that it might be worth looking among the star charts of the Berlin Academy, which were then in course of publication, to verify whether the chart for Hour 21 was amongst those that were finished. It was found that this chart had been printed at the beginning of 1846, but had not yet been distributed; it was therefore availab]e only to the astronomers at the Berlin Observatory. Galle took his place at the telescope, describing the configurations of the stars he saw, while d'Arrest followed them on the map, until Galle said: 'And then there is a star of the 8th magnitude in such a such a position', whereupon d'Arrest exclaimed: 'That star is not on the map.' An observation the following night showed that the object had changed its position and proved that it was the planet. Had this chart been available to Challis, as it would have been but for the delay in distribution, he would undOUbtedly have found the planet at the beginning of August, some weeks before Le Verrier's third memoir was presented to the French Academy. On 1 October, Le Verrier wrote to Airy informing him o~ the discovery of the planet. He mentioned that the Bureau des Longitudes had adopted the name Neptune, the figure a trident, and that the name Janus (which had also been suggested) would have the inconvenience of making it appear that the planet was the last in the solar system, which there was no reason to believe. The discovery of the planet, fol1owing the briUiant researches of Le Verrier, which were known to the scientific world through their publication by the French Academy, was received with admiration and delight,
'lS
and was acclaimed as one of the greatest triumphs of the human intellect. The prior investigations of Adams, his prediction of the position of the planet, the long patient search by Challis were known to only a few people in England. Adams had published nothing; he had communicated his results to Challis and to Airy, but neither of them knew anything of the details of his investigations; his name was unknown in astronomical circles outside his own country. Adams had actually drawn up a paper to be read at the meeting of the British Association at Southampton early in September, but he did not arrive in sufficient time to present it, as Section A closed its meetings one day earlier than he had expected. The first reference in print to the fact that Adams had independently reached conclusions similar to those of Le Verrier was made in a letter from Sir John Herschel, published in the Athenaeum of 3 October. It came as a complete surprise to the French astronomers and ungenerous aspersions were cast upon the work of Adams. It was assumed that his solution was a crude essay which would not stand the test of rigorous examination and that, as he had not pUblished any account of his researches, he could not establish a claim to priority or even to a share in the discovery. Some justification seemed to be afforded by an unfortunate letter from Challis to Arago, of 5 October, stating that he had searched for the planet, in conformity with the suggestions of Le Verrier, and had observed an object on 29 September which appeared to have a disk and which later was proved to have been the planet. No reference was made in this letter to the investigations of Adams or to his own earlier searches during which the planet had twice been observed. Airy, moreover, wrote to Le Verrier on 14 October, mentioning that collateral researches, which had led to the same result as his own, had been made in England. He went on to say: 'If in this I give praise to others I beg that you will not consider it as at all interfering with my acknowledgment of your claims. You are to be recognized, without doubt, as the real predictor of the planet's place. I may add that the EngJish investigations, as I believe, were not so extensive as yours. They were known to me earlier than yours.' It is difficult to understand why Airy wrote in these terms; he had expressed the highest admiration for the manner in which the problem had been solved by Le Verrier, but he was not in a position to express any opinion about the work of Adams, which he had not yet seen. At the meeting of the French Academy on 12 October, Arago made a long and impassioned defence of his protege, Le Verrier, and a violent attack on Adams, referring scornfully to what he described as his clandestine work. 'Le Verrier is to-day asked to share the glory, so loyally, so rightly earned, with a young man who has communicated nothing to the public and whose calculations, more or less incomp1ete, are, with two exceptions, totally unknown in the observatories of Europe! No! no! the
836
friends of science will not allow such a crying injustice to be perpetrated. He concluded by saying that Adams had no claim to be mentioned, in the history of Le Verrier's planet, by a detailed citation nor even by the slightest allusion. National feeling ran very high in France. The paper Le National asserted that the tbree foremost British astronomers (Herschel, Airy and Challis) had organized a miserable plot to steal the discovery from M. Le Verrier and that the researches of Adams were merely a myth invented for this purpose. In England opinion was divided; some English astronomers contended that because Adams's results had not been publicly announced he could claim no share in the discovery. But for the most part it was considered that the credit for the successful prediction of the position of the unknown planet should be shared equally between Adams and Le Verrier. Adams himself took no part in the heated discussions which went on for some time with regard to the credit for the discovery of the new planet; he never uttered a single word of criticism or blame in connexion with the matter. Tbe controversy was lifted to a higher plane by a letter from Sir Jobn Herschel to The Guardian in which he said: 'The history of this grand discovery is that of thought in one of its highest manifestations, of science in one of its most refined applications. So viewed, it offers a deeper interest than any personal question. In proportion to the importance of the step, it is surely interesting to know that more than one mathematician has been found capable of taking it. The fact, thus stated, becomes, so to spea k, a measure of the maturity of our science; nor can I conceive anything better calculated to impress the general mind with a respect for the mass of accumulated facts, laws, and methods, as they exist at present, and the reality and efficiency of the forms into which they have been moulded, than such a circumstance. We need some reminder of this kind in England, where a want of faith in the higher theories is still to a certain degree our besetting weakness.' At the meeting of the Royal Astronomical Society on 13 November 1846, three important papers were read. The first, by the Astronomer Royal, was an 'Account of some Circumstances historically connected with the Discovery of the Planet Exterior to Uranus'. All the correspondence with Adams, Chams and Le Verrier was given, as well as the two memoranda from Adams, the whole being linked together by Airy's own comments. The account made it perfectly clear that Adams and Le Verrier had independently solved the same problem, that the positions which they had assigned to the new planet were in close agreement, and that Adams had been the first to solve the problem. The second was ChalJis's 'Account of Observations undertaken in search of the Planet discovered at Berlin on September 23, 1846', which showed that in the course of the search
Jo,," Couch Adams a"d
Ih~ Discov~'y
01
N~'tu"~
837
for the planet, he had twice observed it before its discovery at Berlin, and that he had observed it a third time before the news of this discovery reached England. The third paper was by Adams and was entitled 'An Explanation of the observed Irregularities in the Motion of Uranus, on the Hypothesis of Disturbances caused by a more Distant Planet~ with a determination of the Mass, Orbit, and Position of the Disturbing Body'. Adams's memoir was a masterpiece; it showed a thorough grasp of the problem; a mathematical maturity which was remarkable in one so young; and a facility in deaJing with complex numerical computations. Lieut. Stratford, Superintendent of the Nautical Almanac, reprinted it as an Appendix to the Nautical Almanac for 1851, then in course of publication, and sent sufficient copies to Schumacher, editor of the Astronomische Nachrichten, for distribution with that periodical. Hansen, the foremost exponent of the lunar theory, wrote to Airy to say that, in his opinion, Adams's investigation showed more mathematical genius than Le Verrier's. Airy, a competent judge, gave his own opinion in a letter to Biot, who had sent to Airy a paper he had written about the new planet. He sent it, he said, with some diffidence because he had expressed a more favourable opinion of the work of Adams than Airy had given. In reply, Airy wrote: 'On the whole I think his [Adams's] mathematical investigation superior to M. Le Verrier's. However, both are so admirable that it is difficult to say.' He went on to state that 'I believe I have done more than any other person to place Adams in his proper position'. With the independent investigations of both men published, there was no difficu1ty in agreeing that each was entitled to an equal share of the honour. The verdict of history agrees with that of Sir John Herschel who, in addressing the Royal Astronomical Society in 1848, said: 'As genius and destiny have joined the names of Le Verrier and Adams, 1 shall by no means put them asunder~ nor will they ever be pronounced apart so long as language shall celebrate the triumphs of science in her sublimest walks. On the great discovery of Neptune, which may be said to have surpassed, by intelligible and legitimate means, the wildest pretensions of clairvoyance, it would now be quite superfluous for me to dilate. That glorious event and the steps which led to it, and the various lights in which it has been placed, are already familiar to everyone having the least tincture of science. I will only add that as there is not, nor henceforth ever can be, the slightest rivalry on the subject of these two illustrious men-as they have met as brothers, and as such will, 1 trust, ever regard each other-we have made, we could make, no distinction between them on this occasion. May they both long adorn and augment our science, and add to their own fame, already so high and pure, by fresh achievements.' Although on 1 October, Le Verrier had informed Airy that the Bureau
838
Sir Harold S",ncer Jones
des Longitudes had assigned the name Neptune to the new planet, Arago announced to the French Academy on 5 October that Le Verrier had delegated to him the right of naming the planet and that he had decided, in the exercise of this right, to call it Le Verrier. 'I pledge myself, he said, 'never to can the new planet by any other name than Le Verrier's Planet.' As though to justify this name, Le Verrier's col1ected memoin on the perturbations of Uranus, which were published in the Connaissance des Temps for 1849 were given the title 'Recherches sur les mouvements de la planete Henchel (dite Uranus)' with a footnote to say that Le Verrier considered it as a strict duty, in future publications, to ignore the name of Uranus entirely and to call the planet only by the name of Henchel! The name Le Verrier for the planet was not welcomed outside France. It was not in accordance with the custom of naming planets after mythological deities, and it ignored entirely the claims of Adams. Moreover, it might set a precedent. As Smyth said to Airy: 'Mythology is neutral ground. Herschel is a good name enough. Le Verrier somehow or other suggests a Fabriquant and is therefore not so good. But just think how awkward it would be if the next planet should be discovered by a German, by a Bugge, a Funk, or your hinute friend BoguslawskU' The widespread feeling against the name of Le Verrier was shared in Germany by Encke, Gauss and Schumacher and in Russia by Struve. Airy therefore wrote to Le Verrier: 'From my conversation with lovers of astronomy in England and from my correspondence with astronomers in Germany, I find that the name assigned by M. Arago is not well received. They think, in the first place, that the character of the name is at variance with that of the names of all the other planets. They think in the next place that M. Arago, as your delegate, could do only what you could do, and that you would not have given the name which M. Arago has given. They are all desirous of receiving a mythological name selected by you. In these feelings I do myself share. It was believed at first that you approved of the name Neptune, and in that supposition we have used the name Neptune when it was necessary to give a name. Now if it was understood that you still approve of the name Neptune (or Oceanus as some English mythologists suggested-or any other of the same class), I am sure that all England and Germany would adopt it at once. I am not sure that they will adopt the name which M. Arago has given.' Airy might have added, but did not, that there was a general feeling, not merely in England, but also in Germany and Russia, that the name Le Verrier by implication denied any credit to the work of Adams and that, for this reason also, it was inappropriate. Le Verrier, in reply, said that since one spoke of Comet Halley, Comet Encke, etc., he saw nothing inappropriate in Planet Le Verrier; that the
1011n Coucll ""dBnu
/Jltd
the Discovery 01 NlplUltl
839
Bureau des Longitudes had given the name Neptune without his consent and had now withdrawn it; 1 and that, since he had delegated the selec· tion of the name to Arago, it was a matter that no longer concerned him. At a later date, when relations between Arago and Le Verrier had become strained, the true story was told by Arago. It appears that Arago had at first agreed to the name Neptune, but Le Verrier had implored him, in order to serve him as a friend and as a countryman, to adopt the name Le Verrier. Arago had in the end agreed, but on condition that Uranus should always be called Planet Herschel, a name which Arago himself had frequently used. The greatest men are liable to human weaknesses and failings; Le Verrier was described by his friends as a mauvais coucheur, an uncomfortable bedfel1ow. By the general consensus of astronomers the name Neptune was adopted for the new planet; the name Le Verrier did not long survive. In the history of the discovery of Neptune so many chances were missed which might have changed completely the course of events, that it is perhaps not surprising to find that the planet might have been discovered SO years earlier. When sufficient observations of Neptune had been obtained to enable a fairly accurate orbit to be computed, a search was made to find out whether the planet had been observed as a star before its discovery. It was discovered that a star recorded in the Histoire Celeste of Lalande as having been observed on 10 May 1795 was missing in the sky; its position was marked as uncertain but was in close agreement with the position to be expected for Neptune. The original manuscripts of Lalande at the Paris Observatory were consulted; it was found that Lalande had, in fact, observed Neptune not only on 10 May but also on 8 May. The two positions, being found discordant and thought to refer to one and the same star, Lalande rejected the observation of 8 May and printed in the Histoire Celeste only the observation of 10 May marking it as doubtful, although it was not so marked in the manuscript. The change in position in the two days agreed closely with the motion of Neptune in the interval. If Lalande had taken the trouble to make a further observation to check the other two, he could scarcely have failed to discover the planet. Airy's comment, when sending the information about the two observations to Adams, was 'Let no one after this blame Challis', 1 This statement by Le Verrier was not correct. The minutes of the Bureau des Longitudes show that the Bureau had not considered assigning a name by 1 October, when Le Verrier had written not only to Airy but also to various other astronomers in Germany and Russia informing them that the Bureau des Longitudes 'had adopted the name Neptune, the figure a trident'. The Bureau neither assigned the name Neptune nor subsequentJy withdrew it. The minutes of the Bureau des Longitudes show that Le Verrier's statements Were repudiated by the Bureau at a subsequent meeting. It seems that the name Neptune was Le Verner's own choice in the first instance but that he sOOn decided that he would like the planet to be named Le Verrier. There is no explanation of his reasons for stating that the name Neptune had been assigned by the Bureau des Longitudes; it was, in fact. outside the competence of the Bureau to assign a name to a newly discovered planet.
COMMENTARY ON
H. G. J. MOSELEY N August 10, 1915, the twenty-eight-year-old British physicist, Henry Gwyn-Jeffreys Moseley, died in the trenches of Gallipoli. This extraordinarily gifted young scientist had left the laboratory of Sir Ernest Rutherford in Manchester to join the army in 1914. He was the victim not only of a Turkish bul1et but of an incomparably stupid recruitment program which permitted a scientist of such proven talent to become a soldier. For Moseley's brilliance was known. He had worked with Rutherford for some years, and at the age of twenty-seven had made discoveries in physics as important as any achieved in this century. Thus was squandered a life of the greatest promise to the history of science. It was from such spectacular instances of waste that both Great Britain and the U. S. learned to keep their best scientists out of the firing lines of the Second World War. Moseley's researches on atomic structure are reported in two communications to the Philosophical Magazine, from which the excerpts below have been selected. Although comparatively easy to explain, they are written in the usual terse and difficult idiom of the physicist. Therefore, I had better summarize their content and say a few words about how Moseley's work fitted into the scene of contemporary physics. After Roentgen discovered X-rays in 1895 a number of scientists devoted themselves to comparing these radiations with light waves. Attempts were made to reflect and refract X-rays, to determine whether they produced interference phenomena as does light. The attempts failed. If a parallel beam of light is allowed to faU on a grating, a surface on which many thousands of fine lines have been drawn to the inch, the transmitted or reflected light is broken up into its component colors, or wave lengths, and appears as a spectrum. When this procedure was applied to X-rays it also fai1ed. These experiments indicated either that X-rays were not waves or that the waves were so short that the methods of study used for light were unsuitable. In 1912 Max von Laue suggested that, while a man-made grating was too crude for the job, a grating provided by nature in the form of crystals might catch waves less than a thousandth as long as the shortest light waves. Von Laue worked out the mathematics of his theory and Friedrich and Knipping confirmed it in a series of beautiful experiments. The array of regular lines of atoms in the crystal served as a supergrating ("of very minute dimensions") and yielded characteristic diffraction spectra for X-rays. These spectra could be photographed and their lines analyzed. Thus, the wave nature of X-rays was established and,
O
840
R. G. 1. MON'"
841
incidentally, a way was opened into the great new field of the structure of crystals. 1 Now we come to Moseley's experiments. X-rays are produced by permitting a stream of electrons emitted by the negative pole (cathode) in a vacuum tube to impinge on a solid target. Roentgen had used platinum for the target but it soon became clear that if other elements were substituted they would also respond to cathode bombardment by emitting X-rays. Moseley successively used forty-two different elements as targets. He passed each set of X-rays through a crystal, photographed the resulting diffraction patterns and analyzed the waves emanating from each sample. He noticed that his figures were falling into a remarkable order. As he moved element by element up the Periodic Table-in which the elements are arranged by atomic weight and bracketed in groups according to their chemical properties-he found a regular increase in the square roots of the frequency of vibration of the characteristic spectrum lines. If this square root is multiplied by an appropriate constant so as to convert the regular increase to unity, one gets What is known as the series of atomic numbers ranging from 1 for hydrogen to 98 for californium. (Moseley went only as far as 79, for gold, when he was called away on more urgent business. The remaining places in the table were gradually filled in by other physicists.) What are these numbers which match so smoothly the order of the Periodic Table? (A few irregularities crop up but they have been explained and do not shake the theory.) The little arithmetic trick of converting the increase to unity should not mislead anyone to believe that the numbers merely represent a superimposed order. For the fact is that the atomic number of each element, representing the square root of frequency of an atom "suitably excited" to emit X radiation, assigns to the element its true place in the Table because it is the number of the unit positive charges in the nucleus of the atom. It is this charge which determines the element's chemical behavior. All subsequent advances in nuclear physics depend upon and are an outgrowth of the fundamental insight gained by Moseley. The beauty and simplicity of his theory, its revelation of the existence of an almost uncanny step-by-step order in the arrangement of the basic structures of matter, would have pleased the ancient Greek philosophers who held that number ruled the universe. "In our own day," says Dampier, "Aston with his integral atomic weights, Moseley with his atomic numbers, Planck with his quantum theory, and Einstein with his claim that physical facts such as gravitation are exhibitions of local spacetime properties, are reviving ideas that, in older, cruder forms, appear in Pythagorean philosophy." 1 For a further discussion of X-rays, von Laue's experiment, and the structure of crystals, see the selections by Bragg and Le Corbeiller, and the introduction preceding them, pp. 851-881.
The most important discoveries of the laws, methods and progress of NatuN have nearly always sprung from lhe examlnallon of lhe smallest objects -J. B. LAUARCK which she contains.
5
Atomic Numbers By H. G. J. MOSELEY THE HIGH FREQUENCY SPECTRA OF THE ELEMENTS
IN the absence of any available method of spectrum analysis, the characteristic types of X radiation, which an atom emits if suitably excited, have hitherto been described in terms of their absorption in aluminium. The interference phenomena exhibited by X-rays when scattered by a crystal have now, however, made possible the accurate determination of the frequencies of the various types of radiation. This was shown by W. H. and W. L. Bragg, who by this method analysed the line spectrum emitted by the platinum target of an X-ray tube. C. G. Darwin and the author extended this analysis and also examined the continuous spectrum, which in this case constitutes the greater part of the radiation. Recently Prof. Bragg has also determined the wave-lengths of the strongest lines in the spectra of nickel, tungsten, and rhodium. The electrical methods which have hitherto been employed are, however, only successful where a constant source of radiation is available. The present paper contains a description of a method of photographing these spectra, which makes the analysis of the X-rays as simple as any other branch of spectroscopy. The author intends first to make a general survey of the principal types of high.frequency radiation, and then to examine the spectra of a few elements in greater detail and with greater accuracy. The results already obtained show that such data have an important bearing on the question of the internal structure of the atom, and strongly support the views of Rutherford and of Bohr. Kaye has shown that an element excited by a stream of sufficiently fast cathode rays emits its characteristic X radiation. He used as targets a number of substances mounted on a truck inside an exhausted tube. A magnetic device enabled each target to be brought in turn into the line of fire. This apparatus was modified to suit the present work. The cathode stream was concentrated on to a small area of the target, and a platinum plate furnished with a fine vertical slit placed immediately in front of the part bombarded. The tube was exhausted by a Gaede mercury pump, charcoal in 1iquid air being also sometimes used to remove water vapour. The X-rays, after passing through the slit marked S in Figure 1, emerged 842
843
A.olfllc NlUJlkr'
FIGURE I
through an aluminum window of ·02 mm. thick. The rest of the radiation was shut off by a lead box which surrounded the tube. The rays fell on the cleavage face, C, of a crystal of potassium ferrocyanide which was mounted on the prism-table of a spectrometer. The surface of the crystal was vertical and contained the geometrical axis of the spectrometer. Now it is known that X-rays consist in general of two types, the heterogeneous radiation and characteristic radiations of definite frequency. The former of these is reflected from such a surface at all angles of incidence, but at the large angles used in the present work the reflexion. is of very little intensity. The radiations of definite frequency, on the other hand, are reflected only when they strike the surface at definite angles. the glancing angle of incidence 8, the wave-length A, and the "grating constant" d of the crystal being connected by the relation nA = 2d sin 8,
................. (1)
where n, an integer, may be called the "order" in which the reflexion occurs. The particular crystal used, which was a fine specimen with face 6 em. square, was known to give strong reflexions in the first three orders, the third order being the most prominent. If then a radiation of definite wave-length happens to strike any part P of the crystal at a suitable angle, a small part of it is reflected. Assuming for the moment that the source of the radiation is a point, the locus of P is obviously the arc of a circle, and the reflected rays will travel along the generating lines of a cone with apex at the image of the source. The effect on a photographic plate L will take the form of the arc of an hyperbola, curving away from the direction of the direct beam. With a fine slit
at S, the arc becomes a fine line which is slightly curved in the direction indicated. The photographic plate was mounted on the spectrometer arm, and both the plate and the slit were 17 cm. from the axis. The importance of this arrangement lies in a geometrical property, for when these two distances are equaJ the point L at which a beam reflected at a definite angle strikes the plate is independent of the position of P on the crystal surface. The angle at which the crystal is set is then immaterial so long as a ray can strike some part of the surface at the required angle. The angle 8 can be obtained from the relation 28 = 180° - SPL = 180° - SAL. The following method was used for measuring the angle SAL. Before taking a photograph a reference line R was made at both ends of the plate by replacing the crystal by a lead screen furnished with a fine slit which coincided with the axis of the spectrometer. A few seconds' expo-sure to the X-rays then gave a line R on the plate, and so defined on it the line joining Sand A. A second line RQ was made in the same way after turning the spectrometer arm through a definite angle. The arm was then turned to the position required to catch the reflected beam and the angles LA P for any lines which were subsequently found on the plate deduced from the known value of RA P and the position of the lines on the plate. The angle LAR was measured with an error of not more than OOel, by superposing on the negative a plate on which reference lines had been marked in the same way at intervals of 10. In finding from this the glancing angle of reflexion two small corrections were necessary in practice, since neither the face of the crystal nor the lead slit coincided accurately with the axis of the spectrometer. Wave-lengths varying over a range of about 30 per cent. could be reflected for a given position of the crystal. In almost all cases the time of exposure was five minutes. Ilford X-ray plates were used and were developed with rodinal. The plates were mounted in a plate-hOlder, the front of whlch was covered with black paper. In order to determine the wave-length from the reflex ion angle 8 it is necessary to know both the order PI in which the reflexion occurs and the grating constant d. PI was determined by photographing every spectrum both in the second order and the third. This also gave a useful check Oil the accuracy of the measurements; d cannot be calculated directly for the complicated crystal potassium ferrocyanide. The grating constant of this particular crystal had, however. previously been accurately compared with d', the constant of a specimen of rock-salt. It was found that ·1988 d=3d'--1985
Now W. L. Bragg has shown that the atoms in a rock-salt crystal are in simple cubical array. Hence the number of atoms per c.c. Na 1 2-=--: M (d') 8 N, the number of molecules in a gram-mo1.. = 6-05 X 1028 assuming the
charge (e) on an electron to be 4-89 X 10-10; a, the density of this crystal of rock-salt, was 2-167, and M the molecular weight = 58-46. This gives d' = 2-814 X 10-8 and d 8-454 X 10- 8 cm. It is seen that the determination of wave-length depends on e21.s so that the effect of uncertainty in the value of this quantity will not be serious. Lack of homogeneity in the crystal is a more likely source of error, as minute inclusions of water would make the density greater than that found experimentally. Twelve elements have so far been examined. . . . The Plate shows the spectra in the third order placed approximately in register. Those parts of the photographs which represent the same angle of reftexion are in the same vertical line. . . . It is to be seen that the spectrum of each element consists of two lines. Of these the stronger has been called a in the table, and the weaker p. The lines found on any of the plates besides a and p were almost certainly all due to impurities. Thus in both the second and third order the cobalt spectrum shows Nia very strongly and Fea faintly. In the third order the nickel spectrum shows Mna!,! faintly. The brass spectra naturally show a and p both of eu and of Zn, but Znp2 has not yet been found. In the second order the ferrovanadium and ferro-titanium spectra show very intense third-order Fe lines, and the former also shows Cuaa faintly. The Co contained Ni and 0-8 per cent. Fe, the Ni 2-2 per cent. Mn, and the V only a trace of Cu. No other lines have been found; but a search over a wide range of wavelengths has been made only for one or two elements. and perhaps prolonged exposures, which have not yet been attempted, will show more complex spectra. The prevalence of lines due to impurities suggests that this may prove a powerful method of chemical analysis. Its advantage over ordinary spectroscopic methods lies in the simplicity of the spectra and the impossibility of one substance masking the radiation from another. It may even lead to the discovery of missing elements, as it will be possible to predict the position of their characteristic Jines.
=
Phil. Mag. (1914), p. 703.
The first part of this paper dealt with a method of photographing X-ray spectra, and included the spectra of a dOZeD elements. More than thirty other elements have now been investigated, and simple laws have been found which govern the results, and make it possible to predict with
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confidence the position of the principal lines in the spectrum of any element from aluminium to SOld. The present contribution is a genera] preliminary survey, which claims neither to be complete nor very accurate. . . . The resuhs obtained for radiations belonging to Barkla's K series are given in Table I. and for convenaence the figures already given in Part I are included. The wave-length .\ has been calculated from the glancing angle of reflexion 6 by meam of the relation n.\ = 2d sin 6. where d has been taken to be 80454 X 10- 11 cm, As before. the strongest line is called a and the next line p. The square root of the frequency of each line is plotted in Figure 3, and the wave-lengths can be read off with the help of the scale at the top of the diagram. 1be spectrum of AI was photographed in the first order only. The very light elements give several other fainter lines, which have nOl yet been fully investiSated, while the results for Mg and Na are qurte complicated. and apparently depart from the simple relations which connect the spectra. of the other elements. In the spectra from yttrium onwards only the a line has so far been mea.sured. and further results in these directions will be
847
Atomic Hum'-"
TABLE. I
Aluminium Silicon Chlorine Potassium Calcium Titanium Vanadium Chromium Manganese Iron Cobalt Nickel Copper Zinc Yttrium Zirconium Niobium Molybdenum Ruthenium Palladium Silver
a line :\. X lOS ern.
011'.
8·364 1·142 4·150 3·159 3·368 2·158 2·519 2·301 2 ·111 1·946 1·198 1·662 1·549 1·445 0·838 0·194 0·150 0·121 0·638 0-584 0·560
12·05 13·04 16·00 11·98 19·00 20·99 21·96 22·98 23·99 24·99 26·00 21·04 28·01 29·01 38·1 39·1 40·2 41·2 43·6 45·6 46·6
NAtomle Number
/lline AX lOS ern.
13 14 11 19 20 22 23 24 25 26 21 28 29 30 39 40 41 42
1·912 6·129 3·463 3·094 2·524 2·291 2·093 1·818 1·165 1·629 1·506 1·402 1·306
44
46 41
TABLE. II a line
Zirconium Niobium Molybdenum Ruthenium Rhodium Palladium Silver Tin Antimony Lanthanum Cerium Praseodymium Neodymium Samarium Europium Gadolinium Holmium Erbium Tantalum Tungsten Osmium Iridium Platinum Gold
A X lOS em.
O£
6·091 5·149 5·423 4·861 4·622 4·385 4·110 3·619 3·458 2·616 2·561 (2·411) 2·382 2·208 2·130 2·051 1·914 1·190 1·525 1·486 1·391 1·354 1·316 1·281
32·8 33·8 34·8 36·1 31·1 38·1 39·6 42·6 43·6 49·5 50·6 51·5 52·5 54·5 55·5 56·5 58·6 60·6 65·6 66·5 68·5 69·6 10·6 11·4
NAtomlc /lline I/J line ')' line Number AX lOS em. AX lOS em. AX lOS em,
40 41 42 44
45 46 41 50 51 51 58 59 60 62 63 64 66 68 13 14 16 11 18 19
5·501 5 ·181 4·660 4·168 3·245 2·411 2·366 2·265 2·115 2·008 1·925 1·853 1·111 1·591 1·330 1·201 1·155 1·121 1·092
3·928
2·424 2·315
2·313 2·209
1·912 1·888 J ·818
1·893 1·814
1·563
1·281 1·112 1·138 1·104 ] ·018
B. O. 1. 110,.141.,
given in a later paper. The spectra both of K and CI were obtained by means of a target of KCI, but it is very improbable that the observed lines have been attributed to the wrong elements. The a line for elements from Y onwards appeared to consist of a very close doublet, an effect previously observed by Bragg in the case of rhodium. The results obtained for the spectra of the L series are given in Table II and plotted in Figure 3. These spectra contain five Jines, a, /3, )', 6, (, reckoned in order of decreasing wave-length and decreasing intensity. There is also always a faint companion a' on the long wave--Iength side of a, a rather faint line • between /3 and )' for the rare earth elements at least, and a number of very faint lines of wave--Iength greater than a. Of these, a, /3, .' and )' have been systematically measured with the object of finding out how the spectrum alters from one element to another. The fact that often values are not given for all these lines merely indicates the incompleteness of the work. The spectra, so far as they have been examined, are so entirely similar that without doubt a, /3, and), at least always exist. Often )' was not included in the limited range of wavelengths which can be photographed on one plate. Sometimes lines have not been measured, either on account of faintness or of the confusing proximity of lines due to impurities. . . . CONCLUSIONS
In Figure 3 the spectra of the elements are arranged on horizontal lines spaced at equal distances. The order chosen for the elements is the order of the atomic weights, except in the cases of A, Co, and Te, where this clashes with the order of the chemical properties. Vacant lines have been left for an element between Mo and Ru, an element between Nd and Sa, and an element between Wand Os, none of which are yet known, while Tm, which Welsbach has separated into two constituents, is given two lines. This is equivalent to assigning to successive elements a series of successive characteristic integers. On this principle the integer N for AI, the thirteenth element, has been taken to be 13, and the values of N then assumed by the other elements are given on the left-hand side of Figure 3. This proceeding is justified by the fact that it introduces perfect regUlarity into the X-ray spectra. Examination of Figure 3 shows that the values of v~ for all the lines examined both in the K and the L series now fall on regular curves which approximate to straight lines. The same thing is shown more clearly by comparing the values of N in Table I with those of
QK=
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the radiations of very short wave-length which gradually diverge from this relation. Again. in Table II a comparison of N with
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where v is the frequency of the La line, shows that QL = N - 7·4 approximately, although a systematic deviation clearly shows that the relation is not accurately linear in this case. Now if either the elements were not characterized by these integers, or any mistake had been made in the order chosen or in the number of places left for unknown elements, these regularities would at once disappear. We can therefore conclude from the evidence of the X-ray spectra alone, without using any theory of atomic structure, that these integers are really characteristic of the elements. Further, as it is improbable that two different stable elements should have the same integer, three, and only three, more elements are likely to exist between AI and Au. As the X-ray spectra of these elements can be confidently predicted, they should not be difficult to find. The examination of keltium would be of exceptional interest, as no place has been assigned to this element. Now Rutherford has proved that the most important constituent of an atom is its central positively charged nucleus, and van den Broek has put forward the view that the charge carried by this nucleus is in all cases an integral multiple of the charge on the hydrogen nucleus. There is every reason to suppose that the integer which controls the X-ray spectrum is the same as the number of electrical units in the nucleus, and these experiments therefore give the strongest possible support to the hypothesis of van den Broek. Soddy has pointed out that the chemical properties of the radio-elements are strong evidence that this hypothesis is true for the elements from thallium to uranium, so that its general validity would now seem to be established.
COMMENTARY ON
The Small Furniture of Earth
T
HE next two selections deal with the wave theory of X-rays and the atomic theory of crystals. The bringing together and verification of these theories in a single famous experiment was one of the major events of the twentieth-century renaissance in physics. Crystals were the subject of much attention in the eighteenth and nineteenth centuries, their optical properties and geometric relations being carefully studied by mineralogists, crystallographers and mathematicians. As early as 1824 the hypothesis was advanced that crystals consist of layers of atoms distributed in regular patterns; 1 in the 1890s mathematicians had worked out fully the number of possible ways the atoms inside a crystal could be distributed. However, experimental proof of these fundamental ideas was still lacking in 1910. Attempts to confirm the theory of crystals by diffraction experiments with light were unsuccessful because the waves were too long for the job; it was as if one were to try to measure the inside of a thimble with a foot-rule. Another important hypothesis, also in a doubtful status at that time, related to X-rays. The majority of physicists were convinced that X-rays were waves of a length much shorter than light rays, but again early attempts to prove the conjecture were unsuccessful. The crucial experiment testing both hypotheses was proposed by the German physicist Von Laue, who was then assistant lecturer in Professor Sommerfeld's d~partment at the University of Munich. 2 He reasoned that if X-rays were short waves and crystals three-dimensional lattices of atoms, a pencil of X-rays passed through a crystal would produce a characteristic pattern on a photographic plate. The X-rays, in other words, could be made to report the arrangement of the tiny furniture encountered inside the crystal. The experiment, performed in 1912, was an extraordinary triumph. S The X-rays portrayed the interior of the crystals, and the crystals repaid the favor by disclosing the form of the X-rays. Von Laue
1 Max von Laue, History oj Physics, New York, 1950, p. 119: "The first scientist to combine the newly created concept of the chemical alom with this idea [of the building-block structure of crystals] and to assume that space lattices are made up of chemical atoms was the physicist Ludwig August Seeber. . . . He published his ideas in 1824, i.e., thirty-two years prior to the entry of atomistics into modem physics in the form of the kinetic theory of gases." 2 See also the introduction to the paper by H. G. J. Moseley, pp. 840-841. 3 The experiment, though conceived by Laue, was actuaJ]y performed by two young Munich research students, Friedrich and Knipping, who had just taken their doctorates under Rontgen. See Kathleen Lonsdale, Crystals and X-Rays, London, 1948. pp. 1-22, for an authoritative historical introduction to the subject; also the standard work for the advanced student: Sir Lawrence Bragg, The Crystalline Slate, a General Survey, London, 1949.
8S1
852
Edlto,'s CO"-lit
described this reciprocal disclosure as "one of those surprising events to which physics owes its power of conviction." The first of the selections below, on Von Laue's experiment and its relation to the geometry of X-rays and crystals, is by the great physicist Sir William Brass. The history of modem crystallography is in large part the history of his researches. Brass was born in Cumberland in 1862 and after a brilliant school career accepted a professorship in physics at the University of Adelaide, Australia. His first research paper did not appear, remarkably enough, until 1904, when he was forty-two. It is unusual for a physicist SO long to defer his original investigations, but the paper itself-it was concerned with alpha particles--was immediately recognized as a first-class achievement and marked the beginning of a prolific output of creative studies. In 1907 Brass was elected a Fellow of the Royal Society, and a year later returned to England to take the Cavendish chair at Leeds. It was there that he became interested in X-rays, which he then regarded, contrary to prevailing opinion, as particles rather than waves. The Laue experiment of 1912 convinced him he was mistaken, and in the same year Bragg and his son took up the research on X-rays and crystals for which in 1915 they jointly received the Nobel prize in physics. Their work "laid the foundation of one of the most beautiful structures of modem science"; it was used for fundamental advances in both inorganic and organic chemistry, metallurgy is deeply indebted to it, as are other branches of pure and applied science. 4 Bragg had a crowded and happy career during which he worked on many other subjects besides X-rays and raised a crop of distinguished pupils who enriched many parts of physics. In 1915 he held a chair at University CoJlege, London; during the First World War he directed acoustic research on submarine detection; in 1923 he became director of the Royal Institution and of the Davy-Faraday Research Laboratory. Not the least of his gifts was in popular exposition. He enjoyed nothing more than to give lectures and experimental demonstrations to youngsters and genera] audiences; his connection with the Royal Institution (he served as president from 1935 to 1940) fortunately facilitated his exercise of this art. The article which fol1ows is a chapter from Bragg's book, The Universe of Light, based on Christmas Lectures delivered in 1931. The second selection is a simple and attractive account by Philippe Le Corbeiller of the mathematics of crystals. A suitable complement to Bragg's discussion, it emphasizes the experimental proof of the theory of space groups. In the development of crystallography the mathematics of group theory and of symmetry has, as I remarked earlier, played a remarkable part. Just as Adams and Leverrier, for example, decreed the motion 4
E. N. daC. Andrade, "Sir William Bragg" (obituary), Nature, March 28, 1942.
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and position of the planet Neptune before it was discovered, so mathematicians by an exhaustive logical analysis of certain properties of space and of the possible transformations (motions) within space, decreed the permissible variations of internal structure of crystals before observers were able to discover their actual structure. Mathematics, in other words, not only enunciated the applicable physical laws, but provided an invaluable syllabus of research to guide future experimenters. The history of the physical sciences contains many similar instances of mathematical prevision. Models, concepts, theories are initially expressed in mathematical form; later they are tested by observation and either confirmed, or disproved and discarded. Not uncommonly, of course, the model is overhau1ed to conform more closely to experimental data and the tests are then repeated. The theory of space groups and crystal classes is among the most successful and striking examples of mathematical model making. Mr. Le Corbeiller is professor of general education and of applied physics at Harvard University. He was born and educated in France, has served as a member of the engineering staff of the Deparlemenl des Communications, as an official of the French Government Broadcasting System and in other administrative and academic posts. His publications include papers on a1gebra, number theory, oscil1ating generators, electroacoustics and other scientific topics. The article on crystals, cast in the congenial framework of a mock-Socratic dialogue, first appeared in Scientific American, January 1953.
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The Rontgen Rays By SIR WILLIAM BRAGG
. . . X-RAYS are Fnerally produced as a consequence of the electric spark or discharse in a space where the pressure of the air or other gas is extremely low. The electric spark has for centuries been a subject of interested observation, but no great step forward was made until it was ananged that the discharge should take place in a &lass tube or bulb from which the air bad been pumped out more or less completely. 'The spark became longer, wider, and more highly c:oJoured as the pressure
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. .. no., = same, P.EpO'> = part). Isomerism is not an exception, it is the rule. The larger the molecule. the more isomeric alternatives are offered. Figure I shows one of the simplest cases, the two kinds of propyl-alcohol, both
981
H ereditl alld the OltalJlllm Theor>,
H~~--{---O-H H
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FIGURE J-The two isomcrcs or propyl-alcohol
consisting of 3 carbons (C), 8 hydrogens (H), I oxygen (0)." The latter can be interposed between any hydrogen and its carbon, but only the two cases shown in our figure are different substances. And they really are. All their physical and chemical constants are distinctly different. Also their energies are different, they represent 'different levels.' The remarkable (act is that both molecules are perfectly stable, both behave as though they were 'lowest states.' There are no spontaneous transitions from either state towards the other. The reason is that the two configurations are not neighbouring configurations. The transition from one to the other can only take place over intermediate configurations which have a greater energy than either of them. To put it crudely, the oxygen has to be extracted from one position and has to be inserted into the other. There does not seem to be a way of doing that without passing through configurations of considerably higher energy. The state of affairs is sometimes figuratively pictured as in Figure 2, in which 1 and 2 represent the two isomeres, 3 the 'threshold' between them, and the two arrows indicate the 'lifts,' that is to say t the energy supplies required to produce the transition from state 1 to state 2 or from state 2 to state I, respectively. Now we can give our 'second amendment: which is that transitions of this 'isomeric' kind are the only ones in which we shaH be interested in our biological application. It was these we had in mind when explaining 'stability' on pp. 978-979. The 'quantum jump' which we mean is the transition from one relatively stable molecular configuration to another. The energy supply required for the transition (the quantity denoted Ii Models. in which C. Hand 0 were represented by black. white and red wooden balls respectively. were exhibited at the lecture. I have not reproduced them here. because their likeness to the actual molecules is not appreciably greater than that of Figure 1.
982
3
I FIGURE 2-Encr1Y threshold
(3)
between the isomeric levels
(I)
and (2) The arrows indicate
the minimum enerlies required for transition
by W) is not the actual level difference. but the step from the initial level up to the threshold (see the arrows in Figure 2). Transitions with no threshold interposed between the initial and the final state are entirely uninteresting, and that not only in our biological application. They have actually nothing to contribute to the chemjcal stability of the molecule. Why? They have no lasting effect, they remain unnoticed. For, when they occur, they are almost immediately followed by a relapse into the initial state, since nothing prevents their return.
DELBROCK'S MODEL DISCUSSED AND TESTED Sane sicut lux seipsam et tenebras manifestat. sic veritas norma sui et falsi est. 7 -SPINOZA, Ethic.'!. P. 11, Prop. 43. THE GENERAL PICTURE OF THE HEREDITARY SUBSTANCE
From these facts emerges a very simple answer to our question, namely: Are these structures, composed of comparatively few atoms, capable of withstanding for long periods the disturbing influence of heat motion to which the hereditary substance is continually exposed? We shall assume the structure of a gene to be that of a huge molecule, capable only of discontinuous change, which consists in a rearrangement of the atoms and leads to an isomeric R molecule. The rearrangement may affect only a small region of the gene, and a vast number of different rearrangements may be possible. The energy thresholds, separating the actual configuration from any possible isomeric ones, have to be high enough (compared 'I Truly, as light manifests itself and darkness, thus truth is the stundard of itself and of error. S For convenience I shall continue to call it un isomeric trunsition. though it would be absurd to exclude the possibility of any exchange with the environment.
Ht'I'~d;/J'
und Iht Quurrlum Thto,,.
98)
with the average heat energy of an atom) to make the change-over a rare event. These rare events we shall identify with spontaneous mutations. The later parts of this chapter will be devoted to putting this general picture of a gene and of mutation (due mainly to the German physicist M. Delbriick) to the test, by comparing it in detail with genetical facts. Before doing so, we may fittingly make some comment on the foundation and general nature of the theory. THE UNIQUENESS OF THE PICTURE
Was it absolutely essential for the biological question to dig up the deepest roots and found the picture on quantum mechanics? The conjecture that a gene is a molecule is to-day, I dare say, a commonplace. Few biologists. whether familiar with quantum theory or not, would disagree with it. In the opening section we ventured to put it into the mouth of a pre-quantum physicist, as the only reasonable explanation of the observed permanence. The subsequent considerations about isomerism, threshold energy, the paramount role of the ratio W:kT in determining the probability of an isomeric transition-all that could very wen be introduced on a purely empirical basis, at any rate without drawing explicitly on quantum theory. Why did I so strongly insist on the quantum-mechanical point of view, though I could not really make it clear in this little book and may well have bored many a reader? Quantum mechanics is the first theoretical aspect which accounts from first principles for all kinds of aggregates of atoms actually encountered in Nature. The Heitler-London bondage is a unique, singular feature of the theory. not invented for the purpose of explaining the chemical bond. It comes in quite by itself, in a highly interesting and puzzling manner, being forced upon us by entirely different considerations. It proves to correspond exactly with the observed chemical facts, and, as I said, it is a unique feature. well enough understood to ten with reasonable certainty that 'such a thing could not happen again' in the further development of quantum theory. Consequently, we may safely assert that there is no ahernative to the molecular explanation of the hereditary substance. The physical aspect leaves no other possibility to account for its permanence. I f the Delbriick picture should fail, we would have to give up further attempts. That is the first point I wish to make. SOME TRADITIONAL MISCONCEPTIONS
But it may be asked: Are there really no other endurable structures composed of atoms except molecules? Does not a gold coin, for example, buried in a tomb for a couple of thousand years, preserve the traits of the portrait stamped on it? It is true that the coin consists of an enormous
Erwin Sc",6difl~'
984
number of atoms, but surely we are in this case not inclined to attribute the mere preservation of shape to the statistics of large numbers. The same remark applies to a neatly developed batch of crystals we find embedded in a rock, where it must have been for geological periods without changing. That leads us to the second point I want to elucidate. The cases of a molecule, a solid, a crystal are not really different. Tn the light of present knowJedge they are virtuaHy the same. Unfortunately, school teaching keeps up certain traditional views, which have been out of date for many years and which obscure the understanding of the actual state of affairs. Indeed, what we have learnt at school about molecules does not give the idea that they are more closely akin to the soJid state than to the liquid or gaseous state. On the contrary, we have been taught to distinguish carefully between a physical change, such as melting or evaporation in which the molecules are preserved (so that, for example, alcohol, whether solid, liquid or a gas, always consis~s of the same molecuJes, C:!H1iO), and a chemical change, as, for example, the burning of alcohol,
where an alcohol molecule and three oxygen molecules undergo a rearrangement to form two molecules of carbon dioxide and three molecules of water. About crystals, we have been taught that they form threefold periodic lattices, in which the structure of the single moJecule is sometimes recognizable, as in the case of alcohol and most organic compounds, while in other crystals, e.g. rock-salt (NaCI) , NaCI molecules cannot be unequivocally delimited, because every Na atom is symmetrically surrounded by six CI atoms, and vice versa. so that it is largely arbitrary what pairs, if any, are regarded as molecular partners. Finally, we have been told that a solid can be crystalline or not, and in the latter case we call it amorphous. DIFFERENT 'STATES' OF MATTER
Now I would not go so far as to say that all these statements and distinctions are quite wrong. For practical purposes they are sometimes useful. But in the true aspect of the structure of matter the limits must be drawn jn an entirely different way. The fundamental distinction is between the two lines of the following scheme of 'equations': moJecule gas
= solid = crystal. = Jiquid = amorphous.
We must explain these statements briefly. The so-called amorphous solids are either not really amorphous or not really solid. In 'amorphous' char-
H",dit.,· IIl1d ''', O"QlItum Tht'or)
coal fibre the rudimentary structure of the graphite crystal has been disclosed by X-rays. So charcoal is a solid, but also crystalline. Where we find no crystalline structure we have to regard the thing as a liquid with very high 'viscosity' (internal friction). Such a substance discloses by the absence of a well-defined melting temperature and of a latent heat of melting that it is not a true solid. When heated it softens gradually and eventually liquefies without discontinuity. (I remember that at the end of the first Great War we were given in Vienna an asphalt-like substance as a substitute for coffee. It was so hard that one had to use a chisel or a hatchet to break the little brick into pieces, when it would show a smooth, shell-like cleavage. Yet, given time, it would behave as a liquid, closely packing the lower part of.a vessel in which you were unwise enough to leave it for a couple of days.) The continuity of the gaseous and liquid state is a well-known story. You can liquefy any gas without discontinuity by taking your way 'around' the so-called critical point. But we shaH not enter on this here. THE DISTINCTION THAT REALLY MATTERS
We have thus justified everything in the above scheme, except the main point, namely, that we wish a molecule to be regarded as a solid crystal. The reason for this is that the atoms forming a molecule, whether there be few or many of them, are united by forces of exactly the same nature as the numerous atoms which build up a true soHd, a crystal. The molecule presents the same solidity of structure as a crystal. Remember that it is precisely this solidity on which we draw to account for the permanence of the gene! The distinction that is really important in the structure of matter is whether atoms are bound together by those 'solidifying' Heitler-London forces or whether they are not. In a solid and in a molecule they all are. In a gas of single atoms (as e.g. mercury vapour) they are not. In a gas composed of molecules, only the atoms within every molecule are ]in ked in this way.
=
THE APERIODIC SOLID
A small molecule might be called 'the germ of a solid.' Starting from such a small solid germ, there seem to be two different ways of building up larger and larger associations. One is the comparatively dull way of repeating the same structure in three directions again and again. That is the way followed in a growing crystal. Once the periodicity is established, there is no definite limit to the size of the aggregate. The other way is that of building up a more and more extended aggregate without the dull device of repetition. That is the case of the more and more complicated
Erwin Schrodllllltr
986
organic molecule in which every atom, and every group of atoms, plays an individual role, not entirely equivalent to that of many others (as is the case in a periodic structure). We might quite properly call that an aperiodic crystal or solid and express our hypothesis by saying: We believe a gene--or perhaps the whole chromosome fibre Il_to be an aperiodic solid. THE VARIETY OF CONTENTS COMPRESSED IN THE MINIATURE CODE
It has often been asked how this tiny speck of material, the nucleus of the fertilized egg, could contain an elaborate code-script involving all the future development of the organism? A well-ordered association of atoms, endowed with sufficient resistivity to keep its order permanently, appears to be the only conceivable material structure, that offers a variety of possible ('isomeric') arrangements. sufficienrly large to embody a complicated system of 'determin~tions' within a small spatial boundary. Indeed, the number of atoms in such a structure need not be very large to produce an almost unlimited number of possible arrangements. For illustration, think of the Morse code. The two different signs of dot and dash in wen· ordered groups of not more than four allow of thirty different specifications. Now, if you allowed yourself the use of a third sign, in addition to dot and dash, and used groups of not more than ten. you could form 29,524 different 'letters'; with five signs and groups up to 25, the number is 372,529,029,846,19IA05. It may be objected that the simile is deficient, because our Morse signs may have different composition (e.g.• - - and" -) and thus they are a bad analogue for isomerisn:t. To remedy this defect, let us pick. from the third example, only the combinations of exactly 25 symbols and only those containing exactly 5 OU! of each of the supposed 5 types (5 dot Ii, 5 dashes, etc.). A rough count gives you the number of combinations as 62,330,000,000,000, where the zeros on the right stand tor figures which I have not taken the trouble to compute. Of course, in the actual case, by no means 'every' arrangement of the group of atoms will represent a possible molecule; moreover, it is not a que~tion of a code to be adopted arbitrarily, for the code-script must itself he the operative factor bringing about the development. But. on the other hand, the number chosen in the example (25) is stiJI very small, and we have envisaged only the simple arrangements in one line. What we wish to illustrate is simply that with the molecular picture of the gene it is no longer inconceivable that the miniature code should preciliely correspond with a highly complicated and specified plan of development and should somehow contain the means to put it inro operation. It
ThaI il is highly flexible b. no objection:
"'0
is a thin copper wire.
987
Htredlty and tit. Quantum TIt,ory
COMPARISON WITH PACTS: DEGREE OP STABILITY; DISCONTINUITY OP MUTATIONS
Now let us at last proceed to compare the theoretical picture with the biological facts. The first question obviously is, whether it can really account for the high degree of permanence we observe. Are threshold values of the required amount-high mu1tiples of the average heat energy kT-reasonabJe, are they within the range known from ordinary chemistry? That question is trivial; it can be answered in the affirmative without inspecting tables. The molecules of any substance which the chemist is able to isolate at a given temperature must at that temperature have a lifetime of at least minutes. (That is putting it mildly; as a rule they have much more.) Thus the threshold values the chemist encounters are of necessity precisely of the order of magnitude required to account for practically any degree of permanence the biologist may encounter; for we recall from the discussion of stability that thresholds varying within a range of about 1: 2 will account for lifetimes ranging from a fraction of a second to tens of thousands of years. But let me mention figures, for future reference. The ratios W /kT mentioned by way of example on page 979, viz. W
- = 30, SO, 60, kT producing lifetimes of
*0 sec., 16 months, 30,000 years, respectively, correspond at room temperature with threshold values of 0·9, l·S, l·S electron-volts.
We must explain the unit 'electron-volt,' which is rather convenient for the physicist, because it can be visualized. For example, the third number (loS) means that an electron, accelerated by a voltage of about 2 volts, would have acquired just sufficient energy to effect the transition by impact. (For comparison, the battery of an ordinary pocket flash-light has 3 volts.) These considerations make it conceivable that an isomeric change of configuration in some part of our molecule, produced by a chance fluctuation of the vibrational energy can actually be a sufficiently rare event to be interpreted as a spontaneous mutation. Thus we account, by the very principles of quantum mechanics, for the most amazing fact about mutations, the fact by which they first attracted de Vries's attention. namely, that they are 'jumping' variations. no intermediate forms occurring.
Erwin Scllriidtn,er
988
STABILITY OF NATURALLY SELECTED GENES
Having discovered the increase of the natural mutation fate by any kind of ionizing rays, one might think of attributing the natural rate to the radio-activity of the soil and air and to cosmic radiation. But a quantitative comparison with the X-ray results shows that the 'natural radiation' is much too weak and could account only for a small fraction of the natural rate. Granted that we have to account for the rare natural mutations by chance fluctuations of the heat motion, we must not be very much astonished that Nature has succeeded in making such a subtle choice of threshold values as is necessary to make mutation rare. For we have, earlier in these lectures. arrived at the conclusion that frequent mutations are detrimental to evolution. Individuals which, by mutation. acquire a gene configuration of insufficient stability, will have little chance of seeing their 'ultra-radical; rapidly mutating descendancy survive long. The species wil1 be freed of them and will thus collect stable genes by natural selection. THE SOMETIMES LOWER STABILITY OF MUTANTS
But, of course, as regards the mutants which occur in our breeding experiments aod which we select, qua mutants, for studying their offspring. there is no reason to expect that they should all show that very high stability. For they have not yet been 'tried out'-or, if they have. they have been 'rejected' in the wild breeds-possibly for too high mutability. At any rate, we are not at all astonished to learn that actually some of these mutants do show a much higher mutability than the normal 'wild' genes. TEMPERATURE INFLUENCES UNSTABLE GENES LESS THAN STABLE ONES
This enables us to test Our mutability formula, which was
(It will be remembered that t is the time of expectation for a mutation with threshold energy W.) We ask: How does t change with the temperature? We easily find from the preceding formula in good approximation the ratio of the value of I at temperature T + 10, to that at temperature T tT+10 __ =
e-lOlf/kT2.
tT
The exponent being now negative, the ratio is, naturally, smaller than 1. The time of expectation is diminished by raising the temperature, the
Heredfo Imd
Ih~
OlUm/llm
Th~orl
989
mutability is increased. Now that can be tested and has been tested with the fly Drosophila in the range of temperature which the insects wiJI stand. The result was, at first sight, surprising. The low mutability of wild genes was distinctly increased, but the comparatively high mutability occurring with some of the already mutated genes was not, or at any rate was much less, increased. That is just what we expect on comparing our two formulae. A large value of W /kT, which according to the first formula is required to make t large (stable gene), wm, according to the second one, make for a small value of the ratio computed there, that is to say for a considerable increase of mutability with temperature. (The actual values of the ratio seem to lie between about 1h and ¥.i. The reciprocal, 2-5, is what in an ordinary chemical reaction we call the van 't Hoff factor.) HOW X-RAYS PRODUCE MUTATION
Turning now to the X-ray-induced mutation rate, we have already inferred from the breeding experiments, first (from the proportionality of mutation rate, and dosage), that some single event produces the mutation; secondly (from quantitative results and from the fact that the mutation rate is determined by the integrated ionization density and independent of the wave-length), this single event must be an ionization, or similar process, which has to take place inside a certain volume of only about 10 atomic-distances-cubed, in order to produce a specified mutation. According to our picture, the energy for overcoming the threshold must obviously be furnished by that explosion-like process, ionization or excitation. I call it explosion-like, because the energy spent in one ionization (spent, incidentally, not by the X-ray itself, but by a secondary electron it produces) is well known and has the comparatively enormous amount of 30 electron-volts. It is bound to be turned into enormously increased heat motion around the point where it is discharged and to spread from there in the form of a 'heat wave,' a wave of intense oscillations of the atoms. That this heat wave should still be able to furnish the required threshold energy of I or 2 electron-volts at an average 'range of action' of about ten atomic distances, is not inconceivable, though it may wel1 be that an unprejudiced physicist might have anticipated a slightly lower range of action. That in many cases the effect of the explosion wi11 not be an orderly isomeric transition but a lesion of the chromosome, a lesion that becomes lethal when, by ingenious crossings, the uninjured partner (the corresponding chromosome of the second set) is removed and replaced by a partner whose corresponding gene is known to be itself morbid -all that is absolutely to be expected and it is exactly what is observed.
THEIR EFFICIENCY DOES NOT DEPEND ON SPONTANEOUS MUTABILITY
Quite a few other features are, if not predictable from the picture, easily understood from it. For example, an unstable mutant does not on the average show a much higher X-ray mutation rate than a stable one. Now, with an explosion furnishing an energy of 30 electron-volts you would certainly not expect that it makes a lot of difference whether the required threshold energy is a little larger or a little smaller, say 1 or 1.3 volta. REVERSIBLE MUTATIONS
In some cases a transition was studied in both directions, say from a certain 'wild' gene to a specified mutant and back from that mutant to the wild gene. In such cases the natural mutation rate is sometimes nearly the same, sometimes very different. At first sight one is puzzled, because the threshold to be overcome seems to be the same in both cases. But, of course, it need not be, because it has to be measured from the energy level of the starting configuration, and that may be different for the wild and the mutated gene. (See Figure 2 on page 982, where 'I' might refer to the wild allele, '2' to the mutant, whose lower stability would be indicated by the shorter arrow.) On the whole, I think, Delbriick's 'model' stands the tests fairly well and we are justified in using it in further considerations. ORDER, DISORDER AND ENTROPY Nec corpus mentem ad cogitandum nec mens corpus ad motum, neque ad quietem nec ad aliquid (si quid est) aliud determinare potest. 10 -SPINOZA, Ethics, P. 111, Prop. 2 A REMARKABLE GENERAL CONCLUSION FROM THE MODEL
Let me refer to the last phrase on page 986, in which I tried to explain that the molecular picture of the gene made it at least conceivable 'that the miniature code should be in one-to-one correspondence with a highly complicated and specified plan of development and should somehow contain the means of putting it into operation.' Very well then, but how does it do this? How are we going to turn 'conceivability' into true understanding? Delbriick's molecular model, in its complete generality, seems to contain no hint as to how the hereditary substance works. Indeed, I do not expect that any detailed information on this question is likely to come from physics in the near future. The advance is proceeding and will, I am 10 Neither can the body determine the mind to think, nor the mind the body to move or to rest nor to anything else, if such there be.
Heredity and 'he Quant/1m Theory
991
sure, continue to do so, from biochemistry under the guidance of physiology and genetics. No detailed information about the functioning of the genetical mechanism can emerge from a description of its structure so general as has been given above. That is obvious. But, strangely enough, there is just one general conclusion to be obtained from it, and that, I confess, was my only motive for writing this book. From Delbriick's general picture of the hereditary substance it emerges that living matter, while not eluding the 'laws of physics' as established up to date, is likely to involve 'other laws of physics' hitherto unknown, which, however, once they have been revealed, wiJ] form just as integral a part of this science as the former. ORDER BAS ED ON ORDER
This is a rather subtle Hne of thought, open to misconception in more than one respect. All the remaining pages are concerned with making it clear. A preliminary insight, rough but not altogether erroneous, may be found in the following considerations: It has been explained 11 that the laws of physics, as we know them, are statisticallaws. 1 :! They have a lot to do with the natural tendency of things to go over into disorder. But, to reconcile the high durability of the hereditary substance with its minute size, we had to evade the tendency to disorder by 'inventing the molecule,' in fact, an unusually large molecule which has to be a masterpiece of highly differentiated order, safeguarded by the conjuring rod of quantum theory. The laws of chance are not invalidated by this 'invention,' but their outcome is modified. The physicist is familiar with the fact that the classical laws of physics are modified by quantum theory, especiaUy at low temperature. There are many instances of this. Life seems to be one of them, a particularly striking one. Life seems to be orderly and lawful behaviour of matter, not based exclusively on its tendency to go over from order to disorder, but based partly on existing order that is kept up. To the physicist-but only to him-J could hope to make my view clearer by saying: The living organism seems to be a macroscopic system which in part of its behaviour approaches to that purely mechanical (as contrasted with thermodynamical) conduct to which all systems tend, as the temperature approaches the absolute zero and the molecular disorder is removed. The non-physicist finds it hard to believe that really the ordinary laws In earlier discussions of the book. ED. To state this in complete generality about 'the laws of physics' is perhaps challengea ble. II
12
Erwill ScltriJt1IlI,er
51512
of physics, which he regards as the prototype of inviolable precision, should be based on the statistical tendency of matter to go over into disorder. The general principle involved is the famous Second Law of Thermodynamics (entropy principle) and its equally famous statistical foundation. In the fo]]owing sections I will try to sketch the bearing of the entropy principle on the large-scale behaviour of a living organismforgetting at the moment all that is known about chromosomes, inheritance, and so on. LIVING MATTER EVADES THE DECAY TO EQUILIBRIUM
What is the characteristic feature of life? When is a piece of matter said to be alive? When it goes on 'doing something,' moving, exchanging material with its environent, and so forth, and that for a much longer period than we would expect an inanimate piece of matter to 'keep going' under similar circumstances. When a system that is not alive is isolated or placed in a uniform environment, all motion usually comes to a standstill very soon as a result of various kinds of friction; differences of electric or chemical potential are equalized, substances which tend to form a chemical compound do so, temperature becomes uniform by heat conduction. After that the whole system fades away into a dead, inert lump of matter. A permanent state is reached, in which no observable events occur. The physicist cans this the state of thermodynamical equilibrium, or of 'maximum entropy! Practical1y, a state of this kind is usually reached very rapidly. Theoretically, it is very often not yet an absolute equiHbrium, not yet the true maximum of entropy. But then the final approach to equilibrium is very slow. It could take anything between hours, years, centuries. . .. To give an e~ample--one in which the approach is still fairly rapid: if a glass filled with pure water and a second one fi]]ed with sugared water are placed together in a hermetical1y closed case at constant temperature, it appears at first that nothing happens, and the impression of complete equilibrium is created. But after a day or so it is noticed that the pure water, owing to its higher vapour pressure, slowly evaporates and condenses on the solution. The latter overflows. Only after the pure water has totally evaporated has the sugar reached its aim of being equally distributed among all the liquid water available. These ultimate slow approaches to eqUilibrium could never be mistaken for life, and we may disregard them here. I have referred to them in order to clear myself of a charge of inaccuracy. IT FEEDS ON 'NEGATIVE ENTROPY'
It is by avoiding the rapid decay into the inert state of 'equilibrium,' that an organism appears so enigmatic; so much so, that from the earliest
Htrndt',. and
th~
993
Quantum ThMT)
times of human thought some special non-physical or supernatural force (vi.f viva, entelechy) was claimed to be operative in the organism, and in some quarters is still claimed. How does the living organism avoid decay? The obvious answer is: By eating, drinking, breathing and (in the case of plants) assimi1ating. The technical term is metabolism. The Greek word (/LE'TafjaA'AEtv) means change or exchange. Exchange of what? Originally the underlying idea is, no doubt. exchange of material. (E.g. the German for metabolism is Stoffwechsel.) That the exchange of material should be the essential thing is absurd. Any atom of nitrogen, oxygen, sulphur, etc., is as good as any other of its kind; what could be gained by exchanging them? For a while in the past our curiosity was silenced by being told that we feed upon energy. In some very advanced country (I don't remember whether it was Germany or the U.S.A. or both) you could find menu cards in restaurants indicating, in addition to the price, the energy content of every dish. Needless to say, taken literally, this is just as absurd. For an adult organism the energy content is as stationary as the material content. Since, surely, any calorie is worth as much as any other calorie, one cannot see how a mere exchange could help. What then is that precious something contained in our food which keeps us from death? That is easily answered. Every process, event, happening--caU it what you will; in a word, everything that is going on in Nature means an increase of the entropy of the part of the world where it is going on. Thus a living organism continually increases its entropy--or, as you may say, produces positive entropy-and thus tends to approach the dangerous state of maximum entropy, which is death. It can only keep aloof from it, i.e. alive. by continually drawing from its environment negative entropy-which is something very positive as we shall immediately see. What an organism feeds upon is negative entropy. Or to put it less paradoxically, the essential thing in metabolism is that the organism succeeds in freeing itself from aU the entropy it cannot help producing while alive. WHAT IS ENTROPY?
What is entropy? Let me first emphasize that it is not a hazy concept or idea, but a measurable physical quantity just like the length of a rod. the temperature at any point of a body. the heat of fusion of a given crystal or the specific heat of any given substance. At the absolute zero point of temperature (roughly -273° C.) the entropy of any substance is zero. When you bring the substance into any other state by slow, reversible little steps (even if thereby the substance changes its physical or chemical nature or splits up into two or more parts of different physical or chemical nature) the entropy increases by an amount which is computed by
dividing every little portion of heat you had to supply in that procedure by the absolute temperature at which it was supplied-and by summing up all these small contributions. To give an example, when you melt a solid, its entropy increases by the amount of the heat of fusion divided by the temperature at the melting-point. You see from this, that the unit in which entropy is measured is cal./o C. (just as the calorie is the unit of heat or the centimetre the unit of length). THE STATISTICAL MEANING OF ENTROPY
I have mentioned this technical definition simply in order to remove entropy from the atmosphere of hazy mystery that frequently veils it. Much more important for us here is the bearing on the statistical concept of order and disorder, a connection that was revealed by the investigations of Boltzmann and Gibbs in statistical physics. This too is an exact quantitative connection, and is expressed by entropy = k log D, where k is the so-called Boltzmann constant (= 3.2983 . 10- 2' cal./o C.), and D a quantitative measure of the atomistic disorder of the body in question. To give an exact explanation of this quantity D in brief nontechnical terms is wen-nigh impossible. The disorder it indicates is partly that of heat motion, partly that which consists in different kinds of atoms or molecules being mixed at random, instead of being neatly separated, e.g. the sugar and water molecules in the example quoted above. Boltzmann's equation is well iI1ustrated by that example. The gradual 'spreading out' of the sugar over all the water available increases the disorder D, and hence (since the 10garithm of D increases with D) the entropy. It is also pretty clear that any supply of heat increases the turmoil of heat motion, that is to say increases D and thus increases the entropy; it is particularly clear that this should be so when you melt a crystal, since you thereby destroy the neat and permanent arrangement of the atoms or molecules and turn the crystal lattice into a continually changing random distribution. An isolated system or a system in a uniform environment (which for the present consideration we do best to include as a part of the system we contemplate) increases its entropy and more or less rapidly approaches the inert state of maximum entropy. We now recognize this fundamental law of physics to be just the natural tendency of things to approach the chaotic state (the same tendency that the books of a library or the piles of papers and manuscripts on a writing desk display) unless we obviate it. (The analogue of irregular heat motion, in this case, is our handling those objects now and again without troubling to put them back in their proper places.)
Hrrrdit, and thr Quantum
Th~or,.
99S
ORGANIZATION MAINTAINED BY EXTRACTING 'ORDER' FROM THE ENVIRONM ENT
How would we express in terms of the statistical theory the marvellous faculty of a living organism, by which it delays the decay into thermIt of the original set corresponds to 'K.'7 in the remaining set. I We have tacitly disrellarded such trivial duplications as are instanced, in the deci· mal system, by 0.1 = 0.09 or 0.8 = 0.79.
1064
Inversely, take out triadic 0.20, meaning, as we made out, %. The corresponding dyadic 0.10 means the infinite series
1
1
1
1
1
-+-+-+-+-+ ... 3 5 9
2 2 2 27 2 If you multiply this by the square of 2, which is 4, you get: 2 + the same series. In other words, three times our series equals 2, the series equals %; that is to say, the number of the 'remaining set' corresponds (or 'is mated') to the number % in the original set.] The remarkable fact about our 'remaining set' is that, though it covers no measurable interval, yet it has still the vast extension of any continuous range. This astonishing combination of properties is, in mathematical language, expressed by saying that our set has stm the 'potency' of the continuum, although it is 'of measure zero.' I have brought this case before you, in order to make you feel that there is something mysterious about the continuum and that we must not be too astonished at the apparent failure of our attempts to use it for a precise description of nature.
*
THE MAKESHIFT OF WAVE MECHANICS
Now I shall try to give you an idea of the way in which physicists at present endeavour to overcome this failure. One might term it an 'emergency exit,' though it was not intended as such, but as a new theory. I mean, of course, wave mechanics. (Eddington called it 'not a physical theory but a dodge-and a very good dodge too.') The situation is about as fonows. The observed facts (about particles and light and all sorts of radiation and their mutual interaction) appear to be repugnant to the classical ideal of a continuous description in space and time. (Let me explain myself to the physicist by hinting at one example: Bohr's famous theory of spectral lines in 1913 had to assume that the atom makes a sudden transition from one state into another state, and that in doing so it emits a train of light waves several feet long, containing hundreds of thousands of waves and requiring for its formation a considerable time. No information about the atom during this transition can be offered.) So the facts of observation are irreconcilable with a continuous description in space and time; it just seems impossible, at least in many cases. On the other hand, from an incomplete description-from a picture with gaps in space and time--one cannot draw clear and unambiguous conclusions; it leads to hazy, arbitrary, unclear thinking-and that is the thing we must avoid at all costs! What is to be done? The method adopted at present may seem amazing to you. It amounts to this: we do give a complete description, continuous in space and time without leaving any gaps,
CllllSlllil,' and Wavt' Mt'clulnic$
1065
conforming to the classical ideal-a description of something, But we do not claim that this 'something' is the observed or observable facts; and still less do we claim that we thus describe what nature (matter, radiation, etc.) really is. ]n fact we use this picture (the so-called wave picture) in full knowledge that it is neither. There is no gap in this picture of wave mechanics, also no gap as regards causation. The wave picture conforms with the classical demand for complete determinism, the mathematical method used is that of fieldequations, though sometimes they are a highly generalized type of fieldequations. But what is the use of such a description, which, as I said, is not believed to describe observable facts or what nature really is like? Well, it is believed to give us in/ormation about observed facts and their mutual dependence. There is an optimistic view, viz. that it gives us all the information obtainable about observable facts and their interdependence. But this view-which mayor may not be correct-is optimistic only inasmuch as it may flatter our pride to possess in principle all obtainable information. It is pessimistic in another respect, we might say epistemologically pessimistic. For the in/ormation we get as regards the causal dependence of observable facts is incomplete. (The cloven hoof must show up somewhere!) The gaps, eliminated from the wave picture, have withdrawn to the connection between the wave picture and the observable facts. The latter are not in one-to-one correspondence with the former. Plenty of ambiguity remains, and, as I said, some optimistic pessimists or pessimistic optimists believe that this ambiguity is essential, it cannot be helped. This is the logical situation at present. I believe I have depicted it correctly, though I am quite aware that without examples the whole discussion has remained a little bloodless-jllst purely logical. I am also afraid that I have given you too unfavourable an impression of the wave theory of matter. I ought to amend both points. The wave theory is not of yesterday and not of 25 years ago. It made its first appearance as the wave theory of light (Huygens 1690). For the better part of 100 years 2 light waves were regarded as an incontrovertible reality, as something of which the real existence had been proved beyond all doubt by experiments on the diffraction and interference of light. I do not think that even today many physicists----certainly not experimentalists-are ready to endorse the statement that 'light waves do not really exist, they are only waves of knowledge' (free quotation from Jeans). If you observe a narrow luminous source L, a glowing Wollaston wire, a few thousandths of a millimetre thick, by a microscope whose objective 2 Not the immediately following hundred years. Newton's authority eclipsed Huygens' theory for about a century.
L
•
FIGURE 7
lens is covered by a screen with a couple of parallel slits. you find (in the image plane conjugate to L) a system of coloured fringes which conform exactly and quantitatively to the idea that light of a given colour is a wave motion of a certain small wave-length, shortest for violet. about twice as long for red light. This is one out of dozens of experiments that clinch the same view. Why. then, has this reality of the waves become doubtful? For two reasons: (a) Similar experiments have been performed with beams of cathode rays (instead of tight); and cathode rays-so it is said-manifestly consist of single electrons, which yield 'tracks' in the Wilson cloud chamber.
L
•
FIGURE 8
( b) There are reasons to assume that light itself also consists of singJe particles-called photons (from the Greek ~ .. = light). Against this one may argue that nevertheless in both cases the concept of waves is unavoidable, if you wish to account for the interference fringes. And one may also argue that the particles are not identifiable objects, they might be regarded as explosion like events within the wavefront-just the events by which the wave-front manifests itself to observation. These events-so one might say-are to a certain extent fortuitous, and that is why there is no strict causal connection between observations. Let me explain in some detail why the phenomena, both in the case of light and in the case of cathode rays, cannot possibly be understood by
1067
(.'allllJlit,· and Wavt Mtt:hanicl
the concept of single, individuaJ, permanently existing corpuscles. This win also afford an example of what I call the 'gaps' in our description and of what I call the 'lack of individuality' of the particles. For the sake
FIGURE 9
of argument we simpJify the experimental arrangement to the utmost. We consider a small, almost point·like source which emits corpuscles in all directions, and a screen with two small holes, with shutters, so that we can open first only the one, then only the other, then both. Behind the screen we have a photographic plate which collects the corpuscles that emerge from the openings. After the plate has been developed, it shows, let me assume, the marks of the single corpuscles that have hit it, each rendering a grain of silver-bromide developable, so that it shows as a black speck after developing. (This is very near the truth.) Now let us first open only one hole. You might expect that after exposing for some time we get a close cluster around one spot. This is not so. Apparently the particles are deflected from their straight path at the
FIGURE 100The lines indicate the places where there are few or no spots. while midway between any two lines the spots would be most frequent The two straight lines in the middle are parallel to the slits.
ErwiN Scll,IdI,.."
1068
opening. You get a fairly wide spreading of black specks, though they are densest in the middle, becoming rarer at greater angles. If you open the second hole alone, you clearly get a similar pattern, only around a different centre. Now let us open both holes at the same time and expose the plate just as long as before. What would you expect-if the idea was correct, that single individual particles fly from the source to one of the holes, are deflected there, then continue along another straight line until they are caught by the plate? Clearly you would expect to get the two former patterns superposed. Thus in the region where the two fans overlap, if near a given point of the pattern you had, say, 25 spots per unit area in the first experiment and 16 more in the second, you would expect to find 25 + 16 = 41 in the third experiment. This is not so. Keeping to these numbers (and disregarding chance-fluctuations, for the sake of argument), you may find anything between 81 and only 1 spot, this depending on the precise place on the plate. It is decided by the difference of its distances from the holes. The result is that in the overlapping part we get dark fringes separated by fringes of scarcity. (N .B. The numbers 1 and 81 are obtained as (y25
+
y) 6) 2
= (5 +
4) 2
== 81,
1.)
If one wanted to keep up the idea of single individual particles flying con-
tinuously and independently either through one or through the other slit one would have to assume something quite ridiculous, namely that in some places on the plate the particles destroy each other to a large extent, while at other places they 'produce offspring: This is not only ridiculous but can be refuted by experiment. (Making the source extremely weak and exposing for a very long time. This does not change the pattern!) The only other alternative is to assume that a particle flying through the opening No. 1 is influenced also by the opening No.2, and that in an extremely mysterious fashion. We must, so it seems, give up the idea of tracing back to the source the history of a particle that manifests itself on the plate by reducing a grain of silver-bromide. We cannot tell where the particle was before it hit the plate. We cannot tell through which opening it has come. This is one of the typical gaps in the description of observable events, and very characteristic of the Jack of individuality in the particle. We must think in terms of spherical waves emitted by the source, parts of each wave-front passing through both openings, and producing our interference pattern on the plate-but this pattern manifests itself to observation in the form of single particles.
COMMENTARY ON
SIR ARTHUR STANLEY EDDINGTON
"I
BELIEVE there are 15.747.724.136,275,002.577.605,653,961,181. 555,468,044, 717,914,527,116,709,366,231,425,076,185,631,031,296 protons in the universe and the same number of electrons." Thus Eddington began one of the Tarner lectures in 1938,1 It is improbable that even the drowsiest member of his audience failed to respond to this opening sentence. That it tells us much about the universe is doubtful, but it tens us a good deal about Eddington. He was the greatest astronomer of his day, commanding a superb mathematical technique and inclined strongly to philosophical speculation. The inventiveness of his imagery, the clarity and felicitousness of his prose in explaining science to the layman were unsurpassed. He was excruciatingly shy but had the self-confidence and boldness of the mystic in advancing his theories. The rather large number above represents the culmination of years of effort by Eddington to formulate a comprehensive theory of the universe. Multiply the number by 2 and you get the total number of particles: 2 X 136 X 22l'iO. Eddington called this N, the Cosmical Number; he regarded it as his supreme achievement. Few scientists today subscribe to his theory, but none depreciates its intellectual grandeur. Arthur Stanley Eddington was born in 1882 at Kendal, Westmorland, of Quaker parents. His father, headmaster of the local Friends school, died of typhoid fever when Arthur was two years old; his mother was left poor, but managed nevertheless to send him to priVate schools. It is reported that he was an exceptional child who could do the 24 X 24 multiplication table at the age of five and was experienced in the use of the telescope before he was ten. A variety of scholarships won in competitive examinations saw Eddington through Owens C01lege, Manchester, and Trinity at Cambridge. In 1906 he was appointed Chief Assistant at the Royal Observatory at Greenwich and in 1907 was elected a Fellow of Trinity. The Plumian chair of astronomy was tendered him when he was only thirty. He held this post until his death, of cancer, in 1944. Eddington's first major contribution to astronomy was an extension of Schwarzschild's theory of the radiative equilibrium of a star's atmosphere to the star's interior. 2 In a paper published on the subject in 1917 Ed· 1 Sir Arthur Eddington, The Philosophy 01 Physical Science, Cambridge, 1939, p.170. S This synopsis of Eddington's scientific researches follows several accounts: J. G. Crowther. British Scientists 01 the Twentieth Century, London. 1952, pp. 140-196;
1069
1070
Ed/lOT'S
Comme,,'
dington provided a beautiful explanation of the general features of stellar structure and stellar evolution. The heart of the problem to which he addressed himself was the relation between the mass and the brightness of a star. A giant star of low density could be supposed to be extremely bright because its "ether-waves," from X-rays to light-rays, are forced to flow out by the steep temperature gradient. Yet there was a huge discrepancy, which conventional theory could not account for. between the star's expected and observed brightness: it fell short of the expected luminosity by a factor of many millions. Eddington showed that the radiation waves are "hindered and turned back by their adventures with the atoms and electrons" and thus the leakage to the surface of the star is considerably reduced. He showed also that the radiation pressure balances the gravitational forces and supports the enormous weight of the star's upper layers. On the basis of this explanation of stellar structure. he demonstrated a surprising correlation, for all ordinary stars; between masses and luminosities. The more massive the star, the more energy it pours out.:"1 The sun's material, he asserted, in spite of being denser than water, really is a perfect gas. Even mOre incredible is the figure of a mean density of 53,000 for a star like Sirius. This "foretold an Einstein shift of spectral lines," actually confirmed by observation shortly thereafter. ]n the Internal Constitution o/the Stars (1926) were summarized the preceding fifteen years of brilliant astronomical investigations:' From the first Eddington was interested in relativity. It appealed to his cosmological bent and he was one of the few quickly to master its mathematical difficulties and to apprehend the full significance of the theory. A copy of one of Einstein's papers on the General Theory was sent to him by the Dutch physicist De Sitter in 1915; for some time it was the only copy available in England. As a Quaker, Eddington was sitting out the war and thus found time to apply himself to the new ideas. His Report on the Relativity Theory 0/ Gravitation made for the Physical Society of London in 1918, contains only ninety-one pages "and is a masterpiece of Sir Harold Spencer Jones. E. A. Milne, E. N. daC. Andrade, Obituaries of Eddington in Nalltre, vol. 154, Dec. 16. 1944. H. C. Plummer in Ohitllan- Notices oj Fellows oj the Royal SOCiel). vol. 5. pp. t t 3 et seq.: Sir Edmund Whittaker. Eddington's Prillciple in the Philosophy oj Science, Cambridge (England), 1951. a The energy output is proportional to something between the cube and the founh power of the mass. Thus, a star only twice the mass of the sun has twelve times its output. 4 Interesting discussions of Eddington's researches on the radiation equilibrium of gaseous stars. of his scientific controversies with Jeans. and of related mallers appear in E. A. Milne. Sir James Jeal/s, A Biography, Cambridge (England), 1952. passim Milne says that Eddington. in calculating the c;tate of equilibrium of a given mass of gas. and the rate of radiation (rom its surface, performed a first-class piece of scientific work. But he failed to realize exactly the nature o( his achievements. What he found was the condition for a star to be gaseous throughout. Whal he claimed to have found--erroneously, says Milne-was the luminosity of the existing stars and their internal opacity. The dispute must be left to the judgment of experts. I abandon it gladly.
Sir A"ltur Slanltl Eddin,ton
1071
concise and elegant exposition." He "not only restated Einstein's work and De Sitter's expositions, he joyfully took wing in flights of physical and mathematical thought and fancy of his own." !'i In 1919 he was one of the leaders of an expedition to the Isle of Principe in the Gulf of Guiana which, during a solar eclipse, tested and confirmed Einstein's prediction of the bending of light rays by matter. A semipopular account of the theory was presented in his famous book Space, Time and Gravitation (1920)-from which an excerpt is given below. In The Mathematical Theory 01 Relativity (1923) he offers both a comprehensive technical analysis and, as his own contribution to the subject, a generalization of Herman Weyl's theory of the electromagnetic and gravitational fields "based on the' notion of parallel displacement." Eddington continued his researches on the structure and composition of the stars, studying such problems as ionization and capture phenomena attending energy interchanges in a diffuse gas, and the pulsations of the Cepheid variables. But in later years he focused his labors on developing the cosmological aspects of relativity theory and on unifying the quantum theory and relativity. What he tried to find was a grand embracing principle derived from a supposed relationship between certain important numbers such as "the radius curvature of the earth, the recession velocity constant of the external galaxies, the number of particles in the universe, and the physical constants such as the ratio of the mass of the proton to that of the electron, the ratio of the gravitational to the electrical force between a proton and an electron, the fine structure constant and the velocity of light." Sir Edmund Whittaker presents an admirably succinct description of the principle which Eddington wished to introduce into the philosophy of science and which is the central theme of his last book, Fundamental Theory.fI One may distinguish between two kinds of assertions in physics: quantitative assertions, e.g., "The masses of the electron, the ". meson, the 1r meson, the T meson, and the proton, are approximately in the ratios 1, 200, 300, 1000, 1836"; or "The ratio of the electron to the gravitational force between a proton and an electron is 2.2714 X 1()39"; and qualitative assertions, e.g., "The velocity of light is independent of the motion of its source," or, "It is impossible to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects." Eddington claimed a relationship between these quantitative and qualitative assertions as follows: "All the quantitative propositions of physics, that is, the exact values of the pure numbers that are constants of science, may be deduced by logical reasoning from qualitative assertions, without making any use of quantitative data derived from observation." 7 A more dramatic statement of the prin; Crowther, op. cit., p. 161. Whittaker, op. cit., pp. 1-3. 'l Whittaker, op. cit., p. 3. 6
Edllo,'s Comme,,'
1072
ciple is given by Eddington: "An intel1igence unacquainted with our universe but acquainted with the system of thought by which the human mind interprets to itself the content of its sensory experience, should be able to attain all the knowledge of physics that we have attained by experiment. He would not deduce the particular events and objects of our experience but he would deduce the generaUzations we have based on them. For example, he would infer the existence and properties of radium but not the dimensions of the earth." His conclusions, it is agreed. have not yet "carried general conviction," The theory is abstruse and complex; many students are repelled by its philosophical framework. Eddington allowed no compromise. He believed his constants to be absolutely precise, his theory as unexceptionable as the syllogism. In a letter he wrote shortly before his death to the astronomer Herbert Dingle, he rejected the criticism of his procedure as "obscure." Einstein, he said, "was once considered obscure . , . I cannot seriously believe that I ever attain the obscurity that Dirac does. But in the case of Einstein and Dirac people have thought it worth while to penetrate the obscurity. I believe they will understand me all right when they realize they have got to do so--and when it becomes the fashion 'to explain Eddington.' " I am drawn to Mnne's opinion. Some of the unified theory he pronounced "beautiful"; some, "palpable fudge," Altogether it was evidently "the production of a man of genius, not excluding tiresomeness; whether or not it will survive as a great scientific work, it is certainly a notable work of art." 8
*
*
*
II
-, so that y
z
=y
z Y
I
1 1 and that - < -, so that z
I
z
> z.
I, i.e. if the velocity of one world relatively to the other is greater than the velocity of Jight. We therefore say that We can have no experience of a body moving with a velocity greater than that of light. And in all our results u must stand for a fraction between + I and -1. It is customary to represent y (1 - u~) by
1
1
fJ
y(1 -
- or to put fJ =
>
so that fJ u2 )
1. In this case, the standard
formuhe may be written: Xl
or X
= fJ(x -
uI); tl
=fJ(t -
ux)
= fJ(Xl + ull); t = fJ(tt + uxI> 1
where fJ =
y(1 -
u2 )
> 1.
1143
Til, Tlleor)' 01 Relativity
And the results on pp. 1138-1139 may be stated as follows: (i) 0 says that the length of a line in the direction of motion which 1 A measures as 1 lux is - luxes.
p
(ii) 0 says that a time-interval which A's clock records as 1 second is
p seconds. 6 6
Exercise
1. The world of A is moving at ¥.i lux per second due east from O. What is O's opinion about (i) the length of A's foot-rule, (ii) the rate of running of A's clock? What is A's opinion about O's foot-rule and O's clock? 2. A and C, who are relatively at rest at a distance apart of 5 luxes, have synchronised their clocks; the world of A, C is moving away from 0 in the direction A -+ C at %... lux per second. A passes 0 at zero hour by O's clock and A's clock. What does 0 say is the difference between A's clock and Cs clock? D is a place in the world of A. C, such that DA = 10 luxes, DC = 15 luxes. What does 0 say is the djfference between D's clock and A's clock? What does 0 say is the time recorded by the clocks of A, C, D when O's clock records 25 seconds past zero? 3. With the data of No.1, A records two events as happening at an interval of 5 seconds and at a distance apart of 3 luxes, the second event being due east of the first event. What are the time and distance-intervals of the events as recorded by 01 4. With the data of No.3, solve the question if the second event is due west of the first event. 5. Given that
Xl
Prove that 11
=
=
x - ut
"t I-
and 2
u ux.
vt - u
+ utl
vi - u
2
and I li
Xl
x=--11 + UXl =---
yr=ut
6. If Event I. is the dispatch of a tight-signal by A and Event II. is the receipt of the light-signal by C. show that with the usual notation (i) XI = II. (H) x = I What does this mean in terms of O's opinion? 7. Using the equations on pp. 1141-1142, prove that X2 /2 is always equal to X1
2
-
1J2 •
PART VI
Mathematics and Social Science I. Gustav Theodor Fechner by EDWIN G. BORING 2. Classification of Men According to Their Natural Gifts by SIR FRANCIS GALTON 3. Mathematics of Population and Food by THOMAS ROBERT MALTHUS 4. Mathematics of Value and Demand by AUGUSTIN COURNOT 5. Theory of Political Economy by WILLIAM STANLEY JEVONS 6. Mathematics of War and Foreign Politics by LEWIS FRY RICHARDSON 7. Statistics of Deadly Quarre1s by LEWIS FRY RICHARDSON 8. The Theory of Economic Behavior by LEONID HURWICZ 9. Theory of Games by s. VAJDA 10. Sociology Learns the Language of Mathematics by ABRAHAM KAPLAN
COMMENTARY ON
The Founder of Psychophysics
G
USTAV FECHNER'S contribution to psychology was to introduce measurement as a tool. While there is some doubt as to what it was that he measured, the importance of his innovation of method is beyond question. For a long time it was believed that mental behavior. being linked to the soul, lay outside the crass reach of arithmetic, let alone of crude physical instruments such as rods and counters. (The prejudice is not entirely without merit;' one might wish that the intrepid little band of surgeons known as lobotomists would succumb to it.) As long as psychology was classed as a branch of philosophy it was a subject for speculation. perhaps for description, but certainly not for experiment. In the nineteenth century this well-established position came under the attack of two advancing fronts of thought. One was Jed by physiologists--Charles Bell, Johannes Milller, E. H. Weber-who investigated the psychological aspects of physiological phenomena. They looked into such matters as the relation between stimuli and their corresponding sensations, the specific quality imposed on the mind by different kinds of nerves, the capacity to discriminate among stimuli. The other front was generated by the growing philosophical belief in a "scientific or physiological psychology." Lotze 1 and Bain,2 each in his own way, supported this point of view; Herbart,8 though opposed to the use of experiments, gave psychology status as a separate branch of learning. "He took it out of both philosophy and physiology and sent it forth with a mission of its own." But it was Fechner who finally demolished the old edifice. He was a physicist, also a fervent and prolific philosopher~ in these spheres. however, his accomplishments were minor. He is remembered for psychophysical experiments that were essentially a side line of his career. His results have not stood the test of research; his famous law is mainly of 1 Rudolf Hermann Lotze (1817-1881) was a German philosopher who in his writings (e.g., Medizillische Psychologie oder Physiologie der Seele) stressed the view that the mind though an immaterial principal could only act on the body and be acted on by it through mechanical means. His opinions were said to have contributed to the destruction of the "phantom of Hegelian wisdom" and to have "vindicated the independent position of empirical philosophy." (EncycloPQediQ BritQnnicQ, Eleventh Edition.) 2 Alexander Bain (1818-1903) was a Scottish philosopher. educationist and psychologist. He was the first in Great Britain to emphasize the necessity of "clearing psychology of metaphysics. of applying the methods of the exact sciences to psychological phenomena and of referring these phenomena to their correlates in the nerves and brain." 3 lohann Friedrich Herbart (1776-1841) was a famous German philosopher and educator who stressed the importance of framing educational method.. on the basis of ethics and psychological knowledge. Psychology, he said. would help explain the mind to be educated: ethics would define the social goals of education, such as the cultivation of good will.
1146
1141
historical interest; but his thumbprint is on every page of modern psychological experiment. Fechner extended to psychology Kelvin's famous dictum that in science you cannot talk about a thing untH you can measure it. In our century, to be sure, this dictum has come to sound a little glib. Mathematics has its paradoxes, astronomy its uncertainties (about what is being measured). physics having suffered certain metaphysical relapses can survive only by swallowing entire jugs of wholly contradictory measurements. As for psychology, its most brilliant and its most scandalous success has been in a realm of theory in which measurement is as welcome as Macduff at Dunsinane. It appears that in psychology, as in other social studies. two distinct methods can be made to yield fruitful results. One is intuitive and nonquantitative--as in psychoanalysis; the other is analytic, formal, experimental, model-making--as in the physical sciences. In the last half century much has been learned about human behavior by developing and applying both methods. but no one has yet constructed a convincing unified field theory in psychology whereby the intuitive and qualitative approach can be reduced to the approach based on mechanism, microstructure and measurement. The divorcement of psychology from metaphysics undoubtedly marked an important step forward; yet it is doubtful that the decree was ever made absolute and final. At any rate one great branch of psychology has taken a new methodological partner much more closely related to philosophy than to physics. To put theories which can be tested by experiment, to discipline experiment by measurement, to evaluate measurement by mathematical apparatus, to advance new theories or models on the basis of what has been learned: these may not represent the only possible steps by which psychological knowledge can be broadened, psychological disorders diagnosed and treated, social behavior better understood. Nevertheless these steps constitute a creative and an indispensable sequence in the study of psychology and it is to the practice of this sequence that Fechner's ideas gave a major impetus. The author of the following selection, Edwin Garrigues Boring, is a noted American scientist who since 1928 has been professor of psychology at Harvard University. His principal researches and writings have been in the fields of theoretical psychology, history of psychology, perception and sensation. Dr. Boring, born in Philadelphia in 1886, was educated at Cornell University from which he received his doctorate; he served for a time as professor of experimental psychology at Clark University, and for twenty-five years (1924-1949) was director of Harvard's Psychological Laboratory. He has been president of the American Psychological Society and his academic honors include membership in the National Academy of Sciences. The essay on Fechner is a chapter from Boring's standard work, A History of Experimental Psychology.
Fechner is a curiosity. His eyelids are strangely fringed and he has IuId a number of holes, square and round, cui, Heaven knows why, in the iris of each eye-and ;s altogether a bundle of oddities in person and manners. He has forgotten aI/ the details of his "Psychophysik"; and is chiefly interested in theorizing how knots can be tied in endless strings, and how words can be written on the inner side of two slates sealed together. -G. S. HALL in a letter (1879) 10 William James
1
Gustav Tlleodor Fechner By EDWIN G. BORING
WE come at last to the formal beginning of experimental psychology, and we start with Fechner: not with Wundt, thirty-one years Fechner's junior, who published his first important but youthful psychological study two years after Fechner's epoch-making work; not with Helmholtz, twenty years younger, who was primarily a physiologist and a physicist but whose great genius extended to include psychology; but with Fechner, who was not a great philosopher nor at all a physiologist, but who performed with scientific rigor those first experiments which laid the foundation for the new psychology and still lie at the basis of its methodology. There had been, as we have seen, a psychological physiology: Johannes MUller, E. H. Weber. There had been, as we have also seen, the development of the philosophical belief in a scientific or a physiological psychology: Herbart, Lotze; Hartley, Bain. Nothing is new at its birth. The embryo had been maturing and had already assumed, in all great essentials, its later form. With Fechner it was born, quite as old, and also quite as young, as a baby. THE DEVELOPMENT OF FECHNER'S IDEAS
Gustav Theodor Fechner (1801-1887) was a versatile man. He first acquired modest fame as professor of physics at Leipzig, but in later life he was a physicist only as the spirit of the Naturforscher penetrated all his work. In intention and ambition he was a philosopher, especially in his last forty years of life, but he was never famous, or even successful, in this fundamental effort that is, nevertheless, the key to his other activities. He was a humanist, a satirist, a poet in his incidental writings and an estheticist during one decade of activity. He is famous, however, for his psychophysics, and this fame was rather forced upon him. He did not wish his name to go down to posterity as a psychophysicist. He did not, Uke Wundt, seek to found experimental psychology. He might have been content to let experimental psychology as an independent science remain 1148
GlUt"" Tlltodor Ftc-lint'
1149
in the womb of time, could he but have established his spiritualistic Tagesansicht as a substitute for the current materialistic Nachtansicht of the universe. The world, however, chose for him; it seized upon the psychophysical experiments, which Fechner meant merely as contributory to his philosophy. and made them into an experimental psychology. A very interesting life to us, who are inquiring how psychologists are made! Fechner was born in ) 80) in the parsonage of a little vilJage in southeastern Germany, near the border between Saxony and Silesia. His father had succeeded his grandfather as village pastor. His father was a man of independence of thought and of receptivity to new ideas. He shocked the villagers by having a lightning-rod placed upon the church tower, in the days when this precaution was regarded as a lack of faith in God's care of his own, and by preaching-as he urged that Jesus must also have donewithout a wig. One can thus see in the father an anticipation of Fechner'S own genius for bringing the brute facts of scientific materialism to the support of a higher spiritualism, but there can have been little, if any, direct influence of this sort, for the father died when Fechner was only five years old. Fechner, with his brother and mother, spent the next nine years with his uncle, also a preacher. Then he went for a short time to a Gymnasium and then for a half year to a medical and surgical academy. At the age of sixteen he was matriculated in medicine at the university in Leipzig, and at Leipzig he remained for the rest of his long life-for seventy years in all. We are so accustomed to associating Fechner's name with the date 1860, the year of the publication of the Elemente der Psychophysik. and with the later years when he lived in Leipzig while Wundt's laboratory was being got under way, that we are apt to forget how old he was and how long ago he was beginning his academic life. In 1817, when Fechner went to Leipzig, Lotze was not even born. Herbart had just published his Lehrbuch, but his Psychologie als Wissenschaft was still seven years away in the future. In England, James Mill had barely completed the History of India and presumably had not even thought of writing a psychology. John Stuart Mill was eleven years old; Bain was not born. Phrenology had only just passed its first climax, and Gall Was still writing on the functions of the brain. Flourens had not yet begun his researches on the brain. BeU, but not Magendie, had discovered the Bel1-Magendie law. It was really, as the history of psychology goes, a very long time ago that Fechner went as a student to Leipzig. It happened that E. H. Weber, the Weber after whom Fechner named "Weber's Law," went to Leipzig in the same year as Dozent in the faculty of medicine and was made in the following year ausserordentlicher Professor of comparative anatomy. After five years of study. Fechner took his degree in medicine, in 1822. Already, however, the humanistic side of the
1150
man was beginning to show itself. His first publication (1821), Beweiss. dass der Mond aus Iodine bestehe, was a satire on the current use of iodine as a panacea. The next year he wrote a satirical panegyric on modern medicine and natural history. Both these papers appeared under the nom de plume 'Dr. Mises: and 'Dr. Mises' was reincarnated in ironical bursts altogether fourteen times from 1821 to 1876. Meanwhile Fechner's association with A. W. Volkmann had begun. Volkmann came to Leipzig as a student in medicine in 1821 and remained, later as Dozent and professor, for sixteen years. After he had taken his degree, Fechner's interest shifted from biological science to physics and mathematics, and he settled down in Leipzig, at first without official appointment, for study in these fields. His means were slender, and he undertook to supplement them by the translation into German of certain French handbooks of physics and chemistry. This work must have been very laborious. for by 1830 he had translated more than a dozen volumes and nearly 9,000 pages; but it was work that brought him into prominence as a physicist. He was also appointed in 1824 to give lectures in physics at the university, and in addition he undertook physical research of his own. It was a very productive period. By 1830 he had published, including the translations, over forty articles in physical science. At this time the properties of electric currents were just beginning to become known. Ohm in 1826 had laid down the famous law that bears his name, the law that states the relation between current, resistance and elec· tromotive force in a circuit. Fechner was drawn into the resulting problem, and in ] 831 he published a paper of great importance on quantitative measurements of direct currents (Massbestimmungen tiber die galvanische Kette), a paper which made his reputation as a physicist. The young Fechner in his thirties was a member of a deJightfuJ intellectual group in the university community at Leipzig. Vo'lkmann, until he went to Dorpat in 1837, was also a member of this group, and it was Volkmann's sister whom Fechner married in 1833. The year after his marriage, the year in which, as we have already seen, Lotze came to Leipzig as a student, Fechner was appointed professor of physics. It must have seemed that his career was already determined. He was professor of physics at only thirty-three, with a program of work ahead of him and settled in a congenial social setting at one of the most important universities. We shall see presently how far wrong the obvious prediction would have been. Fechner for the time being kept on with his physical research, throughout the still very fertile decade of his thirties. 'Dr. Mises: the humanistic Fechner, appeared as an author more than half a dozen times. Toward the end of this period there is. in Fechner's research, the first indication of a quasi-psychological interest: two papers on complementary colors and subjective colors in 1838, and the famous paper on
GustG'II
T"~odo,
F«"nn
1151
subjective after-images in 1840. In general, however. Fechner was a promising younger physicist with the broad intellectual interests of the deutscher Gelehrter. Fechner, however, had overworked. He had developed. as James diagnosed the disease, a 'habit-neurosis! He had also injured his eyes in the research on after-images by gazing at the sun through colored glasses. He was prostrated, and resigned, in 1839, his chair of physics. He suffered great pain and for three years cut himself off from every one. This event seemed like a sudden and incomprehensible ending to a career so vividly begun. Then Fechner unexpectedly began to recover, and, since his malady was so little understood, his recovery appeared miraculous. This period is spoken of as the 'crisis' in Fechner's life. and it had a profound effect upon his thought and after-life. The primary result was a deepening of Fechner's religiOUS consciousness and his interest in the problem of the soul. Thus Fechner. quite natural1y for a man with such an intense intellectual life, turned to philosophy, bringing with him a vivification of the humanistic coloring that always had been one of his attributes. His forties were, of course, a sterile decade as regards writing. 'Dr. Mises' published a book of poems in 1841 and several other papers later. The tirst book that showed Fechner's new tendency was Nanna oder das See/en/eben der Pflanzen, published in 1848. (Nanna was the Norse goddess of flowers.) For Fechner, in the materialistic age of science, to argue for the mental life of plants, even before Darwin had made the mental life of animals a crucial issue, was for him to court scientific unpopUlarity. but Fechner now felt himself possessed of a philosophic mission and he could not keep silence. He was troubled by materiaJism, as his Buchlein yom Leben nach dem Tode in 1836 had shown. His philosophical solution of the spiritual problem lay in his affirmation of the identity of mind and matter and in his assurance that the entire universe can be regarded as readily from the point of view of its consciousness, a view that he later called the Tagesansicht, as it can be viewed as inert matter. the Nachtansicht. Yet the demonstration of the consciousness of plants was but a step in a program. Three years later (1851) a more important work of Fechner's appeared: Zend-Avesta, oder uber die Dinge des Himmels und des lenselts. Oddly enough this book contains Fechner's program of psychophysics and thus bears an ancestral relation to experimental psychology. We shan return to this matter in a moment. Fechner's general intent was that the book should be a new gospel. The title means practically "a revelation of the word." Consciousness, Fechner argued, is in all and through all. The earth, "our mother," is a being like ourselves but very much more perfect than ourselves. The soul does not die, nor can it be exorcised by the priests of materialism when all being is conscious. Fechner's argument was
1152
not rational; he was intensely persuasive and developed his theme by way of p1ausible analogies, which, but for their seriousness, resemble somewhat the method of Dr. Mises' satire, Vergleichende Anatomie der Engel (1825), where Fechner argued that the angels, as the most perfect beings, must be spherical, since the sphere is the most perfect form. Now, however, Fechner was in dead earnest. He said 1ater in Ueber die Seelen/rage (1861) that he had then called four times to a sleeping pubJic which had not yet been aroused from its bed. "I now," he went on, "say a fifth time, 'Steh' au/I' and, if I live, 1 shall yet call a sixth and a seventh time, 'Steh' au/I' and always it will be but the same 'Steh' au/I' " We need not go further into Fechner's philosophy. He did ca1l, or at least so Titchener thought, a sixth and a seventh time, and these seven books with their dates show the persistence and the extent of Fechner's belief in his own gospeL They are: Das Biiehlein vom Leben naeh dem Tode, 1836; Nanna, 1848; Zend-Avesta, 1851; Professor Sehleiden und der Mond, 1856; Ueber die Seelen/rage, 1861; Die drei Motive und Griinde des Glaubens, 1863; Die Tagesansicht gegeniiber der Naehtansieht, 1879. As it happened, the public never "sprang out of bed," not even at the seventh call, as Fechner had predicted it would. His philosophy received some attention; many of these books of his have been reprinted in recent years; but Fechner's fame is as a psychophysicist and not as a philosopher with a mission. His psychophysics, the sole reason for Fechner's inclusion in this book, was a by-product of his philosophy. We return to it. It was one thing to philosophize about mind and matter as two alternative ways of regarding everything in the universe. and another thing to give the idea such concrete empirical form that it might carry weight with the materialistic intellectualism of the times or even be satisfactory to Fechner, the one-time physicist. This new philosophy, so Fechner thought, needed a solid scientific foundation. It was, as he tells us, on the morning of October 22, 1850, while he was lying in bed thinking about this problem, that the general outlines of the solution suggested themselves to him. He saw that the thing to be done was to make "the relative increase of bodily energy the measure of the increase of the corresponding mental intensity," and he had in mind just enough of the facts of this re1ationship to think that an arithmetic series of mental intensities might correspond to a geometric series of physical energies, that a given absolute increase of intensity might depend upon the ratio of the increase of bodily force to the total force. Fechner said that the idea was not suggested by a knowledge of Weber's results. This statement may seem strange, for Weber was in Leipzig and had published the Tastsinn und Gemeinge/uhl in 1846. and it was important enough to be separately reprinted in 1851. We must remember, however, that Weber himself had
GlUt"" Theodor Fechner
1153
not pointed out the general significance of his law and may have seen its most general meaning only vaguely. He had hinted at generality in his manner of talking about ratios as if they were increments of stimulus, and in extending his finding for touch to visual extents and to tones. He had formulated no specific law. It was Fechner who realized later that his own principle was essentially what Weber's results showed, and it was Fechner who gave the empirical relationship mathematical form and called it "Weber's Law." In recent times there has been a tendency to correct Fechner's generosity, and to give the name Fechner's Law to what Fechner called "Weber's Law," reserving the latter term for Weber's simple statement that the just noticeable difference in a stimulus bears a constant ratio to the stimulus. (See formulas 1 and 6 infra, pp. 1159, 1160.) The immediate result of Fechner's idea was the formulation of the program of what he Jater called psychophysics. This program, as we have already observed, was worked out in the Zend-Avesta of 1851. There was still, however, the program to carry out, and Fechner set about it. The methods of measurement were developed, the three psychophysical methods which are still fundamental to much psychological research. The mathematical form both of the methods and of the exposition of the general problem of measurement was established. The classical experiments on lifted weights, on visual brightnesses and on tactual and visual distances were performed. Fechner the philosopher proved to have lost none of the experimental care of Fechner the physicist. His friend and brother-in-law, A. W. Volkmann, then at Halle, helped with many of the experiments. Other data, notably the classification of the stars by magnitude, were brought forth to support the central thesis. For seven years Fechner published nothing of all this. Then in 1858 and 1859 two short anticipatory papers appeared, and then in 1860, full grown, the Elemente der Psychophysik, a text of the "exact science of the functional relations or relations of dependency between body and mind." It would not be fair to say that the book burst upon a sleeping world. Fechner was not popular. Nanna, Zend-Avesta and similar writings had caused the scientists to look askance at him, and he was never accepted as a philosopher. No one suspected at the time what importance the book would come to have. There was no furor; nevertheless the work was scholarly and well grounded on both the experimental and mathematical sides, and, in spite of philosophical prejudice, it commanded attention in the most important quarter of all, namely, with the other scientists who were concerned with related problems. Even before the book itself appeared, the paper of 1858 had attracted the attention of Helmholtz and of Mach. Helmholtz proposed a modification of Fechner's fundamental formula in 1859. Mach began in 1860 tests of Weber's law in the time-
1154
Ed.".. G. BDrln,
sense and published in 1865. Wundt, in his first psychological publications in 1862 and again in 1863, cal1ed attention to the importance of Fechner·s work. A. W. Volkmann published psychophysical papers in 1864. Aubert challenged Weber's Jaw in 1865. Delbreuf, who later did so much for the development of psychophysics, began his experiments on brightness in 1865, inspired by Fechner. Vierordt similarly undertook in 1868 his study of the time-sense in the light of the Elemente. Bernstein, who had just divided with Volkmann the chair of anatomy and physiology at Halle, published in 1868 his irradiation theory, a theory that is based remotely on Herbart's law of the limen, but directly on Fechner's discussion. The Elemente did not take the world by the ears, but it got just the kind of attention that was necessary to give it a basic position in the new psychology. Fechner, however, had now accomplished his purpose. He had laid the scientific foundation for his philosophy and was ready to turn to other matters, keeping always in mind the central philosophical theme. Moreover, he had reached his sixties, the age when men begin to be dominated more by their interests and less by their careers. The next topic, then, that caught the attention of this versatile man was esthetics, and, just as he had spent ten years on psychophysics, so now he spent a decade (18651876) on esthetics, a decade that was terminated when Fechner was seventy-five years old. If Fechner "founded' psychophysics, he also 'founded' experimental esthetics. His first paper in this new field was on the golden section and appeared in 1865. A dozen more papers came out from 1866 to 1872, and most of these had to do with the problem of the two Holbein Madonnas. Both Dresden and Darmstadt possessed Madonnas, very similar although different in detail, and both were reputed to have been painted by Holbein. There was much controversy about them, and Fechner plunged into it. There were several mooted points. The Darmstadt Madonna showed the Christ-child. The Dresden Madonna showed instead a sick child and might have been a votive picture, painted at the request of a famiJy with the image of a child who had died. There was the general question of the significance of the pictures, and there was also the question of authenticity. Which was Holbein's and which was not? Experts disagreed. Fechner, maintaining the judicial attitude, was inclined to believe that they might both be authentic, that if Holbein had sought to portray two similar but different ideas he would have painted two similar but different pictures. And finally, of course, there was the question as to which was the more beautiful. These two latter questions were related in human judgment, for almost every one would be likely to believe that the authentic Madonna must be the more beautiful. Some of these questions Fechner sought to have answered 'experimenta1Jy' by a public opinion poll on the
1155
auspicious occasion when the two Madonnas were exhibited together. He placed an album by the pictures and asked visitors to record their judgments; but the experiment was a failure. Out of over 11,000 visitors, only 113 recorded their opinions, and most of these answers had to be rejected because they did not follow the instructions or were made by art critics or others who knew about the pictures and had formed judgments. Nevertheless the idea had merit and has been looked upon as the beginning of the use of the method of impression in the experimental study of feeling and esthetics. In 1876 Fechner published the Vorschule der A esthetik, a work that closed his active interest in that subject and laid the foundation for experimental esthetics. It goes into the various problems, methods and principles with a thoroughness that rivals the psychophysics, but is too far afield for detailed consideration in this book. There is little doubt that Fechner would never have returned either to psychophysics Or to esthetics, after the publication of his major book in each subject-matter, had the world let him be. The psychophysics, however, had immediately stimulated both research and criticism and, while Fechner was working on esthetics, was becoming important in the new psychology. In 1874, the year of the publication of Wundt's Grundzilge der physiologischen Psychologie. Fechner had been aroused to a brief criticism of Dellxeuf's Etude psychophysique (1873). The next year Wundt came to Leipzig. The following year Fechner finished with esthetics and turned again to psychophysics, publishing in 1877 In Sachen der Psychophysik, a book which adds but little to the doctrine of the Elemente. Fechner was getting to be an old man, and his philosophical mission was stm in his mind. In 1879, the year of Wundt's founding of the Leipzig psychological laboratory, Fechner issued Die Tagesansicht gegenilber der Nachtansicht, his seventh and last call to the somnolent world. He was then seventy-eight years old. Finally, in 1882, he published the Revision der Hauptpunkte der Psychophysik, a very important book, in which he took account of his critics and sought to meet the unexpected demand of experimental psychology upon him. In the following years there were half a dozen psychophysical articles by him, but actually this work was done. He died in 1887 at the age of eighty-six in Leipzig, where for seventy years he had Jived the quiet life of the learned man, faring forth, while keeping his house, on these many and varied great adventures of the mind. This then was Fechner. He was for seven years a physiologist (18171824); for fifteen a physicist (1824-1839); for a dozen years an invalid (1839 to about 1851); for fourteen years a psychophysicist (1851-1865); for eleven years an experimental estheticist (1865-1876); for at least two score years throughout this period, recurrently and persistently, a philoso-
Edwin G. Borin,
1156
pher (1836-1879); and finally, during his last eleven years, an old man whose attention had been brought back by public acclaim and criticism to psychophysics (1876-1887)-all told three score years and ten of varied intellectual interest and endeavor. If he founded experimental psychology, he did it incidentally and involuntarily, and yet it is hard to see how the new psychology could have advanced as it did without an Elemente der Psychophysik in 1860. It is to this book, therefore, that we must now tum our attention. PSYCHOPHYSICS
When Fechner began work on what was eventually to become the Elemente der Psychophysik, he had-beside his philosophical problem, his experience in physical research and his habits of careful experimentation -Herb art's psychology as a background. From Herbart he obtained the conception that psychology should be science, the general idea of mental measurement, the related notion of the application of mathematics to the study of the mind, the concept of the limen (which Herbart got from Leibnitz), the idea of mental analysis by way of the facts of the limen, and probably also a sensationistic cast to all of his work, a cast which resembles Herbart's intellectualism. When Fechner wrote the Zend-A vesta, Lotze had not published his psychology. There was really no psychology at all except the very influential psychology of Herbart and the psychological physiology of Johannes Muller and E. H. Weber. Fechner was, however, too much of an experimentalist to accept Herbart's metaphysicaJ approach or to admit the validity of his denial of the psychological experiment. Instead he set himself to correct Herbart by an experimental measurement of mind. All this, we must not forget, was done in the interests of his philosophical attack upon materialism. There is also to be mentioned Fechner's mathematical background. It will be recal1ed that Fechner had turned in part to the study of mathematics after he had obtained his doctor's degree. Fechner himsel f ac· knowledges debts to "Bernoulli (Laplace, Poisson), Euler (Herbart, Drobisch), Steinheil (Pogson)." He was thinking, however, more of the mathematical and experimental demonstration of Weber's Law. Steinheil had shown that stellar magnitudes follow this law; Euler, that tonal pitch follows it. It is plain, however, that Fechner placed the name of Daniel Bernoulli (1700-1782) first with reason. Bernoulli's interest in the theory of probabilities as applied to games of chance has led to the discussion of fortune morale and fortune physique, mental and physical values which he believed (1738) to be related to each other in such a way that a change in the amount of the 'mental fortune' varies with the ratio that the change in the physical fortune has to the total fortune of its possessor. (Thus in gambling with even stakes, one stands to lose more than one gains, for
Gus",,,
Th~odor
Fechner
lIS7
a given loss after the event bears a larger ratio to the reduced total fortune than wou Id the same physical gain to an increased total fortune-a conclusion with a moral!) In this way fortune morale and fortune physique became mental and physical quantities, mathematica]]y related, quantities that correspond exactly, both in kind and in relationship, to mind and body in general and to sensation and bodily energy in particular, the terms that Fechner sought to relate, in the interests of his philosophy, by way of Weber's law. On the purely mathematical side, Fechner is less clear as to his background, but it is plain that Bernoulli, Laplace and Poisson were important. Nowadays we are apt to think especially of Fechner's use of the normal law of error as representing his mathematical interest. Fechner's method of constant stimuli makes use of this law, and the method has assumed importance because it is closely related to the biological and psychological statistical methods that also make use of normal distributions. The method of constant stimuli was, however, only one of Fechner's three fundamental methods. Nevertheless, it is interesting to answer the question that arises about Fechner's use of the normal law. The principles were an contained in the earlier mathematicians' work on the theory of probabilities, work of which BernouUi's is representative. Laplace, whom Fechner speciaJly mentioned, developed the general law. Gauss gave it its more usual form, and the law ordinarily bears his name. Fechner refers to Gauss in his use of it, but Gauss seems to have been less important than Laplace. There is nothing new in making this practical application of the theory of probabilities. Since 1662 there had been attempts to apply it to the expectation of life, to the evaluation of human testimony and human innocence, to birthrates and sex-ratios, to astronomical observations, to the facts of marriages, sma]]pox and inoculation, to weather forecasts, to annuities, to elections, and finally (Laplace and Gauss) to errors of scientific observation in general. It was in 1835 that Quetelet first thought of using the law of error to describe the distribution of human traits, as if nature, in aiming at an ideal average man, I'homme moyen, missed the mark and thus created deviations on either side of the average. It was Quetelet who gave Francis Galton the idea of the mathematical treatment of the inheritance of genius (1869), but Fechner had nothing of this sort in mind. The older tradition, however, he must have known, at least in part, and it is from it that he took for the method of constant stimuli the normal law of error, now so important to psychologists. It was easier to assume then than it is now that the normal law, as indeed its name implies, is a law of nature which applies whenever variability is uncontrolled. Beside this general background and knowledge, Fechner brought to the problem of psychophysics several very definite things. First, there was
1158
Edwin G. Borin,
the fact of the limen, made familiar by Herbart but also obvious enough in other ways, as, for example, in the invisibility of the stars in daylight. Second, there was Weber's law, a factual principle which, if not verified, could still be expected to persist in modified form. Third, there was the experimental method, which was equally fundamental and which derived from Fechner's own temperament in defiance of Herbart. Fourth, there was Fechner's clear conception of the nature of psychophysics as "an exact science of the functional relations or the relations of dependency between body and mind," This conception was the raison d'etre for the entire undertaking. Finally, there was Fechner's very wise conclusion that he could not attempt the entire program of psychophysics and that he would therefore limit himself, not only to sensation, but further to the intensity of sensation, so that a final proof of his view in one field might, because of its finality, have the weight to 1ead later to extensions into other fields. We must pause here to note that Fechner's view of the relation of mind and body was not that of psychophysical parallelism, but what has been called the identity hypothesis and also panpsychism. The writing of an equation between the mind and the body in terms of Weber's law seemed to him virtually a demonstration both of their identity and of their fundamental psychic character. Nevertheless. Fechner's psycho· physics has played an important part in the history of psychophysical parallelism for the reason that mind and body, sensation and stimulus, have to be regarded as separate entities in order that each can be meas· ured and the relation between the two determined. Fechner's psychology therefore, like sO much of the psychology that came after him, seems at first to be dualistic. It is true that he began with a dualism. but we must remember that he thought he had shown that the dualism is not real and is made to disappear by the writing of the true equation between the two terms. It is so easy nowadays to think that the Weber-Fechner law represents the functional relation between the measured magnitude of stimulus and the measured magnitude of sensation, that it is hard to reaJize what difficulty the problem presented to Fechner. It seemed plain to him, however, that sensation, a mental magnitude, could not be measured directly and that his problem was therefore to get at its measure indirectly. He began by turning to sensitivity. Sensation, Fechner argued, we cannot measure; all we can observe is that a sensation is present or absent, or that one sensation is greater than, equal to, or less than another sensation. Of the absolute magnitude of a sensation we know nothing directly. Fortunately, however, we can measure stimuli, and thus we can measure the stimulus values necessary to give rise to a particular sensation or to a difference between two sensa-
lJS9
GUltav Tht'odor F «hnt"
tions; that is to say, we can measure threshold values of the stimulus. When we do this we are also measuring sensitivity, which is the inverse of the threshold value. Fechner distinguished between absolute and differential sensitivity, which correspond respectively to the absolute and differential Jimens. He recognized the importance of variability in this subject-matter and the necessity of dealing with averages, extreme values, the laws of averages and the laws of variability about the averages-in short, the necessity of using statistical methods. Since Fechner believed that the stimulus, and hence sensitivity, can be measured directly but that sensation can not, he knew that he must measure sensation itself indirectly, and he hoped to do it by way of its differential increments. In determining the differential limen we have two sensations that are just noticeably different, and we may take the just noticeable difference (the jnd) as the unit of sensation, counting up jnd to determine the magnitude of a sensation. There was a long argument later as to whether every liminal increment of sensation (8S) equals every other one, but Fechner assumed that 8S the jnd, and that the sensed differences, being all just noticeably different, are equal and therefore constitute a proper unit. One does not in practice count up units for large magnitudes. One works mathematically on the general case for the general function which can, perhaps, later be applied in measurement. Fechner went to work in the fol1owing manner. In expounding him we shall use the familiar English abbreviations instead of Fechner's symbols: S for the magnitude of the sensation and R for the magnitude of the stimulus (Reiz). Weber's experimental finding may be expressed:
=
8R - = constant, for the jnd.
Weber's Law (1)
R This fact ought to be caned "Weber's law," since it is what Weber found. Fechner, however. used the phrase for his final result. He assumed that. if (1) holds for the jnd. it must also hold for any sma]] increment of S, 8S. and that he could thus express the functional relation between Sand R by writing: 8S
=c
8R Fundamental formula (2)
R
where c = a constant of proportionality. This was Fechner's Fundamentalformel, and we must note that the introduction of 8S into the equation is the mathematical equivalent of Fechner's conclusion that all 8S's are equal and can be treated as units. One has only to integrate to accomplish the mathematical counterpart of counting up units to perform a measurement.
1160
If we can write the fundamental formula, we can certainly measure sensation. Fechner, therefore, integrated the equation, arriving at the result (3)
S=clo&R+C
where C = the constant of integration and e = the base of natural logarithms. In formula (3) we really have the desired result, since it gives the magnitudes of S for any magnitude of R, when the two constants are known. Fechner had thus demonstrated the fundamental point of his philosophy. Nevertheless this formula was unsatisfactory because of the unknown constants, and Fechner undertook to eliminate C by reference to other known facts. He let r = the threshold value of the stimulus, R, a value at which S, by definition, = O. Thus: When R = r, S = 0 Substituting these values of Sand R in (3), we get: 0= clo&r+ C C -c lo&..r
=
Now we can substitute for C in (3): S ::: C )o&R - cloy ::: cOo&R - lo&r) R =c)o&-
(4)
r
We can shift to common logarithms from natural logarithms by an appropriate change of the constant from c to, Jet us say, k: R S ::: k log -
Measurement formula (5)
r
This is the formula for measurement, Fechner's Mass/ormel. The scale of S is the number of jnd that the sensation is above zero, its value at the limen. Beyond this point Fechner went one more step. He suggested that we might measure R by its relation to its liminal value; that is to say, we might take r as the unit of R. If r be the unit of R. then: S::: k log R.
Fechner's Law (6)
This last formula, (6), Fechner called "Weber's Law." It is only as we view the matter now that we see that formula (1) is really Weber's law and that formula (6) should be called Fechner's law. We must remember that S :;:; k log R is true only when the unit of R is the liminal value of the stimulus and in so far as it is valid to integrate S and to assume that S ::: 0 at the limen. Furthermore, the entire conclusion depends on the validity
Gustav Thtodor F'ec'lIItr
1161
of Weber's finding, formula (1), a generalization that further experimentation has verified only approximately in some cases, but not exactly nor for the entire range of stimuli. About this claim of Fechner's that he had measured sensation vigorous controversy raged for forty years or more; and two of the fundamental objections are of sufficient interest to deserve brief mention here. One argument was that Fechner had assumed the equality 0/ all jn.d without sufficient warrant and that he had thus in a sense begged the question, since there is no meaning to the statement that one as equals another unless S is measurable. There is certainly some force to this criticism, but it can be met in two ways. It was actually met in part by Delb(Eurs notion of the sense-distance and the experiments on supraliminal sense-distances. Delb(Euf pointed out that we can judge the size of the interval between two sensations immediately and directly. For example, we can say of three sensations, A, Band C, whether the distance AB is greater than, equal to or less than the distance Be. Thus we perform a mental measurement immediately, and the question is not begged. Now suppose AB = BC psychologically, and suppose that we find that the stimulus for B is the geometric mean of the stimuli for A and C. Then we have shown that the Fllndamental/ormel holds for a large S like AB, and, if the same law holds for large distances judged equal and for jnd, we may assume the jnd must also be equal. As a matter of fact, Weber's law has not been shown to hold generally nor exactly. It depends on what arbitrary scale of stimulus units is being used and it is apt to be wrong for the low values of any convenient measure of stimulus intensity. Modern findings show that the assumption of equality for intensitive jnd is often inconsistent with the direct judgmental comparison of supraliminal intensitive differences. There is evidence, for instance, that jnd for the pitch of tones are equal in this sense, but that jnd for the loudness of tones are not. The other way to meet the objection that all jnd are not equal is to say frankly that the equality of units must be an assumption. Certainly one jnd is equivalent to another in that both are jnd. The issue can be met thus on purely logical grounds, though this solution leaves open still the question of the exact sense in which jnd as such are equa1. So it must be with an units, and even Delb(Euf's sense-distances are not more satisfactory in this regard. The obvious fact is, nevertheless, that the Fechner Law states a relationship between two entities that are not identical. S must be something, and it is not R. Something other than the stimulus has been measured. The other important criticism of Fechner has been called the quantity objection.. It was argued that it is patent to introspection that sensations do not have magnitude. "Our feeling of pink," said James, "is surely not
Edwl" G. Bori".
1162
~
•
en ~ "S
'r
e
( /)
Liminal
ttimulus • r ____~
~S
-4
-3
-2
-t
Negative Sensation FIGURE I-Fechner's Law: S
=
=- 5
0
I
+2
+t
+3
+4
+5
Sensation = + S
k log R. The positions of the equally spaced "enical ordinates
represent an arithmetic series of So their
successi~
heights the corresponding geo-
metric series of R, Thus the CUf"e shows how a logarithmic function represents a correlation between an arithmetic and a geometric series. It also shows why the function requires the theoretical existence of negative sensations. for. when S = O. R = a finite value. r. the limen; and S passes through an infinite number of negati"e values when R varies between rand O. In this diagram R is plotted with r as the unit and k Is arbitrarily chosen as 4· S for common logarithms.
a portion of our feeling of scarlet; nor does the light of an electric arc seem to contain that of a tallow candle within itself." "This sensation of 'gray,' " Kiilpe remarked, "is not two or three of that other sensation of gray." Must not Fechner have tricked us when he proved by his figures something that we an can see is not true? The criticism is not valid, yet Fechner himself was to blame for this turn that criticism took. As we have seen, Fechner had said that stimuli can be measured directly and that sensations can not, that sensations must be measured indirectly by reference to the stimulus and by way of sensitivity. No wonder the critics accused Fechner of measuring the stimulus and calling it sensation. No wonder they argued that his own statement that sensation can not be measured directly is equivalent to saying that it can not be measured at all. Actually the 'quantity objection' was met by being ignored. The experimentalists went on measuring sensation while the objectors complained,
Gustav
1163
T"~odo,. F~C'''"~,,
or at least they went on measuring whatever Fechner's S is. There are, however, two remarks that can be made about this matter. (1) Sensation can indeed be measured as directly as is the stimulus. You can compare directly in judgment two sensory differences. You can say that the difference AB is greater than, or 1ess than or equal to the difference BC, when A, Band C are serial intensities or qualities or extents or durations. Such judgments boil down to the crucial judgment of equal or not-different. Such a comparison is quite as direct for the sensation as it is for the stimulus. Similarly, to compare weights you use a balance and form the judgment equal when the scalepans are not-different in height. Or for length you note on a tape the mark that is not-different in position from the end of the measured object. (2) Contrariwise, we may say that the stimulus is just as unitary and simple as the sensation. A meter is not made up of 100 parts which are called centimeters, or of a thousand parts which are called millimeters, or of 39·37 parts which are called inches. A meter in itself is just as unitary as a scarlet. The magnitude of neither implies complexity but simply a relationship to other objects that is got by the conventional methods of measurement. We must now turn to certain matters that are connected with Fechner's name: Inner psychophysics, the limen of consciousness, negative sensations and the psychophysical methods. Fechner distinguished inner psychophysics from outer psychophysics. Outer psychophysics, he said, deals with the relation between mind and stimulus, and it is in outer psychophysics that the actual experiments are to be placed. Inner psychophysics, however, is the relation between mind and the excitation most immediate to it and thus deals most immediately with the relationship in which Fechner was primari1y interested. S k log R is a relationship in outer psychophysics. Between Rand S, excitation, E, is interposed. Just where is the locus of this logarithmic relationship, between Rand E or between E and S? It is possible that S is simply proportional to E and that the true law is E = k log R, a statement which means that Weber's Jaw does not solve the problem of mind and body as Fechner hoped it would. Fechner, however, maintained that E is probably proportional to R and that Weber's law is the fundamental law of inner psychophysics, S = k log E. This view Fechner supported with five arguments. (1) In the first place, he said in the Elemente, it would be inconceivable that a logarithmic relation should exist between Rand E. Such a statement is hardly an argument, and Fechner took it back in the Revision. (2) Then he observed that the magnitude of S does not change when sensitivity is reduced, whereas it should if S kE and E is involved in the change of sensitivity. (3) Further he noted that Weber's law holds for tonal pitch, and that it would be impossible for the vibrations of E to have other than a propor-
=
=
Edwin G. Bori".
1164
tional relation to the vibrations of R. (Of course he was in error in supposing that nervous excitation is vibratory.) (4) Next he pointed out that a subliminal S probably has an E, that the invisible stars in daytime probably give rise to excitation which is below the limen of consciousness. Such a fact could be true only if S k log E. (5) Finally, he appealed to the distinction between sleep and waking, and between inattention and attention, as indicating the existence of a limen of consciousness rather than a limen of excitation. This last argument is the most cogent. Certainly the mere fact of the selectivity of attention seems to mean that there are many excitations, all prepotent for consciousness, of which only a few become conscious. However, Fechner's entire argument would not be taken very seriously at the present time. It is important for us merely to see why Fechner, working in outer psychophysics, thought he was solving the problem, aU-important to him, of inner psychophysics. From this discussion we see how important the fact of the limen 0/ consciousness was to Fechner. What Fechner called Weber's Law is based upon the limen, for, if S = k log R, then, when S = 0, R is some finite quantity, a liminal value. Herbart's limen of consciousness is thus simply a corollary of this law. In fact, Fechner was further consistent with Herbart in relating the limen to attention: when consciousness is already occupied with other sensations, a new sensation can not enter until it overcomes the "mixture limen." The psychology that depends upon this law also requires the existence of negative sensations. Figure 3 shows graphically the logarithmic curve that gives the relationship of S to R for "Weber's Law:' The function requires that R r, the limen, when S 0, and thus it gives negative sensations for subliminal values of R, for theoretically when R = 0, S is negative and infinite. Fechner believed that "the representation of unconscious psychical values by negative magnitude is a fundamental point for psychophysics," and by way of this mathematical logic he came to hold a doctrine of the unconscious not unlike that of his predecessors, Leibnitz and Herbart. Fechner's claim to greatness within psychology does not, however, derive from these psycho]ogical conceptions of his, nor even from the formulation of his famous law. The great thing that he accomplished was a new kind of measurement. The critics may debate the question as to what it was that he measured; the fact stands that he conceived, developed and established new methods 0/ measurement. and that, whatever interpretation may later be made of their products, these methods are essentially the first methods of mental measurement and thus the beginning of quantitative experimental psychology. Moreover, the methods have stood the test of time. They have proven applicable to all sorts of psychological problems and situations that Fechner never dreamed of, and they are al1
=
=
=
GuSlo" Theodor Fuh"u
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still used with only minor modifications in the greater part of quantitative work in the psychological laboratory today. There were three fundamental methods: (1) the method of just noticeable differences, later caned the method 01 limits; (2) the method of right and wrong cases, later caned the method 01 constant stimuli or simply the constant method; and (3) the method 01 average error, later called the method 01 adjustment and the method 01 reproduction. Each of these methods is both an experimental procedure and a mathematical treatment. Each has special forms. The constant method has been much further developed by G. E. MUlier and F. M. Urban. More recently the method of adjustment has shown certain advantages over the others. Changes and development, however, add to Fechner's distinction as the inventor. There are few other men who have done anything of equal importance for scientific psychology. The storm of criticism that Fechner's work evoked was in general a compliment, but there were also those psychologists who were unable to see anything of value in psychophysics. But three years after Fechner's death, James wrote: "Fechner's book was the starting point of a new department of literature, which it would perhaps be impossible to match for the qualities of thoroughness and subtlety, but of which. in the humble opinion of the present writer, the proper psychological outcome is just nothing." Elsewhere he gave his picture of Fechner and his psychophysics: The Fechnerian Massformel and the conception of it as the ultimate 'psychophysic law' will remain an 'idol of the den: if ever there was one. Fechner himself indeed was a German Gelehrter of the ideal type, at once simple and shrewd, a mystic and an experimentalist, homely and daring, and as loyal to facts as to his theories. But it would be terrible if even such a dear old man as this could saddle our Science forever with his patient whimsies, and, in a world so full of more nutritious objects of attention, compel all future students to plough through the difficulties, not only of his own works, but of the still drier ones written in his refutation. Those who desire this dreadful literature can find it; it has a 'disciplinary value'; but I will not even enumerate it in a foot-note. The only amusing part of it is that Fechner's critics shou1d always feel bound. after smiting his theories hip and thigh and leavin~ not a stick of them standing, to wind up by saying that nevertheless to hIm belongs the imperishable glory, of first formulating them and thereby turning psychology into an exact science. "·And everybody praised the duke Who this great fight did win.' 'But what good came of it at last?' Quoth little Peterkin. ·Why. that I cannot tell,' said he, 'But 'twas a famous victory!' ,. It is plain to the reader that the present author does not agree with
James. Of course, it is true that, without Fechner or a substitute which the times would almost inevitably have raised up, there might still have been
Edwin G. Borin,
an experimental psychology. There would still have been Wundt-and Helmholtz. There would, however, have been little of the breadth of science in the experimental body, for we hardly recognize a subject as scientific if measurement is not one of its tools. Fechner, because of what he did and the time at which he did it, set experimental quantitative psychology off upon the course which it has followed. One may call him the 'founder' of experimental psychology, or one may assign that title to Wundt. It does not matter. Fechner had a fertile idea which grew and brought forth fruit abundantly.
COMMENTARY ON
SIR FRANCIS GALTON IR FRANCIS GALTON was not merely an eminent Victorian but a thoroughly good and attractive man. It has been said of him that he was essentially a social reformer. The implied disparagement of his seien· tific abilities is unjust. He was imbued, it is true, with a strong sense of social responsibility and of mission~ he was neither erudite nor philosophically profound. Yet, judged both by What he accomplished and by the impetus he gave to the researches of others, he was a great scientist. His kindly feelings and earnest social convictions are the more to be valued because they were carried over into his work and shaped its goals. Karl Pearson in his monumental biography of Galton calls him the ·'master builder" of the modern theory of statistics. t Much of his work was handmade and primitive, many of his results were wrong, his theories of heredity and eugenics were grossly oversimplified; yet one is struck by the pioneer character of his labors, by their extraordinary suggestiveness even when, as Pearson says, his methods are "the crude extemporizations of the first settler." His outstanding merit as an investigator was that he blazed the trail. 2 Galton was born near Birmingham on February 16, 1822. The same year saw the birth of Gregor Mendel, with whom Galton said he always felt himself "sentimenta1ly connected." :i On his father's side he came of a Hne of prosperous Quaker businessmen who combined a talent for practical affairs with scientific and statistical interests. His grandfather, Samuel John, was an intimate of Priestley, Watt and Boulton, and a Fellow of the Lunar Society, a provincially famous scientific-philosophical group. His father, Samuel Tertius, wrote a book on currency, filled his home with telescopes. barometers, solar microscopes and similar scientific instruments, was devoted to literature and still found time to be a successful banker. On his mother's side, Galton was related to the eminent Darwin family. Charles Darwin was his cousin and the two men were always on excellent terms. Galton credits his grandmother Darwin, who lived to be eighty-five, for the exceptional longevity of his family:'
S
1 Karl Pearson, The Life, Letters and Labours of Francis Galton, Cambridge, 19141930, Vol. II, p. 424. 2 Pearson, op. cit., Vol. 11, p. V. :I Francis Galton, Memories of My Life, London, 1908, p. 308 . .. "My mother died just short of ninety, my eldest brother at eighty-nine, two sisters, as already mentioned, at ninety-three and ninety-seven respectively; my surviving brother is ninety-three and in good health. My own age is now only eighty-six but may possibly be prolonged another year or more. I find old age thus far to be a' very happy time. on the condition of submitting fran~y to ite; many limitations." Ibid, p. 7.
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Com~n'
Galton's father was eager to give him an excellent education and also to make him se1f-reHant at the earliest possible age. The result was a higgledypigg]edy coune of schooling. At the age of eight he was bundled off to a wretched boarding school at Boulogne, where he remained for two years. Galton remembered it as a fine place for birchings and bullying. He was then sent to sma]] private schools at which. though he was a most gifted child. 5 he claims to have learned nothing. "I had craved for what was denied, namely an abundance of good English reading, wen-taught mathematics. and solid science. Grammar and the dry rudiments of Latin and Greek were abhorrent to me." Both his parents wanted him to take up medicine and he was therefore apprenticed at sixteen to a leading physician. A short time later he became an "indoor pupil" at the Birmingham General Hospital. He completed his education at Kings College, London, and at Trinity College. Cambridge. His scholastic record was undistinguished. The death of his father made Galton financially independent at the age of twenty-two. He thereupon abandoned medicine and set forth on a course of travel. In ] 845 he visited Egypt. the Sudan and Syria. This and other journeys are vividly described in his Memories. For a time he cultivated. in moderation. the pleasures of sporting life; he summered in the Shetlands. went seal-shooting. collected sea-birds, essayed yachting and ballooning. He also published the first of an almost uncountable series of scientific papers. fl It describes an invention of his. the Telotype. which was designed "to print telegraphic messages and to govern heavy machinery by an extremely feeble force." 7 In 1850 he left England for a two-year sojourn in South Africa. His explorations in South-West Africa. conducted under the usual dangerous and harassing conditions, brought out his best qualities as a man and as a scientist. He penetrated for more than 1.000 miles into unexplored territory and recorded a great deal of valuab]e information about the lands through which he passed and their people. For his geographical discoveries and astronomical observations he was awarded the gold medal of the Royal Geographical Society; and in 1856. at the age of thirty-four. was made a Fellow of the Royal Society. His scientific reputation was now established. After his return from Africa Galton was, as he said, "rather used up in health." It happened quite a few times in his life that he suffered physical 1 Galton learned to read at the age of two-and-a-half years, and wrote a letter before he was four. By the age of five he could read "almost any English book," could do multiplication. and could tell time. For an account of his precocity see Lewis M. Terman. "The Intelligence Quotient of Francis Galton in Childhood." American }o"rnal of Pn'chology (1917). Vol. 28. p. 209. also by Terman. "The Psychological Approaches to the Biography of Genius." Sciellce (1940), Vol. XCII, p. 264. 6 Pearson refers to at least fifteen books by Galton. and lists more than 220 papers. He concedes that the register is probably incomplete. 7 Galton. op. cit., p. 119.
Sir F,..ftcls Gillum
1169
breakdowns and periods of severe depression and giddiness-what he called a "sprained brain"--during which he was incapable of working. Invariably, a change of habits, a tour abroad, "plenty of outdoor exercise," completely restored him.s Except for these "tours and cures" the story of the remaining fifty-five years of Galton's life "is to be told less in terms of his movements than of his thoughts." $) Galton was not a mathematician but he was mathematically minded. One of his maxims was: "Whenever you can, count"; he himself was almost obsessed by the need to count and to measure. In his laboratory he measured heads, noses. arms, legs, color of eyes and hair, breathing power, "strength of pull and of squeeze," keenness of sight and of hearing, reaction time, height. weight and so on. He compiled statistics of the weather, of the properties of identical twins. of the frequency of yawns, of the sterility of heiresses, of life span, of the inheritance of physical and mental characters. He counted the number of "fidgets" per minute among persons attending lectures; the purpose of this observation was apparently to derive a coefficient of boredom. Middle-aged persons, he found, are medium fidgets, ~'children are rarely still, while elderly philosophers will sometimes remain rigid for minutes together." He made a "Beauty Map" of the British Isles, classifying the girls he passed in the streets of various towns as "attractive, indifferent or repeHent." The method he employed was to prick h01es in a piece of paper, Htom rudely into a cross with a long leg." which he concealed within his pocket. London ranked highest; Aberdeen, lowest. When 800 visitors to a cattle exhibition at Plymouth tried to guess the weight of an ox, he tabulated the estimates and observed that the "vox populi was correct to within 1 per cent of the real value." This result might be construed, he felt, as evidence of the "trustworthiness of a democratic judgment" except that the proportion of voters capable of assessing the weight of an ox "undoubtedly surpassed that of the voters in ordinary elections who are versed in pol itks." 1 0 Among his most incredible achievements was persuading Herbert Spencer to accompany him to the Derby. Spencer was mildly bored but Galton enjoyed every moment of the excursion. He took the opportunity of recording a scientific observation, viz., the change in "prevalent tint 8 "I was blessed with an abundance of animal spirits and hopefulness. though they were dashed temporarily Over and over again by the great readiness with which my brain became overtaxed; however I always recuperated quickly." Galton, op cit. 9 C. P. Blacker. Ellgellics, Galloll alld Aller, Cambridge (Mass.), 1952, p. 37. This .is a very agreeable and accurate account of Galton's life. of the work he did in eugenics and of later developments in the field. 10 "The judgments were unbiassed by passion and uninfluenced by oratory and the like . . . The average competitor was probably as well fitted for making a just estimate of the dressed weight of the ox. as an average voter .is to judge the merits of most poJitieal issues on which he votes." Galton. "Vox Populi:' Natllre, Vol. LXXV. pp. 450-451, March 7. 1907.
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of the faces in the great stand," under the flush of excitement as the horses approached the finish line. To a benign reasonableness and disinterestedness Galton added an amazing innocence and suggestibility. His "passionate desire to subjugate the body to the spirit" led him to subject himself to some hair-raising experiments; the practice of slow self-suffocation had a strange fascination for him. Having invented an optical underwater device he amused himself by reading while submerged in his bath; on several occasions he almost drowned because he "forgot that [he] was nearly suffocating." He describes various adventures in auto-suggestion, such as pretending that a comic figure in Punch possessed divine attributes or that everyone he met while walking in Piccadilly was a spy. Invariably he managed in these experiments to frighten himself half to death. But they expressed truly his unquenchable curiosity; every experience was acceptable "for one wants to know the very worst of everything as well as the very best." The range of GaIton's scientific interests and pursuits is too broad to describe in this space. One of his major efforts was concentrated on the study of heredity; it was the subject of his most celebrated book, Hereditary Genius (1869), published when he was forty-seven. But this in a sense was the focus and synthesis of other studies. The earlier years were devoted to geography, ethnology and anthropology; later he turned to anthropometry and genetics. Galton formulated the profoundly influential principle of correlation. Others in his century "hovered on the verge of the discovery," and the idea was greatly elaborated and refined by Pearson, Edgeworth and Weldon.I 1 But Galton was the pioneer in evolving the conception of a correlated system of variates, the representation "by a single numerical quantity of the degree of relationship, or of partial causality, between the different variables of our ever-changing universe." 12 The connection between this concept and the problems of inheritance is self-evident; indeed it was his study of heredity that led Galton to correlation. How, for example, did a character possessed by the father influence the like character in the son? It was, of course, a contributing factor, but only one of many, derived from the son's mother, from his grandparents and other forebears. The method of expressing such relations of multiple causality in a single formula came to Galton one morning while waiting at a roadside station for a train, "poring over a small diagram in my notebook." 13 11 Helen M. Walker, Studies in the History 01 Statistical Method, Baltimore. 1931, p. 92. For a general discussion of Gahon's work on correlation, see Pearson, op. cit., Vol. III (a), Chapter XIV. 12 Pearson, as in preceding note, p. 2. 13 Galton's explanation of the correlation concept is a model of clarity and deserves reprinting: "It had appeared from observation, and it was fully confirmed by this theory, that such a thing existed as an ·Index of Correlation'; that is to say, a fraction. now commonly written r, that connects with closer approximation every value
1111
S" FrtII'/ctl Gal'oll
It was an inspiration that changed the course of modem social studies. An amateur in many fields, Galton was never a mere dabbler or dilettante. He deserves to be remembered for his researches in fingerprinting, his studies of imagery, of synesthesia and of color association, his discovery of number~forms (i.e., the images by which we represent numbers to ourselves), his contributions to meteorology (he invented the word "anticyclone"), his valuable experiments in blood transfusion, his origination of composite photography. The selection below is a chapter from Hereditary Genius. The book exemplifies Galton's statistical approach and his primary interest in the problem of mental inheritance, and of improvement of the race by eugenic practices. The seJection itself analyzes the distribution of mental ability and suggests that the pattern resembles that of the distribution of physical traits. Galton's interest in this subject culminated in Inquiries into Human Faculty and Its Development (1883), a book regarded "as the beginning of scientific individual psychoJogy and of the mental tests:' But as Boring points out in his excellent History of Experimental Psychology, Galton's intention regarding the book was different.14 The effect on his mind of the publication of his cousin's Origin of Species was "to demolish a multitude of dogmatic barriers by a single stroke and to arouse a spirit of rebellion against all ancient authorities . . . contradicted by modern science." Heredit'!ry Genius, Human Faculty, his writings on eugenics, even his passion for classification and measurement may be considered as Galton's attempts to further this rebellion. He believed in evolutionary progress and strove to prove that it was at once a more rational and a more promising faith than any offered by current reJigious dogmas. It was a faith for self-respecting, free men who dared to think they could do better for themselves and their ancestors than to rely on the efficacy of prayer. Galton heJd up "as the goal of human effort, not heaven, but the superman." 1.5 His studies of heredity and genius place excessive importance on biological factors as determinants of personality and achievement. Yet the nature-nurture controversy is far from settled, and no thoughtful person would deny the significance of Galton's work or the soundness of many of his principles of eugenics. of deviation [from the median] on the part of the subject. with the ave,.age of all the associated deviations of the Relative as already described. Therefore the closeness of any specified kinship admits of being found and expressed by a single term. If a particular individual deviates so much, the average of the deviations of all his brothers wilJ be a definite fraction of that amount; similarly as to sons, parents, first cousins, etc. Where there is no relationship at aU. r becomes equal to 0; when it is so close 1. Therefore the value of that Subject and Relative are identical in value, then r r lies in every case somewhere between the extreme limits of 0 and 1. Much more could be added. but not without using technical language, which would be inappropriate here:' Memories, p. 303. Edwin G. Borina. A History 01 ExperlmenlaJ Psychology, New York, Second Edition. 19S0, p. 483. lIS Boring. op. cit., p. 483.
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Editor', Commertt
Galton died on January 17, 1911, at his home at Haslemere. Until the last few days he was gay, socially active, busy with correspondence. He had a long and an uncommonly satisfying life. His marriage which lasted for forty-four years (his wife died in 1897) had been very happy, but there were no children. It is true of Galton, as of few men, that he did with his life exactly what he wanted to do and that if he had had it to live over he would not have varied its course. One envies and admires him. He had courage, honesty, imagination, tolerance, sympathy and a sense of humor. He labored and created, he gave friendship and he inspired itboth personally and through his work. This must be my excuse for this perhaps inexcusably long preface to his essay.
MallY 01 the modern bllildings in Italy are historically known to have been built Ollt 01 the pillaged strllctllres 01 older days. Here we may obse"'e a coillmn or a lilltel serving the same purpose a second lime . •.. I will purslle Ihis rough simile jllst olle step Ilirther, which is as much as il will bear. Suppose we were building a hOllse with secolld-hand materials carted Irom a dealer's yard, we should often find COllsiderable portions 01 the same old hOllse 10 be still grollped together. .•. So in the process 01 lrallsmissiOiI by inheritance, elements derived Irom the same ancestor are apt to appear ill large groups. -SIR FRANCIS GALTON
2
Classification of Men According to Their Natural Gifts By SIR FRANCIS GALTON
I HAVE no patience with the hypothesis occasionally expressed, and often implied, especially in tales written to teach children to be good, that babies are born pretty much alike, and that the sole agencies in creating differ· ences between boy and boy, and man and man, are steady application and moral effort. It is in the most unqualified manner that I object to pretensions of natural equality. The experiences of the nursery, the school, the University, and of professional careers, are a chain of proofs to the contrary. 1 acknowledge freely the great power of education and social influences in developing the active powers of the mind, just as I acknowledge the effect of use in developing the muscles of a blacksmith's arm, and no further. Let the blacksmith labour as he will, he will find there are certain feats beyond his power that are well within the strength of a man of herculean make, even although the latter may have led a sedentary life. Some years ago, the Highlanders held a grand gathering in Holland Park, where they challenged all England to compete with them in their games of strength. The chal1enge was accepted, and the well-trained men of the hills were beaten in the foot-race by a youth who was stated to be a pure Cockney, the clerk of a London hanker. Everybody who has trained himself to physical exercises discovers the extent of his muscular powers to a nicety. When he begins to walk, to row, to use the dumb bells, or to run, he finds to his great delight that his thews strengthen, and his endurance of fatigue increases day after day. So long as he is a novice, he perhaps flatters himself there is hardly an assignable limit to the education of his muscles; but the daily gain is soon discovered to diminish, and at last it vanishes altogether. His maximum performance becomes a rigidly determinate quantity. He learns to an inch, how high or how far he can jump. when he has attained the highest state 1173
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Sir
FraNcI~
GaltoN
of training. He learns to half a pound, the force he can exert on the dynamometer, by compressing it. He can strike a blow against the machine used to measure impact, and drive its index to a certain graduation, but no further. So it is in running, in rowing, in walking, and in every other form of physical exertion. There is a definite limit to the muscular powers of every man, which he cannot by any education or exertion overpass. This is precisely analogous to the experience that every student has had of the working of his mental powers. The eager boy, when he first goes to school and confronts intellectual difficulties, is astonished at his progress. He glories in his newly-deve1oped mental grip and growing capacity for application, and, it may be, fondly believes it to be within his reach to become one of the heroes who have left their mark upon the history of the world. The years go by; he competes in the examinations of school and college, over and over again with his fellows, and soon finds his place among them. He knows he can beat such and such of his competitors; that there are some with whom he runs on equal terms, and others whose intellectual feats he cannot even approach. Probably his vanity still continues to tempt him, by whispering in a new strain. It tells him that classics, mathematics, and other subjects taught in universities, are mere scholastic specialties, and no test of the more valuable intellectual powers. It reminds him of numerous instances of persons who had been unsuccessful in the competitions of youth, but who had shown powers in after-life that made them the foremost men of their age. Accordingly, with newly furbished hopes, and with all the ambition of twenty-two years of age, he leaves his University and enters a larger field of competition. The same kind of experience awaits him here that he has already gone through. Opportunities occur-they occur to every man-and he finds himself incapable of grasping them. He tries, and is tried in many things. In a few years more, unless he is incurably blinded by self-conceit, he learns precisely of what performances he is capable, and what other enterprises lie beyond his compass. When he reaches mature life, he is confident only within certain limits, and knows, or ought to know, himself just as he is probably judged of by the world, with all his unmistakable weakness and aU his undeniable strength. He is no longer tormented into hopeless efforts by the fallacious promptings of overweening vanity, but he limits his undertakings to matters below the level of his reach, and finds true moral repose in an honest conviction that he is engaged in as much good work as his nature has rendered him capable of performing. There can hardly be a surer evidence of the enormous difference between the intellectual capacity of men, than the prodigious differences in the numbers of marks obtained by those Who gain mathematical honours at Cambridge. I therefore crave permission to speak at some length upon this subject, although the details are dry and of little general interest.
Classification 01 Men According to The;, Natural Gilts
1175
There are between 400 and 450 students who take their degrees in each year, and of these, about ) 00 succeed in gaining honours in mathematics, and are ranged by the examiners in strict order of merit. About the first forty of those who take mathematical honours are distinguished by the title of wranglers, and it is a decidedly creditable thing to be even a low wrangler; it will secure a fellowship in a small college. It must be carefully borne in mind that the distinction of being the first in this list of honours, or what is called the senior wrangler of the year, means a vast deal more than being the foremost mathematician of 400 or 450 men taken at hap-hazard. No doubt the large bulk of Cambridge men are taken almost at hap-hazard. A boy is intended by his parents for some profession; if that profession be either the Church or the Bar, it used to be almost requisite. and it is still important, that he should be sent to Cambridge or Oxford. These youths may justly be considered as having been taken at hap-hazard. But there are many others who have fairly won their way to the Universities. and are therefore selected from an enormous area. Fully one·half of the wranglers have been boys of note at their respective schools, and, conversely, almost all boys of note at schools find their way to the Universities. Hence it is that among their comparatively sman number of students, the Universities inc1ude the highest youthful scholastic ability of all England. The senior wrangler. in each successive year, is the chief of these as regards mathematics, and this, the highest distinction, is, or was, continually won by youths who had no mathematical training of importance before they went to Cambridge. All their instruction had been received during the three years of their residence at the University. Now, I do not say anything here about the merits or demerits of Cambridge mathematical studies having been directed along a too narrow groove, or about the presumed disadvantages of ranging candidates in strict order of merit, instead of grouping them. as at Oxford, in c1asses, where their names appear alphabetically arranged. All I am concerned with here are the results; and these are most appropriate to my argument. The youths start on their three years' race as fairly as possible. They are then stimu· lated to run by the most powerful inducements, namely, those of competition. of honour, and of future wealth (for a good fellowship is wealth); and at the end of the three years they are examined most rigorously according to a system that they all understand and are equally well prepared for. The examination lasts five and a half hours a day for eight days. An the answers are carefully marked by the examiners, who add up the marks at the end and range the candidates in strict order of merit. The fairness and thoroughness of Cambridge examinations have never had a breath of suspicion cast upon them. Unfortunately for my purposes, the marks are not published. They are not even assigned on a uniform system. since each examiner is permitted
Sir FrtUlCu Galton
1176
to employ his own scale of marks, but whatever scale he uses, the results as to proportional merit are the same. I am indebted to a Cambridge examiner for a copy of his marks in respect to two examinations, in which the scales of marks were so alike as to make it easy, by a slight proportional adjustment, to compare the two together. This was, to a certain degree, a confidential communication, so that it would be improper for me to publish anything that would identify the years to which these marks refer. I simply give them as groups of figures, sufficient to show the enormous differences of merit. The lowest man in the Jist of honours gains less than 300 marks, the lowest wrangler gains about 1,500 marks; and the senior wrangler, in one of the lists now before me, gained more than 7,500 marks. Consequently, the lowest wrangler has more than five times the merit of the lowest junior optime, and less than one-fifth the merit of the senior wrangler. SCALE OF MERIT AMONG THE MEN WHO OBTAIN MATHEMATICAL HONOURS AT CAMBRIDGE
The results of two years are thrown into a single table. The total number of marks obtainable in each year was 17,000. Number of marks obtained by candidates.
Under 500 500 to 1,000 1,000 to 1,500 1,500 to 2,000 2,000 to 2,500 2,500 to 3,000 3,000 to 3,500 3,500 to 4,000 4,000 to 4,500 4,500 to 5,000 5,000 to 5,500 5,500 to 6,000 6,000 to 6,500 6,500 to 7,000 7,000 to 7,500 7,500 to 8,000
Number of candidates in the two years, taken together, who obtained those marks. 241 74 38 21 11 8
11 5 2 1 3 1
o o
o 1
200 The precise number of marks obtained by the senior wrangler in the more remarkable of these two years was 7,634; by the second wrangler in the same year, 4,123; and by the lowest man in the list of honours, only 237. Consequently, the senior wrangler obtained nearly twice as 1 I have included in this table only the first 100 men in each year. The omitted residue is too small to be important. I have omitted it lest, if the precise numbers of honour men were stated, those numbers would have served to identify the years. FOr reasons already given, I desire to afford no data to serve that purpose.
Classification 0/ Men According to Their Natural Gilts
1177
many marks as the second wrangler, and more than thirty-two times as many as the lowest man. I have received from another examiner the marks of a year in which the senior wrangler was conspicuously eminent. He obtained 9,422 marks, whilst the second in the same year-whose merits were by no means inferior to those of second wranglers in general --obtained only 5,642. The man at the bottom of the same honour list had only 309 marks, or one-thirtieth the number of the senior wrangler. I have some particulars of a fourth very remarkable year, in which the senior wrangler obtained no less than ten times as many marks as the second wrangler, in the "problem paper." Now, I have discussed with practised examiners the question of how far the numbers of marks may be considered as proportionate to the mathematical power of the candidate, and am assured they are strictly proportionate as regards the lower places, but do not afford fu]) justice to the highest. In other words. the senior wranglers above mentioned had more than thirty, or thirty-two times the ability of the lowest men on the lists of honours. They would be able to grapple with problems more than thirty-two times as difficult; or when dealing with subjects of the same difficulty, but intelligible to all, would comprehend them more rapidly in perhaps the square root of that proportion. It is reasonable to expect that marks would do some injustice to the very best men, because a very large part of the time of the examination is taken up by the mechanical labour of writing. Whenever the thought of the candidate outruns his pen, he gains no advantage from his excess of promptitude in conception. I should, however, mention that some of the ablest men have shown their superiority by comparatively little writing. They find their way at once to the root of the difficulty in the problems that are set, and, with a few clean, apposite, powerful strokes. succeed in proving they can overthrow it, and then they go on to another question. Every word they write tens. Thus, tire late Mr. H. Leslie Ellis, who was a brilliant senior wrangler in 1840, and whose name is familiar to many generations of Cambridge men as a prodigy of universal genius, did not even remain during the full period in the examination room: his health was weak, and he had to husband his strength. The mathematical powers of the last man on the list of honours, which are so low when compared with those of a senior wrangler, are mediocre, or even above mediocrity. when compared with the gifts of Englishmen generally. Though the examination places 100 honour men above him, it puts no less than 300 "poll men" below him. Even if we go so far as to allow that 200 out of the 300 refuse to work hard enough to get honours. there will remain 100 who, even if they worked hard, could not get them. Every tutor knows how difficult it is to drive abstract conceptions. even of the simplest kind, into the brains of most people--how feeble and hesitating is their mental grasp-how easily their brains are mazed--how in-
1171
$i, FrGltcts Gallo'll
capable they are of precision and soundness of knowledge. It often occurs to persons familiar with some scientific subject to hear men and women of mediocre gifts relate to one another what they have picked up about it from some lecture-say at the Royal Institution, where they have sat for an hour listening with delighted attention to an admirably lucid account, il1ustrated by experiments of the most perfect and beautiful character, in an of which they expressed themselves intensely gratified and highly instructed. It is positively painful to hear what they say. Their recollections seem to be a mere chaos of mist and misapprehension, to which some sort of shape and organization has been given by the action of their own pure fancy, altogether alien to what the lecturer intended to convey. The average mental grasp even of what is called a well-educated audience, will be found to be ludicrously small when rigorously tested. In stating the differences between man and man, let it not be supposed for a moment that mathematicians are necessarily one-sided in their natural gifts. There are numerous instances of the reverse, of whom the following will be fOUdd, as instances of hereditary genius, in the appendix to my chapter on "Science." I would especially name Leibnitz, as being universally gifted, but Ampere, Arago, Condorcet, and 0'Alembert, were all of them very far more than mere mathematicians. Nay, since the range of examination at Cambridge is so extended as to include other subjects besides mathematics, the differences of ability between the highest and lowest of the successful candidates, is yet more glaring than what I have already described. We still find, on the one hand, mediocre men, whose whole energies are absorbed in getting their 237 marks for mathematics; and, on the other hand, some few senior wranglers who are at the same time high classical scholars and much more besides. Cambridge has afforded such instances. Its lists of classical honours are comparatively of recent date, but other evidence is obtainable from earlier times of their occurrence. Thus, Dr. George Butler, the Head Master of Harrow for very many years, including the period when Byron was a schoolboy (father of the present Head Master, and of other sons, two of whom are also head masters of great public schools). must have obtained that classical office on account of his eminent classical ability; but Dr. Butler was also senior wrangler in 1794, the year when Lord Chancellor Lyndhurst was second. Both Dr. Kaye, the late Bishop of Lincoln, and Sir E. Alderson, the late judge, were the senior wranglers and the first classical prizemen of their respective years. Since 1824, when the classical tripos was first established, the late Mr. Goulburn (brother of Dr. Goulburn, Dean of Norwich, and son of the well-known Serjeant Gou]burn) was second wrangler in ]835, and senior classic of the same year. But in more recent times, the necessary labour of preparation, in order to acquire the highest mathematical places, has become so enormous that there has been a wider differentiation
Classification 0/ Mtm According 10 Tlteir Natural Gllu
1179
of studies. There is no longer time for a man to acquire the necessary knowledge to succeed to the first place in more than one subject. There are, therefore, no instances of a man being absolutely first in both examinations, but a few can be found of high eminence in both classics and mathematics, as a reference to the lists published in the "Cambridge Calendar" will show. The best of these more recent degrees appears to be that of Dr. Barry, late Principal of Cheltenham, and now Principal of King's College, London (the son of the eminent architect, Sir Charles Barry, and brother of Mr. Edward Barry, who succeeded his father as architect). He was fourth wrangler and seventh classic of his year. In whatever way we may test ability, we arrive at equally enormous intellectual differences. Lord Macaulay had one of the most tenacious of memories. He was able to recall many pages of hundreds of volumes by various authors, which he had acquired by simply reading them over. An average man could not certainly carry in his memory one thirtysecond-ay, or one hundredth-part as much as Lord MacaUlay. The father of Seneca had one of the greatest memories on record in ancient times. Porson, the Greek scholar, was remarkable for this gift, and, I may add, the "Porson memory" was hereditary in that family. In statesmanship. generalship, Hterature. science. poetry, art, just the same enormous differences are found between man and man, and numerous instances recorded in this book, will show in how small degree, eminence, either in these or any other class of intel1ectual powers, can be considered as due to purely special powers. They are rather to be considered in those instances as the resuh of concentrated efforts, made by men who are widely gifted. People lay too much stress on apparent specialities, thinking overrashly that, because a man is devoted to some particular pursuit, he could not possibly have succeeded in anything else. They might just as well say that, because a youth had fallen desperately in love with a brunette, he could not possibly have faBen in love with a blonde. He mayor may not have more natural liking for the former type of beauty than the latter, but it is as probable as not that the affair was mainly or wholly due to a general amorousness of disposition. It is just the same with special pursuits. A gifted man is often capricious and fickle before he selects his occupation, but when it has been chosen, he devotes himself to it with a truly passionate ardour. After a man of genius has selected his hobby, and so adapted himself to it as to seem unfitted for any other occupation in life, and to be possessed of but one special aptitude, I often notice, with admiration, how well he bears himself when circumstances suddenly thrust him into a strange position. He will display an insight into new conditions, and a power of dealing with them, with which even his most intimate friends were unprepared to accredit him. Many a presumptuous fool has mistaken indifference and neglect for in-
1180
Sfr FraNCis Galton
capacity; and in trying to throw a man of genius on ground where he was unprepared for attack, has himself received a most severe and unexpected fall. I am sure that no one who has had the privilege of mixing in the society of the abler men of any great capital, or who is acquainted with the biographies of the heroes of history, can doubt the existence of grand human animals, of natures pre-eminently noble, of individuals born to be kings of men. I have been conscious of no slight misgiving that I was committing a kind of sacrilege whenever, in the preparation of materials for this book, I had occasion to take the measurement of modern intellects vastly superior to my own, or to criticize the genius of the most magnificent historical specimens of our race. It was a process that constantly recalled to me a once familiar sentiment in bygone days of African travel, when I used to take altitudes of the huge cliffs that domineered above me as I travelled along their bases, or to map the mountainous landmarks of unvisited tribes, that loomed in faint grandeur beyond my actual horizon. I have not cared to occupy myself much with people whose gifts are below the average, but they would be an interesting study. The number of idiots and imbeciles among the twenty milJion inhabitants of England and Wales is approximately estimated at 50,000, or as 1 in 400. Dr. Segiun, a great French authority on these matters, states that more than thirty per cent of idiots and imbeciles, put under suitable instruction, have been taught to conform to social and moral law, and rendered capable of order, of good feeling, and of working like the third of an average man. He says that more than forty per cent have become capable of the ordinary transactions of life, under friendly control; of understanding moral and social abstractions, and of working like two-thirds of a man. And, ]astly, that from twenty-five to thirty per cent come nearer and nearer to the standard of manhood, tiJI some of them wi11 defy the scrutiny of good judges, when compared with ordinary young men and women. In the order next above idiots and imbeciles are a large number of milder cases scattered among private families and kept out of sight, the existence of whom is, however, well known to relatives and friends; they are too siUy to take a part in general society, but are easily amused with some trivia], harmless occupation. Then comes a class of whom the Lord Dundreary of the famous play may be considered a representative; and so, proceeding through successive grades, we gradually ascend to mediocrity. I know two good instances of hereditary siHiness short of imbecility, and have reason to believe I could easily obtain a large number of similar facts. To conclude, the range of menta] power between-I wiU not say the highest Caucasian and the lowest savage-but between the greatest and least of English intellects, is enormous. There is a continuity of natural
Classlficallon 01 ltft!n Accordin,
10
Tht!ir Natural Gilts
tl81
ability reaching from one knows not what height, and descending to one can hardly say what depth. I propose in this chapter to range men according to their natural abilities, putting them into classes separated by equal degrees of merit, and to show the relative number of individuals inc1uded in the several classes. Perhaps some persons might be inclined to make an offhand guess that the number of men included in the several classes would be pretty equa1. If he thinks so, I can assure him he is most egregiously mistaken. The method I shall employ for discovering all this, is an application of the very curious theoretical law of "deviation from an average." First, I will explain the law, and then I will show that the production of natural intellectual gifts comes justly within its scope. The law is an exceedingly general one. M. Quetelet, the AstronomerRoyal of Belgium, and the greatest authority on vital and social statistics, has largely used it in his inquiries. He has also constructed numerical tables, by which the necessary calculations can be easily made, whenever it is desired to have recourse to the law. Those who wish to learn more than I have space to relate, should consult his work, which is a very readable octavo volume, and deserves to be far better known to statisticians than it appears to be. Its title is Letters on Probabilities, translated by Downes. Layton and Co. London: 1849. So much has been published in recent years about statistical deductions, that I am sure the reader will be prepared to assent freely to the foJlowing hypothetical case:-Suppose a large island inhabited by a single race, who intermarried freely, and who had lived for many generations under constant conditions; then the average height of the male adu1ts of that population would undoubtedly be the same year after year. Also-sti11 arguing from the experience of modern statistics, which are found to give constant results in far less carefully-guarded examples-we should undoubtedly find, year after year, the same proportion maintained between the number of men of differ.ent heights. I mean, if the average stature was found to be sixty-six inches, and if it was also found in anyone year that lOOper million exceeded seventy-eight inches, the same proportion of 100 per million would be closely maintained in all other years. An equal constancy of proportion would be maintained between any other limits of height we pleased to specify, as between seventy-one and seventy-two inches; between seventy-two and seventy-three inches; and so on. Statistical experiences are so invariably confirmatory of what I have stated would probably be the case, as to make it unnecessary to describe analogous instances. Now, at this point, the law of deviation from an average steps in. It shows that the number per million whose heights range between seventy-one and seventy-two inches (or between any other limits we please to name) can be predicted from the previous datum of the
Sir Fmnc/s Galton
1182
average, and of anyone other fact, such as that of 100 per million exceeding seventy-eight inches. The diagram on Figure 1 will make this more intelligible. Suppose a million of the men to stand in turns, with their backs against a vertical board of sufficient height, and their heights to be dotted off upon it. The
FIGURE I
board would then present the appearance shown in the diagram. The line of average height is that which divides the dots into two equal parts, and stands, in the case we have assumed, at the height of sixty-six inches. The dots wiH be found to be ranged so symmetrically on either side of the line of average, that the lower half of the diagram wi1l be aJmost a precise reflection of the upper. Next, Jet a hundred dots be counted from above downwards, and let a line be drawn below them. According to the conditions, this line will stand at the height of seventy-eight inches. Using the data afforded by these two lines, it is possibJe, by the help of the law of deviation from an average, to reproduce, with extraordinary closeness, the entire system of dots on the board. M. QueteJet gives tables in which the uppermost Hne, instead of cutting off J00 in a mm ion, cuts off only one in a million. He divides the intervals between that line and the line of average, into eighty equal divisions, and gives the number of dots that fan within each of those divisions. It is easy,
CIMlllfcotIoJl 01 M6n Acco,dlnl 10 Tlu!iI' NtUu,al Gills
1183
by the help of his tables, to calculate what would occur under any other system of classification we pleased to adopt. This law of deviation from an average is perfectly general in its application. Thus, if the marks had been made by bullets fired at a horizontal line stretched in front of the target, they would have been distributed according to the same law. Wherever there is a large number of similar events, each due to the resultant influences of the same variable conditions, two effects will follow. First, the average value of those events will be constant; and, secondly, the deviations of the several events from the average, will be governed by this law (which is, in principle, the same as that which governs runs of luck at a gaming-table). The nature of the conditions affecting the several events must, I say~ be the same. It clearly would not be proper to combine the heights of men belonging to two dissimilar races, in the expectation that the compound results would be governed by the same constants. A union of two dissimilar systems of dots would produce the same kind of confusion as if half the buJlets fired at a target had been directed to one mark, and the other ha1f to another mark. Nay, an examination of the dots would show to a person, ignorant of what had occurred, that such had been the case, and it would be possible, by aid of the law, to disentangle two or any moderate number of superimposed series of marks. The Jaw may, therefore, be used as a most trustworthy criterion, whether or no the events of which an average has been taken are due to the same or to dissimilar classes of conditions. I selected the hypothetical case of a race of men living on an island and freely intermarrying, to ensure the conditions under which they were all supposed to live, being uniform in character. It will now be my aim to show there is sufficient uniformity in the inhabitants of the British Isles to bring them fairly within the grasp of this law. For this purpose, I tirst call attention to an example given in Quetelet's book. It is of the measurements of the circumferences of the chests of a large number of Scotch soldiers. The Scotch are by no means a strictly uniform race, nor are they exposed to identical conditions. They are a mixture of Celts, Danes, Anglo-Saxons, and others, in various proportions, the Highlanders being almost purely Celts. On the other hand, these races, though diverse in origin, are not very dissimilar in character. Consequently, it wil1 be found that their deviations from the average, follow theoretical computations with remarkable accuracy. The instance is as follows. M. Quetelet obtained his facts from the thirteenth volume of the Edinburgh Medical Journal, where the measurements are given in respect to 5,738 soldiers, the results being grouped in order of magnitude, proceeding by differences of one inch. Professor Quetelet compares these
Sir
1184
F,..u thllOII
results with those that his tables give, and here is the result. The marvellous accordance between fact and theory must strike the most unpractised eye. I should say that, for the sake of convenience, both the measurements and calculations have been reduced to per thousandths:-
Measures Number of of the men per 1,000, by chest in inches experience 33 34 35 36 37 38 39 40
5 31 141 322 732 1,305 1,867 1,882
Number of Measures Number 01 Number 01 men per men per men per 01 the 1,000, by chest in 1,000, by 1,000, by calculation inches experience calculation 7 29 110 323 732 1,333 1,838 1,987
1,628 1,148 645 160 87 38 7 2
41 42 43 44 45 46 47 48
1,675 1,096 560 221 69 16 3 I
I will now take a case where there is a greater dissimiJarity in the elements of which the average has been taken. It is the height of 100,000
French conscripts. There is fully as much variety in the French as in the English, for it is not very many generations since France was divided into completely independent kingdoms. Among its peculiar races are those of Normandy, Brittany, Alsatia, Provence, Bearne, Auvergne--each with their special characteristics; yet the following table shows a most striking agreement between the results of experience compared with those derived by calculation. from a purely theoretical hypothesis. The greatest differences are in the lowest ranks. They include the men who were rejected from being too short for the army. M. Quetelet boldly ascribes these differences to the effect of fraudulent returns. It certainly seems that men have been improperly taken out of the second rank and
Number of Men Height of Men
Measured
Calculated
Under 61.8 61.8 to 62.9 62.9 to 63.9 63.9 to 65.0 65.0 to 66.1 66.1 to 67.1 67.1 to 68.2 68.2 to 69.3 Above 69.3
28,620 11,580 13,990 14,410 11,410 8,780 5,530 3,190 2,490
26,345 13,182 14,502 13,982 11,803 8,725 5,527 3,187 2,645
Classlticallor! 01 Me" Accord;lI, 10 Their Nalural Gilts
1185
put into the first, in order to exempt them from service. Be this as it may, the coincidence of fact with theory is, in this instance also, quite close enough to serve my purpose. I argue from the results obtained from Frenchmen and from Scotchmen. that, if we had measurements of the adult males in the British Isles, we should find those measurements to range in close accordance with the law of deviation from an average, although our population is as much mingled as I described that of Scotland to have been, and although Ireland is mainly peopled with Celts. Now, if this be the case with stature, then it wi1t be true as regards every other physical feature-as circumference of head, size of brain, weight of grey matter, number of brain fibres, &c; and thence, by a step on which no physiologist wi]] hesitate, as regards mental capacity. This is what I am driving at-that analogy clearly shows there must be a fairly constant average mental capacity in the inhabitants of the British Isles, and that the deviations from that average-upwards towards genius, and downwards towards stupidity-must follow the law that governs deviations from all true averages. I have, however, done somewhat more than rely on analogy. I have tried the results of those examinations in which the candidates had been derived from the same classes. Most persons have noticed the lists of successful competitors for various pubJic appointments that are published from time to time in the newspapers, with the marks gained by each candidate attached to his name. These lists contain far too few names to faU into such beautiful accordance with theory, as was the case with Scotch soldiers. There are rarely more than 100 names in anyone of these examinations, while the chests of no less than 5,700 Scotchmen were measured. I cannot justly combine the marks of several independent examinations into one fagot, for I understand that different examiners are apt to have different figures of merit~ so I have analysed each examination separately. I give a calculation I made on the examination last before me; it will do as wen as any other. It was for admission into the Royal Military College at Sandhurst. Decem ber 1868. The marks obtained were clustered most thickly about 3,000, so I take that number as representing the average ability of the candidates. From this datum. and from the fact that no candidate obtained more than 6.500 marks, I computed the column B in the following table [see p. 1186] by the help of Quetelet's numbers. It will be seen that column B accords with column A quite as closely as the small number of persons examined could have led us to expect. The symmetry of the descending branch has been rudely spoilt by the conditions stated at the foot of column A. There is, therefore. little room for doubt, if everybody in England had to work up some subject and then to pass before examiners who employed similar figures of merit, that
SIr Francis Galton
1186
Number of Marks obtained by the Candidates
Number of Candidates who obtained those marks
A
B
According to fact According to theory 6,500 and above 5,800 to 6,500 5,100 to 5,800 4,400 to 5,100 3,700 to 4,400 3,000 to 3,700 2,300 to 3,000 1,600 to 2,300 1,100 to 1,600 400 to 1,100 below 400
o 1 3
6 73 11
22 22 8 Either did not venture to complete, or were plucked.
o
1 5 8 13 16 16 13
72
n
their marks would be found to range, according to the law of deviation from an average, just as rigorously as the heights of French conscripts, or the circumferences of the chests of Scotch soldiers. The number of grades into which we may divide ability is purely a matter of option. We may consult our convenience by sorting Englishmen into a few large classes, or into many small ones. I will select a system of classification that shall be easily comparable with the numbers of eminent men, as determined in the previous chapter. We have seen that 250 men per million become eminent; accordingly, I have so contrived the classes in the following table {po 1187] that the two highest, F and 0, together with X (which includes all cases beyond 0, and which are unclassed), shall amount to about that number-namely, to 248 per million:It wiIJ, I trust, be clearly understood that the numbers of men in the several classes in my table depend on no uncertain hypothesis. They are determined by the assured law of deviations from an average. It is an absolute fact that if we pick out of each million the one man who is naturally the ablest, and also the one man who is the most stupid, and divide the remaining 999,998 men into fourteen classes, the average ability in each being separated from that of its neighbours by equal grades, then the numbers in each of those classes will. on the average of many millions, be as is stated in the table. The table may be applied to special, just as truly as to general ability. It would be true for every examination that brought out natural gifts, whether held in painting, in music, or in statesmanship. The proportions between the different classes would be identical in all these cases, although the classes would be made up of different individuals, according as the examination differed in its purport. It will be seen that more than half of each million is contained in the two mediocre classes a and A; the four mediocre classes a, b, A, B, con-
CLASSIPICATION OP MEN ACCORDING TO THEIR. NATURAL GIFTS
Grades of natural ability. separated by equal intervals Below average
Above average
(
Numbers oj men comprised in the several grades of natural ability. whether in respect to their general powers. or to special aptitudes Proportjonate. viz. one in
In each million oj the same age
c:
i
I:)
~ :s
In total male population oj the United Kingdom. viz. J5 millions. of the undermentioned ages:-
:I.-
~
.
~
30-40
20-30
40-50
60-70
50-60
70--80
:;" ~
4 6 16 64 413 4,300 79,000
256,791 162,279 63,563 15,696 2,423 233 14
651,000 409,000 161,000 39,800 6,100 590 35
495,000 312,000 123,000 30,300 4,700 450 27
391,000 246,000 97,000 23,900 3,700 355 21
26~,000
1,000,000
1
3
2
2
2
On either side of average ...... Total, both sides .............
500,000 1,000,000
1,268,000 2,536,000
964,000 1,928,000
761,000 1,522,000
521,000 1,042,000
a b c d e
A B
C
f
D E F
g
G
J68,000 66,000 16,400 2,520 243 15
171,000 107,000 42,000 10,400 1,600 155 9
77,000 48,000 19,000 4,700 729 70 4
..:;-~
ii
i. C'l
~
x
X all grades all grades below above g G
332,000 664,000
149,000 298,000
The proportions of men living at ditJerent ages are calculated from the proportions that are true for England and Wales. (Census 1861, Appendix, p. 107.) Example.-The class F contains 1 in every 4,300 men. In other worda, there are 233 of that class in each million of men. The same is true of class f. In the whole United Kingdom there are 590 men of class F (and the same number of f) between the ages of 20 and 30; 450 between the ages of 30 and 40; and so on.
~
1188
Sf, Francis Galton
tain more than four-fifths, and the six mediocre classes more than nineteen-twentieths of the entire population. Thus, the rarity of commanding ability. and the vast abundance of mediocrity, is no accident, but follows of necessity, from the very nature of these things. The meaning of the word "mediocrity" admits of Httle doubt. It defines the standard of intellectual power found in most provincial gatherings, because the attractions of a more stirring life in the metropolis and elsewhere, are apt to draw away the abler classes of men, and the silly and the imbeciJe do not take a part in the gatherings. Hence, the residuum that forms the bulk of the general society of small provincial places, is commonly very pure in its mediocrity. The class C possesses abilities a trifle higher than those commonly possessed by the foreman of an ordinary jury. D includes the mass of men who obtain the ordinary prizes of life. E is a stage higher. Then we reach F, the lowest of those yet superior classes of intellect, with which this volume is chiefly concerned. On descending the scale, we find by the time we have reached f, that we are already among the idiots and imbeciles. We have seen that there are 400 idiots and imbeciles, to every million of persons living in this country; but that 30 per cent. of their number, appear to be light cases, to whom the name of idiot is inappropriate. There will remain 280 true idiots and imbeciles, to every million of our population. This ratio coincides very closely with the requirements of class f. No doubt a certain proportion of them are idiotic owing to some fortuitous cause, which may interfere with the working of a naturally good brain, much as a bit of dirt may cause a first-rate chronometer to keep worse time than an ordinary watch. But I presume, from the usual smallness of head and absence of disease among these persons, that the proportion of accidental idiots cannot be very large. Hence we arrive at the undeniable. but unexpected conclusion, that eminently gifted men are raised as much above mediocrity as idiots are depressed below it; a fact that is calculated to considerably enlarge our ideas of the enormous differences of intellectual gifts between man and man. I presume the class F of dogs, and others of the more intelligent sort of animals, is nearly commensurate with the f of the human race, in respect to memory and powers of reason. Certainly the class G of such animals is far superior to the g of humankind.
COMMENTARY ON
THOMAS ROBERT MALTHUS T WAS in 1798 that Thomas Robert Malthus (] 766-1834) published anonymously An Essay on the Principle of Population As It Affects the Future Improvement of Society. The main argument of this famous pamphlet was that the world's population increases in a geometrical, and food only in an arithmetical, ratio. Since an arithmetic progression is no match for a geometric progression, man must forever be outstripping his food supplies. Thus, said Malthus, population is necessarily limited by the "checks" of vice and misery. The pamphlet aroused bitter controversy. It offended those who believed in the perfectabilily of man and the gradual advent of the "happy society," and it annoyed conservatives who regarded Malthus as a mischief maker, a remembrancer of unpleasant facts. On the other hand, his views were received with favor in certain conservative circles because of an impression "very welcome to the higher ran ks of society, that they tended to relieve the rich and powerful of responsibility for the condition of the working c1asses, by showing that the latter had chiefly themselves to blame, and not either the negligence of their superiors or the institutions of the country." 1 In 1803, having traveled in Germany, Sweden, Norway, Finland and Russia to collect further information, Malthus issued a second edition of his essay, which was "substantially a new book." Without receding from his central principle, based on the unmistakably persuasive "postulata" that "food is necessary to the existence of man" and that "the passion between the sexes is necessary, and will remain nearly in its present state," he took the position that the checks to population were not to be regarded as "insuperable obstac1es" to social advance but "as defining the dangers which must be avoided if improvement is to be achieved." Malthus' doctrine, despite the furor it created, was not new. Condorcet, the eminent French philosopher, mathematician and revolutionist, had anticipated his ideas as to population, but took the cheerful view that as mankind improved it would avert misery by the practice of birth control. Malthus also acknowledged his indebtedness to Robert Wallace (Various Prospects of Mankind. Nature and Providence, 1761) and to J. P. SUssmilch (Gottliche Ordnung) from whom he got many of his statistical facts.2 Nonetheless, the importance and influence of the Essay must not be underestimated. It gave the first full, systematic exposition to a power-
I
1 2
Encyclopaedia Britannica, Eleventh Edition, article on Malthus, Vol. 17, p. 516. Dictionary 01 Natiollal BioKraphy. article on Malthus. 1189
1190
EditD"$ Comment
ful doctrine, and not the least of its accomplishments was in stimulating the thoughts of Darwin. For as he himself took pains to point out, the phrase "struggle for existence," used by Malthus in relation to social competition, suggested the operation of this principle, with its coronary of survival of the fittest, as the determining factor of evolution in all forms of organic life. For a time in the nineteenth century, as new lands and resources were developed, the Malthusian doctrine fell into disfavor. Today, with the world's population sharply rising and with a scarcity of new frontiers, the truth of his basic assertion is again very much in vogue. It is known that food production can be greatly increased by improvements in agricultural methods, by food synthesis. by ingenuities beyond present conception. It is known also that an unchecked population must soon outrun food sup· plies however much they are augmented. The effect of scientific progress is in some respects to aggravate the difficulty. Epidemic diseases ha'Ve been largely brought under control; the practice of "vice" is no longer so widespread or so lethal as to assure the virtuous of more to eat; even wars, though much more efficient than in Malthus' day. have not yet succeeded in substantially reducing the number of the hungry. The Malthusian governors are a grim company and should obviously be eliminated. But this does not solve the problem of food shortages. The only permanent solution is voluntary birth control, coupled with a rational, worldwide eugenics policy, but many persons despair of its ever being adopted. a In this, as in other aspects of human affairs. man is his own principal adversary. I add a brief biographical note. Malthus was born in Surrey in 1766. His father was welJ off, a cultured and thoughtful man who "lived quietly among his books." He was a friend and disciple of Rousseau and was said to have been his executor. MaJthus was educated at Cambridge, where he studied history, poetry and the classics. and distinguished himself in mathematics. In 1798 he took "holy orders," and the same year published the Essay. which was to some extent a challenge to the social views held by his father, who shared the optimistic theories of Condorcet and Godwin. After spending five years on the revision of the Essay, Malthus became professor of history and political economy at the newly founded college of Hai leybury. In this post he spent the rest of his life. publishing papers on the corn laws, rent and various current economic topics, and several larger treatises on political economy. He was elected a Fellow of the Royal Society in 1819, and an associate of the Royal Society of Literature in 1824. 3 For an unmixedly gloomy but extraordinarily readable discussion of man's future prospects, in light of Malthusian principles, see Sir Charles Galton Darwin, The Next Million Years, New York, ]953. Another valuable study is L. Dudley Stamp, Land for Tomorrow. Indiana University Press, 1952.
Thomas Robert MalthllS
1191
Malthus carried on an extensive correspondence with Ricardo, and was a friend of James Mill, George Grote and other influential thinkers of his period. Politically he was a Whig, moderate in his opinions, favoring most of the contemporary reform legislation yet not altogether convinced of its value. Like other men who have advanced controversial theories, Malthus has been belabored, at times, as if he had personally created the distressing predicament he described. Fortunately he was a "serene and cheerful man," "singularly amiable"-as Harriet Martineau described him-without malice and with a keen sense of humor. Miss Martineau also tells us that although he had "a defect in the palate" which made his speech "hopelessly imperfect," he was the only friend whom she could hear without her trumpet. The world has been hearing him clearly for more than a century. His predictions have not been vindicated by population trends but the central point of his analysis has yet to be disproved. The following selection is from the sixth edition of An E.'isay on the Principle 01 Population.
3
1, 2, 3, 4, 5, 6, 7, 8 . . .
-ARITHMETIC PROGRUSION
1, 2, 4, 8, 16, 32, 64, 128 . . .
--GEOMETRIC PROGRESSION
Mathematics of Population and Food By THOMAS ROBERT MALTHUS
IN an inquiry concerning the improvement of society the mode of conducting the subject which natural1y presents itself is, 1. To investigate the causes that have hitherto impeded the progress of mankind towards happiness; and 2. To examine the probability of the total or partial removal of these causes in the future. To enter funy into this question and to enumerate all the causes that have hitherto influenced human improvement would be much beyond the power of an individual. The principal object of the present essay is to examine the effects of one great cause intimately united with the very nature of man; which, though it has been constantly and powerfully operating since the commencement of society, has been little noticed by the writers who have treated this subject. The facts which establish the existence of this cause have, indeed, been repeatedly stated and acknowledged; but its natural and necessary effects have been almost totally overlooked; though probably among these effects may be reckoned a very considerable portion of that vice and misery, and of that unequal distribution of the bounties of nature, which it has been the unceasing object of Ihe enlightened philanthropist in all ages to correct. The cause to which I allude is the constant tendency in all animated life to increase beyond the nourishment prepared for it. It is observed by Dr. Franklin that there is no bound to the prolific nature of plants or animals but what is made by their crowding and interfering with each other's means of subsistence. Were the face of the earth, he says, vacant of other plants, it might be gradually sowed and overspread with one kind only, as for instance with fennel; and were it empty of other inhabitants, it might in a few ages be replenished from one nation only, as for instance with Englishmen. This is incontrovertibly true. Through the animal and vegetable kingdoms nature has scattered the seeds of life abroad with the most profuse and liberal hand. but has been comparatively sparing in the room and the nourishment necessary to rear them. The germs of existence contained in 1192
Malll,maltcs DI PDpula,io" and Food
1193
this earth, if they could freely develop themselves, would fill millions of worlds in the course of a few thousand years. Necessity, that imperious, aU-pervading law of nature, restrains them within the prescribed bounds. The race of plants and the race of animals shrink under this great restrictive law; and man cannot by any efforts of reason escape from it. In plants and irrational animals the view of the subject is simple. They are all impelled by a powerful instinct to the increase of their species; and this instinct is interrupted by no doubts about providing for their offspring. Wherever therefore there is liberty, the power of increase is exerted; and the superabundant effects are repressed afterwards by want of room and nourishment. The effects of this check on man are more complicated. Impelled to the increase of his species by an equally powerful instinct. reason interrupts his career, and asks him whether he may not bring beings into the world for whom he cannot provide the means of support. If he attend to this natura1 suggestion, the restriction too frequently produces vice. J( he hear it not, the human race will be constantly endeavoring to increase beyond the means of subsistence. But as, by that law of our nature which makes food necessary to the life of man, population can never actual1y increase beyond the lowest nourishment capable of supporting it, a strong check on population, from the difficulty of acquiring food, must be constantly in operation. This difficulty must fall somewhere, and must necessarily be severely felt in some or other of the various forms of misery, or the fear of misery, by a large portion of mankind. That population has this constant tendency to increase beyond the means of subsistence, and that it is kept to its necessary level by these causes will sufficiently appear from a review of the different states of society in which man has existed. But before we proceed to this review the subject will, perhaps, be seen in a clearer light, if we endeavor to ascertain what would be the natural increase of population if left to exert itself with perfect freedom, and what might be expected to be the rate of increase in the productions of the earth under the most favorable circumstances of human industry. It will be allowed that no country has hitherto been known where the manners were so pure and simple, and the means of subsistence so abundant, that no check whatever has existed to early marriages from the difficulty of providing for a family. and that no waste of the human species has been occasioned by vicious customs, by towns, by unhealthy occupations, or too severe labor. Consequently, in no state that we have yet known has the power of population been left to exert itse1f with perfect freedom. Whether the law of marriage be instituted or not, the dictates of nature and virtue seem to be an early attachment to one woman; and where there
1194
rhOnulS Rob", MaEthu$
were no impediments of any kind in the way of an union to which such an attachment would lead, and no causes of depopulation afterwards, the increase of the human species would be evidently much greater than any increase which has hitherto been known. In the Northern States of America, where the means of subsistence have been more ample, the manners of the people more pure, and the checks to early marriages fewer than in any of the modern states of Europe, the population has been found to double itself, for above a century and a half successively, in less than twenty-five years. Yet, even during these periods, in some of the towns the deaths exceeded the births, a circumstance which clearly proves that, in those parts of the country which supplied this deficiency, the increase must have been much more rapid than the general average. In the back settlements, where the sole employment is agriculture, and vicious customs and unwholesome occupations are little known. the population has been found to double itself in fifteen years. Even this extraordinary rate of increase is probably short of the utmost power of population. Very severe labor is requisite to clear a fresh country; such situations are not in general considered as particularly healthy; and the inhabitants, probably, are occasionally subject to the incursions of the Indians, which may destroy some lives, or at any rate diminish the fruits of industry. According to a table of Euler, calculated on a mortality of one to thirty-six, if the births be to the deaths in the proportion of three to one, the period of doubling will be only twelve years and four fifths. And this proportion is not only a possible supposition, but has actually occurred for short periods in more countries than one. Sir William Petty supposes a doubling possible in so short a time as ten years. But, to be perfectly sure that we are far within the truth, we will take the slowest of these rates of increase, a rate in which all concurring testimonies agree, and which has been repeatedly ascertained to be from procreation only. It may safety be pronounced, therefore, that population, when unchecked, goes on doubling itself every twenty-five years, or increases in a geometrical ratio. The rate according to which the productions of the earth may be supposed to increase it will not be so easy to determine. Of this, however, we may be perfectly certain-that the ratio of their increase in a limited territory must be of a totally different nature from the ratio of the increase of population. A thousand miJIions are just as easily dOUbled every twentyfive years by the power of population as a thousand. But the food to support the increase from the greater number will by no means be ob-
Mathemalics Df PDpulation and FDlJd
11"
tained with the same facility. Man is necessarily confined in room. When acre has been added to acre til1 all the fertile land is occupied. the yearly increase of food must depend upon the melioration of the land already in possession. This is a fund which, from the nature of all soils instead of increasing, must be gradually diminishing. But population, could it be supplied with food, would go on with unexhausted vigor; and the increase of one period would furnish the power of a greater increase the next, and this without any limit. From the accounts we have of China and Japan, it may be fairly doubted whether the best-directed efforts of human industry could double the produce of these countries even once in any number of years. There are many parts of the globe, indeed, hitherto uncultivated and almost unoccupied, but the right of exterminating, or driving into a comer where they must starve, even the inhabitants of these thinly-peopled regions, will be questioned in a moral view. The process of improving their minds and directing their industry would necessarily be slow; and during this time, as popUlation would regularly keep pace with the increasing produce, it would rarely happen that a great degree of knowledge and industry would have to operate at once upon rich unappropriated soil. Even where this might take place, as it does sometimes in new colonies, a geometrical ratio increases with such extraordinary rapidity that the advantage could not last long. If the United States of America continue increasing, which they certainly will do, though not with the same rapidity as formerly, the Indians will be driven further and further back into the country, till the whole race is ultimately exterminated and the territory is incapable of further extension. These observations are, in a degree, applicable to all the parts of the earth where the soil is imperfectly cultivated. To exterminate the inhabitants of the greatest part of Asia and Africa is a thought that could not be admitted for a moment. To civilize and direct the industry of the various tribes of Tartars and Negroes would certainly be a work of considerable time, and of variable and uncertain success. Europe is by no means so fully peopled as it might be. In Europe there is the fairest chance that human industry may receive its best direction. The science of agriculture has been much studied in England and Scotland, and there is still a great portion of uncultivated land in these countries. Let us consider at what rate the produce of this island might be supposed to increase under circumstances the most favorable to improvement. If it be allowed that by the best possible policy. and great encouragement to agriculture, the average produce of the island could be doubled in the first twenty-five years, it wiJ) be anowing, probably. a greater increase than could with reason be expected.
1196
Thomas ROM" Mallhus
In the next twenty-five years it is impossible to suppose that the produce could be quadrupled. It would be contrary to all our knowledge of the properties of land. The improvement of the barren parts would be a work of time and labor; and it must be evident, to those who have the sHghtest acquantance with agricultural subjects, that in proportion as cultivation is extended, the additions that could yearly be made to the former average produce must be gradually and regularly diminishing. That we may be the better able to compare the increase of population and food, let us make a supposition which, without pretending to accuracy, is clearly more favorable to the power of production in the earth than any experience we have had of its qualities will warrant. Let us suppose that the yearly additions which might be made to the former average produce, instead of decreasing. which they certainly would do, were to remain the same; and that the produce of this island might be increased every twenty-five years by a quantity equal to what it at present produces. The most enthusiastic speculator cannot suppose a greater increase than this. In a few centuries it would make every acre of land in the island like a garden. If this supposition be applied to the whole earth, and if it be allowed that the subsistence for man which the earth affords might be increased every twenty-five years by a quantity equal to what it at present produces, this will be supposing a rate of increase much greater than we can imagine that any possible exertions of mankind could make it. It may be fairly pronounced, therefore, that considering the present average state of the earth, the means of subsistence, under circumstances the most favorable to human industry, could not possibly be made to increase faster than in an arithmetical ratio. The necessary effects of these two different rates of increase, when brought together, will be very striking. Let us call the population of this island eleven millions; and suppose the present produce equal to the easy support of such a number. In the first twenty-five years the population would be twenty-two millions, and the food being also doubled, the means of subsistence would be equal to this increase. In the next twenty-five years the population would be forty-four millions. and the means of subsistence only equal to the support of thirty-three millions. Tn the next period the population would be eighty-eight millions and the means of subsistence just equal to the support of half that number. And at the conclusion of the first century the population would be a hundred and seventy-six mi11ions. and the means of subsistence only equal to the support of fifty-five millions, leaving a popUlation of a hundred and twenty-one millions totally unprovided for. Taking the whole earth instead of this island, emigration would of course be excluded; and, supposing the present population equal to a
Mllt",matic$ ill Populatiilll alld F oild
1197
thousand milJions, the human species would increase as the numbers, 1,2,4,8, 16,32,64, 128,256, and subsistence as 1,2,3,4,5,6,7,8,9. In two centuries the population would be to the means of subsistence as 256 to 9; in three centuries, as 4096 to 13; and in two thousand years the difference would be almost incalculable. In this supposition no limits whatever are placed to the produce of the earth. It may increase forever, and be greater than any assignable quantity; yet stiJI, the power of population being in every period so much superior, the increase of the human species can only be kept down to the level of the means of subsistence by the constant operation of the strong law of necessity. acting as a check upon the greater power. OF THE GENERAL CHECKS TO POPULATION, AND THE MODE OF THEIR OPERATION
The ultimate check to population appears then to be a want of food arising necessarily from the different ratios according to which population and food increase. But this ultimate check is never the immediate check, except in cases of actual famine. The immediate check may be stated to consist in all those customs, and an those diseases, which seem to be generated by a scarcity of the means of subsistence; and al1 those causes, independent of this scarcity, whether of a moral or physical nature, which tend prematurely to weaken and destroy the human frame. These checks to population, which are constantly operating with more or less force in every society, and keep down the number to the level of the means of subsistence, may be classed under two general heads-the preventive, and the positive checks. The preventive check, as far as it is voluntary. is peculiar to man. and arises from that distinctive superiority in his reasoning faculties which enables him to calculate distant consequences. The checks to the indefinite increase of plants and irrational animals are al1 either positive, or, jf preventive. involuntary. But man cannot look around him and see the distress which frequently presses upon those who have large fami1ies~ he cannot contemplate his present possessions or earnings, which he now near1y consumes himself, and calculate the amount of each share, when with very little addition they must be divided, perhaps, among seven or eight, without feeling a doubt whether, if he follow the bent of his inclina· tions, he may be able to support the offspring which he will probably bring into the world. In a state of equality, if such can exist, this would be the simple question. In the present state of society other considerations occur. Will he not lower his rank in life. and be obliged to give up in great measure his former habits? Does any mode of employment present itself
1198
Thof'lt4s RON,t Matthus
by which he may reasonably hope to maintain a family? Will he not at any rate subject himself to greater difficulties and more severe labor than in his single state? Will he not be unable to transmit to his children the same advantages of education and improvement that he had himself possessed? Does he even feel secure that, should he have a large family, his utmost exertions can save them from rags and squalid poverty, and their consequent degradation in the community? And may he not be reduced to the grating necessity of forfeiting his independence, and of being obliged to the sparing hand of charity for support? These considerations are calculated to prevent, and certainly do prevent, a great number of persons in all civilized nations from pursuing the dictate of nature in an early attachment to one woman. If this restraint does not produce vice, it is undoubtedly the least evil that can arise from the principle of population. Considered as a restraint on a strong natural inclination, it must be allowed to produce a certain degree of temporary unhappiness, but evidently slight compared with the evils which result from any of the other checks to population, and merely of the same nature as many other sacrifices of temporary to permanent gratification, which it is the business of a moral agent continually to make. When this restraint produces vice, the evils which follow are but too conspicuous. A promiscuous intercourse to such a degree as to prevent the birth of children seems to lower, in the most marked manner, the dignity of hUman nature. It cannot be without its effect on men, and nothing can be more obvious than its tendency to degrade the female character and to destroy al1 its most amiable and distinguishing characteristics. Add to which, that among those unfortunate females with which all great towns abound more real distress and aggravated misery are, perhaps, to be found, than in any other department of human life. When a general corruption of morals with regard to the sex pervades all the c1asses of society, its effects must necessarily be to poison the springs of domestic happiness, to weaken conjugal and parental affection, and to lessen the united exertions and ardor of parents in the care and education of their children--effects which cannot take place without a decided diminution of the general happiness and virtue of the society; particularly as the necessity of art in the accompJishment and conduct of intrigues and in the concealment of their consequences necessarily leads to many other vices. The positive checks to popUlation are extremely various, and include every cause, whether arising from vice or misery, which in any degree contributes to shorten the natural duration of human life. Under this head, therefore, may be enumerated all unwholesome occupations, severe labor and exposure to the seasons, extreme poverty, bad nursing of children,
MallU'malks 0/ Poplila/ioll alld Food
1199
great towns, excesses of all kinds, the whole train of common diseases and epidemics, wars, plague, and famine. On examining these obstacles to the increase of population which I have classed under the heads of preventive and positive checks, it will appear that they are all resolvable into moral restraint, vice and misery. Of the preventive checks, the restraint from marriage which is not followed by irregular gratifications may properly be termed moral restraint.1 Promiscuous intercourse, unnatural passions, violations of the marriage bed, and improper arts to conceal the consequences of irregular connections are preventive checks that clearly come under the head of vice. Of the positive checks, those which appear to arise unavoidably from the laws of nature may be called exclusively misery, and those which we obviously bring upon ourselves, such as wars, excesses, and many others which it would be in our power to avoid, are of a mixed nature. They are brought upon us by vice, and their consequences are misery. The sum of all these preventive and positive checks taken together forms the immediate check to population; and it is evident that in every country where the whole of the procreative power cannot be caned into action, the preventive and the positive checks must vary inversely as each other; that is, in countries either natural1y unhealthy or subject to a great morality, from whatever cause it may arise, the preventive check will prevail very little. In those countries, on the contrary, which are naturally healthy. and where the preventive check is found to prevail with consid~ erabJe force, the positive check will prevail very little, or the mortality be very small. In every country some of these checks are with more or less force in constant operation; yet notwithstanding their general prevalence, there are few states in which there is not a constant effort in the population to increase beyond the means of subsistence. This constant effort as constantly tends to subject the lower classes of society to distress, and to prevent any great permanent melioration of their condition. 1 It will be observed that 1 here use the term moral in its most conftned sense. By moral restraint 1 would be understood to mean a restraint from marriage from prudential motives, with a conduct strictly moral during the period of this restraint; and I have never intentionally deviated from this sense. When 1 have wiShed to consider the restraint from marriage unconnected with its consequences, 1 have either caned it prudential restraint, or a part of the preventive check, of which indeed it forms the principal branch. In my review of the different stages of society 1 have been accused of not allowing sufficient weight in the prevention of population to moral restraint; but when the confined sense of the term, which 1 have here explained, is adverted to, 1 am fearful that 1 shall not be found to have erred much in this respect. I should be very glad to believe myself mistaken.
COMMENTARY ON
COURNOT, JEVONS, and the Mathematics of Money ATHEMATICAL economics is old enough to be respectable, but not all economists respect it. It has powerful supporters and impressive testimonials, yet many capable economists deny that mathematics, except as a shorthand or expository device, can be applied to economic reasoning. There have even been rumors that mathematics is used in economics (and in other social sciences) either for the deliberate purpose of mystification or to confer dignity upon commonplaces-as French was once used in diplomatic communications. The value of graphs and symbols for expressing simple facts of economics is widely acknowledged and needs no discussion. A more important use of mathematics is the application of its methods to restricted economic problems (for example, the theory of partial equilibria) selected because of their "particular susceptibility to mathematical treatment." 1 An exponent of this approach was the famous economist Alfred Marshall. 2 The most sweeping and most controversial development in mathematical economics has been the organization of the ''whole body of economic theory into an interdependent set of propositions stated in mathematical terms." Alfredo Pareto was among the leaders of this ambitious school of thought which sought to formulate a theory of genera] equilibrium in which all the economic unknowns-such as prices of consumption goods and of productive services. costs, market quantities of outputs and inputs ~ould be represented in simu1taneous equations, analogous to the procedures of classical physics. "The problem is determined when there can be obtained as many equations as there are unknowns." To be sure, mathematics can be extended to any branch of knowledge, inc1uding economics, provided the concepts are so clearly defined as to permit accurate symbolic representation. That is only another way of saying that in some branches of discourse it is desirable to know what you are talking about. But this extension does not in itself guarantee fruit-
M
1
See Oskar Morgenstern, "Article on Mathematical Economics," in Encyclopaedia
0/ the Social Sciences, 1931, Vol. 5, pp. 364-368, an admirable summary from which the quoted material and other points in this introduction have been taken. 2 ". • • the most helpful applications of mathematics to economics are those which are short and simple, which employ few symbols, and which aim at throwing a bright light on some small part of the great economic movement rather than at representing its endless complexities." Memorials 0/ AI/red Marshall, ed. by A. C. Pigou, London, 1925, p. 313. Quoted by Morgenstern. 1200
1201
ful results. Wittgenstein has described the service mathematics performs for science in a bri1liant passage: "Mathematics is a logical method . . . Mathematical propositions express no thoughts. In life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics/' 8 Mathe· matical methods facilitate inference in some branches of economic thought; their usefulness is therefore beyond dispute. On the other hand, it has not been established-at least for the techniques customarily em· ployed-that mathematics is helpful in handling every type of economic problem~ much less that a fruitful mathematization of the entire body of economic theory is possible.' R.ecently, however, Von Neumann and Morgenstern have introduced into economics a novel analytical apparatus, employing the tools of modem logic, that may profoundly alter the perspective and give new impetus to the process of mathematical-economic generalization.6 The first book dealing with the application of mathematics to economics was written by an Italian engineer, Giovanni Ceva, in 1711. Half a century later, one Cesare Beccaria "employed a1gebra effectively in an essay on the hazards and profits of smuggling": Tentativo analytico sui contrabbandi. Other minor writings followed, but the first treatise to explore the subject systematically was Augustin Cournot's masterpiece, Recherches sur les principes mathematiques de la theorie des richesses (Researches into the Mathematical Principles of the Theory of Wealth), an immensely intluential work published in 1838. 6 Cournot (1801-1877) was a mathematician, a philosopher and a student of probability theory. He served as a university official at Lyons and Grenoble. held high positions in the French government and wrote and translated a large number of books on philosophy, matbematics~ statistics and economics. His treatise, from which two selections appear below, enunciates principles-for example, as to the law of diminishing returns-which have become classic. It presents the original interpretation of supply and demand as "functions or schedules." Cournot's book "seemed a failure when first published. It was too far in advance of the times. Its methods were too 3 Ludwig Wittgenstein, TraClalUS Logico Philosophicus, New York. 1922. p. 169. ""The problem that presents itself in connection with the mathematical method, so long as the question of its applicability IS settled, is one of preference: is it of greater utility than other methods, and are there cases in which it is the only method possible. So far there have been found very few instances in which mathematics is absolutely necessary, but there are more examples in which the application of mathematics facilitates the prosecution of the argument." Morgenstern, op. cit., p. 364. 5 See selection by Leonid Hurwicz, "The Theory of Economic Behavior," pp. )2671284. 81n 1897 it was translated into English by Irving Fisher, a teacher at Yale for forty-five years and one of the first exponents of mathematical economics in the
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"pll"rtinn mlPtf h ..rp ic frnm hit!. tTAnc:IRtinn
1202
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Comm~1II
strange, its reasoning too intricate for the crude and confident notions of political economy then current:' 7 Among the first to recognize the virtues of Cournot's work was William Stanley levons (1835-1882), an English economist and logician noted for developing the theory of utility and for his chief writings, The Theory oj Political Economy and The Principles of Science. As a student at University College in London, levons specialized in mathematics, biology, chemistry and metallurgy-an unusual but apparently stimulating training for an economist. He worked for five years in Australia as assayer to the mint, and then returned to England to devote himself to investigations in economics and logic. The scientific character of his studies in economics, statistics and logic was recognized by his election in 1872 as a Fellow of the Royal Society; he was the first economist so elected, as lohn Maynard Keynes points out, since Sir William Petty. In introducing the material I have taken from Jevons,s I cannot do better than quote a passage from a lecture given by Keynes on the centenary of levons' birth: levons' Theory 01 Political Economy and the place it occupies in the history of the subject are so well known that I need not spend time in describing its content. It was not as uniquely original in 1871 as it would have been in 1862. For, leaving on one side the precursors Coumot, Gossen Dupuit, Von ThUnen and the rest, there were several economists, notably Walras and Marshall. who by 1871, were scribbling equations with X's and Y's, big Deltas and little d's. Nevertheless, Jevons' Theory is the first treatise to present in a finished form the theory of value based on sUbjective valuations, the marginal principle and the now familiar technique of the algebra and the diagrams of the subject. The first modern book on the subject. it has proved singularly attractive to all bright minds newly attacking the subject;-simple. lucid, unfaltering. chiselled in stone where Marshall knits in wool.!) 7 Irving Fisher, in a review of Cournot's work pubJished in Quarterly }o"rnal of EconomiCS, January 1898. reprinted in Henry William Spiegel, The Del'elopmeflt of Economic Though" New York, 1952. pp. 459-469. Cournot himself. it may be noted. look a modest view of his essay. " . . . I believe I he says in his preface I. if this essay is of any practical value, it will be chiefly in making clear how far we are from being abJe to solve. with full knowledge of the case, a multitude of questions which are boldly decided every day." 8 From The Theory of Political Economy, Chaps. T, III. IV. n John Maynard Keynes, Essays in Biography, New York, ]951, p. 284.
What we might call, by way
0/ eminence, the dismal science. -THOMAS CARLYLB
Prolessor Planck 01 Berli", the lamous origillator 01 the quantllm theory, once remarked to me that ill early lile he had thought 01 studying eco<nomiCS, but had lound it too difficult! Prolessor Planck could easily master the whole corpus 01 mathematical economics ;n a lew days, He did not meall that' Bllt the amalgam 01 logic and inwition and the wide knowledge 01 lacts, most 01 which art' not precise, which is required lor economic illterpretation in tis highest lorm, is, quite truly, overwhelmingly difficult lor those whose gilt consists mainly ill the power to imagine and pursue to their II4rthermost points, the implications and prior conditions 01 comparau,·t'ly s;mplt' lacts. which art' known with a high degree 01 preciSion. -JOHN MAYNARD KEYNES (Biography 01 AI/red Marshall) I happened to sit next to Keynes at the High Table 01 King's College a day or two alter Planck had made this observation, and KeY1les told me 01 it. Lowes Dickinson was silting opposite. "That's JllntlY," he said, "because Bertrand Russell once told mt' that ill early Iile he had thought oj studying economics, bill had lound it too easy"! -R. F. HARROD (Lile oj Johll Maynard Keynes)
4
Mathematics of Value and Demand By AUGUSTIN COURNOT OF CHANGES IN VALUE, ABSOLUTE AND RELATIVE
WHENEVER there is occasion to go back to the fundamental conceptions on which any science rests, and to formulate them with accuracy, we almost always encounter difficulties, which come, sometimes from the very nature of these conceptions, but more often from the imperfections of language. For instance, in the writings of economists, the definition of value, and the distinction between absolute and relative value, are rather obscure: a very simple and strikingly exact comparison will serve to throw light on this. We conceive that a body moves when its situation changes with reference to other bodies which we 100k upon as fixed. If we observe a system of material points at two different times, and find that the respective situations of these points are not the same at both times. we necessarily conclude that some, if not alJ, of these points have moved; but if besides this we are unable to refer them to points of the fixity of which we can be sure, it is, in the first instance, impossible to draw any conclusions as to the motion or rest of each of the points in the system. However. if all of the points in the system. except one, had preserved 1203
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AIIIII.stin COllmol
their relative situation, we should consider it very probable that this single point was the only one which had moved, unless, indeed, all the other points were so connected that the movement of one would involve the movement of all. We have just pointed out an extreme case, viz., that in which all except one had kept their relative positions; but, without entering into details. it is easy to see that among all the possible ways of explaining the change in the state of the system there may be some much simpler than others. and which without hesitation we regard as much more probable. If, without being limited to two distinct times, observation should fonow the system through its successive states, there would be hypotheses as to the absolute movements of the different points of the system, which would be considered preferable for the explanation of their relative movements. Thus, without reference to the relative size of the heavenly bodies and to knowledge of the laws of gravitation, the hypothesis of Copernicus would explain the apparent motions of the planetary system more simply and plausibly than those of Ptolemy or Tycho. In the preceding paragraph we have only looked on motion as a geometric relation, a change of position, without reference to any idea of cause or motive power or any knowledge of the laws which govern the movements of matter. From this new point of view other considerations of probability will arise. If, for instance, the mass of the body A is considerably greater than that of the body B, we judge that the change in the relative situation of the bodies A and B is more probably due to the displacemenl of B than of A. Finally, there I are some circumstances which may make it certain that relative or apparent movements come from the displacement of one body and not of another.] Thus the appearance of an animal will show by unmistakable signs whether it is stopping or starting. Thus, to return to the preceding example, experiments with the pendulum. taken in connection with the known laws of mechanics, will prove the diurnal motion of the earth; the phenomenon of the aberration of light will prove its annual motion; and the hypothesis of Copernicus will take its place among established truths. Let us now examine how some considerations perfectly analogous to those which we have just considered, spring from the idea of exchangeable values. Just as we can only assign situation to a point by reference to other points, so we can only assign value to a commodity 2 by reference to See Newton. P,i"cipia, Book I. at the end of the preliminary definitions. It is almost needless to observe that for conciseness the word commodity is used in its most general sense, and that it includes the rendering of valuable services, which can be exchanged either for other services or for commodities proper. and which, like slich commodities. have a definite price or a value in exchange We shall not repeat this remark in the future. as it can easily be supplied from the context 1
:II
Mathematics 01
Va'u~
and Demand
nos
other commodities. Tn this sense there are only relative values. But when these relative values change, we perceive plainly that the reason of the variation may lie in the change of one term of the relation or of the other or of both at once; just as when the distance varies between two points, the reason for the change may lie in the displacement of one or the other or both of the two points. Thus again when two violin strings have had between them a definite musical interval, and when after a certain time they cease to give this interval, the question is whether one has gone up or the other gone down, or whether both of these effects have joined to cause the variation of the interval. We can therefore readily distinguish the relative changes of value manifested by the changes of relative values from the absolute changes of value of one or another of the commodities between which commerce has established relations. Just as it is possible to make an indefinite number of hypotheses as to the absolute motion which causes the observed relative motion in a system of points, so it is also possible to mUltiply indefinitely hypotheses as to the absolute variations which cause the relative variations observed in the values of a system of commodities. However. if an but one of the commodities preserved the same relative values, we should consider by far the most probable hypothesis. the one which would assign the absolute change to this single article; unless there shoUld be manifest such a connection between all the others. that one cannot vary without involving proportional variations in the values of those which depend on it. For instance, an observer who should see by inspection of a table of statistics of values from century to century, that the value of money fell about four-fifths towards the end of the sixteenth century, While other commodities preserved practically the same relative values. would consider it very probable that an absolute change had taken place in the value of money, even if he were ignorant of the discovery of mines in America. On the other hand. if he should see the price of wheat double from one year to the next without any remarkable variation in the price of most other articles or in their relative values, he would attribute it to an absolute change in the value of wheat, even if he did not know that a bad grain harvest had preceded the high price. Without reference to this extreme case, where the disturbance of the system of relative values is expJained by the movement of a single article, it is evident that among all the possible hypotheses on absolute variations some explain the relative variations more simply and more probably than others. If. without being limited to consideration of the system of relative values at two distinct periods, observation follows it through its inter-
Au,ustin COlU'nol
1206
mediate states, a new set of data will be provided to determine the most probable law of absolute variations, from all possibilities for satisfying the observed law of relative variations. Let Ph P2' Pa, etc.,
be the values of certain articles, with reference to a gram of silver; if the standard of value is changed and a myriagram of wheat is substituted for the gram of silver, the values of the same articles will be given by the expressions 1 1 1 - PI' - P2' - Pa, etc.,
a
a
a
a being the price of the myriagram of wheat, or its value with reference to a gram of silver. In general, whenever it is desired to change the standard of value, it will suffice to multiply the numerical expressions of individual values by a constant factor, greater or less than unity; just as with a system of points conditioned to remain in a straight line, it would suffice to know the distances from these points to anyone of their number, to determine by the addition of a constant number, positive or negative, their distances referred to another point of the system, taken as the new origin. From this there results a very simple method of expressing by a mathematical illustration the variations which occur in the relative values of a system of articles. It is sufficient to conceive of a system composed of as many points arranged in a straight line as there are articles to be compared, so that the distances from one of these points to all the others constantly remain proportional to the logarithms of the numbers which measure the values of an these articles with reference to one of their number. All the changes of distance which occur by means of addition and subtraction, from the relative and absolute motions of such a system of movable points, will correspond perfectly to the changes by means of multiplication and division in the system of values which is being compared: from which it foHows that the calculations for determining the most probable hypothesis as to the absolute movements of a system of points, can be applied, by going from logarithms back to numbers, to the determination of the most probable hypothesis for the absolute variations of a system of values. But, in general, such calculations of probability, in view of the absolute ignorance in which we would be of the causes of variation of values, would be of very slight interest. What is rea]]y important is to know the laws which govern the variation of values, or, in other words, the theory of wealth. This theory alone can make it possible to prove to what abso-
MIJtllemlJtics 01 YIJ'ue and Demand
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lute variations are due the relative variations which come into the field of observation; in the same manner (if it is permissible to compare the most exact of sciences with the one nearest its cradle) as the theory of the laws of motion, begun by Galileo and completed by Newton. alone makes it possible to prove to what real and absolute motions are due the relative and apparent motions of the solar system. To sum uP. there are only relative values; to seek for others is to fan into a contradiction with the very idea of value in exchange. which necessarily implies the idea of a ratio between two terms. But also an accomplished change in this ratio is a relative effect, which can and should be explained by absolute changes in the terms of the ratio. There are no absolute values, but there are movements of absolute rise and fan in values. Among the possible hypotheses on the absolute changes which produce the observed relative changes. there are some which the general laws of probability indicate as most probable. Only knowledge of the special laws of the matter in question can lead to the substitution of an assured decision for an opinion as to probability. If theory should indicate one article incapable of absolute variation in its value. and should refer to it an others, it would be possible to immediately deduce their absolute variations from their relative variations~ but very slight attention is sufficient to prove that such a fixed term does not exist, although certain articles approach much more nearly than others to the necessary conditions for the existence of such a term. The monetary metals are among the things which, under ordinary circumstances and provided that too long a period is not considered, only experience slight absolute variations in their value. If it were not so, all transactions would be disturbed, as they are by paper money subject to sudden depreciation. 8 On the other hand, articles such as wheat, which form the basis of the food supply, are subject to violent disturbances; but, if a sufficient period is considered. these disturbances balance each other, and the average value approaches fixed conditions. perhaps even more closely than the monetary metals. This will not make it impossible for the vaJue so determined to vary, nor prevent it from actually experiencing absolute variations on a stiU greater scale of time. Here. as in astronomy. it is necessary to recognize secular variations, which are independent of periodic variations. Even the wages of that lowest grade of labour, which is only considered as a kind of mechanical agent, the element often proposed as the 3 What characterizes a contract of saJe. and distinguishes it essentiaJly from a contract of exchange. is the invariability of the absolute value of the monetary metals, at Jeast for the lapse of time covered by an ordinary business transaction. In a country where the absolute va]ue of the monetary tokens is perceptibly variable. there are. properly speaking. no contracts of sale. This distinction should affect some legal questions.
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standard of value, is subject like wheat to periodic as well as secular variations; and, if the periodic oscillations of this element have generally been less wide than those of wheat, on the other hand we may suspect that in future the progressive changes in the social status will cause it to suffer much more rapid secular variations. But if no article exists having the necessary conditions for perfect fixity, we can and ought to imagine one, which, to be sure, will only have an abstract existence. 4 It will only appear as an auxiliary term of comparison to facilitate conception of the theory, and wi1l disappear in the final applications. In like manner, astronomers imagine a mean sun endowed with a uniform motion, and to this imaginary star they refer, as well the real sun as the other heavenly bodies, and so finally determine the actual situation of these stars with reference to the real sun. It would perhaps seem proper to first investigate the causes which produce absolute variations in the value of the monetary metals, and, when these are accounted for, to reduce to the corrected value of money the variations which occur in the value of other articles. This corrected money would be the equivalent of the mean sun of astronomers. But, on one hand, one of the most delicate points in the theory of wealth is just this analysis of the causes of variation of the value of the monetary metals used as means of circulation, and on the other hand it is legitimate to admit, as has been already said, that the monetary metals do not suffer notable variations in their values except as we compare very distant periods, or else in case of sudden revolutions, now very improbable, which would be caused by the discovery of new metallurgical processes, or of new mineral deposits. It is, to be sure, a common saying, that the price of money is steadily diminishing, and fast enough for the depreciation of value of coin to be very perceptible in the course of a generation; but by going back to the cause of this phenomenon, as we have shown how to do in this chapter, it is plain that the relative change is chiefly due to an absolute upward movement of the prices of most of the articles which go directly for the needs or pleasures of mankind, an ascending movement produced by the increase in population and by the progressive developments of industry and labour. Sufficient explanations on this doctrinal point can be found in the writings of most modern economists. Finally, in what follows, it will be the more legitimate to neglect the absol ute variations which affect the value of the monetary metals, as we do not have numerical applications directly in view. If the theory were sufficiently developed, and the data sufficiently accurate, it would be easy to go from the value of an artic1e in terms of a fictitious and invariable 4
Montesquieu. Esprit des Lois, Book XXII, Chap. 8.
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modulus, to its monetary value. If the value of an article, in terms of this fictitious modulus, was p, at a time when that of the monetary metal was 'If, and if at another time these quantities had taken other values, p' and 'If', it is evident that the monetary value of the article would have varied in the ratio of p
p'
- to-. 'If
'If'
If the absolute value of the monetary metals during long periods only
suffers slow variations, which are hardly perceptible throughout the commercial world, the relative values of these very metals suffer slight variations from one commercial centre to another, which constitute what is known as the rate of exchange, and of which the mathematical formula is very simple. OF THE LAW OF DEMAND
To lay the foundations of the theory of exchangeable values, we shall not accompany most speculative writers back to the cradle of the human race; we shall undertake to explain neither the origin of property nor that of exchange or division of labour. All this doubtless belongs to the history of mankind, but it has no influence on a theory which could only become applicable at a very advanced state of civilization, at a period when (to use the language of mathematicians) the influence of the initial conditions is entirely gone. We shall invoke but a single axiom, or, if you prefer, make but a single hypothesis, i.e. that each one seeks to derive the greatest possible value from his goods or his labour. But to deduce the rational consequences of this principle, we shall endeavour to establish better than has been the case the elements of the data which observation alone can furnish. Unfortunately, this fundamental point is one which theorists, almost with one accord, have presented to u~ we will not say falsely, but in a manner which is really meaningless. It has been said almost unanimously that "the price of goods is in the inverse ratio of the quantity offered, and in the direct ratio of the quantity demanded." It has never been considered that the statistics necessary for accurate numerical estimation might be lacking, whether of the quantity offered or of the quantity demanded, and that this might prevent deducing from this principle general consequences capable of useful application. But wherein does the principle itself consist? Does it mean that in case a double quantity of any article is offered for sale, the price will faU onehalf? Then it should be more simply expressed, and it should only be said that the price is in the inverse ratio of the quantity offered. But the principle thus made intelligible would be false; for, in general, that 100
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units of an article have been sold at 20 francs is no reason that 200 units would sell at 10 francs in the same lapse of time and under the same circumstances. Sometimes less would be marketed; often much more. Furthermore, what is meant by the quantity demanded? Undoubtedly it is not that which is actuaUy marketed at the demand of buyers, for then the generally absurd consequence would result from the pretended principle, that the more of an article is marketed the dearer it is. If by demand only a vague desire of possession of the article is understood, without reference to the limited price which every buyer supposes in his demand, there is scarcely an article for which the demand cannot be considered indefinite; but if the price is to be considered at which each buyer is willing to buy, and the price at which each seller is willing to sell, what becomes of the pretended principle? It is not, we repeat, an erroneous proposition-it is a proposition devoid of meaning. Consequently all those who have united to proclaim it have likewise united to make no use of it. Let us try to adhere to less sterile principles. The cheaper an article is, the greater ordinarily is the demand for it. The sales or the demand (for to us these two words are synonymous, and we do not see for what reason theory need take account of any demand which does not result in a sale)-the sales or the demand generally, we say, increases when the price decreases. We add the word generally as a corrective; there are, in fact, some objects of whim and luxury which are only desirable on account of their rarity and of the high price which is the consequence thereof. If anyone should succeed in carrying out cheaply the crystallization of carbon, and in producing for one franc the diamond which to-day is worth a thousand, it would not be astonishing if diamonds should cease to be used in sets of jewellery, and should disappear as articles of commerce. In this case a great fall in price would almost annihilate the demand. But objects of this nature play so unimportant a part in social economy that it is not necessary to bear in mind the restriction of which we speak. The demand might be in the inverse ratio of the price; ordinarily it increases or decreases in much more rapid proportion-an observation especially applicable to most manufactured products. On the contrary, at other times the variation of the demand is less rapid; which appears (a very singular thing) to be equally applicable both to the most necessary things and to the most superfluous. The price of violins or of astronomical telescopes might fall one-half and yet probably the demand would not doubJe; for this demand is fixed by the number of those who cultivate the art or science to which these instruments belong; who have the disposition requisite and the leisure to cultivate them and the means to pay teachers and to meet the other necessary expenses, in consequences of
Matiremalicl 01 Value and Demand
1111
which the price of the instruments is only a secondary question. On the contrary, firewood, which is one of the most useful articles, could probably double in price, from the progress of clearing land or increase in population, long before the annual consumption of fuel would be halved; as a large number of consumers are disposed to cut down other expenses rather than get along without firewood. Let us admit therefore that the sales or the annual demand D is, for each article, a particular function F(p) of the price p of such article. To know the form of this function would be to know what we call the law of demand or of sales. It depends evidently on the kind of utility of the article. on the nature of the services it can render or the enjoyments it can procure. on the habits and customs of the people, on the average wealth, and on the scale on which wealth is distributed. Since so many moral causes capable of neither enumeration nor measurement affect the law of demand, it is plain that we should no more expect this law to be expressible by an algebraic formula than the law of mortality, and alI the laws whose determination enters into the field of statistics, of what is called social arithmetic. Observation must therefore be depended on for furnishing the means of drawing up between proper limits a table of the corresponding values of D and p; after which, by the wen-known methods of interpolation or by graphic processes, an empiric formula or a curve can be made to represent the function in question; and the solution of problems can be pushed as far as numerical applications. But even if this object were unattainable (on account of the difficulty of obtaining observations of sufficient number and accuracy, and also on account of the progressive variations which the Jaw of demand must undergo in a country which has not yet reached a practically stationary condition). it would be nevertheless not improper to introduce the unknown law of demand into analytical combinations, by means of an indeterminate symbol; for it is well known that one of the most important functions of analysis consists precisely in assigning determinate relations between quantities to which numerical values and even algebraic forms are absolutely unassignable. Unknown functions may none the less possess properties or general characteristics which are known; as, for instance. to be indefinitely increasing or decreasing, or periodical, or only real between certain limits. Nevertheless such data, however imperfect they may seem. by reason of their very generality and by means of analytical symbols, may lead up to relations equaUy general which would have been difficult to discover without this help. Thus without knowing the law of decrease of the capillary forces, and starting solely from the principle that these forces are inappreciable at appreciable distances. mathematicians have demonstrated
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the general laws of the phenomena of capillarity, and these laws have been confirmed by observation. On the other hand, by showing what determinate relations exist between unknown quantities, analysis reduces these unknown quantities to the smallest possible number, and guides the observer to the best observations for discovering their values. It reduces and coordinates statistical documents; and it diminishes the labour of statisticians at the same time that it throws light on them. For instance, it is impossible a priori to assign an algebraic form to the law of mortality; it is equally impossible to formulate the function expressing the subdivision of popUlation by ages in a stationary population; but these two functions are connected by so simple a relation, that, as soon as statistics have permitted the construction of a table of mortality, it will be 'possible, without recourse to new observations, to deduce from this table one expressing the proportion of the various ages in the midst of a stationary population, or even of a population for which the annual excess of deaths over births is known. IS Who doubts that in the field of social economy there is a mass of figures thus mutually connected by assignable relations, by means of which the easiest to determine empirically might be chosen, so as to deduce all the others from it by means of theory? We will assume that the function F(p), which expresses the law of demand or of the market, is a continuous function, i.e. a function which does not pass suddenly from one value to another, but which takes in passing all intermediate values. It might be otherwise if the number of consumers were very limited: thus in a certain household the same quantity of firewood will possibly be used whether wood costs 10 francs or 15 francs the stere, 6 and the consumption may suddenly be diminished if the price of the stere rises above the latter figure. But the wider the market extends, and the more the combinations of needs, of fortunes, or even of caprices, are varied among consumers, the closer the function F(p) will come to varying with p in a continuous manner. However little may be the variation of p, there will be some consumers so placed that the slight rise or fall of the article will affect their consumptions, and wi1l lead them to deprive themselves in some way or to reduce their manufacturing output, or to substitute something else for the article that has grown dearer, as, for instance, coal for wood or anthracite for soft coal. Thus the "exchange" is a thermometer which shows by very slight variations of rates /I The Annuaire du Bureau des Lont:itudes contains these two tables, the second deduced from the first, as above, and calculated on the hypothesis of a stationary population. The work by Duvillard, entitled De l'influence de la petite verole sur la mortalitl, contains many good examples of mathematical connections between essentially em. . al functions. 1 stere = 1 M3 = 35.3 cu. ft. = ~A cord.-TRANSLATOR.]
Mallu!malics oj VatuI' "lid Dt",alld
1'213
the fleeting variations in the estimate of the chances which affect government bonds, variations which are not a sufficient motive for buying or selling to most of those who have their fortunes invested in such bonds. If the function F(p) is continuous, it wiH have the property common to al1 functions of this nature, and on which so many important appUcations of mathematical analysis are based: the variation." of the demand will be sensibly proportional to the variations in price so long as these laJt are small fractions of the original price. Moreover, these variations will be of opposite signs, i.e. an increase in price will correspond with a diminution of the demand. Suppose that in a country like France the consumption of sugar is 100 million kilograms when the price is 2 francs a kilogram, and that it has been observed to drop to 99 millions when the price reached 2 francs 10 centimes. Without considerable error, the consumption which would cor~ respond to a price of 2 francs 20 centimes can be valued at 98 millions, and the consumption corresponding to a price of 1 franc 90 centimes at ] OJ millions. It is plain how much this principle, which is only the mathematical consequence of the continuity of functions. can facilitate applications of theory, either by simplying analytical expressions of the laws which govern the movement of values, or in reducing the number of data to be borrowed from experience, if the theory becomes sufficiently devel~ oped to lend itse1f to numerical determinations. Let us not forget that, strictly speaking, the principle just enunciated admits of exceptions, because a continuous function may have interruptions of continuity in some points of its course; but just as friction wears down roughnesses and softens outlines, so the wear of commerce tends to suppress these exceptional cases, at the same time that commercial machinery moderates variations in prices and tends to maintain them between limits which facilitate the application of theory. To define with accuracy the quantity D, or the function F{p) which is the expression of it, we have supposed that D represented the quantity sold annually throughout the extent of the country or of the market 7 under consideration. In fact, the year is .he natural unit of time, especially for researches having any connection with social economy. An the wants of mankind are reproduced during this term, and all the resources which mankind obtains from nature and by labour. Nevertheless, the price of an article may vary notably in the course of a year. and, strictly speaking, the law of demand may also vary in the same interval. if the country experiences a movement of progress or decadence. For greater accuracy, therefore, in the expression F(p), P must be held to denote the annual ., It is well known that by market economists mean. not a certain place where purchases and sales are carried on, but the entire territory of which the parts are so united by the relations of unrestricted commerce that prices there take the same level throughout, with ease and rapidity.
Augustin
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COUTIIOt
average price, and the curve which represents function F to be in itself an average of all the curves which would represent this function at different times of the year. But this extreme accuracy is only necessary in case it is proposed to go on to numerical applications, and it is superfluous for researches which only seek to obtain a general expression of average results, independent of periodical oscillations. Since the function F(p) is continuous, the function pF(p), which expresses the total value of the quantity annually sold, must be continuous also. This function would equal zero if p equals zero, since the consumption of any article remains finite even on the hypothesis that it is absolutely free; or, in other words, it is theoretically always possible to assign to the symbol p a value so small that the product pF(p) will vary imperceptibly from zero. The function pF(p) disappears also when p becomes infinite, or, in other words, theoretically a value can always be assigned to p so great that the demand for the article and the production of it would cease. Since the function pF(p) at first increases, and then decreases as p increases, there is therefore a value of p which makes this function a maximum, and which is given by the equation, (1)
F(p)
+ pF'(p) =
0,
in which F', according to Lagrange's notation, denotes the differential coefficient of function F.
o
FIGURE I
If we layout the curve anb (Figure 1) of which the abscissas oq and the ordinates qn represent the variables p and D, the root of equation (1) will be the abscissa of the point n from which the triangle ont, formed by the tangent nt and the radius vector on, is isosceles, so that we have oq qt. We may admit that it is impossible to determine the function F(p) empirically for each article, but it is by no means the case that the same obstacles prevent the approximate determination of the value of p which satisfies equation (l) or which renders the product pF(p) a maximum.
=
1215
The construction of a table, where these values could be found, would be the work best calculated for preparing for the practical and rigorous solution of questions relating to the theory of wealth. But even if it were impossible to obtain from statistics the value of p which should render the product pF(p) a maximum, it would be easy to learn, at least for all articles to which the attempt has been made to extend commercial statistics, whether current prices are above or below this value. Suppose that when the price becomes p + &p, the annual consumption as shown by statistics, such as customhouse records, becomes D - &D. According as
&D
D
--, Ap
p
the increase in price, Ap, will increase or diminish the product pF(p); and, consequently, it will be known whether the two values p and p + Ap (assuming Ap to be a small fraction of p) faU above or be10w the value which makes the product under consideration a maximum. Commercial statistics should therefore be required to separate articles of high economic importance into two categories, according as their current prices are above or below the value which makes a maximum of pF(p). We shall see that many economic problems have different solutions, according as the article in question belongs to one or the other of these two categories. We know by the theory of maxima and minima that equation (1) is satisfied as well by the values of p which render pF(p) a minimum as by those which render this product a maximum. The argument used at the beginning of the preceding article shows, indeed, that the function pF(p) necessarily has a maximum, but it might have several and pass through minimum values between. A root of equation (1) corresponds to a maximum or a minimum according as 2 F'(p)
+ pF"(p)
< or> 0,
or, SUbstituting for p its value and considering the essentially negative sign of F'(p), 2 [F'(p)]2 - F(p) X F"(p)
> or < O.
=
In consequence, whenever F"(p) is negative, or when the curve D F(p) turns its concave side to the axis of the abscissas, it is impossible that there should be a minimum, nor more than one maximum. In the contrary case, the existence of several maxima or minima is not proved to be impossible. But if We cease considering the question from an exclusively abstract standpoint, it wi 11 be instantly recognized how improbable it is that the function pF (p) should pass through several intermediate maxima and
1216
minima inside of the limits between which the value of p can vary; and as it is unnecessary to consider maxima which faU beyond these limits, if any such exist, all problems are the same as if the function pF(p) only admitted a single maximum. The essential question is always whether. for the extent of the limits of oscillation of p, the function pF(p) is increas· ing or decreasing for increasing values of p. Any demonstration ought to proceed from the simple to the complex: the simplest hypothesis for the purpose of investigating by what laws prices are fixed, is that of monopoly, taking this word in its most absolute meaning. which supposes that the production of an article is in one man's hands. This hypothesis is not purely fictitious: it is realized in certain cases; and, moreover, when we have studied it, we can analyze more accurately the effects of competition of producers.
The age 01 chivalry is gone. That 01 sophisters, .cono mists and calculators has succeeded. -EDMUND BURKE. (Reflections on the Revolution in France) The egort 01 the economist is to "see," to pict"re the interplay 01 economic elements. The more clearly Cllt these elements appelU ill his "ision, the better; the more elemellts he can grasp and hold in his mind at once, thr beller. The economic world is a misty region. The first explorers used unaided vision. Mathematics is the I4ntern by which what belore was dimly visible now looms up in firm, bold outlines. The old phantasmagoria disappear. We see beller. We also see lurther. -IRVING FISHER (Transactions 01 Conn. Academy. 1892)
5
Theory of Political Economy By WILLIAM STANLEY JEVONS MATHEMATICAL CHARACTER OF THE SCIENCE
IT IS clear that Economics, if it is to be a science at all, must be a mathematical science. There exists much prejudice against attempts to introduce the methods and language of mathematics into any branch of the moral sciences. Many persons seem to think that the physical sciences form the proper sphere of mathematical method, and that the moral sciences demand some other method-I know not what. My theory of Economics, however, is purely mathematical in character. Nay, believing that the quantities with which we deal must be subject to continuous variation, I do not hesitate to use the appropriate branch of mathematical science, involving though it does the fearless consideration of infinitely small quantities. The theory consists in applying the differential calculus to the familiar notions of wealth, utility, value, demand, supply, capital, interest, labour, and aU the other quantitative notions belonging to the daily operations of industry. As the complete theory of almost every other science involves the use of that calculus, so we cannot have a true theory of Economics without its aid. To me it seems that our science must be mathematical, simply because it deals with quantities. Wherever the things treated are capable of being greater or less, there the laws and relations must be mathematical in nature. 1be ordinary laws of supply and demand treat entirely of quantities of commodity demanded or supplied. and express the manner in which the quantities vary in connection with the price. In consequence of this fact the laws are mathematical. Economists cannot alter their nature by denying them tbe name; they might as well try to alter red light by caning it blue. Whether tbe mathematical laws of Economics are stated in words, or in the usual symbols, x. y, z;, p, q, etc., is an accident, or a 1217
1218
William Stanley lel/ons
matter of mere convenience. If we had no regard to trouble and prolixity, the most complicated mathematical problems might be stated in ordinary language. and their solution might be traced out by words. In fact. some distinguished mathematicians have shown a liking for getting rid of their symbols, and expressing their arguments and results in language as nearly as possible approximating to that in common use. In his Systeme du Monde. Laplace attempted to describe the truths of physical astronomy in common language; and Thomson and Tait interweave their great Treatise on Natural Philosophy with an interpretation in ordinary words, supposed to be within the comprehension of general readers. These attempts, however distinguished and ingenious their authors, soon disclose the inherent defects of the grammar and dictionary for expressing complicated relations. The symbols of mathematical books are not different in nature from language; they form a perfected system of language, adapted to the notions and relations which we need to express. They do not constitute the mode of reasoning they embody; they merely facilitate its exhibition and comprehension. If, then, in Economics, we have to deal with quantities and complicated relations of quantities, we must reason mathematica11y; we do not render the science less mathematical by avoiding the symbols of algebra-we merely refuse to employ, in a very imperfect science, much needing every kind of assistance, that apparatus of appropriate signs which is found indispensable in other sciences. CONFUSION BETWEEN MATHEMATICAL AND EXACT SCIENCES
Many persons entertain a prejudice against mathematical language, arising out of a confusion between the ideas of a mathematical science and an exact science. They think that we must not pretend to calculate unless we have the precise data which will enable us to obtain a precise answer to our calculations; but, in reality, there is no such thing as an exact science, except in a comparative sense. Astronomy is more exact than other sciences, because the position of a planet or star admits of close measurement; but, if we examine the methods of physical astronomy, we find that they are all approximate. Every sol ution involves hypotheses which are not really true: as, for instance, that the earth is a smooth, homogeneous spheroid. Even the apparently simpler problems in statics or dynamics are only hypothetical approximations to the truth. We can calculate the effect of a crowbar; provided it be perfectly inflexible and have a perfectly hard fulcrum-which is never the case. The data are almost wholly deficient for the complete solution of anyone problem in natural science. Had physicists waited until their data were perfectly precise before they brought in the aid of mathematics, we should have still been in the age of science which terminated at the time of G alii eo.
TIr~ory
1219
of Polilit:al Et:tmomy
When we examine the less precise physical sciences, we find that physicists are, of all men, most bold in developing their mathematical theories in advance of their data. Let anyone who doubts this examine Airy's "Theory of the Tides:' as given in the Encyclopaedia Metropolitana; he will there find a wonderful1y complex mathematical theory which is can· fessed by its author to be incapable of exact or even approximate application, because the results of the various and often unknown contours of the seas do not admit of numerical verification. In this and many other cases we have mathematical ~heory without the data requisite for precise calculation. The greater or less accuracy attainable in a mathematical science is a matter of accident, and does not affect the fundamental character of the science. There can be but two classes of sciences-those which are simply logical, and those which. besides heing logical. are also mathematical. If there be any science which determines merely whether a thing be or be not-whether an event will happen. or will not happen-it must be a purely logical science; but if the thing may be greater or less, or the event may happen sooner or later, nearer or farther, then quantitative notions enter, and the science must be mathematical in nature, by whatever name we call it. CAPABILITY OF EXACT MEASUREMENT
Many wi1l object, no doubt, that the notions which we treat in this science are incapable of any measurement. We cannot weigh, nor gauge, nor test the feelings of the mind; there is no unit of labour, or suffering, or enjoyment. It might thus seem as if a mathematical theory of Economics would be necessarily deprived for ever of numerical data. I answer, in the first place, that nothing is less warranted in science than an uninquiring and unhoping spirit. In matters of this kind, those who despair are almost invariably those who have never tried to succeed. A man might be despondent had he spent a lifetime on a difficult task without a gleam of encouragement; but the popular opinions on the extension of mathematical theory tend to deter any man from attempting tasks which, however difficult, ought, some day, to be achieved .
•
•
,..
•
,..
MEASUREMENT OF FEELING AND MOTIVES
Many readers may, even after reading the preceding remarks, consider it quite impossible to create such a calculus as is here contemplated, because we have no means of defining and measuring quantities of feeling, like we can measure a mile, or a right angle; or any other physical quantity. I have granted that we can hardly form the conception of a unit of
William Stanley
1120
JtIWIfIS
pleasure or pain, so that the numerical expression of quantities of feeJing seems to be out of the question. But we only employ units of measurement in other things to facilitate the comparison of quantities; and if we can compare the quantities directly, we do not need the units .
...
...
...
...
LOGICAL METHOD OF ECONOMICS
...
III
III
...
...
To return, however, to the topic of the present work, the theory here given may be described as the mechanics 0/ ulility and .'iei/-inleresl. Oversights may have been committed in tracing out its detai Is, but in its main features this theory must be the true one. Its method is as sure and demonstrative as that of kinematics or statics, nay, almost as self-evident as are the elements of Euclid. when the real meaning of the formulae is fuHy seized. I do not hesitate to say, too, that Economics might be gradual1y erected jnto an exact science, if only commercial statistics were far more complete and accurate than they are at present, so that the formulae could be endowed with exact meaning by the aid of numerical data. These data would consist chiefly in accurate accounts of the quantities of goods possessed and consumed by the community. and the prices at which they are exchanged. There is no reason whatever why we should not have those statistics, except the cost and trouble of col1ecting them, and the unwillingness of pe rsons to afford information. The quantities themselves to be measured and registered are most concrete and precise. In a few cases we already have information approximating to completeness. as when a commodity like tea, sugar, coffee, or tobacco is wholly imported. But when articles are untaxed, and partly produced within the country, we have yet the vaguest notions of the quantities consumed. Some slight success is now, at last, attending the efforts to gather agricultural statistics; and the great need felt by men engaged in the cotton and other trades to obtain accurate accounts of stocks. imports, and consumption, will probably lead to the publication of far more complete information than we have hitherto enjoyed. The deductive science of Economics must be verified and rendered useful by the purely empirical science of Statistics. Theory must be invested with the reality and life of fact. But the difficulties of this union are immensely great. and I appreciate them quite as much as does Cairnes in his admirable lectures "On the Character and Logical Method of Political Economy." I make hardly any attempt to employ statistics in this work. and thus I do not pretend to any numerical precision. But, before we
1221
Thror)' of Political £corto,,,,
attempt any investigation of facts, we must have correct theoretical notions; and of what are here presented, J would say, in the words of Hume, in his Essay on Commerce, "If false. let them be rejected: but no one has a right to entertain a prejudice against them merely because they are out of the common road." RELATION OF ECONOMICS TO ETHICS
J wish to say a few words. in this place, upon the relation of Eco-
nomics to Moral Science. The theory which follows is entirely based on a calculus of pleasure and pain; and the object of Economics is to maximise happiness by purchasing pleasure, as it were, at the lowest cost of pain. The language employed may be open to misapprehension, and it may seem as if pleasures and pains of a gross kind were treated as the aUsufficient motives to guide the mind of man. I have no hesitation in accepting the Utilitarian theory of morals which does uphold the effect upon the happiness of mankind as the criterion of what is right and wrong. But I have never felt that there is anything in that theory to prevent our putting the widest and highest interpretation upon the terms used. Jeremy Bentham put forward the Utilitarian theory in the most uncompromising manner. According to him, whatever is of interest or importance to us must be the cause of pleasure or of pain; and when the terms are used with a sufficiently wide meaning, pleasure and pain include aU the forces which drive us to action. They are explicitly or implicitly the matter of all our calculations, and form the ultimate quantities to be treated in aU the moral sciences. The words of Bentham on this subject may require some explanation and qualification, but they are too grand and too full of truth to be omitted. "Nature," he says, "has placed mankind under the governance of two sovereign masters-pain and pleasure. It is for them alone to point out what we oUght to do, as wen as to determine what we shall do. On the one hand the standard of right and wrong, on the other the chain of causes and effects, are fastened to their throne. They govern us in aU we do, in all we say. in aU we think: every effort we can make to throw off Our subjection wiJI serve but to demonstrate and confirm it. In words a man may pretend to abjure their empire; but, in reality, he will remain subject to it all the while. The principle of utility recognises this subjection, and assumes it for the foundation of that system, the object of which is to rear the fabric of felicity by the hands of reason and of law. Systems which attempt to question it deal in sounds instead of sense, in caprice instead of reason, in darkness instead of 1ight."
•
•
•
•
•
William Stanle, leytJns
1222
THE THEORY OF UTILITY UTILITY IS NOT AN INTRINSIC QUALITY
My principal work now lies in tracing out the exact nature and conditions of utility. It seems strange indeed that economists have not bestowed more minute attention on a subject which doubtless furnishes the true key to the problem of economics. In the first place, utility, though a quality of things, is no inherent quality. It is better described as a circumstance of things arising out of their relation to man's requirements. As Senior most accurately says, "Utility denotes no intrinsic quality in the things which we call useful; it merely expresses their relations to the pains and pleasures of mankind." We can never, therefore, say absolutely that some objects have utility and others have not. The ore lying in the mine, the diamond escaping the eye of the searcher. the wheat lying unreaped, the fruit ungathered for want of consumers. have no utility at all. The most wholesome and necessary kinds of food are useless unless there are hands to coHect and mouths to eat them sooner or later. Nor, when we consider the matter closely, can we say that all portions of the same commodity possess equal utility. Water, for instance, may be roughly described as the most useful of all substances. A quart of water per day has the high utility of saving a person from dying in a most distressing manner. Several ga]]ons a day may possess much utility for such purposes as cooking and washing; but after an adequate supply is secured for these uses, any additional quantity is a matter of comparative indifference. All that we can say, then, is that water, up to a certain quantity, is indispensable; that further quantities will have various degrees of utility; but that beyond a certain quantity the utility sinks graduaIJy to zero; it may even become negative, that is to say, further supplies of the same substance may become inconvenient and hurtful. Exactly the same considerations apply more or less c1ear1y to every other article. A pound of bread per day supplied to a person saves him from starvation, and has the highest conceivable utility. A second pound per day has also no slight utility; it keeps him in a state of comparative plenty, though it be not altogether indispensable. A third pound would begin to be superfluous. It is clear, then, that utility is not proportional to commodity: the very same articles vary in utility according as we already possess more or less of the same article. The like may be said of other things. One suit of clothes per annum is necessary, a second convenient, a third desirable, a fourth not unacceptable. but we sooner or later reach a point at which further suppJies are not desired with any perceptible force unless it be for subsequent use.
T1r~ory
1223
of Polilica' Economy
LAW OF THE VARIATION OF UTILITY
Let us now investigate this subject a little more closely. Utility must be considered as measured by or even as actually identical with, the addition made to a person's happiness. It is a convenient name for the t
6 FIOURB 1
aggregate of the favorable ba]ance of feeling produced,-the sum of the pleasure created and the pain prevented. We must now carefully discriminate between the total utility arising from any commodity and the utility attaching to any particular portion of it. Thus the total utility of the food we eat consists in maintaining life, and may be considered as infinitely great; but if we were to subtract a tenth part from what we eat daily, Our loss would be but slight. We should certainly not lose a tenth part of the whole utiJity of food to us. It might be doubtftt] whether we should suffer any harm at all. Let us imagine the whole quantity of food which a person consumes on an average during twenty-four hours to be divided into ten equal parts. If his food be reduced by the last part, he will suffer but little; if a second tenth part be deficient, he will feel the want distinctly; the subtraction of the third tenth part will be decidedly injurious; with every subsequent subtraction of a tenth part his sufferings will be more and more serious, until at length he will be upon the verge of starvation. Now, if we call each of the tenth parts an increment, the meaning of these facts is, that each increment of food is less necessary, or possesses less utiHty, than the previous one. To explain this variation of utility we may make use of space representations. which I have found convenient in i1lustrating the Jaws of economics in my college lectures during fifteen years past (Figure 1). Let the line ox be used as a measure of the quantity of food, and let it be divided into ten equal parts to correspond to the ten portions of food mentioned above. Upon these equal lines are constructed rectangles and
1224
WIIIIGIfI Sltut'ey Je'lOllS
the area of each rectangle may be assumed to represent the utility of the increment of food corresponding to its base. Thus the utility of the last increment is small, being proportional to the small rectangle on x. As we approach towards 0, each increment bears a larger rectangle, that standing upon III being the largest complete rectangle. The utility of the next increment, II, is undefined, as also that of I, since these portions of food would be indispensable to life, and their utility, therefore, infinitely great. We can now form a clear notion of the utility of the whole food, or of any part of it, for we have only to add together the proper rectangles. The utility of the first half of the food will be the sum of the rectangles standing on the Hne oa; that of the second half will be represented by the sum of the smaller rectangles between a and h. The total utility of the food will be the whole sum of the rectangles, and will be infinitely great. The comparative utility of the several portions is, however, the most important. Utility may be treated as a quantity of two dimensions, one dimension consisting in the quantity of the commodity, and another in the intensity of the effect produced upon the consumer. Now the quantity of the commodity is measured on the horizontal line ox, and the intensity of utility will be measured by the length of the upright lines, or ordinates. The intensity of utility of the third increment is measured either by pq, or p'q', and its utility is the product of the units in pp' multiplied by those in pq. But the division of the food into ten equal parts is an arbitrary supposition. If we had taken twenty or a hundred or more equal parts, the same general principle would hold true, namely, that each small portion would be less useful and necessary than the last. The law may be considered to hold true theoretically, however small the increments are made; and in this way we shall at last reach a figure which is undistinguishable from a continuous curve. The notion of infinitely small quantities of food may seem absurd as regards the consumption of one individual; but when we consider the consumption of a nation as a whole, the consumption may well be conceived to increase or diminish by quantities which are, practically speaking, infinitely small compared with the whole consumption. The laws which we are about to trace out are to be conceived as theoretically true of the individual; they can only be practically verified as regards the aggregate transactions, productions, and consumptions of a large body of people. But the laws of the aggregate depend of course upon the laws applying to individual cases. The law of the variation of the degree of utility of food may thus be represented by a continuous curve phq, and the perpendicular height of each point at the curve above the line ox represents the degree of utility of the commodity when a certain amount has been consumed. (See Figure 2.)
122S
Theory 01 Pollt(cal Economy
,
• •
1&
•
FIGURE 2
Thus, when the quantity oa has been consumed, the degree of utility corresponds to the length of the line ab; for if we take a very little more food, aa', its utility wil1 be the product of aa' and ab very nearly, and more nearly the less is the magnitude of aa'. The degree of utility is thus properly measured by the height of a very narrow rectangle corresponding to a very small quantity of food, which theoretical1y ought to be infinitely small. TOTAL UTILITY AND DEGREE OF UTILITY
We are now in a position to appreciate perfectly the difference between the total utility of any commodity and the degree of utility of the commodity at any point. These are, in fact, quantities of altogether different kinds, the first being represented by an area, and the second by a line. We must consider how we may express these notions in appropriate mathematical language. Let x signify, as is usual in mathematical books, the quantity which varies independently-in this case the quantity of commodity. Let u denote the whole utility proceeding from the consumption of x. Then u will be, as mathematicians say, a function of x; that is, it will vary in some continuous and regular, but probably unknown, manner, when x is made to vary. Our great object at present, however, is to express the degree of utility. Mathematicians employ the sign ~ prefixed to a sign of quantity, such as x, to signify that a quantity of the same nature as x, but small in proportion to x, is taken into consideration. Thus ax means a small portion of x, and x + ax is therefore a quantity a little greater than x. Now when x is a quantity of commodity, the utility of x + ax will be more than that of x as a general rule. Let the whole utility of x + ax be denoted by u + ~u; then it is obvious that the increment of utility ~u belongs to the increment of commodity ax; and if, for the sake of argument, we suppose
1226
William Slanle, levon,
the degree of utility uniform over the whole of ax, which is nearly true, owing to its smallness, we shall find the corresponding degree of utility by dividing ~u by ax. We find these considerations fully illustrated by the last figure, in which oa represents x, and ab is the degree of utility at the point a. Now, if we increase x by the small quantity ad, or ax, the utility is increased by the small rectangle abb'tf. or ~u; and since a rectangle is the product of its sides, we find that the length of the line ab, the degree of utility, is represented by the fraction ~ulax. As already explained, however. the utility of a commodity may be considered to vary with perfect continuity, so that we commit a small error in assuming it to be uniform over the whole increment ax. To avoid this, we must imagine ax to be reduced to an infinitely sman size. tlu decreasing with it. The smaller the quantities are the more nearly we shall have a correct expression for abo the degree of utility at the point a. Thus the limit of this fraction ~u/ax, or, as it is commonly expressed, du/dx. is the degree of utility corresponding to the quantity of commodity x. The degree of utility is, in mathematical language, the differential coefficient of u considered as a function of x. and will itself be another function of x. We shan seldom need to consider the degree of utility except as regards the last increment which has been consumed, or, which comes to the same thing, the next increment which is about to be consumed. I shan therefore commonly use the expression final degree of utility, as meaning the degree of utility of the last addition, or the next possible addition of a very small. or infinitely small, quantity to the existing stock. In ordinary circumstances, too, the final degree of utility will not be great compared with what it might be. Only in famine or other extreme circumstances do we approach the higher degrees of utility. Accordingly we can often treat the lower portions of the curves of variation (pbq) which concern ordinary commercial transactions, while we leave out of sight the ponions beyond p or q. It is also evident that we may know the degree of utility at any point while ignorant of the total utility, that is. the area of the whole curve. To be able to estimate the total enjoyment of a person would be an interesting thing, but it would not be real1y so important as to be able to estimate the additions and subtractions to his enjoyment which circumstances occasion. In the same way a very wealthy person may be quite unable to form any accurate statement of his aggregate wealth. but he may nevertheless have exact accounts of income and expenditure, that is, of additions and subtractions. VARIATION OF THE FINAL DEGREE OF UTILITY
The final degree of utility is that function upon which the theory of economics wiIJ be found to turn. Economists, generally speaking, have
Th~o',
1227
of PoUttt'tl1 E('onoml'
failed to discriminate between this function and the total utility, and from this confusion has arisen much perplexity. Many commodities which are most useful to us are esteemed and desired but little. We cannot live without water. and yet in ordinary circumstances we set no value on it. Why is this? Simply because we usually have so much of it that its final degree of utility is reduced nearly to zero. We enjoy every day the almost infinite utility of water. but then we do not need to consume more than we have. Let the supply run short by drought. and we begin to feel the higher degrees of utility, of which we think but little at other times. The variation of the function expressing the final degree of utility is the all-important point in economic problems. We may state, as a general law, that the degree 0/ utility varies with the quantity 0/ commodity, and ultimately decreases as that quantity increases. No commodity can be named which we continue to desire with the same force, whatever be the quantity already in use or possession. All our appetites are capable of satisfaction or satiety sooner or later, in fact. both these words mean, etymologically. that we have had enough. so that more is of no use to us. H does not follow. indeed, that the degree of utility will always sink to zero. This may be the case with some things. especially the simple animal requirements, such as food, water, air. etc. But the more refined and intellectual our needs become, the less are they capable of satiety. To the desire for articles of taste, science, or curiosity, when once excited, there is hardly a limit.
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DlSUTILlTY AND DlSCOMMODITY
A few words will suffice to suggest that as utility corresponds to the production of pleasure, or, at least. a favorable alteration in the balance of pleasure and pain, so negative utility wiH consist in the production of pain, or the unfavorable alteration of the balance. ]n reality we must be almost as often concerned with the one as with the other; nevertheless, economists have not employed any distinct technical terms to express that production of pain which accompanies so many actions of life. They have fixed their attention on the more agreeable aspect of the matter. It will be allowable. however, to appropriate the good English word discommodity, to signify any substance or action which is the opposite of commodity, that is to say, anything which we desire to /let rid 0/, like ashes or sewage. Discommodity is, indeed. properly an abstract form signifying inconvenience, or disadvantage; but as the noun commodities has been used in the English language for four hundred years at least as a concrete term, so we may now convert discommodity into a concrete term, and speak of discommodities as substances or things which possess the quality
William Stanley JellonJ
1228
of causing inconvenience or harm. For the abstract notion, the opposite or negative of utility, we may invent the term disutility, which will mean something different from inutility, or the absence of utiHty. It is obvious that utility passes through inutility before changing into disutility, these notions being related as +, 0, and -. DISTRIBUTION OF COM MODITY IN DIFFERENT USES
The principles of utility may be illustrated by considering the mode in which we distribute a commodity when it is capable of several uses. There are articles which may be employed for many distinct purposes: thus, barley may be used either to make beer, spirits, bread, or to feed cattle; sugar may be used to eat, or for producing alcohol; timber may be used in construction, or as fuel; iron and other metals may be applied to many different purposes. Imagine, then, a community in the possession of a certain stock of barley; what principles will regulate their mode of consuming it? Or, as we have not yet reached the subject of exchange, imagine an isolated family, or even an individual, possessing an adequate stock, and using some in one way and some in another. The theory of utility gives, theoretically speaking, a complete solution of the question. Let s be the whole stock of some commodity, and let it be capable of two distinct uses. Then we may represent the two quantities appropriated s. The perto these uses by Xl and y" it being a condition that XI + YI son may be conceived as successively expending small quantities of the commodity; now it is the inevitable tendency of human nature to choose that course which appears to offer the greatest advantage at the moment. Hence, when the person remains satisfied with the distribution he has made, it follows that no alteration would yield him more pleasure, which amounts to saying that an increment of commodity would yield exactly as much utility in one use as in another. Let AUt, Au:! be the increments of utility which might arise respectively from consuming an increment of commodity in the two different ways. When the distribution is completed, we ought to have AUI = Au:!; or at the limit we have the equation
=
-=-, dx
dy
which is true when x. yare respectively equal to XI. YI' We must, in other words, have the final degrees of utility in the two uses equal. The same reasoning which applies to uses of the same commodity will evidently apply to any two uses, and hence to all uses simultaneously, so that we obtain a series of equations less numerous by a unit than the number of ways of using the commodity. The genera) result is that com-
Theon' 0/ Political Economl'
1229
modity, if consumed by a perfectly wise being, must be consumed with a maximum production of utility. We should often find these equations to fail. Even when x is equal to !IIl 1on of the stock, its degree of utility might stil1 exceed the utility attaching to the remaining ~loll part in either of the other uses. This would mean that it was preferable to give the whole commodity to the first use. Such a case might perhaps be said to be not the exception but the rule; for whenever a commodity is capable of only one use, the circumstance is theoretical1y represented by saying that the final degree of utility in this employment always exceeds that in any other employment. Under peculiar circumstances great changes may take place in the consumption of a commodity. In a time of scarcity the utility of barley as food might rise so high as to exceed altogether its utility, even as regards the smal1est quantity, in producing alcoholic liquors; its consumption in the latter way would then cease. In a besieged town the employment of articles becomes revolutionized. Things of great utility in other respects are ruthlessly applied to strange purposes. ]n Paris a vast stock of horses was eaten, not so much because they were useless in other ways, as because they were needed more strongly as food. A certain stock of horses had, indeed, to be retained as a necessary aid to locomotion, so that the equation of the degrees of utility never whol1y failed.
THE LAW OF INDIFFERENCE
When a commodity is perfectly uniform or homogeneous in quality, any portion may be indifferently used in place of an equal portion: hence, in the same market, and at the same moment, an portions must be exchanged at the same ratio. There can be no reason why a person should treat exactly similar things differently, and the slightest excess in what is demanded for one over the other will cause him to take the latter jnstead of the former. In nicely balanced exchanges it is a very minute scruple which turns the scale and governs the choice. A minute difference of quality in a commodity may thus give rise to preference, and cause the ratio of exchange to differ. But where no difference exists at an, or where no difference is known to exist, there can be no ground for preference whatever. If, in selling a quantity of perfectly equal and uniform barrels of flour, a merchant arbitrarily fixed different prices on them. a purchaser would of course select the cheaper ones; and where there was absolutely no difference in the thing purchased, even an excess of a penny in the price of a thing worth a thousand pounds would be a valid ground of choice. Hence follows what is undoubtedly true. with proper explanations, that in the same open market. at anyone moment. there cannot be
12~
William Stan Ie)' Jelloru
two prices for the same kind of article. Such differences as may practically occur arise from extraneous circumstances, such as the defective credit of the purchasers. their imperfect knowledge of the market, and so on. The principle above expressed is a general law of the utmost importance in Economics. and I propose to can it The Law of IndiDerence, meaning that, when two objects or commodities are subject to no important difference as regards the purpose in view, they win either of them be taken instead of the other with perfect indifference by a purchaser. Every such act of indifferent choice gives rise to an equation of degrees of utility, so that in this principle of indifference we have one of the central pivots of the theory. Though the price of the same commodity must be uniform at anyone moment, it may vary from moment to moment, and must be conceived as in a state of continual change. Theoretically speaking, it would not usuaHy be possible to buy two portions of the same commodity successively at the same ratio of exchange, because, no sooner would the first portion have been bought than the conditions of utility would be altered. When exchanges are made on a large scale, this result will be verified in practice. If a wealthy person invested £100,000 in the funds in the morning, it is hardly likely that the operation could be repeated in the afternoon at the same price. In any market, if a person goes on buying largely, he will ultimately raise the price against himself. Thus it is apparent that extensive purchases would best be made gradually, so as to secure the advantage of a lower price upon the earlier portions. In theory this effect of exchange upon the ratio of exchange must be conceived to exist in some degree, however small may be the purchases made. Strictly speaking, the ratio of exchange at any moment is that of dy to dx. of an infinitely small quantity of one commodity to the infinitely small quantity of another which is given for it. The ratio of exchange is really a differential coefficient. The quantity of any article purchased is a function of the price at which it is purchased, and the ratio of exchange expresses the rate at which the quantity of the article increases compared with what is given for it. We must carefully distinguish, at the same time, between the Statics and Dynamics of this subject. The real condition of industry is one of perpetual motion and change. Commodities are being continually manufactured and exchanged and consumed. If we wished to have a complete solution of the problem in all its natural complexity, we should have to treat it as a problem of motion-a problem of dynamics. But it would surely be absurd to attempt the more difficult question when the more easy one is yet so imperfectly within our power. It is only as a purely statical problem that' can venture to treat the action of exchange. Holders
Th~ory
1231
of PolI"ClI' Econom)'
of commodities will be regarded not as continuously passing on these commodities in streams of trade, but as possessing certain fixed amounts which they exchange until they come to equilibrium. It is much more easy to determine the point at which a pendulum will come to rest than to calculate the velocity at which it wiJ] move when displaced from that point of rest. Just so, it is a far more easy task to lay down the conditions under which trade is completed and interchange ceases, than to attempt to ascertain at what rate trade will go on when equilibrium is not attained. The difference will present itself in this form: dynamically we could not treat the ratio of exchange otherwise than as the ratio of dy and dx, infinitesimal quantities of commodity. Our equations would then be regarded as differential equations, which would have to be integrated. But in the statical view of the question we can substitute the ratio of the finite quantities y and x. Thus, from the self-evident principle, stated earlier, that there cannot, in the same market. at the same moment, be two different prices for the same uniform commodity. it follows that the last increments in an act of exchange must be exchanged in the same ratio as the whole quantities exchanged. Suppose that two commodities are bartered in the ratio of x for y; then every mth part of x is given for the mth part of y, and it does not matter for which of the mth parts. No part of the commodity can be treated differently from any other part. We may carry this division to an indefinite extent by imagining m to be constantly increased, so that, at the 1imit, even an infinitely sma)] part of x must be exchanged for an infinitely small part of y. in the same ratio as the whole quantities. This result we may express by stating that the increments concerned in the process of exchange must obey the equation. dy
y
dx
x
---. The use which we shall make of this equation wiJI be seen in the next section. THE THEORY OF EXCHANGE
The keystone of the whole Theory of Exchange, and of the principal problems of Economics, lies in this proposition-The ratio of exchange of any two commodities will be the reciprocal of the ratio of the final degrees of utility of the quantities of commodity available for consumption alter the exchange is completed. When the reader has reflected a little upon the meaning of this proposition, he will see, I think, that it is necessarily true, if the principles of human nature have been correctly represented in previous pages. Imagine that there is one trading body possessing only corn, and an-
1232
other possessing only beef. It is certain that, under these circumstances, a portion of the com may be given in exchange for a portion of the beef with a considerable increase of utility. How are we to determine at what point the exchange will cease to be beneficial? This question must involve both the ratio of exchange and the degrees of utility. Suppose, for a moment, that the ratio of exchange is approximately that of ten pounds of com for one pound of beef: then if, to the trading body which possesses com, ten pounds of com are less useful than one of beef, that body will desire to carry the exchange further. Should the other body possessing beef find one pound less useful than ten pounds of com, this body will also be desirous to continue the exchange. Exchange win thus go on until each party has obtained all the benefit that is possible, and loss of utility would result if more were exchanged. Both parties, then, rest in satisfac~ tion and equilibrium, and the degrees of utility have come to their level, as it were. This point of equilibrium will be known by the criterion, that an infinitely small amount of commodity exchanged in addition, at the same rate, will bring neither gain nor loss of utility. In other words, if increments of commodities be exchanged at the established ratio, their utilities will be equal for both parties. Thus, if ten pounds of corn were of exactly the same utility as one pound of beef, there would be neither harm nor good in further exchange of this ratio. It is hardly possible to represent this theory completely by means of a diagram, but the accompanying figure may, perhaps, render it clearer. Suppose the line pqr to be a small portion of the curve of utility of one commodity, while the broken line p'qr' is the like curve of another com~
t.•• ••••
r'
r
• FIGURE 3
modity which has been reversed and superposed on the other. Owing to this reversal, the quantities of the first commodity are measured along the base line from a towards b, whereas those of the second must be measured in the opposite direction. Let units of both commodities be represented by equal lengths: then the little line of a'a indicates an increase of the first commodity, and a decrease of the second. Assume the ratio of exchange to be that of unit for unit, or 1 to 1: then, by receiving
TileD"
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of Political EcoNoMY
the commodity a'a the person will gain the utility ad, and lose the utiHty a'c; or he will make a net gain of the utility corresponding to the mixtilinear figure cd. He will, therefore, wish to extend the exchange. If he were to go up to the point b', and were still proceeding, he would, by the next small exchange, receive the utility be. and part with b'f; or he would have a net loss of ef. He would, therefore, have gone too far; and it is pretty obvious that the point of intersection, q, defines the place where he would stop with the greatest advantage. It is there that a net gain is converted into a net loss, or rather where, for an infinitely small quantity, there is neither gain nor loss. To respresent an infinitely small quantity, or even an exceedingly small quantity, on a diagram is, of course, impossible; but on either side of the line mq I have represented the utilities of a smal1 quantity of commodity more or less, and it is apparent that the net gain or loss upon the exchange of these quantities would be trifting. SYMBOLIC STATEMENT OF THE THEORY
To represent this process of reasoning in symbols, let IU denote a small increment of com, and l1y a small increment of beef exchanged for it. Now our Law of Indifference comes into play. As both the corn and the beef are homogeneous commodities, no parts can be exchanged at a different ratio from other parts in the same market: hence, if x be the whole quantity of com given for y the whole quantity of beef received, l1y must have the same ratio to IU as y to x; we have then, l1y
Y
Y
IU
x
x
- =-, or l1y = -IU. In a state of equilibrium, the utilities of these increments must be equal in the case of each party, in order that neither more nor less exchange would Y
be desirable. Now the increment of beef, l1y, is - times as great as the x increment of com, Ax, so that, in order that their utilities shall be equal, x the degree of utility of beef must be - times as great as the degree of y
utility of com. Thus we arrive at the principle that the degrees of utility of commodities exchanged will be in the inverse proportion of the magnitudes of the increments exchanged. Let us now suppose that the first body, A, originally possessed the quantity a of corn, and that the second body, B, possessed the quantity b of beef. As the exchange consists in giving x of com for y of beef, the state of things after exchange wil1 be as follows:-
Williom $'onlll'"
1234
Jilt/OriS
A holds a - x of com, and Y of beef, B ho]ds x of com, and b - y of beef. Let «pda - x) denote the final degree of utility of corn to A, and «P2X the corresponding function for B. Also let l/JIY denote A's final degree of utility for beef, and l/J;t( b - Y) B's similar function. Then, as explained previously A wil1 not be satisfied unless the following equation holds true:«pda - x) ·dx = t/lIY'dy;
«PI (a - x) dy or---l/JIY dx Hence, substituting for the second member by the equation given previously we have
------. x What holds true of A will also hold true of B, mutatis mutandis. He must a]so derive exact]y equal utility from the final increments, otherwise it wil] be for his interest to exchange either more or less, and he will disturb the conditions of exchange. Accordingly the following equation must hold true: l/JIY
or, substituting as before,
------.
t/I;!(b - y)
x
We arrive, then, at the conclusion, that whenever two commodities are exchanged for each other, and more or less can be given or received in infinitely small quantities. the quantities exchanged satisfy two equations, which may be thus stated in a concise form-
----=-= l/J I Y
l/J:! (b - y)
X
The two equations are sufficient to determine the results of exchange; for there are only two unknown quantities concerned, namely, x and y, the quantities given and received. A vague notion has existed in the minds of economical writers, that the conditions of exchange may be expressed in the form of an equation. Thus. 1. S. Mill has said: t "The idea of a ratio, as between demand and 1
Prillciples
0/
PoliTical Ecollomy, book
jjj,
chap. ii, sec. 4.
1235
Tlreory of Polillcal Econom)
supply, is out of place. and has no concern in the matter: the proper mathematical analogy is that of an equation. Demand and supply, the quantity demanded and the quantity supplied, will be made equal." Mill here speaks of an equation as only a proper mathematical analogy. But if Economics is to be a real science at all, it must not deal merely with analogies; it must reason by real equations, like all the other sciences which have reached at al1 a systematic character. Mill's equation. indeed, is not explicitly the same as any at which we have arrived above. His equation states that the quantity of a commodity given by A is equal to the quantity received by B. This seems at first sight to be a mere truism, for this equality must necessarily exist if any exchange takes place at an. The theory of value. as expounded by Mill, fails to reach the root of the matter, and show how the amount of demand or supply is caused to vary. And MiJl does not perceive that, as there must be two parties and two quantities to every exchange, there must be two equations. Nevertheless, our theory is perfectly consistent with the laws of supply and demand; and if we had the functions of utility determined. it would be possible to throw them into a form clearly expressing the equivalence of supply and demand. We may regard x as the quantity demanded on one side and supplied on the other~ similarly, y is the quantity supplied on the one side and demanded on the other. Now, when we hold the two equations to be simultaneously true, we assume that the x and y of one equation equal those of the other. The laws of supply and demand are thus a result of what seems to me the true theory of value or exchange .
•
•
•
•
•
THEORY OF LABOUR QUANTITATIVE NOTIONS OF LABOUR
Let us endeavour to form a clear notion of what we mean by amount of labour. It is plain that duration will be one element of it; for a person labouring uniformly during two months must be allowed to labour twice as much as during one month. But labour may vary also in intensity. In the same time a man may walk a greater or less distance; may saw a greater or less amount of timber; may pump a greater or less quantity of water; in short, may exert more or less muscular and nervous force. Hence amount of labour will be a quantity of two dimensions, the product of intensity and time when the intensity is uniform, or the sum represented by the area of a curve when the intensity is variable. But intensity of labour may have more than one meaning; it may mean the quantity of work done, or the painfulness of the effort of doing it. These two things must be carefully distinguished, and both are of great importance for the theory. The one is the reward, the other the penalty,
William Stanley lellollS
1236
of labour. Or rather, as the produce is only of interest to us so far as it possesses utiHty, we may say that there are three quantities involved in the theory of labour-the amount of painful exertion, the amount of produce, and the amount of utility gained. The variation of utility, as depending on the quantity of commodity possessed, has already been considered; the variation of the amount of produce wiJJ be treated in the next chapter; we wiU here give attention to the variation of the painfulness of labour. Experience shows that as labour is prolonged the effort becomes as a general rule more and more painful. A few hours' work per day may be considered agreeable rather than otherwise; but so soon as the overflowing energy of the body is drained off, it becomes irksome to remain at work. As exhaustion approaches, continued effort becomes more and more intolerable. Jennings has so clearly stated this law of the variation of labour, that I must quote his words. "Between these two points, the point of incipient effort and the point of painful suffering, it is quite evident that the degree of toilsome sensations endured does not vary directly as the quantity of work performed, but increases much more rapidly, like the resistance offered by an opposing medium to the velocity of a moving body." 2
•
*
•
•
•
There can be no question of the general truth of the above statement, although we may not have the data for assigning the exact law of the
FIGURE 4
variation. We may imagine the painfulness of labour in proportion to prod uce to be represented by some such curve as abed (see figure above). 2
"Natural ElemelllS 0/ Political Econom)· ... p. 119.
Theon 0/ POlificld Economy
1237
In this diagram the height of points above the line ox denotes pleasure, and depths below it pain. At the moment of commencing labour it is usually more irksome than when the mind and body are well bent to the work. Thus. at first, the pain is measured by OQ. At b there is neither pain nor pleasure. Between band c an excess of pleasure is represented as due to the exertion itself. But after c the energy begins to be rapidly exhausted, and the resulting pain is shown by the downward tendency of the line cd. We may at the same time represent the degree of utility of the produce by some such curve as pq, the amount of produce being measured along the line ox. Agreeably to the theory of utility, already given, the curve shows that, the larger the wages earned, the less is the pleasure derived from a further increment. There wilt, of necessity, be some point m such that qm = dm, that is to say. such that the pleasure gained is exactly equal to the labour endured. Now, if we pass the least beyond this point, a baJance of pain will result: there will be an ever-decreasing motive in favour of labour, and an ever-increasing motive against it. The labourer will evidently cease, then, at the point m. It would be inconsistent with human nature for a man to work when the pain of work exceeds the desire of possession, including all the motives for exertion. We must consider the duration of labour as measured by the number of hours' work per day. The alternation of day and night on the earth has rendered man essentially periodic in his habits and actions. In a natural and wholesome condition a man should return each twenty-four hours to exact1y the same state; at any rate. the cycle should be closed within the seven days of the week. Thus the labourer must not be supposed to be either increasing or diminishing his normal strength. But the theory might also be made to apply to cases where special exertion is undergone for many days or weeks in succession, in order to complete work, as in collecting the harvest. Adequate motives may lead to and warrant overwork. but, if long continued. excessive labour reduces the strength and becomes insupportable; and the longer it continues the worse it is, the law being somewhat similar to that of periodic labour.
COMMENTARY ON
A Distinguished Quaker and War T EWIS FRY RICHARDSON
(1881-1953) was a British physicist who L for twenty years devoted himself to research in the psychology of peace and war. His earlier work lay in physics, mathematics and meteorology. He wrote a famous paper "The Supply of Energy from and to Atmospheric Eddies" (1920) deaJing with the transformations of radiation in the atmosphere and atmospheric turbulence; he investigated the formidably difficult problems of weather prediction. and though the equations he invented were not altogether successful, "a mighty vision had been displayed"; he was the author of a notable volume, Weather Prediction by N umericaJ Process (1922). t These scientific labors gained wide recognition, including a fellowship in the Royal Society and other academic honors. The shift in interest to social studies, evidenced among otbers by his acquiring a degree in psychology in 1929. at the age of fifty, had its origins during the First World War. Dr. Richardson. a Quaker, served in the Friends' Ambulance Unit attached to the French Army. Thus for two years while being "not paid to think," as he says, he had abundant opportunities for meditation. 2 The publication of Bertrand Russel1's War, The Offspring of Fear, further stimulated his mind. The result was an essay, The Mathematical Psychology of War, which appeared in 1919.3 A subsequent period of "comparative tranquillity" in international affairs turned his mind from the problem, but the failure of the Disarmament Conference in 1935 led him to reconsider his theory. His conclusions were published first as communications to Nature" and were later reprinted in a monograph which drew considerable attention.:; Richardson was one of no more than a handfUl of serious scholars who have in recent years attempted a quantitative treatment of the causes of war, the mechanics of foreign politics, the effects of armament races and kindred matters. His undertaking was bold and unorthodox; its political impJications were often unwelcome. Accordingly, it has not been warmly received by the average specialist in social studies. I suspect that, while Richardson is respected for his "more solid" scientific achievements. his work in the field of international brawling is regarded much as was the 1 A most interesting sketch of Richardson's meteorological ideas is given by O. G. Sutton in "MethOds of Forecasting the Weather," Th~ List~n~r, March 2S, 19S4, pp. S22-24. II Preface to Gen~raliud For~ign Politics: A Study in Group Psychology. by Lewis P. Richardson, The British Journal 0/ Psychology. Monograph Supplement XXII, Cambridge. 1939. 3 Oxford, 1919 (William Hunt). 4 May 18 and December 2S, 1935. !) Lewis F. Richardson. op. cit.
1238
.4 Distinguishcod Quakcor alld War
12)9
psychic research of Sir Oliver Lodge. In his monumental treatise, A Study 0/ War, Professor Quincy Wright-an uncommonly able student of the problem-acknowledges the importance of Richardson's contribution. He analyzes the theory and calls it "suggestive, though not in all respects convincing." *' That is a fair appraisal. The selections below furnish a clear summary of Richardson's results. The presentation is for a general audience and the mathematics is not only held to a minimum but explained simply, step by step. In the first essay, "Threats and Security," Richardson expresses the main thesis that the more a nation arms---even granting that those in power sincerely desire to preserve peace-the better are its chances for war. If the best way to preserve peace is to prepare for war, it is not clear. as C. E. M. load has pointed out, "why all nations should regard the armaments of other nations as a menace to peace. However, they do so regard them and are accordingly stimulated to increase their armaments to overtop the armaments by which they conceive themselves to be threatened." T This ruinous, deadly competition is what Richardson examines. Among the Kwakiutl Indians, competition between tribes took the form of each attempting to outdo the other in destroying its own property. But this curious practice. roughly analogous to that of more civilized peoples, was held within prudent limits. A chief might destroy a copper pot, bought for "four thousand blankets," in order to vanquish his rival by wasting more than the other could afford to waste. But he "was not free to destroy property to the utter impoverishment of his people or to engage in contests ruinous to them. Overdoing was always dangerous. . . ." 8 The second essay, "Statistics of Deadly Quarrels," deals with such topics as the distribution of wars in time, the number of big and little wars since the sixteenth century, the nations most involved, the possibility of forecasting future wars on the basis of past records and so on. It is full of fresh insights. Richardson wrote with humor and becoming diffidence; he advanced no sweeping claims for his theories but neither did he undervalue them in false modesty. He made no effort to conceal his hatred of war. I suppose the time has come when such behavior must be counted as remarkable. 6 Quincy Wright, A Study 01 War, Chicago, 1942. pp. 1482 et seq. See also Wright's interesting exposition. some of it mathematical, of his own theories of the probability. causes and prevention of war. (Chapters 36, 37, 39, 40, op. cit.) '7 Why War, Penguin Special, Harmondsworth, 1939. p. 69; quoted by Richardson. 8 Ruth Benedict. PaUems 01 Clllture. New York, 1934.
Though this may be play to you, 'tis death to
US.--SIR ROGER L'ESTRANOE
Man, a child in understanding of himself, has placed ill his hands physical tools of incalcillable power. He plays with them like a child. . . . The instrumentality becomes a master and works fatally ... not because it MS a will but because man MS nol. -JOHN DEWEY
6
Mathematics of War and Foreign Politics By LEWIS FRY RICHARDSON THE DIVERSE EFFECTS OF THREATS
THE reader has probably heard a mother say to her child: "Stop that noise, or I'll smack you." Did the chi.ld in fact become quiet? A threat from one person, or group of people, to another person or group has occasionally produced very little immediate effect, being received with contempt. Effects, when conspicuous, may be classified as submission at one extreme, negotiation or avoidance in the middle, and retaliation at the other extreme. The fol1owing incidents are classified in that manner; otherwise they are purposely miscel1aneous.
Contempt EXAMPLE 1. About fifty states, organized as the League of Nations, tried in 1935 and 1936, by appeals and by cutting off supplies. to restrain Mussolini's Italy from making war on Abyssinia. At the time Mussolini disregarded the League, and went on with the conquest of Abyssinia. He did not however forget. Four years later in his speech to the Italian people on the occasion of Italy's declaration of war against Britain and France. Mussolini said, "The events of quite recent history can be summarized in these words-half promises. constant threats, blackmail and finally, as the crown of this ignoble edifice, the League siege of the fifty-two States." 1
Submission EXAM PLE 2.2 In 1906 the British Government had a disagreement with the Sultan of Turkey about the exact location of the frontier between Egypt and Turkish Palestine. A British battleship was sent on 3rd May with an ultimatum, and thereupon the Sultan accepted the British view. Glas/:ow Herald, 11th June, 1940. Ellcy. Brit. XIV, ed. 14, 2, 156. Grey, Viscount. 1925. Twenty-Fi, e Yean. Hodder and Stoughton, London. 1
2
1240
MQ'''~mQllcs
01 War and
Fo'~lt",
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Polilics
Some resentment may perhaps have lingered, for after the First World War had been going on for three months Turkey joined the side opposite to Britain. EXAMPLE 3. In August 1945 the Japanese, having suffered several years of war, having lost by defeat their Italian and German allies, being newly attacked by Russia, having had two atomic bombs dropped on them, and being threatened with more of the same, surrendered unconditionally.
Negotiation Followed by Submission 4. 3 After the Germans annexed Austria on 13th March, 1938, the German minority in Czechoslovakia began to agitate for selfgovernment, and both the German and Czech Governments moved troops towards their common frontier. On ] 3th August there began German Army manreuvres on an unprecedented scale. On 6th September France called up reservists. In September the cession of the Sudetenland by Czechoslovakia to Germany was discussed, to the accompaniment of threats by Hitler on 12th, the partial mobilization of France on 24th, and the mobilization of Czechoslovakia and of the British Navy on 27th. Finally, at Munich on the 29th the French and British agreed to advise the Czechs to submit to the German demand for the Sudetenland, partly because its population spoke German, and partly because of the German threat to take it by armed force. Intense resentment at this humiliation Hngered. EXAMPLE
/
Negotiation Followed by a Bargain 5. 4 In the spring of 1911 French troops entered Fez, in Morocco, the German Government protesting. On lst July, 1911, the German Government notified those of France and Britain that a German gunboat was being dispatched to Agadir on the southern coast of Morocco in order there to protect some German firms from the local tribesmen. The French and British interpreted the movement of the gunboat as a threat against themselves, like a 'thumping of the diplomatic table.' On 21st July Mr. Lloyd George made a speech containing the sentence "I say emphatically that peace at that price would be humiliation intolerable for a great country like ours to endure. Much indignation was expressed in the newspapers of France, Britain and Germany. Negotiations ensued. By 4th November France and Germany had agreed on a rearrangement of their West African territories and rights. The arms race continued. EXAM PLE
It
a Keesing's Contemporary Archives. Bristol. 4 Ency. Brit. XIV, ed. 14, 23, 349, 352. Morel, E. D., 1915, Ten Years 01 Secret Diplomacy, National Labour Press, London. Grey, Viscount, 1925, Twenty-Five Years, Hodder and Stoughton, London.
1241
Lewis Fry RlclttJrtuon
Avoidance
EXAMPLE 6. The normal behaviour of the armed personnel guarding any frontier in time of peace is to avoid crossing the frontier lest they should be attacked. EXAM PLE 7. During 1941 British shipping mostly went eastward via the Cape of Good Hope avoiding the Mediterranean where the enemy threat was too strong. EXAMPLE 8. Criminals are usually said to avoid the police: that is, when the criminals are decidedly outnumbered by the police. Retaliation
ExAMPLE 9. After the Agadir incident had been settled by the FrancoGerman Agreement of 4th November, 1911, the 'defence' expenditures of both France and Germany nevertheless continued to increase. See the tabular statement on page ] 246. EXAMPLE 10. Within six weeks after the Munich Agreement of 29th September, 1938, rearmament was proceeding more rapidly in France,» Britain,6 Germany 7 and U.S.A.R EXAM PLE 11. On 5th November, 1940, the British armed merchantship Jervis Bay, being charged with the defence of a convoy, saw a German pocket-battleship threatening it. The Jervis Bay, although of obviously lesser power, attacked the battleship, and continued to fight until sunk, thus distracting the battleship's attention from the convoy, and giving the latter a chance to escape. EXAMPLE 12. On 10th November, 1941, Mr. Winston Churchill warned the Japanese that "should the United States become involved in war with Japan the British declaration will follow within the hour." This formidable threat did not deter the Japanese from attacking Pearl Harbour within a month. These miscellaneous illustrations may serve to remind the reader of many others. Is there any understandable regularity about the wider phenomena which they represent? The present question is about what happens in fact; the other very important question as to whether the fact is ethically good or bad, is here left aside. Some conc1usions emerge: (a) People, when threatened, do not always behave with coldly calculated self-interest. They sometimes fight back, taking extreme risks. (Examples 11 and 12.) (b) There is a notable disti nction between fresh and tired nations. in the sense that a formidable threat to a fresh nation was folJowed by Athlone broadcast, 5th October and 12th October. ParJiamemar)' Debates, 3rd to 6th October. ., GlasRow Herold, 10th October. S Glasgow Herold, 7th November. IS
6
Mdt"~mdti(:s
of War alld For~iB" PoUtks
1243
retaliation (Example 12) whereas an even more severe threat to the same nation, when tired, produced sUbmission. (Example 3.) (c) A group of people, having a more or less reasonable claim, has sometimes quickly obtained by a threat of violence, more than it otherwise would. But that may not have been the end of the matter. (Examples 2 and 4.) (d) There have often been two contrasted effects, one immediate, the other delayed. An immediate effect of contempt or submission or negotiation or avoidance has been folJowed by resentful plans for retaliation at some later opportunity. (Example I, Example 2, Examples 4 and 10, Examples 5 and 9.) (e) What nowadays is euphemisticalJy called national 'defence,' in fact always includes preparations for attack, and thus constitutes a threat to some other group of people. This type of 'defence' is based on the assumption that threats directed towards other people will produce in them either submission, or negotiation, or avoidance; and it neglects the possibility that contempt or retaliation may be produced instead. Yet in fact the usual effect between comparable nations is retaliation by counterpreparations. thus leading on by way of an arms race towards another war. SCHISMOOENESIS
In his study of the Iatmul tribe in New Guinea. Gregory Bateson {) noticed a custom whereby, at a meeting in the ceremonial hall two men would boast alternately, each provoking the other to make bolder claims, until they reached extravagant extremes. He also noticed a process whereby a man would have some control over a woman. Then her acceptance of his leadership would encourage him to become domineering. This in turn made her submissive. Then he became more domineering and she became more abject, the process running to abnormal extremes. Bateson called both these processes ·schismogenesis.' which may be translated as 'the manner of formation of cleavages.' When both parties developed the same behaviour, for example, both boasting, Bateson called the schismogenesis 'symmetrical.' When the parties developed contrasted behaviour, say, one domineering and the other submissive, he called the schismogenesis 'comp1ementary: In this terminology an arms race between two nations is properly described as a case of symmetrical schismogenesis. In the year 1912 Germany was allied with Austria-Hungary, while France was allied with Czarist Russia. Britain was loosely attached to the latter group, thus forming the Triple Entente. while Italy was nominally 9 Bateson, G., 1935, in the periodical Man, p. 199. Bateson, G., 1936, Naven. University Press, Cambridge.
1244
attached to the former group, thus making the Triple Alliance. The warlike preparations of the AlIiance and of the Entente were both increasing. The usual explanation was then, and perhaps still is, that the motives of the two sides were quite different, for we were only doing what was right, proper and necessary for our own defence, whilst they were disturbing the peace by indulging in wild schemes and extravagant ambitions. There are severa] distinct contrasts in that omnibus statement. Firstly that their conduct was morally bad, ours morally good. About so national a dispute it would be difficult to say anything that the world as a whole would accept. But there is another alleged contrast as to which there is some hope of general agreement. It was asserted in the years 1912-14 that their motives were fixed and independent of our behaviour, whereas our motives were a response to their behaviour and were varied accordingly. In ]914 Bertrand Russe1l 1O (now Earl Russell) put forward the contrary view that the motives of the two sides were essentially the same, for each was afraid of the other; and it was this fear which caused each side to increase its armaments as a defence against the other. Russell's pamphlet came at a time when a common boast in the British newspapers was that the British people 'knew no fear.' Severa] conspicuous heroes have since explained that they achieved their aims in spite of fear. When we analyse arms races it is, however, unnecessary to mention fear, or any other emotion; for an arms race can be recognized by the characteristic outward behaviour. which is shown in the diagram on page 1246. The valuable part of Russe1l's doctrine was not his emphasis on fear, but his emphasis on mutual stimulation. This view has been restated by another philosopher, C. E. M. load: 11 . . . if. as they maintain, the best way to preserve peace is to prepare war, it is not altogether clear why a1l nations should regard the armaments of other nations as a menace to peace. However, they do so regard them and are accordingly stimulated to increase their armaments to overtop the armaments by which they conceive themselves to be threatened. . . . These increased arms being in their turn regarded as a menace by nation A whose allegedly defensive armaments have provoked them, are used by nation A as a pretext for accumulating yet greater armaments wherewith to defend itself against the menace. These yet greater armaments are in their turn interpreted by neighbouring nations as constituting a menace to themselves and so on. . . . This statement is, I think, a true and very clear description but needs two amendments. The competition is not usually between every nation and every other nation, but rather between two sides; so that a nation looks with moderate favour on the armaments of other nations on its own side, and with strong dislike on those of the opposite side. load's descrip10 Rusc;ell. B. A W., 1914. "War the Offspring of Fel.lr," Union of Democratic Control. London. 11 Joad, C. E. M .. 19J9, Will Wm' Penguin Special, p. 69.
Matlll'mal#t:s 01 War and FOT(:'itm PoUth-'
124!
tion applies to an arms race which has become noticeable. Motives other than defence may have been important in starting the arms race. It may be well to translate these ideas into the phraseology of 'operational research' which began to he used during the Second World War. Professor C. H. Waddington 1:.l explains that "The special characteristic which differentiates operational research from other branches of applied science is that it takes as the phenomenon to be studied the whole executive problem and not the individual technical parts . . . ." Surely the maintenance of world peace is an executive problem large enough to be called an operation and to require an appropriate background of operational research. Sir Charles Goodeve,l~ in a survey of operational research, distinguishes between 'self-compensating and self-aggravating systems,' and he mentions, as an example of the latter, the system composed of the public and of the store-keepers; a system such that a rumour of scarcity can make a real scarcity. Tn this phraseology it can be said that a system of two great powers, not in the presence of any common enemy, is a 'selfaggravating system' such that a rumour of war can make a real war. It wi11 be shown in the next section that arms races are best described in quantitative terms; but, for those who do not like mathematics, Bateson's word 'schismogenesis' may serve as an acceptable summary of a process which otherwise requires a long verbal description such as those given by Russell, Bateson, or load. THE QUANTITATIVE THEORY OF ARMS RACES
The facts for the years ] 909 to 1914 are interesting. The 'defence' budgets of France, Germany and Russia were taken from a digest by Per Jacobsson; 14 those for Austria-Hungary from the Statesman's Year Books. To make them comparable, they were all reduced to sterling. In those years the exchange rates between national currencies were held steady by the shipment of gold, so that the conversion to sterling is easy and definite. Because France was allied to Russia it is reasonable to consider the total of their 'defence' expenditures. Let it be U. For a similar reason let V be the total for Germany and Austria-Hungary. In the accompanying diagram the rate of increase (U + V) is plotted against (U + V). See Figure 1. The accuracy with which the four observed points are fitted by a straight line is remarkable. especially as one of the co-ordinates is a difference, Similar diagrams drawn for other years, for other countries, and from other sources of information, are not so straight; but still they are straight enough to suggest that the explanation of the phenomenon is Waddington, C. H., in NO/lire. Vol. CLXI, p. 404. Goodeve, Sir Charles in Nafllrt>. Vol. CLXl, p. 384. 14 Jacobsson. Per (1929"). Armaments Expenditures of the World. published by the Economist. London. 12 J3
uwls Fry Rlcltard.um
1246
hardly likely to be found in the caprice of a few national leaders; the financial facts suggest either regular planning, or the regularity which results from the average of many opinions. The main feature shown by the diagram is that the more these nations spent, the more rapidly did they increase their expenditure. Athletic races are not like that, for in them the speed of the contestants does not in~ crease so markedly with the distance that they have run. TABLE. THE ARMS RACE OF
1909-14
Defence budgets expressed in millions of £ sterling 1909
1910
19/1
1912
J 913
France ................ 48·6 50·9 57·1 63·2 74·7 Russia................ 66·7 68·5 70·7 81·8 92·0 Germany .............. 63·1 62·0 62·5 68·2 95·4 Austria-Hungary........ 20·8 23·4 24·6 25·5 26·9 Total U + V .....••.. 199·2 204·8 214·9 238·7 289·0 Time rate I1(U + V)/111 5·6 10·1 23·8 50·3 (U + V) at same date.... 202·0 209·8 226·8 263·8
=
=
Here .:1 signifies 'take the annual increase of whatever symbol fo11ows next. From Monog. Supplt. No. 23 of Brit. Joum. Psychol., by permission of the British Psychological Society.
The sloping line when produced backwards cuts the horizontal, where I1(U + V)/111 vanishes, at the point where U + V 194 million £. This point may suitably be calJed a point of equilibrium. To explain how it could be a point of equilibrium we can suppose that the total expenditure of 194 million was regarded as that which would have been so ordinary as not to constitute any special threat. H was a theory 1:1 which led L. F. Richardson to make a diagram
=
FIGURE I-The financial facts in tht' table dbovc are here plolted 1~ Richardson, L. F., 1919. Mathematical Pn'clwlogl of War In Briti.. h copyright libraries. The diagram firsl appeared in Nature of 1938, Vol. CXLJ1. p 792.
Malh~malks
of Well' and
For~i'H
1247
Politics
having those co-ordinates U + V and b.( U + V) Ib.t. This theory wi11 now be explained. The opening phase of the First World War afforded a violent ilJustration of Russell's doctrine of mutuality, for it was evident that warlike activity, and the accompanying hatred, were both growing by tit for tat. alias mutual reprisals. Tit for tat is a jerky alternation; but apart from details, the general drift of mutual reprisals was given a smoothed quantitative expression in the statement that the rate of increase of the warlike activity of each side was proportional to the warlike activity of the other side. This statement is equivalent to the following pair of simultaneous differential equations dx -=ky, dt
dy -=Ix dt
(1), (2)
where t is the time, x is the warlike activity of one side. y that of the other side, k and I are positive constants. dxldt is the excellent notation of Leibniz for the time rate of increase of x. and dyldt is the time rate of increase of y. In accordance with modem custom, the fraction-line is set sloping when it occurs in a line of words. If k were equal to /, then the relation of x to y would be the same as the relation of y to x, so that the system of x and y would be strictly mutual. Strict mutuality is, however, not specially interesting. The essential idea is that k and I, whether equal or not, are both positive. They are caned 'defence coefficients' because they represent a pugnacious response to threats. The reader may here object that anything so simple as the pair of equations (l) and (2) is hardly likely to be true description of anything so compHcated as the politics of an arms race. In reply appeal must be made to a working rule known as Occam's Razor whereby the simplest possible descriptions are to be used until they are proved to be inadequate. The meaning of (l) and (2) will be further iI1ustrated by deducing from them some simple consequences. If at any time both x and y were zero, it follows according to (1) and (2) that x and y would always remain zero. This is a mathematical expression of the idea of permanent peace by all-round total disarmament. Criticism of that idea will fonow, but for the present let us continue to study the meaning of equations (1) and (2). Suppose that x and y being zero, the tranquillity were disturbed by one of the nations making some very slightly threatening gesture, so that y became slightly positive. According to (1) x would then begin to grow. According to (2) as soon as x had become positive, y would begin to grow further. The larger x and y had become the faster would they increase. Thus the system defined by (1) and (2) represents a possible equilibrium at the point where x and yare both zero, but this equi1ibrium is unstable, because any slight deviation from it tends to increase. If any historian or politician reads these words, I beg him or her to notice that
1248
in the mechanical sense, which is used here, stability is not the same as equilibrium; for on the contrary stable and unstable are adjectives qualifying equilibrium. Thus an equilibrium is said to be stable, or to have stability, if a small disturbance tends to die away; whereas an equilibrium is said to be unstable, or to have instability, if a small disturbance tends to increase. In this mechanical sense the system defined by (l) and (2) has instability. It describes a schismogenesis. "It is an old proverb," wrote William Penn in 1693: "Maxima bella ex levissimis causis: The greatest Feuds have had the smallest Beginnings." One advantage of expressing a concept in mathematics is that deductions can then be made by re1iable techniques. Thus in (1) and (2) the nations appear as entangled with one another, for each equation involves both x and y. These variables can, however, be separated by repeating the operation dldt which signifies 'take the time rate of.' Thus from (1) it follows that -d (dX) dt dt dy Simultaneously from (2) kdt
dy = kdt
= klx
On elimination of dy/dt between these two equations there remains an equation which does not involve y, namely
.:!.- (dX) = klx. dt
Similarly
Y !-. (d ) = kly dt dt
(3)
dt
(4)
In (3) and (4) each nation appears as if sovereign and independent, managing its own affairs, until we notice that the constant kl is a property of the two nations jointly. Another advantage of a mathematical statement is that it is so definite that it might be definitely wrong; and if it is found to be wrong, there is a plenteous choice of amendments ready in the mathematicians' stock of formulm. Some verbal statements have not this merit; they are so vague that they could hardly be wrong, and are correspondingly useless. The formulm (1) and (2) do indeed require amendment, for they contain no representation of any restraining influences; whereas it is well known that, after a war, the victorious side, no longer feeling threatened by its defeated enemy, proceeds to reduce its armed forces in order to save expenditure, and because the young men are desired at home. The simplest mathematical representation of disarmament by a victor is dx
-=-ax dt
(5)
1249
where 0. is a positive constant. For, so long as x is positive, equation (5) asserts that dx/dt is negative, so that x is decreasing. Equation ( 5) is commonly used in physics to describe fading away. In accountancy, depreciation at a fixed annual percentage is a rule closely similar to (5). As a matter of fact Ul (5) is a good description of the disarmaments of Britain, France, U.S.A., or Italy during the years just after the First World War. In equation (5), which represents disarmament of the victor, there is no mention of y, because the defeated nation no longer threatens. It seems reasonable to suppose that restraining influences of the type represented by (5) are also felt by both of the nations during an arms race. so that equations (l) and (2) should be amended so as to become
dx
- = ley -
dy dt
ax.
dt
= Ix -
fJy
(6), (7)
in which fJ is another positive constant. At first 11 0. and fJ were caned 'fatigue and expense coefficients,' but a shorter and equally suitable name is restraint coefficients, These restraining influences may, or may not, be sufficient to render the equilibrium stabJe. The interaction is easily seen in the special, but important, case of simi1ar nations, such that 0. = p. and k
= I. For then the subtraction of (7)
from (6) gives
d(x - y)
--- = -
(k
dt
+ o.)(x -
y)
(8)
In this (k + 0.) is always positive. If at any time (x - y) is positive, equation (8) shows that then (x - y) is decreasing; and that moreover (x - y) wilJ continue to decrease until it vanishes, leaving x = y. If on the contrary (x - y) is initiaUy negative, (8) shows that (x - y) will increase towards zero. Thus there is a stabJe drift from either side towards equality of x with y. That is more or less in accord with the historical facts about arms races between nations which can be regarded as similar. To see the other aspect, let (7) be added to (6) giving
d(x
+ y)
- - - = (Ie dt
o.)(x
+ y)
(9)
The meaning of (9) can be discussed in the same manner as that of (8). The result is that (x + y) will drift towards zero if (Ie - 0.) is negative, that is if 0. > Ie. We may then say that restraint overpowers 'defence,' and that the system is thoroughly stable. Unfortunately that is not what has happened in Europe in the present century. The other case is that in which Ie > 0. so that 'defence' overpowers restraint, and (x + y) drifts away from zero. That is like an arms race. When Ie > 0., the system is footnote 21.) Richardson. L. F., in Nature of 18th May, 1935, p. 830.
'8 (See 1'1
L~wls
1150
Fr'f RtclttutUoli
stable as to (x - y), but unstable as to (x + y). It is tbe instability whicb has the disastrous consequences. The owner of a ship which has capsized by rolling over sideways can derive little comfort from the knowledge that it was perfectly stable for pitching fore and aft. People who trust in the balance of power should note this combination of stability with instability. If at any time x and y were both zero, it would follow from (6) and (7) that x and y would always remain zero. So that the introduction of the restraining terms still leaves the theoretical possibility of permanent peace by universal total disarmament. Sma1l-scale experiments on absence of armament have been tried with success between Norway and Sweden, between Canada and U.S.A., between the early settlers in Pennsylvania and the Red Indians. The experiment of a genera) world-wide absence of arms has never been tried. Many people doubt if it would result in permanent peace; for, they say, grievances and ambitions would cause various groups to acquire arms in order to assert their rights, or to domineer over their unarmed neighbours. The theory is easily amended to meet this objection. Let two constants 9 and h be inserted respectively into (6) and (7) thus dx
dy
dt
dt
- = ky -ax+g; - = Ix -
fly
+h
(10), (11)
=
If x and y were at any time both zero, then would dxldt K, and dy/dt = h, which do not indicate a permanent condition. There may be still an equiJibrium, but it is not at the point x 0, y O. To find the new point of equilibrium, let dx1dl 0, and dyldl O. Then by (10) and (11)
=
=
0= ky - ax
+ g;
0= Ix - fly
+h
= =
(12). (13)
These equations represent two straight lines in the plane of x and y. If these lines are not parallel, their intersection is the point of equilibrium. It may be stable or unstable. The assertion that the defence-coefficients k and I are positive is equivalent to supposing that the effect of threats is always retaliation. The reader may object that in the opening section of this chapter other effects were also mentioned, namely, contempt, submission, negotiation, or avoidance. The most important of these objections relates to submission, because it is the direct opposite of retaliation. The answer is that the scope of the present theory is restricted to the interaction of groups which style themselves powers, which are proud of their so-called sovereignty and independence, are proud of their armed might, and are not exhausted by combat. This theory is not about victory and defeat. In different circumstances k or I might be negative. A theory of submissiveness showing this has
1251
Mathematics 01 War a"d Foreign Politics
been published. tR As to contempt, negotiation, or avoidance, they have sometimes gone on concurrently with an arms race, as in Examples (1), (5) and (9). (4) and ()O) of the diverse effects of threats. Let us now return to the 'defence' budgets of France, Russia, Germany and Austria-Hungary. The diagram on page 1246 relates to the total U + V of the warlike expenditures of the two opposing sides. The equilibrium point, at U + V = ]94 millions sterling, presumably represents the expenditure which was excused as being customary for the maintenance of internal order, and harmless in view of the treaty situation. In the theory the treaty situation is represented by the constants g and h. Their effects can be regarded as included in the 194. together with the general goodwiJl between the nations. It appears suitable therefore to compare fact with theory by setting simultaneously x + y = U + V - 194, together with g 0 and h O. (14). (15), (16) From (14) one derives
=
=
d(x
+ y) dt
d(U
+ V)
( 17)
dt
The two opposing alHances were about of equal size and civilization, so that it seems permissible to simplify the formulz by setting a fJ and k I. Addition of the formulz (10) and (11) then gives, as before in (9)~
=
=
d(x
+ y)
---=
(k-a)(x+y)
(18)
dt
Now x + y can be thoroughly eliminated from (18); from its first member by (17), and from its second member by (14), with the result that d(U
+ V)
- - - = (k-a)(U+ V-194)
(19)
dt
This is a statement about the expenditures of the nations in a form comparabJe with Figure) on page 1246. For d( U + V) /dt is a close approximation to 11( U + V) /111. The assumed constancy of k a agrees with the fact that the sloping line on the diagram is straight. Moreover the absence from (6) and (7) of squares, reciprocals, or other more complicated functions, which was at first excused on the plea of simplicity, is now seen to be so far justified by comparison with historical fact. The slope of the line on the diagram, when compared with (19) gives k - a III
See footnotes 19, 21.
O· 7 per year
(20)
uwis Fry Richardson
1252
Further investigations of this sort have dealt with demobilization,19.21 with the arms race of 1929-39 between nine nations,IO.21 with war-weariness and its fading,:!l with submissiveness in general,lo.:n and with the submission of the defeated in particular. 2o • 21 All those investigations had to do with warlike preparations. an outward or behaviouristic manifestation. The best measure of it was found to be a nation's expenditure on defence divided, in the same currency, by the annual pay of a semi-skilled engineer. This conception may be caned 'war-finance per salary,' a phrase which can be packed into the new word 'warfinpersa]. ' MOODS, FRI ENOL Y OR HOSTILE, PRIOR TO A WAR
What of the inner thoughts, emotions, and intentions which accompany the growth of warfinpersal? Lloyd George, who was Chancellor of the Exchequer in 19]4, describes some of them in his War Memoirs, revised in 1938. He relates that although there had been naval rivalry between Britain and Germany during the previous six years, yet as late as 24th July, 1914, only a very small minority of Britons wished for war with Germany. Eleven days later the British nation had changed its mind. Another brilliant description of the moods occurs in H. G. Wells's novel Mr. Britling Sees It Through. The contrast between the comparatively slow growth of irritation over years, and the sudden outbreak of war, can be explained by the well-established concept of the subconscious. Suppose, for simplicity, that in a person there are only two mental levels, the overt and the subconscious, and that the moods in these levels are not necessarily the same. In Britain in the year 1906 the prevailing mood towards Germany was friendly openly, and friendly also in the subconscious. The arms race during 1908 to ] 913 did not prevent the King from announcing annually to Parliament that "My relations with foreign powers continue to be friendly"; and the majority of British citizens continued to speak in friendly terms of their German acquaintances. It is reasonable to suppose, however, that during those same years there was a growing hostility to Germany in the British subconscious mind, caused by the arms race and by diplomatic crises. The hostile mood, having been thus slowly prepared in the subconscious, was ready suddenly to take open control at the beginning of August] 9] 4. A quantitative theory of such changes of mood is offered by L. F. Richardson.!!!! Here is a simplified specimen of that theory: d",,/dt
= C :!~1"'!! 1
In Richardson, L. F., 1939. Gellerali:.ed ForeiRII Po/ilin. Monog. Supplt No. 23 of the B, i*h /01(111C11 01 PV) c"oIO~l' 20 Richardson. L. F., 1944. Jetter in Nall(1e of 19th August. p 240. 21 Richardson. L. F .. 1947. Awn alld /1IIecmill on 35 mm. punched safety microfilm. sold by the author. 22 Richardson. L. F., 1948. PII chomelrif..a. Vol. XIII, pp 147-74 and 197-232.
1253
Mutht'maticJ 01 WD, Dlld Fort'i,1I I'ori,ies
where "It is the fraction of the British population that was eager for war at time t. while "I.! is the corresponding fraction for Germany, C 12 is a is the fraction of the British population that was in the constant, and susceptible mood: overtly friendly, but subconsciously hostile. An equation of this type is used in the theory of epidemics of disease. 23 Eagerness for war can be regarded analogously as a mental disease infected into those in a susceptible mood by those who already have the disease in the opposing country. In this theory, as in Russell's War the Offspring of Fear, the relations between the two nations are regarded as mutual. Accordingly the same letter, , say, is used in relation to either, but is distinguished by suffix I for the British, suffix 2 for the German quantity. Also there is another equation obtainable from that above by interchanging the suffixes.
'1
CONCLUSION
This chapter is not about wars and how to win them, but is about attempts to maintain peace by a show of armed strength. Is there any escape from the disastrous mutuul stimulation by threat and counter-threat? Jonathan Griffin:!4 argued that each nation should confine itself to pure defence which did not include any preparation for attack. while aggressive weapons should be controlled by a supranational authority. The difficulty is that. when once a war has started, attack is more effective than defence. Gandhi's remarkable discipline and strategy of non-violent resistance is explained and discussed by Gregg. 2 ;; The pacifying influence of intermarriage has been considered by Richardson. 2o 23 Kermack. W.O. and McKendrick. A. G .. 1927. Proc. ROj', Soc. Lond. A Vol CXV, pp. 700 22. 24 Griffin. J. 1916, A/lema/ile 10 ReQlmamem, Macmillan, London. 2. Gregg. R, B.. 1936. The POlter 01 NOll·Violellce, Routledge, London. :!'l Richardson. L. F .• 1950. The Eugm;cs Rei iew, Vol. xur, pp. 25-36.
They be larre more in number, that love to read 01 great Armies. bloudy Batte/s, and many thousands slaine at once, then that minde the Art by which the ADairs, both 01 Armies, and Cities, be conducted to their ends. -THOMAS HOBBES (Pre/ace to Thllcydides) Men grow tired 0/ sleep, love, sillging and dancing sooner than 01 war. -HOMER (Iliad) The Sllccess 0/ a war is gauged by the amOll1lt 0/ damage it does. -VICTOR HUGO
There mustn't be any mOre war. It disturbs too many people. -AN OLD FRENCH PEASANT WOMAN (To Aristide Briand, 1917)
7
Statistics of Deadly Quarrels By LEWIS FRY
~ICHARDSON
THERE are many books by military historians deaHng in one way or another with the general theme 'wars, and how to win them.' The theme of the present chapter is different, namely, 'wars, and how to take away the occasions for them,' as far as this can be done by inquiring into general causes. But is there any scope for such an inquiry? Can there be any general causes that are not well known, and yet of any importance? Almost every individual in a be1ligerent nation explains the current war quite simply by giving particulars of the abominable wickedness of his enemies. Any further inquiry into general causes appears to a belligerent to be futile. comic, or disloya1. Of course an utterly contradictory explanation is accepted as obviously true by the people on the other side of the war; while the neutrals may express chilly cynicism. This contradiction and variety of explanation does provide a prima-facie case for further investigation. Any such inquiry should be so conducted as to afford a hope that critical individuals belonging to all nations will ultimately come to approve of it. National alliances and enmities vary from generation to generation. One obvious method of beginning a search for general causes is therefore to collect the facts from the whole world over a century or more. Thereby national prejudices are partly eliminated. COLLECTIONS OF FACTS FROM THE WHOLE WORLD
Professor Quincy Wright 1 has published a collection of Wars 0/ Modern Civilization extending from A.D. 1482 to A.D. 1940, and including 278 wars, together with their dates of beginning and ending, the name of any treaty of peace, the names of the participating states, the number of battles. and a classification into four types of war. This extensive summary J
Wright, Q., 1942, A Stlldy 0/ War, Chicago University Press, Chicago. 1254
Stlllistics 01 [Hadb- Qllarrtlls
J2SS
of fact is very valuable, for it provides a corrective to those frequent arguments which are based on the few wars which the debater happens to remember, or which happen to support his theory. Wright explains his selection by the statement that his list "is intended to include all hostilities involving members of the family of nations, whether international, civil, colonial, or imperial. which were recognized as states of war in the legal sense or which involved over 50.000 troops. Some other incidents are included in which hostilities of considerable but lesser magnitude, not recognized at the time as legal states of war, led to important legal results such as the creation or extinction of states, territorial transfers, or changes of government." Another world-wide collection has been made by L F. Richardson for a shorter time interval, only A.D. 1820 onwards, but differently selected and classified. No attention was paid to legality or to important legal results, such concepts being regarded as varying too much with opinion. Instead attention was directed to deaths caused by quarrelling, with the idea that these are more objective than the rights and wrongs of the quarrel. The wide class of 'deadly quarrels' includes any quarrel that caused death to humans. This class was subdivided according to the number of deaths. For simplicity the deaths on the opposing sides were added together. The size of the subdivisions had to be suited to the uncertainty of the data. The casualties in some fightings are uncertain by a factor of three. It was found in practice that a scale which proceeded by factors of ten was suitable, in the sense that it was like a sieve which retained the reliable part of the data, but let the uncertainties pass through and away. Accordingly the first notion was to divide deadly quarrels into those which caused about J0,000,000 or J,000,000 or 100,000 or 10,000 or 1,000 or 100 or 10 or J deaths. These numbers are more neatly written respectively as 107, 106, 10:1, 10"', 103, 102 , 101, 100 in which the index is the logarithm of the number of deaths. The subsequent discussion is abbreviated by the introduction of a technical term. Let the 'magnitude' of any deadly quarrel be defined to be the logarithm. to the base ten, of the number of persons who died because of that quarrel. The middles of the successive classes are then at magnitudes 7, 6, 5, 4, 3, 2, I, O. To make a clean cut between adjacent classes it is necessary to specify not the middles of the classes, but their edges. Let these edges be at 7·5, 6·5, 5·5, 4·5, 3·5, 2·5 . . . on the scale of magnitude. For example magnitude 3·5 lies between 3,162 and 3, t 63 deaths, magnitude 4·5 lies between 31,622 and 31,623 deaths, magnitude 5·5 lies between 316,227 and 316,228 deaths, and so on. THE DISTRIBUTION OF WARS IN TIME
This aspect of the collections is taken first, not because it is of the most immediate political interest, but almost for the opposite reason, namely, that it is restfully detached from current controversies.
uwl. Ff7 Ric""""
1256
Before beginning to build, I wish to clear three sorts of rubbish away from the site. 1. There is a saying that "If you take the date of the end of the Boer War and add to it the sum of the digits in the date, you obtain the date of the beginning of the next war, thus 1902 + 1 + 9 + 0 + 2 = 1914," Also 1919 + 1 + 9 + J + 9 = 1939. These are merely accidental coincidences. If the Christian calendar were reckoned from the birth of Christ in 4 B.C. then the first sum would be 1906 + 1 + 9 + a + 6 = 1922, not 1914 + 4. 2. There is a saying that "Every generation must have its war." This is an expression of a belief, perhaps well founded, in latent pugnacity. As a statistical idea, however. the duration of a generation is too vague to be serviceable. 3. There is an assertion of a fifty·year period in wars which is attributed by Wright (l942, p. 230) to Mewes in 1896. Wright mentions an explanation by Spengler of this supposed period, thus: "The warrior does not wish to fight again himself and prejUdices his son against war, but the grandsons are taught to think of war as romantic." This is certainly an inter~ting suggestion, but it contradicts the other suggestion that "Every generation must have its war." Moreover the genuineness of the fifty-year period is challenged. Since 1896, when Mewes published, the statisticians have de-veloped strict tests for periodicity (see for example Kendall's Advanced Theory of Statistics, Part II, 1946). These tests have discredited various periods that were formerly believed. In particular the alleged fifty-year period in wars is mentioned by Kendall 2 as an example of a lack of caution. Having thus cleared the site, let us return to Wright's collection as to a quarry of building material. The Distribution of Years in Their Relation to War and Peace A list was made of the calendar years. Against each year was set a mark for every war that began in that year. Thus any year was characterized by the number, x, of wars that began in it. The number, y, of years having the character x was then counted. The resu1ts were as follows. 3 YEARS FROM A.D. 1500, TO A.D. J931 INCLUSIVE. WRIGHT'S COLLECTION. Number, x, of outbreaks in a year . Number. y, of such years. . . . . Y, as defined below .
o
I
2
3
223 216·2
142 149·7
48 51·8
15 12·0
4
4 2·1
>4
Totals
a
432 432·1
O· 3
It is seen that there is some regularity about the progression of the numbers y. Moreover they agree roughly with the numbers Y. These are I 3
Kendall, M. G .• J945. I. Roy. Statistical Soc.• 108, 122. Richardson, L. F., 1945. I. Roy. Statistical Soc., It7, 242.
SlaliSlics 01
1157
D~adb OIUm'~1s
of interest because they are calcu1ated from a well-known formula, called by the name of its discoverer the 'Poisson Distribution' and specified thus N'AlI
y=----(2-7183»). x!
in which N is the whole number of years, 'A is the mean number of outbreaks per year, and x! is caned 'factorial x' and is equal respectively to 1, 1, 2, 6, 24, when x equals 0, 1, 2. 3, 4. Similar results were obtained from Richardson's collection both for the beginnings and for the ends of fatal quarrels in the range of magnitude extending from 3·5 to 4·5, thus: YEARS A.D.
x outbreaks in a year y for war.
Poisson . . for peace Poisson . y
.
.
1820
0 65 64·3 63 63·8
TO
1929
1 35 34·5 35 34·8
INCLUSIVE
2 6 9·3 11 9-S
3 4 1·7 1 1- 7
4 >4 0 0 0·2 0·0 0 0 0·2 0·0
Total 110 110·0 110 110·0
The numbers in the rows beginning with the word 'Poisson' were calculated from the formula already given, in which N and 'A have the same verbal definitions as before, and therefore have appropriately altered numerical values. Such adjustable constants are called parameters. If every falal quarrel had the same duration, then the Poisson distribution for their beginnings would entail a Poisson distribution for their ends; but in fact there is no such rigid connection. The durations are scattered: Spanish America took fourteen years to break free from Spain, but the siege of Bharatpur was over in two months. Therefore the Poisson distributions for war and for peace may reasonably be regarded as separate facts. Observed numbers hardly ever agree perfectly with the formulae that are accepted as representing them. In the paper cited 4 the disagreement with Wright's coUection is examined by the x 2 test and is shown to be unimportant. It should be noted, however, that the application of this standard Xi test involves the tacit assumption that there is such a thing as chance in history. There is much available information about the Poisson distribution; about the theories from which it can be derived; and about the phenomena which are approximately described by it.1i The latter include the distribution of equal time intervals classified according to the number of alpha particles emitted during each by a film of radioactive substance . .. I. Ro)'. Statistical Soc., 101, 242. I) Jeffreys, H .• 1939, Theor), oj Probabilit)'. Oxford University Press. Kendall. M. G., 1943, The Advanced Theory 0/ Statistics. Griffin, London. Shilling, W., 1947, I. A mer. Statistical Assn., 42, 407-24. Cramer, H., 1946. Mathematical Methods oj Statistics, Princeton University Press.
1258
uwis F" Rtclr.rtborr
In order to bring the idea home, an experiment in cookery may be suggested. Take enough flour to make N buns. Add AN currants, where A is a small number such as 3. Add also the other usual ingredients, and mix all thoroughly. Divide the mass into N equal portions, and bake them. When each bun is eaten, count carefully and record the number of currants which it contains. When the record is complete, count the number y of buns, each of which contains exactly x currants. Theory would suggest that y will be found to be nearly equal to Y, as given by the Poisson formula. I do not know whether the experiment has been tried. A more abstract, but much more useful, summary of the relations, is to say that the Poisson distribution of years, follows logically from the hypothesis that there is the same very small probability of an outbreak of war, or of peace, somewhere on the globe on every day. In fact there is a seasonal variation, outbreaks of war having been commoner in summer than in winter. as Q. Wright shows. But when years are counted as wholes, this seasonal effect is averaged out; and then A is such that the probability of a war beginning, or ending, during any short time dr years is Adt. This explanation of the occurrence of wars is certainly far removed from such explanations as ordinarily appear in newspapers, including the protracted and critical negotiations, the inordinate ambition and the hideous perfidy of the opposing statesmen, and the suspect movements of their armed personnel. The two types of explanation are, however, not necessarily contradictory; they can be reconciled by saying that each can separately be true as far as it goes, but cannot be the whole truth. A similar diversity of explanation occurs in regard to marriage: on the one hand we have the impersonal and moderately constant marriage rate; on the other hand we have the intense and fluctuating personal emotions of a love-story; yet both types of description can be true. Those who wish to abolish war need not be discouraged by the persistent recurrence which is described by the Poisson formula. The regularities observed in social phenomena are seldom like the unalterable laws of physical science. The statistics, if we had them, of the sale of snuff or of slaves, would presumably show a persistence during the eighteenth century; yet both habits have now ceased. The existence of a descriptive formula does not necessarily indicate an absence of human control, especially not when the agreement between formula and fact is imperfect. Nevertheless, the Poisson distribution does suggest that the abolition of war is not likely to be easy, and that the League of Nations and its successor the United Nations have taken on a difficult task. In some other flelds of human endeavour there have been long lags between aspiration and achievement. For example Leonardo da Vinci drew in detail a flying
Statistics tJl Dt!adh' Oliarrt!'s
.2S9
machine of graceful appearance. But four centuries of mechanical re· search intervened before flight was achieved. Much of the research that afterwards was applied to aeroplanes was not at first made specifical1y for that object. So it may be with social science and the abolition of war. The Poisson distribution is not predictive; it does not answer such questions as 'when wi)) the present war end?' or 'when will the next war begin?' On the contrary the Poisson distribution draws attention to a persistent probabiHty of change from peace to war, or from war to peace. Discontent with present weather has been cynically exaggerated in a comic rhyme; As a rule a man's a fool: When it's hot he wants it cool, When it's cool he wants it hot, Always wanting what is not. A suggestion made by the Poisson Jaw is that discontent with present circumstances underlies even the high purposes of peace and war. There is plenty of psychological evidence in support. This is not the place to attempt a general review of it; but two illustrations may serve as pointers. In 1877 Britain had not been engaged in any considerable war since the end of the conflict with China in 1860. During the weeks of national excitement in 1877 preluding the dispatch of the British Mediterranean squadron to Ga1lipoli, in order to frustrate Russian designs on Constanti· nople, a bellicose music-hall song with the refrain: 'We don't want to fight, but, by Jingo, if we do: We've got the men, we've got the ships, we've got the money too.' was produced in London and instantly became very popular.s Contrast this with the behaviour of the governments of Britain, China, USA, and USSR in 1944, after years of severe war, but with victory in sight. who then at Dumbarton Oaks officially described themselves as ·peace-Ioving.' 1 Chance in history, The existence of a more or less constant A, a probability per time of change, plainly directs our attention to chance in history. Thus the question which statisticians are accustomed to ask about any sample of people or things. namely "whether the sample is large enough to justify the conclusions which have been drawn from it" must also be asked about any set of wars. Have wars become more frequent? In particular the discussion of any alleged trend towards more or fewer wars is a problem in sampling. No definite conclusion about trend can be drawn from the occurrence of two world wars in the present century, because the sample is too small. When, 6
7
Ency Brit., XIV, ed. 13, 69. H.M. Stationery Office, London, Cmd. 6666.
1260
however, the sample was enlarged by the inclusion of aU the wars in Wright's collection, and the time was divided into two equal intervals, the following result was obtained. Dates of beginning N umbers of wars
A.D.
1500 to 1715 143
A.D.
1716 to 1931 156
The increase from 143 to 156 can be explained away as a chance effect. This was not so for all subdivisions of the time. When the interval from A.D. 1500 to A.D. 1931 was divided into eight consecutive parts of fifty-four years each, it was found that the fluctuation, from part to part, of the number of outbreaks in Wright's collection was too large to be explained away as chance. The extremes were fifty-four outbreaks from A.D. 1824 to 1877, and sixteen outbreaks from A.D. 1716 to 1769. Other irregular fluctuations of A were found, although less definitely, for parts of twenty-seven and nine years.K AU these results may, of course, depend on Wright's selection rules. The problem has been further studied by Moyal. 9 THE LARGER, THE FEWER
When the deadly quarrels in Richardson's collection were counted in units ranges of magnitude, the following distribution was found. 10 The numbers are those of deadly quarrels which ended from A.D. 1820 to 1929 inclusive. Ends of range of magnitude Quarrel-dead at centre of range Number of deadly quarrels
} } }
6+%
5±lh
4 ± 1h
10,000,000
] ,000,000
JOO,ooo
10,000
I
3
]6
62
7 ± 'h
Although Wright's list is not classified by magnitudes, yet some support for the observation that the smaller incidents were the more numerous is provided by his remark (p. 636) that uA list of all revolutions, insurrections, interventions, punitive expeditions, pacifications, and explorations involving the use of armed force would probably be more than ten times as long as the present list." Deadly quarrels that cause few deaths are not in popular language called wars. The usage of the word 'war' is variable and indefinite; but perhaps on the average the customary boundary may be at about 3,000 deaths. From the scientific point of view it would be desirable to extend the above tabular statement to the ranges of magnitude ending at 3 + ~~, 2 ± %, 1 ± %, by collecting the corresponding II !l
J. Roy. Statistical Soc. 107,246-7. Moyal, J. E., 1950. J ROJ. Stali\ticai Soc .• 111,446- 9.
10
Letter in Natwe of 15th November. 1941.
1261
Statlltics of Deadh' Quarrels
numbers of deadly quarrels from the whole world. There is plenty of evidence that such quarrels. involving about 1,000 or 100 or 10, deaths, have existed in large numbers. They are frequently reported in the radio news. Wright alludes to them in the quotation above. M any are briefly mentioned in history books. But it seems not to have been anyone's professional duty to record them systematically. For the range of magnitude between 3·5 and 2·5 I have made a card index for the years A.D. 1820 to 1929 which recently contained 174 incidents, but was still growing. This number 174, though an underestimate, notably exceeds 62 fatal quarrels in the next unit range of larger magnitude, and is thus in accordance with ·the larger the fewer.' Between magnitudes 2·5 and 0·5 the world totals are unknown. Be~ yond this gap in the data are those fatal quarrels which caused 3, 2, or 1 deaths, which are mostly called murders, and which are recorded in criminal statistics. For the murders it is possible to make a rough estimate of the world total in the following manner. Different countries are first compared by expressing the murders per million of population during a year. This 'murder rate' has varied from 6 J 0 for ChiJe 11 in A.D. 1932, to O· 3 for Denmark I:! A.D. 1911-20. The larger countries had middling rates. From various sources, including a governmental report 13 it was estimated that the murder rate for the whole world was of the order of 32 in the interval A.D. J 820 to 1929. As the world population 14 averaged about J ,358 million for the same interval. it follows that the whole number of murders in the world was about 110 X 32 X 1358
= 5 million
This far exceeds the number of small wars in the whole world during the same 110 years. Thus 'the larger, the fewer' is a true description of all the known facts about world totals of fatal quarrels. In the gap where world totals are lacking there are local samples: one of banditry in Manchukuo,l;; and one of ganging in Chicago. I " Before these can be compared with the world totals it is essential that they should be regrouped according to equal ranges ot quarrel-dead or of magnitude; for the maxim 'the larger the fewer' relates to statistics arranged in either of those manners. When thus transformed the statistics of banditry and of ganging fit quite wen with the gradation of the world totals, on certain assumptions. A thorough statistical discussion will be found elsewhere. 17 Keesill/1's Contemporar}' Archil'es, p. 1052. Bristol. Corrected by a factor of ten. Calvert, E. R., 1930, Capital Punishment in the Twentieth Celltur}', Putnam's, London. 13 Select Commillee 011 Capital PUllishment, 1931. H.M.S.O .. London, for reference to which I am indebted to Mr. John Paton. 14 Carr-Saunders. A. M., 1936, World Population, Clarendon Press, Oxford. 15 Japan and Mallchukllo Year Book. 1938, Tokio. 16 Thrasher. F. M., 1927. The Gam:. Chicago University Press. 1'1 Richardson, L. E. 1948, JOIlTlt. Amer. Statistical Assn., Vol. XLIII. pp. 523-46. 11
12
lAwll Fry Rtc1uU'dtoJt
1262
The suggestion is that deadly quarrels of aU magnitudes, from the world wars to the murders, are suitably considered together as forming one wide class, gradated as to magnitude and as to frequency of occurrence. This is a statistical chapter; and for that reason the other very important gradations, legal, social, and ethical, between a world war and a murder are not discussed here. WHICH NATIONS WERE MOST INVOLVED?
This section resembles quinine: it has a bitter taste, but medicinal virtues. The participation of some well-known states in the 278 'wars of modern civilization' as listed by Wright is summarized and discussed by him.18 Over the whole time interval from A.D. 1480 to 1941 the numbers of wars in which the several nations participated were as follows: England (Great Britain) 78, France 71, Spain 64, Russia (USSR) 61, Empire (Austria) 52. Turkey 43, Poland 30, Sweden 26. Savoy (Italy) 25. Prussia (Germany) 23, Netherlands 23. Denmark 20. United States 13, China II, Japan 9. It may be felt that the year 1480 has not much relevance to present-day affairs. So here are the corresponding numbers for the interval A.D. 1850 to 1941. almost within living memory: Great Britain 20, France 18, Savoy (Italy) 12, Russia (USSR) 11, China 10, Spain 10, Turkey 10. Japan 9, Prussia (Germany) 8, USA 7. Austria 6, Poland 5, Netherlands 2. Denmark 2, Sweden O. It would be difficult to reconcile these numbers of wars in which the various nations ~ave participated. with the claim made in 1945 by the Charter of the United Nations 19 to the effect that Britain, France. Russia, China, Turkey, and USA. were 'peace-loving' in contrast with Italy, Japan, and Germany. Some special interpretation of peace-lovingness would be necessary: such as either 'peace-lovingness' at a particular date; or else that 'peace-loving' states participated in many wars in order to preserve world peace. It would be yet more difficult to reconcile the participations found by Wright with the concentration of Lord Vansittart's invective against Germans, as though he thought that Germans were the chief, and the most persistent, cause of war.20 In fact no one nation participated in a majority of the wars in Wright'S list. For the greatest participation was that of England (Great Britain) namely in seventy-eight wars; leaving 200 wars in which England did not Wright, Q., 1942, A Stud}' of War. Chicago University Press, pp. 220 3 and 650. H.M. Stationery Office. London, Cmd. 6666. Articles 3 and 4 together with the list of states represented at t he San Francisco Conference. 20 Vansittart. Sir Robert (now Lord), ]941, Black Record. Hamish Hamilton, London. )8
19
Statistics 01 Drad" Qllarrrls
1263
participate. The distinction between aggression and defence is usually controversial. Nevertheless, it is plain that a nation cannot have been an aggressor in a war in which it did not participate. The conclusion is. there· fore, that no one nation was the aggressor in more than 28 per cent of the wars in Wright's list. Aggression was widespread. This result for wars both civil and external agrees broadly with Sorokin's findings after his wide investigation of internal disturbance. He attended to Ancient Greece, Ancient Rome, and to the long interval A.D. 525 to 1925 in Europe. Having compared different nations in regard to internal violence, Sorokin concluded that 'these results are enough to dissipate the legend of "orderly" and "disorderly" peoples.' . . . 'All nations are orderly and disorderly according to the times.' 21 There does not appear to be much hope of forming a group of permanently peace·loving nations to keep the permanently aggressive nations in subjection; for the reason that peace·lovingness and aggressiveness are not permanent qualities of nations. Instead the facts support Ranyard West's 22 conception of an international order in which a majority of momentarily peace·loving nations. changing kaleidoscopically in its membership, may hope to restrain a changing minority of momentarily aggressive nations. 21 22
Sorokin, Pitirim A., 1937, Social and ClIltliral D}'namics, American Book Co. West, R .. 1942, Comciellce and Society, Methuen, London.
COMMENTARY ON
The Social Application of Mathematics N 1928 John von Neumann reported a curious discovery to the Mathematical Society of Gottingen: he had worked out a rational strategy for matching pennies. This may not strike you as a momentous achievement, but it was the beginning of a new branch of science. The Theory of Games is today regarded as the most promising mathematical tool yet devised for the analysis of man's social relations. Von Neumann's proof, which extended to other and more polite amusements such as chess, cards, backgammon, showed that there is in each case a mathematically determinable "best-possible" method of play. The "best-possible" or "rational" strategy is that which assures a player the maximum advantage, regardless of what his opponents may do. It does not promise him good fortune; it does not insure him against ruin. Its best office is to minimize the maximum loss he can sustain "not in every play of the game, but in the long run." To be sure, the rational strategy indicated for a given game may not always be practicable. Von Neumann's version of chess, for example, is a no-move contest in which the opponents work out their secret strategies in advance, each player specifying the moves he will make under all possible circumstances, and then leaving it to an umpire, on the basis of these schedules, to declare the winner. The calculations required for this dismal affair might take centuries and wear out batteries of electronic computers. Even a simplified version of poker, involving a three-card deck. a one-card, no-draw hand, and two participants, would require for its strategic determination the performance of at least two billion multiplications and additions.l But these limitations are of secondary concern. What is important is that in each case an optimum strategy exists, that the game has what might be called a "solution." By demonstrating this fact Von Neumann introduced several novel and far-reaching concepts to the field of mathematics. The name of this branch of science is perhaps too restricted. The Theory of Games has as much to do with players as with games. Von Neumann was not interested in helping people to play winning bridge nor in proving that only croupiers can safely depend on roulette for their living. He was led to his researches by the conjecture that an analysis of the general structure of games would in itself be of mathematical value;
I
1 Oskar Morgenstern, "The Theory of Games," Scientific American, May 1949, p.23.
1264
Tile Social Appllcatltm 01
MQlh~mQtlcs
1265
but beyond that, he hoped that the solution of certain problems of games might throw light on problems of economic theory. It is evident that games of strategy have many elements in common with "real-life" situations. Decisions must be faced and choices must be made; some issues can be solved in advance by pure reasoning (e.g. in chess). others involve chance elements (e.g. in poker) and require a different treatment; rarely does a p1ayer have sole control of the variables determining the final outcome. Games vary as to the information available to each participant regarding the past actions and resources of his opponents. It may be essential for a player to conceal his strategy; in that case, he must be prepared for the contingency that his plans will become known. Above all, games are pervaded by conflicts of interest: one player cannot win unless another loses and it is plausible to assume that most players want to win. These and other resemb1ances justify the belief that the study of "rational behavior" in games is a fruitful approach to an understanding of rational behavior in social and economic processes. The c1assic work on the subject is Von Neumann and Morgenstern's great treatise Theory of Games and Economic Behavior. Its aim is not mere1y to demonstrate an analogy between the competitive dynamics of games and economics; but to prove "that the typical problems of economic behavior [are] strictly identical with the mathematical notions of suitable games of strategy." 2 The authors do not claim to have formulated a fun·fledged mathematical theory of society; their book treats only of economic problems, and even in this area both theory and applications are in infancy. But it is a vigorous and promising infancy; the theory of games can fairly be said to have 1aid the foundation for a systematic and penetrating mathematicaJ treatment of a vast range of prob1ems in social science. The three papers which fonow illustrate different aspects of this exciting subject. The first is a review of the Von Neumann book by Leonid Hurwicz, Research Professor of Economics and Mathematical Statistics at the University of flIinois. Hurwicz's essay. whiCh appeared in the American Economic Review. is an admirable simplification of some of the main ideas presented by Von Neumann and Morgenstern. The second selection discusses a few elementary applications of the theory to a few simple games-matching pennies. the three-boxes game and so on. The author, Dr. S. Vajda, is a mathematician and physicist, now on the staff of the British Admiralty as a member of the Royal Naval Scientific Service. Dr. Abraham Kaplan, head of the department of philosophy of the University of California at Los Angeles, presents a survey of recent attempts to extend the uses of mathematics to social phenomena. It is a 2 John von Neumann and Oskar Morgcostern, Theory 0/ Gamts and Economic Bthavior. Princeton, 1947, p. 2.
1266
Editor's COMment
readable and succinct essay, affording the reader an opportunity to com· pare several different approaches, of which the theory of games is obviously the most fruitful. The theory of probability and statistics. which made earlier important contributions in this sphere, also originates in the study of games. Pastimes are apparently an inexhaustible source of knowledge about the outside world and society.
I WQme yow wei, it is no chi/des pley.
-CHAUCER
8 The Theory of Econonlic Behavior
1
By LEONID HURWICZ HAD it merely called to our attention the existence and exact nature of certain fundamental gaps in economic theory, the Theory of Games and Economic Behavior by von Neumann and Morgenstern would have been a book of outstanding importance. But it does more than that. It is essentially constructive: where existing theory is considered to be inadequate, the authors put in its place a highly novel analytical apparatus designed to cope with the problem. It would be doing the authors an injustice to say that theirs is a contribution to economics only. The scope of the book is much broader. The techniques applied by the authors in tackling economic problems are of sufficient generality to be valid in political science, sociology. or even military strategy. The applicability to games proper (chess and poker) is obvious from the title. Moreover, the book is of considerable interest from a purely mathematical point of view. This review, however, is in the main confined to the purely economic aspects of the Theory of Games and Economic Behavior. To a considerable extent this review is of an expository nature. This seems justified by the importance of the book, its use of new and un· familiar concepts and its very length which some may find a serious obstacle. The existence of the gap which the book attempts to fill has been known to the economic theorists at least since Cournot's work on duopoly, although even now many do not seem to realize its seriousness. There is no adequate solution of the problem of defining "rational economic behavior" on the part of an individual when the very rationality of his actions depends on the probable behavior of other individuals: in the case of oJigopoly, other sel1ers. Coumot and many after him have attempted to sidetrack the difficulty by assuming that every individual has a definite idea as to what others wi'll do under given conditions. Depending on the nature of this expected behavior of other individuals, we have the special, well-known solutions of Bertrand and Cournot, as well as the more general Bowley concept of the "conjectural variation."!! Thus, the individual's I The tables and figures used in this article were drawn by Mrs. D. Friedlander of Ihe University of Chicago. 2 More recent investigations have led to the idea of a kinked demand curve. This. however, is a special-though very interesting-case of the conjectural variation.
)267
1268
"rational behavior" is determinate i/ the pattern of behavior of "others" can be assumed a prior; known. But the behavior of "others" cannot be known a priori if the "others," too, are to behave rationally! Thus a logical impasse is reached. The way, or at least a way,S out of this difficulty had been pointed out by one of the authors 4 over a decade ago. It lies in the rejection of a narrowly interpreted maximization principle as synonymous with rational behavior. Not that maximization (of utility 5 or profits) would not be desirable if it were feasible, but there can be no true maximization when only one of the several factors which decide the outcome (of, say, oligopolistic competition) is controlled by the given individual. Consider, for instance, a duopolistic situation 6 where each one of the duopolists A and B is trying to maximize his profits. A's profits will depend not only on his behavior ("strategy") but on B's strategy as well. Thus, i/ A could control (directly or indirectly) the strategy to be adopted by B, he would select a strategy for himself and one for B so as to maximize his own profits. But he cannot select B's strategy. Therefore, he can in no way make sure that by a proper choice of his own strategy his profits will actually be unconditionally maximized. It might seem that in such a situation there is no possibility of defining rational behavior on the part of the two duopolists. But it is here that the novel solution proposed by the authors comes in. An example will illustrate this. Suppose each of the duopolists has three possible strategies at his disposaJ.1 Denote the strategies open to duopolist A by AI' A 2 , and Aa, and those open to duopolist B by B1, B2• and Ba. The profit made by A, to be denoted by a, obviously is determined by the choices of strategy made by the two duopolists. This dependence wi11 be indicated by subscripts attached to a, with the first subscript referring to A's strategy and the second subscript to that of B; thus, e.g., al3 is the profit which will be made by A if he chooses strategy Al while B chooses the strategy Bs. Similarly, bl3 would denote the profits by B under the same circumstances. :) C/. reference to von Stackelberg in footnote 16 and some of the work quoted by von Stackelberg, op. cit. 4 J. von Neumann, "Zur Theorie der Gesellschaftsspiele," Math. Annalen (1928). 6 A side-issue of considerable interest discussed in the Theory 0/ Games is that of measurability of the utility function. The authors need measurability in order to be able to set up tables of the type to be presented later in the case where utility rather than profit is being maximized. The proof of measurability is not given; however, an article giving the proof is promised for the near future and it seems advisable to postpone comment until the proof appears. But it should be emphasized that the validity of the core of the Theory 0/ Games is by no means dependent on measurability or transferabiJity of the utilities and those who feel strongly on the subject would perhaps do best to substitute "profits" for "utility" in most of the book in order to avoid jUdging the achievements of the Theory 0/ Games from the point of view of an unessential assumption. 6 It is assumed that the buyer's behavior may be regarded as known. 'I Actually the number of strategies could be very high. perhaps infinite.
1269
Tile Theory of Eeonomic' Behavior
TABLE IB
TABLE lA
B'. Profit.
A'. Profit.
~
..... e.
-."".." ~
B.
AI
0 ..
011
0,.
A,
A,
oZ,
Ou
On
A,
0,.
OR
°D
ofJ~
tf
B.
B,
A',
n,.,.
B.
B,
bu
bl'
bl'
At
btl
bat
btl
As
bit
bR
bD
'=:::'iH
... '....a
The possible outcomes of the "duopoJistic competition" may be represented in the following two tables: Table IA shows the profits A will make depending on his own and D's choice of strategies. The first row corresponds to the choice of At, etc.; columns correspond to B's strategies. Table 1B gives analogous infonnation regarding D's profits. In order to show how A and B will make decisions concerning strategies we shan avail ourselves of a numerical example given in Tables 2A and 2B. TABLE 28
TA8LE 2A
............
A $ Profits
-
8'5 Profits
~.:.:... "
~~ I'...."
6.
6,
6,
I~
B.
Bz
6,
A.
2
8
I
AI
11
2
20
AI
4
3
9
Az
9
15
3
A,
5
6
7
AI
6
7
6
"' .....
A',
'.1" of nflt"""
Now let us watch A's thinking processes as he considers his choice of strategy. First of aU, he will notice that by choosing strategy A:J he will be sure that his profits cannot go down below S, while either of the remaining alternatives would expose him to the danger of going down to 3 or even to J. But there is another reason for his choosing A:i • Suppose there is a danger of a "leak": B might learn what A's decision is before he makes his own. Had A chosen. say, AT' B-if he knew about this-would obviously choose D~ so as to maximize his own profits; this would leave
1270
A with a profit of only 1. Had A chosen A 2 , B would respond by selecting B2 , which again would leave A with a profit below 5 which he could be sure of getting if he chose As. One might perhaps argue whether A's choice of As under such circumstances is the only way of defining rational behavior, but it certainly is a way of accomplishing this and, as will be seen later, a very fruitful one. The reader will verify without difficulty that similar reasoning on B's part will make him choose Bl as the optimal strategy. Thus, the outcome of TABLE 3B
TABU 3A
-. I~-
A's Profits
-
,,-
S's Profits
-
BI
Bz
B~
~ =.:!'
B.
B.
B,
AI
2
B
1
AI
8
2
9
A.
4
3
9
Az
6
7
I
As
5
6
7
As
5
4
3
A'I
-"
A'I
the duopolistic competition is determinate and can be described as follows: A will choose Aa, B will choose Bit A's profit will be 5, B's 8. An interesting property of this solution is that neither duopolist would be inclined to alter his decision, even if he were able to do so, after he found out what the other mail's strategy was. To see this, suppose B has found out that A's decision was in favor of strategy Aa. Looking at the third row of Table 2B, he will immediately see that in no case could he do better than by choosing BIt which gives him the highest profit consistent with A's choice of Aa. The solution arrived at is of a very stable nature, independent of finding out the other man's strategy. But the above example is artificial in several important respects. For one thing, it ignores the possibility of a "collusion or, to use a more neutral term, coalition between A and B. In our solution, yielding the strategy combination (Aa, B1 ), the joint profits of the two duopolists amount to 13; they could do better than that by acting together. By agreeing to choose the strategies Al and Bs respectively, they would bring their joint profits up to 21; this sum could then be so divided that both would be better off than under the previous solution. A major achievement of the Theory 0/ Games is the analysis of the conditions and nature of coalition formation. How that is done will be H
1271
Tltt! 1ltt!01', 01 Economic Bt!ltal'ior
shown below. But, for the moment, let us eliminate the problem of coalitions by considering a case which is somewhat special but nevertheless of great theoretical interest: the case of constant sum profits. An example of such a case is given in Tables 3A and 3B. Table 3A is identical with Table 2A. But figures in Table 3B have been selected in such a manner that the joint profits of the two duopolists always amount to the same (10), no matter what strategies have been chosen. In such a case, A's gain is B's loss and vice versa. Hence, it is intuitively obvious (although the authors take great pains to show it rigorously) that no coalition will be formed. The solution can again be obtained by reasoning used in the previous case and it will again tum out to be (Aa, B1 ) with the respective profits Sand S adding up to 10. What was said above about stability of solution and absence of advantage in finding the opponent 8 out stm applies, There is, however, an element of artificiality in the example chosen that is responsible for the determinateness of the solution. To see this it TABLE 4
-
--... ~
A'S Profifs
o f _•
of
B,
BI
8.
A,
2
8
1
Az
4
3
9
As
6
5
7
will suffice to interchange 5 and 6 in Table 3A. The changed situation is portrayed in Table 4 which gives A's profits for different choices of strategies.9 There is no solution now which would possess the kind of stability found in the earlier example. For suppose A again chooses As; then if B should find that out, he would obviously "play" B2 which gives him the highest possible profit consistent with As. But then As would no longer be A's optimum strategy: he could do much better by choosing AI; but if he does so, B's optimum strategy is Bs, not B2, etc. There is no solution 8 In this case the interests of the two duopolists are diametricaJJy opposed and the term "opponents" is fuDy justified; in the previous example it would not have been. 9 The table for 8's profits is omitted because of the constant sum assumption. Clearly, in the constant sum case, 8 may be regarded as minimizing A's profits since this implies maximization of his own.
1272
which would not give at least one of the opponents an incentive to change his decision if he found the other man out! There is no stabiJity.l0 What is it in the construction of the table that insured determinateness in the case of Table 3 and made it impossible in Table 41 The answer is that Table 3 has a saddle point ("minimax") while Table 4 does not. The saddle point has the following two properties: it is the highest of all the row minima and at the same time it is lowest of the column maxima. Thus, in Table 3a the row minima are respectively I, 3, and 5, the last one being highest among them (Maximum Minimorum); on the other hand, the column maxima are respectively 5, 8, and 9 with 5 as the lowest (Minimum Maximorum). Hence the combination (A:i' B.) yields both the highest row minimum and the lowest column maximum, and, therefore, constitutes a saddle point. It is easy to see that Table 4 does not possess a saddle point. Here 5 is still the Maximum Minimorum, but the Minimum Maximorum is given by 6; the two do not coincide, and it is the absence of the saddle point that makes for indeterminateness in Table 4. Why is the existence of a unique saddle point necessary (as welt as sufficient) to insure the determinateness of the solution? The answer is inherent in the reasoning used in connection with the earlier examples: if A chooses his strategy so as to be protected in case of any leakage of information concerning his decision, he will choose the strategy whose row in the table has the highest minimum value, i.e., the row corresponding to the Maximum Minimorum-A s in case of Table 4-for then he is sure he will not get less than 5, even if B should learn of this decision. B, following the same principle, will choose the column (i.e., strategy) corresponding to the Minimum Maximorum-B J in Table 4-thus making sure he will get at least 4. even if the information does leak out. In this fashion both duopolists are sure of a certain minimum of profit -5 and 4, respectively. But this adds up to only 9. The residual-lis still to be allocated and this allocation depends on outguessing the opponent. It is this residual that provides an explanation, as wen as a measure, of the extent of indeterminacy. Its presence will not surprise economists familiar with this type of phenomenon from the theory of bilateral monopoly. But there are cases when this residual does equal zero, that is. when the Minimum Maximorum equals the Maximum Minimorum, which (by definition) implies the existence of the saddle point and complete determinacy. At this stage the authors of the Theory 0/ Games had to make a choice. to There is. however. a certain amount of determinateness, at least in the negative sense, since certain strategy combinations are excluded: e g. (A J • BI); A would never choose A:! if he knew B had chosen B,. and \"ice versa.
n.. TlltD'Y DI Economic Btlulllior
1273
They could have accepted the fact that saddle points do not always exist so that a certain amount of indeterminacy would. in general, be present. They preferred. however. to get rid of the indeterminacy by a highly ingenious modification of the process which leads to the choice of appropriate strategy. So far our picture of the duopolist making a decision on strategy was that of a man reasoning out which of the several possible courses of action is most favorable ("pure strategy"). We now change this picture and put in his hands a set of dice which he will throw to determi ne the strategy to be chosen. Thus, an element of chance is introduced into decision making ("mixed strategy").ll But not everything is left to chance. The duopolist A must in advance formulate a rule as to what results of the TABLE .5
A's Prorits
..._.. I~ 01-
B.
~
A,
5
3
A.
I
5
1
'y,." ...UMUII ."III.ORUM
throw-assume that just one die is thrown-would make him choose a given strategy. In order to illustrate this we shall use a table that is somewhat simpler, even if less interesting than those used previously. In this new table (Table 5) 12 each duopolist has only two strategies at his disposal. 11 The authors' justification for introducing "mixed strategies" is that leaving one's decision to chance is an effective way of preventing "leakage" of information since the individual making the decision does not himself know which strategy he will choose• • 8 In Table 5 there is no saddle point.
uonld H""wlcr.
1274
An example of a rule A might adopt would be: If the result of the throw is 1 or 2, choose AI;
if the result of the throw is 3_ 4, 5_ or 6, choose A:!. If this rule were followed, the probability that A will chose Al is ;~, that of his choosing A2 is %. If a different rule had been decided upon (say,
one of choosing Al whenever the result of the throw is 1, 2, or 3), the probability of choosing A1 would have been ¥.!. Let us call the fraction giving the probability of choosing Al A's chance coefjiciem; in the two examples, A's chance coefficients were % and ¥.! respectively. III As a special case the value of the chance coefficient might be zero (meaning, that is, definitely choosing strategy A;!) or one (meaning that A is definitely choosing strategy AI); thus in a sense "pure strategies" may be regarded as a special case of mixed strategies. However. this last stateTABLE 6
Mothematical
_0.
.... ...-_. ~ -0
0
Expectations I
1
of
1
A's
Profits
ROW
Mun ...
5
M.N.MU .. M"II.MORUlil
ment is subject to rather important qualifications which are of a complex nature and will not be given here. Now instead of choosing one of the available strategies the duopolist A must choose the optimal (in a sense not yet defined) chance coefficient. t:1 Since the probability of choosing AJ is always equal to one minus that of choosing AI> specification or the probability of choosing AI is sufficient to describe a given rule. However, when the number of available strategies exceeds two, there are several such chance coefficients to be specified.
1275
How is the choice of the chance coefficient made? The answer ties in constructing a table which differs in two important respects from those used earlier. Table 6 provides an example. Each row in the table now corresponds to a possible value of A's chance coefficient; similarly, columns correspond to possible values of 8's chance coefficient. Since the chance coefficient may assume any value between zero and one (inc1uding the latter two values), the table is to be regarded merely as a "sample." This is indicated by spaces between rows and between colum ns. The numbers entered in the table are the average values (mathematical expectations) corresponding to the choice of chance coefficients indicated by the row and column .... (One should mention that Table 6 is on1y an expository device: the actual procedures used in the book are algebraic and much simpler computationally.) If we now assume with the authors that each duopolist is trying to maximize the mathematical expectation of his profits (Table 6) rather than the profits themselves (Table 5), it might seem that the original source of difficulty remains if a saddle point does not happen to exist. But the mixed strategies were not introduced in vain! Jt is shown (the theorem was originally proved by von Neumann in 1928) that in the table of mathematical expectations (like Table 6) a saddle point must exist; the problem is always determinate. 11i The reader who may have viewed the introduction of dice into the decision-making process with a certain amount of suspicion will probably agree that this is a rather spectacular result. Contrary to the initial impression, it is possible to render the problem determinate. But there is a TABLE 7
~"'''IU "'It.. • n
B.
Bz
...... ~
l'
l'
A.
l'
5
3
A,
2
1
5
A's '''I(t'
0'
str&tf'9i t\
,Hffl. d .....
11',
2
COMPUTATION or THE MATHEMATICAL EXPECTATION FOa THE 2ND ItOW, 3aD COLUMN IN TABLE 6
I
... ft ......
,
3'
IJ..X%XS+JI.sX~XJ
+%X%Xl+%XY.tXS =:.
2% =
l
14 To see this we shall show how, e.K., we have obtained the value in the second row and third column of Table 5 (\';;0;" 3). We construct an auxiliary table valid only for this particular combination of chance coefficients (A's Vol, D's ~t). This table differs from Table 5 only by the omission of row maxima and column minima and by the insertion of the probabilities of choosing the available strategie" corresponding to the second row third column of Table 6. The computation of the mathematical expectation is indicated in Table 6. t'S In Table 6 the saddle point is in the third row second column: it io,; to be stre'tsed that Table 5 has no saddle point.
1176
L,ollld HurwiC'%
price to be paid: acceptance of mixed strategies, assumption that only the mathematical expectation of profit (not its variance, for instance) matters, seem to be necessary. Many an economist will consider the price too high. Moreover, one might question the need for introducing determinateness into a problem of this nature. Perhaps we should consider as the "solution" the interval of indeterminacy given by the two critical points: the Minimum Maximorum and Maximum Minimorum. As indicated earlier in this review, one should not ignore, in general, the possibility of a collusion. This is especially evident when more complex economic situations are considered. We might, for instance, have a situation where there are two sellers facing two buyers. Here a "coalition" of buyers, as well as one of sellers, may be formed. But it is also conceivable that a buyer would bribe a seller into some sort of cooperation against the other two participants. Several other combinations of this type can easily be found. When only two persons enter the picture, as in the case of duopoly (where the role of buyers was ignored), it was seen that a coalition would not be formed if the sum of the two persons' profits remained constant. But when the number of participants is three or more, subcoalitions can profitably be formed even if the sum of all participants' profits is constant; in the above four-person example it might pay the sellers to combine against the buyers even if (or, perhaps, especialJy if) the profits of all four always add to the same amount. Hence, the formation of coalitions may be adequately treated without abandoning the highly convenient constant-sum assumption. Tn fact, when the sum is known to be non-constant, it is possible to introduce (conceptually) an additional fictitious participant who, by definition, loses what all the reaJ participants gain and vice versa. In this fashion a non-constant sum situation involving, say, three persons may be considered as a special case of a constant-sum four-person situation. This is an additional justification for confining most of the discussion (both in the book and in the review) to the constant-sum case despite the fact that economic problems are as a rule of the non-constant sum variety. We shall now proceed to study the simplest constant-sum case which admits coalition formation, that involving three participants. The technique of analysis presented earlier in the two-person case is no longer adequate. The number of possibilities increases rapidly. Each of the participants may be acting independently; or e1se, one of the three possible two-person coalitions (A and B vs. C, A and C vr. B, Band C vs. A) may be formed. Were it not for the constant-sum restriction, there would be the additional possibility of the coalition comprising all three participants. Here again we reaHze the novel character of the authors' approach to
1277
the problem. In most HJ of traditiona1 economic theory the formation--or absence--of specific coalitions is postulated. Thus, for instance, we discuss the economics of a cartel without rigorously investigating the necessary and sufficient conditions for its formation. Moreover, we tend to exclude CI priori such phenomena as collusion between buyers and sellers- even if these phenomena are known to occur in practice. The Theory 0/ Games, though seemingly more abstract than economic theory known to us, approaches reality much more closely on points of this nature. A complete solution to the problems of economic theory requires an answer to the question of coalition formation, bribery, collusion, etc. This answer is now provided. even though it is of a somewhat formal nature in the more complex cases; and even though it does not always give sufficient insight into the actual workings of the market. Let us now return to the case of three participants. Suppose two of them are sellers. one a buyer. Traditional theory would tell us the quantity sold by each seller and the price. But we know that in the process of bargaining one of the sellers might bribe the other one into staying out of the competition. Hence the seller who refrained from market operations would make a profit; on the other hand. the nominal profit made by the man who did make the sale would exceed (by the amount of bribe) the actual gain made. It is convenient, therefore, to introduce the concept of gain: the bribed man's gain is the amount of the bribe, the seller's gain is the profit made on a sale minus the bribe. etc. A given distribution of gains among the participants is called an imputation. The imputation is not a number: it is a set of numbers. For instance, if the gains of the participants in a given situation were g.j. gil. gr. it is the set of 'these three g's that is caned the imputation. The imputation summarizes the outcome of the economic process. In any given situation there are a great many possib1e imputa· tions. Therefore. one of the chief objectives of economic theory is that of finding tI.ose among all the possible imputations which will actua]]y be observed under rational behavior. In a situation such as that described (three participants, constant-sum) each man will start by asking himself how much he could get acting inde· pendently, even if the worst should happen and the other two formed a coalition against him. He can determine this by treating the situation as a two-person case (the opposing coalition regarded as one person) and finding the relevant Maximum Minimorum, or the saddle point, if that 16 In his Grll1uilagell eiuer reil1eu Koslelllheorie (Vienna. 1932) H. von Stackelberg does point out (p. 89) that "the competitors rduopolistsl must somehow unite; they must supplement the economic mechanics. which in this case is inadequate. by economic politics. But no rigorous theory is developed for such situations (although an outline of possible developments is given). This is where the Theory of Game.v has made real progress.
t.
1278
point does exist; tbe saddle point would, of course, exist if "mixed strate~ gies" are used. Next, tbe participant will consider tbe possibility of forming a coalition witb one of tbe otber two men. Now comes the crucial question: under what conditions might sucb a coalition be formed? Before discussing tbis in detail, let us summarize, in Table 8, all the relevant information. It will be noted that under imputation # 1, Band C are eacb better oft than if tbey bad been acting individuaUy: they get respectively 8.3 and 10.2 instead of 7 and 10. Hence, there is an incentive for Band C to form TABLB 8
5 7 10.
I. If A acts alone, he can get If B acts alone, be can get If C acts alone, be can get
II. If A and B form a coalition, tbey can get If A and C form a coalition, tbey can get If Band C form a coalition, they can get
15 18 20,
III. If A, B, and C act togetber, tbey can get
25.
Among the many possible imputations, let us now consider tbe tbree given in Table 9. TABLB 9
#1 #2
#3
A
6.5 5.0
4.0
B 8.3 9.5 10.0
C
10,2 10.5 11.0
a coalition since without sucb a coalition imputation # 1 would not be possible. But once tbe coalition is formed, tbey can do better than under # 1; viz., under #2, wbere each gets more (9.5 and 10.5 instead of 8.3 and 10.2, respectively). In sucb a case we say that imputation #2 dominates imputation # 1. It might seem that #3, in turn, dominates #2 since it promises still more to botb Band C. But it promises too much: the sum of B's and e's gains under # 3 is 21, which is more than their coalition could get (c/. Table 8)! Thus #3 is ruled out as unrealistic and cannot be said to dominate any other imputation. Domination is an exceptionally interesting type of relation. For one thing, it is not transitive: we may have an imputation i I dominating the imputation ;2 and is dominating i.1' without thereby implying that ;1 domi-
7'M
neo" of Economic
1279
BellGvio,.
nates i3; in fact, il might be dominated by i3. 17 Moreover, it is easy to construct examples of, say, two imputations, neither of which dominates the other one.18 To get a geometric picture of this somewhat unusual situation one may turn to Figure 1, where points on the circle represent different possible imputations. (The reader must be cautioned that this is merely a geometrical analogy, though a helpful one.) Let us now say that point #1 dominates point #2 if #2 is less than 90° (clockwise) from # 1. It is easy to see in Figure 1 that # 1 dominates #2 and #2 dominates #3, but in spite of that, # 1 does not dominate #3. This geometrical picture will help define the very fundamental concept of a solution. Consider the points (imputations) #1, 3, 5 and 7 in Figure 1. None of them dominates any other since any two are either exactly or more than 90° apart. But any other point on the circle is dominated by at 1east (in this case: exactly) one of them: an points between #1 and #3 are dominated by #1, etc. There is no point on the circle which is not dominated
FlGUKE 1
by one of the above four points. Now we define a solution as a set of points (imputations) with two properties: (l) no element of the set dominates any other element of the set, and (2) any point outside the set must be dominated by at least one element within the set. We have seen that the points # 1, 3, 5, 7 do have both of these proper1'7 I.e., domination may be a cyclic relation. For instance, consider the following three imputations in the above problem: #1 and #2 as in Table 9, and #4, where #4
A
B
6.0
7.0
C 12.0.
Here #2 (as shown before) dominates #1 (for coalition B, C), #4 dominates #2 (for coalition A, C), but at the same time #1 dominates #4 (for the coalition A, B): the cycle is completed. 18 For inst8nce, #2 and #3 in Table 9.
l280
ties; hence, the four points together form a solution. It is important to see that none of the individual points by itself can be regarded as a solution. In fact, if we tried to leave out anyone of the four points of the set, the remaining three would no longer form a solution; for instance, if # 1 were left out, the points between #1 and #3 are not dominated by any of the points #3, 5, 7. This violates the second property required of a solution and the three points by themselves are not a solution. On the other hand, if a fifth point were added to #1, 3, 5, 7, the resulting five element set would not form a solution either; suppose #2 is the fifth point chosen; we note that #2 is dominated by #1 and it also dominates #3. Thus, the first property of a solution is absent. Contrary to what would be one's intuitive guess, an element of the solution may be dominated by points outside the solution: # 1 is dominated. by #8, etc. There can easily be more than one solution. The reader should have no trouble verifying the fact that #2, 4, 6, 8 also form a solution, and it is clear that infinitely many other solutions exist. Does there always exist at least one solution? So far this question remains unanswered. Among the cases examined by the authors none has been found without at least one solution. But it has not yet been proved that there must always be a solution. To see the theoretical possibility of a case without a solution we shall redefine slightly our concept of domination (ct. Figure 2): # ] dominates #2 if the angle between them (measured clockwise) does not exceed 180 0 • Hence, in Figure 2 point # 1 dominates #3, but not #4, etc. It can now be shown that in this case no solution exists. For suppose there is one; then we may, without loss of generality, choose # 1 as one of its points. Clearly, # 1 by itself does not constitute a solution, for there are points
-3 FIGURE 2
Th" Theo", oj Economic B"havlor
1281
on the circle (e.g., #4) not dominated by # 1; thus the solution must have at least two points. But any other point on the circle either is dominated by #1 (e.g., #2), or it dominates #1 (e.g., #4), or both (#3). which contradicts the first requirement for the elements of a solution. Hence there is no solution consisting of two points either. A fortiori. there are no solutions containing more than two points. Hence we have been able to construct an example without a solution. But whether this type of situation could arise in economics (or in games, for that matter) is still an open question. Now for the economic interpretation of the concept of solution. Within the solution there is no reason for switching from one imputation to another since they do not dominate each other. Moreover, there is never a good reason for going outside a given solution: any imputation outside the solution can be "discredited" by an imputation within the solution which dominates the one outside. But, as we have seen, the reverse is also usually true: imputations within the solution may be dominated by those outside. If we are to assume that the latter consideration is ignored, the given solution acquires an institutional, if not accidental, character. According to the authors, a solution may be equivalent to what one would call the "standards of behavior" which are accepted by a given community. The multiplicity of solutions can then be considered as corresponding to alternative institutional setups; for a given institutional framework only one solution would be relevant. But even then a large number of possibilities remains since, in general, a solution contains more than one imputation. More indeterminacy yet would be present if we had refrained from introducing mixed strategies. It would be surprising, therefore, if in their applications von Neumann and Morgenstern should get no more than the classical results without discovering imputations hitherto neglected or ignored. And there are some rather interesting "unorthodox" results pointed out, especially in the last chapter of the book. In one case, at least, the authors' claim to generality exceeding that of economic theory is not altogether justified in view of the more recent literature. That is the case of what essentially corresponds to bilateral monopoly (p. 564, proposition 61 :C). The authors obtain (by using their newly developed methods) a certain interval of indeterminacy for the price; this interval is wider than that indicated by Bohm-Bawerk, because (as the authors themselves point out) of the dropping of Bohm-Bawerk·s assumption of a unique price. But this assumption has been abandoned, to give only one example, in the theories of consumer's surplus. with analogous extension of the price interval.
1282
Laollid Hurwlcz.
It will stand repeating, however, that the Theory 0/ Games does offer a greater generality of approach than could be attained otherwise. The existence of "discriminatory" solutions, discovered by purely analytical methods, is an instance of this. Also, the possibility of accounting for various types of deals and collusions mentioned earlier in connection with the three-person and four-person cases go far beyond results usuaUy obtained by customarily used methods and techniques of economic theory. The potentialities of von Neumann's and Morgenstern's new approach seem tremendous and may, one hopes, lead to revamping, and enriching in realism, a good deal of economic theory. But to a large extent they are only potentialities: results are still largely a matter of future developments. The difficultjes encountered in handling, even by the more powerful mathematical methods, the situations involving more than three persons are quite formidable. Even the problems of monopoly and monopsony are beyond reach at the present stage of investigation. The same is true of perfect competition, though it may turn out that the latter is not a "legitimate" solution since it excludes the formation of coatitions which may dominate the competitive imputations. A good deal of light has been thrown on the problem of oligopoly. but there again the results are far from the degree of concreteness desired by the economic theorist. The reviewer therefore regards as somewhat regrettable some of the statements made in the initial chapter of the book attacking (rather indiscriminately) the analytical techniques at present used by the economic theorists. True enough, the deficiencies of economic theory pointed out in the Theory 0/ Games are very real; nothing would be more welcome than a model giving the general properties of a system with, say, m sel1ers and n buyers, so that monopoly, duopoly, or perfect competition could simply be treated as special cases of the general analysis. Unfortunately. however, such a model is not yet in sight. In its absence less satisfactory, but still highly useful, models have been and no doubt will continue to be used by economic theorists. One can hardly afford to ignore the social need for the results of economic theory even if the best is rather crude. The fact that the theory of economic fiuctuations has been studied as much as it has is not a proof of "bow much the attendant difficulties have been underestimated" ( p. 5). Rather it shows that economics cannot afford the luxury of developing in the theoretically most "logical" manner when the need for the resuJts is as strong as it happens to be in the case of the ups and downs of the employment level! Nor is it quite certain, though of course conceivable, that, when a rigorous theory developed along the lines suggested by von Neumann and Morgenstern is available, the results obtained in the important problems will be sufficiently remote from those obtained with the help of the current (admittedly imperfect) tools 10 justify some of the harsher accusations
1283
to be found in the opening chapter of the book. It must not be forgotten, for instance. that, while theoretical derivation of coalitions to be formed is of great value, we do have empirical knowledge which can be used as a substitute (again imperfect) for theory. For example. cartel formation may be so clearly "in the cards" in a given situation that the economic theorist will simply include it as one of his assumptions while von Neumann and Morgenstern would (at least in principle) be able to prove the formation of the cartel without making it an additional (and logically unnecessary) assumption. The authors criticize applications of the mathematical methods to economics in a way which might almost, in spite of protests to the contrary, mislead some readers into thinking that von Neumann and Morgenstern are not aware of the amount of recent progress in many fields of economic theory due largely to the use of mathematical tools. They also seem to ignore the fact that economics deveJoped in literary form is, implicitly, based on the mathematical techniques which the authors criticize. (Thus it is not the methods of mathematical economics they are really questioning, but rather those elements of economic theory which literary and mathematical economics have in common.) While it is true that even mathematical treatment is not always sufficiently rigorous. it is as a rule more so than the corresponding literary form, even though the latter is not infrequently more realistic in important respects. There is little doubt in the reviewer's mind that nothing could have been further from the authors' intentions than to give aid and comfort to the opponents of rigorous thinking in economics or to increase their complacency. Yet such may be the effect of some of the vague criticisms contained in the first chapter; they hardly seem worthy of the constructive achievements of the rest of the book, Economists will probably be surprised to find so few references to mOl'e recent economic writings. One might almost form the impression that economics is synonymous with Bohm-Bawerk plus Pareto. Neither the nineteenth century pioneers (such as Cournot) nor the writers of the last few decades (Chamberlin, Joan Robinson, Frisch, Stackelberg) are even alluded to. But. perhaps, the authors are entitled to claim exemption from the task of relating their work to that of their predecessors by virtue of the tremendous amount of constructive effort they put into their opus, One cannot but admire the audacity of vision, the perseverance in details, and the depth of thought displayed on almost every page of the book. The exposition is remarkably lucid and fascinating, no matter how involved the argument happens to be. The authors made an effort to avoid the assumption that the reader is familiar with any but the more elementary parts of mathematics; more refined tools are forged "on the spot" whenever needed.
1284
uDntd Hurwtcz
One should also mention, though this transcends the scope of the review. that in the realm of strategic games proper (chess. poker) the results obtained are more specific than some of the economic applications. Those interested in the nature of determinacy of chess, in the theory of "bluffing" in poker, or in the proper strategy for Sherlock Holmes in his famous encounter with Professor Moriarty. wi1l enjoy reading the sections of the book which have no direct bearing on economics. The reader's views on optimum military or diplomatic strategies are also Hkely to be affected. Thus, the reading of the book is a treat as wen as a stage in one's intellectual development. The great majority of economists should be able to go through the book even if the going is slow at times; it is well worth the effort. The appearance of a book of the caliber of the Theory 01 Games is indeed a rare event.
Heads I win; tails you lose.
-ENGLISH SAYING,
II is a silly game where 1I0body wins.
-THOMAS FULLER
171h century. (Gnomolog;a)
II has long been all axiom oj mille Ihal the lillie things are infinitely the mostlmporlaIPI.-SIR ARTHUR CONAN DOYLE (The Advenlures 0/ Sherlock Holmes. A Case o/Identity)
9
Theory of Games By S. VAJDA
FROM a variety of considerations which, not very long ago, would have seemed in no way to be connected, a new branch of science has emerged: the Theory of Games. The origins of this theory go back to 1928, when John von Neumann read a paper to the Mathematical Society of Gottingen, introducing new and unorthodox concepts. During the last war Professors von Neumann and Morgenstern published their monumental and painstaking treatise Theory 01 Games and Economic Behavior, and scientists concerned with operational research on both sides of the Atlantic investigated and developed various theoretical aspects of tactics and strategy. The theory does not help one to become proficient in any specific games. It does not tabulate chess openings or advise on poker biddings. It is concerned, not with any particular game, but rather with general aspects applicable to all games, and with processes which obtain a special significance when a long succession of plays is being considered. Such conclusions as that you cannot win at Monte Carlo in the long run may emerge as a trivial consequence of the theory. but statistics is not of its essence. Being a mathematical theory. it draws on the results of several branches of mathematics, such as algebra and measure theory. but entirely new concepts had to be created as wen, and it is in these that the particular attraction of the subject lies. The essential feature of any game is the fact that one has to do with one or more opponents and that, therefore, only some of the relevant variables are under the control of any single player. It is clear that a theory which takes account of this peculiarity can be appJied to the analysis of warfare, to economic problems, and even to decisions which must be taken where no specific opponent appears to exist, but where all variables outside one's one control are dependent on 'Chance' or on 'Laws of Nature.' It is the aim of this article to introduce the reader to some of the new concepts and to show how they emerge naturally from a consideration of typical features of games. 1285
1286
MATCHING PENNIES
To begin with, let us consider one of the simplest games imaginable, that of 'Matching Pennies.' Two players put down a penny each, either head or tail up, unknown to the opponent. They then uncover their coins and A receives. his own and also B·s penny, if both coins show the same side. Otherwise B collects both pennies. Clearly it is A·s aim to show the same side of the penny as B does t and all his decisions connected with the game depend on this aim of his and on his ignorance of B's procedure. It is convenient to introduce here a few definitions. There are two persons to play Matching Pennies and one person wins what the other loses. In other words, the gains and losses of the two players are balanced. Such a game is called a zero-sum two-person game. By a 'game' we understand in this context the aggregate of rules which set out the possible behaviour of the players and their gains and losses. Thus 'game' is an abstract concept. One particular instance of it, as it is actually performed, is called a 'play" consisting of a .set of 'moves.' Matching Pennies exhibits an extreme simpJicity because it finishes after only one move of each player. In more complicated games one imagines, as a rule, that the players decide on their own moves as the play develops. This is, however, not essential. One could equal1y wen imagine that each player decides before the play begins what 'strategy' he wants to apply. Such a strategy determines in advance what moves the player would make under al1 conceivable circumstances. One could imagine that each player chooses his strategy unknown to the other player and informs an umpire of his choice. The latter would then consult a list containing all possible pairs of strategies of the two players and would read out the result. It would not be necessary to play any more. Fortunately the number of possible strategies in such sophisticated games as, for instance, chess is so enormously large that. there is no danger that their complete enumeration and evaluation will ever kill the interest in the game. Reverting to Matching Pennies, we can construct a table showing the payments which A receives and how they depend on the possible outcomes of a play. Such a table is referred to as the 'pay-off table' (for A) and looks as follows: TABLE I: PAY-OFF TABLE FOR MATCHING PENNIES
B's coin
A's coin
Head Tail
Head
I Tail
1
-1
-1
1
1287
Tlleory of Games
It is clearly impossible for either player to choose his move or his strategy
(the two concepts are equivalent in this particular game) in such a way that he can be sure of winning. On the other hand, any player who knows the opponent's move can win. It would be very bad policy to stick to any particular move in a succession of games, since this would soon be noticed. It is therefore natural to suggest that a player should change his strategies at random, i.e., in a manner that does not give any indication of the strat· egy to be used in the next play. He can still decide on the over·all propor· tions in which his strategies should be used in the long run. It is, of course, also possible that the opponent finds out what these proportions are. Each player should therefore choose them in such a way that his position does not become worse when he is found out. It is clear that if A chooses his originally given 'pure strategies' in the proportion V:!: ¥l, then, whatever B does, A's gains will equal his losses in the long run. But this is, in fact, the best A can hope to achieve. If he chooses his strategies in any other proportion, say once head to three times tail, then B, provided he knew it, would be shrewd enough to choose always head, because then A would lose more often in the long run than he would win. Reversing the argument, it is seen that B should, in ignorance of A's strategy, once again choose head and tail with equal frequencies, in order to view with· out concern the danger of being found out. (This principle has been described as expecting the best in the worst possible world. We leave it to the reader's temperament to decide whether this description, if justified, should induce the player to gamble excessively rather than to play safe.) If, then, both players use their pure strategies in the proportion ~: ¥.!, a certain stability will be reached in the sense that no player will change his behaviour, even if he finds out what his adversary does, since by doing so he would not improve his prospects. The pair of frequencies, which leads to such a stability, is called a 'solution,' and the decision of a player to use his (pure) strategies in given proportions is itself a strategy and is called 'mixed.' We have just seen that in Matching Pennies no pure, but only a mixed, strategy leads to a solution. However, this is by no means always so, as can be shown by considerin2 a simple modification of the game. MODIFIED MATCHING PENNIES
We assume now that any player is allowed to 'call off' the play and that he receives, if he is the only one to do this, a ha1fpenny from his opponent, whereas no payment is made if both players call off. The corresponding pay-off table (again for A) is shown as Table II. An inspection of the table shows that now both players can choose pure strategies which give a solution. To achieve this, they must both call off.
1211
s. ". . TABLE II: PAY-oFF TABLE FOR MODIFIED MATCHING PENNIES
8 ts coin Head Head Ats coin
Tail Call off
Column maxima
Tail
Call off
Row minima
1
-1
-1
-1
1
-* -*
* 1
* 1
0
\
-1
0
0
In any other case the player who finds out his opponent's intention can win a penny. The entry 0, corresponding to the pair of strategies which form the solution, is called the value of the game. It is the smallest number in its row and the highest in its column. Therefore its position is called a 'saddJe point,' by an obvious analogy. It can easily be proved that if there are more saddle points in a pay-off table, then the numbers in all of them must be equal. A simple way of finding a saddle point, if one exists, is to write down the column maxima and the row minima. as we have done in the margins of the table above. In this example the maximum of the row minima equals the minimum of the column maxima, viz. 0 (the value of the game). In more abbreviated terms, the 'maximin' equals the 'minimax: The common value indicates a saddle point, and the corresponding solution is called a 'Minimax Solution: Now one can easily prove that the maximin can never exceed the minimax, but as we have already seen (see Matching Pennies), the two need not be equal. However, it has been proved that the maximin of aU possible strategies, including mixed ones. is always equal to the minimax of all possible strategies of the opponent, again including mixed ones. Once more, the resulting pay-off is caned the value of the game. The equality of the maximin and the minimax, extended to refer to mixed strategies, was proved first by John von Neumann in t 928. He called it the Main Theorem. His proof has since been simplified in various ways, but it is still too involved to be included in this merely expository article. THE THREE BOXES GAME
The game of Matching Pennies led to an obvious best mixed strategy and its modification had a Minimax Solution of pure strategies. We shall now deal with a game whose solution is again given by mixed strategies,
Tluory of Games
1219
but not by such obvious ones. Let three boxes marked 1, 2, and 3 be given, containing these numbers of shinings respectively. The 'banker' removes the bottom of one of the boxes, but this is not discernible from the outside. A 'player' puts into two of the boxes the amounts of shil1ings marked on them. He then receives all the money in those two boxes. He will, of course. lose the money that he happened to put into a box without bottom. The pay-off table for the player is easily constructed (Table III). For instance. if the bottom of box 3 is missing and if the player puts ts and 2s into the appropriate boxes. then he wins 3s altogether. However, if he puts his money into boxes 1 and 3, then he wins 1s and loses 3s. so that, on balance, he suffers a loss of 2s. In this way the following table has been constructed: TABLE III: PAy-oPF TABLE FOR THE THREE BOXES GAME
Bottom removed from box
Money put into boxes
Row minima
1
2
3
1. 2
1
-1
3
-1
I, 3
2
4
-2
-2
2,3
5
1
-1
-1
5
4
3
Column maxima
This table has no saddle point, since the maximum of the row minima is -1, whereas the minimum of the column maxima is 3. Therefore the solution (which is known to exist by virtue of the Main Theorem) must contain mixed strategies. In order to find the solution, one could first try to see whether three positive numbers at b, and c, adding up to unity, can be found such that every row gives the same value if the banker follows a mixed strategy in which the bottoms of the three boxes are removed in those proportions. This is the condition that the player will gain no benefit from discovering the strategy used by the banker. We could then repeat the procedure for every column, and hope in that way to obtain the player's mixed strategy such that the banker gains nothing from its discovery. In the present case, the conditions for the banker's mixed strategy are (for the rows) that a + b + c =1, a - b + 3c = 2a + 4b - 2c = 5a + b - c. If the banker chooses the proportions a 5/22, b 8/22, and c 9/22, which are consistent with these conditions, then the player wins 12/11, whatever he does, whereas if the banker chose any other proportions, the player could win more than this amount. By the same argument,
=
=
=
we must try to find three numbers x, y, and z, say, sO that x + y + z = 1 and (applying these proportions to the columns) x + 2y + 5z = -x + 4y + z 3x - 2y - z. However, this results in x = 7/11, y = 5/11, and z = -1/11. Since z is negative, this is not a possible frequency, and we must find some other method for finding a solution. It would be too long to explain here how this is done systematically. The solution turns out to be (3/5, 2/5, 0) for the player and (0, 1/2, 1/2) for the banker. When the banker uses the proportions (0, 1/2, 1/2), the gains of the player for his pure strategies are -1/2 + 3;2 1, 4/2 - 2/2::::: 1, and 1/2 - 1/2 ::::: 0 respectively. The latter will, therefore, be wise to use mixed strategies, in which only two pure strategies are combined-those in which he puts money into the 1sand 2s, and the I sand 3s boxes respectivelysince by using the 2s, 3s strategy as well he would invariably reduce his expected average gain. The proportions in which he should use those strategies are 3 : 2. as already stated. Doing this, he gains 1 and the banker loses]. (We may assume that the latter will demand a fee amounting to I, the value of the game, for participation.) The player cannot improve on his gain of 1, as long as the banker adheres to the proportions (0, 1/2. 1/2) on his part. For any other choice of the banker's the player could gain more. Thus, for instance. if the banker used (0, ] /3, 2/3), the player could obtain a gain of 5/3. by choosing the pure strategy (J s, 2s). Conversely, if the player chose proportions different from those given for him, he could be made to gain less than 1, if he were found out. Thus. for instance, if he chose (4/5, ) 15, 0). then the banker might choose pure strategy 2, and the gain of the player would thereby be reduced. in the long run, to zero. So far we have always assumed that all proceedings depend only on the decisions of the players. However, this would not cover aJl possible games. We imagine now a slightly modified Three Boxes Game, where the banker does not decide on his own moves, but where they are chosen for him by a chance mechanism, which makes every pure strategy equally likely. In other words, his mixed strategy is (1/3. ] /3. 1/3). and this is known to the player. who must choose the pure strategy of always putting his money in the 2s and 3s boxes to gain in the proportion 5: 3 over a series of plays; every other choice would give him less in the long run. This is an example of a game with so-called 'chance moves.' Obviously. these can also occur in combination with deliberate moves, and they may be avail· able to both players. Card games, for instance, contain always both types of moves, because at least some of the cards are dealt at random (or should be). In the Three Boxes Game botb participants make their choice without knowledge of the other's moves, as was also the case in Matching Pennies, where indeed otherwise the game would have been senseless. However,
=
=
1291
T"eorr of Gamtf
games exist where the moves occur one at a time alternately and where. whenever a move is about to be made, all pJayers know all previous moves. whether their own or those of the opponent (though not, of course. the opponent's strategy, which includes also all his future intentions for any possible contingency). Such games are said to have 'perfect information: whereas Matching Pennies. for instance. is a game with 'imperfect information! It has been proved by von Neumann that a game with perfect information has always a saddle point and that, in a probability sense, this is also true of games with chance moves (and perfect information). With regard to games without chance moves, the theorem has, in effect, also been proved. 1 We shall now introduce a 'Modified Three Boxes Game' with perfect information which will, therefore. have a solution consisting of pure strategies for both players. MODJFIED THREE BOXES GAME
The rules of the Modified Three Boxes Game are as foUows: (1) the player puts his money into one of the boxes, (2) the banker, who has watched him, removes the bottom of one of the boxes, (3) the player puts his money into one of the two boxes not previously chosen. The player has thus two moves. As to the banker, who has only one move, we can take it for granted that his best strategy is to remove the bottom of that box into which the player has put the money on his first move. Thus the move of the banker is known to the player in advance. The latter, however, has a choice of the following six strategies, with the gains as given (provided the banker uses his best strategy. as he will): First move of the player Second move of the player Gain
1
2 1
1 3
2 1
2
-I
233 312
1
-2
-1
It emerges, then, that the player should first put Is into box 1 (knowing
that he will lose his money) and then 3s into box 3, obtaining an overall gain of 2s. This is. of COUTSe, also clear from common-sense considerations. The Modified Three Boxes Game. besides being somewhat artificial, is also of rather restricted generality in that the banker has only one move and that. if this is known, his whole strategy is thereby known as wen. But it will be clear by now that the same methods are applicable also to more complicated games. It is perhaps worth mentioning that there is no connexion between the division of games according to information and according to chance moves. This can be seen from Table IV: 1 I By D. W. Davies in an article on "A Theory of Chess and Noughts and Crosses" in Science News 16 (Penguin Books. Harmondsworth). ED.'
S. VII/da
1292
TABLE IV: THE DIVISION OF GAMES ON THE BASIS OF INFORMATION AND OF CHANCE MOVES
Games with perfect information
Games with imperfect information
Games with chance moves
Backgammon
Poker
Games without chance moves
Chess 2
Matching Pennies
THE TWO GENERALS
SO far, we have considered games with saddle points, or games where
the solution consists of a single mixed strategy for both players. We shall now introduce a game which has more than one single solution. The example will also illustrate how the theory of games may be applied to military considerations. We imagine the foUowing simple strategic situation. Two generals, A and B, face one another. A has three companies at his disposal and B has four. It is the aim of A to reach a town on one of two possible roads. B tries to prevent him from doing it. General A can send all three companies on the same road, or he may split up their number, but he may not divide up any single company. General B again, may send al1 his companies on the same road, or on different roads, but he must keep 'any single company undivided. We assume that A has achieved his object if he has on any road a number of companies exceeding that of his adversary. If he reaches the town, he wins (indicated by 1), if he does not, the issue remains undecided (indicated by 0). The pay-off table (for A) is shown here as Table V: TABLE
v:
PAY-OFF TABLE FOR THE 'TWO GENERALS' GAME
Number of companies of B on the two roads, respectively
o and 4 o and 3
1 and 3 2 and 2 3 and 1 4 and 0
0
0
1
1
1
Number of companies of A on 1 and 2 the two roads, 2 and 1 respectively
1
0
0
1
1
1
1
0
0
1
3 and 0
1
1
1
0
0
2 It is assumed that the strategies of chess-players are chosen when it is known who plays White.
1293
Theory 01 Games
The roW minima are all 0, the column maxima are a]] 1. Hence this table has no saddle point. It is now possible to see, by inspection, that if A uses strategies mixed (%,0, %,0), or S:! (%, 0, 0, in any of the following proportions: S, %), or S;I (0, %, 0, 1.~), he will gain on the average at least % in the long run, and that 8 can keep him down to this if he uses strategy (0, %, 0, %, 0) = T, say, but cannot reduce A's gain below %. On the other hand, if 8 uses strategy T, then A cannot obtain more than %, whatever he does. This means that any of the following pai rs of mixed strategies, (S" T), (So!, T), and (S:h TL is a solution. The value of the game is the same for al1 solutions (namely Y:'!), and it can be proved generally that whenever a game has more than one solution, the value is the same for all of them. General A has the three 'best' strategies S I, S:!, and S:I at his disposal, and it is obvious that, if he uses any combination of these mixed strate· gies, the resulting mixed strategy will also form together with T for General 8, a solution in the sense (as before) that neither can lose from the discovery by the other of their respective strategies. Thus if r, s, and t are three positive numbers which add up to ], but are otherwise quite arbitrary, then A may use the foHowing mixed strategy: (r + s)/2, t/2, r12, (s + t) 12. For example, taking r .'i t 1/3, the following strategy emerges: (1/3, 1:'6, 116, 1'3). Together with (0, 1~, 0, 1k, 0) for 8, we obtain again the value of the game, namely one·half.
=
=
=
= = =
MORE PERSON GAMES
Games in which more than two players take part have not been so intensively studied as twa.person games, though the great hook by J. von Neumann and O. Morgenstern contains chapters on this subject as well. Pay-off tables can again be constructed, but no theorem similar to the Main Theorem exists. An essentially new situation arises from the fact that two players can now combine in 'coalitions' and play against the remaining player or players. This is quite a common situation in real·life problems, and sometimes. as at an auction, the coalition may be agreed upon in advance hetween some of the players. Only if this condition is satisfied can the effects of coalitions be treated by the theory in its present form. Coalitions can also arise spontaneously in many·person games, which ostensibly are all·against-all, at a stage when some one individual is too evidently winning. Coalitions of the latter kind are unpredictable, and may be changed and re·formed, according to the mood of the players, as the play proceeds. They cannot be treated by the theory, as at present developed, and illustrate one type of limitation in its application to reallife problems.
Never too late to learn.
-ScOTTISH PROVBD
With just enough oj learning to misquote.
-BYRON
10 Sociology Learns the Language of Mathematics By ABRAHAM KAPLAN A TROUBLING question for those of us committed to the widest application of intelligence in the study and solution of the problems of men is whether a general understanding of the social sciences will be possible much longer. Many significant areas of these disciplines have already been removed by the advances of the past two decades beyond the reach of anyone who does not know mathematics; and the man of letters is increasingly finding, to his dismay, that the study of mankind proper is passing from his hands to those of technicians and specialists. The aesthetic effect is admittedly bad: we have given up the belletristic "essay on manu for the barbarisms of a technical vocabulary. or at best the forbidding elegance of mathematical syntax. What have we gained in exchange? To answer this question we must be able to get at the content of the new science: But when it is conveyed in mathematical formulas, most of us are in the position of the medieval layman confronted by clerical Latin -with this difference: mathematical language cannot be forced to give way to a vernacular. Mathematics, if it has a function at all in the sciences, has an indispensable one; and now that so much of man's relation to man has come to be treated in mathematical terms, it is impossible to ignore or escape the new language any longer. There is no completely satisfactory way out of this dilemma. All this article can do is to grasp either hom. sometimes oversimplifying. sometimes taking the reader out of his depth: but hoping in the end to suggest to him the significance of the growing use of mathematical language in socia] science. To complicate matters even further, the language has severa] dialects. "Mathematics" is a plural noun in substance as weIJ as form. Geometry. algebra. statistics. and topology use distinct concepts and methods. and are applicable to characteristically different SOTts of problems. The role of mathematics in the new social science cannot be discussed in a general way: as we shaH see, everything depends on the kind of mathematics being used. 1294
.295 I
The earliest and historically most influential of the mathematical sciences is geometry. Euclid's systematization of the results of Babylonian astronomy and Egyptian surveying set up a model for scientific theory that remained effective for two thousand years. Plato found in Euclid's geometry the guide to the logical analysis of all knowledge, and it was the Renaissance's "rediscovery" of Plato's insistence on the fundamentally geometric structure of reality that insured the triumph of the modern world view inaugurated by Copernicus. Scientists like Kepler accepted the Copernican hypothesis because it made the cosmos more mathematically elegant than Ptolemy'S cumbersome epicycles had been able to do. The study of man-to say nothing of God!-enthusiastically availed itself of mathematical method: witness Spinoza's Ethic:;. which claimed that it "demonstrated according to the geometrical manner." But Spinoza'8 Ethics failed, as demonstrative science. because the 17th century did not clearly understand the geometry it was applying with such enthusiasm. The discovery of non-Euclidean geometries two hundred years later revealed tilat the so-cal1ed axioms of geometry are not necessary truths, as SpinozaI land his fellow rationalists had always supposed, but merely postulates: propositions put forward for the sake of subsequent inquiry. It is only by deducing their consequences and comparing these with the perceived facts that we can decide whether or not the postulates are true. Geometry is a fully developed example of a set of undefined terms, prescribed operations, and the resulting postulates and theorems which make up a postulational system; it is in this form that it is influential in some of the recent developments in the social sciences. Perhaps the most elaborate postulational system for dealing with the data of social and psychological science was constructed in 1940 by Clark Hull and associates of the Yale Institute of Human Relations (C. L. Hu]) et at, Mathematico-Deductive Theory of Rote Learning, Yale University Press, 1940). "Rote Jeaming" is a very specialized branch of psychology that studies the learning of series of nonsense syUables; presumably, this telJs us something about the act of learning in its "purest" form, with no admixture of influence from the thing learned. The problems of the fie1d revolve around ways of explaining the patterns of learning that are in fact discovered; why the first syllables in any series are learned first, the 1ast soon after, and why the syllables a little past the middle of the given series take longest to memorize, and so on. There is a vast number of observations of this sort to be made, and Hull's ideal was to set up a postulational system which would allow observed patterns of learning to be logically deduced from relative1y few postulates. The system consists of 16 undefined terms, 86 definitions, and ] 8 postu-
12M
lates. From these, S4 theorems are deduced. The deductions can, in principle, be checked against the results of direct observation, in experimental situations or elsewhere. In many cases, as the book points out, existing evidence is as yet inadequate to determine whether the theorems hold true; in the great majority of cases, experimental evidence is in agreement with logically deduced theorems; in others. there is undoubted disagreement between deduced expectation and observed fact. Such disagreements point to basic defects in the postulate system, which, however, can be progressively improved. The authors consider their book to be principally important as an example of the proper scientific method to be used in the study of behavior. And certainly, as a formal demonstration of the handling of definitions. postulates, and theorems. the book is unexceptionable. However, science prides itself on its proper method because of the fruitfulness of its results; and it is to the fruitfulness of this effort that we must address ourselves. One example of the method may suggest better than general criticism the problem raised. Hull proves in one of his theorems that the greater the "inhibitory potential" at a given point in the learnin,l of a rote series.. the greater will be the time elapsing between the stimulus (the presentation of one nonsense syllable in the list) and the reaction (the pronouncing of the next memorized nonsense syllable). "Inhibitory potential" is one of the undefined terms of the system; it denotes the inability of the subject, before his involvement in the 1earning process, to pronounce the appropriate syllable on the presentation of the appropriate stimulus. (It may be pictured as a force that wanes as the stimuli are repeated and the syllable to be uttered is 1earned.) Now this theorem certainly follows logically from three postulates of the system (they involve too many special terms to be enlightening if quoted). However, on examining these postulates, the theorem is seen to be so directly impJied by them that one wonders what additional knowledge has been added by formally deducing it. A certain amount must have been known about rote learning to justify the selection of those postulates in the first place. To deduce this theorem from them has added very little -if anything-to what was already known. In short: the geometric method used by Hull, correct as it is formally, does not, for this reader, extend significantly what we already knew about rote learning from his and others' work. In the course of HuU's book "qualitative postulates," by which is meant the "unquantified" ideas of thinkers like Freud and Darwin, are condemned because they have "so little deductive fertilityU-because so few theorems may be deduced from them. In the narrowest logical sense of the phrase, this may be true. But fertility in the sense of yielding precisely determinable logical consequences is one thing; in the sense of yielding further
1297
insights into the subject matter-whether or not these can be presented as strict consequences of a system of postulates-it is quite another. The ideas of Darwin and Freud can hardly be condemned as lacking in fertility, even though they leave much to be desired from the standpoint of logical systematization. This is not to deny that the postulational method can playa valuable role in science. But it is a question of the scientific context in which the method is appJied. Newton and Euclid both had available to them a considerable body of fairly well-established knowledge which tbey could systematize, and in the process derive other results not apparent in the disconnected materials. Hull recognizes this condition, but supposes it to be already fulfilled in the area of learning theory. The results of the particular system he has constructed raise serious doubts that his supposition is true. Science, basically, does not proceed by the trial-and-error method to which Hull, as a student of animal learning, is so much attached. It em· ploys insight, itseJf SUbject to logical analysis, but too SUbtle to be caught in the coarse net of any present-day system of postulates. The geometric method in the new social science can be expected to increase in value as knowledge derived from other methods grows. But for the present, it is an elegantly written check drawn against insufficient funds. D
If the 17th century was the age of geometry, the 18th was tbat of
algebra. The essential idea of algebra is to provide a symbolism in which relations between quantities can be expressed without a specification of the magmtudes oj the quantities. An equation simply formulates the equality between two quantities in a way that shows how the magnitude of one depends on the magnitude of certain constituents of the other. The characterization of mathematics as a language is nowhere more appropriate than in algebra; the notation is everything. The power of algebra consists in that it allows the symbolism to think for us. Thought enters into the formulation of the equations and tbe establishing of the rules of manipulation by which they are to be solved, but the rest is mechanicalso much so that more and more the manipulation is being done, literally, by machines. The postulational method characteristic of classical geometry no longer plays as important a part here. Derivation is replaced by calculation; we proceed, not from postulates of uncertain truth, but from arithmetical propositions whose truth is a matter of logic following necessarily from the definitions of the symbols occurring in them. The equations that express relations between real quantities are, of course, empirical hypotheses; but ij the equations correctly formulate the function relating two
Abraham Kalllllll
1298
quantities, then certain numerical values for one necessarily imply certain numerical values for the other. Again, as in geometry, the facts are the final test. In this spirit, the mathematicians of the 18th century replaced Newton's geometrizing by the methods of algebra, and the culmination was La· place's system of celestial mechanics. Laplace's famous superman, given the position and momentum of every particle in the universe, could com· pute, it was thought, the entire course of world history, past and future. The development of quantum mechanics made this program unrealizable even in principle, just as the non·Euclidean geometries were fatal to the aspirations of the 17th-century rationalists. Nevertheless, this scientific ideal, so nearly realized in physics, has exerted enormous inftuence on the study of man, and much of the new social science is motivated by the attempt to take advantage of the powerful resources of algebra. Among the most ambitious, but also most misguided, of such attempts is a 900-page treatise by the sociologist Stuart C. Dodd, Dimensions of Society (Macmillan, ] 942). The author's ambition, declared in the subtitle, is to provide "a quantitative systematics for the social sciences." What is misguided is the failure to realize that a system is not provided by a symbolism alone: it is necessary also to have something to say in the symbolism that is rich enough to give point to the symbolic manipulation. Dodd presents a dozen symbols of what he regards as basic sociological concepts, together with four others for arithmetical operations. They stand for such ideas as space, time, population. and characteristics (the abstract idea "characteristics," and not the specific characteristics to be employed in sociological theory). In addition to these sixteen basic symbols, there are sixteen auxiliary symbols, compounds or special cases of the basic symbols, or supplements to them-for instance. the question mark, described as one of four "new operators," used to denote a hypothesis, or a questioned assertion. With this notation. every situation studied hy sociologists is to be defined by an expression of the form: S=
s s
(T; I ; L ; P)
s s
The capital "S" stands for the situation, and the letters within the parentheses for time, indicators of the characteristics specified, length or spatial regions, and populations, persons. or groups. The semicolon stands for an unstated form of mathematical combination, and the small "s" for various ways of particularizing the four major kinds of characterizations. Thus "TO" stands for a date, "TI" for a rate of change. both of these being different sorts of time specifications. Instructions for the use of the notation require one hundred distinct rules for their formulation. But this whole notational apparatus is. as Dodd recognizes. "a syste-
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matic way of expressing societal data, and not, directly, a system of the functionings of societal phenomena." "Facts," however, are data only for hypotheses; without the latter, they are of no scientific significance. Certainly a notational system can hardly be caned a "theory," as Dodd constantly designates it, unless it contains some statements about the facts. But Dimensions of Society contains only one socia] generalization: "This theory . . . generalizes societal phenomena in the statement: 'People, Environments, and Their Characteristics May Change.' This obvious generalization becomes even more obvious if the time period in which the change is observed is prolonged." The last sentence may save Dodd, but not his "theory." from hopeless naivety. Dodd's hope that his system of "quantic classification" wi1l "come to playa role for the social sciences comparable to the classification of the chemical atoms in Mendelyeev's periodic table" is groundless. The periodic tabJe, after a11, told us something about the elements; more, it suggested that new elements that we knew nothing of existed, and told us what their characteristics would be when discovered. The fundamenta1 point is that we have, in the case of Dodd, only a notation; when he speaks of the "verification" of his "theory," he means only that it is possible to formulate societal data with his notation. Dodd's basic error is his failure to realize that after "Let x equal suchand-such:' the important statement is still to come. The question is how to put that statement into mathematical form. An answer to this question is provided in two books of a very different sort from Dodd's, both by the biophysicist N. Rashevsky: Mathematical Theory of Human Relations (The Principia Press, 1947) and Mathematical Biology of Social Behavior (University of Chicago Press, 1951). In these two books the author does not merely talk about mathematics; he actually uses it. In the earlier one, the methods of mathematical biology are applied to certain social phenomena on the basis of formal postulates -more simply, assumptions-about these phenomena. In the later book, these assumptions are interpreted in terms of neurobiophysical theory, and are derived as first approximations from that theory. The results, according to Rashevsky, are "numerous suggestions as to how biological measurements, made on large groups of individuals, may lead to the prediction of some social phenomena." As a natural scientist, Rashevsky is not seduced, like so many aspirants to a social science. by the blandishments of physics. Scientific method is the same everywhere: it is the method of logical inference from data provided and tested by experience. But the specific techniques of science are different everywhere. not only as between science and science but even as between problem and problem in the same field. The confusion of method and technique, and the resultant identification of scientific method
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with the techniques of physics (and primarily 19th-century physics at that) has hindered the advance of the social sciences not a little. For the problems of sociology are different from those of physics. There are DO CODcepts in social phenomena comparable in simplicity and fruitfulness to the space, time, and mass of classical mechanics; experiments are difficult to perform, and even harder to control; measurement in sociological situations presents special problems from which physics is relatively free. Yet none of these differences, as Rashevsky points out, prevents a scientific study of man. That social phenomena are complex means only that techniques must be developed of corresponding complexity: today's schoolboy can solve mathematical problems beyond the reach of Euclid or Archimedes. Difficulties in the way of experimentation have not prevented astronomy from attaining full maturity as a science. And the allegedly "qualitative" character of social facts is, after all, only an allegation; such facts have their quantitative aspects too. And what is perhaps more to the point, mathematics can also deal with qualitative characteristics, as we shall see. Rashevsky addresses himself to specific social subject matters: the formation of closed social classes, the interaction of military and economic factors in international relations, "individualistic" and "collectivistic" tendencies, patterns of social influence, and many others. But though the problems are concrete and complex, he deals with them in a deliberately abstract and oversimplified way. The problems are real enough, but their formulation is idealized, and the equations solved on the basis of quite imaginary cases. Both books constantly repeat that the treatment is intended to "illustrate" how mathematics is applicable "in principlett to social science: for actual solutions, the theory is admitted to be for the most part too crude and the data too scarce. What this means is that Rashevsky's results cannot be interpreted as actual accounts of social phenomena. They are, rather, ingenious elaborations of what would be the case if certain unrealistic assumptions were granted. Yet this is not in itself objectionable. As he points out in his own defense, physics made great strides by considering molecules as rigid spheres, in complete neglect of the complexity of their electronic and nuclear internal structures. But the critical question is whether Rashevsky's simpJifications are, as he claims, "a temporary expedient!' An idealization is useful only if an assumption that is approximately true yields a solution that is approximately correct; or at any rate, if we can point out the ways in which the solution must be modified to compensate for the errors in the assumptions. It is in this respect that Rashevskfs work is most questionable. Whatever the merits of his idealizations from the standpoint of "iI1ustrating," as he says, the potentiaHties of mathematics, from the standpoint of the
IIOJ
study of man they are so idealized as almost to lack all purchase on reality. Rashevsky's treatment of individual freedom, for example, considers it in two aspects: economic freedom and freedom of choice. The former is defined mathematically as the fraction obtained when the amount of work a man must actually do is subtracted from the maximum amount of work of which he is capable, and this remainder is divided by the original maximum. A person's economic freedom is 0 when he is engaged in hard labor to the point of daily exhaustion; it is 1 when he does not need to work at all. This definition equates increase in economic freedom with shortening of the hours of work: and an unemployed worker, provided he is kept alive on a dole, enjoys complete economic freedom. Such critical elements of economic freedom as real wages, choice of job, and differences in level of aspirations are all ignored. Freedom of choice, the other aspect of individual freedom, is analyzed as the proportion borne by the amount of time an individual is not in contact with others who might interfere with his choices, to the time be spends alone plus time that is spent in the company of others with the same preferences. This makes freedom of choice decrease with increasing population, so that by this definition one is freer in a small village than a large city. Nothing is said about prying neighbors, or the presence or absence of a secret police, a most important determinant of "freedom of choice." The whole matter of tbe "introjection" of other persons' standards, as discussed for instance in Erich Fromm's Escape from Freedom, is ignored, as are such fundamental considerations as knowledge of the choices available, or the opportunity to cultivate skills and tastes. On current social issues Rashevsky betrays that he suffers from the same confusions and rationalizations as afflict students without the advantages of a mathematical training. To explain the Lysenko case, for example. be suggests that it is possible tbat tbe facts of genetics u may be interpreted from two different points of view,t' tbus naively substituting a scientific question (if tbere be one) for tbe real issue, wbicb is tbe political control of science. His assumptions encourage bim to attempt the conc1usion, "even at the present state of our knowledge:' tbat after World War n peace will be most strongly desired by tbe Soviet Union, least by the United States, with England and Germany in between. And be confesses tbat be finds it ndifficult to understand wby the Soviet Union insists on repatriating individuals who left the Soviet Union during World War n and do not desire to return." Matbematics is not yet capable of coping witb tbe naivety of tbe mathematician bimself.
m The 19tb century saw tbe rise of mathematical statistics. From its origins in the treatment of games of chance, it was expanded to cope with
UOl
the new problems posed by the development of insurance, and finally came to be recognized as fundamental to every science deaJing with repetitive phenomena. This means the whole of science, for knowledge everywhere rests on the cumulation of data drawn from sequences of situations on whose basis predictions can be made as to the recurrence of similar situations. Mathematical statistics is the theory of the treatment of repeated--or multiple--observations in order to obtain al1 and only those conclusions for which such observations are evidence. This, and not merely the handling of facts stated in numerical terms, is what distinguishes statistics from other branches of quantitative mathematics. The application of statistics to social phenomena far exceeds, in fruitfulness as well as extent, the use of mathematics of a fundamentally geometrical-Le., postulational--or algebraic character in the social sciences. Social scientists themselves have made important contributions to mathematical statistics-which is a good indication of its usefulness to them in their work. Only two of the most recent contributions in the application of statistics to social phenomena can be dealt with here. The first is a rather remarkable book by a Harvard linguist, G. K. Zipf, Human Behavior and the Principle of Least Effort (Addison-Wesley Press, 1949). Its basic conception is that man is fundamentally a user of tools confronted with a variety of jobs to do. Culture can be regarded as constituting a set of tools, and human behavior can be analyzed as the use of such tools in accord with the principle of minimizing the probable rate of work which must be done to perform the jobs that arise. It is this principle that Zipf cans "the law of least effort.n As a consequence of it, he claims, an enormous variety of social phenomena exhibit certain reglr larities of distribution, in accordance with the principle that the tools nearest to hand, easiest to manipulate, and adapted to the widest variety of purposes are those which tend to be used most frequently. These regularitie5 often take the form, according to Zipf, of a constant rank-frequency relationship, according to which the tenth most frequently used word, for instance, is used one-tenth as often as the most frequently used one of all. This is the case, for example, with James Joyce's Ulysses as well as with clippings from American newspapers. A large part of Zipfs book deals with linguistic phenomena, since he is most at home in this field, and it is there that his results seem most fully established. But an enormous range of other subjects is also treated: evolution, sex, schizophrenia, dreams, art, population, war, income, fads, and many others. Many of these topics are dealt with statistically, as likewise conforming to the law of least effort; and all are discussed with originality and insight. For example. the cities in any country, according to Zip!, tend to show the same regularity-the tenth most populous city will have one-tenth as many people as the most populous, the one-hundredth most
S«iolo,y wa".s tlte
un,fla,e of Mathematics
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populous city will have one-hundredth as many. Where this pattern does not prevail, we have an indication of serious potential conflict. It seems that starting about 1820 the growing divisions between Northern and Southern economies in the United States could be seen by a break in this pattern, which reached a peak of severity around 1840, and disappeared after the Civil War! But while this breadth of topic endows the book with a distinctive fascination, it also makes the reader skeptical of the validity of the theory it puts forward. In the human sciences, the scope of a generalization is usuaJly inversely proportional to its precision and usefulness-at any rate, in the present state of our knowledge. Zipf's law, or something like it, is well known in physics as Maupertuis' principle of least action. But Zipf applies it to a much wider field of phenomena than physical ftows of energy. It is understandable that action will follow some sort of 1eastaction pattern if economy enters into its motivation and if the action is sufficiently rational. But that the law of least effort should be manifested everywhere in human conduct, as Zipf holds-indeed, "in a11 living process"-is difficult to believe. That a theory is incredible is, of course, no logical objection whatever. And Zipf does not merely speculate; he presents an enormous mass of empirical evidence. The question is what it proves. It does not show, as he claims, the existence of "natural social laws," but, at best, only certain regularities. Brute empiricism is not yet science. Unless observed regularities can be brought into logical relation with other regularities previously observed. we remain at the level of description rather than explanation; and without explanation, we cannot attach much predictive weight to description. As a collection of data that deserve the careful attention of the social scientist, Ziprs work will have interest for some time to come. But something more precise and less universal than his principle of least effort will be requited to transform that data into a body of scientific knowledge. The importance of clear conceptualization in giving scientific significance to observed fact is admirab1y expounded in the recently published fourth volume of the monumental A merican Soldier series (S. A. Stouffer, L. Guttman, et a1., Measurement and Prediction: Studies in Social Psychology in World War II, Vol. IV, Princeton University Press, 1950). Measurement and Prediction deals with the study of attitudes. It is concerned with the development of methodological rather than substantive theory. It deals with the way in which attitudes-any attitudes-should be studied and understood, but says little about attitudes themselves. However, methodology here is not an excuse for irresponsibility in substantive assumptions, or for confusion as to the ultimate importance of a substantive theory of attitudes.
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Abl'Glulm ICGPItut
The major problem taken up is this: how can we tell whether a given set of characteristics is to be regarded as variations of some single underlying quality? Concretely. when do the responses to a questionaire constitute expressions of a single attitude, and when do they express a number of attitudes? If there is such a single quality. how can we measure how much of it is embodied in each of the characteristics? If all the items on a questionaire do deal with only one attitude, can we measure, in any meaningful sense. how favorable or unfavorable that attitude is, and how intensely it is held by the particular respondent? This problem arises directly out of the practical-as weD as the theo-retical-side of opinion study. Consider the case of a poll designed to test the extent and intensity of anti-semitism. Various questions are included: "Do you think the Jews have too much power?" "Do you think we should allow Jews to come into the country?" "Do you approve of what Hitler did to the Jews?" Some people give anti-Semitic answers to an the questions, some to a few, some to none. Is this because they possess varying amounts of a single quality that we may call "anti-Semitism"? Or is there really a mixture of two or more very different attitudes in the individual, present in varying proportions? If a person is against Jewish immigration. is this because he is against immigration or against Jews, and to what extent? And if there is a single quality such as anti-Semitism, what questions wiU best bring it out for study? It is problems such as these that the research reported on in Measurement and Prediction permits us to solve. The approach taken stems from the work done in the past few decades by L. L. Thurstone and C. Spearman. Their problems were similar. but arose in a different field, the study of intelligence and other psychological characteristics. Their question was: do our intelligence tests determine a single quality called inte1ligence? Or do they actually tap a variety of factors, which though combining to produce the total intelligence score, are reaUy quite different from each other? Thurstone and Spearman developed mathematical methods that in effect determined which items in a questionnaire were interdependent-that is, if a person answered a, he would tend to answer b, but not c. On the basis of such patterns, various factors of intelJigence were discovered .. In opinion study. one inquires whether items of a questionnaire "hang together"--or, to use the technical term, whether they scale. A complex of attitudes is said to be scalable if it is possible to arrange the items in it in such a way that every person who answers "yes" to any item also answers "yes" to every item coming after it in the arrangement. In the case of anti-Semitism, we would consider the complex of anti-Semitism scalable--and therefore referring to a single factor in a person's attitudes, rather than including a few distinct attitudes-if we could order the
Soc/olol" utlrru 'h~ l.a"'lItl'~ 01 I14tlth~mtJllcl
UO!
questions in such a way that if someone answered any question in the list Uanti-Semitically," he would answer all those following it Uanti-Semitically." Attitudes on anti-Semitism would then have the same cumulative character as a series of questions on height-if a person answers "yes" to "Are you more than six feet taU?" we know he wi)) answer "yes" to the question "Are you more than five and a half feet taU?" and all others down the list. This type of reduction of an apparently complex fieJd of attitudes to the simple scheme of a series of cumulative questions is of great value. In Measurement and Prediction Louis Guttman describes how to determine whether a group of attitudes does uscale"-that is, does measure a single underlying reality. Guttman developed, for example, such a scale for manifestations of fear in battle-vomiting, trembling, etc.: soldiers who vomit when faced with combat also report trembling, and so on down the scale; while those who report trembling do not necessarily report vomiting too. On the other hand, it turned out. when he studied paratroopers, various kinds of fear of very different types had to be distinguished. for the paratroopers' symptoms were not scalable in terms of a single variable. One of the most direct applications of scaling methods is in the detection of spurious "causal" connections. We may find, for instance. that the attitude to the continuation of OPS by the government correlates closely with the attitude to farm subsidies. Scale analysis now makes it possible for us to provide an explanation for this fact by testing whether these two items do not in fact express a single attitude-say, the attitude to governmental controls. Scale analysis permits us to handle another important problem. Suppose we find that 80 per cent of a group of soldiers tested agreed that "the British are doing as good a job as possible of fighting the war, everything considered," While only 48 per cent disagree with the statement that "the British always try to get other countries to do their fighting for them." How many soldiers are "favorable" toward the British? It is clear that we can get different percentages in answer to this question, depending on how we word our question. Scale analysis provides a method which yields what is called an "objective zero point": a division between the numbers of those "favorable" and those "unfavorable" that remains constant no matter what questions we ask about the British. The method demands that, besides asking a few questions testing the attitude, we also get a measure of the "intensity" with which the respondent holds his opinion-we ask for example, whether the respondent feels strongly, not so strongly~ or is relatively indifferent about the matter. With this method, it turns out that jf we asked a group of entirely different questions about the British, the application of the procedure for measuring the ··objective zero point" would show the same result. This limited area of attitude comes to have
Abraham Kal1l/Vf
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the same objectivity as the temperature scale, which shows the same result whether we use an alcohol or a mercury thermometer. Measurement and Prediction also presents, for the first time, a full description of Lazarsfeld's "latent structure" analysis. This is in effect a mathematical generalization of the scaling method of Guttman, which permits us to extend the type of inquiry that scale analysis makes possible into other areas. Scale analysis and latent structure analysis together form an important contribution to the development of more reliable methods of subjecting attitudes-and similar "qualities"-to precise and meaningful measurement. The prediction part of Measurement and Prediction does not contain any comparable theoretical developments. For the most part, prediction in the social sciences today is not above the level of "enlightened common sense," as the authors recognize. IV
The distinctive development in mathematics in the last one hundred years is the rise of a discipline whose central concept is neither number nor quantity, but structure. The mathematics of structure might be said to focus on qualitative differences. Topology, as the new discipline is called. is occupied with those properties of ftgures-conceived as sets of pointsthat are independent of mere differences of quantity. Squares, circles, and rectangles, of whatever size, are topologically indistinguishable from one another, or from any simple closed figure, however irregular. On the other hand, if a "hole" is inscribed in any of these figures, it thereby becomes qualitatively different from the rest. Topology, more than most sectors of higher mathematics, deals with questions that have direct intuitive meaning. But intuition often misleads us as to the answers, when it does not fail us altogether. For instance, how many different colors are required to color any map so that no two adjoining countries have the same color? It is known. from topological analyses. that five colors are quite enough; but no one has been able to produce a map that needs more than four. or even to prove whether there could or could not be such a map. It is paradoxical that the field of mathematics which deals with the most familiar subject matter is the least familiar to the non-mathematician. A smattering of geometry, algebra, and statistics is very widespread; topology is virtually unknown, even among social scientists sympathetic to mathematics. To be sure, the late Kurt Lewin's "topological psychology" has given the name much currency. But topology, for Lewin, provided only a notation. The rich content of his work bears no logical relation to the topological framework in which it is presented. as is clear from the
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posthumously published collection of his more important papers, Field Theory in Social Science (Harper. 1951). In these papers. talk about the "'ife space," and "paths," "barriers," and "regions" in it. are elaborately sustained metaphors. Such figures of speech are extraordinarily suggestive. but do not in themselves constitute a strict mathematical treatment of the ulife space" in a topological geometry. The actual application of topology to psychology remained for Lewin a program and a hope. One further development must be mentioned as playing an important role in the new approaches: the rise of symbolic logic. As in the case of topology, this discipline can be traced back to the 17th century. to Leibniz's ideas for a universal language; but not till the late 19th century did it undergo any extensive and precise development. Boolean algebra provided a mechanical method for the determination of the consequences of a given set of propositions (in another form, this is called the "calculus of propositions"). A few years later. De Morgan, Schroeder, and the founder of Pragmatism, Charles Peirce, investigated the formal properties of relations. leading to an elaborately abstract theory of relations. These results, together with work on the foundations of mathematics (like Peano's formulation of five postulates sufficing for the whole of arithmetic). were extended and systematized shortly after the turn of the century in Russell and Whitehead's monumental Principia Mathemalica. The word cybernetics (N. Wiener. Cybernetics, Wiley. 1948) is from the Greek for "steersman," from the same root as the word "governor." It is Wiener's coinage for the new science of "control and communication in the animal and the machine." The element of control is that exemplified in the mechanical device known as a governor, operating on the fundamental principle of the feed-back: the working of the machine at a given time is itself a stimulus for the modification of the future course of its working. Communication enters into cybernetics by way of an unorthodox and ingeniously precise concept of "information" as the degree to which the impUlses entering a machine reduce the uncertainty among the set of alternatives available to it. Thus if a machine contains a relay which has two possible positions, an impulse which puts the relay into one of these two has conveyed to the machine exactly one "bit" of information. All communication can be regarded as made up of such bits, as is suggested by the dots and dashes of the Morse code. for example. Modern machines actually use a binary arithmetic, i.e.• a notation in which an numbers are expressed in terms of powers of 2 (rather than 10), and are therefore formulable as strings of the cyphers for ·'zero" and "one." One bit of information is conveyed by each choice of either a zero or a one, the word "bit" being an abbreviation for "binary digit," Working with these concepts of communication and control, cybernetics becomes relevant to the study of man because human behavior is
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paralleled in many respects by the communication machines. This paral1el is no mere metaphor, but consists in a similarity of structure between the machine processes and those of human behavior. The Darwinian continuity between man and the rest of nature has now been carried to completion: man's rationality marks only a difference of degree from other animals, and fundamentally, no difference at all from the machine. For modern computers are essentiaHy logical machines: they are designed to confront propositions and to draw from them their logical conclusions. With communication and control as the key, a similarity of structure can also be traced between an individual (whether human or mechanical) and a society (again, whether human or machines in a well-designed factory). The metaphors of Plato and Hobbes can now be given a literalist interpretation. Thus cybernetics bears on the study of human behavior in a variety of ways: most directly, by way of neurology and physiological psychology; and by simple extension, to an improved understanding of functional mental disorders, which Wiener finds to be primarily diseases of memory, thus arriving at Freudian conclusions by a totally different route. Of particular interest are the implications, definitely present though not explicitly drawn, for such classical philosophical puzzles as those concerning free will and the mind-body relation. The analysis of mind and individual personality as a structure of certain information processes renders obsolete not only the "mind substance" of the idealist, but mechanistic materialism as well. Mind is a patterning of information and not spirit, matter, or energy. In a more recent book (The Human Use of Human Beings, Houghton Mifflin, 1950) Wiener considers the relationship between cybernetics and the study of man at quite a different level. The sorts of control mechanisms with which cybernetics is concerned are creating, Wiener argues, a fundamental social transformation which he calls the "cybernetic revolution." This is the age of such mechanisms, he says, as the 18th century was of the clock and the 19th of the steam engine. It is now possible to construct machines for almost any degree of elaborateness of performance --chess-playing machines, for example, are no longer the hoaxes of Edgar Allan Poe's day. It is even possible to arrange machines so that they can communicate with one another, and in no merely figurative sense. And in addition to electronic brains, machines can be equipped with sensory receptors and efferent channels. The social problems posed by the "cybernetic revolution" are basically those of the industrial one, but on an enlarged scale: whether the new technology is to be organized so as to produce leisure or unemployment, the ennoblement or degradation of man. The Human Use of Human Beings consists mainly in a forceful statement of this problem. The Indus-
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trial Revolution at least allowed for the localization of human dignity in man's reason, which stm p1ayed an indispensable role in the operation of the technology, even though his muscle could be increasingly dispensed with. This last line of defense is in process of being undermined. Wiener accordingly devotes his book to a vigorous protest against "any use of a human being in which less is demanded of him and less is attributed to him than his ful1 status." In the concrete, this inhuman use of man he finds in the increasing control over the individual in social organization. particularly in the control over information, as in scientific research under military sponsorship. But cybernetics itself only al10ws us to understand something of the technological developments that have posed this social problem; it does not contain, nor does it pretend to contain, any scientific theory-mathematical or otherwise-of how the problem can be resolved. Among the recent applications to the study of man of this whole generaJ body of ideas. one is especially celebrated; the theory of games presented by J. von Neumann and O. Morgenstern in Theory of Games and Economic Behavior (Princeton University Press, 1947). Here the focus is confined to problems of economics, but it is hoped that it will be extended to the whole range of man's social relations. It may seem, superficia]]y. that von Neumann and Morgenstern, in selecting games as a way of approaching the study of social organization. fall into the trap of oversimplification. But unlike Rashevsky, von Neumann and Morgenstern do not so much introduce simplifying assumptions in order to deal artificially with the whole complex social order as select relatively simple aspects of that order for analysis. Only after a theory adequate to the simple problems has been developed can the more com~ plicated problems be attacked fruitfully. To be sure, the decision-maker cannot suspend action to satisfy the scruples of a scientific conscience; but neither can the scientist pretend to an advisory competence that he does not have, in order to satisfy practical demands. While the theory of games does not deal with the social process in its full complexity, it is not merely a peripheral aspect of that process which it studies, but its very core. The aim is "to find the mathematically complete principles which define 'rational behavior' for the participants in a soCial economy, and to derive from them the general characteristics of that behavior." Games are analyzed because the pattern of rational behavior that they exhibit is the same as that manifested in social action, insofar as the latter does in fact involve rationality. The theory is concerned with the question of what choice of strategy is rational when all the relevant probabilities are known and the outcome is not determined by one's choice alone. It is in the answer to this question that the new mathematics enters. And with this kind of mathematics, the
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social sciences finally abandon the imitation of natural science that has dogged so much of their history. The authors first present a method of applying a numerical scale other than the price system to a set of human preferences. A man might prefer a concert to the theater, and either to staying at home. If we assign a utility of "I" to the last alternative, we know that going to the theater must be assigned a higher number, and the concert a higher one still. But how much higher? Suppose the decision were to be made by tossing two coins, the first to settle whether to go to the theater or not, the second (if necessary) to decide between the remaining alternatives. If the utility of the theater were very little different from that of staying at home, most of the time (three-fourths, to be exact) the outcome would be an unhappy one; similarly, if the theater and concert had comparable utilities, the outcome would be usually favorable. Just how the utilities compare, therefore, could be measured by allowing the second coin to be a loaded one. When it is a matter of indifference to the individual whether he goes to the theater or else tosses the loaded coin to decide whether he hears the concert or stays home, the loading of the coin provides a numerical measure of the utilities involved. Once utility can be measured in a way that does not necessarily correspond to monetary units, a theory of rational behavior can be developed which takes account of other values than monetary ones. A game can be regarded as being played, not move by move, but on the basis of an overall strategy that specifies beforehand what move is to be made in every situation that could possibly arise in the game. Then, for every pair of strategies seJected--one by each player in the game--the rules of the game determine a value for the game: namely, what utility it would then have for each player. An optimal strategy is one guaranteeing a certain value for the game even with best possible play on the part of the opponent. Rational behavior, that is to say, is characterized as the selection of the strategy which minimizes the maximum loss each player can sustain. If a game has only two players and is "zero-sum"-whatever is won by one player being lost by the other, and vice versa-then, if each player has "perfect information" about all the previous moves in the game, there always exists such an optimal strategy for each player. The outcome of a rationally played game can therefore be computed in advance; in principle, chess is as predictable as ticktacktoe. Not every game, however, is as completely open as chess. In bridge and poker, for example, we do not know what cards are in the other players' hands; and this is ordinarly the situation in the strategic interplay of management and labor, or in the relations among sovereign states. In such cases rationality consists in playing several strategies, each with a
SocioloV' LADrn! t"~ l.Q"'UD~ 01 MDt"~mDtlcs
1311
mathematically determined probability. Consider a type of matching pennies in which each player is permitted to place his coin as he chooses. If we were to select heads always, we should be quickly found out; even if we favored heads somewhat, our opponent (assuming it is he who wins when the coins match) could always select heads and thereby win more than half the time. But if we select heads half the time-not in strict alternation, to be Sllre, but at random-then, no matter what our opponent does, we cannot lose more than half the time. Rational play consists in what is actually done in the game: we toss the coin each time to determine, as it were, whether we select heads or tails. Of course, in more complex games our strategies are not always to be "mixed" in equal proportions. The fundamental theorem of the theory of games is that for every two-person zerO-sum game, no matter how complex Ithe rules or how great the range of possible strategies open to eacli player, there always exists some specific pattern of probabilities of strategies which constitutes rational play. It minimizes the maximum loss that each player can sustain, not in every play of the game, but in the long run. And there is a mathematical solution that tells us what this strategy is. Unfortunately, many games are not uzero-sum": in the game between management and labor, utilities are created in the process of play; in war they are destroyed. It is simply not true in such cases that what one side loses the other gains, or vice versa. In such cases, the mathematics of the uzero-sum" game will not apply. Moreover, many games have more than two-players: a number of manufacturers may be competing for a single market. Here the mathematics of a two-person game wi11 not hold. The first difficulty, however, can be absorbed into the second. A nonuzero-sum" two-person game can be regarded as a three-person game, where the third person wins or loses whatever is lost or won by the other two together. But how are we to solve games of more than two persons? Only if coalitions are formed, in effect making the game a two-person one. This is not an unrealistic assumption, since, obviously, if two players can coordinate their strategies against a third, they wil1 enjoy a special advantage: odd-man-out is the simple but fundamental principle in such situations. For such games, however, the theory does not provide a detailed solution, for it cannot determine what is a rational division of the spoils between the members of a coalition in the way it can determine what is a rational strategy for the coalition as a whole. And here, of course, is the great difficulty in politics. The United States and Russia may be conceived of, in this theory, as playing a two-person non"zero-sum" game with nature as the third player: only nature wins from atomic destruction, only nature loses if resources need not be diverted to military purposes. But the coalition of men against nature still leaves open
1312
how the utilities acquired in the game are to be divided between the participants. And here conflicting interests stand in the way of the joint interests that would make rational a coalition strategy. From our present standpoint, the important outcome of the theory is to show that there exists a rigorously mathematical approacb to precisely those aspects of the study of man that have seemed in tbe past to be least amenable to mathematical treatment--questions of conflicting or parallel interest, perfect or imperfect information, rational deoision or chance effect. Mathematics is of importance for the social sciences not merely in the study of those aspects in which man is assimilable to inanimate nature, but precisely in his most human ones. On this question the theory of games leaves no room for doubt. But a mathematical theory of games is one thing, and a mathematical theory of society another. Von Neumann and Morgenstern, it must be said, never confuse the two. Many fundamental difficulties remain, even within the limitations of the games framework. The theory of games involving many players is in a very unsatisfactory state; there is no way at present of comparing the utilities of different persons; and the whole theory is so far a static one. unable to take account of the essential learning process which (it may be hoped) takes place in the course of the selection of real-life strategies. Yet the theory has already provided enormous stimulation to mathematical and substantive research, and much more can be expected from it. Above all, it has shown that the resources of the human mind for the exact understanding of man himself are by nO means already catalogued in the existing techniques of the natural sciences. Thus the application of mathematics to the study of man is no longer a programmatic hope, but an accomplished fact. The books we have surveyed are by no means the only mathematical approaches to problems of social science published even within the past few years. For instance, K. Arrow's Social Choice and Individual Values (Wiley, 1951) is a penetrating application of postulational method and the logical theory of relations to the problem of combining, in accord with democratic principles. individual preferences into a representative set of preferences for the group as a whole. Harold Lasswel1 and his associates report in The Language of Politics (0. W. Stewart, 1949) on the procedures and results of the application of the now widely familiar content-analysis techniques to political discourse, in order to objectify and quantify the role of ideologies and utopias in politics. Shannon and Weaver's Mathematical Theory of Communication (University of Illinois Press. 1950) reprints the classic papers in which Claude Shannon first developed the precise theory of information now being applied in cybernetics, linguistics, and elsewhere. There are a number of other books; and dozens of papers have appeared
1313
in the professional journals of a wide variety of disciplines, including such pre-eminently "qualitative" fields as social organizations, psychiatry. and even literary history. But if the new language is widely spoken. there are also wide local differences in dialects, and many individual pecuHarities of usage. Yet on the scientific scene, this is in itself scarcely objectionable. New problems call for new methods. and if for a time these remain ad hoc, it is only from such a rich variety of approaches that systematic and general theories can emerge. No such theories of society or even of part of the social process have yet been developed. though, as we have seen, there have been some premature attempts. But the situation is promising. If tbe mathematics employed is not merely notational, if it is not merely an "illustration" of an abstract methodology, if it does not outstrip the availability of data and especially of ideas, there is reason to hope that it will eventually contribute as much to the study of man as it already has to the understanding of the rest of nature.
PART VII
The Laws of Chance 1. Concerning Probability by PIERRE SIMON DE LAPLACE 2. The Red and the Black by CHARLES SANDERS PEIRCE 3. The Probability of Induction by CHARLES SANDERS PEIRCE 4. The Application of ProbabiJity to Human Conduct by JOHN MAYNARD KEYNES 5. Chance by HENRI POINCARE 6. The Meaning of Probability by ERNEST NAGEL
COMMENTARY ON
PIERRE SIMON DE LAPLACE
H
ISTORIANS of science and biographers have no difficulty pronouncing a verdict on the Marquis de Laplace. As a scientistmathematical astronomer and mathematician-he was second only to Newton; as a person, his qualities were mixed. He was ambitious t but not unamiable; disposed to borrow without acknowledgment from the works of others~ but not ungenerous. Above all he was a virtuoso in the art of rapid adaptation to a changing social and political environment. He is condemned for this-probably because it is such a common frailty that mere mention of it makes everyone uncomfortable. Pierre Simon de Laplace was born at Beaumont-en-Auge, a Normandy viJIage in sight of the English Channel, on March 23 t 1749. The facts of his Hfe, of the earHer years especiaUy, are both sparse and in dispute.1 Most of the original documents essential to an accurate account were burned in a fire which in J 925 destroyed the chateau of his great-greatgrandson the Comte de Colbert-Laplace; others were lost during World War II in the hombardment of Caen. Many errors about Laplace's life have gained currency: that his father was a poor peasant, that he owed his education to the generosity of prosperous neighbors, that after he became famous he sought to conceal his "humble origins." Recent researches hy the mathematician Sir Edmund Whittaker seem to show that whatever Laplace's reasons for reticence about his chHdhood, poverty of his parents was not among them. His father owned a small estate and was a syndic of the parish; his family belonged to the "good boUrgeoisie of the land." One of Laplace's uncles was a surgeon, another a priest. The Jatter. a member of the teaching staff of the Benedictine Priory at Beaumont. where Laplace had his first schooling, is said to have awakened the hoy's interest in mathematics. For a time it was thought that Laplace would follow his uncle's profession as a priest, but at the Uni· versity of Caen, which he entered at the age of sixteen, he soon demonstrated his mathematical inclinations. He wrote a paper on the calculus of finite differences which was published in a journal edited by Joseph Louis I The le~ding older biographical sources are Baron Fourier's tlo1(t. Mlmoires de J'imtillll. X. LXXXI (1831). Rellie £J/n c/opidiqlle. XLIII (1829); S. D. Poisson's Funeral Oration (COIIII. de~ T(!mp~. 1830. p. 19): Fram.ois Arago, Biographief of Di~/;"r:lIl\1l('tI S, ieJIliJ;c Mell. London. 1857: Agnes Mary Clerke. article on Laplace, Encyclopaedia Britannica. Eleventh Edition. Also brief read~ble accounts in E. T. Bell's Mt'l1 of MUlhelUaliCl. New York. 1937. Sir Oliver Lodge. Piol/eers of Science, London. 1928; E J. C. Morton. Hero~~ (If Science. London (n.d.): Sir Robert S. Ball. Greal A IIrmwmel\. London. 1901. The recent re~ x-rays "Roentgen Rays, The" (Bragg), 854-870 ROmer, Olaus, 786 Roos, Charles F., 888 roots, quadratic equation, 1087, 1089 Roscoe, Henry Enfield. 916 rote learning. 1295-1297 Roujon, Henry, 786/n. roulette, initial impulse of needle, 1383, 1387 rowing on running water, 111~1117. 1119 Royal Society of London, 786 Royce, Josiah, 1378 Russell, Bertrand A. W.: acceleration, falling bodies, 729, 730 /n.; economiCS, 1203; Human Knowledge, Its Scope and Limits. 1049/n.; probabilities, 1357; Re/a,hlty, T"~ ABC 0/. 1105: uncertainty principle, 1049; "War the Offspring of Fear," 1238, 1244, 1247, 1153; quoted, 1356 Russia, oil deposits, 929 Rutherford, Ernest: atom, nucleus, 842, 850 RYlands. G. H. W., 1358/n.
S Sachs, Julius von, 1032, 1034; saddle point, 1272-1273. 1275. 1277-1278, 1288, 1289 Soggiatore, II (GalDeo), 731 salt, crystal structure, 878 Salusbury, Thomas, 732/n. salvation, attainment, 1372 Salvio, Alfonso de, 733 In. Sandler, Christian, 795/n. Santa Cruz, Alonso de, SOS SantDlana, Giorgio de, 732/n.; GalU~o, The Crim~ 0/, 733/n. Saturn: motions, 1317, 1320. 1326; obserVations, 791 Saunderson, Nicholas. 807 Savart, F'lix, 1041 "saving the phenomena," 727 Sayer, A. B., 1039/n. scale, eRect of, 1003 scale analysis. 1304-1305, 1306 Schaeffer, G •• 1032/n. Schaud1nn, Fritz, 1044, 1045 Scheinfeld, Amram. New You ond Heredity, Th~, 933-934/n. 8Chismogenesis, 1243-1244 Schoenfties, space groups, 878 Schrl.ider, theory of relations, 1307 Schrl.idlnger, Erwin: "Causality and Wave MechaniCS," 1049, 105~1068: commentary on, 973-974; "Heredity and the Quantum Theory," 975-995; Science and Humanism, 1049 /n .. 1056; wave mechanics, 974, 1050; What Is U/e?, 973, 974/n. Schumacher, Heinrich Christian, 837 Schwann, Theodor, cell theory, 1043 In. Schwarzddld, astronomy, 1069 science: logical, 1219; mathematical and exact, confusion between, 1218-1219; political control, 1301; trial-and-error me.hod, 1297 SCience, A History 0/ (Dampier), 726/n., 1048/n. Science. O,.igins 0/ Modern (Butterfield), 726 In. Science, Pioneers 0/ (Lodge), 732/n. Science, The Principles 0/ (Jevons), 1202 Science and Humonlsm (Schrddinger), 1049 /n., 1056
Science and lhe Modern World (Whitehead), quoted. 726/n. Sciences, Acodlmie Royale des, 785, 78~789, 792 Sci~nlific Adl·entrlr~.
The (Dingle), 728 scientific observation, probabDities, theory of, 1157 Scientific Rel·olution. Th~ (Hall), 726/n. scientific thought, evolution of, 726 ~dileau, planisphere, 794 Seeber, Ludwig August, 851/n. Seguin, tdouard, 1180 selection, measurement of intensity, 965-966 selfishness, 1339 semi-parabola, path of projectDe, 755-758 Seneca, comets, 1326 Senior, Nassau William, quoted, 1222 sensation, measuring, 1159, 1161-1163 sense-distance, 1161 sensitivity, measuring, I1S9 Shakespeare, William, quoled, 1093 Shannon, Claude E., information theory of, 1312 Shapley, HarlOW, t039/n. Shelley, Percy Bysshe, quoted. 1325 Short, James, 815/n. shortsightedness, sight, cause, %0-961 SidgWick, Henry, 1355 Siedenlopf, Henry Friedrich W., microns, 1039
In. similitude, prinCiple of. 1004, 1005, 1007-1008, 1012, 1030, 1039 Sirius (star): density, 1070; radius, 1077 Sitter, Willem de, 1070, 1075 size, weight adjustment, 952-953 "Size, On Being the Right" (Haldane), 952957 size and weakness concomitant, 1004 Siui, Francesco, quoted, 822 skeletal efficiency, animals, 1007 skeletal proportions, 1006 Slosson, Edwin E., 1374/n. Smeaton. John, 1007 Smi~h, Kennelh, 1036/n. Smyth, Charles P' J 838 Snell, Wiliebrod. 792 soap bubble: breaking, 894-896; commentary on, 882-885; composite, 897-900; divided, volume, 1031; interface. 897, 898, 900; size limit, 893, spherical, 909; surface concentration, 893; tension, 894; thickness, 1040 "Soap-bubble, The" (Boys). 891-900 Soap B.bbles: Their Colou,.s and th~ FOrc~s Which Mould Them (Boys), 884/n.; quoted, 886 soap film experiments, 901-909 "Soap Film Experiments with Minimal Surfaces" (Courant), 884/n. soap solution, surface tension, 891 Social ChOice and Individual Values (Arrow), 1312 Social Science. Field Theo,.y in (Lewin), 1307 social sciences, quantitative systematics, 12981299 socioilism, biolopts, 956 society, improvement of, 1192 Society, Dimensions of (Dodd), 1198. 1299 Soddy, Frederick, 850 solar system, changes, 1318 solids: amorphous, 984; aperiodic, 985-986; five regular, 873 Solids Naturally Contaln~d within Solids (Steno). 873
lruln:
Xl"
solution: concept, 1279--1280, 1281; discrimi. natory, 1282; surface tension, recipe, 902 Sommerfeld, Arnold, 8SI, 10SO, 1106/11. Sorokin, Pllirim A .• 1263 sound: reftc
:: c
.. z
HEleHT
IN
INCHES
FIGURE 13-Hlstollram corresponding to the ogive of Figure 12.
Supposing the giant to be 9 feet 7 inches and the dwarf 3 feet 2 inches, we should have obtained for our range the value 6 feet 5 inches. It is obviously undesirable to have a measure which will depend entirely on the value of any freaks that may occur. It is impossible for a measure based on freaks to speak as the representative of the ordinary population. The range, then, although it is used in certain circumstances, is not ideal as a measure of dispersion. 4 It would be better to have a parameter less likely to be upset by extreme values. We may tackle this problem by devising a measure for dispersion along the same line that we took for the median when we were discussing measures of central tendency. The median was the value above which 50% of the population fell and below which the other 50% fell. Suppose, now, we divide the population, after it has been set out in order of size, into four equal groups. The value above which only 25 % of the popUlation falls we call the upper quartile, and the value below which only 25 % of the population falls we call the lower quartile. Evidently, 50% of the population falls between the upper and lower quartile values. The reader may care to check for himself that the upper and lower quartiles, for the table of heights we are using as an example, are roughly 5 feet 9 inches and 5 feet 6 inches respectively. Thus, we may see at once that roughly 50% of the population differ in height by amounts not exceeding three inches, despite the fact that the tallest man observed was no less than 2 feet 9 inches taller than the shortest man. This, of course, is a consequence of the way in which the large majority of heights cluster closely to the average. This is a very common effect. Intelligence Quotients be.. The range is very efficient when the samples contain very few items.
1507
On the A verare and Scatter
have in the same sort of way. Most people are little removed from average intelligence, but geniuses and morons tend to occur in splendid isolation. (We may recall here that the modal ('fashionable') value tends to coincide with the arithmetic mean when the distribution is fairly symmetrical.) Thus the interquartile range, i.e., the difference between the upper and lower quartile values, makes a good measure of dispersion. It is immune from the disturbances occasioned by the incidence of extreme values. It is easy to calculate. It has a simple and meaningful significance in that it tells us the range of variability which is sufficient to contain 50% of the popUlation. The interquartile range is frequently used in economic and commercial statistics for another reason. Often, data are collected in such a way that there are indetenninate ranges at one or both
... ...ow ...0 po L
..,Z
...0 •... •Z ~
Z
...u •...• ,.c
U45
""14
u-.,
144'
so- .. 100 -,. .
zoo .. z••
771
_.64
4],
:S00-3" .7S 400 " OYIJII
_In
FIGURE 14-Showmg numbers of firms wIth the stated number of employees
1n
the food. drink.
and tobacco trades of Great Britain. (Based on Census of Production 1930, quoted by M. G. Kendall. "Advanced StatistIcs," Vol. I.)
ends of the table. An example is shown in Figure 14. The largest group is labelled '400 and over.' This is vague, and it would obviously be impossible to do a precise calculation for any measure depending on arithmetical processes involving the actual values in the unbounded upper class. (We shall show in the next chapter how the limited vagueness in the other bounded classes is dealt with.) The median and the interquartile range provide us with measures of central tendency and scatter respectively in such cases. Median and quartiles are simply special cases of a quite general scheme for dividing up a distribution by quantiles. Thus, we may arrange our distribution in order of size and split it up into ten groups containing equal numbers of the items. The values of the variable at which the divisions occur are known then as the first, second, third, and so on, deciles. This idea is used by educational psychologists to divide pupils into 'top 10%, second 10%, third 10%,' and so on, with regard to inherent intelligence in so far as that characteristic may be measured by tests.
M. 1. Moroney
lS08
Yet another measure of dispersion, which depends on all the measurements, is the mean deviation. In order to calculate this parameter, we first of all find the arithmetic mean of the quantities in the distribution. We then find the difference between each of the items and this average, calling all the differences positive. We then add up all the differences thus obtained and find the average difference by dividing by the number of differences. Thus the mean deviation is the average difference of the several items from their arithmetic mean. In mathematical form we have :£Ix -
Mean Deviation
xl
= --n
where as before the symbol x stands for the arithmetic mean of the various values of x. The sign Ix - xl indicates that we are to find the difference between x and the average of the x values, ignoring sign. The sign:£ means 'add'up all the terms like.' Example. Find the arithmetic mean and mean deviation for the set of numbers; 11, 8,6, 7, 8. Here we have n = S items to be averaged. As previously shown, the average of the items is :£x
11
+8 + 6 +7 +8
40
x=-=--------=-= S
n
5
8
In order to get the mean difference, we calculate the several differences of the items from their average value of 8 and sum them, thus: tIl -
3
81 + 18 - 81 + 16 - 81 + 17 - 81 + 18 - 81
+
0
+
2
+
1
~
0
=6
We then ca1culate the mean deviation by dividing this total of the deviations by n = 5, and so find the mean deviation as % = 1 ·2. The mean deviation is frequently met with in economic statistics. The measures so far suggested are often used in elementary work on account of their being easy to calculate and easy to understand. They are, however, of no use in more advanced work because they are extremely difficult to deal with in sampling theory, on which so much of advanced work depends. The most important measure of dispersion is the standard deviation, which is a little more difficult to calculate and whose significance is less obvious at first sight. Calculation and interpretation, however, soon become easy with a little practice, and then the standard deviation is the most illuminating of all the parameters of dispersion. The standard deviation will be familiar to electrical engineers and mathematicians as the root-mean-square deviation.~ The general reader will do well to remember this phrase as it will help him to remember exactly how the 3
It is strictly analogous to radius of gyration in the theory of moments of inertia.
1509
On the A \Ierage and Scatter
standard deviation is calculated. We shall detail the steps for the calculation of the standard deviation of a set of values thus: Step I. Calculate the arithmetic average of the set of values. Step 2. Calculate the differences of the several values from their arithmetic average. Step 3. Calculate the squares of these differences (the square of a number is found by multiplying it by itself. Thus the square of 4 is written 42 and has the value 4 X 4:= 16). Step 4. Calculate the sum of the squares of the differences to get a quantity known as the sample sum of squares. Step 5. Divide this 'sample sum of squares' by the number of items, n. in the set of values. This gives a quantity known as the sample variance. Step 6. Take the square root of the variance and so obtain the standard deviation. (The square root of any number. x, is a number such that when it is multiplied by itself it gives the number x. Thus, if the square root of x is equal to a number y then we shall have y2 = Y X Y = x.) This sounds much more complicated than it really is. Let us work out an example, step by step. Example. Find the standard deviation of the set of values 11, 8, 6, 7, 8. Step 1. We calculated the arithmetic average previously as x:= 8. Step 2. The differences of the items from this average (sign may be ignored) are: 3, 0, 2, 1, O. Step 3. The squares of these differences are:
3X3=9
OXO=O
2X2=4
Ixl=1
OXO=O
Step 4. The sample sum of squares is: 9 + 0 + 4 + 1 + 0:= 14. Step 5. Dividing the sample sum of squares by the number of items, n:= 5, we get the sample variance as s2 := 1% := 2·8 (S2 is the accepted symbol for sample variance). Step 6. The standard deviation is found as the square root of the sample variance thus: s y'278 1· 673. The formula for the standard deviation is:
=
=
We have seen how to calculate the standard deviation. What use is it to us in interpretation? Actually it is very easy to visualize. If we are given any distribution which is reasonably symmetrical about its average and which is unimodal (i.e., has one single hump in the centre, as in the histogram shown in Figure 13) then we find that we make very little error in assuming that two-thirds of the distribution lies less than one standard deviation away from the mean, that 95 % of the distribution lies less than two standard deviations away from the mean, and that less than
M. 1. Moroney
1510
1% of the distribution lies more than three standard deviations away from the mean. This is a rough rule, of course, but it is one which is found to work very well in practice. Let us suppose, for example, that we were told no more than that the distribution of intelligence, as measured by lntelliMental Age gence Quotients (a person's LQ. is defined as X 100) Chronological Age has an average value :x == 100, with standard deviation s == 13. Then we might easily picture the distribution as something like the rough sketch shown in Figure 15. The reader may care to compare the rough picture thus formed from a simple knowledge of the two measures x and s with the histogram shown in Figure 16 which is based on results obtained by L. M. Terman and 1=.00 S-IJ
140
J STD. DIYN.
1 ITI). OEVN'
t
AVERAGI , Q.
FIGURE IS-Knowing only that we have a fairly symmetrical. unimodal distrtbutlon whose mean value is I.Q. 100 units and whose standard deviation is I.Q. 13 units. we can at once pIcture In our minds that the distnbutlon looks somethmg as shown.
quoted by 1. F. Kenney from his book The Measurement of Intelligence. This is typical of the use of measures of central tendency and dispersion in helping us to carry the broad picture of a whole distribution (provided it be reasonably symmetrical and unimodal) in the two values x and s. Such measures properly may be said to represent the distribution for which they were calculated. The measures of dispersion which we have so far dealt with are all expressed in terms of the units in which the variable quantity is measured. It sometimes happens that we wish to ask ourselves whether one distribu-
1511
011 the A verage and Scatter
tion is relatively more variable than another. Let us suppose, for example, that for the heights of men in the British Isles we find a mean value 57 inches with standard deviation 2·5 inches, and that for Spaniards the mean height is 54 inches with standard deviation 2·4 inches. It is evident that British men are taller than Spaniards and also slightly more variable in height. How are we to compare the relative variability bearing in mind :2= 100
Z w
-u ... Z z -c w til:.
S
=13
V C
lilt
w""
,..
..
'C
....
u V
z W
0
;:)w
ow ...-c .:
...
"'" ..,.
1.0.
FIGURE 16-Dlstributlon of Intelligence Quotient (compare with Figure is). DistributIOn of 1.Q. With
x-100,
$
== 13. Based on data by L. M. Terman and quoted by J. F. Kenney,
"Mathernaucs of Statistics," Vol. I).
that .the Spaniards are shorter in height than the British? Karl Pearson's coefficient of variation is the most commonly used measure in practice for such a case.
100s It is defined as: v::::X
If we calculate the coefficient of variation for our two cases, we get:
100 British
X 2·5
v ::::
67 100 X 2·4
Spaniards v ::::
64
=37·3 %
=37 ·5%
We conclude that, though the British are more variable in an absolute sense, the variability of the Spaniards, expressed as a percentage of the mean height, is just slightly greater.
6
For her own breakjast she'll project a scheme -EDWARD Nor take her tea wIthout a stratagem.
YOUNG
"Come little girl, you seem To want my cup oj tea And will you take a little cream? Now teU the truth to me" She had a rustic woodland grin Her cheek was sojt as silk, And she replied, "Sir, please put in A little drop oj milk"
(The Poets at Tea)
-BARRY PAIN
(1683-1765)
Mathematics of a Lady
Tasting Tea By SIR RONALD A. FISHER
STATEMENT OF EXPERIMENT
A LADY declares that by tasting a cup of tea made with milk she can discriminate whether the milk or the tea infusion was first added to the cup. We will consider the problem of designing an experiment by means of which this assertion can be tested. For this purpose let us first lay down a simple form of experiment with a view to studying its limitations and its characteristics, both those which appear to be essential to the experimental method, when well developed, and those which are not essential but auxiliary. Our experiment consists in mixing eight cups of tea, four in one way and four in the other, and presenting them to the subject for judgment in a random order. The subject has been told in advance of what the test will consist, namely that she will be asked to taste eight cups, that these shall be four of each kind, and that they shall be presented to her in a random order, that is in an order not determined arbitrarily by human choice, but by the actual manipulation of the physical apparatus used in games of chance, cards, dice, roulettes, etc., or, more expeditiously, from l published collection of random sampling numbers purporting to give the actual results of such manipulation. Her task is to divide the 8 cups into two sets of 4, agreeing, if possible, with the treatments received. INTERPRETA TlON AND ITS REASONED BASIS
In considering the appropriateness of any proposed experimental design, it is always needful to forecast all possible results of the experiment, and 1512
Mathematics 01 a Lady Tasting Tea
IS13
to have decided without ambiguity what interpretation shall be placed upon each one of them. Further, we must know by what argument this interpretation is to be sustained. In the present instance we may argue as follows. There are 70 ways of choosing a group of 4 objects out of 8. This may be demonstrated by an argument familiar to students of "permutations and combinations," namely. that if we were to choose the 4 objects in succession we should have successively 8, 7, 6, 5 objects to choose from, and could make our succession of choices in 8 X 7 X 6 X 5, or 1680 ways. But in doing this we have not only chosen every possible set of 4, but every possible set in every possible order; and since 4 objects can be arranged in order in 4 X 3 X 2 xl, or 24 ways, we may find the number of possible choices by dividing 1680 by 24. The result, 70, is essential to our interpretation of the experiment. At best the subject can judge rightly with every cup and, knowing that 4 are of each kind, this amounts to choosing, out of the 70 sets of 4 which might be chosen, that particular one which is correct. A subject without any faculty of discrimination would in fact divide the 8 cups correctly into two sets of 4 in one trial out of 70, or, more properly, with a frequency which would approach 1 in 70 more and more nearly the more often the test were repeated. Evidently this frequency, with which unfailing success would be achieved by a person lacking altogether the faculty under test, is calculable from the number of cups used. The odds could be made much higher by· enlarging the experiment, while, if the experiment were much smaller even the greatest possible success would give odds so low that the result might, with considerable probability, be ascribed to chance.
THE TEST OF SIGNIFICANCE
It is open to the experimenter to be more or less exacting in respect of the smallness of the probability he would require before he would be willing to admit that his observations have demonstrated a positive result. It is obvious that an experiment would be useless of which no possible result would satisfy him. Thus, if he wishes to ignore results having probabilities as high as 1 in 20-the probabilities being of course reckoned from the hypothesis that the phenomenon to be demonstrated is in fact absent-then it would be useless for him to experiment with only 3 cups of tea of each kind. For 3 objects can be chosen out of 6 in only 20 ways, and therefore complete success in the test would be achieved without sensory discrimination, i.e., by "pure chance," in an average of 5 trials out of 100. It is usual and convenient for experimenters to take 5 per cent. as a standard level of significance, in the sense that they are prepared to ignore all results which fail to reach this standard, and, by this means, to eliminate from further discussion the greater part of the fluctu-
1514
Sir RDnald ..4.. Fisher
alions which chance causes have introduced into their experimental results. No such selection can eliminate the whole of the possible effects of chance coincidence, and if we accept this convenient convention, and agree that an event which would occur by chance only once in 70 trials is decidedly "significant," in the statistical sense, we thereby admit that no isolated experiment, however significant in itself, can suffice for the experimental demonstration of any natural phenomenon; for the "( ne chance in a million" will undoubtedly occur, with no less and no more than its appropriate frequency, however surprised we may be that it should occur to us. In order to assert that a natural phenomenon is experimentally demonstrable we need, not an isolated record, but a reliable method of procedure. In relation to the test of significance, we may say that a phenomenon is experimentally demonstrable when we know how to conduct an experiment which will rarely fail to give us a statistically significant result. Returning to the possible results of the psycho-pbysical experiment, having decided that if every cup were rightly classified a significant positive result would be recorded, or, in otber words, that we should admit that the lady had made good her claim, what should be our conclusion if, for each kind of cup, her judgments are 3 right and 1 wrong? We may take it, in the present discussion, that any error in one set of judgments will be compensated by an errOr in the other, since it is known to the subject that there are 4 cups of each kind. In enumerating the number of ways of choosing 4 things out of 8, such that 3 are right and 1 wrong, we may note that the 3 right may be chosen, out of the 4 available, in 4 ways and, independently of this choice, that the 1 wrong may be chosen, out of the 4 available, also in 4 ways. So that in all we could make a selection of the kind supposed in 16 different ways. A similar argument shows that, in each kind of judgment, 2 may be right and 2 wrong in 36 ways, 1 right and 3 wrong in 16 ways and none right and 4 wrong in 1 way only. It should be noted that the frequencies of these five possible results of the experiment make up together, as it is obvious they should, the 70 cases out of 70. It is obvious, too, that 3 successes to 1 failure, although showing a bias, or deviation, in the right direction, could not be judged as statistically significant evidence of a real sensory discrimination. For its frequency of chance occurrence is 16 in 70, or more than 20 per cent. Moreover, it is not the best possible result, and in judging of its significance we must take account not only of its own frequency, but also of the frequency for any better result. In the present instance "3 right and 1 wrong" occurs 16 times. and "4 right" occurs once in 70 trials, making 17 cases out of 70 as good as or better than that observed. The reason for including cases better than that observed becomes obvious on considering what our con-
MathematIcs of a Lady Tasting Tea
ISIS
clusions would have been had the case of 3 right and 1 wrong only 1 chance, and the case of 4 right 16 chances of occurrence out of 70. The rare case of 3 right and 1 wrong could not be judged significant merely because it was rare, seeing that a higher degree of success would frequently have been scored by mere chance. THE NULL HYPOTHESIS
Our examination of the possible results of the experiment has therefore r led us to a statistical test of significance, by which these results are divided into two classes with opposed interpretations. Tests of significance are of many different kinds, which need not be considered here. Here we are only concerned with the fact that the easy calculation in permutations which we encountered, and which gave us our test of significance, stands for something present in every possible experimental arrangement; or, at least, for something required in its interpretation. The two classes of results which are distinguished by our test of significance are, on the one hand, those which show a significant discrepancy from a certain' hypothesis; namely, in this case, the hypothesis that the judgments given are in no way influenced by the order in which the ingredients have been added; and on the other hand, results which show no significant discrepancy from this hypothesis. This hypothesis, which mayor may not be impugned by the result of an experiment, is again characteristic of all experimentation. Much confusion would often be avoided if it were explicitly formulated when the experiment is designed. In relation to any experiment we may speak of this hypothesis as the "null hypothesis," and it should be noted that the null hypothesis is never proved or established, but is possibly disproved, in the course of experimentation. Every experiment may be said to exist only in order to give the facts a chance of disproving the null hypothesis. It might be argued that if an experiment can disprove the hypothesis that the subject possesses no sensory discrimination between two different sorts of object, it must therefore be able to prove the opposite hypothesis, that she can make some such discrimination. But this last hypothesis, however reasonable or true it may be, is ineligible, as a null hypothesis to be tested by experiment, because it is inexact. If it were asserted that the subject would never be wrong in her jUdgments we should again have an exact hypothesis, and it is easy to see that this hypothesis could be disproved by a single failure, but could never be proved by any finite amount of experimentation. It is evident that the null hypothesis must be exact, that is free from vagueness and ambiguity, because it must supply the basis of the "problem of distribution," of which the test of significance is the solution. A null hypothesis may, indeed, contain arbitrary elem.ents,
1516
Sir Ronald A. Fisher
and in more complicated cases often does so: as, for example, if it should assert that the death-rates of two groups of animals are equal, without specifying what these death-rates usually are. In such cases it is evidently the equality rather than any particular values of the death-rates that the experiment is designed to test, and possibly to disprove. In cases involving statistical "estimation" these ideas may be extended to the simultaneous consideration of a series of hypOthetical possibilities. The notion of an error of the so-called "second kind," due to accepting the null hypothesis "when it is false" may then be given a meaning in reference to the quantity to be estimated. It has no meaning with respect to simple tests of significance, in which the only available expectations are those which flow from the null hypothesis being true. RANDOMISATION; THE PHYSICAL BASIS OF THE VALIDITY OF THE TEST
We have spoken of the experiment as testing a certain null hypothesis, namely, in this case, that the subject possesses no sensory discrimination whatever of the kind claimed; we have, too, assigned as appropriate to this hypothesis a certain frequency distribution of occurrences, based on the equal frequency of the 70 possible ways of assigning 8 objects to two classes of 4 each; in other words, the frequency distribution appropriate to a classification by pure chance. We have now to examine the physical conditions of the experimental technique needed to justify the assumption that, if discrimination of the kind under test is absent, the result of the experiment will be wholly governed by the laws of chance. It is easy to see that it might well be otherwise. If all those cups made with the milk first had sugar added, while those made with the tea first had none, a very obvious difference in ft.avour would have been introduced which might well ensure that all those made with sugar should be classed alike. These groups might either be classified all right or all wrong, but in such a case the frequency of the critical event in which all cups are classified correctly would not be 1 in 70, but 35 in 70 trials, and the test of significance would be wholly vitiated. Errors equivalent in principle to this are very frequently incorporated in otherwise well-designed experiments. It is no sufficient remedy to insist that "all the cups must be exactly alike" in every respect except that to be tested. For this is a totally impossible requirement in our example, and equally in all other forms of experimentation. In practice it is probable that the cups will differ perceptibly in the thicknes~ or smoothness of their material, that the quantities of milk added to the different cups will not be exactly equal, that the strength of the infusion of tea may change between pouring the first and the last cup, and that the temperature also at which the tea is tasted will change during the course of the experiment. These arc only examples
MathematIcs of a Lady Tasting Tea
1517
of the differences probably present; it would be impossible to present an exhaustive Hst of such possible differences appropriate to anyone kind of experiment, because the uncontrolled causes which may influence the result are always strictly innumerable. When any such cause is named, it is usually perceived that, by increased labour and expense, it could be largely eHminated. Too frequently it is assumed that such refinements constitute improvements to the experiment. Our view, which will be much more fully exemplified in later sections, is that it is an essential characteristic of experimentation that it is carried out with limited resources, and an essential part of the subject of experimental design to ascertain how these should be best applied; or, in particular, to which causes of disturbance care should be given, and which ought to be deliberately ignored. To ascertain, too, for those which are not to be ignored, to what extent it is worth While to take the trouble to diminish their magnitude. For our present purpose, however, it is only necessary to recognise that, whatever degree of care and experimental skill is expended in equalising the conditions, other than the one under test, which are liable to affect the result, this equalisation must always be to a greater or less extent incomplete, and in many important practical cases will certainly be grossly defective. We are concerned, therefore, that this inequality, whether it be great or small, shall not impugn the exactitude of the frequency distribution, on the basis of which the result of the experiment is to be appraised. THE EFFECTIVENESS OF RANDOMISAT,ION
The element in the experimental procedure which contains the essential safeguard is that the two modifications of the test beverage are to be prepared "in random order." This, in fact, is the only point in the experimental procedure in which the laws of chance, which are to be in exclusive control of our frequency distribution, have been explicitly introduced. The phrase "random order" itself, however, must be regarded as an incomplete instruction, standing as a kind of shorthand symbol for the full procedure of randomisation, by which the validity of the test of sig· nificance may be guaranteed against corruption by the causes of dis· turbance which have not been eliminated. To demonstrate that, with satisfactory randomisation, its validity is, indeed, wholly unimpaired, let us imagine all causes of disturbance-the strength of the infusion, the quantity of milk, the temperature at which it is tasted, etc.-to be predetermined for each cup; then since these, on the null hypothesis, are the only causes influencing classification, we may say that the probabilities of each of the 70 possible choices or classifications which the subject can make are also predetermined. If, now, after the disturbing causes are fixed, we assign, strictly at random, 4 out of the 8 cups to each of our
S
R
IdAFih
expenmental treatments then every set of 4 whatever Its probablhty of bemg so clasSified will certamly have a probabihty of exactly 1 m 70 of bemg the 4 for example to which the mIlk IS added first However 1m portant the causes of dIsturbance may be even If they were to make It certam that one partIcular set of 4 should receive thiS classdicatlOn the probablhty that the 4 so classified and the 4 whIch ought to have been so classified should be the same must be rIgorously m accordance with our test of slgmficance It IS apparent therefore that the random choIce of the objects to be treated m dIfferent ways would be a complete guarantee of the valIdity of the test of sIgmficance If these treatments were the last m hme of the stages In the physIcal history of the objects which mIght affect theIr ex penmental reaction The cIrcumstance that the expenmental treatments cannot always be apphed last and may come relatively early m their history causes no practical mconvemence for SUbsequent causes of dlf ferentlatlOn If under the experimenter S control as for example the chOIce of dtfferent pipettes to be used wIth different flasks can either be predetermmed before the treatments have been randomlsed or If thIS has not been done can be randomised on theIr own account and other causes of differentiation Will be either (a) consequences of differences already randomlsed or (b) natural consequences of the ddference m treatment to be tested of whIch on the null hypothesIs there will be none by defi mtlon or (c) effects supervemng by chance mdependently from the treatments apphed Apart therefore from the aVOIdable error of the expenmenter himself mtroducmg With hiS test treatments or subsequently other dIfferences m treatment the effects of whIch the experiment IS not mtended to study It may be said that the sImple precautIon of randomlsa bon WIll suffice to guarantee the valIdity of the test of slgmficance by whIch the result of the expenment IS to be Judged THE SENSITIVENESS OF AN EXPERIMENT EFFECTS OF ENLARGEMENT AND REPETITION
A probable objection which the subject mIght well make to the expen ment so far descnbed IS that only If every cup IS classified correctly WIll she be Judged successful A smgle mistake wIll reduce her performance below the level of sIgmficance Her clalm however might be not that she could draw the dlstmctlon WIth mvanable certamty but that though sometimes mIstaken she would be nght more often than not and that the experiment should be enlarged sufficIently or repeated suffiCIently often for her to be able to demonstrate the predommance of correct classIficattons m spite of occasIOnal errors An extensIOn of the calculation upon whIch the test of sIgmficance was
MathematIcs of a Lady Tasting Tea
1519
based shows that an experiment with 12 cups, six of each kind, gives, on the null hypothesis, 1 chance in 924 for complete success, and 36 chances for 5 of each kind classified right and 1 wrong. As 37 is less than a twen· tieth of 924, such a test could be counted as significant, although a pair of cups have been wrongly classified; and it is easy to verify that, using larger numbers still, a significant result could be obtained with a still higher proportion of errors. By increasing the size of the experiment, we can render it more sensitive, meaning by this that it will allow of the detection of a lower degree of sensory discrimination, or, in other words, of a quantitatively smaller departure from the null hypothesis. Since in every case the experiment is capable of disproving, but never of proving this hypothesis, we may say that the value of the experiment is increased whenever it permits the null hypothesis to be more readily disproved. The same result could be achieved by repeating the experiment, as originally designed, upon a number of different occasions, counting as a success all those occasions on which 8 cups are correctly classified. The chance of success on each occasion being 1 in 70, a simple application of the theory of probability shows that 2 or more successes in 10 trials would occur, by chance, with a frequency below the standard chosen for testing significance; so that the sensory discrimination would be demonstrated, although, in 8 attempts out of 10, the subject made one or more mistakes. This procedure may be regarded as merely a second way of enlarging the experiment and, thereby, increasing its sensitiveness, since in our final calculation we take account of the aggregate of the entire series of results, whether successful or unsuccessful. It would clearly be illegitimate, and would rob our calculation of its basis, if the unsuccessful results were not all brought into the account. QUALITATIVE METHODS OF INCREASING SENSITIVENESS
Instead of enlarging the experiment we may attempt to increase its sensitiveness by qualitative improvements; and these are, generally speaking, of two kinds: (a) the reorganisation of its structure, and (b) refinements of technique. To illustrate a change of structure we might consider that, instead of fixing in advance that 4 cups should be of each kind, determining by a random process how the subdivision should be effected, we might have allowed the treatment of each cup to be determined inde· pendently by chance, as by the toss of a coin, so that each treatment has an equal chance of being chosen. The chance of classifying correctly 8 cups randomised in this way, without the aid of sensory discrimination, is 1 in 28 , or 1 in 256 chances, and there are only 8 chances of classifying 7 right and 1 wrong; consequently the sensitiveness of the experiment has been increased, while still using only 8 cups, and it is possible to score a
1520
Sir Ronald A.. Fisher
significant success, even if one is classified wrongly. In many types of experiment, therefore, the suggested change in structure would be evidently advantageous. For the special requirements of a psycho-physical experiment, however, we should probably prefer to forego this advantage, since it would occasionally occur that all the cups would be treated alike, and this, besides bewildering the subject by an unexpected occurrence, would deny her the real advantage of judging by comparison. Another possible alteration to the structure of the experiment, which would, however, decrease its sensitiveness, would be to present determined, but unequal, numbers of the two treatments. Thus we might arrange that 5 cups should be of the one kind and 3 of the other, choosing them properly by chance, and informing the subject how many of each to expect. But since the number of ways of choosing 3 things out of 8 is only 56, there is now, on the null hypothesis, a probability of a completely correct classification of 1 in 56. It appears in fact that we cannot by these means do better than by presenting the two treatments in equal numbers, and the choice of this equality is now seen to be justified by its giving to the experiment its maximal sensitiveness. With respect to the refinements of technique, we have seen above that these contribute nothing to the validity of the experiment, and of the test of significance by which we determine its result. They may, however, be important, and even essential, in permitting the phenomenon under test to manifest itself. Though the test of significance remains valid, it may be that without special precautions even a definite sensory discrimination would have little chance of scoring a significant success. If some cups were made with India and some with China tea, even though the treatments were properly randomised, the subject might not be able to discriminate the relatively small difference in flavour under investigation, when it was confused with the greater differences between leaves of different origin. Obviously, a similar difficulty could be introduced by using in some cups raw milk and in others boiled, or even condensed milk, or by adding sugar in unequal quantities. The subject has a right to claim, and it is in the interests of the sensitiveness of the experiment, that gross differences of these kinds should be excluded, and that the cups should, not as far as possible, but as far as is practically convenient, be made alike in all respects except that under test. How far such experimental refinements should be carried is entirely a matter of judgment, based on experience. The validity of the experiment is not affected by them. Their sole purpose is to increase its sensitiveness, and this object can usually be achieved in many other ways, and particularly by increasing the size of the experiment. If, therefore, it is decided that the sensitiveness of the experiment should be increased, the experi-
Mathematics of a Lady Tastmg Tea
1521
menter has the choice between different methods of obtaining equivalent results; and will be wise to choose whichever method is easiest to him, irrespective of the fact that previous experimenters may have tried, and recommended as very important, or even essential, various ingenious and troublesome precautions.
COMMENTARY ON
The Scientific Aptitude of Mr. George Bernard Shaw ERNARD SHAW was not at his best as a scientific thinker. Science interested him but he was inclined to be erratic. Though science offered a fertile field for the exercise of his talents as a controversialist, a foe of pretense and a joker, he was often unable to distinguish between the stuffed robe and the honest scientist, between theories that merited serious attention and theories that were pure humbug. Moreover he himself espoused the most incredible nonsense. He fought vivisection and vaccination; he had a low opinion of medical knowledge and an even lower opinion of its practitioners; he had his own astonis~ing theories of biology, physiology, bacteriology and hygiene, and nothing would persuade him that the sun was burning itself out (since he expected to live longer than Methuselah he felt he had a personal stake in the catastrophe); he dismissed laboratory experiments generally as mere "put-up jobs," performances rigged for the purpose of proving preconceived theories regardless of the weight of evidence. But for all his prejudices and eccentric notions, Shaw did not close his mind to the important works of science. He followed the advances of research in fields as varied as Pavlov's work on dogs and the MichelsonMorley interferometer experiments on ether drift. He "liked visiting laboratories and peeping at bacteria through the microscope." 1 He was curious about how things work: automobiles, radios, machine tools, motorcycles, phonographs. He was an enthusiastic photographer and camera tinkerer. Every efficient labor-saving device won his admiration but "for old-fashioned factory machinery his contempt was boundless: he said a louse could have invented it all if it had been keen enough on profits." 2 Shaw and Jonathan Swift were much alike in their attitudes to science. Both men lived in periods of great scientific advance; both respected science; neither had any special aptitude for it. Both approached the subject as social reformers and satirists; both despised pretentiousness; neither had much use for science as a purely speculative activity. Swift aimed his wit at mathematics, which in its advanced forms seemed to him completely trivial; Shaw waged war on biological practices which he thought
B
1 Hesketh.Pe~rson, G.B.S., A Full Length Portrait, New York, 1942, p. 270. I have drawn on this bIography for many of the details of this sketch' also on Bernard Shaw Sixteen Self-Sketches. New York. 1949. ' , II Pearson, op. cit., p. 270.
1522
The Scientific Aptitude of Mr. George Bernard Shaw
1523
cruel and stupid. That he exaggerated is understandable. He enjoyed exaggeration and he regarded it as an essential tool of reform. "If you do not say a thing in an irritating way, you may just as well not say it at all, since nobody will trouble themselves about anything that does not trouble them." (The grammar is bizarre, even for G.B.S.) There was, however, one branch of science which Shaw neither tilted at nor enlarged with theories of his own. The subject he spared was mathematics. He did not minimize its importance and he admitted, dropping at least this once the pose of omniscience, that he knew very little about it. He blamed his ignorance on the wretched instruction he received at the Wesleyan Connexional School. "Not a word was said to us about the meaning or utility of mathematics: we were simply asked to explain how an equilateral triangle could be constructed by the intersection of two circles, and to do sums in a, b, and x instead of in pence and shillings, leaving me so ignorant that I concluded that a and b must mean eggs and cheese and x nothing, with the result that I rejected algebra as nonsense, and never changed that opinion until in my advanced twenties Graham Wallas and Karl Pearson convinced me that instead of being taught mathematics I had been made a fool of." The influence of these distinguished men was highly beneficial. To be sure Shaw never became unduly proficient as a calculator: "1 never used a logarithm in my life, and could not undertake to extract the square root of four without misgivings." But he learned to appreciate the importance of at least one division of higher mathematics, the theory of probability and statistics. The following selection presents a Shavian version of the development and practical application of the calculus of chance. It is a delightful account and very sensible. No one else, so far as 1 know, has treated the history of mathematics in this way. I suspect that if there were more Shaws teaching the subject it would become popular. But the mathematical probability of this compound circumstance is admittedly small.
Let the king prohibit gambling and beltlng in his kingdom, for these are -THE CODE OF MANU (c. 100)
vices that destroy the kingdoms of prmces.
In play there are two pleasures for your choo,singThe one is winning, and the other losing.
7
-BYRON
The Vice of Gambling and the Virtue of Insurance By GEORGE BERNARD SHAW
INSURANCE, though founded on facts that are inexplicable, and risks that are calculable only by professional mathematicians called actuaries, is nevertheless more' congenial as a study than the simpler subjects of bank~ ing and capital. This is because for every competent politician in our country there must be at least a hundred thousand gamblers who make bets every week with turf bookmakers. The bookmaker's business is to bet against any horse entered for a race with anybody who thinks it will win and wants to bet that it will. As only one horse can win, and all the rest must lose, this business would be enormously lucrative if all the bets were for even money. But the competition among bookmakers leads them to attract customers by offering "odds," temptingly "long," against horses unlikely to win: whilst giving no odds at all on the most likely horse, called the favorite. The well-known cry, puzzling to novices, of "two to one bar one" means that the bookmaker will bet at odds of two to one against any horse in the race except the favorite. Mostly, however, he will bet at odds of ten to one or more against an "outsider." In that case, if, as sometimes happens, the outsider wins, the bookmaker may lose on his bet against it all that he gained on his bets against the favorites, On the scale between the possible extremes of gain and loss he may come out anywhere according to the number of horses in the race, the number of bets made on each of them, and the accuracy of his jUdgment in guessing the odds he may safely offer. Usually he gains when an outsider wins, because mostly there is more money laid on favorites and fancies than on outsiders; but the contrary is possible; for there may be several outsiders as well as several favorites; and, as outsiders win quite often, to tempt customers by offering too long odds against them is gambling; and a bookmaker must never gamble, though he lives by gambling. There are practically always enough variable factors in the game to tax the bookmaker's :financial ability to the utmost. He must budget so as to come out at worst still solvent. A bookmaker who gambles will ruin himself as cer-
The VIce of Gambling and the VIrtue of Insurance
1525
tainly as a licensed victualler (publican) who drinks, or a picture dealer who cannot bear to part with a good picture. The question at once arises, how is it possible to budget for solvency in dealing with matters of chance? The answer is that when dealt with in sufficient numbers matters of chance become matters of certainty, which is one of the reasons why a million persons organized as a State can do things that cannot be dared by private individuals. The discovery of this fact nevertheless was made in the course of ordinary private business. In ancient days, when travelling was dangerous, and people before starting on a journey overseas solemnly made their wills and said their prayers as if they were going to die, trade with foreign countries was a risky business, especially when the merchant, instead of staying at home and consigning his goods to a foreign firm, had to accompany them to their destination and sell them there. To do this he had to make a bargain with a ship owner or a ship captain. Now ship captains, who live on the sea, are not subject to the terrors it inspires in the landsman. To them the sea is safer than the land; for shipwrecks are less frequent than diseases and disasters on shore. And ship captains make money by carrying passengers as well as cargo. Imagine then a business talk between a merchant greedy for foreign trade but desperately afraid of being shipwrecked or eaten by savages, and a skipper greedy for cargo and passengers. The captain assures the merchant that his goods will be perfectly safe, and himself equally so if he accompanies them. But the merchant, with his head full of the adventures of Jonah, St. Paul. Odysseus, and Robinson Crusoe, dares not venture. Their conversation will be like this; CAPTAIN. Come! I will bet you umpteen pounds that if you sail with me you will be alive and well this day year. MERCHANT. But if I take the bet I shall be betting you that sum that I shall die within the year. CAPTAIN. Why not if you lose the bet, as you certainly will? MERCHANT. But if I am drowned you will be drowned too; and then what becomes of our bet? CAPTAIN. True. But I will find you a landsman who will make the bet with your wife and family. MERCHANT. That alters the case of course; but what about my cargo? CAPTAIN. Poohl The bet can be on the cargo as well. Or two bets: one on your life, the other on the cargo. Both will be safe, I assure you. Nothing will happen; and you will see all the wonders that are to be seen abroad. MERCHANT. But if I and my goods get through safely I shall have to pay you the value of my life and of the goods into the bargain. If I am not drowned I shall be ruined.
1526
George Ber,rrard Shaw
That also is very true. But there is not so much for me in it as you think. If you are drowned I shall be drowned first; for I must be the last man to leave the sinking ship. Still, let me persuade you to venture. I will make the bet ten to one. Will that tempt you? MERCHANT. Oh, in that case-The captain has discovered insurance just as the goldsmiths discovered banking. It is a lucrative business; and, if the insurer's judgment and information are sound, a safe one. But it is not so simple as bookmaking on the turf, because in a race, as all the horses but one must lose and the bookmaker gain, in a shipwreck all the passengers may win and the insurer be ruined. Apparently he must therefore own, not one ship only, but several, so that, as many more ships come safely to port than sink, he will win on half a dozen ships and lose on one only. But in face the marine insurer need no more own ships than the bookmaker need own horses. He can insure the cargoes and lives in a thousand ships owned by other people without his having ever owned or even seen as much as a canoe. The more ships he insures the safer are his profits; for half a dozen ships may perish in the same typhoon or be swallowed by the same tidal wave; but out of a thousand ships most by far will survive. When the risks are increased by war the odds on the bets can be lowered. When foreign trade develops to a point at which marine insurers can employ more capital than individual gamesters can supply, corporations like the British Lloyds are formed to supply the demand. These corpora~ tions soon perceive that there are many more risks in the world than the risk of shipwreck. Men who never travel nor send a parcel across the seas, may lose life or limb by accident, or have their houses burnt or robbed. Insurance companies spring up in all directions; and the business extends and develops until there is not a risk that cannot be insured. L10yds will bet not only against shipwreck but against almost any risk that is not specifically covered by the joint stock companies, provirfed it is an insurable risk: that is, a safe one. This provision is a contradiction in terms; for how can a safe transaction involve a risk or a risk be run safely? The answer tlikes us into a region of mystery in which the facts are unreasonable by any method of ratiocination yet discovered. The stock example is the simplest form of gambling, which is tossing a coin and betting on which side of it will be uppermost when it falls and comes to rest. Heads or tails they call it in England, head or harp in Ireland. Every time the coin is tossed, each side has an equal chance with the other of winning. If head wins it is just as likely to win the next time and the next and so on to the thousandth; so that on reasonable grounds a thousand heads in succession are possible~ or a thousand tails; for the fact that CAPTAIN.
The Vice of Gambling and the Virtue of Insurance
1527
head wins at any toss does not raise the faintest reasonable probability that tails will win next time. Yet the facts defy this reasoning. Anyone who possesses a halfpenny and cares to toss it a hundred times may find the same side turning up several times in succession; but the total result will be fifty-fifty or as near thereto as does not matter. I happen to have in my pocket ten pennies; and I have just spilt them on the floor ten times. Result: forty nine heads and fifty one taIls, though five-five occurred only twice in the ten throws, and heads won three times in succession to begin with. Thus though as between any two tosses the result is completely uncertain, in ten throws it may be six-four or seven-three often enough to make betting a gamble; but in a hundred the result will certainly be close enough to fifty-fifty to leave two gamblers, one crying heads and the other tails every time, exactly or very nearly where they were when they started, no richer and no poorer, unless the stakes are so high that only players out of their senses would hazard them. An insurance company, sanely directed, and making scores of thousands of bets, is not gambling at all; it knows with sufficient accuracy at what age its clients will die, how many of their houses-will be burnt every year, how often their houses will be broken into by burglars, to what extent their money 'will be embezzled by their cashiers, how much compensation they will have to pay to persons injured in their employment, how many accidents will occur to their motor cars and themselves, how much they will suffer from illness or unemployment, and what births and deaths will cost them: in short, what will happen to every thousand or ten thousand or a million people even when the company cannot tell what will happen to any individual among them. In my boyhood I was equipped for an idle life by being taught to play whist, because there were rich people who, having nothing better to do, escaped from the curse of boredom (then called ennui) by playing whist every day. Later on they played bezique instead. Now they play bridge. Every gentleman's club has its card room. Card games are games of chance; for though the players may seem to exercise some skin and judgment in choosing which card to play, practice soon establishes rules by which the stupidest player can learn how to choose correctly: that is, not to choose at all but to obey the rules. Accordingly people who play every day for sixpences or shillings find at the end of the year that they have neither gained nor lost sums of any importance to them, and have killed time pleasantly instead of being bored to death. They have not really been gambling any more than the insurance companies. At last it is discovered that insurers not only need not own ships or horses or houses or any of the things they insure, but that they need not exist. Their places can be taken by machines. On the turf the bookmaker, flamboyantly dressed and brazenly eloquent, is superseded by the Total-
1528
Geqrge Bernard Shaw
izator (Tote for short) in which the gamblers deposit the sums they are prepared to stake on the horses they fancy. After the race all the money staked on the winner is divided among its backers. The machine keeps the rest. On board pleasure ships young ladies with more money than they know what to do with drop shillings into gambling machines so constructed that they occasionally return the shilling ten or twentyfold. These are the latest successors of the roulette table, the "little horses," the dice casters, and all other contraptions which sell chances of getting money for nothing. Like the Tote and the sweepstake, they do not gamble: they risk absolutely nothing, though their customers have no certainty except that in the lump they must lose, every gain to Jack and Jill being a loss to Tom and Susan. How does all this concern the statesman? In this way. Gambling, or the attempt to get money without earning it, is a vice which is economically (that is, fundamentally) ruinous. In extreme cases it is a madness which persons of the highest intelligence are unable to resist: they will stake all they possess though they know that the chances are against them. When they have beggared themselves in half an hour or half a minute, they sit wondering at the folly of the people who are doing the same thing, and at their own folly in having done it themselves. Now a State, being able to make a million bets whilst an individual citizen can afford only one, can tempt him or her to gamble without itself running the slightest risk of losing financially; for, as aforesaid, what will happen in a million cases is certain, though no one can foresee what will happen in anyone case. Consequently governments, being continually in pressing need of money through the magnitude of their expenses and the popular dislike of taxation, are strongly tempted to replenish the Treasury by tempting their citizens to gamble with them. No crime against society could be more wickedly mischievous. No public duty is more imperative than the duty of creating a strong public conscience against it, making it a point of bare civic honesty not to spend without earning nor consume without producing, and a point of high civic honor to earn more than you spend, to produce more than you consume, and thus leave the world better off than you found it. No other real title to gentility is conceivable nowadays. Unfortunately our system of making land and capital private property not only makes it impossible for either the State or the Church to inculcate these fundamental precepts but actually drives them to preach just the opposite. The system may urge the energetic employer to work hard and develop his business to the utmost; but his final object is to become a member of the landed gentry or the plutocracy, living on the labor of others and enabling his children to do the same without ever having worked at all. The reward of success in Hfe is to become a parasite and
The Vice of Gambltllg and the Virtue of Insurance
1529
found a race of parasites. Parasitism is the linchpin of the Capitalist applecart: the main Incentive without which, we are taught, human society would faU to pieces. The boldest of our archbishops, the most democratic of our finance ministers, dares not thunder forth that parasitism, for peers and punters alike, is a virus that will rot the most powerful civilization, and that the contrary doctrine is diabolical. Our most eminent churchmen do not preach very plainly and urgently against making selfishness the motive power of civilization; but they have not yet ventured to follow Ruskin and Proudhon in insisting definitely that a citizen who is neither producing goods nor performing services is in effect either a beggar or a thief. The utmost point yet reached in England is the ruling out of State lotteries and the outlawing of the Irish sweepstakes. But here again the matter is not simple enough to be disposed of by counsels of Socialist perfection in the abstract. There are periods in every long lifetime during which one must consume without producing. Every baby is a shamelessly voracious parasite. And to turn the baby into a highly trained producer or public servant, and make its adult life worth living. its parasitism must be prolonged well into its teens. Then again old people cannot produce. Certain tribes who lay an excessive stress on Manchester School economics get over this difficulty easily by killing their aged parents or turning them out to starve. This is not necessary in modern civilization. It is quite possible to organize society in such a manner as to enable every ablebodied and ableminded person to produce enough not only to pay their way but to repay the cost of twenty years education and training, making it a first-rate investment for the community. besides providing for the longest interval between disablement by old age and natural death. To arrange this is one of the first duties of the modern statesman. Now childhood and old age are certainties. What about accidents and illnesses, which for the individual citizen are not certainties but chances? Well, we have seen that what are chances for the individual are certainties for the State. The individual citizen can share its certainty only by gambling with it. To insure myself against accident or illness I must make a bet with the State that these mishaps will befall me; and the State must accept the bet, the odds being fixed by the State actuaries mathematically. I shall at once be asked Why with the State? Why not with a private insurance company? Clearly because the State can do what no private company can do. It can compel every citizen to insure, however improvi~ dent or confident in his good luck he may happen to be, and thus, by making a greater number of bets, combine the greatest profit with the greatest certainty, and put the profit into the public treasury for the gen~ eral good. It can effect an immense saving of labor by substituting a single organization for dozens of competing ones. Finally it can insure at cost
1530
George Bernard Shaw
price, and, by including the price in the general rate of taxation, pay for all accidents and illnesses directly and simply without the enormous cIeri· cal labor of collecting specific contributions or having to deal in any way with the mass of citizens who lose their bets by having no accidents nor illnesses at any given moment. The oddity of the situation is that the State, to make insurance certain and abolish gambling, has to compel everyone to gamble, becoming a Supertote and stakeholder for the entire population. As ship insurance led to life insurance, life insurance to fire insurance and so on to insurance against employer's liability, death duties, and un· employment, the list of insurable risks will be added to, and insurance policies will become more comprehensive from decade to decade, until no risks that can worry a reasonably reckless citizen are left uncovered. And when the business of insurance is taken on by the State and lumped into the general taxation account, every citizen will be born with an unwritten policy of insurance against all the common risks, and be spared the painful virtues of providence, prudence, and self-denial that are now so oppressive and demoralizing, thus greatly lightening the burden of middle-class morality. The citizens will be protected whether they like it Or not, just as their children are now educated and their houses now guarded by the police whether they like it or not, even when they have no children to be educated nor houses to be guarded. The gain in freedom from petty cares will be immense. Our minds will no longer be crammed and our time wasted by uncertainty as to whether there will be any dinners for the family next week or any money left to pay for our funerals when we die. There is nothing impossible or even unreasonably difficult in all this. Yet as I write, a modest and well thought-out plan of national insurance by Sir William Beveridge, Whose eminence as an authority on political science nobody questions, is being fiercely opposed, not only by the private insurance companies which it would supersede, but by people whom it would benefit; and its advocates mostly do not understand it and do not know how to defend it. If the schooJing of our legislators had included a grounding in the principles of insurance the Beveridge scheme would pass into law and be set in operation within a month. As it is, if some mutilated remains of it survive after years of ignorant squabbling we shall be lucky, unless, indeed, some war panic drives it through Parlia~ ment without discussion or amendment in a few hours. However that may be, it is clear that nobody who does not understand insurance and com~ prehend in some degree its enormous possibilities is qualified to meddle in national business. And nobody can get that far without at least an acquaintance with the mathematics of probability, not to the extent of making its calculations and filling examination papers with typical equa-
The Vice of Gamblmg and the Vmue of Insurance
1531
tions, but enough to know when they can be trusted, and when they are cooked. For when their imaginary numbers correspond to exact quantities of hard coins unalterably stamped with heads and tails, they are safe within certain limits; for here we have solid certainty' and two simple possibilities that can be made practical certainties by an hour's trial (say one constant and one variable that does not really vary); but when the calculation is one of no constant and several very capricious variables, guesswork, personal bias, and pecuniary interests, come in so strongly that those who began by ignorantly imagining that statistics cannot lie end by imagining, equally ignorantly, that they never do anything else.
PAR T IX
The Supreme Art of Abstraction: Group Theory 1. The Group Concept by CASSIUS J. KEYSER 2. The Theory of Groups by SIR ARTHUR STANLEY
EDDINGTON
COMMENTARY ON
Certain Important Abstractions HE Theory of Groups is a branch of ma~hematics in which one does something to something and then compares the result with the result obtained from doing the same thing to something else, or something else to the same thing. This is a broad definition but it is not triviaL The theory is a supreme example of the art of mathematical abstraction. It is concerned only with the fine filigree of underlying relationships; it is the most powerful instrument yet invented for illuminating structure. The term group was first used in a technical sense by the French mathematician £'Variste Galois in 1830. He wrote his brilliant paper on the subject at the age of twenty, the night before he was killed in a stupid due1. 1 The concept was strongly developed in the nineteenth century by leading mathematicians, among them Augustin-Louis Cauchy (17891857),2 Sir Arthur Cayley (see p. 341), Camille Jordan (1838-1922), and two eminent Norwegians, Ludwig Sylow (1832-1918) and Marius Sophus Lie (1842-1899). In a little more than a century it has effected a remarkable unification of mathematics, revealing connections between parts of algebra and geometry that were long considered distinct and unrelated. "Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos." 3 Group theory has also helped physicists penetrate to the basic structure of the phenomenal world, to catch glimpses of innermost pattern and relationship. This is as deep, it should be observed, as science is likely to get. Even if we do not accept the idea that the ultimate essence of things is pattern, we may conclude with Bertrand Russell that any other essence is an individuality "which always eludes words and baffles description, but Which, for that very reason, is irrelevant to science." Let us return briefly to our somewhat dreamy definition and make it more concrete and explicit. The best plan, perhaps, is to give a specific example of a group and then to erase most of its details until nothing is left but essentials. This is the famous Carrollian method of defining a grin as what remains after the Cheshire cat, the vehicle of the grin, has vanished. The class or set of all the positive and negative integers, including zero,
T
1 The story of his tragic life is dramatically told by E. T. Bell in Men of Mathematics, New York, 1937. 2 "To Cau~hy has been given the credit of being the founder of the theory of groups of fiDlte order, even though fundamental results had been previously reached by 1. L. Lagrange, PIetro Abbati, P. Ruffini, N. H. Abel, and Galois." Florian Cajori ' A History of Mathematics, New York, 1919, p. 352. S E. T. Bell, Mathematics, Queen and Servant of Science. New York, 1951, p. 164.
1534
Certain Important Abstractions
1535
in conjunction with the ordinary arithmetic operation of addition, constitutes a familiar group. Its defining properties are these: (1) The sum of any two integers of the set is an integer of the set; (2) in adding three (or more) integers, any set of them may be added first without varying the result; you will recognize this as the associative rule of arithmetic (e.g., (3 + 7) + 9 = 3 + (7 + 9) ); (3) the set contains an "identity" or "unit" element (namely zero) such that the sum of this element and any other element in the set is again the latter element (e.g., 4 + 0 4, 0 + 8 8, etc.); (4) every integer in the set has an inverse or reciprocal, such that the sum of the two is the identity element (e.g., 2 + (-2) = 0, -77 + 77::::: 0, etc.). These are the attributes of our particular group. Now for some erasures. (1) The elements of the set may be arithmetic objects (e.g., numbers), geometric objects (e.g., points), physical entities (e.g., atoms), or they may be undefined; 4 (2) their number may be finite or infinite; (3) the operation or rule of combining the elements may be an arithmetic process (e.g., addition, multiplication), a geometric process (e.g., rotation, translation), or it may be undefined. Two further conditions are essential: (4) the combining rule must be associative; (5) every element of the set must have an inverse. Besides these five conditions, a set may be Abelian or non-Abelian according as the combining rule is commutative or noncommutative (i.e., for addition, either 2 + 3 == 3 + 2 or 2 + 3 7" 3 + 2, and, for multiplication, either 2·3 3 ·2 or 2·3 7" 3 . 2). These are the bare bones of the group concept. It is hard to believe how' much unification of bewildering details has been achieved by the theory; "what a wealth, what a grandeur of thought may spring from what slight beginnings." 5 A few words should be added concerning two other fundamental mathematical terms which arise frequently in group theory. The first is transformation, which embodies the idea of change or motion. An algebraic expression is transformed by changing it to another having different form, by substituting for the variables their values in terms of another set of variables; a geometric figure is transformed by changing its co~ordinates, by mapping one space on another, by moving the figure pursuant to a procedural rule, e.g., projection, rotation, translation. 6 More generally,
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4 It should be emphasized that while both the elements and operations of a group may theoretically be undefined, if the group is to be useful in science they must in some way correspond to elements and operations of observable experience. Otherwise manipulating the group amounts to nothing more than a game, and a pretty vague and arid game at that, suitable only for the most withdrawn lunatics. IS The British geometer H. F. Baker, as quoted by Florian Cajori in A History of Mathematics, New York, 1919, p. 283. ' 6 This definition of transformation suffices for our purposes, but the concept is much more comprehensive than I have indicated. Any problem, process or operation. as Keyser says (see next note), having to do with ordinary functions is a problem, process or operation having to do with relations or transformations. A pairing or coupling is a transformation; so is a relation, a function, a mathematical calculation, a deductive inference.
£diJor's Comment
1536
any object of thought may be transformed by associating it with or converting it into another object of thought. The second important term is invariance. The invariant properties of an algebraic expression, geometric figure, class, or other object of thought are those which remain the same under transformations. Suppose the elements of the class of positive integers (1, 2, 3 . . .) are transformed by the rule of doubling each element
123 4
.!, J, J, J,
2 4 6 8 .
The transforms constitute the class of even integers: 2, 4, 6, . . '. Since the integers are transformed into integers, the property of being an integer is evidently preserved; which is to say it is an invariant under the doubling transformation. However, the value of each integer of the original class is not preserved; it is doubled; thus value is not an invariant under this transformation. Another example of an invariant in our case are the ratios of the elements of each class: if the elements of the first class are called x's and those of the transformed class y's then under the rule y = 2x, Yl
2Xl
Xl
Y2
2X2
X2
_=_=_.7 A geometric invariant is similar. Take a rigid object such as a glass paper-weight and move it by sliding from one end of the table to the other. This is a transformation. The paper-weight retains its shape and dimensions; the retained properties are therefore invariants. The mathematician describes these facts by saying that the metric properties of rigid bodies are invariant under the transformation of motion. Since the paperweight's position and distance from an object such as the mirror on the wall or the Pole Star are changed by the transformation, they are not invariant. If the object moved were a blob of mercury it is unlikely that it would retain its shape or dimensions, but its mass would probably be invariant and certainly its atomic structure. (For a further discussion of this point see pp. 581-598, selection on topology.) Group theory has to do with the invariants of groups of transformations. One studies the properties of an object, the features of a problem unaffected by changes of condition. The more drastic the changes, the fewer the invariants. What better way to get at the fundamentals of structure than by successive transformations to strip away the secondary properties. It is a method analogous to that used by the archaeologist who clears away hills to get at cities, digs into houses to uncover ornaments, utensils and potsherds, tunnels into tombs to find sarcophagi, the winding '1
The example is from Cassius J. Keyser, Mathematical Philosophy, New York,
1922, pp. 183-185.
Certain Important Abstrachons
1537
sheets they hold and the mummies within. Thus he reconstructs the features of an unseen society; and so the mathematician and scientist create a theoretical counterpart of the unseen structure of the phenomenal world. Whitehead has characterized these efforts in a famous observation: "To see what is general in what is particular and what is permanent in what is transitory is the aim of scientific thought."
I have selected two essays to illustrate group theory. The first is a chapter from Cassius J. Keyser's lectures on mathematical philosophy.s Keyser, a prominent American mathematician, was born in Rawson, Ohio, in 1862. He was educated in Ohio schools and at Missouri University~ for a time he studied law but then turned to mathematics and earned his graduate degrees at Columbia. After five years as superintendent of schools in Ohio and Montana (1885-90), he became professor of mathematics at the New York State Normal School, and in 1897 joined the staff of Columbia. He was appointed Adrain professor in 1904, serving in this post unti11927 when he was made emeritus. He died in 1947. aged eightyfive. Keyser had broad interests in mathematics, as a geometer, historian and philosopher. He was much admired as a teacher for the care he took to his lectures, their breadth, clarity and honesty. He is a little oldfashioned in his style and a trifle long-winded. Keyser was not the man to drop a point until he had squeezed it dry both as to its scientific content and cultural bearings; the reader may also come to feel a little squeezed. But he had an unfailingly interesting and reflective mind, and I have nowhere found a better survey of the group concept than in the selection below. The second essay, by Sir Arthur Eddington, is one of the Messenger Lectures given at Cornell University, appearing in a book titled New Pathways of Science. The discussion of groups exhibits the usual dazzling Eddington virtuosity; it is one of his best pieces of popularization in one of his best books. iii
Mathematical Philosophy, A Study of Fate and Freedom, New York, 1922.
Mazes intricate. Eccentric, interwov'd, yet regular Then most, when most irregular they seem.
1
-MILTON
The Group Concept By CASSIUS J. KEYSER
I INVITE your attention during the present hour to the notion of group. Even if I were a specialist in group theory,-which 1 am not,-I could not in one hour give you anything like an extensive knowledge of it, nor facility in its technique, nor a sense of its intricacy and proportions as known to its devotees, the priests of the temple. But the hour should suffice to start you on the way to acquiring at least a minimum of what a respectable philosopher should know of this fundamental subject; and such a minimum will include: a clear conception of what the term "group" means; ability to illustrate it copiously by means of easily understood examples to be found in all the cardinal fields of interest-number, space, time, motion, relation, play, work, the world of sense-data and the world of ideas; a glimpse of its intimate connections with the ideas of transformation and invariance; an inkling of it both as subject-matter and as an instrument for the delimitation and discrimination of doctrines; and discernment of the concept as vaguely prefigured in philosophic speculation from remote antiquity down to the present time. I believe that the best way to secure a firm hold of the notion of group is to seize upon it first in the abstract and then, by comparing it with concrete examples, gradually to win the sense of holding in your grasp a living thing. In presenting the notion of group in the abstract, it is convenient to use the term system. This term has many meanings in mathematics and so at the outset we must clearly understand the sense in which the term is to be employed here. The sense is this: as employed in the definition of group, the term system means some definite class of things together with some definite rule, or way, in accordance with which any member of the class can be combined with any member of it (either with itself or any other member). For a simple example of such a system we may take for the class the class of ordinary whole numbers and for the rule of combination the familiar rule of addition. You should note that there are three and only three respects in which two systems can differ: by having different classes, by having different rules of combination, and by differing in both of these ways. IS38
1539
The Group Concept
The definition of the term "group" is as follows. Let S denote a system consisting of a class C (whose members we will denote by a, b, c and so on) and of a rule of combination (which rule we will denote by the symbol 0, so that by writing, for example, aob, we shall mean the result of combining b with a). The system S is called a group if and only if it satisfies the following four conditions: (a) If a and b are members of C, then aob is a member of C; that is, aob = c, where c is some member of C. (b) If a, b, c are members of C, then (aob)oc ao(boc); that is, combining c with the result of combining b with a yields the same as com~ bining with a the result of combining c with b; that is, the rule of combina~ tion is associative. (c) The class C contains a member i (called the identical member or element) such that if a be a member of C, then aoi ioa a; that is, C has a member such that, if it be combined with any given member, or that member with it, the result is the given member. (d) If a be a member of C, then there is a member a' (called the reciprocal of a) such that aoa' a'oa i; that is, each member of C is matched by a member such that combining the two gives the identical member. Other definitions of the term "group" have been proposed and are sometimes used. The definitions are not all of them equivalent but they all agree that to be a group a system must satisfy condition (a). Systems satisfying condition (a) are many of them on that account so important that in the older literature of the subject they are called groups, or closed systems, and are now said to have "the group property," even if they do not satisfy conditions (b), (c) and (d). The propriety of the term "closed system" is evident in the fact that a system satisfying (a) is such that the result of combining any two of its members is itself a member-a thing in the system, not out of it. Various Simple Examples of Groups and of Systems that Are Not Groups.-You observe that by the foregoing definition of group every group is a system; groups, as we shall see, are infinitely numerous; yet it is true that relatively few systems are groups or have even the group property-so few relatively that, if you select a system at random, it is highly probable you will thus hit upon one that is neither a group nor has the group property. Take, for example, the system S1.whose class C is the class of integers from 1 to 10 inclusive and whose rule of combination is that of ordinary mUltiplication X; 3 X 4 12; 12 is not a member of C, and so S1 is not closed-it has not the group property. Let S2 have for its C the class of all the ordinary integers, 1, 2, 3, . . . ad infinitum, and let 0 be X as before; as the product of any two integers
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1540
CassiJ18 1. Keyser
is an integer, (a) is satisfied-S2 is closed, has the group property; (b), too, is evidently satisfied, and so is (c), the identity element being 1 for, if n be any integer, n Xl::: 1 X n ::: n; but (d) is not satisfied-none of the integers (except 1) composing C has a reciprocal in C-there is, for example, no integer n such that 2 X n ::: n X 2 ::: 1j and so S2, though it has the group property, is not a group. Let Ss be the system consisting of the class C of all the positive and negative integers including zero and of addition as the rule of combination; you readily see that Sa is a group, zero being the identical element, and each element having its own negative for reciprocal. A group is said to be finite or infinite according as its C is a finite or an infinite class and it is said to be Abelian or non-Abelian according as its rule of combination is or is not commutative-according, that is, as we have or do not have aob::: boa, where a and b are arbitrary members of C. You observe that the group Sa is both infinite and Abelian. For an example of a group that is finite and Abelian it is sufficient to take the system S4 whose C is composed of the four numbers, 1, -1, I, -I, where i is y!=1, and whose rule of combination is multiplication; you notice that the identical element is 1, that 1 and -1 are each its own reciprocal and that i and -i are each the other's reciprocal. Let S5 have the same C as Sa and suppose 0 to be subtraction instead of addition; show that S5 has the group property but is not a group. Show the,like for S6 in which C is the same as before and 0 denotes mUltiplication. Show that 8 7 where C is the same as before and 0 means the rule of division, has not even the group property. Consider S8 where C is the class of all the rational numbers (that is, all the integers and all the fractions whose terms are integers, it being understood that zero can not be a denominator) and where 0 denotes +; you will readily find that S8 is a group, infinite and Abelian. Examine the systems obtained by keeping the same C and letting 0 denote subtraction, then multiplication, then division. Devise a group system where 0 means division. If 8 and 8' be two groups having the same rule of combination and if the class C of 8 be a proper part of the class C' of S' (i.e., if the members of C are members of C' but some members of C' are not in C), then 8 is said to be a sub-group of 8'. Observe that 8s is a sub-group of S8' Show that 8 9 is a group if its C is the class of all real numbers and its o is +; note that 8 8 is a sub-group of 8 9 and hence that Sa is a sub-group of a sub-group of a group. Is S9 itself a sub-group? If so, of what group or groups? Examine the systems derived from S9 by altering the rule of combination. The most difficult thing that teaching has to do is to give a worthy sense of the meaning and scope of a great idea. A great idea is always
1541
The Group Concept
generic and abstract but it has its living significance in the particular and concrete-in a countless multitude of differing instances or examples of it; each of these sheds only a feeble light upon the idea, leaving the infinite range of its significance in the dark; whence the necessity of examining and comparing a large number of widely differing examples in the hope that many little lights may constitute by union something like a worthy illustration; but to present these numerous examples requires an amount of time and a degree of patience that are seldom at one's disposal, and so it is necessary to be content with a selected few. And now here is the difficulty: if the examples selected be complex and difficult, they repel; if tliey be simple and easy, they are not impressive; in either case, the significance of the general concept in question remains ungrasped and unappreciated. I am going, however, to take the risk-to the foregoing illustrations of the group concept I am going to add a few further ones,some of them very simple, some of them more complex,-trusting that the former may not seem to you too trivial nor th~ latter too hard. Everyone has seen the pretty phenomenon of a grey squirrel rapidly rotating a cylindrical wire cage enclosing it. It may rotate the cage in either of two opposite ways, senses or directions. Let us think of rotation in only one of the ways, and let us call any rotation, whether it be much or little, a turn. Each turn carries a point of the cage along a circle-arc of some length, short or long. Denote by R the special turn (through 360°) that brings each point of the cage back to its starting place. Let S10 be the system whose C is the class of all possible turns and whose 0 is addition of turns so that aob shall be the whole turn got by following tum a by tum b. You see at once that S has the group property for the sum of any two turns is a turn; it is equally evident that the associative lawcondition (b) -is satisfied. Note that R is equivalent to no turn,----equivalent to rest,-equivalent to a zero turn, if you please; note that, if a be a turn greater than R and less than 2R, then a is equivalent to a's excess over R; that, if a be greater than 2R and less than 3R, then a is equivalent to ds excess over 2R; and so on; thus any tum greater than R and not equal to a multiple of R is equivalent to a turn less than R; let us regard any turn that is thus greater than R as identical with its equivalent less than R; we have, then, to consider no turns except R and those less than R----of which there are infinitely many; you see immediately that, if a be any turn, aoR Roa a, which means that condition (c) is satisfied with R for identical element. Next notice that for any tum a there is a tum if such that aoa' a'oa R. Hence S10 is, as you see, a group. Show it to be Abelian. You will find it instructive to examine the system derived from S10 by letting C be the class of all turns (forward or backward). Perhaps, you will consider the system suggested by the familiar spectacle of a ladybug- or a measuring-worm moving round the rim or edge of a
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1542
Cassius 1. Keyser
circular tub; or the system suggested by motions along the thread of an endless screw; or that suggested by the turns of the earth upon its axis; or that suggested by the motions of a planet in its orbit. Do such examples give the meaning of the group concept? Each one gives it somewhat as a water-drop gives the meaning of ocean, or a burning match the meaning of the sun, or a pebble the meaning of the Rocky Mountains. Are they, therefore, to be despised? Far from it. Taken singly, they tell you little; but taken together, if you allow your imagination to play upon them, noting their differences, their similitudes, and the variety of fields they represent, they tell you much. Let us pursue them further, having a look in other fields. Consider the field of the data of sense,-a field of universal interest,and fundamental. We are here in the domain of sights and sounds and motions among other things. Are there any groups to be found here? Who, except the blind-born, are not lovers of color? Do the colors constitute a group? I mean sensations of color,-color sensations,-including all shades thereof and white and black. Denote by 8 11 the system whose C is the class of all such sensations and whose rule of combination is, let us say, the mixing of such sensations. But what are we to understand by the mixing of two color sensations? Suppose we, have two small boxes of powder,-say of finely pulverized chalk,-a box of, say, red powder and a box of blue; one of the powders gives us the sensation red, the other the sensation blue; let us thoroughly mix the powders; the mixture gives us a color sensation; we agree to say that we have mixed the sensations and that the new sensation is the result of mixing the old ones. As the combination of any two color sensations is a color sensation, 8] 1 has, you see, the group property. Is it a group? Evidently condition (b) is satisfied. Are conditions (c) and (d) also satisfied? Let us pass from colors to figures or shapes,-to figures or shapes, I mean, of physical or material objects,-rocks, chairs, trees, animals and the like,-as known to sense-perception. No doubt what we ordinarily call perception of an object's figure or shape is genetically complex, a result of experience contributed to by two or more senses, as sight, touch, motion; let us not, however, try to analyze it thus; let us take it at its face value-let us regard it as being, what it appears to be before analytic reflection upon it, a sense-given datum; and let us confine ourselves to the sense of sight. Here is a dog; its ears have shape; so, too, its eyes, its nose and the other features of its head; these shapes combine to make the shape or figure of the head; each other one of its visible organs has a shape of its own; these shapes all of them combine to make that thing which we call the shape or figure of the dog. Yonder is a table; it has a shape, and this is due to some sort of combination of the shapes of its parts-legs, top, and so on; upon it are several objects-a picture frame,
TIlt! Group Concept
1543
a candlestick, some vases; each has a shape; the table and the other things together constitute one object--disclosed as such to a single glance of the eye; this object has a figure or shape due to the combined presence of the other shapes. In speaking of the dog and the table, I have been using the word "combination" in a very general sense. Can it, in this connection, be made precise enough for our use? Is it possible to find or frame a rule by which, any two visible shapes being given, these can be combined? If so, is the result of the combination a visible shape? If so, the system consisting of the rule and the total class of shapes has the group property. Does the system satisfy the remaining three conditions for a group? And what of sounds-sensations of sound? Are sounds combinable? Is the result always a sound or is it sometimes silence? If we agree to regard silence as a species of sound,-as the zero of sound,-has the system of sounds the property of a group? There is the question of thresholds: sound is a vibrational phenomenon; if the rate of vibration be too slow or too great,-say, 100,000 per second,-no sound is heard. If you disregard the thresholds, has the sound system the group property? Is it a group? If so, what is the identical element? And what would you say is the reciprocal of a given sound or tone? Consider other vibrational phenomena-as those of light or electricity. Can you so conceive them as to get group systems? Sharpen your questions and then carry them to physicists. You need have no hesitance-the service is apt to be mutual. The Infinite Abelian Group of Angel Flights.-We are accustomed to think of ourselves as being in a boundless universe of space tilled with what we call points any two of which are joined by what we call a straight line. Imagine one of those curious creatures which are to-day for most of us hardly more than figures of speech but which for many hun· dreds of years were very real and very lovely or very terrible things for millions of men, women and children and were studied and discoursed about seriously by men of genius: I mean angels. Angels can fly. Let us confine their flights to straight lines but impose no other restrictions. I am going to ask you to understand a :8.ight as having nothing but length, direction and sense; if it is parallel to a given straight line, it has that line's direction; if it goes from A towards B, it has that sense; if from B toward A, the opposite sense; a :8.ight from A to B and one from C to D are the same if they agree in length, direction and sense. Consider a flight (J, from point A to point B followed by a flight b from B to C; you readily see that a and b are two adjacent sides of a parallelogram, one of whose diagonals is the direct flight d from A to C; d is called the resultant or flight-sum of a and b because d tells us how far the angel has finally got from the starting place; and so we write aob = d. H flight b' goes from P to Q but agrees with b in length, direction and sense, we write aob' = d
Cassius J. Keyser
1544
as before for, as already said, band b' are one and the same. Now let S12 denote the system whose C is the class of all possible angel flights including rest, or zero flight, and whose rule of combination is flight summation as above explained; you see at once that 8 12 is a closed system, has the group property, for the combination of any two flights is a flight; if a, band c be three flights, we may suppose them to go respectively from A to B, from B to C, and from C to D; consider (aob) oc; aob d, the flight from A to C; doc e, the flight from A to D; now consider ao(boc); boc d', flight from B to D; aod' e', flight from A to D; so e e' and (aob) oc = ao(boc); hence summation of flights is associative -condition (b) is satisfied. Condition (c) is satisfied with zero (0) flight for i; for, if a be any flight, it is plain that aoO Ooa a. And condition (d) is satisfied for it is evident that aoa' = a'oa 0 where a' and a agree in length and direction but are opposite in sense. Hence the system of angel flights is a group. And it is easy to see that it is both infinite and Abelian. What I have here called an angel flight is known in mathematics and in physics as a vector; a vector has no position-it has its essential and complete being in having a length, a direction and a sense. And so, you see, the system composed of the vectors of space and of vector addition as a rule of combination is an infinite Abelian group. Connection of Groups with Transformations and Invariants.-Let us have another look at our angel flights, or vectors. I am going to ask you to view them in another light. Let V be any given vector-that is, a vector of given length, sense and direction; where does it begin and where does it end? A moment's reflection will show you that every point in the universe of space is the beginning of a vector identical with V and the end of a vector identical with V. Though these vectors are but one, it is convenient to speak of them as many equal vectors-having the same length, direction and sense. Let the point P be the beginning of a V and let the point Q be its end. Let us now associate every such P with its Q(P ~ Q); we have thus transformed ou~ space of points into itself in such wise that the end of each V is the transform of its beginning; call the transformation T; let us follow it with a transformation T' converting the beginnings of all vectors equal to a given vector V' into their corresponding ends. What is the result? Notice that T converted Pinto Q and that T' then converted Q into Q', the end of the V' beginning at Q; now there is a vector beginning at P and running direct to Q'; and so there is a transformation Til converting Pinto Q'; it is this Til that we shall mean by ToT'. Without further talk, you see that our group of angel flights, or vectors, now appears as an infinite Abelian group of transformations (of our space of points into itself). Such transformations do not involve motion in fact; it is customary, however, for mathematicians to call them
=
=
=
=
=
= = =
The Group Concept
1545
motions, or translations, of space; T, for example, being thought of and spoken of as a translation of the whole of space (as a rigid body) in the direction and sense of V and through a distance equal to V's length. In accordance with this stimulating, albeit purely figurative, way of speaking, the group in question is the group of the translations of our space. We are now in a good position to glimpse the very intimate connection between the idea of group and the idea of invariance. Suppose we are given a group of transformations; one of the big questions to be asked regarding it is this: what things remain unaltered,-remain invariant,under each and all the transformations of the group? In other words, what property or properties are common to the objects transformed and their transforms? Well, we have now before us a certain group of space transformations-the group of translations. Denote it by G. Each translation in G converts (transforms, carries, moves) any point into a point, and so converts any configuration F of points,-any geometric figure,-into some configuration F'. What remains unchanged? What are the invariants? It is obvious that one of the invariants,-a very important one,-is distance; that is, if P and Q be any two points and if their transforms under some translation be respectively P' and Q', then the distance between P' and Q' is the same as that between P and Q; another is order among points-if Q is between P and R, Q' is between P' and R'; you see at once that angles, areas, volumes, shapes are all of them unchanged: in a word, congruence is invariant-if a translation convert a figure F into a figure F', F and F' are congruent. Of course congruence is invariant under all the translations having a given direction. Do these constitute a group? Obviously they do, and this group G' is a sub-group of G. Congruence is invariant under G'; it is also invariant under G; G' is included in G; it is natural, then, to ask whether G itself may not be included in a still larger group having congruence for an invariant. The question suggests the inverse of the one with which we set out. The former question was: given a group, what are its invariants? The inverse question is: given an invariant, what are its groups and especially its largest group? This question is as big as the other one. Consider the example in hand. The given invariant is congruence. Is G,-the group of translations,-the largest group of space transformations leaving congruence unchanged? Evidently not; for think of those space transformations that consist in rotations of space (as a rigid body) about a fixed line (as axis); if such a rotation converts a figure F into F', the two figures are congruent. It is clear that the same is true if a transformation be a twist-that is, a simultaneous rotation about, and translation along, a fixed line. All such rotations and twists together with the translations constitute a group called the group of displacements of space; it includes all transformations leaving congruence invariant. This group, as a little reflection will show you, has many sub-groups, infinitely many;
1546
Cassius 1. Keyser
the set of displacements leaving a specified point invariant is such a subgroup; the set leaving two given points unchanged is another. How is the latter related to the sub-group leaving only one of the two points invariant? Is there a displacement leaving three non-collinear points invariant? Do the displacements leaving a line unchanged constitute a group? Such questions are but samples of many that you will find it profitable to ask and to try to answer. For the sake of emphasis, permit me to repeat the two big questions: ( 1) Given a group of transformations, what things are unchanged by them? (2) Given something-an object or property or relation, no matter what-that is to remain invariant, what are the groups of transformations, and especially the largest group, that leave the thing unaltered? You may wish to say: I grant that the questions are interesting, and I do not deny that they are big-big in the sense of giving rise to innumerable problems and big in the sense that many of the problems are difficult; but I do not see that they are big with importance. Why should I bother with them? In reply I shall not undertake to say why you should bother with them; it is sufficient to remind you that as human beings you cannot help it and you do not desire to do so. In the preceding lecture, we saw that the sovereign impulse of Man is to find the answer to the question: what abides? We saw that Thought,-taken in the widest sense to embrace art, philosophy, religion, science, taken in their widest sense,-is the quest of invariance in a fluctuant world. We saw that the craving and search for things eternal is the central binding thread of human history. We saw that the passion for abiding reality is itself the supreme invariant in the life of reason. And we saw that the bearings of the mathematical theory of in variance upon the universal enterprise of Thought are the bearings of a prototype and guide. It is evident that the same is true of the mathematicai theory of groups. Our human question is: what abides? As students of thought and the history of thought, we have learned at length that the question can not be answered fully at once but only step by step in an endless progression. And now what are the steps? You can scarcely fail to see, if you reflect a little, that each of them,-whether taken by art or by science or by philosophy,-consists virtually in ascertaining either the invariants under some group of transformations or else the groups of transformations that leave some thing or things unchanged. Groups as Instruments for Defining. Delimiting. Discriminating and Classifying Doctrines.-The foregoing question (2) has another aspect, which I believe to be of profound interest to all students except those. if there be such, who are insensate to things philosophical. I mean that, if and whenever you ascertain the group of all the transformations that leave invariant some specified object or objects of thought, you thereby define perfectly some actual (or potential) branch of science-some actual (or
lS47
The Group Concept
potential) doctrine. I will endeavor to make this fact evident by a few simple examples, and I will choose them from the general field of geometry, though, as you will perceive, such examples might be taken from other fields. For a first example, consider the above-mentioned group D of the displacements of our space. I say that this group defines a geometry of space, which may be called the geometry of displacements. It defines it by defining, or delimiting, its subject-matter. What is its subject~matter? What does the geometry study? The two questions are not equivalent. It studies all the figures in space but it does not study all their properties. Its subjectmatter consists of those properties which it does study. What are these? They are those and only those properties (of figures) that remain invariant under all displacements but under no other transformations of space. The geometry of displacements might be called congruence geometry. It includes the greater part of the ordinary geometry of high school but not all of it, for the latter deals, for example. with similarity of figures; similarity is indeed invariant under displacements, but it is also invariant under other transformations-the so-caned similitude transformations. to be mentioned presently. For a second example, consider the following. I may wish to confine my study of spatial figures to their shape. The doctrine thus arising may be called the geometry of shape, or shape geometry. If I tell you that I am studying shape geometry and you ask me what I mean by the geometry of shape, there are two ways in which I may answer your question. One of the ways requires me to define the term shape-shape of a geometric figure; the other way,-the group way,-does not. Let us examine them a little. I have never seen a mathematical definition of shape, but it may, I believe, be precisely defined as follows. We must distinguish the three things = sameness of shape; shape of a given figure; and shape of a figure. I will define the first; then the second in terms of the first; and, finally, the third in terms of the second. Two figures F and F' will be said to have the same shape if and only if it is possible to set up a one-to-one correspondence between the points of F and those of F', such that, A Band CD being any distances between points of F, and A'B' and C'D' being the distances between the corresponding points of F', AB/CD A'B'/C'D'. Two figures having the same shape will be said to be similar, and conversely. Having defined sameness of shape, or similarity, of figures, I will define the term "shape of a given figure" as fonows: if F be a given, or specific, figure, the shape of F is the class er of all figures similar to F; it is evident that, if F and F' are not similar, the class er and the class .r.the shape of F'-have no figures in common; it is evident, moreover, that there are as many er'S as there are figure shapes. And now what do we mean by the general term shape, or-what is tantamount-shape of a
=
1548
figure? What the answer must be is pretty obvious: shape is the class l of all the o-'s. Note that ~ is a class of classes and that any 0- is a class of (similar) figures. Having defined the general term shape, I have, you see, virtually answered your question: what is the geometry of shape? Let us now see how the question may be answered by means of the group concept. Two congruent figures are clearly similar, and so similarity is invariant under the group of displacements. But you readily see that there are many other transformations under which similarity is invariant. Let 0 be a point; consider the bundle of straight lines,-all the lines O,-having 0 for its vertex; every point of space is on some line of the bundle; let k be any real number (except zero); let P be any point and let pi be such a point on the line OP that the segment OP' = k X segment OP; you see that each point P is transformed into a point P'; the transformation is called homothetic; its effect, if k be positive and exceed 1, is a uniform expansion of space from 0 outward in all directions; if k be positive and less than 1, the effect is contraction toward 0; if k be negative, the effect is such an expansion or contraction, followed or preceded by reflection in 0 as in a mirror; distances are clearly not preserved, but distance ratios are; that is, if A, B, C be any three points and if their respective transforms under some homothetic transformation be A', B', C', it is evident that AB/BC = A'B'/B'C'; accordingly, if F' be the transform of a figure P, the figures are similar,- they have the same shape but not the same size,-they are not congruent: similarity is, then, invariant under all homothetic transformations, and hence under combinations of them with one another and with displacements; the displacements and the homothetic transformations together with all such combinations constitute a group called the group of similitude transformations; it contains all and only such space transformations as leave similarity unchanged. Here, then, is our group definition of shape geometry: namely, the geometry of shape is the study of that property of figures which is common to every figure and its transforms under each and all transjormations oj the similitude group. Observe that this definition, unlike the former one, employs
neither the notion of shape in general, nor that of the shape of a given figure; it employs only the notion of similarity-sameness of shape. We ought, I think, to consider one more example of how a group of transformations serves to determine the nature and limits of a doctrine and hereby to discriminate the doctrine from all others. I will again take a geometric example, but for the sake of simplicity I will choose it from the geometry of the plane (instead of space). Before presenting it, let me adduce a yet simpler example of the same kind taken from the geometry of points in a straight line. In a previous lecture I explained what is meant by a projective line-an ordinary line endowed with an "ideal" point, or point at infinity. where the line meets all lines parallel to it. Let L
1549
The Group Concept
be a projective line. In the preceding lecture, we gained some acquaintance with the transformations of the form
ax+ b
(1)
x'=---
cx+d where the coefficients are any real numbers such that ad - bc =F 0; we saw that there are co 8 such transformations and that each of them converts the points of L into the points of L in such a way that the anharmonic ratio of any four points is equal to the anharmonic ratio of their transforms. Distances are not preserved; neither are the ordinary ratios of distances preserved; hence neither congruence nor similarity is invariant; no relation among points-that is, no property of figures (for here a figure is simply a set of points on L)-is invariant except anharmonic ratio and properties expressible in terms of the latter; no other transformations leave these properties invariant. By a little finger work you can prove in a formal way that these transformations constitute a group. I will merely indicate the procedure, leaving it to you to carry it out if you desire to do so. The transformations differ only in their coefficients. Let (a l , b 1, CI, d1 ), (a~, b 2, C2, d 2 ), (as, bs, cs, ds) be any three of the transformations; consider the first and second; the rule 0 of combination is to be: operate with the first and then on the result with the second. The first converts point x into point x': (2)
x' =
a1x
+ b1
clx
+ d1
;
the second converts xl' into point x": (3) x"
Q2
x' + b2
= ----'
c2x' + d2 in (3) replace x' by its value given by (2), simplify and then notice that you have a transformation of form (1) converting x directly into x", This shows that the set of transformations have the group property. To show that they obey the associative law, it is sufficient to perform the operations (4) (ai' b1, Cl. d l )0[(a2. b 2•
C2.
d2)0(a~h b a, CSt
ds )],
(5) [(a1. bi> Cl' d 1)0(a2. b 2• C2. d 2)]0(as. b s, CSt ds),
and then to observe that the results are the same. The identical element i is (a, 0, 0, a)-that is. the transformation, x' = x. The inverse of any transformation (a, b, c, d) is (-d, b, c, -a) for you can readily show that combination of these gives (a, 0, 0, a). The fact to be specially noted is that this group of so-called homographic transformations defines a certain kind of geometry in the line L-
Cassius J. Keyser
1550
namely, its projective geometry. In a line there are various geometries; among these the projective geometry is characterized by its subject-matter, and its subject-matter consists of such properties of point sets, or figures, as remain invariant under its homographic group. And now I come to the example alluded to a moment ago-the one to be taken from geometries in (or of) a plane. The foregoing homographic group-in a line, a one-dimensional space-has an analogue in a projective plane, another in ordinary 3-dimensional projective space, another in a projective space of four dimensions, and so on ad infinitum. What is the analogous group for a plane? In a chosen plane take a pair of axes and consider the pair of equations
Ax+By+C
x=----Gx+Hy+l
(1)
Dx+Ey+F y'=-----,
Gx+Hy+l where the coefficients are any real numbers such that (I')
ABC DEF G HI
~O;
i.e., such that
(1') AEI - CEG
+ CDR -
BDI + BFG - AFR -F O.
The coefficients furnish eight independent ratios,--called "parameters,"and so we have 00 8 equation pairs of form (1); choose anyone of them and notice that it is a transformation converting the number pair (x, y) into a number pair (x', y'), and so converting the point (x, y) into a point (x', y'); owing to the inequality (1'), any point (x, y) is transformed into a definite point (x', y'). In any line ax' + by' + c = 0, replace x and y' by their values given by (1), and simplify; the resulting equation is of first degree in x and y and hence represents a line; hence, you see, points of a straight line are converted into points of a straight line--the relation, collinearity, is preserved; so is copunctality-a pencil of lines has a pencil for its transform; you can readily show that order is not preserved, nor distances nor ordinary distance-ratios, nor angles; hence, if the figure F' be the transform of F, the two figures are, in general, neither congruent nor similar; we say, however, that F and F' are projective because, as can be proved, the anharmonic ratio of any 4 points (or lines) of either is equal to that of the corresponding (transform) points (or lines) of the other. By the method indicated for the homographic transformations of a line, you can prove that the 00 8 transformations of form (1) constitute a group.
The Group Concept
1551
Just as a point of the plane has two coordinates (x, y), so a line depends on two coordinates; there are various ways to see that such is the case; an easy way is this: the line, ax + by + c 0, depends solely upon the ratios (a; b : c) of the coefficients; these three ratios are not independent-two of them determine the third one; you thus see that the line depends upon only two independent variables-it has, like the point, two coordinates; let us denote them by (u, v). Now consider the transformations
=
Ju +Kv+L
u' =
,
Pu+Qv+R
(2) v' =
Mu +Nv+ 0
,
Pu+Qv+R where the coefficients are subject to a relation like (1'). We saw that a transformation (1) converts points into points directly and lines into lines indirectly; just so, a transformation (2) converts lines into lines directly and points into points indirectly; hence the group of line-to-line transformations (2) is essentially the same as the foregoing group of pointto-point transformations (1). This latter group is called the group of collineations of (or in) the plane. I am going now to ask you to notice an ensemble of transformations (of the plane) that are neither point-to-point nor line-to-line transformations .but are at once point-to-line and line-to-point transformations. These are represented by the pair of formulas
+ by + c , gx + hy + i
ax
u= (3)
dx+ey+f v= gx
+ hy + i
,
where the coefficients are again subject to a relation like (1'). Any such transformation converts a point (x, y) into a line (u, v); now operate on the points of this line by the same transformation or another one of form (3); the points are converted into lines constituting a pencil having a vertex, say (x', y'); thus the combination converts point (x, y) into point (x', y')-it is a point-to-point transformation and hence belongs to the group of coIlineations; you thus see that the set of transformations (3) is not a group; but this set and the collineations together constitute a group including the collineations as a subgroup_ This large group is called the Group of Projective Transformations of the Plane. Why? Because every transformation in it and no other transformation leaves all anharmonic ratios unchanged.
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Casdus 1. Keyser
What is the projective geometry of the plane? The group now in hand enables us to answer the question perfectly. The answer is: Projective plane geometry is that geometry which studies such and only such properties of plane figures as remain invariant under the group of projective transformations. In reading the essays of the late Henri Poincare you have met the statement: "Euclidean space is simply a group." The foregoing examples should enable you to understand its meaning. And they should lead you to surmise -what is true-that answers like the foregoing ones are available for similar questions regarding all geometries of a space of any number of dimensions and-what is more-regarding mathematical doctrines in general. Whatsoever things are invariant under all and only the transformations of some group constitute the peculiar subject-matter of some (actual or potential) branch of knowledge. And you see that every such groupdefined science views its subject matter under the aspect of eternity. A Question for Psychologists.-Before closing this lecture, I desire to speak briefly of two additional matters connected with the notion of group: one of the matters is psychological: the other is historicaL Being students of philosophy, you are obliged to have at least a good secondary interest in psychology. I wish to propose for your future consideration a psychological question--one which psychologists (I believe) have not considered and which, though it has haunted me a good deal from time to time in recent years, I am not yet prepared to answer confidently. The question is: Is mind a group? Let us restrict the question and ask: Is mind a closed system-that is, has it the group property? Some of the difficulties are immediately obvious. In order that the question shall have definite meaning, it is necessary to think of mind as a system composed of a class of things and a rule, or law. of combination by which each of the things can be combined with itself and with each of the other things. We may make a beginning by saying that the required class is the class of mental phenomena. But what does the class include? What phenomena are members of it? Some phenomena,-feeling, for example, seeing, hearing, tasting, thinking, believing, doubting, craving, hoping, expecting,-are undoubtedly mental; others seem not to be-as what I see, for example, what I taste, what I believe, and the like. Here are difficulties. You will find a fresh and suggestive treatment of them in Bertrand Russell's The Ultimate Constituents of Matter, found in the author's Mysticism and Logic and especially in his Analysis of Mind. Let us suppose that, despite the difficulties in the way, you have decided what you are going to call mental phenomena. You have then to consider the question of their combination. We do habitually speak of combining mental things: hoping, for example, is, in some sense, a union, or combination, of desiring and expecting; the feeling called patriotism is evidently a combination of a
The Group Concept
lSS3
pretty large variety of feelings; in the realm of ideas,-which you will probably desire to include among mental phenomena,-we have seen that, for example, the idea named "vector" is a union, or combination, of the ideas of direction, sense and length; and so on---examples abound. But does combination as a process or operation have the same meaning in all such cases? It seems not. What, then, is your rule 0 to be? Possibly the difficulty could be surmounted as follows: if you discovered that some mental phenomena are combinable by a rule 010 others by a rule 02, still others by a rule Os, and so on, thus getting a finite number of particular rules, you could then take for a more general rule ° the disjunction, or so-called logical sum, of the particular ones; that is, you could say that rule ° is to be: 01 or 02 or Os or . . . or 011,; so that two phenomena would be combinable by ° if they were combinable by one or more of the rules 01' 02, . • ., 0",. If you thus found a rule by which every two of the phenomena you had decided to call mental admitted of combination, then your final question would be: is the result of every such combination a mental phenomenon? That is not quite the question; for under the rule two phenomena might be combinable in two or more ways, and some of the results might (conceivably) be mental and the others not; so your question would be: can every two mental phenomena be combined under your rule so as to yield a mental phenomenon? If so, then mind, as you had defined it, would have the group property under some rule of combination. If you found mind to have the group property under some rule or rules but not under others, you would be at once confronted with a further problem, which I will not tarry to state. We have been speaking of mind--of mind in general. Similar questions, -perhaps easier if not more fruitful questions,-can be put respecting particular minds-your mind, mine, John Smith's. Has every individual mind the group property? Has no such mind the property? Have some of them the property and others not? It seems very probable that the answer to the first of the questions must be negative. There are at all events some minds having (presenting, containing) mental phenomena that are definitely combinable in a way to yield mental phenomena that nevertheless do not belong to those minds. What is meant is this: a given mind may possess certain ideas which are combinable so as to form another idea; it may happen that the mind in question is incapable of grasping the new idea. Such minds have no doubt come under the observation of every experienced teacher. I myself have seen many such cases and remember one of them very vividly: that of a young woman who had made a brilliant record in undergraduate collegiate mathematics including the elements of analytical geometry and calculus; and who, encouraged by this initial success, aspired to the mathematical doctorate and entered seriously upon higher studies essential
1554
Cassius 1. Keyser
thereto; it was necessary for her to grasp more and more complicated concepts formed by combining ideas she already possessed; after no long time she reached the limit of her ability in this matter,-a fact first noticed by her instructors and then by herself,-and being a woman of good sense, she abandoned the pursuit of higher mathematics. I may add that subsequently she gained the doctorate in history. It may be that some minds are not thus limited. It may be that a genius of the so-called universal type,-an Aristotle, for example, or a Leibniz or a Leonardo da Vinci,-is one whose mind has the group property. May I leave the questions for your consideration in the days to come? The Group Concept Dimly Adumbrated in Early Philosophic Speculation.-The mathematical theory of groups is immense and manifold; in the main it is a work of the last sixty years; even the germ of it seems not to antedate Ruffini and Lagrange. Why so modern? Why did not the con cept of a closed system,---of a system having the group property,--come to birth many centuries earlier? The elemental constituents of the concept,-the idea of a class of things, the idea of anything being or not being a member of a class, the idea of a rule or law of combination,-all these were as familiar thousands of years ago as they are now. The question is one of a host of similar questions whose answers, if ever they be found, will constitute what in a previous lecture I called the yet unwritten history of the development of intellectual curiosity: Who will write that history? AI1d when? The fact that the precise formation of the mathematical concept of group is of so recent date is all the more curious because an idea closely resembling that of group has haunted the minds of a long line of thinkers and is found stalking like a ghost in the mist of philosophic speculation from remote antiquity down even to Herbert Spencer. I refer to those worldwide, age-long, philosophic specUlations which, because of their peculiar views of the universe, may be fitly called the Philosophy oj the Cosmic Cycle or Cosmic Year. This philosophy. despite the spell of a certain beauty inherent in it, has lost its vogue. To-day we are accustomed to thinking of the universe as undergoing a beginningless and endless evolution in course of which no aspect or event ever was or ever will be exactly repeated. In sharpest contrast with that conception, the philosophy of the cosmic cycle regards all the changes of which the universe is capable as constituting an immense indeed but finite and closed system of transformations, which follow each other in definite succession, like the spokes of a gigantic revolving wheel, until all possible changes have occurred in the lapse of a long but finite period of time-called a cosmic cycle Or cosmic year-whereupon everything is repeated precisely, and so on and on without end. This philosophy, I have said, has lost its vogue; but, if the philosophy be true, it will regain it, for, if true, it belongs to
The Group Concept
lSSS
the cosmic cycle and hence will recur. The history of the philosophy of the cosmic year is exceedingly interesting and it would, I believe, be an excellent subject for a doctor's dissertation. The literature is wide-ranging in kind, in place and in time. Let me cite a little of it as showing how closely its central idea resembles the mathematical concept of a cyclic group. In his Philosophie der Griechen (Vol. III, 2nd edition) Zeller, speaking of the speculations of the Stoics, says: Out of the original substance the separate things are developed according to an inner law. For inasmuch as the first principle, according to its definition, is the creative and formative power, the whole universe must grow out of it with the same necessity as the animal or the plant from the seed. The original fire, according to the Stoics and Heraclitus, first changes to "air" or vapor, then to water; out of this a portion is precipitated as earth, another remains water, a third evaporates as atmospheric air, which again kindles the fire, and out of the changing mixture of these four elements there is fonned,-from the earth as center,-the world. . . . Through this separation of the elements there arises the contrast of the active and the passive principle: the soul of the world and its body. . . . But as this contrast came in time, so it is destined to cease; the original substance gradually consumes the matter, which is segregated out of itself as its body, till at the end of this world-period a universal world conflagration brings everything back to the primeval condition. . . . But when everything has thus returned to the original unity, and the great world-year has run out, the formation of a new world begins again, which is so exactly like the former one that in it all things, persons and phenomena, return exactly as before; and in this wise the history of the world and the deity . . . moves in an endless cycle through the same stages. A similar view of cosmic history is present in the speculations of Empedocles, for whom a cycle consists of four great periods: Predominant Love-a state of complete aggregation; decreasing Love and increasing Hate; predominant Strife-complete separation of the elements; decreasing Strife and increasing Love. At the end of this fourth period, the cycle is complete and is then repeated-the history of the universe being a continuous and endlessly repeated vaudeville performance of a single play. Something like the foregoing seems to be implicit in the following statement by Aristotle in the Metaphysics: Every art and every kind of philosophy have probably been invented many times up to the limits of what is possible and been again destroyed. And in Ecclesiastes (III, 15); That which hath been is now; and that which is to be hath already been. Even Herbert Spencer at the close of his First Principles speaks as follows:
1556
Cassiru J. Keyser
Thus we are led to the conclusion that the entire process of things, as displayed in the aggregate of the visible universe, is analogous to the entire process of things as displayed in the smallest aggregates. Motion as well as matter being fixed ,in quan~ity, it would se~m that t~e. c~ange in the distribution of matter whlch motion effects, commg to a hnut m whatever direction it is carried, the indestructible motion necessitates a reverse redistribution. Apparently the universally coexistent forces of attraction and repulsion, which necessitate rhythm in all minor changes throughout the umverse, also necessitates rhythm in the totality of changes-alternate eras of evolution and dissolution. And thus there is suggested the conception of a past during which there have been successive evolutions analogous to that which is now going on; and a future during which successive other evolutions may go on-ever the same in principle but never the same in concrete result. Spencer was, you know, but poorly informed in the history of thought and he was probably not aware that the main idea in the lines just now quoted was andent two thousand years before he was born. You should note that the Spencerian universe of transformations narrowly escapes being a closed system-escapes by the last six words of the foregoing quotation. The cosmic cycles do indeed follow each other in an endless sequence-"ever the same in principle but never the same in concrete result." The repetitions are such "in principle" only, not in result-there is always something new. One of the very greatest works of man is the De Rerum Natura of Lucretius~immorta1 exposition of the thought of Epicurus, "who surpassed in intellect the race of man and quenched the light of all as the ethereal sun arisen quenches the stars." Neither a student of philosophy nor a student of natural science can afford to neglect the reading of that book. For, although it contains many,-very, very many,--errors of detail,-some of them astonishing to a modern reader ,-yet there are at least four great respects in which it is unsurpassed among the works that have come down from what we humans, in our ignorance of man's real antiquity, have been wont to call the ancient world: it is unsurpassed, I mean, in scientific spirit; in the union of that spirit with literary excellence; in the magnificence of its enterprise; and in its anticipation of concepts among the most fruitful of modem science. For such as can not read it in the original there are, happily, two excellent English translations of it-one by H. A. J. Munro and a later one by Cyril Bailey. Of this work I hope to speak further in a subsequent lecture of this COurse. My purpose in citing it here is to signalize it as being perhaps the weightiest of all contributions to what I have called the philosophy of the cosmic year. The Lucretian universe though not a finite system, is indeed a closed system, of transformations: any event, whether great or small, that has occurred in course of the beginningless past has occurred infinitely many times and will recur infinitely many times in course of an unending future; and nothing can occur that has not occurred,-there never has been and
The Group Concept
1557
there never will be aught that is new,--every occurrence is a recurrence. Let me say parenthetically, in passing, that such a concept of the universe is damnably depressing but not more so than the regnant mechanistic hypothesis of modem natural science. In relation to this hypothesis you should by no means fail to read and digest Professor W. B. Smith's great address: "Push or Pull?" (Monist, VoL XXIII, 1913). See also Smith's "Are Motions Emotions?" (Tulane Graduates' Magazine, Jan., 1914). And you should read 1. S. Haldane's Life, Mechanism and Personality. Should you desire to pursue the matter further either with a view to noting speculative adumbrations of the group concept or, as I hope, with the larger purpose of writing a historical monograph on the philosophy of the Cosmic Cycle, the following references may be of some service as a clue. "The Dream of Scipio" in Cicero's Republic (Hardingham's translation). Michael Foster's Physiology. Lyell's Principles of Geology. The fourth Eclogue of Virgil (verses 31-36). Ruckert's poem Chidher. Moleschott's Kreislauf des Lebens. Clifford's "The First and Last Catastrophe" in his Leclures and Essays. Inee's The Idea of Progress (being the Romanes Lecture, 1920).
Les mathematiciens n'etudient pas des obJets, mais des relations entre Jell objets,' iI leur est done indifferent de remplacer ces objets par d'autres, pourvu que les relations ne changent pas. La matiere ne leur importe pas, Ia forme seule les interesse. -HENRI POINCAR~
2
The Theory of Groups By SIR ARTHUR STANLEY EDDINGTON
There has been a great deal of speculation in traditional philosophy which might have been avoided if the importance of structure, and the difficulty of getting behind it, had been realised. For example, it is often said that space and time are -subjective, but they have objective counterparts; or that phenomena are subjective, but are caused by things in themselves, which must have differences inter se corresponding with the differences in the phenomena to which they give rise. Where such hypotheses are made, it is generally supposed that we can know very little about the objective counterparts. In actual fact, however, if the hypotheses as stated were correct, the objective counterparts would form a world having the same structure as the phenomenal world. . . . In short, every proposition having a communicable significance must be true of both worlds or of neither: the only difference must lie in just that essence of individuality which always eludes words and baffles description, but which, for that very reason, is irrelevant to science. BERTRAND RUSSELL, Introduction to Mathematical Philosophy, p. 61. I
LET us suppose that a thousand years hence archaeologists are digging over the sites of the forgotten civilisation of Great Britain. They have come across the following literary fragment, which somehow escaped destruction when the abolition of libraries was decreed'Twas brillig, and the slithy toves Did gyre and gimble in the wabe, All mimsy were the borogoves And the mome raths outgrabe. This is acclaimed as an important addition to the scanty remains of an interesting historical period. But even the experts are not sure what it means. It has been ascertained that the author was an Oxford mathematician; but that does not seem wholly to account for its obscurity. It is certainly descriptive of some kind of activity; but what the actors are, and what kind of actions they are performing, remain an inscrutable mystery. It would therefore seem a plausible suggestion that Mr. Dodgson was expounding a theory of the physical universe. Support for this explanation might be found in a further fragment of the same poem1558
The Theory of Groups
1559
One, two! One, two! and through and through The vorpal blade went snicker-snack! "One, two! One, two!" Out of the unknown activities of unknown agents mathematical numbers emerge. The processes of the external world cannot be described in terms of familiar images; whether we describe them by words or by symbols their intrinsic nature remains unknown. But they are the vehicle of a scheme of relationship which can be described by numbers, and so give rise to those numerical measures (pointerreadings) which are the data from which all knowledge of the external universe is inferred. Our account of the external world (when purged of the inventions of the story teller in consciousness) must necessarily be a "Jabberwocky" of unknowable actors executing unknowable actions. How in these conditions can we arrive at any knowledge at all? We must seek a knowledge which is neither of actors nor of actions, but of which the actors and actions are a vehicle. The knowledge we can acquire is knowledge of a structure or pattern contained in the actions. I think that the artist may partly understand what I mean. (Perhaps that is the explanation of the J abberwockies that we see hung on the walls of Art exhibitions.) In mathematics we describe such knowledge as knowledge of group structure. It does not trouble the mathematician that he has to deal with unknown things. At the outset in algebra he handles unknown quantities x and y. His quantities are unknown, but he subjects them to known operationsadditign, multiplication, etc. Recalling Bertrand Russell's famous definition, the mathematician never knows what he is talking about, nor whether what he is saying is true; but, we are tempted to add, at least he does know what he is doing. The last limitation would almost seem to disqualify him for treating a universe which is the theatre of unknowable actions and operations. We need a super-mathematics in which the operations are as unknown as the quantities they operate on, and a supermathematician who does not know what he is doing when he performs these operations. Such a super-mathematics is the Theory of Groups. The Theory of Groups is usually associated with the strictest logical treatment. I doubt whether anyone hitherto has committed the sacrilege of wrenching it away from a setting of pure mathematical rigour. But it is now becoming urgently necessary that it should be tempered to the under~ standing of a physicist, for the general conceptions and results are beginning to playa big part in the progress of quantum theory. Various mathematical tools have been tried for digging down to the basis of physics, and at present this tool seems more powerful than any other. So with rough argument and make-shift illustration I am going to profane the temple of rigour. My aim, however, must be very limited. At the one end we have the
Sir Arthur Stanley Eddington
1560
phenomena of observation which are somehow conveyed to man's consciousness via the nerves in his body; at the other end we have the basal entities of physics--electrons, protons, waves, etc.-which are believed to be the root of these phenomena. In between we have theoretical physics, now almost wholly mathematical. In so far as physical theory is complete it claims to show that the properties assigned to, and thereby virtually defining, the basal entities are such as to lead inevitably to the laws which we see obeyed in the phenomena accessible to our senses. If further the properties are no more than will suffice for this purpose and are stated in the most non-committal form possible, we may take the converse point of view and say that theoretical physics has analysed the universe of observable phenomena into these basal entities. The working out of this connection is the province of the mathematician, and it is not our business to discuss it here. What I shall try to show is how mathematics first gets a grip on the basal entities whose nature and activities are essentially unknowable. We are to consider where the material for the mathematician comes from, and not to any serious extent how he manipulates the material. This limitation may unfortunately give to the subject an appearance of triviality. We express mathematically ideas which, so far as we develop them, might just as well have been expressed non-mathematically. But that is the only way to begin. We want to see where the mathematics jumps off. As soon as the mathematics gets into its stride, it leaves the non-technical author and reader panting behind. I shall not be altogether apologetic if the reader begins to pant a little towards the end of the chapter. It is my task to show how a means of progress which begins with trivialities can work up momentum sufficient for it to become the engine of the expert. So in the last glimpse we shall have of it, we see it fast disappearing into the wilds.
u In describing the behaviour of an atom reference is often made to the jump of an electron from one orbit to another. We have pictured the atom as consisting of a heavy. central nucleus together with a number of light and nimble electrons circulating round it like the planets round the sun. In the solar system any change of the orbit of a planet takes place gradually, but in the atom the electron can only change its orbit by a jump. Such jumps from one orbit to an entirely new orbit occur when an atom absorbs or emits a quantum of radiation. You must not take this picture too literally. The orbits can scarcely refer to an actual motion in space, for it is generally admitted that the ordinary conception of space breaks down in the interior of an atom; nor
The Theory of Groups
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is there any desire nowadays to stress the suddenness or discontinuity conveyed by the word "jump." It is found also that the electron cannot be localised in the way implied by the picture. In short, the physicist draws up an elaborate plan of the atom and then proceeds critically to erase each detail in turn. What is left is the atom of modem physics! I want to explain that if the erasure is carefully carried out. our conception of the atom need not become entirely blank. There is not enough left to form a picture; but something is left for the mathematician to work on. In explaining how this happens, I shall take some liberties by way of simplification; but if I can show you the process in a system having some distant resemblance to an actual atom, we may leave it to the mathematician to adapt the method to the more complex conditions of Nature. For definiteness, let us suppose that there are nine main roads in the atom-nine possible orbits for the electron. Then on any occasion there are nine courses open to the electron; it may jump to any of the other eight orbits, or it may stay where it is. That reminds us of another well· known jumper-the knight in chess. He has eight possible squares to move to, or he may stay where he is. Instead of picturing the atom as containing a particle and nine roads or orbits, why should we not picture it as containing a knight and a chess-board? "You surely do not mean that literally!" Of course not; but neither does the physicist mean the particle and the orbits to be taken literally. If the picture is going to be rubbed out, is it so very important that it should be drawn one way rather than another? It turns out that my suggestion would not do at all. However metaphorical our usual picture may be, it contains an essential truth about the behaviour of the atom which would not be preserved in the knight-chessboard picture. We have to formulate this characteristic in an abstract or mathematical way, so that when we rub out the false picture we may still have that characteristic-the something which made the orbit picture not so utterly wrong as the knight picture-to hand over to the mathematician. The distinction is this. If the electron makes two orbit jumps in succession it arrives at a state which it could have reached by a single jump; but if a knight makes two moves it arrives at a square which it could not have reached by a single move. Now let us try to describe this difference in a regular symbolic way. We must find invent a notation for describing the different orbit jumps. The simplest way is to number the orbits from 1 to 9, and to imagine the numbers placed consecutively round a circle so that after 9 we come to 1 again. Then the jump from orbit 2 to orbit 5 will be described as moving on 3 places, and from orbit 7 to orbit 2 as moving on 4 places. We shall call the jump or operation of moving on one place PI' of moving
Sir Arthur Stanley Eddington
1562
on two places P2. and so on. We 'shall then have nine different operators P~ including the stay-as-you-were or identical operator Po· , We shall use the symbol A to denote the atom in some initial state, which we need not specify. Suppose that it undergoes the jump P 2• Then we shall call the atom in the new state PzA; that is to say, the atom in the new state is the result of performing the operation P2 on the system described as A. If the atom makes another jump P4' the atom in the resulting state will be described as P4P2,A, since that denotes the result of the operation P4 on the system described as P2A. If we do not want to mention the particular jumps, but to describe an atom which has made two jumps from the original state A, we shall call it correspondingly PbPa,A; a and b stand for two of the numbers 0, 1, 2, . . . 8, but we do not disclose Which. We have seen that two orbit jumps in succession give a state which could have been reached by a single jump. If the state had been reached by a single jump we should have called the atom in that state PeA, where c is one of the numbers 0, 1, 2, . . . 8. Thus we obtain a characteristic property of orbit jumps, viz. they are such that
Since it does not matter what was the initial state of the atom, and we do not pretend to know more about the atom than that it is the theatre of the operations P, we will divide the equation through by 4, leaving PbPa, = Pc·
This division by A may be regarded as the mathematical equivalent of the rubbing out of the picture. To treat the knight's moves similarly we may first distinguish them as directed approximately towards the points of the compass N.N.E., E.N.E., E.S.B., and so on, and denote them in this order by the operators Qi' Q2' . . . Q s' Qo will denote stay-as-you-were. Then since two knight's moves are never equivalent to one knight's move, our result will be 1 Qb Qa, ¥= Qc (unless a, b or c = 0).
=
We have to exclude c 0, because two moves might bring the knight back where it was originally. Let us spend a few moments contemplating this first result of our activities as super-mathematicians. The P's represent activities of an unknown kind occurring in an entity (called an atom) of unknown nature. It is true that we started with a definite picture of the atom with electrons jumping from orbit to orbit and showed that the equation Pa,P b = Pc was true of it. But now we have erased the picture; A has disappeared from 1 The sign ¥= means "is not equal to."
1563
Tlte Theory of Groul's
the formula. Without the picture, the operations P which we preserve are of entirely unknown nature. An ordinary mathematician would want to be doing something definite--to multiply, take square roots, differentiate, and so on. He wants a picture with numbers in it so that he can say for example that the electron has jumped to an orbit of double or n times the former radius. But we super-mathematicians have no idea what we are doing to the atom when we put the symbol P before A. We do not know whether we are extending it, or rotating it, or beautifying it. Nevertheless we have been able to express some truth or hypothesis about the activities of the atom by our equation PbPa = Pc. That our equation is not merely a truism is shown by the fact that when we start with a knight moving on a chess-board and make similar erasures we obtain just the opposite result QbQa =F Q c' It happens that the property expressed by PbPa Pc is the one which has given the name to the Theory of Groups. A set of operators such that the product of any two of them always gives an operator belonging to the set is called a Group. Knight's moves do not form a Group. I am not going to lead you into the ramifications of the mathematical analysis of groups and subgroups. It is sufficient to say that what physics ultimately finds in the atom, or indeed in any other entity studied by physical methods, is the structure of a set of operations. We can describe a structure without specifying the materials used; thus the operations that compose the structure can remain unknown. Individually each operation might be anything; it is the way they interlock that concerns us. The equation PbPa Pc is an example of a very simple kind of interlocking. The mode of interlocking of the operations, not their nature, is responsible for those manifestations of the external universe which ultimately reach our senses. According to our present outlook this is the basal principle in the philosophy of science. I must not mislead you into thinking that physics can derive no more than this one equation out of the atom, or indeed that this is one of the most important equations. But whatever is derived in the actual (highly difficult) study of the atom is knowledge of the same type, i.e., knowledge of the structure of a set of unknown operators.
=
=
m A very useful ki?d of operator is the selective operator. In my schooldays a foolish riddle was current-"How do you catch lions in the desert?" Answer: "In the desert you have lots of sand and a few lions;- so you take a sieve and sieve out the sand, and the lions remain." I recall it because it describes one of the most usual methods used in quantum theory for obtaining anything that we wish to study.
15ti4
Sir Arthur StQ1lley Eddington
Let Z denote the zoo, and SI the operation of sieving out or selecting lions; then StZ == L, where L denotes lions-or, as we might more formally say, L denotes a pure ensemble having the leonine characteristic. These pure selective operators have a rather curious mathematical property, viz.
(A). For S12 (an abbreviation for S~Sz) indicates that having selected all the lions, you repeat the operation, selecting all the lions from what you have obtained. Putting through the sieve a second time makes no difference; and in fact, repeating it n times, you have Sr == S" The property expressed by equation (A) is called idempotency. Now let St be the operation of selecting tigers. We have (B).
For if you have first selected all the lions, and go on to select from these all the tigers, you obtain nothing. Now suppose that the different kinds of animals in the zoo are numbered in a catalogue from 1 to n and we introduce a selective operator for each; then (C),
where 1 is the stay-as-you-were operator. For if you sieve out each constituent in tum and add together the results, you get the mixture you started with. A set of operators which satisfies (A), (B) and (C) is called a spectral set, because it analyses any aggregation into pure constituents in the same way that light is analysed by a prism or grating into the different pure colours which form the spectrum. The three equations respectively secure that the operators of a spectral set are idempotent, non-overlapping and exhaustive. Let us compare the foregoing method of obtaining lions from the zoo with the method by which "heavy water" is obtained from ordinary water. In the decomposition of water into oxygen and hydrogen by electrolysis, the heavy water for some reason decomposes rather more slowly than the ordinary water. Consequently if we submit a large quantity of water to electrolysis, so that the greater part disappears into gas, the residue contains a comparatively high proportion of heavy water. This process of "fractionating" is a selective operation, but it is not pure selection such as we have been considering. If taking the residue we again perform the operation of electrolysis we shall still further concentrate the heavy water. A fractionating operator F is not idempotent (F2"", F), and this distinguishes it from a pure selective operator S.
The TheOfY of Groups
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The idea of analysing things into pure constituents and of distinguishing mixtures from pure ensembles evidently plays an important part in physical' conceptions of reality. But it is not very easy to define just what we mean by it. We think of a pure ensemble as consisting of a number of individuals all exactly alike. But the lions at the zoo are not exactly alike; they are only alike from a certain point of view. Are the molecules of heavy water all exactly alike? We cannot speak of their intrinsic nature, because of that we know nothing. It is their relations to, or interactions with, other objects which define their physical properties; and in an interrelated universe no two things can be exactly alike in all their relations. We can only say then that the molecules of heavy water are alike in some common characteristic. But that is not sufficient to secure that they form a pure ensemble; the molecules which form any kind of mixture are alike in one common characteristic, viz. that they are molecules. If we have a difficulty in defining purity of things for which we have more or less concrete pictures, we find still more difficulty with regard to the more recondite quantities of physics. Nevertheless it is clear that the idea of distinguishing pure constituents from mixtures contains a germ of important truth. It is the duty of the mathematician to save that germ out of the dissolving picture; and he does this by directing attention not to the nature of what we get by the operation but to the nature of the selective operation itself. He shows that those observational effects which reach our perceptions, generally attributed to the fact that we are dealing with an assembly of like individuals, are deducible more directly from the fact that the assembly is obtainable by a kind of operation which, once performed, can be repeated any number of times without making any difference. He thus substitutes a perfectly definite mathematical property of the operator, viz. S~2 Sz, for a very vaguely defined property of the result of the operation, viz. a certain kind of likeness of the individuals which together form L. He thus frees his results from various unwarrantt!d hypotheses that may have been introduced in trying to form a picture of this property of L. ' I In the early days of atomic theory, the atom was defined as an indivisible particle of matter. Nowadays dividing the atom seems to be the main occupation of physicists. The definition contained an essential truth; only it was wrongly expressed. What was really meant was a property typically manifested by indivisible particles but not necessarily confined to indivisible particles. That is the way with all models and pictures and familiar descriptions; they show the property that we are interested in, but they connect it with irrelevant properties which may be erroneous and for which at any rate we have no warrant. You will see that the mathematical method here discussed is much more economical of hypothesis. It says no more about the system than that which it is actually going t(
=
1566
Sir Arthur Stanley Eddington
embody in the formulae which yield the comparison of theoretical physics with observation. And, in so far as it can surmount the difficulties of investigation, its assertions about the physical universe are the exact systematised equivalent of the observational results on which they are based. I think it may be said that hypotheses in the older sense are banished from those parts of physical science to which the group method has been extended. The modern physicist makes mistakes, but he does not make hypotheses. One effect of introducing selective operators is that it removes the distinction between operators and operands. In considering the "jump" operators P, we had to introduce an operand A, for them to work on. We must furnish some description of A, and A is then whatever answers to that description. Let S4 be the operation of selecting whatever answers to the description A, and let U be the universe. Then evidently A SaU; and instead of PbPa,A we can write PbPaSaU. Thus special operands, as distinct from operators, are not required. We have a large Variety of operators, some of them selective, and just one operand-the same in every formula-namely the universe. This mathematical way of describing everything with which we deal emphasises, perhaps inadvertently, an important physical truth. Usually when we wish to consider a problem about a hydrogen atom, we take a blank sheet of paper and mark in first the proton and then the electron. That is all there is in the problem unless or until we draw something else that we suppose to be present. The atom thus presents itself as a work of creation-a creation which can be stopped at any stage. When we have created our hydrogen atom, we mayor may not go on to create a universe for it to be part of. But the real hydrogen atoms that we experiment on are something selected from an always present universe, often selected or segregated experimentally, and in any case selected in our thoughts. And we are learning to recognise that a hydrogen atom would not be what it is, were it not the result of a selective operation performed on that maze of interrelatedness which we call the universe. In Einstein's theory of relativity the observer is a man who sets out in quest of truth armed with a measuring-rod. In quantum theory he sets out armed with a sieve.
=
IV
I am now going to introduce a set of operations with which we can accomplish something rather more ambitious. They are performed on a set of four things which I will represent by the letters A, B, C, D. We begin with eight operations; after naming (symbolically) and describing each operation, I give the result of applying it to ABCD:
Tlze Theory af Groups
1567
Sa: Interchange the first and second, also the third and fourth. BADC. S~. Interchange the first and third, also the second and fourth. CDAB. S'Y' Interchange the first and fourth, also the second and third. DCBA. S13' Stay as you were. ABCD. Do.. Turn the third and fourth upside down. AB:::>O D~.
Turn the second and fourth upside down. AaCa.
D'Y' Turn the second and third upside down. Aa:)D. D 13 • Stay as you were. ABCD.
We also use an operator denoted by the sign - which means "tum them all upside down." We can apply two or more of these operations in succession. For example, SaS{3 means that, having applied the operation S{3 which gives CDAB, we perform on the result the further operation So. which inter~ changes the first and second and also the third and fourth. The result is DeBA. This is the same as the result of the single operation S'Y; consequently SaS{J
= S'Y'
Sometimes, but not always, it makes a difference which of the two operations is performed first. For example, Taking the result of the operation D'Y' viz. Aa:JD, and performing on it the operation Sa, we obtain HADJ. But taking the result of the operation Sa' viz. BADC, and performing on it the operation D'Y' we obtain BVaC. Thus the double operation SaD'Y is not the same as D'YSa' There is, however, a simple relation. aAD:) is obtained by inverting each letter in BVaC, that is to say, by applying the operation which we denote by the
sign -. Thus SaD'Y
= -D'YSa'
Operators related in this way are said to anticommute. On examination we find that SQ.' S{J commute, and so do Do., D{J; so also do Sa. and Da.' It is only a combination of an S and a D with different suffixes a, {3, ')I (but not 8) which exhibits anticommutation. We can make up sixteen different operators of the form SaDb' where a and b stand for any of the four suffixes a., {3, ')I, 8. It is these combined operators which chiefly interest us. I will call them E-operators and denote them by E 1 , E 2 , E s,' . . E 16• They form a Group, which (as we have seen) means that the result of applying two operations of the Group in succession can equally be obtained by applying a single operation of the Group.
Sir Artlwl' Stanley Eddinrton
1568
I should, however, mention that the operation - is here regarded as thrown in gratuitously.2 We may not by a single operation Ee be able to get the letters into the same arrangement as that given by EbEa,; but if not, we can get the same arrangement with all the letters inverted. This property of the E-operators is accordingly expressed by Ea,Eb:= ±Ec·
We now pick out five of the E-operators. Our selection at first sight seems a strange one, because it has no apparent connection with their constitution out of S- and D-operators. It is as followsE1 := SaPo, which gives BADe. ,.
AgCO.
:= S,,/D,,/
"
DJgA.
E4 := SaP"/
"
gADJ.
E2
:=
E3
E5
S'PfJ
:= S,,/Dfj
.
OJ8A.
These five are selected because they all anticommute with one another; that is to say, E1E2 = -E2E1' and so on for all the ten pairs. You can verify this by operating with the four letters, though, of course, there are mathematical dodges for verifying it more quickly. We call a set like this a pentad. There are six different ways of choosing our pentad, obtained by ringing the changes on the suffixes IX, {3, 'Y' But it is not possible to find more than five E-operators each of which anticommutes with all the others. That is why we have to stop at pentads. Another important property must be noticed. You will see at once that £1 2 1; for E1 is the same as Sq., and a repetition of the interchange expressed by Sa. restores the original arrangement. But consider El. In the operation E5 we tum the second and fourth letters upside down, and then reverse the order of the letters. The letters left right way up are thereby brought into the second and fourth places, so that in repeating the operation they become turned upside down. Hence the letters come back to their original order, but are all upside down. This is equivalent to the operation -. So that we have E52 := -1. In this way we find that
=
E 12-E2-E2-1 2 3 , E2-E24 5 - -1 .
A pentad always consists of three operators whose square is 1, and two operators whose square is -1. With regard to the symbols 1 and -1, I should explain that 1 here 2 To obtain a Group according to the strict definition we should have to take 32 operators, VIZ. the above 16, and the 16 obtained by prefixing -.
15159
The Theory of Groups
stands for the stay-as-you-were operator. Since that is the effect of the number 1 when it is used as an operator (a multiplier) in arithmetic, the notation is appropriate. (We have also denoted the stay-as-you-were operator by S6 and Do, so that we have So Do 1.) Since the operator 1 makes no difference, the operators "-" and "-1" are the same; so we sometimes put in a 1, when - by itself would look lonely. Repetition of the operation - restores the original state of things; consequently ( - ) 2 or (-1)2 is equal to 1. Although the symbol, as we have here defined it, has no connection with "minus," it has in this respect the same and -1 in algebra. property as I have told you that the proper super-mathematician never knows what he is doing. We, who have been working on a lower plane, know what we have been doing. We have been busy rearranging four letters. But there is a super-mathematician within us who knows nothing about this aspect of what we have been studying. When we announce that we have found a group of sixteen operations, certain pairs of which commute and the remaining pairs anticommute, some of which are square roots of 1 and the others square roots of -1, he begins to sit up and take notice. For he can grasp this kind of structure of a group of operations, not referring to the nature of the operations but to the way they interlock. He is interested in the arrangement of the operators to form six pentads. That is his ideal of knowledge of a set of operations--knowledge of its distinctive kind of structure. A great many other properties of E-operators have been found, which I have not space to examine in detail. There are pairs of triads, such that members of the same triad all anticommute but each commutes with the three members of the opposite triad. There are antitriads composed of three mutually commuting operators, which become anti-tetrads if we include the stay-as-you-were operator. All this knowledge of structure can be expressed without specifying the nature of the operations. And it is through recognition of a structure of this kind that we can have knowledge of an external world which from an ordinary standpoint is essentially unknowable. Some years ago I worked out the structure of this group of operators in connection with Dirac's theory of the electron. I afte~ards learned that a great deal of what I had written was to be found in a treatise on Kummer's quartic surface. There happens to be a model of Kummer's quartic surface in my lecture-room, at which I had sometimes glanced with curiosity, wondering what it was all about. The last thing that entered my head was that I had written (somewhat belatedly) a paper on its structure. Perhaps the author of the treatise would have been equally surprised to learn that he was dealing with the behaviour of an electron. But then, you see, we super-mathematicians never do know what we are doing
= =
Sir Arthur Stanley Eddington
1570
v As the result of a game with four letters we have been able to describe a scheme of structure, which can be detached from the game and given other applications. When thus detached, we find this same structure occur~ ring in the world of physics. One small part of the scheme shows itself in a quite elementary way, as we shall presently see; another part of it was brought to light by Dirac in his theory of the electron; by further search the whole structure is found, each part having its appropriate share in physical phenomena. When we seek a new application for our symbolic operators E, we cannot foresee what kind of operations they will represent; they have been identified in the game, but they have to be identified afresh in the physical world. Even when we have identified them in the familiar story of consciousness, their ultimate nature remains unknown; for the nature of the activity of the external world is beyond our apprehension. Thus armed with our detached scheme of structure we approach the physical world with an open mind as to how its operations will manifest themselves in our experience. I shall have to refer to an elementary mathematical result. Consider the square of (2E 1 + 3E2 ), that is to say the operation which is equivalent to twice performing the operation (2E 1 + 3E:). We have not previously mixed numbers with our operation; but no difficulty arises if we understand that in an expression of this kind 2 stands for the operation of multiplying by 2, 3 for the operation of multiplying by 3, as in ordinary algebra. We have (2E 1 + 3E2 ) (2E 1 + 3E2 )
= 4E12 + 6E1E 2 + 6E2E 1 + 9E'i.
We have had to attend to a point which does not arise in ordinary algebra. In algebra we should have lumped together the two middle terms and have written 12E1E2 instead of 6E 1E 2 + 6E2E 1• But we have seen (p. 1568) that the operation E2 followed by the operation E1 is not the same as the operation E1 followed by the operation E 2 ; in fact we deliberately chose these operators so that E2El = -E1E 2 • Consequently the two middle terms cancel one another and we are left with (2E 1
+ 3E)2 = 4E12 + 9E22.
= 1, E22 = 1. Thus + 3E =4 + 9 = 13.
But we have also seen that E12 (2E 1
2 )2
In other words (2E 1 + 3E2 ) is the square root of 13, or rather a square root of 13. Suppose that you move to a position 2 yards to the right and 3 yards forward. By the theorem of Pythagoras your resultant displacement is
The Theory of Groups
\1'(2 2
1571
+ 32 )
or y13 yards. It suggests itself that when the super-mathematician (not knowing what kind of operations he is referring to) says that (2E1 + 3E2 ) is a square root of 13, he may mean the same thing as the geometer who says that a displacement 2 yards to the right and 3 yards forward is square-root-of-13 yards. Actually the geometer does not know what kind of operations he is referring to either; he only knows the familiar story teller's description of them. He can render himself independent of the imaginations of the familiar story teller by becoming a super-mathematician. He will then say: What the familiar story teller calls displacement to the right is an operation whose intrinsic nature is unknown to me and I will denote it by E 1 ; what he calls displacement forward is another unknown operation which I will denote by E 2• The kind of knowledge of the properties of displacement which I have acquired by experience is contained in such statements as "a displacement 2 yards to the right and 3 yards forward is square-root-of-13 yards." In my notation this becomes "2E1 + 3E2 is a square-root of 13." Super-mathematics enables me to boil down these statements to the single conclusion that displacement to the right and displacement forward are two operations of the set whose group structure has been investigated in Section IV.s Similarly we can represent a displacement of 2 units to the right, 3 units forward and 4 units upward by (2E1 + 3E2 + 4Es). Working out the square of this expression in the same way, the result is found to be 29, which agrees with the geometrical calculation that the resultant displacement is y(2 2 + 32 + 42) = \1'29. The secret is that the super-mathematician expresses by the anticommutation of his operators the property which the geometer conceives as perpendicularity of displacements. That is why on p. 1568 we singled out a pentad of anticommuting operators, foreseeing that they would have an immediate application in describing the property of perpendicular directions without using the traditional picture of space. They express the property of perpendicularity without the picture of perpendicularity. Thus far we have touched only the fringe of the structure of our set of sixteen E-operators. Only by entering deeply into the theory of electrons could I show the whole structure coming into evidence. But I will take you one small step farther. Suppose that you want to move 2 yards to the right, 3 yards forward, 4 yards upward, and 5 yards perpendicular to all three-in a fourth dimension. By this time you will no doubt have learned the trick, and will write down readily (2E1 + 3E2 + 4E8 + 5E4 ) as the operator which symbolises this displacement. But there is a break-down. S He will, of course, require more than a knowledge relating to two of the operators to infer the group structure of the whole set. The immediate inference at this stage is such portion of the group structure as is revealed by the equations E12 E,.!? = 1, E;El = -E1Es-
=
Sir Arthur Stanley Eddington
1572
The trouble is that we have exhausted the members of the pentad whose square is 1, and have to fall back on E4 whose square is -1 (p. 270). Consequently ( 2El
+ 3E2 + 4Es + 5E4 ) 2 =22 + 32 + 42 -
52
=4.
Thus our displacement is a square root of 4, whereas Pythagoras's theo~ rem would require that it should be the square root of 22 + 32 + 42 + 52, or 54. Thus there is a limitation to our representation of perpendicular directions by E-operators; it is only saved from failure in practice because in the actual world we have no occasion to consider a fourth perpendicular direction. How lucky! It is not luck. The structure which we are here discussing is claimed to be the structure of the actual world and the key to its manifestations in our experience. The structure does not provide for a fourth dimension of space, so that there cannot be a fourth dimension in a world built in that way. Our experience confirms this as true of the actual universe. If we wish to introduce a fourth direction of displacement we shall have to put up with a minus sign instead of a plus sign, so that it will be a displacement of a somewhat different character. It was found by Minkowski in 1908 that "later" could be regarded in this way as a fourth direction of displacement, differing only from ordinary space displacements in the fact that its square combines with a minus instead of a plus sign. Thus 2 yards to the right, 3 yards forward, 4 yards upward and 5 "yards" later 4 is represented by the operator (2El + 3E2 + 4Es + 5E4 ). We have calculated above that it is a square root of 4, so that it amounts to a displacement of 2 yards. When, as here, we consider displacement in time as well as in space, the resultant amount is called the interval. The value of the interval in the above problem according to Minkowski's formula is 2 yards, so that our results agree. Minkowski introduced the minus instead of the plus sign in the fourth term, regarding the change as expressing the mathematical distinction between time and space; we introduce it because we cannot help it-it is forced on us by the group structure that we are studying. Minkowski's interval afterwards became the starting point of the general theory of relativity. Thus the distinction between space and time is already foretold in the structure of the set of E-operators. Space can have only three dimensions, because no more than three operators fulfil the necessary relationship of perpendicular displacement. A fourth displacement can be added, but it has a character essentially different from a space displacement. Calling it a time displacement, the properties of its associated operator secure that the relation of a time displacement to a space displacement shall be precisely that postulated in the theory of relativity. 4.
A "yard" of time is to be interpreted as the time taken by light to travel a yard.
The Theory of Groups
1573
I do not suggest that the distinction between the fourth dimension and the other three is something that we might have predicted entirely by a priori reasoning. We had no reason to expect a priori that a scheme of structure which we found in a game with letters would have any importance in the physical universe. The agreement is only impressive if we have independent reason to believe that the world-structure is based on this particular group of operators. We must recall therefore that the E-operators were first found to be necessary to physics in Dirac's wave equation of an electron. Dirac's great achievement in introducing this structure was that he thereby made manifest a recondite property of the electron, observationally important, which is commonly known as its "spin." That is a problem which seems as far removed as possible from the origin of the distinction of space and time. We may say that although the distinction of space and time cannot be predicted for a universe of unknown nature, it can be predicted for a universe whose elementary particles are of the character described in modern wave mechanics.
'PART X L
-
•
I
Mathematics of Infinity ....
,.
, . 1. Mathematics and the Met~physiCi~s by BERTRAND RUSSELL 2. Infinity by HANS HAHN ' \
See skulking Truth to her old cavern fled, Mountains of Casuistry heap'd o'er her head! Philosophy, that lean'd on Heav'n bejore, Shrmks to her second cause, and is no more. PhYsic of Metaphysic begs defence, And Metaphysic calls jor aid on Sense! See Mystery to MathematIcs fly!
1
-ALBXANDEIl POPE
Mathematics and the Metaphysicians By BERTRAND RUSSELL *
THE nineteenth century, which prided itself upon the invention of steam and evolution, might have derived a more legitimate title to fame from the discovery of pure mathematics. This science, like most others, was baptised long before it was born; and thus we find writers before the nineteenth century alluding to what they called pure mathematics. But if they had been asked what this subject was, they would only have been able to say that it consisted of Arithmetic, Algebra, Geometry, and so on. As to what these studies had in common, and as to what distinguished them from applied mathematics, our ancestors were completely in the dark. Pure mathematics was discovered by Boole, in a work which he called the Laws of Thought (1854). This work abounds in asseverations that it is not mathematical, the fact being that Boole was too modest to suppose his book the first ever written on mathematics. He was also mistaken in supposing that he was dealing with the laws of thought: the question how people actually think was quite irrelevant to him, and if his book had really contained the laws of thought, it was curious that no one should ever have thought in such a way before. His book was in fact concerned with formal logic, and this is the same thing as mathematics. Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. Both these points would belong to applied mathematics. We start, in pure mathematics, from certain rules of inference, by which we can infer that if one proposition is true, then so is some other proposition. These rules of inference consti-
* For a biographical note about Bertrand Russell, see p. 1576
377.
Mathematics and the Metaphysicians
1517
tute the major part of the principles of formal logic. We then take any hypothesis that seems amusing, and deduce its consequences. If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate. As one of the chief triumphs of modern mathematics consists in having discovered what mathematics really is, a few more words on this subject may not be amiss. It is common to start any branch of mathematics-for instance, Geometry-with a certain number of primitive ideas, supposed incapable of definition, and a certain number of primitive propositions or axioms, supposed incapable of proof. Now the fact is that, though there are indefinables and indemonstrables in every branch of applied mathematics, there are none in pure mathematics except such as belong to general logic. Logic, broadly speaking, is distinguished by the fact that its propositions can be put into a form in which they apply to anything whatever. All pure mathematics-Arithmetic, Analysis, and Geometry -is built up by combinations of the primitive ideas of logic, and its propositions are deduced from the general axioms of logic, such as the syllogism and the other rules of inference. And this is no longer a dream or an aspiration. On the contrary, over the greater and more difficult part of the domain of mathematics, it has been already accomplished; in the few remaining cases, there is no special difficulty, and it is now being rapidly achieved. Philosophers have disputed for ages whether such deduction was possible; mathematicians have sat down and made the deduction. For the philosophers there is now nothing left but graceful acknowledgments. The subject of formal logic, which has thus at last shown itself to be identical with mathematics, was, as every one knows, invented by Aristotle, and formed the chief study (other than theology) of the Middle Ages. But Aristotle never got beyond the syllogism, which is a very small part of the subject, and the schoolmen never got beyond Aristotle. If any proof were required of our superiority to the medireval doctors. it might be found in this. Throughout the Middle Ages, almost all the best intellects devoted themselves to formal logic, whereas in the nineteenth century only an infinitesimal proportion of the world's thought went into this subject. Nevertheless, in each decade since 1850 more has been done to advance the subject than in the whole period from Aristotle to Leibniz. People have discovered how to make reasoning symbolic, as it is in Algebra, so that deductions are effected by mathematical rules. They have discovered many rules besides the syllogism, and a new branch of logic,
1578
Bertrand Russell
called the Logic of Relatives, l has been invented to deal with topics that wholly surpassed the powers of the old logic, though they form the chief contents of mathematics. It is not easy for the lay mind to realise the importance of symbolism in discussing the foundations of mathematics, and the explanation may perhaps seem strangely paradoxical. The fact is that symbolism is useful because it makes things difficult. (This is not true of the advanced parts of mathematics, but only of the beginnings.) What we wish to know is, what can be deduced from what. Now, in the beginnings, everything is self-evident; and it is very hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we invent some new and difficult symbolism, in which nothing seems obvious. Then we set up certain rules for operating on the symbols, and the whole thing becomes mechanical. In this way we find out what must be taken as premiss and what can be demonstrated or defined. For instance, the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions. But without a symbolism it would have been very hard to find this out. It is so obvious that two and two are four, that we can hardly make ourselves sufficiently sceptical to doubt whether it can be proved. And the same holds in other cases where self-evident things are to be proved. But the proof of self-evident propositions may seem, to the uninitiated, a somewhat frivolous occupation. To this we might reply that it is often by no means self-evident that one obvious proposition follows from another obvious proposition; so that we are really discovering new truths When we prove what is evident by a method which is not evident. But a more interesting retort is, that since people have tried to prove obvious propositions, they have found that many of them are false. Self-evidence is often a mere will-o'-the-wisp, which is sure to lead us astray if we take it as our guide. For instance, nothing is plainer than that a whole always has more terms than a part, or that a number is increased by adding one to it. But these propositions are now known to be usually false. Most numbers are infinite, and if a number is infinite you may add ones to it as long as you like without disturbing it in the least. One of the merits of a proof is that it instils a certain doubt as to the result proved; and when what is obvious can be proved in some cases, but not in others, it becomes possible to suppose that in these other cases it is false. The great master of the art of fo~mal reasoning, among the men of our own day, is an Italian, Professor Peano. of the University of Turin. 2 He This subject is due in the main to Mr. C. S. Peirce. I ought to have added Frege, but hi!> writings were unknown to me when this article was written. [Note added in 1917. ] 1
2
Mathematics and the Metaphysicians
1579
has reduced the greater part of mathematics (and he or his followers will, in time, have reduced the whole) to strict symbolic form, in which there are no words at all. In the ordinary mathematical books, there are no doubt fewer words than most readers would wish. Still, little phrases occur, such as therefore, let us assume, consider, or hence it follows. AU these, however, are a concession, and are swept away by Professor Peano. For instance, if we wish to learn the whole of Arithmetic, Algebra, the Calculus, and indeed all that is usually called pure mathematics (except Geometry), we must start with a dictionary of three words. One symbol stands for zero, another for number, and a third for next after. What these ideas mean, it is necessary to know if you wish to become an arithmetician. But after symbols have been invented for these three ideas, not another word is required in the whole development. All future symbols are symbolically explained by means of these three. Even these three can be explained by means of the notions of relation and class; but this requires the Logic of Relations, which Professor Peano has never taken up. It must be admitted that what a mathematician has to know to begin with is not much. There are at most a dozen notions out of which all the notions in all pure mathematics (including Geometry) are compounded. Professor Peano, who is assisted by a very able school of young Italian disciples, has shown how this may be done; and although the method which he has invented is capable of being carried a good deal further than he has carried it, the honour of the pioneer must belong to him. Two hundred years ago, Leibniz foresaw the science which Peano has perfected, and endeavoured to create it. He was prevented from succeeding by respect for the authority of Aristotle, whom he could not believe guilty of definite, formal fallacies; but the subject which he desired to create now exists, in spite of the patronising contempt with which his schemes have been treated by all superior persons. From this "Universal Characteristic," as he called it, he hoped for a solution of all problems, and an end to all disputes. "If controversies were to arise," he says, "there would be no more need of disputation between two philosophers than between two accountants. For it would suffice to take their pens in their hands, to sit down to their desks, and to say to each other (with a friend as witness, if they liked), 'Let us calculate.' " This optimism has now appeared to be somewhat excessive; there still are problems whose solution is doubtful, and disputes which calculation cannot decide. But over an enormous field of what was formerly controversial, Leibniz's dream has become sober fact. In the whole philosophy of mathematics, which used to be at least as full of doubt as any other part of philosophy, order and certainty have replaced the confusion and hesitation which formerly reigned. Philosophers, of course, have not yet discovered this fact,
1580
Bertrand. Russell
and continue to write on such subjects in the old way. But mathematicians, at last in Italy, have now the power of treating the principles of mathematics in an exact and masterly manner, by means of which the certainty of mathematics extends also to mathematical philosophy. Hence many of the topics which used to be placed among the great mysteries-for example, the natures of infinity, of continuity, of space, time and motion-are now no longer in any degree open to doubt or discussion. Those who wish to know the nature of these things need only read the works of such men as Peano or Georg Cantor; they will there find exact and indubitable expositions of all these quondam mysteries. In this capricious world, nothing is more capricious than posthumous fame. One of the most notable examples of posterity's lack of judgment is the Eleatic Zeno. This man, who may be regarded as the founder of the philosophy of infinity, appears in Plato's Parmenides in the privileged position of instructor to Socrates. He invented four arguments, all immeasurably subtle and profound, to prove that motion is impossible, that Achilles can never overtake the tortoise, and that an arrow in flight is really at rest. After being refuted by Aristotle, and by every subsequent philosopher from that day to our own, these arguments were reinstated, and made the basis of a mathematical renaissance, by a German professor, who probably never dreamed of any connection between himself and Zeno. Weierstrass,s by strictly banishing from mathematics the use of infinitesimals, has at last shown that we live in an unchanging world. and that the arrow in its flight is truly at rest. Zeno's only error lay in inferring (if he did infer) that, because there is no such thing as a state of change, therefore the world is in the same state at anyone time as at any other. This is a consequence which by no means follows; and in this respect, the German mathematician is more constructive than the in~ genious Greek. Weierstrass has been able, by embodying his views in mathematics, where familiarity with truth eliminates the vulgar prejudices of common sense, to invest Zeno's paradoxes with the respectable air of platitudes; and if the result is less delightful to the lover of reason than Zeno's bold defiance. it is at any rate more calculated to appease the mass of academic mankind. Zeno was concerned, as a matter of fact, with three problems, each presented by motion, but each more abstract than motion, and capable of a purely arithmetical treatment. These are the problems of the infinitesimal, the infinite, and continuity. To state clearly the difficulties involved, was to accomplish perhaps the hardest part of the philosopher's task. This was done by Zeno. From him to our own day, the finest intellects of each generation in turn attacked the problems, but achieved, broadly speaking, nothing. In our own time, however, three men-Weier3
Professor of Mathematics in the University of Berlin. He died in 1897.
Mathematics and t1le Metaphysicians
1581
strass, Dedekind, and Cantor-have not merely advanced the three problems, but have completely solved them. The solutions, for those acquainted with mathematics, are so clear as to leave no longer the slightest doubt or difficulty. This achievement is probably the greatest of which our age has to boast; and I know of no age (except perhaps the golden age of Greece) which has a more convincing proof to offer of the transcendent genius of its great men. Of the three problems, that of the infinitesimal was solved by Weierstrass; the solution of the other two was begun by Dedekind, and definitively accomplished by Cantor. The infinitesimal played formerly a great part in mathematics. It was introduced by the Greeks, who regarded a circle as differing infinitesimally from a polygon with a very large number of very small equal sides. It gradually grew in importance, until, when Leibniz invented the Infinitesimal Calculus, it seemed to become the fundamental notion of all higher mathematics. Carlyle tells, in his Frederick the Great, how Leibniz used to discourse to Queen Sophia Charlotte of Prussia concerning the infinitely little, and how she would reply that on that subject she needed no instruction-the behaviour of courtiers had made her thoroughly familiar with it. But philosophers and mathematicians-who for the most part had less acquaintance with courts-continued to discuss this topic, though without making any advance. The Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be. It was plainly not quite zero, because a sufficiently large number of infinitesimals, added together, were seen to make up a finite whole. But nobody could point out any fraction which was not zero, and yet not finite. Thus there was a deadlock. But at last Weierstrass discovered that the infinitesimal was not needed at all, and that everything could be accomplished without it. Thus there was no longer any need to suppose that there was such a thing. Nowadays, therefore, mathematicians are more dignified than Leibniz: instead of talking about the infinitely small, they talk about the infinitely great-a subject which, however appropriate to monarchs, seems, unfortunately, to interest them even less than the infinitely little interested the monarchs to whom Leibniz discoursed. The banishment of the infinitesimal has all sorts of odd consequences, to which one has to become gradually accustomed. For example, there is no such thing as the next moment. The interval between one moment and the next would have to be infinitesimal, since, if we take two moments with a finite interval between them, there are always other moments in the interval. Thus if there are to be no infinitesimals-, no two moments are quite consecutive, but there are always other moments between any two. Hence there must be an infinite number of moments between any two; because if there were a finite number one would be nearest the first of the
1582
Bertrand RUSSt!11
two moments, and therefore next to it. This might be thought to be a difficulty; but, as a matter of fact, it is here that the philosophy of the infinite comes in, and makes all straight. The same sort of thing happens in space. If any piece of matter be cut in two, and then each part be halved, and so on, the bits will become smaller and smaller, and can theoretically be made as small as we please. However small they may be, they can still be cut up and made smaller still. But they will always have some finite size, however small they may be. We never reach the infinitesimal in this way, and no finite number of divisions will bring us to points. Nevertheless there are points, only these are not to be reached by successive divisions. Here again, the philosophy of the infinite shows us how this is possible, and why points are not infinitesimal lengths. As regards motion and change, we get similarly curious results. People used to think that when a thing changes, it must be in a state of change, and that when a thing moves, it is in a state of motion. This is now known to be a mistake. When a body moves, all that can be said is that it is in one place at one time and in another at another. We must not say that it will be in a neighbouring place at the next instant, since there is no next instant. Philosophers often tell us that when a body is in motion, it changes its position within the instant. To this view Zeno long ago made the fatal retort that every body always is where it is; but a retort so simple and brief was not of the kind to which philosophers are accustomed to give weight, and they have continued down to our own day to repeat the same phrases which roused the Eleatic's destructive ardour. It was only recently that it became possible to explain motion in detail in accordance with Zeno's platitude, and in opposition to the philosopher's paradox. We may now at last indulge the comfortable belief that a body in motion is just as truly where it is as a body at rest. Motion consists merely in the fact that bodies are sometimes in one place and sometimes in another, and that they are at intermediate places at intermediate times. Only those who have waded through the quagmire of philosophic speculation on this subject can realise what a liberation from antique prejudices is involved in this simple and straightforward commonplace. The philosophy of the infinitesimal, as we have just seen, is mainly negative. People used to believe in it, and now they have found out their mistake. The philosophy of the infinite, on the other hand, is wholly positive. It was formerly supposed that infinite numbers, and the mathematical infinite generally, were self-contradictory. But as it was obvious that there were infinities-for example, the number of numbers--the contradictions of infinity seemed unavoidable, and philosophy seemed to have wandered into a "cul-de-sac." This difficulty led to Kant's antinomies, and hence, more or less indirectly, to much of Hegel's dialectic method.
Mathematics and the MetaphysIcIans
1583
Almost all current philosophy is upset by the fact (of which very few philosophers are as yet aware) that all the ancient and respectable contradictions in the notion of the infinite have been once for all disposed of. The method by which this has been done is most interesting and instructive. In the first place, though people had talked glibly about infinity ever since the beginnings of Greek thought, nobody had ever thought of asking, What is infinity? If any philosopher had been asked for a definition of infinity, he might have produced some unintelligible rigmarole, but he would certainly not have been able to give a definition that had any meaning at all. Twenty years ago, roughly speaking, Dedekind and Cantor asked this question, and, what is more remarkable, they answered it. They found, that is to say, a perfectly precise definition of an infinite number or an infinite collection of things. This was the first and perhaps the greatest step. It then remained to examine the supposed contradictions in this notion. Here Cantor proceeded in the only proper way. He took pairs of contradictory propositions, in which both sides of the contradiction would be usually regarded as demonstrable, and he strictly examined the supposed proofs. He found that all proofs adverse to infinity involved a certain principle, at first sight obviously true, but destructive, in its consequences, of almost all mathematics. The proofs favourable to infinity, on the other hand, involved no principle that had evil consequences. It thus appeared that common sense had allowed itself to be taken in by a specious maxim, and that, when once this maxim was rejected, all went well. The maxim in question is, that if one collection is part of another, the one which is a part has fewer terms than the one of which it is a part. This maxim is true of finite numbers. For example, Englishmen are only some among Europeans, and there are fewer Englishmen than Europeans. But when we come to infinite numbers, this is no longer true. This breakdown of the maxim gives us the precise definition of infinity. A collection of terms is infinite when it contains as parts other collections which have just as many terms as it has. If you can take away some of the terms of a collection, without diminishing the number of terms, then there are an infinite number of terms in the collection. For example, there are just as many even numbers as there are numbers altogether, since every number can be doubled. This may be seen by putting odd and even numbers together in one row, and even numbers alone in a row below:1, 2, 3, 4, 5, ad infinitum. 2, 4, 6, 8, 10, ad infinitum. There are obviously just as many numbers in the row below as in the row above, because there is one below for each one above. This property, which was formerly thought to be a contradiction, is now transformed into
1584
B~rtrtJnd
Russell
a harmless definition of infinity, and shows, in the above case, that the number of finite numbers is infinite. But the uninitiated may wonder how it is possible to deal with a number which cannot be counted. It is impossible to count up all the numbers, one by one, because, however many we may count, there are always more to follow. The fact is that counting is a very vulgar and elementary way of finding out how many terms there are in a collection. And in any case, counting gives us what mathematicians call the Qrdinal number of our terms; that is to say, it arranges our terms in an order or series, and its result tells us what type of series results from this arrangement. In other words, it is impossible to count things without counting some first and others afterwards, so that counting always has to do with order. Now when there are only a finite number of terms, we can count them in any order we like; but when there are an infinite number, what corresponds to counting will give us quite different results according to the way in which we carry out the operation. Thus the ordinal number, which results from what, in a general sense may be called counting, depends not only upon how many terms we have, but also (where the number of terms is infinite) upon the way in which the terms are arranged. The fundamental infinite numbers are not ordinal, but are what is called cardinal. They are not obtained by putting our terms in order and counting them, but by a different method, which tells us, to begin with, whether two collections have the same nwnber of terms, or, if not, which is the greater.' It does not tell us, in the way in which counting does, what number of terms a collection has; but if we define a number as the nwnber of terms in such and such a collection, then this method enables us to discover whether some other collection that may be mentioned has more or fewer terms. An illustration will show how this is done. If there existed some country in which, for one reason or another, it was impossible to take a census, but in which it was known that every man had a wife and every woman a husband, then (provided polygamy was not a national institution) we should know. without counting, that there were exactly as many men as there were women in that country, neither more nor less. This method can be applied generally. If there is some relation which, like marriage, connects the things in one collection each with one of the th4tgs in another collection, and vice versa, then the two collections have the same number of terms. This was the way in which we found that there are as many even numbers as there are numbers. Every number can be doubled, and every even number can be halved, and each process gives just one number corresponding to the one that is doubled or halved. And 4- [Note added in 1917.] Although some infinite numbers are greater than some others, it cannot be proved that of any two infinite numbers one must be the greater.
Mathematics and the Metaphysicians
1585
in this way we can find any number of collections each of which has just as many terms as there are finite numbers. If every term of a collection can be hooked on to a number, and all the finite numbers are used once, and only once, in the process, then our collection must have just as many terms as there are finite numbers. This is the general method by which the numbers of infinite collections are defined. But it must not be supposed that all infinite numbers are equal. On the contrary, there are infinitely more infinite numbers than finite ones. There are more ways of arranging the finite numbers in different types of series than there are finite numbers. There are probably more points in space and more moments in time than there are finite numbers. There are exactly as many fractions as whole numbers, although there are an infinite number of fractions between any two whole numbers. But there are more irrational numbers than there are whole numbers or fractions. There are probably exactly as many points in space as there are irrational numbers, and exactly as many points on a line a millionth of an inch long as in the whole of infinite space. There is a greatest of all infinite numbers, which is the number of things altogether, of every sort and kind. It is obvious that there cannot be a greater number than this, because, if everything has been taken, there is nothing left to add. Cantor has a proof that there is no greatest number, and if this proof were valid, the contradictions of infinity would reappear in a sublimated form. But in this one point, the master has been guilty of a very subtle fallacy, which I hope to explain in some future work. I) We can now understand why Zeno believed that Achilles cannot overtake the tortoise and why as a matter of fact he can overtake it. We shall see that aU the people who disagreed with Zeno had no right to do so, because they all accepted premises from which his conclusion followed. The argument is this: Let Achilles and the tortoise start along a road at the same time, the tortoise (as is only fair) being allowed a handicap. Let Achilles go twice as fast as the tortoise, or ten times or a hundred times as fast. Then he will never reach the tortoise. For at every moment the tortoise is somewhere and Achilles is somewhere; and neither is ever twice in the same place while the race is going on. Thus the tortoise goes to just as many places as Achilles does, because each is in one place at one moment, and in another at any other moment. But if Achilles were to catch up with the tortoise, the places where the tortoise would have been would be only part of the places where Achilles would have been. Here, we must suppose, Zeno appealed to the maxim that the whole has !S Cantor was not guilty of a fallacy on this point. His proof that there is no greatest number is valid. The solution of the puzzle is complicated and depends upon the theory of types, which is explained in PrincipIa Mathematlca, Vol. I (Camb. Univ. Press, 1910). [Note added in 1917.J
1586
Bertrand Russell
more terms than the part. 6 Thus if Achilles were to overtake the tortoise, he would have been in more places than the tortoise; but we saw that he must, in any period, be in exactly as many places as the tortoise. Hence we infer that he can never catch the tortoise. This argument is strictly correct, if we allow the axiom that the whole has more terms than the part. As the conclusion is absurd, the axiom must be rejected, and then all goes well. But there is no good word to be said for the philosophers of the past two thousand years and more, who have all allowed the axiom and denied the conclusion. The retention of this axiom leads to absolute contradictions, while its rejection leads only to oddities. Some of these oddities, it must be confessed, are very odd. One of them, which I call the paradox of Tristram Shandy, is the converse of the Achilles, and shows that the tortoise, if you give him time, will go just as far as Achilles. Tristram Shandy, as we know, employed two years in chronicling the first two days of his life, and lamented that, at this rate, material would accumulate faster than he could deal with it, so that, as years went by, he would be farther and farther from the 68d of his history. Now I maintain that, if he had lived for ever, and had not wearied of his task, then, even if his life had continued as eventfully as it began, no part of his biography would have remained unwritten. For consider: the hundredth day will be described in the hundredth year, the thousandth in the thousandth year, and so on. Whatever day we may choose as so far on that he cannot hope to reach it, that day will be described in the corresponding year. Thus any day that may be mentioned will be written up sooner or later, and therefore no part of the biography will remain permanently unwritten. This paradoxical but perfectly true proposition depends upon the fact that the number of days in all time is no greater than the number of years. Thus on the subject of infinity it is impossible to avoid conclusions which at first sight appear paradoxical, and this is the reason why so many philosophers have supposed that there were inherent contradictions in the infinite. But a little practice enables one to grasp the true principles of Cantor's doctrine, and to acquire new and better instincts as to the true and the false. The oddities then become no odder than the people at the antipodes, who used to be thought impossible because they would find it so inconvenient to stand on their heads. The solution of the problems concerning infinity has enabled Cantor to solve also the problems of continuity. Of this, as of infinity, he has given a perfectly precise definition, and has shown that there are no contradic6 ~ ~ust not .be regarded as a historicaI1:y correct account of what Zeno actually had m nund. It IS a new argument for his conclusion not the argument which influenced him. On this point, see e.g., C. D. Broad ':Note on Achilles and the TO!1oise," Mind, N.S., Vol. XXII, pp. 318-19. Much ;aluable work on the interpretatIOn of Zeno has been done since this article was written. [Note added in 1917.]
Mathematics and the MetaphyslCla.ns
1587
tions in the notion so defined. But this subject is so technical that it is impossible to give any account of it here. The notion of continuity depends upon that of order, since continuity is merely a particular type of order. Mathematics has, in modern times, brought order into greater and greater prominence. In former days, it was supposed (and philosophers are still apt to suppose) that quantity was the fundamental notion of mathematics. But nowadays, quantity is banished altogether, except from one little corner of Geometry, while order more and more reigns supreme. The investigation of different kinds of series and their relations is now a very large part of mathematics, and it has been found that this investigation can be conducted without any reference to quantity, and, for the most part, without any reference to number. All types of series are capable of formal definition, and their properties can be deduced from the principles of symbolic logic by means of the Algebra of Relatives. The notion of a limit, which is fundamental in the greater part of higher mathematics, used to be defined by means of quantity, as a term to which the terms of some series approximate as nearly as we please. But nowadays the limit is defined quite differently, and the series which it limits may not approximate to it at all. This im· provement also is due to Cantor, and it is one which has revolutionised mathematics. Only order is now relevant to limits. Thus, for instance, the smallest of the infinite integers is the limit of the finite integers, though all finite integers are at an infinite distance from it. The study of different types of series is a general subject of which the study of ordinal numbers (mentioned above) is a special and very interesting branch. But the unavoidable technicalities of this subject render it impossible to explain to any but professed mathematicians. Geometry, like Arithmetic, has been subsumed, in recent times, under the general study of order. It was formerly supposed that Geometry was the study of the nature of the space in which we live, and accordingly it was urged, by those who held that what exists can only be known empirically, that Geometry should really be regarded as belonging to applied mathematics. But it has gradually appeared, by the increase of nonEuclidean systems, that Geometry throws no more light upon the nature of space than Arithmetic throws upon the population of the United States. Geometry is a whole collection of deductive sciences based on a corresponding collection of sets of axioms. One set of axioms is Euclid's; other equally good sets of axioms lead to other results. Whether Euclid's axioms are true, is a question as to which the pure mathematician is indifferent; and, what is more, it is a question which it is theoretically impossible to answer with certainty in the affirmative. It might possibly be shown, by very careful measurements, that Euclid's axioms are false; but no measurements could ever assure us (owing to the errors of observation) that
1588
Bertrand Russdl
they are exactly true. Thus the geometer leaves to the man of science to decide, as best he may, what axioms are most nearly true in the actual world. The geometer takes any set of axioms that seem interesting, and deduces their consequences. What defines Geometry, in this sense, is that the axioms must give rise to a series of more than one dimension. And it is thus that Geometry becomes a department in the study of order. In Geometry, as in other parts of mathematics, Peano and his disciples have done work of the very greatest merit as regards principles. Formerly, it was held by philosophers and mathematicians alike that the proofs in Geometry depended on the figure; nowadays, this is known to be false. In the best books there are no figures at all. The reasoning proceeds by the strict rules of formal logic from a set of axioms laid down to begin with. If a figure is used, all sorts of things seem obviously to follow, which no formal reasoning can prove from the explicit axioms, and which, as a matter of fact, are only accepted because they are obvious. By banishing the figure, it becomes possible to discover all the axioms that are needed; and in this way all sorts of possibilities, which would have otherwise remained undetected, are brought to light. One great advance, from the point of view of correctness, has been made by introducing points as they are required, and not starting, as was formerly done, by assuming the whole of space. This method is due partly to Peano, partly to another Italian named Fano. To those unaccustomed to it, it has an air of somewhat wilful pedantry. In this way, we begin with the following axioms: (1) There is a class of entities called points. (2) There is at least one point. (3) If a be a point, there is at least one other point besides a. Then we bring in the straight line joining two points, and begin again with (4), namely, on the straight line joining a and b. there is at least one other point besides a and b. (5) There is at least one point not on the line abo And so we go on, till we have the means of obtaining as many points as we require. But the word space, as Peano humorously remarks, is one for which Geometry has no use at all The rigid methods employed by modern geometers have deposed Euclid from his pinnacle of correctness. It was thought, until recent times, that, as Sir Henry Savile remarked in 1621, there were only two blemishes in Euclid, the theory of parallels and the theory of proportion. It is now known that these are almost the only pOints in which Euclid is free from blemish. Countless errors are involved in his first eight propositions. That is to say, not only is it doubtful whether his axioms are true, which is a comparatively trivial matter, but it is certain that his propositions do not follow from the axioms which he enunciates. A vastly greater number of axioms, which Euclid unconsciously employs, are required for the proof of his propositions. Even in the first proposition of all, where he constructs an equilateral triangle on a given base, he uses two circles
Mathematics and the MetaphYSICIans
1589
which are assumed to intersect. But no explicit axiom assures us that they do so, and in some kinds of spaces they do not always intersect. It is quite doubtful whether our space belongs to one of these kinds or not. Thus Euclid fails entirely to prove his point in the very first proposition. As he is certainly not an easy author, and is terribly longwinded, he has no longer any but an historical interest. Under these circumstances, it is nothing less than a scandal that he should still be taught to boys in England. 1 A book should have either intelligibility or correctness; to combine the two is impossible, but to lack both is to be unworthy of such a place as Euclid has occupied in education. The most remarkable result of modern methods in mathematics is the importance of symbolic logic and of rigid formalism. Mathematicians, under the influence of Weierstrass, have shown in modern times a care for accuracy, and an aversion to shipshod reasoning, such as had not been known among them previously since the time of the Greeks. The great inventions of the seventeenth century-Analytical Geometry and the Infinitesimal Calculus-were so fruitful in new results that mathematicians had neither time nor inclination to examine their foundations. Philosophers, who should have taken up the task, had too little mathematical ability to invent the new branches of mathematics which have now been found necessary for any adequate discussion. Thus mathematicians were only awakened from their "dogmatic slumbers" when Weierstrass and his followers showed that many of their most cherished propositions are in general false. Macaulay, contrasting the certainty of mathematics with the uncertainty of philosophy, asks who ever heard of a reaction against Taylor's theorem? If he had lived now, he himself might have heard of such a reaction, for this is precisely one of the theorems which modem investigations have overthrown. Such rude shocks to mathematical faith have produced that love of formalism which appears, to those who are ignorant of its motive, to be mere outrageous pedantry. The proof that all pure mathematics, including Geometry, is nothing but formal logic, is a fatal blow to the Kantian philosophy. Kant, rightly perceiving that Euclid's propositions could not be deduced from Euclid's axioms without the help of the figures. invented a theory of knowledge to account for this fact; and it accounted so successfully that, when the fact is shown to be a mere defect in Euclid, and not a result of the nature of geometrical reasoning, Kant's theory also has to be abandoned. The whole doctrine of a priori intuitions, by which Kant explained the possibility of pure mathematics, is wholly inapplicable to mathematics in its present form. The Aristotelian doctrines of the schoolmen come nearer in spirit to 7 Since the above was written, he has ceased to be used. as a textbook. But I fear many of the books now used are so bad that the change is no great improvement. [Note added in 1917.]
1590
Bertrand Russell
the doctrines which modern mathematics inspire; but the schoolmen were hampered by the fact that their formal logic was very defective, and that the philosophical logic based upon the syllogism showed a corresponding narrowness. What is now required is to give the greatest possible development to mathematical logic, to allow to the full the importance of relations, and then to found upon this secure basis a new philosophical logic, which may hope to borrow some of the exactitude and certainty of its mathematical foundation. If this can be successfully accomplished, there is every reason to hope that the near future will be as great an epoch in pure philosophy as the immediate past has been in the principles of mathematics. Great triumphs inspire great hopes; and pure thought may achieve, within our generation, such results as will place our time, in this respect, on a level with the greatest age of Greece. 8 8 The greatest age of Greece was brought to an end by the Peloponnesian War. [Note added in 1917.]
COMMENTARY ON
HANS HAHN INCE ancient times philosophers, theologians and mathematicians have occupied themselves with the subject of infinity. Zeno of Elea invented a group of famous paradoxes whose difficulties are connected with the concept; in their time such leading thinkers as Aristotle, Descartes, Leibniz and Gauss grappled with the infinity problem without making any notable contributions to its clarification. The subject is admittedly complex and undeniably important. A firm grasp of the problems of infinity is essential to an understanding of the revolution in ideas that paved the way for the triumphant advance of modem mathematics, with important consequences to physics, cosmology and related sciences. The selection following is a slightly condensed version of a lecture on infinity by a noted Austrian mathematician, Hans Hahn, delivered some years ago before a general audience in Vienna. This is the first translation of the lecture into English. Hahn was born in Vienna in 1879 and after receiving his doctorate from the University of Vienna taught there and also at the universities of Innsbruck (where he took over Otto Stolz's post) and Czernowitz. During this period (1902-1915) he published a number of papers on the calculus of variations, function theory and the theory of sets, and worked in seminars with leading scientists, among them Ludwig Boltzmann, David Hilbert, Felix Klein and Hermann Minkowski. In 1915 Hahn entered the Austrian Army and soon thereafter was severely wounded in action and decorated for bravery. For a time he was professor at Bonn but in the spring of 1921 he returned to the University of Vienna to accept a chair in mathematics, which he held until he died of cancer in 1934 at the age of fifty-five. Hahn was a gifted and prolific investigator and a brilliant teacher. His main interest, pursued with great energy especially during the second part of his career, was in philosophy and the foundations of mathematics. In philosophical outlook he stood at many points close to Bertrand Russell and Ludwig Wittgenstein. He believed that experience and observation provide the only sound basis for knowledge about the physical universe while rational thought itself consists of nothing but "tautological transformations." Mathematics and logic are transformations of this kind and thus differ from a science such as physics whose propositions are rooted in the circumstances of the outside world. Mathematical propositions, he said, are like the propositions of logic in that neither is concerned with the things we wish to discuss
S
15!n
1592
Editor's Comment
but rather with the way we wish to discuss them. 1 One recognizes in these opinions Hahn's inclination to the philosophy of logical positivism and he was in fact a leading member of the celebrated Vienna Circle, a group of positivistic philosophers and scientists whose founders included Moritz Schlick) Karl Menger, Kurt Goedel, Otto Neurath and Rudolf Camap. The Circle annually presented popular lectures on science, and this survey by Hahn of the concept of infinity is one of the best of the series. Hahn began his lecture with a historical resume (here omitted) and then launched his discussion with a description of the work of the founder of the modern mathematical theory of infinity, Georg Cantor. 1 Hans Hahn, "Die Bedeutung der Wissenschattlichen Weltauffassung, Insbesondere Jilr Mathematik und Physik," Erkenntnis 1, 1930, p. 97. Hahn's mathematical work and philosophical opinions are described in an obituary by K. Mayrhofer, Monatshefte fiir Mathematik und Physik, vol. 41 (1934), pp. 221-238.
In two words:
IM-possmLE.-5AMUEL GOLDWYN
(Quoted in Alva Johnson's The Great Goldwyn)
I have found you an argument: but I am not obliged to find you an under-SAMUEL JOHNSON standing.
The infinzte! No other question has ever moved so profoundly the spirit of man. -DAVID Hn.BERT (1921)
2
Infinity By HANS HAHN
IT was Georg Cantor, who in the years 1871-84 created a completely new and very special mathematical discipline, the theory of sets, in which was founded, for the first time in a thousand years of argument back and forth, a theory of infinity with all the incisiveness of modern mathematics. Like so many other new creations this one began with a very simple idea. Cantor asked himself: "What do we mean when we say of two finite sets that they consist of equally many things, that they have the same number, that they are equivalent?" Obviously nothing more than this, that between the members of the first set and those of the second a correspondence can be effected by which each member of the first set matches exactly a member of the second set, and likewise each member of the second set matches one of the first. A correspondence of this kind is called "reciprocally unique," or simply "one-to-one." The set of the fingers of the right hand is equivalent to the set of fingers of the left hand, since between the fingers of the right hand and those of the left hand a one-toone pairing is possible. Such a correspondence is obtained, for instance, when we place the thumb on the thumb, the index finger on the index finger, and so on. But the set of both ears and the set of the fingers of one hand are not equivalent, since in this instance a one-to-one correspondence is obviously impossible; for if we attempt to place the fingers of one hand in correspondence with our ears, no matter how we contrive there will necessarily be some fingers left over to which no ears correspond. Now the number (or cardinal number) of a set is obviously a characteristic that it has in common with all equivalent sets, and by which it distinguishes itself from every set not equivalent to itself. The number 5, for instance, is the characteristic which all sets equivalent to the set of the fingers of one hand have in common, and which distinguishes them from all other sets. 1593
1594
Bans Hahn
Thus we have the following definitions. Two sets are called equivalent if between their respective members a one-to-one correspondence is possible; and the characteristic that one set has in common with all equivalent sets, and by which it distinguishes itself from all other sets not equivalent to itself is called the (cardinal) number of that set. And now we make the fundamental assertion that in these definitions the finiteness of the sets considered is in no sense involved; the definitions can be applied as readily to infinite sets as to finite sets. The concepts "equivalent" and "cardinal number" are thereby transferred to sets of infinitely many objects. The cardinal numbers of finite sets, i.e., the numbers 1, 2, 3, . . . , are called natural numbers; the cardinal numbers of infinite sets Cantor calls "transfinite cardinal numbers" or "transfinite powers." But are there really any infinite sets? We can convince ourselves of this at once by a very simple example. There are obviously infinitely many different natural numbers; hence the set of all the natural numbers contains infinitely many members; it is an infinite set. Now then, those sets that are equivalent to the set of all natural numbers, whose members can be paired in one-to-one correspondence with the natural numbers, are called denumerably infinite sets. The meaning of this designation can be explained as follows. A set with the cardinal number "five" is a set whose members can be put in one-to-one correspondence with the first five natural numbers, that is, a set that can be numbered by the integers 1, 2, 3, 4, 5, or counted off with the aid of the first five natural numbers. A denumerably infinite set is a set whose members can be put in one-toone correspondence with the totality of natural numbers. According to our definitions all denumerably infinite sets have the same cardinal number; this cardinal number must now be given a name, just as the cardinal number of the set of the fingers on one hand was earlier given the name 5. Cantor gave this cardinal number the name "aleph-null," written ~o. (Why he gave it this rather bizarre name will become clear later.) The number No is thus the first example of a transfinite cardinal number. Just as the statement "a set has the cardinal number 5" means that its members can be put in one-to-one correspondence with the fingers of the right hand, or-what amounts to the same thing-with the integers 1, 2, 3, 4, 5, so the statement "a set has the cardinal number ~o" means that its elements can be put in one-to-one correspondence with the totality of natural numbers. If we look about us for examples of denumerably infinite sets we arrive immediately at some highly surprising results. The set of all natural numbers is itself denumerably infinite; this is self-evident, for it was from this set that we defined the concept "denumerably infinite." But the set of all even numbers is also denumerably infinite, and has the same cardinal
1595
Infinity
number ~o as the set of all natural numbers, though we would be inclined to think that there are far fewer even numbers than natural numbers. To prove this proposition we have only to look at the correspondence diagrammed in Figure 1, that is, to put each natural number opposite its double. It may clearly be seen that there is a one-to-one correspondence between all natural and all even numbers, and thereby our point is established. In exactly the same way it can be shown that the set of all odd numbers is denumerably infinite. Even more surprising is the fact that the
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Hans Hahn
1596
set of all pairs of natural numbers is denumerably infinite. In order to understand this we have merely to arrange the set of all pairs of natural numbers "diagonally" as indicated in Figure 2, whereupon we at once obtain the one-to-one correspondence shown in Figure 3 between all natural numbers and all pairs of natural numbers. From this follows the conclusion, which Cantor discovered while still a student, that the set of all rational fractions (i.e., the quotients of two whole numbers, like :!h, %, etc.) is also denumerably infinite, or equivalent to the set of all natural numbers, though again one might suppose that there are many, many more fractions than there are natural numbers. What is more, Cantor was able to prove that the set of all so-called algebraic numbers, that is, the set of all numbers that satisfy an algebraic equation of the form lzoX1I> + a1xtz'-1
+ ... + an._lx + a1l! :: 0
with integral coefficients ao, aI' . . . a1l!' is denumerably infinite. At this point, the reader may ask whether, in the last analysis, all infinite sets are not denumerably infinite-that is, equivalent? If this were so, we should be sadly disappointed; for then, alongside the finite sets there would simply be infinite ones which would all be equivalent, and there would be nothing more to say about the matter. But in the year 1874 Cantor succeeded in proving that there are also infinite sets that are not denumerable; that is to say. there are other infinite numbers, transfinite cardinal numbers differing from aleph-null. Specifically, Cantor proved that the set of all so-called real numbers (i.e., composed of all whole numbers, plus all fractions, plus all irrational numbers) is nondenumerably infinite. The proof is so simple that I can give his reasoning here. It is sufficient to show that the set of all real numbers between o and 1 is not denumerably infinite, for then the set of all real numbers obviously cannot be denumerably infinite. In the proof we make use of the familiar fact that every real number between 0 and 1 can be expressed as an infinite decimal fraction. The statement that is to be proved, that the set of all real numbers between 0 and 1 is not denumerably infinite, can also be phrased as follows: "a denumerably infinite set of real numbers between 0 and 1," or thus: "Given a denumerably infinite set of real numbers between 0 and 1, there will always be a real number between o and 1 that does not belong to the given set." To prove this let us imagine a denumerably infinite set of real numbers between 0 and 1 as given; then, since this set is denumerably infinite, the real numbers that occur in it can be put in one-to-one correspondence with the natural numbers. Let us write down as a continuing decimal fraction the real number that in this arrangement corresponds to the natural number 1, and under it ,the real number that corresponds to the natural number 2, under it again
1597
1nMit}'
the real number that corresponds to the natural number 3, and so on. We would get something like this: 0.20745 0.16238
0.97126
Now one can, in fact, at once write down a real number between 0 and 1 which does not occur in the given denumerably infinite set of real numbers between 0 and 1. Take as its first digit one differing from the first digit of the decimal in the first row, say 3; as its second digit one differing from the second digit of the decimal in the second row, say 2; as its third digit one differing from the third digit of the decimal in the third row, say 5; and so on. It is clear that by this procedure we obtain a real number between 0 and 1 which differs from all the given infinitely many real numbers of our set, and this is precisely what we sought to prove was possible.! It has thus been shown that the set of natural numbers and the set of real numbers are not equivalent; that these two sets have different cardinal numbers. The cardinal number of the set of real numbers Cantor called the "power of the continuum"; we shall designate it by c. Earlier it was noted that the set of all algebraic numbers is denumerably infinite, and we just now saw that the set of all real numbers is not denumerably infinite, hence there must be real numbers that axe not algebraic. These are the so-called transcendental numbers, whose existence is demonstrated in the simplest way conceivable by Cantor's brilliant train of reasoning. It is well known that the real numbers can be put in one-to-one correspondence with the points of a straight line; hence c is also the cardinal number of the set of all points of a straight line. Surprisingly Cantor was also able to prove that a one-to-one pairing is possible between the set of 1 A few words further to clarify this point may be permitted. The essence of Cantor's proof is that no comprehe1l9ive counting procedure can be devised for the entire set of real numbers, nor even for one of its proper subsets, such as all the real numbers lying between 0 and 1. By various ingenious methods certain infinite sets such as all rational fractions or all algebraic numbers can be paired off with the natural numbers; every attempt, however, to construct a formula for counting the all-inclusive set of real numbers is invariably frustrated. No matter what counting scheme is adopted it can be shown that some of the real numbers in the set so considered remain uncounted, which is to say that the scheme fails. It follows that an infinite set for which no counting method can be devised is noncountable, in other words nondenumerably infinite. ED.
1598
Hans Hahn
all points of a plane and the set of all points of a ,straight line. These two sets are thus equivalent, that is to say, C IS also the cardinal number of the set of all points of a plane, though here too we should have thought that a plane would contain a great many more points than a straight line. In fact, as Cantor has shown, c is the cardinal number of all points of three-dimensional space, or even of a space of any number of dimensions. We have discovered two different transfinite cardinal numbers, ~o and c, the power of the denumerably infinite sets and the power of the continuum. Are there yet others? Yes, there certainly are infinitely many different transfinite cardinal numbers; for given any set M, a set with a higher cardinal number can at once be indicated, since the set of all possible subsets of M has a higher cardinal number than the set M itself. Take, for example, a set of three elements, such as the set of the three figures 1, 2, 3. Its partial sets are the following: 1; 2; 3; 1, 2; 2, 3; 1, 3; thus the number of the partial sets is more than three. Cantor has shown that this is generally true,2 even for infinite sets. For example, the set of all possible point-sets of a straight line has a higher cardinal number than the set of all points of the straight line, that is to say, its cardinal number is greater than c. What is now desired is a general view of all possible transfinite cardinal numbers. As regards the cardinal numbers of finite sets, the natural numbers, the following simple situation prevails: Among such sets there is one that is the smallest, namely 1; and if a finite set M with the cardinal number m is given, a set with the next-larger cardinal number can be formed by adding one more object to the set M. What is the rule in this respect with regard to infinite sets? It can be shown without difficulty that among the transfinite cardinal numbers, as well as among the finite ones, there is one that is the smallest, namely ~o, the power of denumerably infinite sets (though we must not think that this is self-evident, for among all positive fractions, for instance, there is none that is the smallest). It is, however, not so easy as it was in the case of finite sets to form the nextlarger to a transfinite cardinal number; for whenever we add one more member to an infinite set we do not get a set of greater cardinality, only one of equal cardinality. But Cantor also solved this difficulty, by showing that there is a next-larger to every transfinite cardinal number (which again is by no means self-evident: there is no next-larger to a fraction, for instance) and by showing how it is obtained. We cannot go into his proof here, since this would take us too far into the realm of pure mathematics. It is enough for us to recognize the fact that there is a smallest transfinite cardinal number, namely No; after this there is a next-larger, 2 For sets, M, c~nsisting of one or two members, the rule holds only if one adds to the proper partial sets the empty set (null-set) possessing no members and the set M itself.
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which is called t-h; after this there is again a next~larger, which is N2; and so on. But this still does not exhaust the class of transfinite cardinals; for if it be assumed that we have formed the cardinal numbers No. N I, N2• . . . Nlo•... NlOO, •.. NIooo, . . . that is. all alephs (Nn) whose index n is a natural number, then there is again a first transfinite cardinal number larger than any of these-Cantor called it Nw-and a next-larger successor NW+I' and so on and on. The successive alephs formed in this manner represent all possible transfinite cardinal numbers, and hence the power c of the continuum must occur among them. The question is which aleph is the power of the continuum? This is the famous problem of the continuum. We already know that it cannot be No, since the set of all real numbers is nondenumerably infinite, that is to say, not equivalent to the set of natural numbers. Cantor took N1 to be the power of the continuum. This question, however, remains open, and for the present we see no trace of a path to its solution. I must here call attention to a detail of logic that was first noticed by Ernst Zermelo some time after the formulation of the theory of sets by Cantor. In the proof that No is the smallest transfinite cardinal number, as well as in the proof that every transfinite cardinal number occurs in the aleph series, one makes an assumption that is also used in other mathematical proofs without its being explicitly recognized. This assumption. which, as it seems, cannot be derived from the other principles of logic, is known as the postulate, or the axiom of choice, and may be stated as follows: Given a set of M sets, no two of which have a common member, then there exists a set that has exactly one member in common with each of the sets M. The postulate raises no difficulty when dealing with a finite collection of sets, for one can select a member from each of these sets in a finite number of operations. When one member has been selected from each of the given sets, our task is done for we then have formed a new set that has exactly one member in common with each of the given sets. Neither is there any difficulty if infinitely many sets are given and at the same time a rule is provided that distinguishes one member of each of these sets. The set composed of the objects denoted by the rule is the one we seek, since it also has exactly one member in common with each of the given sets. But if infinitely many sets are given and no rule is formulated that distinguishes one member in each of them, we cannot proceed as we did in the case of a finite number of sets. For if we started to select arbitrarily one member from each of the given sets, we could of course never complete the task and thus never could obtain a set having one member in common with each of the given sets. s Hence the assertion 3 The defining rule, in other words, makes it possible to form an infinite set in a fimte operation; in the absence of such a rule the selection requires an infinite number of steps and is therefore impossible. ED.
1600
Hans Hahn
that such a set exists in every case represents a special logical postulate. The famous English logician and philosopher, Bertrand Russell, has made this clear in an ingenious and amusing illustration. In civilized countries shoes are so designed that the right and left shoe of each pair can readily be distinguished; but the distin~tion between right and left cannot be made for pairs of stockings. Let us imagine an infinitely rich man (a millionaire or a billionaire will not do, he must be an "infinitillionaire") who is infinitely eccentric and owns a collection of infinitely many pairs of shoes and infinitely many pairs of stockings. This fortunate man, it may be observed, has at his disposal a set' of shoes that contains exactly one shoe from each pair; for instance, the infinite set composed of the right shoes, one from each pair. But how is he to obtain a set that contains exactly one stocking from each pair of stockings? This is, of course, only a facetious illustration, yet the principle itself represents a serious and important logical discovery, which bears on many problems. For the moment, however, let us leave it; we shall return to it. On the basis of this rather sketchy description of the structure of the theory of sets the answer to the question "Is there an infinity?" appears to be an unqualified "Yes." There are not only, as Leibniz had already asserted, infinite sets, but there are even what Leibniz had denied, infinite numbers, and it can also be shown that it is quite possible to operate with them, in a manner similar to, if not identical with that used for finite natural numbers. But now we must look with a critical eye at what has been accomplished. When existence is asserted of some entity in mathematics by the words "there is" or "there are," and non-existence by the words "there is are not," evidently some thing entirely different is meant than when these expressions are used in everyday life, Or in geography, say, or natural history. Let us make this clear by a few examples. Since about the time of Plato it has been known that there are regular bodies with four sides (tetrahedrons), with six sides (hexahedrons or cubes), with eight sides (octahedrons), with twelve sides (dodecahedrons), and with twenty sides (icosahedrons). There are no regular bodies other than these five. Now it is obvious, that the expressions "there are" and "there are no" have a different meaning as used in the two preceding sentences about mathematics than in a geographical statement, such as: there are mountains over 25,000 feet high, but there are no mountains 35,000 feet high. For even though mathematics teaches that there are cubes and icosahedrons, yet in the sense that there are mountains over 25,000 feet high, that is, in the sense of physical existence, there are no cubes and no icosahedrons. The most beautiful rock-salt crystal is not an exact mathematical cube, and a model of an icosahedron, however well constructed, is not an icosahedron in the mathematical sense. While it is fairly clear
Infinity
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what is meant by the expressions "there is" or "there are" as used in the sciences dealing with the physical world, it is not at all clear what mathematics means by such existence statements. On this point indeed there is no agreement whatever among scholars, whether they be mathematicians or philosophers. So many different interpretations are represented among them that one might almost say: "quot capita, tot sententiae" ("as many meanings as individuals"). However, if we stick to essentials we can perhaps distinguish three basically different points of view on this subject. These I shall describe briefly. The first can be designated the realistic or the Platonic position. It ascribes to the objects of mathematics a real existence in the world of ideas; the physical world we may note is merely an imperfect image of the world of ideas. Thus, there are no perfect cubes in the physical world, only in the world of ideas; through our senses we can comprehend only the physical world, but it is in thought that we comprehend the world of ideas. A mathematical concept derives its existence from the real object corresponding to it in the world of ideas; a mathematical statement is true if it correctly represents the real relationship of the corresponding objects in the world of ideas. The second view which we can call the intuitionistic or the Kantian position, is to the effect that we possess pure intuition; mathematics is construction by pure intuition, and a mathematical concept 4'exists" if it is constructable by pure intuition. The philosophical formulation of this idea might be popularly expressed somewhat as follows: "If there is indeed no perfect cube in the physical world I can at least imagine a perfect cube." The third view is best called the logistic position. If one sought to trace its historical roots one could perhaps connect it with the nominalist school of scholastic philosophy; thus it may be called the nominalist position. According to this view mathematics is a purely logical discipline and, like logic, is carried on entirely within the confines of language; it has nothing whatever to do with reality, or with pure intuition; on the contrary it deals exclusively with the use of signs or symbols. These signs or symbols can be used as we like, in conformity with rules that we have set. The only restriction on our freedom is that we may under no circumstances contradict the self-established rules. The final criterion of mathematical existence thus becomes freedom from contradiction; that is; mathematical existence can be ascribed to every concept whose use does not enmesh us in contradiction. Let us now examine a little more critically these three points of view. The realistic position is, in my opinion, untenable since it ascribes to the mind abilities that it does not possess; our thinking consists of tautological transformation~ it is incapable of comprehending a reality. Plato assumed a mystical recollection (anamnesis) by the soul of a state in which it beheld the ideas face to face, as it were. In any event, this first position
1602
Hans Hnhn
is entirely metaphysical and seems wholly unsuitable as the foundation of mathematics. Nonetheless, it is still the source, though often unsuspected, of a great deal of confusion in research on the foundations of mathematics. With regard to the second, the intuitionistic position, I have attempted on other occasions to explain that there is no such thing as pure intuition. To be sure, Kant assigned it a very broad role, but in light of the develop~ ment of mathematics since his day this view cannot possibly be main~ tained. Hence the recent supporters of the second position have, in fact, become more modest in their claims. But as to what this allegedly pure intuition can and cannot do, what is consistent with it and what is notabout these matters there is no agreement whatever among the supporters of the intuitionistic position. This is shown very clearly by their answers to the question that concerns us here, "Are there infinite sets and infinite numbers?" Some intuitionists would say that arbitrarily large numbers can perhaps be constructed by pure intuition, but not the set of all natural numbers. This group, in other words, would flatly deny the existence of infinite sets. Others of this school hold that While intuition suffices for the construction of the set of all natural numbers, nonMdenumerably infinite sets are beyond intuition's reach; which is to say they deny the existence of the set of all real numbers. Still others ascribe constructability, and thereby existence, to certain non-denumerable sets. The intuitionist doctrine is thus seen to rest on very uncertain ground; in glaring contrast to this uncertainty is the gruffness with which the supporters of this position declare meaningless everything that in their opinion is not constructable by pure intuition. Having rejected the two first positions we must then turn to the third, the logistic interpretation. But before discussing the question "Are there infinite sets and infinite numbers?" from the logistic point of view, it may be useful to point out the difference between the three positions with respect to the axiom of choice, mentioned earlier. A representative of the realistic stand would say: "Whether We are to accept or reject the axiom of choice in logic depends on how reality is constituted; if it is constituted as the axiom of choice asserts, then we must accept it, but if reality is not so constituted we shall have to reject the axiom. Unfortunately we do not know which is the case, and because of the inadequacy of our means of perception we shall-again unfortunately-never know." An intuitionist would perhaps say: "We must consider whether a set of the kind required by the axiom of choice (that is to say, a set having exactly one member in common with each set of a given system of sets) can be constructed by pure intuition. For this purpose one would have to select from each set of the given system one member; if the system consisted of infinitely many sets the task would involve infinitely many separate operations;
Infinity
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since these cannot possibly be carried out. the axiom of choice is to be rejected." A consistent representative of the logistic position would say: "If the axiom of choice is in truth independent of the other principles of logic, that is to say, if the statement of the axiom of choice as well as the contrary statement is consistent with the other principles of logic, then we are free to accept it or replace it with its contradictory. That is, we can as well operate with a mathematics in which the axiom of choice is taken as a basic principle-a 'Zermelo mathematics'-as with a mathematics in which a contrary axiom is taken as the basis-a 'non-Zermelo mathematics.' The entire question has nothing to do with the nature of reality, as the realists think, or with pure intuition, as the intuitionists think. The question is rather in what sense we decide to use the word 'set'; it is a matter of determining the syntax of the word 'set.' " Let us return to the problem we are mainly concerned with, and consider what answer an adherent of the logistic school would give to the question "Are there infinite sets and infinite numbers?" He would perhaps reply: "Yes, infinite sets and infinite numbers can be said to exist, provided it is possible to operate with them without contradiction." What is the situation, then, with regard to this freedom from contradiction? Various philosophers have, in fact, repeatedly raised the objection against Cantor's theory that it would lead to contradictions. The objection might be phrased as follows: "According to Cantor the set of all natural numbers is equivalent to the set of even numbers; this however contradicts the axiom that the whole can never be equal to one of its parts," To refute the objection we must examine the meaning of this alleged axiom. Certainly it cannot mean that reality itself is constituted as the axiom asserts; for that would be a reversion to the metaphysical realistic position. Its meaning must rather be in a syntatic determination of how we are to use the words "whole," "part," "equaL" We must establish, in other words, that this determination does not correspond completely with the usage of everyday speech or with the language of science. Yet note that even in ordinary linguistic usage it is meaningful to say that the whole may be equal to a part-as, for example, with respect to color; why, then, should it be forbidden to say that the whole is equal to one of its parts with respect to quantity? The truth is, however, this axiom is used neither in the construction of logic nor of mathematics. Hence no contradiction confronts us here, but merely the fact that as to certain aspects of behavior infinite sets differ from finite sets. A finite set cannot be equivalent to one of its parts, but an infinite set can be. It can be shown that every infinite set has parts to which it is equivalent, and one can indeed make use of this very fact-as Dedekind did-in defining the concept "infinite set." It has been suggested that many other contradictions lurk in the con-
1604
Hans Hahn
cepts of infinite sets and infinite numbers. These objections, as in the case above, usually consist of showing that certain properties that necessarily belong to finite sets and finite numbers do not belong to infinite sets and infinite numbers. For example, every number must be either even or odd: but Cantor's transfinite cardinal numbers are neither even nor odd; or, every number becomes larger when 1 is added to it: but this is not true of Cantor's transfinites. That transfinite numbers have properties differing from those of finite numbers affords no contradiction; it is not even to be wondered at; it must be so. For if transfinites were in every respect indistinguishable from finite numbers the need for a separate transfinite category would vanish; transfinite numbers could be classed simply as oversized finite numbers. It is the same when we discover a new species of animal-it must differ in some way from the known ones, otherwise it would not be a new species. After disposing of these arguments a single meaningfuI question remains: "Since they differ so greatly from ordinary numbers is it perhaps not inappropriate to designate transfinites as numbers?" Like many so-called philosophical problems, we are free to consider this one as turning on a simple issue of terminology; though it was for the very purpose of avoiding such purely terminological controversies that Cantor gave his transfinite cardinals the neutral and relatively non-committal name of "powers:' The controversy, however, intrigues us in much the same way as the once celebrated dispute over whether "one" is a number, or numbers only begin with "two." Let us content ourselves with the indisputable statement that the term "transfinite numbers" has shown itself wholly suited to its purpose. Cantor had no difficulty in dealing with objections of this sort; they could not endanger his edifice. But the refutation of a few inadequate proofs of supposed contradictions clearly does not demonstrate that none exist; and serious contradictions have, in fact, been discovered in Cantor's structure. Contradictions have appeared in certain set formations that are sweepingly inclusive, such as the set of all objects, the set of all sets, and the set of all infinite cardinal numbers. Note, however, that the concept of infinity was not the source of these contradictions; instead they arose from certain deficiencies of classical logic. Thus it became evident that what was needed was a reform in logic. This reform consisted mainly in a more careful use of the word "all," as taught in Russell's theory of logical types. 4 Thereupon the contradictions that had appeared in the theory of sets were successfully explained and eliminated. There is no longer any known contradiction in the present formulation of the theory of sets. But from the fact that no contradiction is known, it does not necessarily 4 See A. N. Whltehead and B. Russell, PrinCIpia Mathematica; Cambridge 1925, Vol. I, second edition, p. 37 et seq. '
Infinity
1605
follow that none exist, any more than the fact that in 1900 no okapi was known proved that none existed. We face the question, then, "Can any proof be given of freedom from contradiction?" On the basis of present knowledge it may be said that an absolute proof of freedom from contradiction is probably unattainable; every such proof is relative; we can do no more than to relate the freedom from contradiction of one system to that of another. But is not this concession fatal to the logistic position, according to which mathematical existence depends entirely on freedom from contradiction? I think not. For here, as in every sphere of thought, the demand for absolute certainty of knowledge is an exaggerated demand: in no field is such certainty attainable. Even the evidence adduced by many philosophers-the evidence of immediate inner perception exhibited in a statement such as "I now see something white" affords no example of certain knowledge. For even as I formulate and utter the statement "I see something white" it describes a past event and I can never know whether in the period of time that has elansed, however short, my memory has not deceived me. There is, then, no absolute proof of freedom from contradiction for the theory of sets and thus no absolute proof of the mathematical existence of infinite sets and infinite numbers. But neither is there any such proof for the arithmetic of finite numbers, nor for the simplest parts of logic. It is a fact, however, that no contradiction is known in the theory of sets, and not a trace of evidence can be found that such a contradiction may turn up. Hence we can ascribe mathematical existence to infinite sets and to Cantor's transfinite numbers with approximately the same certainty as we ascribe existence to finite numbers. So far we have dealt only with the question whether there are infinite sets and infinite numbers; but no less important, it would appear, is the question whether there are infinite extensions. This is usually phrased in the form: "Is space infinite?" Let us begin by treating this question also from a purely mathematical standpoint. We must recognize at the outset that mathematics deals with very diverse kinds of space. Here, however, we are interested only in the so-called Riemann spaces, and in particular, in the three-dimensional Riemann spaces. Their exact definition does not concern us; it is sufficient to make the point that such a Riemann space is a set of elements, or points, in which certain subsets, called lines, are the objects of attention. By a process of calculation there can be assigned to every such line a positive number, called the length of the line, and among these lines there are certain ones of which every sufficiently small segment AB is shorter than every other line joining the points A,B. These lines are called the geodesics, or the straight lines of the space in question. Now it may be that in any particular Riemann space there are s.traight lines of arbitrarily
1606
Hans Hahn
great length; in that case we shall say that this space is of infinite extension. On the other hand, it may also be that In this particular Riemann space the length of all straight lines remains less than a fixed number; then we say that the space is of finite extension. Until the end of the 18th century only a single mathematical space was known and hence it was simply called "space." This is the space whose geometry is taught in school and which we call Euclidean space, after the Greek mathematician Euclid who was the first to develop the geometry of thIS space systematically. And from our definition above, this Euclidean space is of infinite extension. There are, however, also three-dimensional Riemann spaces of finite extension; the best known of these are the so-called spherical spaces (and the closely related elliptical ones); which are three-dimensional analogues of a spherical surface. The surface of a sphere can be conceived as twodimensional Riemann space, whose geodesics, or "straight" lines, are arcs of great circles. (A great circle is a circle cut on the surface of a sphere by a plane passing through the center of the sphere, as for instance, the equator and the meridians of longitude on the earth.) If r is the radius of the sphere, then the full circumference of a great circle is 2'77'r; that is to say, no great circle can be longer than 2'77'r. Hence the sphere considered as two-dimensional Riemann space is a space of finite extension. With regard to three-dimensional spherical space the situation is fully analogous; this also is a space of finite extension. Nevertheless, it has no boundaries; it is unbounded as the surface of a sphere is unbounded; one can keep walking along one of its straight lines without ever being stopped by a boundary of the space. After a finite time one simply comes back to the starting point, exactly as if one had kept moving farther and farther along a great circle of a spherical surface. In other words. we can make a circular tour of spherical space, just as easily as we can make a circular tour of the earth. Thus we see that in a mathematical sense there are spaces of infinite extension (e.g., Euclidean space) and spaces of finite extension (e.g., spherical and elliptical spaces). Yet, this is not at all what most persons have in mind when they ask "Is space infinite?" They are asking, rather, "Is the space in which our experience and in which physical events take place of finite or of infinite extension?" So long as no mathematical space other than Euclidean space was known, everyone naturally believed that the space of the physical world was Euclidean space, infinitely extended. Kant, who explicitly formulated this view, held that the arrangement of our observations in Euclidean space was an intuitional necessity; the basic postulates of Euclidean geometry are synthetic, a priori judgments. But when it was discovered that in a purely mathematical sense spaces
Infinity
1607
other than Euclidean also "existed," (that is, led to no logical contradictions) it became possible to question the view that the space of the physical world must be Euclidean space. And the idea developed that it was a question of experience, that is, a question that must be decided by experiment, whether the space of the physical world was Euclidean or not. Gauss actually made such experiments. But after the work of Henri Poincare, the great mathematician of the end of the 19th century, we know that the question expressed in this way has no meaning. To a considerable extent we have a free choice of the kind of mathematical space in which we arrange our observations. The question does not acquire meaning until it is decided how this arrangement is to be carried out. For the important thing about Riemann space is the manner in which each of its lines is assigned a length, that is, how lengths are measured in it. If we decide that measurements of length in the space of physical events shall be made in the way they have been made from earliest times, that is, by the application of "rigid" measuring rods, then there is meaning in the question whether the space of physical events, considered as a Riemann space, is Euclidean or non-Euclidean. And the same holds for the question whether it is of finite or of infinite extension. The answer that many perhaps are prompted to give, "Of course, by this method of measurement physical space becomes a mathematical space of infinite extension," would be somewhat too hasty. As background for a brief discussion of this problem we must first give a short and very simpJe statement of certain mathematical facts. Euclidean space is characterized by the fact that the sum of the three angles of a triangle in such space is 180 degrees. In spherical space the sum of the angles of every triangle is greater than 180 degrees, and the excess over 180 degrees is greater the larger the triangle in relation to the sphere. In the twodimensional analogue of spherical space, the surface of a sphere, this point is presented to us very clearly. On the surface of a sphere, as already mentioned, the counterpart of the straight-line triangle of spherical space is a triangle whose sides are arcs of great circles, and it is a wellknown proposition of elementary geometry that the sum of the angles of a spherical triangle is greater than 180 degrees, and that the excess over 180 degrees is greater the larger the surface area of the triangle. If a further comparison be made of spherical triangles of equal area on spheres of different sizes, it may be seen at once that the excess of the sum of the angles over 180 degrees is greater the smaller the diameter of the sphere, which is to say, the greater the curvature of the sphere. This gave rise to the adoption of the following terminology (and here it is simply a matter of terminology, behind which nothing whatever secret is hidden): A mathematical space is called "curved" if there are triangles in it the sum of whose angles deviates from 180 degrees. It is "positively curved" if
1608
Hom Hahn
the sum of the angles of every triangle in it (as in elliptical and spherical spaces) is greater than 180 degrees, and "negatively curved" if the sum is less than 180 degrees-as is the case in the "hyperbolic" spaces discovered by Bolyai and Lobachevsky. From the mathematical formulations of Einstein's General Theory of Relativity it now follows that, if the previously mentioned method of measurement is used as a basis, space in the vicinity of gravitating masses must be curved in a "gravitational field." The only gravitational field immediately accessible, that of the earth, is much too weak for us to be able to test this assertion directly. It has been possible, however, to prove it indirectly by the deflection of light rays-as determined during total eclipses-in the much stronger gravitational field of the sun. So far as our present experience goes, we can say that if, by using the measuring methods mentioned above, we turn the space of physical events into a mathematical Riemann space, this mathematical space will be curved, and its curvature will, in fact, vary from place to place, being greater in the vicinity of gravitating masses and smaller far from them. To return to the question that concerns us: Can we now say whether this space will be of finite or infinite extension? What has been said so far is not sufficient to give the answer; it is still necessary to make certain rather plausible assumptions. One such assumption is that matter is more or less evenly distributed throughout the entire space of the universe: that is to say, in the universe as a whole there is a spatially uniform density of mass. The observations of astronomers to date can, at least with the help of a little good will, be brought into harmony with this assumption. Of course it can be true only when taken in the sense of a rough average, in somewhat the same sense as it can be said that a piece of ice has on the whole the same density throughout. Just as the mass of the ice is concentrated in a great many very small particles, separated by intervening spaces that are enormous in relation to the size of these particles, so the stars in world-space are separated by intervening spaces that are enormous in relation to the size of the stars. Let us make another quite plausible assumption, namely, that the universe, taken by and large, is stationary in the sense that this average constant density of mass remains unchanged. We consider a piece of ice stationary, even though we know that the particles that constitute it are in active motion; we may likewise deem the universe to be stationary, even though we know the stars to be in active motion. With these assumptions, then, it follows from the principles of the General Theory of Relativity that the mathematical space in which we are to interpret physical events must on the whole have the same curvature throughout. Such a space, however, like the surface of a sphere in two dimensions, is necessarily of finite extension. In other words, if we use as a basis the usual method of measuring length and wish to
Infinity
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arrange physical events in a mathematical space, and if we make the two plausible assumptions mentioned above, the conclusion follows that this space must be of finite extension. I said that the first of our assumptions, that of the equal density of mass throughout space, conforms somewhat with observations. Is this also true of the second assumption, as to the constant density of mass with respect to time? Until recently this opinion was tenable. Now, however, certain astronomical observations seem to indicate-again speaking in broad terms-that all heavenly bodies (fixed stars and nebulae) are moving away from us with a velocity that increases the greater their distance from us, the velocity of those that are the farthest away from us, but still within reach of study, being quite fantastic. But if this is so, the average density of mass of the universe cannot possibly be constant in time; instead it must continually become smaller. Then if the remaining features of our picture of the universe are maintained, it would mean that we must assume that the mathematical space in which we interpret physical events is variable in time. At every instant it would be a space with (on the average) a constant positive curvature, that is to say, of finite extension, but the curvature would be continually decreasing while the extension would be continually increasing. This interpretation of physical events occurring in an expanding space turns out to be entirely workable and in accord with the General Theory of Relativity. But is this the only theory consistent with our experience to date? I said before that the assumption that the space of the universe was on the whole of uniform density could fairly well be brought into harmony with astronomical observations. At the same time these observations do not contradict the entirely different assumption that we and our system of fixed stars are situated in a region of space where there is a strong concentration of mass, while at increasing distances from this region the distribution of mass keeps getting sparser. This would lead us-still using the ordinary method of measuring length, to conceive of the physical world as situated in a space that has a certain curvature in the neighborhood of our fixed star system, a curvature, however, that grows smaller and smaller further away. Such a space can of course be of infinite extension. Similarly the phenomenon that the stars are in general receding from us, with greater velocity the farther away they are, can be quite simply explained as follows: Assume that at some time many masses with completely different velocities were concentrated in a relatively small region of space, let us say in a sphere K. In the course of time these masses will then, each with its own particular velocity, move out of this region. of the space. After a sufficient time has elapsed, those that have the greatest velocities will have moved farthest away from the sphere K, those with lesser velocities will be nearer to K, and those with the lowest veloci-
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Htuu Hahn
ties will still be very close to K or even within K. Then an observer within K, or at least not too far removed from K, will see the very picture of the stellar world that we have described above. The masses will on the whole be moving away from him, and those farthest away will '6e moving with the greatest velocities. We would thus have an interpretation of the physical world in an entirely different kind of mathematical space-that is to say, in an infinitely extended space. 5 In summary we might very well say that the question: "Is the space of our physical world of infinite or of finite extension 1" has no meaning as it stands. It does not become meaningful until we decide how we are to go about getting the observed events of the physical world into a mathematical space, that is, what assumptions must be made and what logical requirements must be satisfied. And this in turn leads to the question: "Is a finite or an infinite mathematical space better adapted for the arrangement and interpretation of physical events?" At the present stage of our knowledge we cannot give any reasonably well-founded answer to this question. It appears that mathematical spaces of finite and of infinite extension are almost equally well suited for the interpretation of the observational data thus far accumulated. Perhaps at this point confirmed "finitists" will say: "If this is so, we prefer the scheme based on a space of finite extension, since any theory incorporating the concept of infinity is wholly unacceptable to us," They are free to take this view if they wish, but they must not imagine thereby to have altogether rid themselves of infinity. For even the finitely extended Riemann spaces contain infinitely many points, and the mathematical treatment of time is such that each time-interval, however small, contains infinitely many time-points. Must this necessarily be so? Are we in truth compelled to lay the scene of our experience in a mathematical space or in a mathematical time that consists of infinitely many points? I say no. In principle one might very well conceive of a physics in which there were only a finite number of space points and a finite number of time points-in the language of the theory of relativity, a finite number of "world points." In my opinion neither logic nor intuition nor experience can ever prove the impossibility of such a truly finite system of physics. It may be that the various theories of the atomic structure of matter, or today's quantum physics, are the first foreshadowings of a future finite physics. If it ever comes, then we shall have returned after a prodigious circular journey to one of the starting points of western thought, that is, to the Pythagorean doctrine that everything in the world is governed by the natural numbers. If the famous theorem of the right-angle triangle rightly bears the name of Pythagoras S This interpretation was suggested by the late E. A. Milne, noted British physicist and founder of the theory of Kinematic Relativity. ED.
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then it was Pythagoras himself who shook the foundations of his doctrine that everything was governed by the natural numbers. For from the theorem of the right triangle there follows the existence of line segments that are incommensurable, that is, whose relationship with one another cannot be expressed by the natural numbers. And since no distinction was made between mathematical existence and physical existence, a finite physics appeared impossible. But if we are clear on the point that mathematical existence and physical existence mean basically different things; that physical existence can never follow from mathematical existence; that physical existence can in the last analysis only be proved by observation; and that the mathematical difference between rational and irrational forever transcends any possibility of observation-then we shall scarcely be able to deny the possibility in principle of a finite physics. Be that as it may, whether the future produces a finite physics or not, there will remain unimpaired the possibility and the grand beauty of a logic and a mathematics of the infinite.
PART XI
Mathematical Truth and the Structure of Mathematics 1. 2. 3. 4. 5.
On the Nature of Mathematical Truth by CARL G. HEMPEL Geometry and Empirical Science by CARL G. HEMPEL The Axiomatic Method by RAYMOND L. WILDER Goedel's Proof by ERNEST NAGEL and JAMES R. NEWMAN A Mathematical Science by OSWALD VEBLEN and JOHN WESLEY YOUNG 6. Mathematics and the World by DOUGLAS GASKING 7. Mathematical Postulates and Human Understanding by RICHARD VON MISES
COMMENTARY ON
The Foundations of Mathematics HE seven selections which follow deal with the nature of mathematical propositions, the structure of mathematical systems, mathematical inference and the concept of mathematical truth. There is no more interesting and important branch of the literature of mathematics than the study of foundations, to which these topics belong. If one were to attempt to explain to someone who had never heard of mathematics what the subject is about it would be easy to give examples of mathematical statements but very difficult to give a general definition. The difficulty arises not only from the abstract character of the subject but also from the generality and lack of content of its propositions. It is hard to know what you are talking about in mathematics~ yet no one questions the validity of what you say. There is no other realm of discourse half so queer. Let us take an illustration. If I say "1 + 1 = 2," 1 am making a statement which everyone will agree is "true." (I cloak the word in quotes to concede that opinions differ as to the meaning of truth and that mathematicians and philosophers are no less troubled on this score, and no closer to a final conclusion, than was Pilate.) Why does this statement win universal acceptance? Obviously its power of conviction does not derive from my reputation as an expert. My authority on this point is neither smaller than Isaac Newton's nor greater than that of a performing crow. The statement is not to be proved true by appeals to experience. John Stuart Mill and certain other philosophers have suggested that mathematical propositions do not differ from other scientific laws, but are merely exceptionally "well-founded empirical generalizations." I regard this opinion as false and you will see it discredited in the selections below. The fact that one apple added to one apple invariably gives two apples helps in the teaching of arithmetic but has no bearing on the truth of the proposition about the sum of 1 + 1. Is the persuasiveness of the proposition adequately explained by pointing out that it is self-evident? This view does not lack supporters, but it is a purely subjective criterion and we caimot seriously argue that mathematics owes its persuasiveness to fervor of conviction or to everyman's common sense. Moreover, there are many mathematical propositions which, though as unassailable as our simple proposition, not only affront any sober man's common sense but even strain the uncommon sense of the specialist. To say that a proposition is believed because it is self-evident is only to baptize the difficulty-
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The Foundations of Mathematics
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as Poincare remarked-'not to solve it. Why is the statement self-evident? We are circling in the forest. Mathematical statements are compelling, but their force is of a special kind; they are true, but their truth is uniquely defined. Mathematical reasoning is rigorous and deductive and mathematical propositions are simply the consequences of applying this reasoning to certain primitive axioms. Yet this ingrown, self-contained, iron-disciplined method is unlimited in its creativeness, unbounded in its freedom. Neither arithmetic, algebra and analysis on the one hand, nOr geometry on the other hand, are empirical sciences. Mathematics cannot be validated by physical facts, nor its authority impugned or subverted by them. Yet there is a vital connection between the propositions of mathematics and the facts of the physical world. Even the symbols of pure mathematics correspond to some aspect of reality. Abstractions are after all made by men, not by other abstractions. Mathematical ideas are born of experience and are in many cases borne out by experience. It is no accident that a 25-cent piece covers the cost exactly of five five-cent candy bars, and that 5 X 5 25; nor that a straight-edge ruler is an accurate measure of length and that a straight line is a geodesic of a surface of zero curvature. Counting and measuring in the everyday world invariably parallel mathematical propositions but it is essential to distinguish between mathematical propositions and the results of counting and measuring. Pr,pfessor Carl Hempel's two essays are concerned with the nature of mathematical truth and the relation between geometry and empirical science. They treat a subtle and complex problem with uncommon clarity. Hempel, born in Germany in 1905, and a graduate of the University of Berlin, has since 1948 been professor of philosophy at Yale; earlier he taught at the College of the City of New York and Queens College. His special interest is in logic, philosophy of science, probability theory and the foundations of mathematics. He has published many papers on these topics. From Raymond Wilder's excellent book, Introduction to the Foundations of Mathematics, I have taken two chapters which examine mathematical reasoning and the axiomatic method. They are first-rate pieces of exposition. Mr. Wilder, Research Professor of Mathematics at the University of Michigan has also taught at Brown, the University of Texas and Ohio State, and has done research in the foundations of mathematics and topology at the Institute for Advanced Study and at the California Institute of Technology. (It is a commentary on the curious times in which we live that Wilder's most unmilitary study was financed by a grant from the Office of Naval Research.) The essay by Ernest Nagel and myself on Goedel's Proof may be
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Editor's Comment
regarded as a companion to the Wilder selection: It is an attempt to explain to the nonspecialist what is generally regarded as the most brilliant, most difficult, and most stunning sequence of reasoning in modern logic. Goedel set out to show that the axiomatic method which has served mathematics so long and so well has limitations; in particular, that it is impossible within the framework of even a relatively simple mathematical system--ordinary whole-number arithmetic, for example--to demonstrate the internal consistency (non-contradictoriness) of the system without using principles of inference whose own consistency is as much open to question as that of the principles of the system being tested. In this endeavor he was successful; thus we reach a dead end so far as one of the major branches of mathematical research is concerned. Formal deduction has as its crowning achievement proved its own incapacity to make certain formal deductions. In a sense, therefore, formal deduction may be said to have refuted itself. If a mathematical system is to be proved flawless, other methods than the axiomatic will have to be devised for the task. But as the essay points out, Goedel's proof is occasion for neither despair nor mystery-mongering; on the contrary, it justifies "a renewed appreciation of the powers of creative reason." For the more advanced reader I have included a selection from a classic book by Oswald Veblen and John Wesley Young, Projective Geometry. Veblen, one of the country's leading mathematicians, was born in Iowa in 1880, studied at Harvard and the University of Chicago, taught at Chicago and Princeton, and in 1932 became professor at the Institute for Advanced StUdy. He has made notable contributions to differential and projective geometry, foundations of geometry and topology. John Wesley Young (1879-1932), a native of Columbus, Ohio, was educated in German and American schools, receiving his doctorate at Cornell University. He was on the staff of Northwestern University, Princeton, and the universities of Illinois and Chicago, and for many years served as professor of mathematics at Dartmouth. His two best-known works are the volume on projective geometry with' Veblen, and his Lectures on Fundamental Concepts of Algebra and Geometry (1911), an exemplary introduction to the subject, accessible to the general reader as well as to advanced students. The discussion in the Veblen-Young treatise of a simple example of a mathematical science offers an entry into the structure of mathematical thinking and the deductive method. Professor Douglas Gasking of the University of Melbourne presents a quite charming philosophical analysis of the relation of mathematical propositions to the everyday world of counting and measuring. He :finds an ingenious way of demonstrating that mathematics cannot be validated by experience by showing that the particular system we use is not dictated by physical events but rather
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selected for convenience. We could, for example, count and measure entirely differently than we usually do and use a queer multiplication table-a system, say, in which 2 X 4 = 6, 2 X 8 10, 4 X 4 = 9, 4 X 6 = 12; yet every activity in which these methods and this table were employed could be executed accurately. Our books would balance, our floors would be level, our bridges would stand, our bombs would explode as prettily as ever. The conventional system is of course simpler than many others and, having grown up with the human race, fits its psychological needs. But it is not the only possible system. And as we delve into strange things the need arises for strange tools. It might happen "that we found our physical laws getting very complicated indeed, and might discover that, by changing our mathematical system, we could effect a very great simplification in our physics. In such a case we might decide to use a different mathematical system." This, as Gasking reminds us, is exactly what we have done in certain important, disagreeably complex branches of contemporary physics. What is worth remembering is that mathematics has its pragmatic aspects: we should choose the mathematics that helps us understand the world; and we are free to choose it because physical events do not impose a system upon us. The final selection is by the late Richard E. von Mises, a distinguished contemporary mathematician and a foremost exponent of the philosophy of positivism. From his Positivism: A Study in Human Understanding I have excerpted three sections dealing with axiomatics, logistics and the foundations of mathematics. l The material overlaps to some extent other selections in the group and elsewhere in the book, but I do not think you will find this objectionable. It is desirable to hear from a positivist on these matters, especially from one so moderate and knowledgeable as Von Mises. His book considers in tum the application of positivistic theory to the problems of language and communication, to mathematics and logic, to the physical sciences, philosophy and metaphysics, social studies, literature, law, ethics, poetry, religion and the fine arts. In each case his position is open and reasonable; in each case he makes a contribution to understanding. The selections given here sweep away a good deal of muddle and confusion in mathematical thinking. Von Mises merits particularly close attention when he speaks of the axiomatic formulation of a discipline and the practical purposes the axioms should serve; the relation between the so-called tautologies of mathematics (e.g., 2 + 2 = 4) and the common experience of mathematical truths (e.g., 2 cows + 2 cows = 4 cows) ; the inadequacies of the three principal interpretations of the foundations of mathematics. For him, "there is no difference in principle between the disciplines of arithmetic, geometry, mechanics,
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1 The comment here follows on one or two points a brief review by me of Von Mises' book in Scientific American, February 1952, p. 79.
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thermodynamics, optics, electricity, etc. . The foundations and basic assumptions of arithmetic are debatable ill the same sense as those of any part of physics, i.e., on the one hand, as to the internal questions of tauto~ logical structure, and on the other hand, as to the relations with the world of experience." In these sections, as elsewhere in the book, the pointedness and polish of the arguments make Von Mises' brand of positivism almost irresistible. It is an outlook at once more logical and less positive (as another writer has characterized the "more chastened" form of this philosophy) than the logical positivism of the founders. No one who reads the book, says Ernest Nagel, "will credit the frequent but tiresome charge that positivism marks a failure of nerve or a decadence in thought." 2 Dr. von Mises was born in Austria in 1883 and received his graduate degree from the University of Vienna in 1908. He taught applied mathe· matics at the Brno German Technical University and at the universities of Strassburg and Dresden, and was director (1920-33) of the Berlin Institute of Applied Mathematics. During the First World War he served with the Austro-Hungarian Air Force and in 1918 built the first 600 h.p. military plane. Mter holding a chair in mathematics at Istanbul from 1933 to 1939, he became a lecturer at Harvard and in 1943 was appointed Gordon McKay Professor of Aerodynamics and Applied Mathematics. In 1946 he became an American citizen. His research extended into many fields including the theory of flight, mechanics, theory of fluids, elasticity, probability and statistics; his writings comprise more than 100 scientific papers, two books, Theory oj Flight and Probability, Statistics and Truth, essays on the philosophy of science, and a bibliography of Rainer Maria Rilke. Von Mises died after an operation in 1953. 2 In a review of the book appearing in the New Republic, November 26, 1951, p.20.
Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this ,s so only because mathematlcs does not attempt to draw absolute conclusions. All mathematical truths are relatzve, condi-CHARLES PROTEUS STEINMETZ (1923) tional. (Quoted by E. T. Bell, Men of Mathematics)
1
On the Nature of Mathematical Truth By CARL G. HEMPEL 1.
THE PROBLEM
IT is a basic principle of scientific inquiry that no proposition and no theory is to be accepted without adequate grounds. In empirical science, which includes both the natural and the social sciences, the grounds for the acceptance of a theory consist in the agreement of predictions based on the theory with empirical evidence obtained either by experiment or by systematic observation. But what are the grounds which sanction the acceptance of mathematics? That is the question I propose to discuss in the present paper. For reasons which will become clear subsequently, I shall use the term "mathematics" here to refer to arithmetic, algebra, and analysis-to the exclusion, in particular, of geometry. 2.
ARE THE PROPOSITIONS OF MATHEMATICS
SELF-EVIDENT TRUTHS?
One of the several answers which have been given to our problem asserts that the truths of mathematics, in contradistinction to the hypotheses of empirical science, require neither factual evidence nor any other justification because they are "self-evident." This view, however, which ultimately relegates decisions as to mathematical truth to a feeling of selfevidence, encounters various difficulties. First of all, many mathematical theorems are so hard to establish that even to the specialist in the particular field they appear as anything but self-evident. Secondly, it is well known that some of the most interesting results of mathematics--especially in such fields as abstract set theory and topology-run counter to deeply ingrained intuitions and the customary kind of feeling of selfevidence. Thirdly. the existence of mathematical conjectures such as those of Goldbach and of Fermat, which are quite elementary in content and yet undecided up to this day, certainly shows that not all mathematical 1619
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Carl G. Hempel
truths can be self-evident. And finally, even if self-evidence were attributed only to the basic postulates of mathematics, from which all other mathematical propositions can be deduced, it would be pertinent to remark that judgments as to what may be considered as self-evident are subjective; they may vary from person to person and certainly cannot constitute an adequate basis for decisions as to the objective validity of mathematical propositions. 3.
IS MATHEMATICS THE MOST GENERAL EMPIRICAL SCIENCE?
According to another view, advocated especially by John Stuart Mill, mathematics is itself an empirical science which differs from the other branches such as astronomy, physics, chemistry, etc., mainly in two respects: its subject. matter is more general than that of any other field of scientific research, and its propositions have been tested and confirmed to a greater extent than those of even the most firmly established sections of astronomy or physics. Indeed, according to this view, the degree to which the laws of mathematics have been borne out by the past experiences of mankind is so overwhelming that-unjustifiably-we have come to think of mathematical theorems as qualitatively different from the wellconfirmed hypotheses or theories of other branches of science: we consider them as certain, while other theories are thought of as at best "very probable" or very highly confirmed. But this view, too, is open to serious objections. From a hypothesis which is empirical in character-such as, for example, Newton's law of gravitation-it is possible to derive predictions to the effect that under certain specified conditions certain specified observable phenomena will occur. The actual occurrence of these phenomena constitutes confirming evidence, their non-occurrence disconfirming evidence for the hypothesis. It follows in particular that an empirical hypothesis is theoretically disconfirmable; i.e., it is possible to indicate what kind of evidence, if actually encountered, would disconfirm the hypothesis. In the light of this remark, consider now a simple "hypothesis" from arithmetic: 3 + 2 == 5. If this is actually an empirical generalization of past experiences, then it must be possible to state what kind of evidence would oblige us to concede the hypothesis was not generally true after all. If any disconfirming evidence for the given proposition can be thought of, the following illustration might well be typical of it: We place some microbes on a slide: putting down first three of them and then another two. Afterwards we count all the microbes to test whether in this instance 3 and 2 actually added up to 5. Suppose now that we counted 6 microbes altogether. Would we consider this as an empirical disconfirmation of the given proposition, or at least as a proof that it does not apply to microbes?
On the Nature of Mathematical Truth
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Clearly not; rather, we would assume we had made a mistake in counting or that one of the microbes had split in two between the first and the second count. But under no circumstances could the phenomenon just described invalidate the arithmetical proposition in question; for the latter asserts nothing whatever about the behavior of microbes; it merely states that any set consisting of 3 + 2 objects may also be said to consist of 5 objects. And this is so because the symbols "3 + 2" and "5" denote the same number: they are synonymous by virtue of the fact that the symbols "2," "3," "5," and "+" are defined (or tacitly understood) in such a way that the above identity holds as a consequence of the meaning attached to the concepts involved in it. 4.
THE ANALYTIC CHARACTER OF MATHEMATICAL PROPOSITIONS
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The statement that 3 + 2 5, then, is true for similar reasons as, say, the assertion that no sexagenarian is 45 years of age. Both are true simply by virtue of definitions or of similar stipUlations which determine the meaning of the key terms involved. Statements of this kind share certain important characteristics: Their validation naturally requires no empirical evidence; they can be shown to be true by a mere analysis of the meaning attached to the terms which occur in them. In the language of logic, sentences of this kind are called analytic or true a priori, which is to indicate that their truth is logically independent of, or logically prior to, any experiential evidence. l And while the statements of empirical science, which are synthetic and can be validated only a posteriori, are constantly subject to revision in the light of new evidence, the truth of an analytic statement can be established definitely, once and for all. However, this characteristic "theoretical certainty" of analytic propositions has to be paid for at a high price: An analytic statement conveys no factual information. Our statement about sexagenarians, for example, asserts nothing that could possibly conflict with any factual evidence: it has no factual implications, no empirical content; and it is precisely for this reason that the statement can be validated without recourse to empirical evidence. Let us illustrate this view of the nature of mathematical propositions by reference to another, frequently cited, example of a mathematical---or 1 The objection is sometimes raised that without certain types of experience, such as encountering several objects of the same kind, the integers and the arithmetical operations with them would never have been invented, and that therefore the propositions of arithmetic do have an empirical basis. This type of argument, however, involves a confusion of the logical and the psychological meaning of the term "basis." It may very well be the case that certain experiences occasion psychologically the formation of arithmetical ideas and in this sense form an empirical "basis" for them; but this point is entirely irrelevant for the logical questions as to the grounds on which the propositions of arithmetic may be accepted as true. The point made above is that no empirical "basis" or evidence whatever is needed to establish the truth of the 'propositions of arithmetic.
Carl G. Hempel
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rather logical-truth, namely the proposition that whenever a band b c then a-c. On what grounds can this so-called "transitivity of identity" be asserted? Is it of an empirical nature and hence at least theoretically disconfirmable by empirical evidence? Suppose, for example, that a, b, c, are certain shades of green, and that as far as we can see, a b and b c, but clearly a c. This phenomenon actually occurs under certain conditions; do we consider it as disconfirming evidence for the proposition under consideration? Undoubtedly not; we would argue that if a =F c, it is impossible that a b and also b c; between the terms of at least one of these latter pairs, there must obtain a difference, though perhaps only a subliminal one. And we would dismiss the possibility of empirical disconfirmation, and indeed the idea that an empirical test should be relevant here, on the grounds that identity is a transitive relation by virtue of its definition or by virtue of the basic postulates governing it. 2 Hence, the principle in question is true a priori.
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MATHEMATICS AS AN AXIOMATIZED DEDUCTIVE SYSTEM
I have argued so far that the validity of mathematics rests neither on its alleged self-evidential character nor on any empirical basis, but derives from the stipulations which determine the meaning of the mathematical concepts, and that the propositions of mathematics are therefore essentially "true by definition:' This latter statement, however, is obviously oversimplified and needs restatement and a more careful justification. For the rigorous development of a mathematical theory proceeds not simply from a set of definitions but rather from a set of non~definitional propositions which are not proved within the theory; these are the postulates or axioms of the theory.s They are formulated in terms of certain basic or primitive concepts for which no definitions are provided within the theory. It is sometimes asserted that the postulates themselves represent "implicit definitions" of the primitive terms. Such a characterization of the postulates, however, is misleading. For while the postulates do limit, in a specific sense, the meanings that can possibly be ascribed to the primitives, any self-consistent postulate system admits, nevertheless, many different interpretations of the primitive terms (this will soon be illustrated), whereas a set of definitions in the strict sense of the word determines the meanings of the definienda in a unique fashion. Once the primitive terms and the postulates have been laid down, the entire theory is completely determined; it is derivable from its postulational basis in the following sense: Every term of the theory is definable 2 A precise account of the definition and the essential characteristics of the identity relation may be found in A. Tarski, Introduction to Logic, New York, 1941, ch. III. 3 For a lucid and concise account of the axiomatic method, see A. Tarski, loco clI., ch. VI.
On the Nature of Mathematical Truth
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in terms of the primitives, and every proposition of the theory is logically deducible from the postulates. To be entirely precise, it is necessary also to specify the principles of logic which are to be used in the proof of the propositions, i.e., in their deduction from the postulates. These principles can be stated quite explicitly. They fall into two groups: Primitive sentences, or postulates, of logic (such as: If p and q is the case, then p is the case), and rules of deduction or inference (including, for example, the familiar modus ponens rule and the rules of substitution which make it possible to infer, from a general proposition, anyone of its substitution instances). A more detailed discussion of the structure and content of logic would, however, lead too far afield in the context of this article. 6.
PEANO'S AXIOM SYSTEM AS A BASIS FOR MATHEMATICS
Let us now consider a postulate system from which the entire arithmetic of the natural numbers can be derived. This system was devised by the Italian mathematician and logician G. Peano (1858-1932). The primitives of this system are the terms 0" "number'" and "successor." While, of course, no definition of these terms is given within the theory, the symbol "0" is intended to designate the number 0 in its usual meaning, while the term "number" is meant to refer to the natural numbers 0, 1, 2, 3 . . . exclusively. By the successor of a natural number n, which will sometimes briefly be called n', is meant the natural number immediatel) following n in the natural order. Peano's system contains the followin! 5 postulates: Pl. 0 is a number P2. The successor of any number is a number P3. No two numbers have the same successor P4. 0 is not the successor of any number P5. If P is a property such that (a) 0 has the property P, and (b) whenever a number n has the property P, then the successor of n also has the property P, then every number has the property P. The last postulate embodies the principle of mathematical induction and illustrates in a very obvious manner the enforcement of a mathematical "truth" by stipulation. The construction of elementary arithmetic on this basis begins with the definition of the various natural numbers. 1 is defined as the successor of 0, or briefly as 0'; 2 as 1', 3 as 2', and so on. By virtue of P2, this process can be continued indefinitely; because of P3 (in combination with P5), it never leads back to one of the numbers previously defined, and in view of P4, it does not lead back to 0 either. As the next step, we can set up a definition of addition which expresses in a precise form the idea that the addition of any natural number to some given number may be considered as a repeated addition of 1; the 41
Carl G. Hempel
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latter operation is readily expressible by means of the successor relation. This definition of addition runs as follows:
01. (a) n + 0 == n;
(b) n + k' == (n
+ k) '.
The two stipulations of this recursive definition completely determine the sum of any two integers. Consider, for example, the sum 3 + 2. According to the definitions of the numbers 2 and 1, we have 3 + 2 == 3 + l' == 3 + (0'),; by Dl (b), 3 + (0')' == (3 + 0')' == «3 + 0)'),; but by 01 (a), and by the definitions of the numbers 4 and 5, «3 0)'), == (3'), == 4' 5. This proof also renders more explicit and precise the comments made earlier in this paper on the truth of the proposition that 3 + 2 == 5: Within the Peano system of arithmetic, its truth flows not merely from the definition of the concepts involved, but also from the postulates that govern these vafious concepts. (In our specific example, the postulate& PI and P2 presupposed to guarantee that 1, 2, 3, 4, 5, are numbers in Peano's system; the general proof that Dl determines the sum of any two numbers also makes use of P5.) If we call the postulates and definitions of an axiomatized theory the "stipulations" concerning the concepts o:Vthat theory, then we may say now that the propositions of the anthmetic of the natural numbers are true by virtue of the stipulations which have been laid down initially for the arithmetical concepts. (Note, incidentally. that our proof of the formula "3 + 2 == 5" repeatedly made use of the transitivity of identity; the latter is accepted here as one of the rules of logic which may be used in the proof of any arithmetical theorem; it is, therefore, included among Peano's postulates no more than any other principle of logic.) Now, the multiplication of natural numbers may be defined by means of the following recursive definition, which expresses in a rigorous form the idea that a product nk of two integers may be considered as the sum of k terms each of which equals n.
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02. (a) n'O = 0;
(b) n·k'=n·k+n.
It now is possible to prove the familiar general laws governing addition
and multiplication, such as the commutative, associative, and distributive laws + k == k + n; n' k k ·n; n + (k + 1) == (n + k) + 1; n· (k ·1) (n· k) ·1; n' (k + 1) == (n· k) + (n·1) ). In terms of addition and multiplication, the inverse operations of subtraction and division can then be defined. But it turns out that these "cannot always be performed"; i.e., in contradistinction to the sum and the product, the difference and the quotient are not defined for every couple of numbers; for example, 7 - 10 and 7 -;- 10 are undefined. This situation suggests an enlargement of thenumber system by the introduction of negative and of rational numbers.
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On the Nature 0/ MathematIcal Truth
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It is sometimes held that in order to effect this enlargement, we have to "assume" or else to "postulate" the existence of the desired additional kinds of numbers with properties that make them fit to fill the gaps of subtraction and division. This method of simply postulating what we want has its advantages; but, as Bertrand Russell 4 puts it, they are the same as the advantages of theft over honest toil; and it is a remarkable fact that the negative as well as the rational numbers can be obtained from Peano's primitives by the honest toil of constructing explicit definitions for them, without the introduction of any new postulates or assumptions whatsoever. Every positive and negative integer (in contradistinction to a natural number which has no sign) is definable as a certain set of ordered couples of natural numbers; thus, the integer +2 is definable as the set of all ordered couples (m, n) of natural numbers where m n + 2; the integer -2 is the set of all ordered couples (m, n) of natural numbers with n m + 2. (Similarly, rational numbers are defined as classes of ordered couples of integers.) The various arithmetical operations can then be defined with reference to these new types of numbers, and the validity of all the arithmetical laws governing these operations can be proved by virtue of nothing more than Peano's postulates and the definitions of the various arithmetical concepts involved. The much broader system thus obtained is still incomplete in the sense that not every number in it has a square root, and more generally, not every algebraic equation whose coefficients are all numbers of the system has a solution in the system. This suggests further expansions of the number system by the introduction of real and finally of complex numbers. Again, this enormous extension can be effected by mere definition, without the introduction of a single new postulate. 5 On the basis thus obtained, the various arithmetical and algebraic operations can be defined for the numbers of the new system, the concepts of function, of limit, of derivative and integral can be introduced, and the familiar theorems pertaining to thes~ concepts can be proved, so that finally the huge system ' of mathematics as here delimited rests on the narrow basis of Peano's system: Every concept of mathematics can be defined by means of Peano's three primitives, and every proposition of mathematics can be deduced from the five postulates enriched by the definitions of the non-primitive
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4 Bertrand Russell, Introduction to Mathematical Philosophy, New York and London, 1919, p. 71. ~ For a more detailed account of the construction of the number system on Peano's basis, cf. Bertrand Russell, loco cit., esp. cbs. I and VII. A rigorous and concise presentation of that construction, beginning, however, with the set of all integers rather than that of the natural numbers, may be found in G. Birkhoff and S. MacLane, A Survey of Modern Algebra, New York, 1941, cbs. I, II, III, V. For a general survey of the construction of the number system, cf. also J. W. Young, Lectures on the Fundamental Concepts of Algebra and Geometry, New York, 1911, esp. lectures X, XI, XII.
Carl G. Hempel
1626
terms. 6 These deductions can be carried out, in most cases, by means of nothing more than the principles of formal logic; the proof of some theorems concerning real numbers, however, requires one assumption which is not usually included among the latter. This is the so-called axiom of choice. It asserts that given a class of mutually exclusive classes, none of which is empty, there exists at least one class which has exactly one element in common with each of the given classes. By virtue of this principle and the rules of formal logic, the content of all of mathematics can thus be derived from Peano's modest system-a remarkable achievement in systematizing the content of mathematics and clarifying the foundations of its validity. 7.
INTERPRETATIONS OF PEANO'S PRIMITIVES
As a consequence of this result, the whole system of mathematics might be said to be true by virtue of mere definitions (namely, of the nonprimitive mathematical terms) provided that the five Peano postulates are true. However, strictly speaking, we cannot, at this juncture, refer to the Peano postulates as propositions which are either true or false, for they contain three primitive terms which have not been assigned any specific meaning. All we can assert so far is that any specific interpretation of the primitives which satisfies the five postulates-i.e., turns them into true statements-will also satisfy all the theorems deduced from them. But for Peano's system, there are several-indeed, infinitely many-interpretations which will do this. For example, let us understand by 0 the origin of a half-line, by the successor of a point on that half-line the point 1 cm. behind it, counting from the origin, and by a number any point which is either the origin or can be reached from it by a finite succession of steps each of which leads from one point to its successor. It can then readily be seen that all the Peano postulates as well as the ensuing theo6 As a result of very deep-reaching investigations carried out by K. Godel it is known that arithmetic, and a jortlOn mathematics, is an incomplete theory in the following sense: While all those propositions which belong to the classical systems of arithmetic, algebra, and analysis can mdeed be derived, in the sense characterized abore, from the Peano postulates, there exist nevertheless other propositions which Can be expressed in purely arithmetical terms, and WhICh are tnte, but which cannot be derived from the Peano system. And more generally: For any postulate system of arithmetic (or of mathematics for that matter) which is not self-contradictory, there exist propositions which are tnte, and which can be stated in purely arithmetical terms, but which cannot be derived from that postulate system. In other words, it is impossible to construct a postulate system which is not self-contradictory, and which contains among its consequences all true propositions which can be formulated wIthin the language of arithmetic. T~is fact does not, however, affect the result outlined above, namely. that it is possIble to deduce, from the Peano postulates and the additional definitions of nonprimitive terms, all those propositions which constitute the classical theory of arithmetic, algebra, and analysis; and it is to these propositions that I refer above and subsequently as the propositions of mathematics.
On tile Nature of MathematIcal Truth
1627
rems turn into true propositions, although the interpretation given to the primitives is certainly not the customary one, which was mentioned earlier. More generally, it can be shown that every progression of elements of any kind provides a true interpretation, or a "model," of the Peano system. This example illustrates our earlier observation that a postulate system cannot be regarded as a set of "implicit definitions" for the primitive terms: The Peano system permits of many different interpretations, whereas in everyday as well as in scientific language, we attach one specific meaning to the concepts of arithmetic. Thus, e.g., in scientific and in everyday discourse, the concept 2 is understood in such a way that from the statement "Mr. Brown as well as Mr. Cope, but no one else is in the office, and Mr. Brown is not the same person as Mr. Cope." the conclusion "Exactly two persons are in the office" may be validly inferred. But the stipulations laid down in Peano's system for the natural numbers, and for the number 2 in particular, do not enable us to draw this conclusion; they do not "implicitly determine" the customary meaning of the concept 2 or of the other arithmetical concepts. And the mathematician cannot acquiesce in this deficiency by arguing that he is not concerned with the customary meaning of the mathematical concepts; for in proving, say, that every positive real number has exactly two real square roots, he is himself using the concept 2 in its customary meaning, and his very theorem cannot be proved unless we presuppose more about the number 2 than is stipulated in the Peano system. If therefore mathematics is to be a correct theory of the mathematical concepts in their intended meaning, it is not sufficient for its validation to have shown that the entire system is derivable from the Peano postulates plus suitable definitions; rather, we have to inquire further whether the Peano postulates are actually true when the primitives are understood in their customary meaning. This question, of course, can be answered only after the customary meaning of the terms "0," "natural number," and "successor" has been clearly defined. To this task we now turn. 8.
DEFINITION OF THE CUSTOMARY MEANING OF THE CONCEPTS OF ARITHMETIC IN PURELY LOGICAL TERMS
At first blush, it might seem a hopeless undertaking to try to aefine these basic arithmetical concepts without presupposing other terms of arithmetic, which would involve us in a circular procedure. However, quite rigorous definitions of the desired kind can indeed be formulated, and it can be shown that for the concepts so defined. all Peano postulates tum into true statements. This important result is due to the research of the German logician G. Frege (1848-1925) and to the subsequent systematic and detailed work of the contemporary English logicians and philos-
Carl G. Hempel
1628
ophers B. Russell and A. N. Whitehead. Let us consider briefly the basic ideas underlying these definitions.7 A natural number-or, in Peano's term, a number-in its customary meaning can be considered as characteristic of certain classes of objects. Thus, e.g., the class of the apostles has the number 12, the class of the Dionne quintuplets the number 5, any couple the number 2, and so on. Let us now express precisely the meaning of the assertion that a certain class C has the number 2, or briefly, that n(C) 2. Brief reflection will show that the following definiens is adequate in the sense of the customary meaning of the concept 2: There is some object x and some object y such that (1) XEC (Le., x is an element of C) and YEC, (2) xo;=y, and (3) if z is any object such that ZeC, then either z x or z y. (Note that on the basis of this definition it becomes indeed possible to infer the statement "The number of persons in the office is 2" from "Mr. Brown as well as Mr. Cope, but no one else is in the office, and Mr. Brown is not identical with Mr. Cope"; C is here the class of persons in the office.) Analogously, the meaning of the statement that n(C) -= 1 can be defined thus: There is some x such that xeC, and any object y such that yeC, is identical with x. Similarly, the customary meaning of the statement that n(C) = 0 is this: There is no object such that XEC. The general pattern of these definitions clearly lends itself to the definition of any natural number. Let us note especially that in the definitions thus obtained, the definiens never contains any arithmetical term, but merely expressions taken from the field of formal logic, including the signs of identity and difference. So far, we have defined only the meaning 2," but we have given no definition for the of such phrases as "n (C) numbers 0, 1, 2, . . . apart from this context. This desideratum can be met on the basis of the consideration that 2 is that property which is common to all couples, i.e.~ to all classes C such that n( C) == 2. This common property may be conceptually represented by the class of all those classes which share this property. Thus we arrive at the definition: 2 is the class of all couples, i.e., the class of all classes C for which n(C) 2.-This definition is by no means circular because the concept of couple-in other words, the meaning of "n(C) == 2"-has been previously defined without any reference to the number 2. Analogously, 1 is the class of all unit classes, i.e., the class of all classes C for which n(C) == 1. Finally,
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7 For a more detailed discussion, cf. Russell, loco Cit., chs. II, III, IV. A complete techmcal development of the idea can be found in the great standard work in mathematical logic, A. N. Whitehead and B. Russell, Principia Mathematica, Cambridge, England, 1910-1913. For a very precIse development of the theory, see W. V. O. Quine, Mathematical Logic, New York, 1940. A specIfic discussion of the Peano system and its interpretations from the viewpoint of semantics is included in R. Carnap, Foundations of Logic and Mathematics, InternatlOnal Encyclopedia of Unified Science, Vol. I, no. 3, Chicago, 1939; especially sections 14, 17, 18.
On the Nature of Mathematical Truth
1629
o is the class of all null classes, i.e., the class of all classes without elements. And as there is only one such class, 0 is simply the class whose only element is the null class. Clearly, the customary meaning of any given natural number can be defined in this fashion,s In order to characterize the intended interpretation of Peano's primitives, we actually need, of all the definitions here referred to, only that of the number O. It remains to define the terms "successor" and "integer." The definition of "successor," whose precise formulation involves too many niceties to be stated here, is a careful expression of a simple idea which is illustrated by the following example: Consider the number 5, i.e., the class of all quintuplets. Let us select an arbitrary one of these quintuplets and add to it an object which is not yet one of its members. 5', the successor of 5, may then be defined as the number applying to the set thus obtained (which, of course, is a sextuplet). Finally, it is possible to formulate a definition of the customary meaning of the concept of natural number; this definition, which again cannot be given here, expresses, in a rigorous form, the idea that the class of the natural numbers consists of the number 0, its successor, the successor of that successor, and so on. If the definitions here characterized are carefully written out-this is one of the cases where the techniques of symbolic, or mathematical, logic prove indispensable-it is seen that the definiens of everyone of them contains exclusively terms from the field of pure logic. In fact, it is possible to state the customary interpretation of Peano's 'primitives, and thus also the meaning of every concept definable by means of them-and that includes every concept of mathematics-in terms of the following seven expressions (in addition to variables such as "x" and "C"): not, and, ifthen; jor every object x it is the case that . . .; there is some object x such that . . . ; x is an element of class C; the class oj all things x such that . . . And it is even possible to reduce the number of logical concepts needed to a mere four: The first three of the concepts just mentioned are all definable in terms of "neither-nor," and the fifth is definable by means of the fourth and "neither-nor." Thus, all the concepts of mathe matics prove definable in terms of four concepts of pure logic. (The w
8 The assertion that the defimtions given above state the "customary" meaning of the arithmetical terms involved is to be understood in the logical, not the psycho~ logical sense of the term "meaning." It would obviously be absurd to claim that the above definitions express "what everybody has in mind" when talking about numbers and the various operations that can be performed with them. What is achieved by those definitions is rather a "logical reconstruction" of the concepts of arithmetic in the sense that if the definitions are accepted, then those statements in science and everyday discourse which involve arithmetical terms can be interpreted coherently and systematically in such a manner that they are capable of objective validation. The statement about the two persons in the office provides a very elementary illustration of what is meant here.
Carl G. Hempel
1630
definition of one of the more complex concepts of mathematics in terms of the four primitives just mentioned may well fill hundreds or even thousands of pages; but clearly this affects in no way the theoretical importance of the result just obtained; it does, however, show the great convenience and indeed practical indispensabiilty for mathematics of having a large system of highly complex defined concepts available.) 9.
THE TRUTH OF PEANO'S POSTULATES IN THEIR CUSTOMARY INTERPRETATION
The definitions characterized in the preceding section may be said to render precise and explicit the customary meaning of the concepts of arithmetic. Moreover-and this is crucial for the question of the validity of mathematics-it can be shown that the Peano postulates all turn into true propositions if the primitives are construed in accordance with the definitions just considered. Thus, PI (0 is a number) is true because the class of all numbers-i.e., natural numbers-was defined as consisting of 0 and all its successors. The truth of P2 (the successor of any number is a number) follows from the same definition. This is true also of P5, the principle of mathematical induction. To prove this, however, we would have to resort to the precise definition of "integer" rather than the loose description given of that definition above. P4 (0 is not the sUccessor of any number) is seen to be true as follows: By virtue of the definition of "successor," a number which is a successor of some number can apply only to classes which contain at least one element; but the number 0, by definition, applies to a class if and only if that class is empty. While the truth of PI, P2, P4, P5 can be inferred from the above definitions simply by means of the principles of logic, the proof of P3 (no two numbers have the same successor) presents a certain difficulty. As was mentioned in the preceding section, the definition of the successor of a number n is based on the process of adding, to a class of n elements, one element not yet contained in that class. Now if there should exist only a finite number of things altogether then this process could not be continued indefinitely, and P3, which (in conjunction with PI and P2) implies that the integers form an infinite set, would be false. This difficulty can be met by the introduction of a special "axiom of infinity" 9 which asserts, in effect, the existence of infinitely many Objects, and thus makes P3 demonstrable. The axiom of infinity does not belong to the generally recognized laws of logic; but it is capable of expression in purely logical terms and may be considered as an additional postulate of modern logical theory. 9
Cf. Bertrand Russell, loco cit., p. 24 and ch. XIII.
1631
On the Nature of Mathematical Truth
10.
MATHEMATICS AS A BRANCH OF LOGIC
As was pointed out earlier, all the theorems of arithmetic, algebra, and analysis can be deduced from the Peano postulates and the definitions of those mathematical terms which are not primitives in Peano's system. This deduction requires only the principles of logic plus, in certain cases, the axiom of choice. By combining this result with what has just been said about the Peano system, the following conclusion is obtained, which is also known as the thesis oj logicism concerning the nature of mathematics: Mathematics is a branch of logic. It can be derived from logic in the following sense: a. All the concepts of mathematics, i.e., of arithmetic, algebra, and analysis, can be defined in terms of four concepts of pure logic. h. All the theorems of mathematics can be deduced from those definitions by means of the principles of logic (including the axioms of infinity and choice) .10 In this sense it can be said that the propositions of the system of mathematics as here d~limited are true by virtue of the definitions of the mathematical concepts involved, or that they make explicit certain characteristics with which we have endowed our mathematical concepts by definition. The propositions of mathematics have, therefore, the same unquestionable certainty which is typical of such propositions as "All bachelors are unmarried," but they also share the complete lack of empirical content which is associated with that certainty: The propositions of mathematics are devoid of all factual content; they convey no information whatever on any empirical subject matter. 11.
ON THE APPLICABILITY OF MATHEMATICS TO EMPIRICAL SUBJECT MATTER
This result seems to be irreconcilable with the fact that after all mathematics has proved to be eminently applicable to empirical subject matter, and that indeed the greater part of present-day scientific knowledge has been reached only through continual reliance on and application of the propositions of mathematics. Let us try to clarify this apparent paradox by reference to some examples. Suppose that we are examining a certain amount of some gas, whose 10 The principles of logic developed in modern systems of formal logic embody certain restrictions as compared with those logical rules which had been rather generally accepted as sound until about the turn of the 20th century. At that time, the discovery of the famous paradoxes of logic, especially of Russell's paradox (cf. Russell, loco cit., ch. XIII), revealed the fact that the logical principles implicit in customary mathematical reasoning involved contradictions and therefore had to be curt8.1led in one manner or another.
Carl G H~mp~l
1632
volume v, at a certain fixed. temperature, is found to be 9 cubic feet when the pressure p is 4 atmospheres. And let us assume further that the volume of the gas for the same temp'erature and p 6 at., is predicted by means of Boyle's law. Using elementary' arithmetic we reason thus: For corresponding values of v and p? vp c, and v 9 when p 4; hence c 36: Therefore, when p ::: 6, then v 6. Suppose that this prediction is borne out by subsequent test. Does that show that the arithmetic used has a predictive power of its own, that its proposi,ions have factual implications? Certainly not. All the predictive p'ower here deployed, all the empirical content exhibited stems from the initial data and from Boyle's law, which asserts that vp :::;: c for any two corresponding values of v and p, hence also for v = 9, p ::: 4, and for p 6 and the corresponding value of v.n The function of the mathematics here applied is not predictive at all; rather, it is analytic or explicative: it renders explicit certain assumptions or assertions which are included in the content of the premises of the argument (in our case, these consist of Boyle's law plus the additional data); mathematical reasoning reveals that those premises contain-hidden in them, as it were,-an assertion about the case as yet unobserved. In accepting our premises-so arithmetic reveals-we have-knowingly or unknowingly-already accepted the implication that the p-value in question is 6. Mathematical as well as logical reasoning is a conceptual technique of making explicit What is implicitly contained in a set of premises. The conclusions to which this technique leads assert nothing that is theoretically new in the sense of not being contained in the content of the premises. But the results obtained may well be psychologically new: we may not have been aware, before using the techniques of logic and mathematics, what we committed ourselves to in accepting a certain set of assumptions or assertions. A similar analysis is possible in all other cases of applied mathematics, including those involving, say, the calculus. Consider, for example, the hypothesis that a certain object, moving in a specified electric field, will undergo a constant acceleration of 5 feet/sec 2 • For the purpose of testing this hypothesis, we might derive from it, by means of two successive integrations. the prediction that if the object is at rest at the beginning of the motion, then the distance covered by it at any time t is %t2 feet. This conclusion may clearly be psychologically new to a person not acquainted with the subject, but it is not theoretically new; the content of the conclusion is already contained in that of the hypothesis about the constant acceleration. And indeed, here as well as in the case of the compression of a gas, a failure of the prediction to come true would be considered as indicative of the factual incorrectness of at least one of the premises in-
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11 Note that we may say "hence" by virtue of the rule of substitution, which is one of the rules of logical inference.
On the Nature of MathematIcal Truth
1633
volved (I. ex. , of Boyle's law in its app1ication to the particular gas), but never as a sign that the logical and mathematical principles involved might be unsound. Thus, in the establishment of empirical knowledge, mathematics (as well as logic) has, so to speak, the function of a theoretical juice extractor: the techniques of mathematical and logical theory can produce no more juice of factual information than is contained in the assumptions to which they are applied; but they may produce a great deal more juice of this kind than might have been anticipated upon a first intuitive inspection of those assumptions which form the raw material for the extractor. At this point, it may be well to consider briefly the status of those mathematical disciplines which are not outgrowths of arithmetic and thus of logic; these include in particular topology, geometry, and the various branches of abstract algebra, such as the theory of groups, lattices, fields, etc. Each of these disciplines can be developed as a purely deductive sys· tern on the basis of a suitable set of postulates. If P be the conjunction of the postulates for a given theory, then the proof of a proposition T of that theory consists in deducing T from P by means of the principles of formallogic. What is established by the proof is therefore not the truth of T, but rather the fact that T is true provided that the postulates are. But since both P and T contain certain primitive terms of the theory, to which no specific meaning is assigned, it is not strictly possible to speak of the truth of either P or T; it is therefore more adequate to state the point as follows: If a proposition T is logically deduced from P, then every specific interpretation of the primitives which turns all the postu~ lates of P into true statements, will also render T a true statement. Up to this point, the analysis is exactly analogous to that of arithmetic as based on Peano's set of postulates. In the case of arithmetic, however, it proved possible to go a step further, namely to define the customary meanings of the primitives in terms of purely logical concepts and to show that the postulates-and therefore also the theorems--of arithmetic are unconditionally true by virtue of these definitions. An analogous procedure is not applicable to those disciplines which are not outgrowths of arithmetic: The primitives of the various branches of abstract algebra have no specific "customary meaning"; and if geometry in its customary interpretation is thought of as a theory of the structure of physical space, then its primitives have to be construed as referring to certain types of physical entities, and the question of the truth of a geometrical theory in this interpretation turns into an empirical problem. 12 For the purpose of applying anyone of these non-arithmetical disciplines to some specific field of mathematics or empirical science, it is therefore necessary first to assign to the primitives some specific meaning and then to ascertain whether in this interpretation l2
For a more detailed discussion of this point, d. the next article in this collection.
1634
Carl G. Hempel
the postulates turn into true statements. If this is the case, then we can be sure that al1 the theorems are true statements too, because they are logically derived from the postulates and thus simply explicate the content of the latter in the given interpretation. In their application to empirical subject matter, therefore, these mathematical theories no less than those which grow out of arithmetic and ultimately out of pure logic, have the function of an analytic tool, which brings to light the implications of a given set of assumptions but adds nothing to their content. But while mathematics in no case contributes anything to the content of our knowledge of empirical matters, it is entirely indispensable as an instrument for the validation and even for the linguistic expression of such knowledge: The majority of the more far~reaching theories in empirical science-including those which lend themselves most eminently to prediction or to practical application-are stated with the help of mathematical concepts; the formulation of these theories makes use, in particular, of the number system, and of functional relationships among different metrical variables. Furthermore, the scientific test of these theories, the establishment of predictions by means of them, and finally their practical application, all require the deduction, from the general theory, of certain specific consequences; and such deduction would be entirely impossible without the techniques of mathematics which reveal what the given general theory implicitly asserts about a certain special case. Thus, the analysis outlined on these pages exhibits the system of mathe~ matics as a vast and ingenious conceptual structure without empirical content and yet an. indispensable and powerful theoretical instrument for the scientific understanding and mastery of the world of OUf experience.
That all our knowledge begins wIth experience, there IS mdeed no doubt . . . but although our knowledge ongmates with experience, It does not all -IMMANUEL KANT arise out of experience.
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Geometry and Empirical Science By CARL G. HEMPEL 1.
INTRODUCTION
THE most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which accounts for its fame as the queen of the sciences, is no doubt the peculiar certainty and necessity of its re~ults. No proposition in even the most advanced parts of empirical science can ever attain this status; a hypothesis concerning "matters of empirical fact" can at best acquire what is loosely called a high probability or a high degree of confirmation on the basis of the relevant evidence available; but however well it may have been confirmed by careful tests, the possibility can never be precluded that it will have to be discarded later in the light of new and disconfirming evidence. Thus, all the theories and hypotheses of empirical science share this provisional character of being established and accepted "until further notice," whereas a mathematical theorem, once proved, is established once and for all; it holds with that particular certainty which no subsequent empirical discoveries, however unexpected and extraordinary, can ever affect to the slightest extent. It is the purpose of this paper to examine the nature of that proverbial "mathematical certainty" with special reference to geometry, in an attempt to shed some light on the question as to the validity of geometrical theories, and their significance for our knowledge of the structure of physical space. The nature of mathematical truth can be understood through an analysis of the method by means of which it is established. On this point I can be very brief: it is the method of mathematical demonstration, which consists in the logical deduction of the proposition to be proved from other propositions, previously established. Clearly, this procedure would involve an infinite regress unless some propositions were accepted without proof; such propositions are indeed found in every mathematical discipline which is rigorously developed; they are the axioms or postulates (we shall use these terms interchangeably) of the theory. Geometry provides the historically first example of the axiomatic presentation of a mathematical discipline. The classical set of postulates, however, on which Euclid based his system, has proved insufficient for the deduction of the well-known 1635
Carl G. Hempel
1636
theorems of so-called euclidean geometry; it has therefore been revised and supplemented in modern times, and at present various adequate sys· terns of postulates for euclidean geometry are available; the one most closely related to Euclid's system is probably that of Hilbert. 2.
THE INADEQUACY OF EUCLID'S POSTULATES
The inadequacy of Euclid's own set of postulates illustrates a point which is crucial for the axiomatic method in modern mathematics: Once the postulates for a theory have been laid down, every further proposition of the theory must be proved exclusively by logical deduction from the postulates; any appeal, explicit or implicit, to a feeling of self-evidence, or to the characteristics of geometrical figures, or to our experiences concerning' the behavjor of rigid bodies in physical space, or the like, is strictly prohibited; such devices may have a heuristic value in guiding our efforts to find a strict proof for a theorem, but the proof itself must contain absolutely no reference to such aids. This is particularly important in geometry, where our so-called intuition of geometrical relationships, supported by reference to figures or to previous physical experiences, may induce us tacitly to make use of assumptions which are neither formulated in our postulates nor provable by means of them. Consider, for example, the theorem that in a triangle the three medians bisecting the sides intersect in one point which divides each of them in the ratio of 1 : 2. To prove this theorem, one shows first that in any triangle ABC (see figure) the line segment MN which connects the centers of AB and AC is parallel to BC and therefore half as long as the latter side. Then the lines BN and CM are drawn, and an examination of the triangles MON and BOC leads to the proof of the theorem. In this procedure, it is usually taken for granted that BN and CM intersect in a point 0 which lies between Band N as well as between C and M. This assumption is based on geometrical
A
B------------~·C intuition, and indeed, it cannot be deduced from Euclid's postulates; to make it strictly demonstrable and independent of any reference to intuition, a special group of postulates has been added to those of Euclid; they are the postulates of order. One of these-to give an example-asserts
1637
Geometry alld Empirical SClellce
that if A, B, C are points on a straight line [, and if B lies between A and C, then B also lies between C and A. Not even as "trivial" an assumption as this may be taken for granted; the system of postulates has to be made so complete that all the required propositions can be deduced from it by purely logical means. 'Another illustration of the point under consideration is provided by the proposition that triangles which agree in two sides and the enclosed angle, are congruent. In Euclid's Elements, this proposition is presented as a theorem; the alleged proof, however, makes use of the ideas of motion and superimposition of figures and thus involves tacit assumptions which are based on our geometric intuition and on experiences with rigid bodies, but which are definitely not warranted by-Le., deducible from-Euclid's postulates. In Hilbert's system, therefore, this proposition (more precisely: part of it) is explicitly included among the postulates. 3.
MATHEMATICAL CERTAINTY
It is this purely deductive character of mathematical proof which forms the basis of mathematical certainty: What the rigorous proof of a theorem -say the proposition about the sum of the angles in a triangle-establishes is not the truth of the proposition in question but rather a conditional insight to the effect that that proposition is certainly true provided that the postulates are true; in other words, the proof of a mathematical proposition establishes the fact that the latter is logically implied by the postulates of the theory in question. Thus, each mathematical theorem can be cast into the form (P1·P2'P:~·
. . . . Pn )
~
T
where the expression on the left is the conjunction (joint assertion) of all the postulates, the symbol on the right represents the theorem in its customary formulation, and the arrow expresses the relation of logical implication or entailment. Precisely this character of mathematical theorems is the reason for their peculiar certainty and necessity, as I shall now attempt to show. It is typical of any purely logical deduction that the conclUSIOn to which it leads simply re-asserts (a proper or improper) part of what has already been stated in the premises. Thus, to illustrate this point by a very elementary example, from the premise, "This figure is a right triangle," we can deduce the conclusion, "This figure is a triangle"; but this conclusion clearly reiterates part of the information already contained in the premise. Again, from the premises, "All primes different from 2 are odd" and "n is a prime different from 2," we can infer logically that n is odd; but this consequence merely repeats part (indeed a relatively small part) of the
Carl G. Hempel
1638
information contained in the premises. The same situation prevails in all other cases of logical deduction; and we may, therefore, say that logical deduction-which is the one and only method of mathematical proof-is a technique of conceptual analysis: it discloses what assertions are concealed in a given set of premises, and it makes us realize to what we committed ourselves in accepting those premises; but none of the results obtained by this technique ever goes by ~)lle iota beyond the information already contained in the initial assumptions. Since all mathematical proofs rest exclusively on logical deductions from certain postulates, it follows that a mathematical theorem, such as the Pythagorean theorem in geometry, asserts nothing that is objectively or theoretically new as compared with the postulates from which it is derived, although its content may well be psychologically new in the sense that we were not aware of its being implicitly contained in the postulates. The nature of the peculiar certainty of mathematics is now clear: A mathematical theorem is certain relatively to the set of postulates from which it is derived; i.e., it is necessarily true if those postulates are true; and this is so because the theorem, if rigorously proved, simply re-asserts part of what has been stipulated in the postulates. A truth of this condi~ tional type obviously implies no assertions about matters of empirical fact and can, therefore, never get into conflict with any empirical findings, even of the most unexpected kind; consequently, unlike the hypotheses and theories of empirical science, it can never suffer the fate of being disconfirmed by new evidence: A mathematical truth is irrefutably certain just because it is devoid of factual, or empirical content. Any theorem of geometry, therefore, when cast into the conditional form described earlier, is analytic in the technical sense of logic, and thus true a priori; Le., its truth can be established by means of the formal machinery of logic alone, without any reference to empirical data. 4.
POSTULATES AND TRUTH
Now it might be felt that our analysis of geometrical truth so far tells only half of the relevant story. For while a geometrical proof no doubt enables us to assert a proposition conditionally-namely on condition that the postulates are accepted-is it not correct to add that geometry also unconditionally asserts the truth of its postulates and thus, by virtue of the deductive relationship between postulates and theorems, enables us unconditionally to assert the truth of its theorems? Is it not an unconditional assertion of geometry that two points determine one and only one straight line that connects them, or that in any triangle, the sum of the angles equals two right angles? That this is definitely not the case, is evidenced
Geomet17 and Empzrical Science
1639
by two important aspects of the axiomatic treatment of geometry which will now be briefly considered. The first of these features is the well~known fact that in the more recent development of mathematics, several systems of geometry have been constructed which are incompatible with euclidean geometry, and in which, for example, the two propositions just mentioned do not necessarily hold. Let us briefly recollect some of the basic facts concerning these noneuclidean geometries. The postulates on which euclidean geometry rests include the famous postulate of the parallels, which, in the case of plane geometry, asserts in effect that through every point P not on a given line l there exists exactly one parallel to 1, i.e., one straight line which does not meet 1. As this postulate is considerably less simple than the others, and as it was also felt to be intuitively less plausible than the latter, many efforts were made in the history of geometry to prove that this proposition need "not be accepted as an axiom, but that it can be deduced as a theorem from the remaining body of postulates. All attempts in this direction failed, however; and finally it was conclusively demonstrated that a proof of the parallel principle on the basis of the other postulates of euclidean geometry (even in its modern, completed form) is impossible. This was shown by proving that a perfectly self-consistent geometrical theory is obtained if the postulate of the parallels is replaced by the assumption that through any point P not on a given straight line 1 there exist at least two parallels to 1. This postulate obviously contradicts the euclidean postulate of the parallels, and if the latter were actually a consequence of the other postulates of euclidean geometry, then the new set of postulates would clearly involve a contradiction, which can be shown not to be the case. This first non-euclidean type of geometry, which is called hyperbolic geometry, was discovered in the early 20's of the last century almost simultaneously, but independently by the Russian N. 1. Lobatschefskij, and by the Hungarian J. Bolyai. Later, Riemann developed an alternative geometry, known as elliptical geometry, in which the axiom of the parallels is replaced by the postulate that no line has any parallels. (The" acceptance of this postulate, however, in contradistinction to that of hyperbolic geometry, requires the modification of some further axioms of euclidean geometry, if a consistent new theory is to result.) As is to be expected, many of the theorems of these non-euclidean geometries are at variance with those of euclidean theory; thus, e.g., in the hyperbolic geometry of two dimensions, there exist, for each straight line 1, through any point P not on 1, infinitely many straight .lines which do not meet 1; also, the sum of the angles in any triangle is less than two right angles. In elliptic geometry, this angle sum is always greater than two right angles; no two straight lines are parallel; and while two different points usually determine exactly one straight line connecting tht"m (as they always do in euclidean
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geometry), there are certain pairs of points which are connected by infinitely D1any different straight lines. An illustration of this latter type of geometry is provided by the geometrical structure of that curved twodimensional space which is represented by the surface of a sphere, when the concept of straight line is interpreted by that of great circle on the sphere. In this space, there are no parallel lines since any two great circles intersect; the endpoints of any diameter of the sphere are points connected by infinitely many different "straight lines," and the sum of the angles in a triangle is always in excess of two right angles. Also, in this space, the ratio between the circumference and the diameter of a circle (not necessarily a great circle) is always less than 27T. Elliptic and hyperbolic geometry are not the only types of non-euclidean geometry; various other types have been developed; we shall later have occasion to refer to a much more general form of non-euclidean geometry which was likewise devised by Riemann. The fact that these different types of geometry have been developed in modern mathematics shows clearly that mathematics cannot be said to assert the truth of any particular set of geometrical postulates; all that pure mathematics is interested in, and all that it can establish, is the deductive consequences of given sets of postulates and thus the necessary truth of the ensuing theorems relatively to the postulates under consideration. A second observation which likewise shows that mathematics does not assert the truth of any particular set of postulates refers to the status of the concepts in geometry. There exists) in every axiomatized theory, a close parallelism between the treatment of the propositions and that of the concepts of the system. As we have seen, the propositions fall into two classes: the postulates, for which no proof is given, and the theorems, each of which has to be derived from the postulates. Analogously, the concepts fall into two classes: the primitive or basic concepts, for which no definition is given, and the others, each of which has to be precisely defined in terms of the primitives. (The admission of some undefined concepts is clearly necessary if an infinite regress in definition is to be avoided.) The analogy goes farther: Just as there exists an infinity of theoretically suitable axiom systems for one and the same theory-say, euclidean geometry -so there also exists an infinity of theoretically possible choices for the primitive terms of that theory; very often-but not always-different axiomatizations of the same theory involve not only different postulates, but also different sets of primitives. Hilbert's axiomatization of plane geometry contains six primitives: point, straight line, incidence (of a point on a line), betweenness (as a relation of three points on a straight line), congruence for line segments, and congruence for angles. (Solid geometry, in Hilbert's axiomatization, requires two further primitives, that of
Geometry and Empirical Science
1641
plane and that of incidence of a point on a plane.) All other concepts of geometry, such as those of angle, triangle, circle, etc., are defined in terms of these basic concepts. But if the primitives are not defined within geometrical theory, what meaning are we to assign to them? The answer is that it is entirely unnecessary to connect any particular meaning with them. True, the words "point," "straight line," etc., carry definite connotations with them which relate to the familiar geometrical figures, but the validity of the propositions is completely independent of these connotations. Indeed, suppose that in axiomatized euclidean geometry, we replace the over-suggestive terms "point," "straight line," "incidence," "betweenness," etc., by the neutral terms "object of kind 1," "object of kind 2," "relation No.1," "relation No.2," etc., and suppose that we present this modified wording of geometry to a competent mathematician or logician who, however, knows nothing of the customary connotations of the primitive terms. For this logician, all proofs would clearly remain valid, for as we saw before, a rigorous proof in geometry rests on deduction from the axioms alone without any reference to the customary interpretation of the various geometrical concepts used. We see therefore that indeed no specific meaning has to be attached to the primitive terms of an axiomatized theory; and in a precise logical presentation of axiomatized geometry the primitive concepts are accordingly treated as so-called logical variables. As a consequence, geometry cannot be said to assert the truth of its postulates, since the latter are formulated in terms of concepts without any specific meaning; indeed, for this very reason, the postulates themselves do not make any specific assertion which could possibly be called true or false! In the terminology of modern logic, the postulates are not sentences, but sentential functions with the primitive concepts as variable arguments. This point also shows that the postulates of geometry cannot be considered as "self-evident truths," because where no assertion is made, no self-evidence can be claimed. 5.
PURE AND PHYSICAL GEOMETRY
Geometry thus construed is a purely formal discipline; we shall]efer to it also as pure geometry. A pure geometry, then,-no matter whether it is of the euclidean or of a non-euclidean variety-deals with no specific subject-matter; in particular, it asserts nothing about physical space. All its theorems are analytic and thus true with certainty precisely because they are devoid of factual content. Thus, to characterize the import of pure geometry, we might use the standard form of a movie-disclaimer: No portrayal of the characteristics~ of geometrical figures or of the spatial properties or relationships of actual physical bodies is intended, and any
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similarities between the primitive concepts and their customary geometri~ cal connotations are purely coincidental. But just as in the case of some motion pictures, so in the case at least of euclidean geometry, the disclaimer does not sound quite convincing: Historically speaking, at least, euclidean geometry has its origin in the generalization and systematization of certain empirical discoveries which were made in connection with the measurement of areas and volumes, the practice of surveying, and the development of astronomy. Thus understood, geometry has factual import; it is an empirical science which might be called, in very general terms, the theory of the structure of physical space, or briefly, physical geometry. What is the relation between pure and physical geometry? When the physicist uses the concepts of point, straight line, incidence, etc., in statements about physical objects, he obviously connects with each of them a more or less definite physical meaning. Thus, the term "point" serves to designate physical points, i.e., objects of the kind illustrated by pin-points, cross hairs, etc. Similarly, the term "straight line" refers to straight lines in the sense of physics, such as illustrated by taut strings or by the path of light rays in a homogeneous medium. Analogously, each of the other geometrical concepts has a concrete physical meaning in the statements of physical geometry. In view of this situation, we can say that physical geometry is obtained by what is called, in contemporary logic, a semandcal interpretation of pure geometry. Generally speaking, a semantieal interpretation of a pure mathematical theory, whose primitives are not assigned any specific meaning, consists in giving each primitive (and thus, indirectly, each defined term) a specific meaning or designatum. In the case of physical geometry, this meaning is physical in the sense just illustrated; it is possible, however, to assign a purely arithmetical meaning to each -concept of geometry; the possibility of such an arithmetical interpretation of geometry is of great importance in the study of the consistency and other logical characteristics of geometry, but it falls outside the scope of the present discussion. By virtue of the physical interpretation of the originally uninterpreted primitives of a geometrical theory, physical meaning is indirectly assigned also to every defined concept of the theory; and if every geometrical term is now taken in its physical interpretation, then every postulate and every theorem of the theory under consideration turns into a statement of physics, with respect to which the question as to truth or falsity may meaningfully be raised-a circumstance which clearly contradistinguishes the propositions of physical geometry from those of the corresponding uninterpreted pure theory. Consider, for example, the following postulate of pure euclidean geometry: For any two objects x, y of kind 1, there exists exactly one object I of kind 2 such that both x and y stand in rela-
Geomt!try and Empirical Science
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tion No.1 to 1. As long as the three primitives occurring in this postulate are uninterpreted, it is obviously meaningless to ask whether the postulate is true. But by virtue of the above physical interpretation, the postulate turns into the following statement: For any two physical points x, y there exists exactly one physical straight line 1 such that both x and y lie on I. But this is a physical hypothesis, and we may now meaningfully ask whether it is true or false. Similarly, the theorem about the sum of the angles in a triangle turns into the assertion that the sum of the angles (in the physical sense) of a figure bounded by the paths of three light rays equals two right angles. Thus, the physical interpretation transforms a given pure geometrical theory-euclidean or non-euclidean-into a system of physical hypotheses which, if true, might be said to constitute a theory of the structure of physical space. But the question whether a given geometrical theory in physical interpretation is factually correct represents a problem not of pure mathematics but of empirical science; it has to be settled on the basis of suitable experiments or systematic observations. The only assertion the mathematician can make in this context is this: If all the postulates of a given geometry, in their physical interpretation, are true, then all the theorems of that geometry, in their physical interpretation, are necessarily true, too, since they are logically deducible from the postulates. It might seem, therefore, that in order to decide whether physical space is euclidean or non-euclidean in structure, all that we have to do is to test the respective postulates in their physical interpretation. However, this is not directly feasible; here, as in the case of any other physical theory, the basic hypotheses are largely incapable of a direct experimental test; in geometry, this is particularly obvious for such postulates as the parallel axiom or Cantor's axiom of continuity in Hilbert's system of euclidean geometry, which makes an assertion about certain infinite sets of points on a straight line. Thus, the empirical test of a physical geometry no less than that of any other scientific theory has to proceed indirectly; namely, by deducing from tbe basic hypotheses of the tbeory certain consequences, or predictions, which are amenable to an experimental test. If a test bears out a prediction, then it constitutes confirming evidence (though, of course, no conclusive proof) for the theory; otherwise, it disconfirms the theory. If an adequate amount of confirming evidence for a theory has been estab~ lished, and if no disconfirming evidence has been found, then tbe theory may be accepted by the scientist "until further notice," It is in the context of this indirect procedure that pure mathematics and logic acquire their inestimable importance for empirical science: While formal logic and pure mathematics do not in themselves establish any assertions about matters of empirical fact, they provide an efficient and entirely indispensable machinery for deducing, from abstract theoretical
Carl G Hempel
1644
assumptions, such as the laws of Newtonian mechanics or the postulates of euclidean geometry in physical interpretation, consequences concrete and specific enough to be accessible to direct experimental test. Thus, e.g., pure euclidean geometry shows that from its postulates there may be de~ duced the theorem about the sum of the angles in a triangle, and that this deduction is possible no matter how the basic concepts of geometry are interpreted; hence also in the case of the physical interpretaion of euclidean geometry. This theorem, in its physical interpretation, is accessible to experimental test; and since the postulates of elliptic and of hyperbolic geometry imply values different from two right angles for the angle sum of a triangle, this particular proposition seems to afford a good opportunity for a crucial experiment. And no less a mathematician than Gauss did indeed perform this test; by means of optical methods-and thus using the interpretation of physical straight lines as paths of light rays-he ascertained the angle sum of a large triangle determined by three mountain tops. Within the limits of experimental error, he found it equal to two right angles. 6.
ON POINCARE'S CONVENTIONALISM CONCERNING GEOMETRY
But suppose that Gauss had found a noticeable deviation from this value; would that have meant a refutation of euclidean geometry in its physical interpretation, or, in other words, of the hypothesis that physical space is euclidean in structure? Not necessarily; for the deviation have been accounted for by a hypothesis to the effects that the paths of the light rays involved in the sighting process were bent by some disturbing force and thus were not actually straight lines. The same kind of reference to deforming forces could also be used if, say, the euclidean theorems of congruence for plane figures were tested in their physical interpretation by means of experiments involving rigid bodies, and if any violations of the theorems were found. This point is by no means trivial; Henri Poincare, the great French mathematician and theoretical physicist, based on considerations of this type his famous conventionalism concerning geometry. It was his opinion that no empirical test, whatever its outcome, can conclusively invalidate the euclidean conception of physical space; in other words, the validity of euclidean geometry in physical science can always be preserved-if necessary, by suitable changes in the theories of physics, such as the introduction of new hypotheses concerping deforming or deflecting forces. Thus, the question as to whether physical space has a euclidean or a non-euclidean structure would become a matter of convention, and the decision to preserve euclidean geometry at all costs would recommend itself, according to Poincare, by the greater simplicity of euclidean as compared with non-euclidean geometrical theory.
Geometry lind Empirical Science
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It appears, however, that Poincare's account is an oversimplification. It
rightly calls attention to the fact that the test of a physical geometry G always presupposes a certain body P of non-geometrical physical hypotheses (including the physical theory of the instruments of measurement and observation used in the test), and that the so-called test of G actually bears on the combined theoretical system G· P rather than on G alone. Now, if predictions derived from G· P are contradicted by experimental findings, then a change in the theoretical structure becomes necessary. In classical physics, G always was euclidean geometry in its physical interpretation, GE; and when experimental evidence required a modification of the theory, it was P rather than GE which was changed. But Poincare's assertion that this procedure would always be distinguished by its greater simplicity is not entirely correct; for what has to be taken into consideration is the simplicity of the total system G· P, and not just that of its geometrical part. And here it is clearly conceivable that a simpler total theory ih accordance with all the relevant empirica1 evidence is obtainable by going over to a non-euclidean form of geometry rather than by preserving the euclidean structure of physical space and making adjustments only in part P. And indeed, just this situation has arisen in physics in connection with the development of the general theory of relativity: If the primitive terms of geometry are given physical interpretations along the lines indicated before, then certain findings in astronomy represent good evidence in favor of a total physical theory with a non-euclidean geometry as part G. According to this theory, the physical universe at large is a three-dimensional curved space of a very complex geometrical structure; it is finite in volume and yet unbounded in all directions. However, in comparatively small areas, such as those involved in Gauss' experiment, euclidean geometry can serve as a good approximative account of the geometrical structure of space. The kind of structure ascribed to physical space in this theory may be illustrated by an ana10gue in two dimensions; namely, the surface of a sphere. The geometrical structure of the latter, as was pointed out before, can be described by means of elliptic geometry, if the primitive term "straight line" is interpreted as meaning "great circle," and if the other primitives are given analogous interpretations. In this sense, the surface of a sphere is a two-dimensional curved space of non-euclidean structure t whereas the plane is a two-dimensional space of euclidean structure. While the plane is unbounded in all directions, and infinite in size, the spherical surface is finite in size and yet unbounded in all directions: a two-dimensional physicist, travelling along "straight lines" of that'space would never encounter any boundaries of his space; instead, he would finally return to his point of departure, provided that his life span and his technical facilities were sufficient for such a trip in consideration of the size of his
Carl G. Hempel
"universe." It is interesting to note that the physicists of that world, even if they lacked any intuition of a three-dimensional space, could empirically ascertain the fact that their two-dimensional space was curved. This might be done by means of the method of travelling along straight lines; another, simpler test would consist in determining the angle sum in a triangle; again another in determining, by means of measuring tapes, the ratio of the circumference of a circle (not necessarily a great circle) to its diameter; this ratio would turn out to be less than '1i'. The geometrical structure which relativity physics ascribes to physical space is a three-dimensional analogue to that of the surface of a sphere, or, to be more exact, to that of the closed and finite surface of a potato, whose curvature varies from point to point. In our physical universe, the curvature of space at a given point is determined by the distribution of masses...in its neighborhood; near large masses such as the sun, space is strongly curved, while in regions of low mass-density, the structure of the universe is approximately euclidean. The hypothesis stating the connection between the mass distribution and the curvature of space at a point has been approximately confirmed by astronomical observations concerning the paths of light rays in the gravitational field of the sun. The geometrical theory which is used to describe the structure of the physical universe is of a type that may be characterized as a generalization of elliptic geometry. It was originally constructed by Riemann as a purely mathematical theory, without any concrete possibility of practical application at hand. When Einstein, in developing his general theory of relativity, looked for an appropriate mathematical theory to deal with the structure of physical space, he found in Riemann's abstract system the conceptual tool he needed. This fact throws an interesting sidelight on the importance for scientific progress of that type of investigation which the "practicalminded" man in the street tends to dismiss as useless, abstract mathematical speCUlation. Of course, a geometrical theory in physical interpretation can never be validated with mathematical ce~ainty, no matter how extensive the experimental tests to which it is subjected; like any other theory of empirical science, it can acquire only a more or less high degree of confirmation. Indeed, the considerations presented in this article show that the demand for mathematical certainty in empirical matters is misguided and unreasona:"'le; for, as we saw, mathematical certainty of knowledge can be attained only at the price of analyticity and thus of complete lack of factual content. Let me summariz~this insight in Einstein's words: "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality,"
The errors of definitions multiply themselves according as the reckoning proceeds; and lead men mto absurdIties, which at last they see but cannot -THOMAS HOBBES avoid, without reckoning anew from the beginning. Mathematicians are lIke lovers ... Grant a mathematician the least prmciple, and he will draw from it a consequence whIch you must also grant -FONTENELLE him, and from this consequence another.
3
The Axiomatic Method By RAYMOND L. WILDER I
1.
EVOLUTION OF THE METHOD
IF the reader has at hand a copy of an elementary plane geometry, of a type frequently used in high schools, he may find two groupings of fundamental assumptions, one entitled "Axioms," the other entitled "Postulates." The intent of this grouping may be explained by such accompanying remarks as; "An axiom is a self-evident truth." "A postulate is a geometrical fact so simple and obvious that its validity may be assumed." The "axioms" themselves may contain such statements as: "The whole is greater than any of its parts." "The whole is the sum of its parts." "Things equal to the same thing are equal to one another." "Equals added to equals yield equals." It will be noted that such geometric terms as "point" or "line" do not occur in these statements; in some sense the axioms are intended to transcend geometry-to be "universal truths." In contrast, the "postulates" probably contain such statements as: "Through two distinct points one and only one straight line can be drawn." "A line can be extended indefinitely." "If L is a line and P is a point not on L, then through P there can be drawn one and only one line parallel to L." (Some so-called "definitions" of terms usually precede these statements.) This grouping into "axioms" and "postulates" has its roots in antiquity. Thus we find in Aristotle (384-321 B.C.) the following viewpoint: \ Every demonstrative science must start from indemonstrable principles; otherwise, the steps of demonstration .}Vould be endless. Of these -indemonstrable principles some are (a) common to all sciences, otherBirare (b) particular, or peculiar to the particular science; (a) the common principles are the axioms, most commonly illustrated by the aiiom_ that, if equals be subtracted from equals, the remainders are equal. In (b) we have first the genus or SUbject-matter, the existence of which must be assumed. 1 As summarized by T. L. Heath, The Thirteen Books of Euclid's Elements, Cambridge (England), 1908, p. 119. The reader is referred to this book for citations from Aristotle, Procins, et aI. .
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11 In EuclId s Elements (wntten about 300 Be) the two groups occur respectIvely labeled Common notIOns and Postulates From these and a collection of defimtIons Euchd deduced 465 propositions m a logIcal cham Although the actual background for Euchd s work IS not clear apparently he dId not ongmate thIS method of deducmg logIcally from certam unproved propOSItIons gIven at the start all the remammg propo SItlOns As we have Just noted Anstotle and probably other scholars of the penod had a well conceIved notion of the nature of a demonstrative SCIence and the logIcal deductIOn of mathematIcal propOSItions was com mon m Plato s Academy and perhaps among the Pythagoreans Neverthe less the mfluence of EuclId s work has been tremendous probably no other document has had a greater mfluence on SCIentific thought For example modern hIgh school geometnes are usually modeled after EuclId s famous work (m England Eucbd IS stIll used as a textbook) thus ex plammg the still common groupmg mto aXIOms and postulates Also the use m non mathematIcal wntmgs of such phrases as It IS aXIOmatic that and It IS a fundamental postulate of In the sense of somethmg bemg umversal or beyond oppOSItIon IS explamed by thIS tradItIonal use of the terms m mathematlcs The method featured m EuclId s work was employed by ArchImedes (287-212 Be) In hiS two books wmch prOVided a foundatIOn for the SCIence of theoretIcal mechanICS m Book I of thIS treatIse ArchImedes proved 15 propOSItions from 7 postulates Newton s famous Prmclpza first publIshed In 1686 IS orgamzed as a deductIve lIystem m Which the well known laws of motIon appear as unproved propOSItions or postulates gIven at the start The treatment of analytic mechamcs publIshed by Lagrange m 1788 has been conSIdered a masterpIece of logical perfectIOn movmg from exphcIt1y stated pnmary propOSItions to the other propOSI tIons of the system 1 2 There eXIsts a large lIterature devoted to the dISCUSSIOn of the nature of axIOms and postulates and theIr phIlosophIcal background Most of thIS IS mfluenced by the fact that only withm comparatIVely recent years have aXIoms and postulates been very generally employed m parts of mathe matiCS other than geometry Even though the method populanzed by Euchd IS acknowledged now as a fundamental part of the sCIentrfic method m every realm of human endeavor our modern understandmg of axIOms and postulates as well as our comprehensIOn of deductive methods In general has resulted to a great extent from studIes In the field of geometry And smce geometry was conceIved to be an attempt to descnbe the actual phYSIcal space 10 whIch we hve there arose a convIction that aXIoms and postulates possessed a character of logIcal necesszty For example Euchd s fifth postulate (the parallel postulate ) was Let the follOWIng be postu lated that If a straIght hne fallmg on two straIght hnes make the mterIor
The AXIomatic Method
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angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." 2 Proclus (A.D. 410-485) described vividly in his writings the controversy that was taking place in connection with this postulate even in his time; in fact, he argued in favor of the elimination "from our body of doctrine this merely plausible and unreasoned statement." 2 With the renewal of interest in Greek learning during the Renaissance, controversy in regard to the fifth postulate was renewed. Attempts were made to prove the "parallel postulate," often from logical-nongeometrical-principles alone. Surely if a statement is a "logical necessity" . the assumption of its invalidity should lead to contradiction-such was the motivation of much of the work on the postulates of geometry. With the invention of non-euclidean geometries the futility of such attempts became clear. 1.3. The development of the non-euclidean geometries was evidence of a growing recognition of the independent nature of the fifth postulate; that is, this postulate cannot be demonstrated as a logical consequence of the other axioms and postulates in the euclidean system. By a suitable replacement of the fifth postulate, one may obtain the alternative and logically consistent geometry of Bolyai. Lobachevski. and Gauss in which the fifth postulate of Euclid fails to hold. In it appears, for example, the proposition that the sum of the interior angles of a triangle is less than two right angles. Riemann in 1854 developed another non-euclidean geometry, likewise composed of a non-contradictory collection of propositions, in which allUnes are of finite length and the sum of the interior angles of a triangle is greater than two right angles. The invention of the non-euclidean geometries was only part of the rapidly moving developments of the nineteenth century that were to lead to the acceptance of formal geometries apart from those that might be regarded as constituting definitive sciences of space or extension. Grassmann's Ausdehnungslehre, published in 1844 and a critical landmark during this era of changing ideas, was described by its author in these terms: "My Ausdehnungslehre is the abstract foundation for the doctrine of space, i.e., it is free from all spatial intuition, and is a purely mathematical discipline whose application to space yields the science of space. This latter science, since it refers to something given in nature (i.e., space), is no branch of mathematics, but is an application of mathematics to nature." 3 In explanation of Grassmann's concept of a formal science, Nagel writes: "Formal science~ are characterized by the fact that their sole principles of procedure are the rules of logic as well as by the further Quoted from T. L. Heath, op. cit., pp. 154-155, 203. As quoted by E. Nagel, "The formation of modem conceptions of formal logic in the development of geometry," Osiris, vol. 8 (1939), pp. 142-222, pp. 169, 172. 2
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Raymond L. Wilder
fact that their theorems are not 'about' some phase of the existing world but are 'about' whatever is postulated by thought." 3 1.4. The idea expressed by Grassmann is essentially the one held at the present time; that is, a mathematical system called "geometry" is not necessarily a description of actual space. One must distinguish, of course, between the origin of a theory and the form to which it evolves. Geometry, like arithmetic, originated in things "practical," but to assert that any particular type of geometry is a description of physical space is to make a physical assertion, not a mathematical statement. In short, the modern viewpoint is that one must distinguish between mathematics and applications of mathematics. A natural consequence of this change in viewpoint on the significance of a mathematical system was a re-examination of the nature of the basic, unproved propositions. It became clear, for instance, that the euclidean "common notion" that "the whole is greater than the part" has no more of an absolute character than the "parallel postulate" but is contingent upon the meaning of "greater than"; in fact, the proposition may even fail to hold, as in the theory of the infinite. Although there was much discussion as to whether the parallel postulate should be listed as a "postulate" or as a "common notion" (axiom), it was finally realized that neither had any more universality than the other and the distinction might as well be deleted. 4 Accordingly we find in the classical work of Hilbert on the foun~ dations of geometry, 5 published in 1899, that only one name, "axioms," is applied to the fundamental statements or assumptions, and that certain basic terms such as "point" and "line" are left completely undefined. To be sure, Hilbert made a grouping of his axioms-into five groups-but this pertained only to the technical character of the statements, and not to their relative status of "trueness" or "commonness." 1.5. Although this work of Hilbert has come to be regarded by many as the first to display the axiomatic method in its modern form, it should be recognized that similar ideas were appearing in works of his contemporaries. . . . 1.6. . . . Such studies as those of Pasch, Peano, HUbert, and Pieri in euclidean geometry provided a tremendous impetus for investigations of possible formal organizations of the subject matter of this old discipline; these considerations, in turn, provided new understanding of mathematical systems in general and were partly responsible for the remarkable mathematical advances of the twentieth century. . . . 4 For an excellent non-technical description of this "revolution" in thought see E. T. Bell, The Search jor Truth, Baltimore, 1934, chap. XIV. ' 5 Hilbert, Grundlagen der Geometrie, Leipzig, 1899 (published in Festschrift zur Feler der Enthilllung des Gauss-Weber-Denkmals In Gottingen); The Foundations of Geometry, Chicago, 1902. See also the seventh edition of Grundlagen der GeQmetrie Leipzig, 1930. '
The AxiomatIc Method
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It is noteworthy that these early studies in the field of geometry were revealing the great generality that was inherent in formal mathematical systems. Mathematics was evolving in a direction that was to compel the development of a method which could encompass in a single framework of undefined terms and basic statements concepts like group and abstract space that were appearing in seemingly unrelated branches of mathematics. . . . The economy of effort so achieved is one of the characteristic features of modern mathematics. 2. DESCRIPTION OF THE METHOD; THE UNDEFINED TERMS AND AXIOMS
As commonly used in mathematics today, the axiomatic method consists in setting forth certain basic statements about the concept (such as the geometry of the plane) to be studied, using certain undefined technical terms as well as the terms of classical logic. Usually no description of the meanings of the logical terms is given, and no rules are stated about their use or the methods allowable for proving theorems; perhaps these omissions form a weakness of the method. 6 The basic statements are called axioms (or, synonymously, postulates). It is assumed that in proving theorems from the axioms the rules of classical logic regarding contradictions and "excluded middle" may be employed; hence the "reductio ad absurdum" type of proof is in common use. The statements of both the axioms and the theorems proved from them are said to be implied by or deduced from the axioms. An example might be instructive: 2.1. Let us consider again the subject of plane geometry. It will be unnecessary to recall many details. We may perhaps assume, however, that the reader recalls from his high school course that points and straight lines, and such notions as that of parallel lines, were fundamental. Now, if we were going to set forth an axiomatic system for plane geometry in rigorous modem form, we would first of all select certain basic terms that we would leave undefined; perhaps "point" and "line" would be included here (the adjective "straight" can be omitted, since the undefined character of the term "line" enables us to choose to mean "straight line" in our thinking as well as in the later selection of statements for the axioms). Next we would scan the propositions of geometry and try to select certain basic ones with an eye to both their simplicity and their adequacy for proving the ones not selected; these we would call our primary propositions or axioms, to be left unproved in our system. 2.2. To be more explicit, let us proceed as though we were actually carrye We are not here describing the method as used in modern mathematical logic or the formalistic treatises of Hilbert and his followers, where the rules for operations with the basic symbols and formulas are (of necessity) set forth in the language of ordinary discourse.
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ing out the above procedure; although we do not intend to give a complete system of axioms, a miniature sample of what the axioms and secondary propositions or theorems might be like, together with sample proofs of the latter, follows: Undefined terms: Point; line. Axiom 1. Every line is a collection of points. Axiom 2. There exist at least two points. Axiom 3. If p and q are points, then there exists one and only one line containing p and q. Axiom 4. If L is a line, then there exists a point not on L. Axiom 5. If L is a line, and p is a point not on L, then there exists one and only one line containing p that is parallel to L. These axioms would not by any means suffice as a basis for proof of all the theorems of plane geometry, but they will be sufficient to prove a certain number of 'the theorems found in any organization of plane geometry. Their selection is motivated as follows: In the first place, the undefined terms "point" and "line" are to playa role like that of the variables in algebra. Thus, in the expression x 2 - y2
= (x -
y)(x
+ y)
the x and yare undefined, in the sense that they may represent any individual numbers in a certain domain of numbers (as for instance the domain of ordinary integers). In the present instance, "point" may be any individual in a domain sufficiently delimited as to satisfy the statements set forth in the axioms. On the other hand, "line," as indicated in Axiom 1, has a range of values (= meanings) limited to certain collections of the individuals that are selected as "points." Thus Axiom 1 is designed to set up a relationship between the undefined entities point and line. It is not a definition of line, since (if the study is carried through) there will be other collections of points (circles, angles, etc.) that are not Jines. Furthermore, it enables us, as we shall see presently, to define certain terms needed in the statements of the later axioms. Axiom 2 is the first step toward introducing lines into our geometry, and this is actually accomplished by adding Axiom 3. Before the latter can have meaning, however, we need the following formal definition: 2.3. Definition. If a point p is an element of the collection of points which constitutes a line L (cf. Axiom 1), then we say, variously, that L contains p, p is on L, or L is a line containing p. Having stated Axioms 2 and 3, we would have that there exists a line in our geometry, but in order to have plane geometry and not merely a line or "one-dimensional" geometry we would have to say something to insure that not all points lie on a single line; Axiom 4 is designed to accomplish this. We would now imagine, intuitively (since we have a line L,
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a point p not on that line, and also a line through p and each point q of L), that we have practically a plane; however, so far as euclidean geometry is concerned, we have not provided, in Axioms 1-4, for the parallel to L through p until we have stated Axiom 5. And of course Axiom 5 is not significant until we have the definition: 2.4. Definition. Two lines Ll and L z are called parallel if there is no point which is on both Ll and L 2 • (We may also call Ll "parallel to" L 2 • or conversely.) 2.S. Let us denote the above set of five axioms, together with the undefined terms point, line, by r and call it the axiom system r. (We shall also frequently use the t.erm "axiom system" in a broader sense to include the theorems, etc., implied by the axioms.) For future purposes we note two aspects of r, but we shall not go into these fully at this point: (1) In addition to the geometrical ("technical") undefined terms point, line, we have used logical ("universal") undefined terms such as collection, there exist, one, every, and not. (2) That r is far from being a set of axioms adequate for plane geometry may be shown as follows: Since point and line are left undefined, we are at liberty to consider possible meanings for them, subject of course to the restriction that we take into account the statements made in the axioms. If we have been educated in the American or English school systems, our reactions to these terms will no doubt immediately be specialized, our geometric experience in the schools having the upper hand in our response. But let us imagine that the terms are entirely unfamiliar, although the logical terms used in the axioms are not unfamiliar, so that we may consider other possible meanings for point and line. Unquestionably this will involve considerable experimentation before suitable meanings are found. For example, we might first try letting "point" mean book and "line" mean library; we know from the statement in Axiom 1 that a line is a collection of points, and libraries form one of the most familiar col1ections in our daily experience. We can imagine that we live in a city, C, which has two distinct libraries, and that by library we mean either one of the libraries of C, and by book anyone of the books in these two libraries. Axiom 2 becomes a valid statement: "There exist at least two books." However, Axiom 3 fails, since, if p and q designate books in different libraries, then there is no library that contains p and q. However, before trying other meanings for point and line, we notice that Axioms 4 and 5 are valid, becoming, respectively, "If L is a library, then there exists a book not on (i.e., in) L," and "If L is a library and p a book not on (i.e., in) L, then there exists one and only one library containing p that is • parallel to (has no books in common with) L." 7 '1 In parentheses we have placed the terms commonly employed in connection with libraries and books that are indicated by our definition of "on" and "parallel."
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Now, impressed by our failure to satisfy Axiom 3 on our first attempt at meanings for "point" and "line," we may, with an eye on Axiom 3, try to imagine a community, which we denote by Z, of people in which everyone belongs to some club, but in such a manner that, if p and q are two persons in Z, then there is one and only one club of which p and q are both members. In other words, we may try letting "point" mean a person in Z and "line" mean a club in Z, and imagine that the club situation in Z is such that the statement just made is valid, so that Axiom 3 is satisfied. We would then have no difficulty in seeing that Axioms 1, 2, and 4 are satisfied: "A club in Z is a collection of people in Z"; "There exist at least two people in Z"; etc. However, Axiom 5 becomes (with suitable change of wording to suit the new meanings): "If L is a club in Z, and p is a person in Z not in the club L, then there exists one and only one club in Z of which p is a member and which has no members in common with L." This is a statement which apparently makes a rather strong convention regarding the club situation in Z, and which may conceivably fail to apply; in any case, the stipUlation that only one club have a given pair of persons as members can hardly be expected to suffice for Axiom 5! To clinch the matter suppose that Z is a "ghost" community, there being only three persons, whom we shall designate by a, band e respectively, living in Z; and that as a result of certain circumstances each pair ab, be, and ae shares a secret from the third member of the community, so that we may consider this bond between each two as forming them into a club ("secret society") excluding the third member. Now, with the meaning of point and line as before, we see that Axioms 1-4 hold but Axiom 5 does not hold. Before rejecting the latter attempt as impossible, however, let us imagine that Z has four citizens: a, b, c, and d. And suppose that each pair of these people forms a club excluding the other two members of the community; that is, there are six club consisting of ab, ae, ad, be, bd, and ed. Now all the axioms of r are satisfied with the meanings person in Z for "point" and club in Z for "line"! And we may then notice that we could arrive at a similar example by taking any collection Z of four things a, b, e, and d, and, by letting "point" mean a member or element of the collection Z, and "line" mean any pair of elements of Z, satisfy the statements embodied in the axioms of r. 2.6. Although we may experience no particular thrill at this discoverymay, rather, begin to feel that it is a rather trivial game we are playing in toying with possible meanings for the system r-we might conceivably be beguiled into seeking an answer to questions such as: How many "points" must a collection have in order to serve as the basis for an example satisfying the statements in r? For a given collection at hand, how many "points" must a "line" have in order to satisfy r? (For example, a "line" above could not have consisted of three persons in Z in the case
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where Z has exactly four citizens.) Furthermore, if we have already a general knowledge of, or experience with, plane geometry, the above example shows us that r is far from being a sufficient basis for euclidean geometry; certainly an adequate set of axioms for plane geometry would exclude the possibility of the geometry permitting a set of only four points satisfying all the axioms. Before proceeding any further with this general discussion, however, let us notice how theorems would be proved from such a system as r.
3.
DESCRIPTION OF THE METHOD; THE PROVING OF THEOREMS
Having set down a system, such as r for instance, we then proceed to see what statements are implied, or can be proved or deduced from the system. Contrary to the manner in which we proceeded in high school, when we brought in all kinds of propositions and assumptions not included in the fundamental terms and axioms (such as "breadth"; "a line has no breadth"), and even drew diagrams and pictures embodying properties that we promptly accepted as part of our equipment,S we take care to use only points and lines, and those relations and properties of points and lines that are given in the axioms. (Of course, after we have proved a statement, we may use it in later proofs instead of going back to the axioms and proving it all over again.) There is no objection to drawing diagrams, provided they are used only to aid in the reasoning process and do not trick us into making assumptions not implied by the axioms; indeed, the professional mathematician uses them constantly. . . . 3.1. Consider the following formal theorem and proof: Theorem 1. Every point is on at least two distinct lines. Proof. Let p denote any point. Since by Axiom 2 there exist at least two points, there must exist a point q distinct from p. And by Axiom 3 there exists a line L containing p and q. Furthermore, by Axiom 4 there exists a point r not on L, and (again by Axiom 3) a line K containing p and r. Now by Axiom 1 every line is a collection of points. Hence, for two lines to be distinct (i.e., different), the two collections which constitute them must be different; or, what amounts to the same thing, one of them must contain a point that is not on the other. The lines Land K are distinct, then, because K contains the point r which is not on L. As p is on both Land K, the theorem is proved. 3.2. Now it will be noticed that we have used Axioms 1-4 in the proof, 8 A classical example may be found in the well-known "proof" that all triangles are isosceles, which IS based on a diagram that deceives the eye by placing a certain point within an angle instead of outside, where rigorous reasoning about the situation would place it. This may be found in J. W. Young, Lectures on Fundamental Concepts of Algebra and Geometry, New York, 1916, pp. 143-145.
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but not Axiom 5. We could, then, go back to the example of the community Z, let "point" mean person in Z and "line" mean pair of persons in Z, rephrase Axioms 1-4 in these terms, and carry through the proof of Theorem 1 in these terms. That is, Theorem 1 is a "true" statement about any example, such as Z, w:qich satisfies the statements embodied in Axioms 1-4 of r. In proving Theorem 1, then, we have in one step proved many different statements about many different examples, namely, the statements corresponding to Theorem 1 as they appear in the different examples that satisfy Axioms 1-4 of r. This [is an important] aspect of the "economy" achieved in using the axiomatic method. . . . If, because of some diagram or other aid to thought, we had used some property of point or line not stated in Axioms 1-4, we could not expect to make the above assertions, and the "economy" cited would be lost! Note, too, that Theorem 1 will remain valid in any axiom system (such as r) that contains the undefined terms point and line as well as Axioms 1-4. In particular, it is valid for euclidean plane geometry, which is only one of the possible geometries embodying these four axioms, and which, as we stated before, would require many more axioms than those stated above. 3.3. Now consider the following statement, which we call a corollary of Theorem 1: Corollary. Every line contains at least one point. 3.4. Before considering a proof, we hasten to meet an objection which the "uninitiated" might make at this point; to wit, since Axiom 1 explicitly states that a line is a collection of points, of course every line contains at least one point, so why should this be repeated as a corollary of Theorem 1? This is not a trivial matter, and it leads directly to a question which causes considerable concern in modern mathematics, namely, what is meant by collection? We said above that "collection" is an undefined logical term, and as such we took it for granted that its use is universally understood and employed, just as the word "the" is universally understood and used by anyone familiar with the English language. But now we find ourselves almost immediately in need of explaining the use of the term in the above corollary. However, there is nothing so very astonishing about this if we reflect that, whenever we try to make very precise a term in ordinary use, it is usually necessary to adopt certain conventions. For example, such terms as vegetable, fruit, animal are commonly "understood" and used by anyone who habitually uses the English language, but, when we come to apply them to certain special objects, it is frequently necessary to agree on some convention; as, for example, that a certain type of living substance shall be called "animal" rather than "fish" (e.g., whale). So, for instance, we
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may want to make the convention that, if person A wishes to talk about "the collection of all coins in B's pockets," he may do so even though person B is literally penniless! In other words, no matter whether there actually are coins in B's pockets or not, the collection of all such coins is to be regarded as an existing entity; we call the collection empty if B has no coins. (In case B is penniless, we may also talk about "the coin in B's pocket," but in this case there is no existing entity to which the phrase refers.) And this is the convention that is generally agreed on throughout mathematics and logic, namely, that a collection may "exist," as in the case of the collection of an coins in B's pockets, even though it is empty. . . . 3.5. Proof of corollary to Theorem 1. There exists a point p by Axiom 2, and by Theorem 1 there exist two distinct lines Ll and L2 containing p. Now, if there exists a line L that contains no points, then both L1 and L2 are parallel to L (by definition). As this would stand in contradiction to Axiom 5, it follows that there cannot exist such a line L. 3.6. A statement "stronger" than the above corollary is embodied in the next theorem: Theorem 2. Every line contains at least two points. Proof. Let L be any line. By the above corollary, L contains a point p. To show p not the only point on L, we shan use a "proof by contradic· tion." Suppose p is the only point that L contains. By Theorem 1 there is another line L t containing p. Now Ll must contain at least one other point, q; for otherwise Land L1 would each contain only p, hence be the same collection of points and ergo the same line (Axiom 1). By Axiom 4 there is a point x which is not on L 1 • and by Axiom 5 there is a line L2 containing x and parallel to Lt. But both Land Ll are lines con· taining p and parallel to L 2, in violation of Axiom 5. We must conclude, then, that the supposition that p is the only point on L cannot hold and hence that L contains at least two points. Now, since by Theorem 2 every line contains at least two points, and since by Axiom 3 two given points can lie simultaneously on only one line, we can state: Coronary (to Theorem 2). Every line is completely determined by any two of its points that are distinct. 3.7. Theorem 3. There exist at least four distinct pOints. Proof. By Axiom 2 there exist at least two distinct points p and q. By Axiom 3 there exists a line L containing p and q, and by Axiom 4 there exists a point x not on L. By Axiom 5 there exists a line L1 containing x and parallel to L, and by Theorem 2 L1 contains at least two distinct points (cf. Definition 2.4). 3.8. Theorem 4. There exist at least six distinct lines.
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Before proving Theorem 4, we perhaps need to make sure that the meaning of another one of our "common" terms is agreed upon, namely the word "distinct." As we are using the term, two collections are distinct if they are not the same. Thus, the lines Land L1 which figure in the proof of Theorem 2 are distinct, although under the supposition made there L1 contains L, for they are not the same line (L1 contains q and L does not). Proof of Theorem 4. We proceed, as in the proof of Theorem 3, to obtain the line L containing the points p and q, and the line L1 parallel to L containing two distinct points (Theorem 2) x and y. By Axiom 3 there exist lines K and K1 determined respectively by the pairs (p, x), (q, y). Now the point q is not on K, else by Axiom 3 K and L would Be the same line (which is impossible since x is not on L). Also, y is not on K, else K and Ll would be the same line. Similarly, p is not on K1 and x is not on K 1• Now there also exist lines M and Ml determined respectively by the pairs (p, y), (q, x); and we can show that q is not on M, x is not on M, p is not on M 1 , and y is not on MI' It follows that no two of the lines L. L 1 • K, K I , M, M 1 are the same. 4.
COMMENT ON THE ABOVE THEOREMS AND PROOFS
If the reader has followed the proofs given above, he has probably resorted to the use of figures by this time! This would be quite natural, since in high school geometry he used figures; and they help to keep the various symbols (L, P. q, ... ) and their significance in mind. However, as we stated above, no special meanings have been assigned to "point" and "line," and consequently the above proofs should, and do, hold just as well if the reader uses coins for "points" and pairs of coins for "lines." As a matter of fact, if any collection of four objects is employed, "point" meaning any object of the collection and "line" any pair of the objects, then the reader may follow the above proofs with these meanings in mind. Of course, the theorems we have stated in the preceding sections are not by any means all the theorems that we might state. For example, we can show that any collection of objects satisfying the axioms of the system r must, if not infinite as in ordinary geometry, satisfy certain conditions regarding the number of points (there cannot be just 5 points in the collection, for instance), and that there must be a relation between the number of lines and the number of points in the collection. In fact, we can continue the above study to a surprising extent; we could hardly expect to reach a point where we could confidently assert that no more theorems could be proved. It is not our intention to extend the number of theorems, however, since we believe that we have already obtained
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enough theorems and proofs to serve as specimens for our later purposes. 4.1. As a useful terminology in what follows, let us agree that, when we use the term "statement" in connection with an axiom system ~, we shall mean a sentence phrased, or phrasable, in the undefined terms and universal terms of ~; such a statement may be called a "$.-statement. Thus the axioms of r are r-statements (Axiom 5 contains the word "parallel," but this is "phrasable" in the undefined terms and universal terms), as are also the theorems. 4.2. In conformance with the conventions made in Section 2, we shall say that an axiom system ~ implies a statement S if S follows by logical argument, such as used above, from ~. In particular, each axiom is itself implied, trivially. We shall also say S is logically deducible from ~ if t implies S. 4.3. In the course of our work above we had to pause in two instances to explain the conventions we were making in regard to the use of two words commonly used in ordinary discourse, namely "collection" and "distinct." These words were left undefined, to be sure, in the sense that they are supposedly universally understood non-technical terms; but, as we discovered, not so "universal" but that it was felt advisable to give some conventions we were making in regard to their use here! On the other hand, the words ''point'' and "line" we left strictly undefined, saying that any meaning whatsoever could be assigned to them as long as these meanings were consistent with the statements embodied in the axioms. We saw that the "collection library," "point book" meanings were not permissible, but that, if C is any collection of four objects, then "point object of C," "line pair of objects of C" are permissible meanings. The terms "point," "line," "parallel," etc., we may call technical terms of the system, the terms "point" and "line" being the undefined technical terms. The terms "collection," "distinct" might be called universal terms or logical terms. Other examples of universal terms in r are "exist" (in Axiom 2), "one" (Axiom 3), "two" (Theorem 1), "four" (Theorem 3), "six" (Theorem 4), "and" (Axiom 3), "or" (Definition 1), "not" (Axiom 4), and "every" (Theorem 1). However, if we were setting up an axiom system for the elementary arithmetic of integers (1 + 2 = 3, 2 X 2 = 4, etc.), we might use a term like "one" as an undefined technical term. Thus the same term may have different roles in different axiom systems! . . . As the term "exist" is used above, it is chiefly permissive so far as proofs are concerned, and stipulative for examples; thus in the proof of Theorem 3 we were permitted to introduce the line Ll by virtue of Axiom 5, and the example of the "ghost community" containing only three persons failed because it could not meet the stipulation concerning the existence of a certain line parallel to another line which is made in Axiom 5.
=
=
=
=
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5.
SOURCE OF THE AXIOMS
Let us consider more fully the source of the statements embodied in the axioms. We chose axioms for geometry in our example r since we felt that we could assume the reader had studied some elementary geometry in high school. That is, we were careful to pick a subject already familiar! The undefined technical terms "point" and "line" already have a meaning of some sort for us. And . . . this is the usual way in which axioms are obtained; they are statements about some concept with which we already have some familiarity. Thus, if we are already familiar with arithmetic, we might begin to set down axioms for arithmetic. Of course the method is not restricted to mathematics. If we are familiar with some field such as physics, philosophy, chemistry, zoology, economics, for instance, we might choose to set down some axioms for it, or a portion of it, and see what theorems we might logically deduce from them. 9 We may say, then, that an axiom, as used in the modem way, is a statement which seems to hold for an underlying concept, an axiom system being a collection of such statements about the concept.
Thus, in practice, the concept comes first, the axioms later. Theoretically this is not necessary, of course. Thus, we may say "Let us take as undefined terms aba and daba, and set down some axioms in these and universal logical terms." With no concept in mind, it is difficult to think of anything to say! That is, unless we first give some meanings to "aba" and "daba"-that is, introduce some concept to talk about-it is difficult to find anything to say at all. And, if we finally do make some statements without first fitting a suitable concept to "aba" and "daba," we shall, very likely, make statements that contradict one another! The underlying concept is not only a source of the axioms, but it also guides us to consistency (about which we shall speak directly). Thus, we select the concept; then we select the terms that are to be left undefined and the statements that are to form our axioms; and finally we prove theorems as we did above. This is a simplification of the process, to be sure, but in a general way it describes the method. It is to be noted how the procedure, as so formulated, differs from the classical use of the method. In the classical use the axioms were regarded as absolute truthsabsolutely true statements about material space-and as having a certain character of necessity. To have stated the parallel axiom, Axiom 5 above, was to have stated something "obviously true," something one took for granted if one had thought about the character of the space in which he lived. It would have been inconceivable before the nineteenth century to state an axiom such as "If L is a line and p a point not on L, then there I
9 As an example in genetics and embryology, see J. H. Woodger, The Axiomatic Method in Biology, Cambridge (England), 1937.
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exist at least two distinct lines containing p and parallel to L." To have in mathematics, simultaneously, two axiom systems r 1 and r 2 with axioms in r 1 denying axioms in r 2 as is the case in mathematics today with the euclidean and non-euclidean geometries, would also have been inconceivable! But, if we take the point of view that an axiom is only a statement about some concept,lO so that axioms contradicting one another in different systems only express basic differences in the concepts from which they were derived, we see that no fundamental difficulty exists. What is important is that axioms in the same system should not contradict one another. This brings us to the point where we should discuss consistency , and other characteristics of an axiom system. 5.1.
REMARK
The derivation of an axiom system for non-euclidean geometry from axioms for euclidean geometry, using the device of replacing the parallel . axiom by one of its denials, is an example of another manner in which new axiom systems may be obtained. In general, we may select a given axiom system and change one or more of the axioms therein in suitable manner to derive a new axiom system. II
ANALYSIS OF THE AXIOMATIC METHOD [When the undefined terms and primary propositions or axioms of a system have been selected at least three relevant questions suggest themselves: (1) Is the system suited to the purposes for which it was set up? (2) Are the axioms truly independent, Le., are any of them provable from the others (in which case they should perhaps be deleted from the system and transferred to the body of theorems to be proved)? (3) Does the system imply any contradictory theorems (if so, this defect must be eliminated if the theorems are to be relied on)? Of these questions, the third, relating to contradictoriness, is by far the most fundamental and' critical. In the selection below, a continuation of the preceding discussion, I have excerpted Wilder's analysis of the consistency and independence of an axiom system. ED.]
1.
CONSISTENCY OF AN AXIOM SYSTEM
From a logical point of view we can make the following definition: 1.1. Definition. An axiom system ~ is called consistent if contradictory statements are not implied by ~. Now this definition gives rise to certain questions and criticisms. In the first place, given an axiom system ~, how are we going to tell whether 10 It is only in this sense--that an axiom is a statement true of some concept-that the word "true" can be used of an axiom.
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it is consistent or not? Conceivably we might prove two theorems from "S which contradict one another, and hence conclude that ~ is not con-
sistent. For example, if we added to the system r, discussed above, the new axiom, "There exist at most three points," it would become apparent, as soon as Theorem 3 of r was proved, that the new system of axioms is not consistent. But, supposing that this does not happen, are we going to conclude that ~ is consistent? How can we tell that, if we continued stating and proving theorems, we might not ultimately arrive at contradictory statements and hence inconsistency? We remarked about the system r that we could hardly expect to reach a point where we could say with confidence that no more theorems could be stated. And, unless we could have all possible theorems in front of our eyes, capable of being scanned for contradictions, how could we assert that the system is consistent? We are immediately faced with the problem: Is there any procedure for proving a system of axioms consistent? And, if so, on what basis does the proof rest, since the proof cannot be conducted within the system as in the case of the theorems of the system? Another difficulty would arise from the fact that it might be very hard to recognize that a contradiction is implied, when such is the case. There are examples in mathematical literature of cases where considerable material was published concerning systems which later were found to be inconsistent. Until someone suspected the inconsistency and set out to prove it, or (as in some cases) stumbled upon it by chance, the systems seemed quite valid and worth while. It Can also happen, for example, that the theorems become so numerous and complicated that we fail to detect a pair of contradictory ones. For example, although two theorems might really be of the form "S" and "not S" respectively, because of the manner in which they are stated it might escape our attention that they contradict one another. In short, the usefulness of the above definition is limited by our ability to recognize a contradiction even when it is staring us in the face, so to speak. The former objection, that hinging upon the probable impossibility of setting down all theorems implied by the system, is the more serious from the point of view of the working mathematician. And as a consequence the mathematician usually resorts to the procedure described below: Let us make the definition: 1.2. Definition. If ~ is an axiom system, then an interpretation of ~ is the assignment of meanings to the undefined technical terms of ~ in such a way that the axioms become true statements for all values of the variables (such as p and q of Axiom 3, I 2.2, for instance). This definition requires some explanation. First, as an example we can
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cite the system r above and the meanings "point" == anyone of a collection of four coins and "line" = any pair of coins in this collection. The axioms now become statements about the collection of coins and are easily seen to be true thereof. Hence this assignment of meanings is an interpretation of r. As the axioms stand, with "point" and "line" having no assigned meanings, they cannot be called either true or not true. (Similarly, we cannot speak of the expression "x 2 - y2 = (x - y) (x + y)" as being either true or false until meanings, such as "x and y are integers," are assigned.) But, with the meanings assigned above, they are true statements about a "meaningful" concept. As a rule,' we shall use the word "model" to denote the result of the assignment of meanings to the undefined terms. Thus the collection of four coins, considered a collection of points and lines according to the meanings assigned above, is a model of r. Generally, if an interpretation I is made of an axiom system, we shall denote the model resulting from I by £'@ (I) . For some models of an axiom system !, certain axioms of ! may be vacuously satisfied. That is, axioms of the form "If . . ., then . . . ," such as Axiom 3 of r, which we might call "conditional axioms," may be true as interpreted only because the conditional "If . . ." part is not fulfilled by the model. Suppose, for example, we delete Axioms 2 and 4 from r and denote the resulting system by I". Then a collection of coins containing just one coin is a model for r', if we interpret "point" to mean coin and "line" to mean a collection containing just one coin. For in this model the "If ..." parts of Axioms 3 and 5 are not fulfilled. (Note that, for Axiom 3 to be false of a model £'@, there must be two points p and q in 9lfJ such that either no line of 9lfJ contains p and q or more than one line of 9lfJ contains p and q.) This may be better illustrated, perhaps, by the following digression: Suppose boy A tells girl B, "If it happens that the sun shines Sunday, then I will take you boating." And let us suppose that on Sunday it rains all day, the sun not once peeping out between the clouds. Then, no matter whether A takes B boating or not, it cannot be asserted that he made her a false promise. For his promise to have been false, (1) the sun must have shone Sunday and (2) A must not have taken B boating. And, in general, for a statement of the form "If . . . , then . . ." to be false, the "If ... " condition must be fulfilled and the "then . . ." not be fulfilled. Now we did not have in mind a collection of four coins when we set down the axioms of r. We were thinking of something entirely different, namely euclidean geometry as we knew it in high school. "Point" had for us then an entirely different meaning-something ''without length, breadth, or thickness"; and "line" meant a "straight" line that had "length, but no breadth or thickness." Do not these meanings also yield a model of r-what we might perhaps call an "ideal" model? We may admit that
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this is so, and we resort frequently in mathematics to such ideal modelsalways, of course, when it happens that every collection of objects serving as a model must of necessity be infinite in number. (Such is the case, for instance, when we have enough axioms in a geometry to insure an infinite number of lines.) We return to this discussion later (see 2.3); at present, let us go on to the so-called "working definition" of consistency: 1.3. Definition. An axiom system l: is satisfiable if there exists an interpretation of~. Now what is the relation between the two definitions 1.1 and 1.31 What we actually want of any axiom system is that it be consistent in the sense of 1.1. But we saw that 1.1 was not a practicable definition except in cases where contradictory statements are actually found to be implied by the system and inconsistency is thus recognized. Where a system is consistent, we are usually unable to tell the fact from 1.1. But, as in the case of the four-coin interpretation of r above, we have a very simple test showing "satisfiability" in the sense of 1.3. Does this imply consistency in the sense of 1.1? The mathematician and the logician take the point of view that it does, and, in order to explain why, we have to go into the domain of iogic for a few moments. 2.
THE PROOF OF CONSISTENCY OF AN AXIOM SYSTEM
The Law of Contradiction and the Law of the Excluded Middle. First let us recall two basic "laws" of classical (i.e., Aristotelian) logic, namely the Law of Contradiction and the Law of the Excluded Middle; the latter is also called the Law of the Excluded Third ("tertium non datur"). These are frequently, and loosely, described as follows: If S is any statement, then the Law of Contradiction states that S and a contradiction (Le., any denial) of S cannot both hold. And the Law of the Excluded Middle states that either S holds or a denial of S holds. For example, let S be the statement "Today is Tuesday." The Law of Contradiction certainly holds here, for today cannot be both Tuesday and Wednesday, for example. And the Law of the Excluded Middle states that either today is Tuesday or it is not Tuesday. But "things are not so simple as they seem" here. Unless one limits himself to a specified point on the earth (or parallel of longitude), it can be both Tuesday and Wednesday at the same time! And, without including in S such a geographical provision, the statement "Either today is Tuesday or it is not Tuesday" can hardly be accepted. As a matter of fact, whenever such statements are made there usually exists a tacit understanding between speaker and listener that their locale at the time is the place being referred to. Or consider the statement "The king of the United States wears bow
The AXIomatic Method
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ties." Does the Law of the Excluded Middle hold here? Or let S be the statement "All triangles are green." The upshot of this is merely that, although these "laws" are called "universally valid," some sort of qualifications have to be made with regard to their applicability in order for them to have validity. In so far as axiomatic systems are concerned, the problem is not so great, since we can restrict our use of the term "statement" to the convention already made in section 1 4.1 ("~-statement"). And this will be our understanding from now on. 2.2. As soon as an interpretation of a system ~ is made, the statements of the system become statements about the resulting model. Let us assume the following, which may be considered basic principles of applied logic. 2.2.1. All statements implied by an axiom system ~ hold true for all models of ~; 2.2.2. The Law of Contradiction holds for all statements about a model of an axiom system !, provided they are !-statements whose technical terms have the meanings given in the interpretation. We can mit~e this clearer and more precise by introducing the notion of an I·~·statement: 2.2.3. If ~ is an axiom system and 1 denotes an interpretation of !, then the result of assigning to the technical terms in a ~-statement their meanings in 1 will be called an I-~-statement. Then 2.2.1 and 2.2.2 become respectively: 2.2.1. Every I-!-statement, such that the corresponding !-statement is implied by!, holds true for £@(I) (ct. 1.2); 2.2.2. Contradictory I -l,-statements cannot both hold true of 9@(1). Under the assumption that 2.2.1 and 2.2.2 hold, satisfiability implies consistency. For, if an axiom system! implies two contradictory !-statements, then by 2.2.1 these statements as I-!-statements hold true for the model ~ (I); but the latter is impossible by 2.2.2. Hence we must conclude that, if 2.2.1 and 2.2.2 are valid, then the existence of an interpretation for an axiom system ! guarantees the consistency of ~ in the sense of 1.1. And this is the basis for the "working definition" 1.3. For example, the exi~tence of the "four-coin interpretation" of the system r above guarantees the consistency of r if we grant 2.2.1 and 2.2.2. The reader will have noticed that we have not proved that consistency in the sense of 1.1 implies satisfiability. To go into this question would be impractical, since it would necessitate going into detail concerning formal logical systems and is too complicated to describe here. 2.3. In Section 1 we used the term "ideal" model, by way of contrast to such models as that of the four coins for r; the latter might be termed a "concrete" model. It was pointed out that. whenever an axiom system ~ requires an infinite collection in each of its models, then of necessity the models are "idea1."
Raymond L. Wilder
This raises not only the question as to how reliable are "ideal" models, but also the question as to what constitutes an allowable model. What we should like, of course, is a criterion which would allow only models that satisfy assumptions 2.2.1, 2.2.2, and especially the latter. If there is any danger that an ideal model may require such a degree of abstraction that it harbors contradictions in violation of 2.2.2, then clearly the use of models is no general guarantee of consistency in spite of what we have said above. Further light can be shed on this matter by a consideration of wellknown examples. It is not an uncommon practice, for instance, to obtain a model of an axiom system l in another branch of mathematics-even in a branch of mathematics that is, in its turn, based on an axiom system l'. How valid are such models? Do they necessarily satisfy 2.2.21 For example, to establish the consistency of a non-euclidean geometry we give a model of it in euclidean geometry. (See Richardson [Fundamentals of Mathematics, New York, 1941, pp. 4] 8-19] for instance.) But suppose that the euclidean geometry harbors contradictions; what then? Evidently all we can conclude here is that, if euclidean geometry is consistent, then so is the non·euclidean geometry whose model we have set up in the euclidean framework. We are forced to admit that in such cases we have no absolute test for consistency, but only what we may call a relative consistency proof. The axiom system l' may be one in whose consistency we have great confidence, and then we may feel that we achieve a high degree of plausibility for consistency, but in the final analysis we have to admit that we are not sure of it.ll, .. 3. INDEPENDENCE OF AXIOMS Earlier we mentioned "independence" of axioms. By "independence" we mean essentially that we are "not saying too much" in stating our axioms. For example, if to the five axioms of the system r (I 2.2) we added a sixth axiom stating "There exist at least four points,O' we would provide no new information inasmuch as the axiom is already implied by r (see Theorem 3 of I 3.7). Of course the addition of such an axiom would not destroy the property of consistency inherent in r. 3.1. In order to state a formal definition of independence, let l denote an axiom system and let A denote one of the axioms of ~. Let us denote some denial of A by ,....,A, and let ~ - A denote the system l with A deleted. If S is any l>statement, ~ + S will mean the axiom system con11 In one well-known case, the system 1;' is a subsystem of 1;; viz" the Godel proof (Godel, The Conszstency of the Axiom of Choice and the Generalized Contmuum Hy-pothesis with the Axioms of Set Theory, Princeton, 1940) of the relative consistency of the axiom of choice when adjomed to the set theory axioms.
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1667
taining the axioms of :s and the statement S as a new axiom. Then we define: 3.1.1. Definition. If ! is an axiom system and A is one of the axioms of I, then A is called independent in !, or an independent axiom of :S, if both ! and the axiom system (~ - A) + -A are satisfiable. 3.2. Just which of the many forms of -'A is used is immaterial. Thus Axiom 5 is independent in r (1 2.2) if r is satisfiable and if the first four axioms of r togther with a "non-euclidean" form of the axiom constitute a satisfiable system. For example, for a denial of Axiom 5 take the statement: "There exist a line L and point p not on L, such that there does not exist a line containing p and paral1el to L." To show that the system r with Axiom 5 replaced by this statement forms a satisfiable system, let us take a collection of three coins, let "point" mean a coin of this collection, and "line" mean any pair of points of this collection. Then we have an interpretation of the new system, showing it to be satisfiable. We have already ascertained (2.2) that r is satisfiable, and so we conclude that Axiom 5 is independent in r. 3.3. The reader will probably gather by this time that the reason for specifying the satisfiability of I. in Definition 3.1.1, is to insure that some -A is not a necessary consequence of the axioms of t - A; for, if it were, we would not wish to call A "independent." And, as the definition is phrased, it insures that neither A nor any denial, -A, of A is implied by the system! - A, so that the addition of A to ! - A is really the supplying of new information. 3.4. Actually, however, we do not place the same emphasis on independence as we do on consistency. Consistency is always desired, but there may be cases where independence is not desired. . . . Generally speaking, of course,.it is preferable to have all axioms independent; but, if some axiom turns out not to be independent, the system is not invalidated. As a matter of fact, some well-known and important axiom systems, when first published, contained axioms that were not independent (a fact unknown at the time to the authors, of course). An example of this is the original formulation of the set of axioms for geometry given by Hilbert in 1899. This set of axioms contained two axioms which were later discovered to be implied by the other axioms. 12 This in no way invalidated the system; it was only necessary to change the axioms to theorems (supplying the proofs of the latter, of course). . . . 12 See, for example, E. H. Moore, "On the projective axioms of geometry," Trans. A mer. Math. Soc .• ......1 i (1902), pp. 142-158.
-LUDWIG WlTTGENSTElN There can never be surprises in logic. Do I contradict myself? Very well then I contradict myself. -WA'LT WHrrMAN (1 am large, 1 contain multitudes.) Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the "logic piano" imagined by Stanley Jevons; or, if you choose, a machine mzght be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in allve and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does. -HENRI POINCARE
4
Goedel's Proof By ERNEST NAGEL and JAMES R. NEWMAN
IN 1931 there appeared in a German scientific periodical an exceptionally difficult and brilliant paper entitled "Ueber formal unentscheidbare Saetze der Principia Mathematica und verwandter Systeme" ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems"). The author of the paper was Kurt Goedel, then a young mathematician of 25 at the University of Vienna, now a member of the Institute for Advanced Study at Princeton. When at a convocation in 1952 Harvard University awarded Goedel an honorary degree, the citation described his achievement as the most important advance in mathematical logic in a quarter century. "On Formally Undecidable Propositions of Principia Mathematica and Related Systems" is a milestone in the history of modern logic and mathematics, yet probably neither its title nor its contents were at the time of its appearance intelligible to the great majority of professional mathematicians. This is not surprising. The term "undecidable propositions" may for the moment be briefly identified as the name of propositions which can be neither proved nor disproved within a given system; the Principia Mathematica, to which the paper referred, is the monumental three-volume treatise by Alfred North Whitehead and Bertrand Russell on mathematical logic and the foundations of mathematics. Now familiarity with the thesis and the techniques of the Principia, let alone with some of the questions it raised, was not in 1931 (and is not now) a prerequisite to successful research in most bran9hes of mathematics. There were, to be sure, a number of mathematicians, chiefly under the influence of the outstanding German mathematician David IDlbert, who 1668
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were profoundly interested in these matters; but the group was small. Logico~mathematical problems have never attracted a wide audience even among those who are partial to abstract reasoning. On the other hand, to those who were able to read Goedel's paper with understanding, its conclusions came as an astounding and a melancholy revelation. For the central theorems which it demonstrated challenged deeply rooted precon~ ceptions concerning mathematical method, and put an end to one great hope that motivated decades of research on the foundations of mathematics. Goedel showed that the axiomatic method, which mathematicians had been exploiting with increasing power and rigor since the days of Euclid, possesses certain inherent limitations when it is applied to suf~ ficiently complex systems-indeed, when it is applied even to relatively simpJe systems such as the familiar arithmetic of cardinal numbers. He also proved, in effect, that it is impossible to demonstrate the internal consistency (non-contradictoriness) of such systems, except by employing principles of inference which are at least as powerful (and whose own internal consistency is therefore as much open to question) as are the logical principles employed in developing theorems within the systems themselves. Goedel's paper was not, however, exclusively negative in import. It introduced a novel technique of analysis into the foundations of mathematics that is comparable in fertility with the power of the algebraic method which Descartes introduced into the study of geometry. It suggested and initiated new problems and branches of logico-mathematical research. It provoked a critical reappraisal, not yet completed, of widely held philosophies of knowledge in general, and of philosophies of mathematics in particular. Despite the novelty of the techniques Goedel introduced, and the complexity of the details in his demonstrations, the major conclusions of his epoch-making paper can be made intelligible to readers with even limited mathematical preparation. The aim of the present article is to make the substance of Goedel's findings generally understandable. This aim will perhaps be most easily achieved if the reader is first briefly reminded of certain relevant developments in the history of mathematics and modem formal logic. I
The nineteenth century witnessed a tremendous expansion and intensification of mathematical research. Many fundamental problems that had long withstood the best efforts of earlier thinkers received definitive solutions; new areas of mathematical study were created; and the foundations for various branches of the discipline were either newly laid, or were recast with the help of more rigorous techniques of analysis. In particular, the development of the non-Euclidean geometries stimulated the revision
1670
Ernest Nagel
Illld
James R Newman
and completion of the axiomatic basis for many mathematical systems; and axiomatic foundations were supplied for fields of inquIrY which hitherto had been cultivated in a more or less intuitive manner. One Important conclusion that emerged from this critical examination of the foundations of mathematics was that the traditional conception of matht matics as the "science of quantity" is both inadequate and misleading. For it became evident that mathematics is the discipline par excellence which draws necessary conclusions from any given set of axioms (or postulates), and that the validity of the inferences drawn does not depend upon any particular interpretation which may be assigned to the postulates. Mathematics was thus recognized to be much more "abstract" and "formal" than had been traditionally supposed. The postulates of any branch of demonstrative mathematics are not inherently "about" space, quantity, or anything else; and any special meaning which may be associated with the "descriptive" terms (or predicates) in the postulates plays no essential role in the process of deriving theorems. The sale question which confronts the pure mathematician (as distinct from the scientist who employs mathematics in investigating a special subject matter) is not whether the postulates he assumes or the conclusions he deduces from them are true, but only whether the alleged conclusions are in fact the necessary logical consequences of the initial assumptions. For example, among the undefined terms employed by Hilbert in his famous axiomatization of geometry are the following; "point," "hne," "plane," "lies on," and "between." The customary meanings attributed to these (predicate) expressions undoubtedly promote the cause of discovery and learning. That is, because of the very familiarity of these notions, they not only motivate and facilitate the formulation of axioms, but they also suggest the goals of inquiry, i.e., the statements one wishes to establish as theorems, Nevertheless, as Hilbert states explicitly, for mathematical purposes familiar connotations are to be banished and the "meanings" of the expressions are to be taken as completely described by the axioms into which they enter. In more technical language the expressions are "implicitly defined" by the axioms and whatever is not embraced by the implicit definitions is irrelevant to the demonstration of theorems. The procedure recalls Russell's famous epigram: pure mathematics is the subject in which we do not know what we are talking about, nor whether what we are saying is true, This land of rigorous abstraction, empty of all familiar landmarks, was certainly not easy to get around in. But it offered compensations in the form of a new freedom of movement and fresh vistas. The intensified formalization of mathematics emancipated men's minds from the restrictions which the standard interpretation of expressions placed on the construction of novel systems of postulates. As the meaning of certain terms became more general, less explicit, their use became broader, the inferu
Goedel's Proof
1671
ences to be drawn from them less confined. Formalization led in fact to a great variety ofaxiomatized deductive systems of considerable mathe~ matical interest and value. Some of these systems, it must be admitted, did not lend themselves to an intuitively obvious interpretation, but this fact caused no alarm. Intuition, for one thing, is an elastic faculty; our children will have no difficulty in accepting as intuitively obvious the paradoxes of relativity,just as we do not boggle at ideas which were regarded as wholly unintuitive a couple of generations ,ago. Moreover, intuition, as we all know, is not a dependable guide: it cannot be used safely as a criterion of either truth or fruitfulness in scientific explorations. A more serious problem, however, was raised by the increased abstractness of mathematics. This turned on the question whether a given set of postulates underlying a new system was internally consistent, so that no mutually contradictory theorems could be deduced from the set. The problem does not seem pressmg when a set of axioms is taken to be "about" a definite and familiar domain of objects; for then it is not only significant to ask, but it may be possible to ascertain. whether the axioms are indeed true of these objects. Thus, since the Euclidean axioms were generally supposed to be true statements about space (or objects in space), apparently no mathematician prior to the nineteenth century ever entertained the question whether a pair of contradictory theorems might not some day be deduced from the axioms. The basis for this confidence in the consistency of Euclidean geometry was the sound principle that logically incompatible statements cannot be simultaneously true; accordingly, if a set of statements are true (and this was generally assumed to be the case for the Euclidean axioms), they are also mutually consistent. But the non-Euclidean geometries were clearly in a different case. For since their axioms were initially regarded as being plainly false of space, and, for that matter, doubtful1y true of anything, the problem of establishing the internal consistency of non-Euclidean systems was recognized to be both substantial and serious. In Riemannian geometry, for example, the famous parallel postulate of Euclid (which is equivalent to the assumption that through a given point in a plane just one parallel can be drawn to a given line in the plane) is replaced by the assumption that through a given point in a plane no parallel can be drawn to a given line in the plane. Now suppose the question: is the Riemannian set of postulates consistent? They are evidently not true of the ordinary space of our experience. How then is their consistency to be tested? How can one prove they will not lead to contradictory theorems? A general method was devised for solving this problem. The under~ lying idea is to find a "model" (or interpretation) for the postulates so that each postulate is converted into a true statement about the model. In the case of Euclidean geometry. as we have seen, the model was ordi-
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Ernest Nagel and lames R. Newman
nary space. Now the method was extended to tind other models, the elements of which could be used as crutches for the abstractions of the postulates. The procedure goes something like this. Suppose the following set of postulates is given concerning two classes K and L, whose special nature is left undetermined except as "implicitly" defined in the postulates: (1) Any two members of K are contained in just one member of L. (2) No member of K is contained in more than two members of L. (3) The members of K are not all contained in a single member of L. (4) Any two members of L contain just one member of K. (5) No member of L contains more than two members of K. From this little set, using customary rules of inference, theorems can be derived. For example, it can be shown that K contains just three members. But is the set a consistent one, so that mutually contradictory theorems can never be derived from it? The fact that no one has as yet deduced such theorems does not settle the question, because this does not prove that contradictory theorems may not eventually be deduced. The question is readily resolved, however, with the help of the following model. Let K be the vertices of a triangle, and L its sides. Each of the five abstract postulates is then converted into a true statement-for example, the first asserts that any two of the vertices are contained on just one side. Thereby the set is proved to be consistent. In a similar fashion the consistency of plane Riemannian geometry can be established. Let us interpret the expression "plane" in the Riemannian postulates to signify the surface of a Euclidean sphere, the expression "point" to signify a point on this surface, the expression "straight line" to signify an arc of a great circle on this surface, and so on. Each Riemannian postulate is then converted into a truth of Euclid. For example, on this interpretation the Riemannian parallel postulate reads as follows: Through a point on the surface of a sphere, no arc of a great circle can be drawn parallel to a given arc of a great circle. All this is very tidy, no doubt, but we must not become complacent. For as any sharp eye will have seen by now we are not so much answering the problem as removing it to familiar ground. We seek to settle the question of Riemannian consistency by appealing. in effect, to the authority of Euclid. But what about his system of geometry-are its axioms consistent? To say that they are uself-evidently true," and therefore consistent, is today no longer regarded as an acceptable reply. To describe the axioms as inductive generalizations from experience would be to claim for them only some degree of probable truth. A great mass of evidence might be adduced to support them, yet a single contrary item would destroy their title of universality. Induction therefore will not suffice to establish the consistency of Euclid's geometry as logically certain. A different approach was tried by David Hilbert. He undertook to interpret the
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1673
Euclidean postulates in a manner made familiar by Cartesian co-ordinate geometry, so that they are transformed into algebraic truths. Thus, in the axioms for· plane geometry, construe the expression "point" to signify a pair of real numbers, the expression "straight line" to signify the relation between real numbers which is expressed by a first degree equation with two unknowns, the expression "circle" to signify the relation between numbers expressed by a quadratic equation of a certain form, and so on. The geometric statement that two distinct points uniquely determine a straight line is then transformed into the algebraic truth that two pairs of real numbers uniquely determine a linear form; the geometric theorem that a straight line intersects a circle in at most two points, is transformed into the algebraic theorem that a linear form and a quadratic form of a certain type determine at most two pairs of real numbers; and so on. In brief, the consistency of the Euclidean postulates is established by showing that they are satisfied by an algebraic model. This method for establishing consistency is powerful and effective. Yet it too remains vulnerable to the objections set forth above. In other words the problem has again been solved in one domain only by transferring it to another. Hilbert's proof of the consistency of his postulates simply shows that if algebra is consistent, then so is his geometric system. The proof is merely relative to the assumed consistency of some other system and is not an "absolute" proof. In attempting to solve the problem of consistency one notices a recurrent source of difficulty. It is encountered whenever a nonfinite model is invoked for purposes of interpretation. It is evident that in making generalizations about space only a very limited portion-that which is accessible to our senses-serves as the basis of grand inferences; we extrapolate from the small to the universal. But where the model has a finite number of elements the difficulty is minimized, if it does not completely vanish. The vertex-triangle model used above to show the consistency of the five abstract K and L class postulates is finite; it was therefore comparatively simple to determine by actual inspection whether all the elements in the model actually satisfied the postulates. If this condition is fulfilled they are "true" and hence consistent. To illustrate: by examining in turn al1 the vertices of the model triangJe one can learn whether any two of them lie on one side-so that the first postulate is established as true. UnfortunatelY, however, most of the postulate systems that constitute the foundations of important branches of mathematics cannot be mirrored in finite models and can be satisfied only by non-finite ones. One of the postulates, for example, in a well known axiomatization of elementary arithmetic asserts that every integer has an immediate successor which differs from any integer preceding it in the progression. It is evident that the set of postulates containing this one cannot be interpreted
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Ernest Nagel and James R Newman
by means of a finite model; the model itself will have to mirror the infinity of elements postulated by the axioms. The truth (and so the consistency) of the set cannot therefore be established by inspection and enumeration. Apparently then we have reached an impasse. Finite models suffice to establish the consistency of certain sets of postulates, but these are of lesser importance. Non-finite models, necessary for the interpretation of most postulate systems, can be described only in general terms, and we are not warranted in concluding as a matter of course that the descriptions themselves are free from a concealed contradiction. It may be tempting to suggest at this point that we can be assured of the consistency of descriptions which postulate non-finite models, if the basic notions employed in such descriptions are transparently "clear" and "certain." But the history of thought has not dealt kindly with the doctrine of intuitive knowledge which is implicit in the suggestion. In certain areas of mathematical research, in which assumptons about infinite domains play central roles, radical contradictions (or "antinomies") have turned up, despite the "intuitive" clarity of the notions involved· in the assumptions, and despite the seemingly consistent character of the intellectual constructions performed. Such antinomies have emerged in the theory of transfinite numbers developed by Georg Cantor in the nineteenth century; and the occurrence of these contradictions has made plain that the apparent clarity of even such an elementary notion as that of class, does not guarantee the consistency of the system built on it. Now the theory of classes (or aggregates) is often made the foundation for other branches of mathematics, and in particular for elementary arithmetic. It is therefore pertinent to ask whether antinomies similar to those encountered in the theory of transfinite numbers may not infect other parts of mathematics. In point of fact, Russell constructed a contradiction within the framework of elementary logic itself, a contradiction which is the precise analogue of the antinomy first developed in the Cantorian theory of transfinite numbers. Russell's antinomy can be stated as follows: Classes may be divided in two groups: those which do not, and those which do contain themselves as members. A class will be called "normal" if, and only if, it does not contain itself as a member. Otherwise it is "non-norma1." An example of a normal class is the class of mathematicians, for patently the class itself is not a mathematician and is therefore not a member of itself. An example of a non-normal class is the class of all thinkable concepts; for the class of an thinkable concepts is itself a concept and is therefore a member of itself. Now let "N°' by definition stand for the class of all normal classes. We ask whether N itself is a normal class. If N is normal, it is a member of itself for, by definition of "N," N is to include all normal classes; but in that case also N is non-normal because by definition of
1615
Goedel's Proof
"non-normal," non-normal classes are those which contain themselves as members. On the other hand, if N is non-normal, then again it is a member of itself by definition of "non-normal," but then also it is normal because it belongs to N which is defined as normal. N, in other words, is normal if and only if N is non-normal. This fatal contradiction results from an uncritical use of the apparently pellucid notion of class. Moreover, additional antinomies were found subsequently, each of them constructed by means of familiar and seemingly cogent modes of reasoning. But the intellectual construction and formulation of non-finite models generally involves the use of possibly inconsistent sets of postulates. Accordingly, although the classical method for establishing the consistency of axioms continues to be an invaluable mathematical tool, that method does not supply a final answer to the problem it was designed to resolve. n
The inadequacies of the model method of demonstrating consistency, and the growing apprehension, based on the discovery of the antinomies, that established mathematical systems were infected by contradictions, led to new attacks upon the problem. An alternative to relative proofs of consistency was proposed by Hilbert. He sought to construct so-called "absolute" proofs of freedom from contradiction. These we must explain briefly as a further preparation for discussing Goedel's proof. The first requirement of an absolute proof as Hilbert conceived it is the complete formalization of the system. This, the reader will recall, means draining the expressions occurring within the system of any meaning whatever; they are to be regarded simply as empty, formal signs. How these signs are to be manipulated is then to be set forth explicitly in a set of rules. The purpose of this procedure is to construct a calculus which conceals nothing, which has in it only that which we intended to put in it. When theorems of this calculus are derived from the postulates by the combination and transformation of its meaningless signs in accord with precisely stated rules of operation, the danger is eliminated of the use of any unavowed principles of reasoning. Formalization is a difficult and tricky business, but it serves a valuable purpose. It reveals structure and function in naked clarity as does a cut-away working model of a' machine. When a system has been formalized the logical relations between mathematical propositions are exposed to view; one is able to see the structures of con:fi.guration~ of certain "strings" (or sequences) of "meaningless" signs, how they hang together, are syntactically combined, nest in one another and so on. A page covered with the "meaningless" marks of this calculus speaks for itself, as does a mosaic, an abstract design, a geometric diagram; it
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Ernest Nagel and James R. Newman
has its own language. But suppose we as observers wish to make statements about a given configuration in the calculus. Such statements will be meaningful in quite a different sense, and will have to be expressed in a language belonging not to the calculus (or to mathematics) but to what Hilbert called "meta-mathematics" (or the language about mathematics). Meta-mathematical statements are statements about the signs in a calculus. They describe the kinds and arrangements of such signs when they are combined to form longer strings of marks called "formulas," and the relations between formulas in consequence of the rules of manipulation that have been specified for them. The following table illustrates some of the differences between expressions within arithmetic (mathematics) and statements about such expressions (meta-mathematics). Mathematics For every x) if x is a prime and x > 2, then x is odd.
Meta-mathematics 'x' is a numerical variable. '2' is a numerical constant. 'prime' is a predicate expression. '>' is a binary predicate.
2+3=5
If the sign '=' occurs in an expression which is a formula of arithmetic, the sign must be flanked on both its left and right sides by numerical expressions. '2 + 3 = 5' is a formula.
x=x 0=0
The formula '0 0' is derivable from the formula 'x = x' by substituting the numeral '0' for the numerical variable 'x'.
o~o
'0 ~ 0' is not a theorem. Arithmetic is consistent-that is, it is not possible to derive from the axioms of arithmetic both the formula '0 0' and the formula '0 ~ 0'.
=
=
It is worth observing that, despite appearances to the contrary, the meta-
mathematical statements in the right-hand column do not actually contain any of the mathematical expressions listed in the left-hand column. The right-hand column contains only the names of some of the arithmetical expressions in the left-hand column. This is so, because the rules of English grammar require that no English sentence shall contain the objects to which it refers, but only their names. The rule is enforced in the above table through the convention of enclosing an expression within single quotation marks in order to obtain a name for that expression. In consonance with this convention, it is correct to say that 2 + 3 is identical with 5, but it is false to say that '2 3' is identical with '5.' The importance of the division between the mathematical and the metamathematical language cannot be overemphasized. By erecting a separate, formal calculus whose symbols are free of all hidden assumptions and
Goedel's Proof
1677
intuitive associations, and each of whose operations are precisely and rigidly defined, we have an instrument which exposes to plain view the nature of mathematical reasoning. But as human beings who wish to analyze this stark symbolism and to communicate our findings, we must construct another language which will enable us to describe, discuss, explain and theorize about the more formal system. Thus we separate the theory of the thing from the thing itself and devise the discourse of meta-mathematics. It was by the application of this meta-mathematical language that Hilbert hoped to prove the consistency of the formalized calculus itself. Specifically, he sought to develop a theory of proof (Beweistheorle) that would yield demonstrations of consistency by an analysis of the purely structural features of expressions in uninterpreted calculi. Such an analysis consists exclusively of noting the kinds and arrangements of signs in formula, and of showing whether a given combination of signs can be obtained from others in accordance with the explicitly stated rules of operation. An essential requirement for demonstrations of consistency, as propounded in the original version of Hilbert's program, is that they employ only finitary notions, and make no reference either to an infinite number of formulas or to an infinite number of operations upon them. A proof of the consistency of a set of postulates which conforms to these requirements is called "absolute." Such a proof achieves its objective by means of a bare minimum of inferential principles, without assuming the consistency of some other set of axioms. An absolute proof of the consistency of arithmetic, if one could be devised, would afford a demonstration, by finitary meta-mathematical means, that two "contradictory" formulas, such as '(0 = 0)' and ',...., (0 = O),-where the sign '-', called a tilde, signifies negation-are not both derivable from the axioms or initial formulas of the system, when the derivations conform to the stated rules of inference. It may be useful, by way of illustration, to compare meta-mathematics as a theory of proof with the theory of some game, such as chess. Chess is a game played with 32 pieces of specified design on a square board containing 64 square subdivisions, where the pieces may be moved in accordance with fixed rules. The game can obviously be played without assigning any "interpretation" to the pieces or to their various positions on the board, although it is clear that such interpretations could be supplied if desired. There is thus an analogy between the game and a formalized mathematical calculus. The pieces and the squares of the board correspond to the elementary signs of the calculus; the permitted configurations of pieces on the board correspond to the formulas of the calculus; the initial positions of pieces on the board correspond to the axioms or initial formulas of the calculus; the subsequent configurations of pieces on the
Ernest Nagel and James R. Newman
1678
board correspond to formulas derived from the axioms (i.e., to the theorems); and the rules of the game correspond to the rules of derivation for the calculus. Again, although configurations of pieces on the board, like the formulas of the calculus, are "meaningless," statements about these configurations, like meta-mathematical statements about formulas, are quite meaningful. A meta-chess statement may assert, for example, that there are 20 possible opening moves for White, or that, given a certain configuration of pieces on the board with White to move, Black is mate in three moves. It is pertinent to note, moreover, that general metachess theorems can be established, whose proof involves the consideration of only a finite number of permissible configurations on the board. The meta-chess theorem about the number of possible opening moves for White can be established in this way; and so can the meta-chess theorem that if White has only two Knights, it is impossible for White to mate Black. These and other meta-chess theorems can thus be proved by finitary methods of reasoning, consisting in the examination in tum of each of a finite number of configurations that can occur under stated conditions. The aim of Hilbert's theory of proof, similarly, was to demonstrate by such finitary methods the impossibility of deriving certain formulas in a calculus. m
There are two more bridges to cross before entering upon Goedel's proof itself. Something needs be said about how and why the Principia Mathematica came into being; also we must give a short illustration of the formalization of a deductive system-we shall take a fragment of Principia-and how its consistency can be established. Ordinarily, even when mathematical proofs conform to accepted standards of professional rigor, they suffer from one important omission. They employ principles (or rules) of inference which are not explicitly formu· lated, and of which mathematicians are frequently unaware. Take as example Euclid's proof that there is no greatest prime number. This is cast in the form of a reductio ad absurdum argument and runs as follows. Suppose there is a greatest prime x. Then: (1) x is the greatest prime number. (2) Form the product of all primes less than or equal to x and add 1 to the product. This yields a new number y, where y = (2 X 3 X 5 X 7 . . . X x)
+ 1.
(3) Now if y is itself a prime, then x is not the greatest prime, for y is greater than x. ( 4) But suppose y is composite, i.e. not a prime; then again x is not the greatest prime. For if y is composite, it must have a prime divisor z,
Goedel's Proof
1679
which is different from each of the primes 2, 3, 5, 7 . . . x; hence z itself is a prime greater than x. (5) But y is either prime or composite, and in either case x is not the greatest prime. (6) Hence, since x is not the greatest prime, and x can be any prime number, there is no greatest prime. We have shown the essential steps of this proof, and we could show also--though we cannot here take the time--that a number of elementary rules of inference are essential to its development, (e.g., the "Rule of Substitution," the "Rule of Detachment") and even rules and theorems belonging to more advanced parts of logical theory (e.g., the theory of "quantification," having to do with the proper use of expressions such as "all," "every," "some" and their synonyms). It has been pointed out that the use of these rules and theorems is an all but unconscious process; however, even more noteworthy is the fact that the analysis of Euclid's proof which uncovers the use of these logical props depends upon ad· vances in the theory of logic which have occurred only within the past century. Like Moliere's M. Jourdain, who spoke prose without knowing it, mathematicians have been reasoning without knowing their reasons. Modern students have had to show them the real nature of the tools of their craft. For almost 2,000 years Aristotle's codification of valid forms of deduction was widely regarded as complete and as incapable of essential im· provement. As late as 1787, the German philosopher Immanuel Kant was able to say that since Aristotle, formal logic "has not been able to adva'i.ce a single step, and is to all appearances a closed and completed body of doctrine." But the fact is that the traditional logic is seriously incomplete and fails to give an account of many principles of inference employed in even quite elementary mathematical reasoning, such as the above proof of Euclid. In any event, a renaissance of logical studies in modem times began with the publication in 1847 of George Boole's The Mathematical Analysis of Logic. The primary concern of Boole and his immediate successors was to develop a non-numerical algebra of logic, which would provide a precise algorithm for handling more general and more varied types of deductions than were covered by traditional logical principles. Another line of inquiry, intimately related to the work of 19th century mathematicians on the foundations of analysis, became associated eventually with the Boolean program. This new development sought to exhibit all of pure mathematics as simply a chapter of formal logic; and it received its classical embodiment in the Principia Mathematica of Whitehead and Russell in 1910. Mathematicians of the 19th century
Ernest Nagel and James R. Newman
1680
succeeded in "arithmetizing" algebra and the so-called "infinitesimal calculus,;' by showing that the various notions employed in mathematical analysis are definable exclusively in arithmetical terms (i.e., in terms of the integers and the arithmetical operations upon them). What Russell (and, before him, the German mathematician Gottlob Frege) sought to show was tItat all arithmetical notions are in turn definable in terms of purely Jogical ideas, and that, furthermore, the axioms of arithmetic are all deducible from a small number of basic propositions certifiable as purely logical truths. For instance, the notion of class belongs to general logic. Two classes are defined to be "similar," if there is a one-to-one correspondence between their members, the notion of such a correspondence being specifiable in terms of other logical ideas. A class which has no members (e.g., the class of satellites of the planet Venus) is said to be "empty." Then the cardinal number 0 can be defined as the class of all classes which are similar to an empty class. Again, a class which has a single member is said to be a "unit" class (e.g., the class of satellites of the planet Earth); and the cardinal number 1 can be defined as the class of all classes similar to a unit class. Analogous definitions can be given of the other cardinal numbers, and the various arithmetical operations can also be defined in terms of the notions of formal logic. An arithmetical statement, e.g., 1 + 1 2, can then be exhibited as a condensed transcription of a statement containing only expressions belonging to general logic; and such purely logical statements can be shown to be deducible from certain logical axioms, some of which will be mentioned presently. Principia Mathematica thus appeared to advance the final solution of the problem of consistency of mathematical systems, and of arithmetic in particular, by reducing that problem to the question of the consistency of formal logic. For if the axioms of arithmetic are simply transcriptions of theorems in logic, then the question whether those axioms are consistent is immediately transposed into the problem whether the fundamental axioms of logic are consistent. The Frege-Russell thesis that mathematics is but a chapter of logic has not won universal acceptance from mathematicians, for various reasons of detai1. Moreover, as we pointed out earlier, the antinomies of the Cantorian theory of transfinite numbers can be dUplicated within logic itself, unless special measures are taken to prevent such an outcome. But are the measures adopted in Principia Mathematica to outflank these antinomies sufficient to exclude all forms of self-contradictory constructions? This cannot be asserted as a matter of course. It follows that the Frege-Russell reduction of arithmetic to logic does not provide a final answer to the consistency problem-indeed, the problem simply emerges in a more general form. On the other hand, irrespective of the
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Goedel's Proof
1681
validity of the Frege-Russell thesis, two features of Principia have proved to be of inestimable value for the further study of the problem. Principia supplies an inclusive system of notation, with the help of which all statements of pure mathematics can be codified in a standard manner; and Principia makes explicit most of the rules of formal inference (eventually these rules were made more precise and complete) which are employed in mathematical demonstrations. In short, Principia provides the essential instrument for investigating the entire system of formal logic as an uninterpreted calculus, whose formulas are combined and transformed in accordance with explicitly stated rules of operation. We turn now to the formalization of a small portion of Principia, namely, the elementary logic of propositions. The task is to convert this fragment into a "meaningless" calculus of uninterpreted signs and to show how its freedom from contradiction can be proved. Four steps are involved. First a complete catalogue is presented of the signs to be employed in the calculus. These are its vocabulary. Second, the "Formation Rules" are laid down. These indicate the permissible combinations of the elementary signs which are acceptable as formulas (or sentences). The rules may be said to constitute the grammar of the system. Third, the "Transformation Rules" are specified. They describe the precise structure of formulas from which some other formula is derivable. Finally, certain formulas are selected as axioms (or as "primitive formulas"). They serve as foundation for the entire system. By the expression "theorems of the system" we denote all the formulas, including the axioms, which can be derived from the axioms by successively applying the Transformation Rules. By "proof" we mean a finite sequence of legitimate formulas, each of which is either an axiom or is derivable from preceding formulas in the sequence by the Transformation Rules. For the elementary logic of propositions (often also called the "sentential calculus") the vocabulary is extremely simple. It consists of sentential variables (which stand for sentences) and are written 'p', 'q: 'r', etc.,
of sentential connectives '.-' is short 'v' is short ':J' is short '.' is short
for 'not' for 'or' for 'if . . . then' for 'and'
and of parentheses, used as signs of punctuation. It is convenient to define the last two connectives in terms of the first two, so that expressions containing ':J' or '.' can be replaced by expressions containing only 'v'
1682
Ernest Nagel and James R. Newman
and '.-'. For example 'p ::> q' is defined as being simply shorthand for the slightly longer expression '.- p v q'.l The Formation Rules are so laid down that combinations of the ele mentary signs which would normally be called "sentences" are designated as "formulas." Accordingly, each sentential variable will count as a formula. Moreover, if S is a formula, so is its negation -- (S); and if S1 and S2 are formulas, so is (Sl) v (S2)' with similar conventions for the other connectives. Two Transformation Rules are adopted. One of them, the Rule of Substitution, says that if a sentence containing sentential variables has been accepted as logically true, any formulas may be uniformly substituted for these variables, whereupon the new sentence will also be logically true. The other rule, that of Detachment, simply says that if we have two logically true sentences of the form S1' and S1 ::> S2' we may also accept as logically true the sentence S2' The axioms of the calculus (essentially those of Principia) are the following: M
1. (p v p) ::> P v q) 3. (p v q) :J (q v p) 4. (p:J q) :J [(rv p) :J (rvq)] 2. P ::> (p
Their meaning is easily understood. The second, for instance, says that a proposition (or sentence) implies that either it or some other proposition (or sentence) is true. Our purpose is to show that this set of axioms is not contradictory; in other words that, by using the stated Transformation Rules, it is impossible to derive from the axioms any formula S together with its ~egation .- S. Now it happens that 'p :J (,...., P :J q)' is a theorem in the calculus. (We shall simply accept this as a fact without exhibiting the derivation.) Suppose, then, that some formula S, as well as .- S were deducible from the axioms. (The reader will recognize the reductio ad absurdum approach of Euclid's proof.) By substituting S for 'p' in the theorem (as permitted by the Rule of Substitution), and applying the Rule of Detachment twice, the formula 'q' would be deducible. But this immediately has the consequence that by substituting any formula whatsoever for 'q', any formula whatsoever would be deducib1e from the axioms. It is thus clear that if both some formula S and its contradictory .- S were deducible from the axioms, then any formula would be deducible. In short, if the calculus is not consistent, every formula is a theorem. And likewise, if not every formula is a theorem (i.e., if there is at least one formula which is not 1 That is, "if p then q" is defined as short for "either not-p or q." In view of this definition, the statement "If GableD played the lute then Galileo was a musician" is simply a slightly more compact way of rendering what is expressed by the statement "Either Galileo did not play the lute or Galileo was a musician."
Goedel's Proof
1683
derivable from the axioms), then the calculus is consistent. The task, therefore, is to exhibit some formula which cannot be derived from the axioms. The way this is done is to employ meta-mathematical reasoning upon the system to be tested. We place ourselves, so to speak, outside the calculus and consider how theorems are generated within it. The actual procedure is pretty. (1) We try to find a characteristic common to all four axioms; (2) we try to show that this characteristic is "hereditary" under the Transformation Rules-i.e., that if all the axioms have this characteristic, any formula derived from them by the rules (which is to say, any theorem) also has it; (3) we try to exhibit a formula that does not have this characteristic. If we succeed in this triple task, we will have an absolute proof of consistency. For if the common characteristic exists and is hereditary, so that it is transmitted to all properly derived formulas, then any array of symbols which in appearance conforms to the requirements of a formula but nevertheless does not possess the characteristic in question cannot be a theorem. That is to say, structurally it may be a formula, yet not one which could have been derived from the axioms; or to put it yet another way, since the suspected offspring (formula) lacks an invariably inherited trait of the forbears (axioms) it cannot in fact be their descendant (theorem). Furthermore, if we can find such a formula we will have established the consistency of the calculus; because, as we noted a moment ago, if the calculus were not consistent, every formula could be derived from the axioms, i.e., every formula would possess the characteristic and therefore be a theorem. Let us specify a common characteristic. The trait we have in mind is that of being a tautology. In common parlance tautology is defined as the saying of a thing twice over in different words, e.g., "John is the father of Charles and Charles is the son of John." In logic a tautology is not even required to use different words, e.g., "the moon is the moon" or, "either it is raining ('r it is not raining." The essence of a tautology is that it is "true in all possible worlds," whence it is a truth of logic. Now it can be shown (though we shall not turn aside to give the demonstration) with the aid of an ingenious device known as a "truth-table," that each of the four axioms of our little set is a tautology. That is to say, if each axiom is regarded as a formula made up of simpler formulas (e.g., the compound formula or sentence p :) (p v q) is constituted of the simple formulas 'po and 'q') , it must be accepted as true irrespective of the truth or falsity of its elementary constituents. Even the skeptical reader will have no difficulty accepting the fact, for example, that axiom 1: (p v p) ::> p is "true in all possible worlds," if he substitutes the elementary sentence "2 is a prime number" for the sentential variable p and derives the sentence "If 2 is a prime number or 2 is a prime number. then 2 is a prime number," It iq ::11 4, and so on.) The importance of this arithmetization of meta-mathematics stems from the fact that, since each of its statements can be uniquely represented in the formal system by an expression tagged with a Goedel number, relations of logical dependence between meta-mathematical statements can be explored by examining relations between integers and their factors. To take a trivial analogue: if customers in a supermarket are given tickets with numbers determining the order in which they are to be waited on when buying meat, it is a simple matter, merely by scrutinizing the numbers themselves to discover (a) how many persons have been served, (b) how many
Goedel's Proof
1691
are waiting, (C) who precedes whom and by how many customers, etc. Consider the meta-mathematical statement: 'The sequence of formulas whose Goedel number is x is a demonstration for the formula whose Goede! number is z.' This statement is represented (mirrored) by a definite formula in the arithmetical calculus, a formula which expresses a purely arithmetical relation between x and z. (In the above example of assigning the Goedel number k to a demonstration, we found that k 2m X 3n ; and a little reflection shows that there is a definite though complex arithmetical relation between k, the Goedel number of the proof, and n, the Goede1 number of the conclusion.) We write this arithmetical relation between x and z as the formula 'Dem (x,z),' to remind ourselves of the meta-mathematical statement to which it corresponds. Similarly, the meta-mathematical statement: "The sequence of formulas with Goedel number x is not a demonstration for the formula with the Goedel number z," is also represented by a definite formula in the arithmetical formalism. This formula we shall write as '-- Dem (x,z).' We shall need one additional bit of special notation for stating the crux of Goedel's argument. Begin with an example. The formula '(Sx) (x sy)' has the Goedel number m, and the variable 'y' has the Goedel number 16. Substitute in this formula for the variable with Goedel number 16 (Le., for 'y') the numeral for m. We then obtain the formula '(Rx) (x 8m).' This latter formula obviously also has a Goedel number-a number which can be actually calculated, and which, in fact, is a certain complex arithmetical function of the two numbers m and 16. However, instead of calculating this Goedel number, we can give an unambiguous meta-mathematical characterization for it: it is the Goede! number of the formula which is obtained from the formula with Goedel number m, by substituting for the variable with Goedel number 16 the numeral for m. Accordingly, this meta-mathematical characterization corresponds to a definite arithmetical function of the numbers m and 16, a function which can be expressed within the arithmetical calculus. We shall write this function as 'sub (m, 16, m),' to remind ourselves of the meta-mathematical description which it represents. More generally, the expression 'sub (y, 16, y)' is the mirrorimage within the arithmetical formalism of the meta-mathematical characterization: "the Goedel number of the formula which is obtained from the formula with Goedel number y, by substitutin'g for the variable with Goedel number 16 the numeral for y." It should be noted that, when a definite numeral is substituted for 'y' in 'sub (y, 16, y)/ sub (y, 16~ y) is a definite integer which is the Goedel number of a certain formula. We are now equipped to follow in outline Goede!'s argument. Consider the formula '(x) ,.... Dem (x,z).' This represents, in the arithmetical calculus, the meta-mathematical statement "For every x, where x is the Goedel number of a demonstration, x is not the number of a demonstra-
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1691
Ernest Nagel and James R. Newman
tion for the formula whose Goedel number is z." This formula may therefore be regarded as a formal paraphrase of the statement "The formula with Goedel number z is not demonstrable." What Goedel was able to show was that a c;ertain special case of this formula itself is in fact not formally demonstrable. To construct this special case we start with a formula which we shall display as line (1): (1)
(x) -- Oem (x, sub (y, 16, y)
It corresponds to the meta-mathematical statement that the formula with the Goedel number sub (y, 16, y) is not demonstrable. Moreover, since line (1) is a formula within the arithmetical calculus, it has its own Goedel number, say n. Let us now obtain another formula from the one on line (1) by substituting the numeral for n for the variable with Goedel number 16 (i.e., for 'y'). We thus arrive at the special case we wished to construct, and display it as line (2):
(2)
(x) -- Oem (x, sub (n, 16, n)
Since this last formula occurs within the arithmetical calculus, it must have a Goedel number. What is its Goedel number? A little reflection shows that it is sub (n, 16, n). To see this, we must recall that sub (n, 16, n) is the Goedel number of the formula which is obtained from the formula with Goedel number n, by substituting for the variable with Goedel number 16 (Le., for 'y') the numeral for n. But the formula (2) has indeed been obtained from the formula with Goedel number n (Le., from the formula on line (I») by substituting for the variable 'y' the numeral for n. Let us also remind ourselves, however, that the formula '(x) -- Dem (x~ sub (n, 16, n»' is the mirror-image within the arithmetical calculus of the meta-mathematical statement: "The formula whose Goedel number is sub (n, 16, n) is not demonstrable." It follows that the arithmetical formula '(x) ,...., Oem (x, sub (n, 16, n»' represents the metamathematical statement: "The formula '(x) -- Oem (x, sub (n, 16, n»' is not demonstrable." In a sense, therefore, this arithmetical formula can be construed as saying that it itself is not demonstrable. Goedel is now able to show, in a manner reminiscent of the Richard Paradox, but free from the fallacious reasoning involved in that puzzle, that this arithmetical formula is indeed not demonstrable. The argument from this point on is relatively simple and straightforward. He shows that if the formula were demonstrable, then its formal contradictory (Le., '.- (x) ,..." Dem (x, sub (n, 16, n),' which in effect says that the formula is in fact demonstrable) would also be demonstrable; and conversely, if the formal contradictory of the formula were demonstrable, the formula itself would also be demonstrable. But as was noted earlier, if a formula as well as its contradictory can both be derived from a set of axioms, the
Goedel';s Proof
1693
axioms are not consistent. Accordingly, if the axioms are cons,istent, neither the formula nor its contradictory is demonstrable. In short, if the axioms are consistent, the formula is "undecidable"-neither the formula nor its contradictory can be formally deduced from the axioms. Very welL Yet there is a surprise coming. For although the formula is undecidable if the axioms are consistent, it can nevertheless be shown by meta-mathematical reasoning to be true. That is to say, the formula is a true arithmetical statement which expresses a complex but definite numerical property of integers-just as the formula '(x).- (x + 3 = 2)' (in words, "There is no positive integer which when added to 3 will equal 2") expresses another but much simpler property of integers. The reasoning that shows the truth of the undecidable formula is rather simple. In the first place, on the assumption that arithmetic is consistent, we have already established the meta-mathematical statement: "The formula '(x) '""'" Dem (x, sub (n, 16, n»' is not demonstrable." It must be accepted, then, that this meta-mathematical statement is true. Secondly, the statement is represented within arithmetic by that very formula itself. Third, we recall that meta-mathematical statements have been mapped upon the arithmetical formalism in such a way that true-mathematical statements always correspond to true arithmetical formulas. (Indeed, this is the whole point of the mapping procedure-just as in analytic geometry, geometric statements are mapped onto algebra in such a way that true geometric statements always correspond to true algebraic ones.) Accordingly, the formula in question must be true. We have thus established an arithmetical truth, not by deducing it formally from the axioms of arithmetic, but by a meta-mathematical argument. When we were discussing the sentential calculus, we explained that the axioms of that system are "complete," since all the logical truths expressible in the system are formally derivable from the axioms. More generally, we can say that the axioms of any formalized system are "complete" if every true statement expressible in the system is formally deducible from the axioms. A set of axioms is therefore "incomplete" if not every true statement expressible in the system is formally derivable from them. It follows, since we have now established as true an arithmetical formula which is not derivable from the axioms of arithmetic, the system is incomplete. Moreover, the system is essentially incomplete, which means that even if we added this true but undemonstrable formula to the axioms as a further axiom, the augmented system would still not suffice to yield formally all arithmetical truths: another true arithmetical formula could be constructed, such that neither the formula nor its contradictory would be demonstrable within the enlarged system. This remarkable conclusion would hold, no matter how often w~ enlarged the system by adding further axioms to it.
1694
Ernest Nagel and James R. Newman
We come then to the coda of Goedel's amazing and profound intellectual symphony. It can be shown that the meta-mathematical statement just established, namely, "If arithmetic is consistent, then it is incomplete," itself corresponds to a demonstrable formula in the arithmetical system. But the antecedent clause of this formula (the one corresponding to the meta-mathematical statement "arithmetic is consistent") is not demonstrable within the system. For if it were, the consequent clause of the formula (the one corresponding to the statement "arithmetic is incomplete," and which in fact turns out to be our old friend '(x) ,.... Dem (x, sub (n, 16, n)) ') would also be demonstrable. This conclusion would, however, be incompatible with the previously obtained result that the latter formula is not demonstrable. The grand final step is now before us: we must conclude that the consistency of arithmetic cannot be established by any meta-mathematical reasoning which can be represented within the formalism of arithmetic! A meta-mathematical proof of the consistency of arithmetic is not excluded by this capital result of Goedel's analysis. In point of fact, metamathematical proofs of the consistency of arithmetic have been constructed, notably by Gerhard Gentzen, a member of the Hilbert school, in 1936. But such proofs are in a sense pointless if, as can be demonstrated, they employ rules of inference whose own internal consistency is as much open to doubt as is the formal consistency of arithmetic itself. Thus, Gentzen used the so-called "principle of transfinite mathematical induction" in his proof. But the principle in effect stipulates that a formula is derivable from an infinite class of premises. Its use therefore requires the employment of nonftnitistic meta-mathematical notions, and so raises once more the question which Hilbert's original program was intended to resolve. The import of Goedel's conclusions is far-reaching, though it has not yet been fully fathomed. They seem to show that the hope of finding an absolute proof of consistency for any deductive system in which the whole of arithmetic is expressible cannot be realized, if such a proof must satisfy the finitistic requirements of Hilbert's original program. They also show that there is an endless number of true arithmetical statements which cannot be formally deduced from any specified set of axioms in accordance with a closed set of rules of inference. It follows, therefore, that an axiomatic approach to number theory, for example, cannot exhaust the domain of arithmetic truth, and that mathematical proof does not coincide with the exploitation of a formalized axiomatic method. Just in what way a general notion of mathematical or logical truth is to be defined which is adequate to the fact here stated, and whether, as Goedel himself appears to believe, only a thoroughgoing Platonic realism can supply such a definition, are problems still under debate and too difficult for more than mention here.
Goedel's Proof
1695
Goedel's conclusions also have a bearing on the question whether calculating machines can be constructed which would be substitutes for a living mathematical intelligence. Such machines, as currently constructed and planned, operate in obedience to a fixed set of directives built in, and they involve mechanisms which proceed in a step-by-step manner. But in the light of Goedel's incompleteness theorem, there is an endless set of problems in elementary number theory for which such machines are inherently incapable of supplying answers, however complex their built-in mechanisms may be and however rapid their operations. It may very well be the case that the human brain is itself a "machine" with built-in limitations of its own, and that there are mathematical problems which it is incapable of solving. Even so, the human brain appears to embody a structure of rules of operation which is far more powerful than the structure of currently conceived artificial machines. There is no immediate prospect of replacing the human mind by robots. None of this is to be construed, however, as an invitation to despair, or as an excuse for mystery mongering. The discovery that there are formally indemonstrable arithmetic truths does not mean that there are truths which are forever incapable of becoming known, or that a mystic intuition must replace cogent proof. It does mean that the resources of the human intellect have not been, and cannot be, fully formalized, and that new principles of demonstration forever await invention and discovery. We have seen that mathematical propositions which cannot be established by formal deduction from a given set of axioms, may nevertheless be established by "informal" meta-mathematical reasoning. It would be an altogether irresponsible claim to maintain that the formally indemonstrable truths Goedel established by lneta-mathematical arguments are asserted in the absence of any proof or by appeals simply to an uncontrolled intuition. Nor do the inherent limitations of calculating machines constitute a basis for valid inferences concerning the impossibility of physico-chemical explanations of living matter and human reason. The possibility of such explanations is neither precluded nor affirmed by Goedel's incompleteness theorem. The theorem does indicate that in structure and power the human brain is far more complex and subtle than any nonliving machine yet envisaged. Goedel's own work is a remarkable example of such complexity and subtlety. It is an occasion not for dejection because of the limitations of formal deduction but for a renewed appreciation of the powers of creative reason.
Mathematics is the science which draws necessary conclusions. -BENJAMIN PEIRCE
5
A Mat11ematical Science By OSWALD VEBLEN and JOHN WESLEY YOUNG 1.
UNDEFINED ELEMENTS AND UNPROVED PROPOSITIONS
GEOMETRY deals with the properties of figures in space. Every such figure is made up of various elements (points, lines, curves, planes, sur~ faces, etc.), and these elements bear certain relations to each other (a point lies on a line, a line passes through a point. two planes intersect, etc.). The propositions stating these properties are logically interdependent, and it is the object of geometry to discover such propositions and to exhibit their logical interdependence. Some of the elements and relations, by virtue of their greater simplicity, are chosen as fundamental, and all other elements and relations are defined in terms of them. Since any defined element or relation must be defined in terms of other elements and relations, it is necessary that one Or more of the elements and one or more of the relations between them remain entirely undefined; otherwise a vicious circle is unavoidable. Likewise certain of the propositions are regarded as fundamental, in the sense that all other propositions are derivable, as logical consequences, from these fundamental ones. But here again it is a logical necessity that one or more of the propositions remain entirely unproved; otherwise a vicious circle is again inevitable. The starting point of any strictly logical treatment of geometry (and in~ deed ot any branch of mathematics) must then be a set of undefined elements and relations, and a set of unproved propositions involving them; and from these all other propositions (theorems) are to be derived by the methods of formal logic. Moreover, since we assumed the point of view of formal (Le., symbolic) Jogic, the undefined elements are to be regarded as mere symbols devoid of content, except as implied by the fundamental propositions. Since it is manifestly absurd to speak of a proposition in~ volving these symbols as self-evident, the unproved propositions referred to above must be regarded as mere assumptions. It is customary to refer to these fundamental propositions as axioms or postulates, but we prefer to retain the term assumption as more expressive of their real logical character. 1696
1697
A MathematIcal Science
We understand the term a mathematical science to mean any set oj propositions arranged according to a sequence of logical deduction. From the point of view developed above such a science is purely abstract. If any concrete system of things may be regarded as satisfying the fundamental assumptions, this system is a concrete application or representation of the abstract SCIence. The practical importance or triviality of such a science depends simply on the importance or triviality of its possible applications. These ideas will be illustrated and further discussed in the next section, where it will appear that an abstract treatment has many advantages quite apart from that of logical rigor. 2.
CONSISTENCY~ CATEGORICALNESS, INDEPENDENCE. EXAMPLE OF
A MATHEMATICAL SCIENCE
The notion of a class 1 of objects is fundamental in logic and therefore in any mathematical science. The objects which make up the class are called the elements of the class. The notion of a class, moreover, and the relation of belonging to a class (being included in a class, being an element of a class, etc.) are primitive notions of logic, the meaning of which is not here called in question. 2 The developments of the preceding section may now be illustrated and other important conceptions introduced by considering a simple example of a mathematical science. To this end let 5 be a class, the elements of which we will denote by A, B, C, . . . Further, let there be certain undefined subclasses 8 of 5, anyone of which we will call an m-class. Concerning the elements of S and the m-classes we now make the following Assumptions: I. If A and B are distinct elements of 5, there is at least one m-class containing both A and B. II. If A and B are distinct elements of 5, there is not more than one m-class containing both A and B. III. A ny two m-classes have at least one element of 5 in common. IV. There exists at least one m-class. V. Every m-class contains at least three elements of 5. VI. A II the elements of 5 do not belong to the same m-class. VII. No m-class contains more than three elements of S. The reader will observe that in this set of assumptions we have just two undefined terms, viz., element of 5 and m-class, and one undefined rela1 Synonyms for class are set, aggregate, assemblage, totality; in German, Menge; in French, ensemble. 2 Cf~ B. Russell, The Pnnciples of Mathematics, Cambridge, 1903; and L. Couturat, us principes des mathimatiques, Paris, 1905. 3 A class Sf is said to be a subclass of another class S, if every element of SI is an element of S.
Oswald Veblen and John Wesley YOWlg
1698
tion, belonging to a class. The undefined terms, moreover, are entirely devoid of content except such as is implied in the assumptions. Now the first question to ask regarding a set of assumptions is: Are they logically consistent? In the example above, of a set of assumptions, the reader will find that the assumptions are all true statements, if the class S is interpreted to mean the digits 0, 1, 2, 3,4, 5, 6 and the m-classes to mean the columns in the following table:
o
5 6 1 2 3 4 6 0 1 2 3 4 5 1 2 34560 This interpretation is a concrete representation of our assumptions. Every proposition derived from the assumptions must be true of this system of triples. Hence none of the assumptions can be logically inconsistent with the rest; otherwise contradictory statements would be true of this system of triples. Thus, in general, ~ set of assumptions is said to be consistent if a single concrete representation of the assumptions can be given. 4 Knowing our assumptions to be consistent, we may proceed to derive some of the theorems of the mathematical science of which they are the basis: Any two distinct elements of 5 determine one and only one m-class containing both these elements (Assumptions I, II). The m-class containing the elements A and B may conveniently be denoted by the symbol AB. Any two m-classes have one and only one element of 5 in common (Assumptions II, III). There exist three elements oj S which are not all in the same m-class (Assumptions IV, V, VI). In accordance with the last theorem, let A, B, C be three elements of S not in the same m-class. By Assumption V there must be a third element in each of the m-classes AB, Be, CA, and by Assumption II these elements must be distinct from each other and from A, B, and C. Let the new elements be D, E, G, so that each of the triples ABD, BCE, CAG belongs to the same m-class. By Assumption III the m-classes AE and BG, which are distinct from all the m-classes thus far obtained, have an element of S in common, which, by Assumption II, is distinct from those hitherto mentioned; let it be denoted by F, so that each of the triples AEF and BFG belong to the same m-class. No use has as yet been made of Assumption VII. We have, then, the theorem: (1)
4 It will be noted that this test for the consistency of a set of assumptions merely shifts the difficulty from one domain to another It is, however, at present the only test known. On the question as to the pOSSibility of an absolute test of consistency, cf. Hilbert, GrundJagen der Geometrie, 2d ed., Leipzig (1903), p. 18, and Verhandlungen d. III. ill tern. math. Kongresses zu Heidelberg, Leipzig (1904), p. 174; Padoa, L'Enseignement mathhnatique, Vol. V (1903), p. 85.
A Mathematical Science
1699
Any class S subject to Assumptions I-VI contains at least seven elements. Now, making use of Assumption VII, we find that the m-classes thus far obtained contain only the elements mentioned. The m-c1asses CD and AEF have an element in common (by Assumption III) which cannot be A or E, and must therefore (by Assumption VII) be F. Similarly, ACG and the m-class DE have the element G in common. The seven elements A, B, C, D, E, F, G have now been arranged into m-classes according to the table (1')
ABC BCD D E F
D E
G
E
F G F G A ABC
in which the columns denote m-c1asses. The reader may note at once that this table is, except for the substitution of letters for digits, entirely equivalent to Table (1); indeed (1') is obtained from (1) by replacing 0 by A, I by B, 2 by C, etc. We can show, furthermore, that S can contain no other elements than A, B, C, D, E, F, G, For suppose there were another element, T. Then, by Assumption III, the m-classes TA and BFG would have an element in common. This element cannot be B, for then ABTD would belong to the same m-class; it cannot be P, for then AFTE would all belong to the same m-class; and it cannot be G, for then A GTC would all belong to the same m-c1ass. These three possibilities all contradict Assumption VII. Hence the existence of T would imply the existence of four elements in the m-c1ass BFG, which is likewise contrary to Assumption VII. The properties of the class S and its m-classes may also be represented vividly by the accompanying figure (Figure 1). Here we have represented • the elements of S by points (or spots) in a plane, and have joined by a
FIGURE 1
line every triple of these points which form an m~class. It is seen that the points may be so chosen that all but one of these lines is a straight line. This suggests at once a similarity to ordinary plane geometry. Suppose we interpret the elements of S to be the points of a plane, and interpret the
Oswald Veblen and John Wesley Young
1700
m-classes to be the straight lines of the plane, and let us reread our assumptions with this interpretation. Assumption VII is false, but all the others are true with the exception of Assumption III, which is also true except when the lines are parallel. How this exception can be removed we will discuss in the next section, so that we may also regard the ordinary plane geometry as a representation of Assumptions I-VI. Returning to our miniature mathematical science of triples, we are now in a position to answer another important question: To what extent do Assumptions I-VII characterize the class 5 and the m-classes? We have just seen that any class 5 satisfying these assumptions may be represented by Table (1") merely by properly labeling the elements of 5. In other words, if 51 and 52 are two classes 5 subject to these assumptions, every element of 51 may be made to correspond £) to a unique element of 52, in such a way that every element of 52 is the correspondent of a unique element of 51' and that to every m-c1ass of 51 there corresponds an m-class of 52' The two classes are then said to be in one-to-one reciprocal correspondence, or to be simply isomorphic. s Two classes 5 are then abstractly equivalent; Le., there exists essentially only one class 5 satisfying Assumptions I-VII. This leads to the following fundamental notion: A set of assumptions is said to be categorical, if there is essentially only one system, for which the assumptions are valid; i.e., if any two such systems may be made simply isomorphic. We ha-.ve just seen that the set of Assumptions I-VII is categorical. If, however, Assumption VII be omitted, the remaining set of six assumptions is not categorical. We have already observed the possibility of satisfying Assumptions I-VI by ordinary plane geometry. Since Assumption III, however, occupies as yet a doubtful position in this interpretation, we give another, which, by virtue of its simplicity, is peculiarly adapted to make clear the distinction between categorical and noncategoricaL The reader will find, namely, that each of the first six assumptions is satisfied by interpreting the class 5 to consist of the digits O~ 1, 2, "', 12, arranged according to the following table of m-classes, every column constituting one m-c1ass:
o (2)
1
1
2
4
5
2
345
6
3
4
9
10
3
567
8
6 7 9
12
1
2
11
0
7 8 10
8 9 11
3
4
9 10 10 11 12
5
0
11 12 12 0 1 2
678
Hence Assumptions I-VI are not sufficient to characterize completely the class $, for it is evident that Systems (1) and (2) cannot be made iso5 The notion of correspondence is another primitIve notion which we take over without discussIOn from the general logic of classes. 6 The. iso;norphlsm of Systems (1) and (1') IS clearly exhibited In Figure 1, where each PRint IS labeled both with a digit and With a letter. This isomorphism may, moreover, he estabhshed in 7·6· 4 dIfferent ways.
A Matllematical Science
1701
morphic. On the other hand, it should be noted that all theorems derivable from Assumptions I-VI are valid for both (1) and (2). These two systems are two essentially different concrete representations of the same mathematical science This brings us to a third question regarding our assumptions: Are they independent? That is, can anyone of them be derived as a logical consequence of the others? Table (2) is an example which shows that Assumption VII is independent of the others. because it shows that they can all be true of a system in which Assumption VII is false. Again, if the class S is taken to mean the three letters A; B, e, and the m-classes to consist of the pairs AB, Be, eA, then it is clear that Assumptions I, II, III, N, VI, VII are true of this class S, and therefore that any logical consequence of them is true with this interpretation. Assumption V, however, is false for this class, and cannot, therefore, be a logical consequence of the other assumptions. In like manner, other examples can be constructed to show that each of the Assumptions I-VII is independent of the remaining ones. 3.
IDEAL ELEMENTS IN GEOMETRY
The miniature mathematical science which we have just been studying suggests what we must do on a larger scale in a geometry which describes our ordinary space. We must first choose a set of undefined elements and a set of fundamental assumptions. This choice is in no way prescribed a priori, but, on the contrary, is very arbitrary. It is necessary only that the undefined symbols be such that all other elements and relations that occur are definable in terms of them; and the fundamental assumptions must satisfy the prime requirement of logical consistency, and be such that all other propositions are derivable from them by formal logic. It is desirable, further, that the assumptions be independent 7 and that certain sets of assumptions be categorical. There is, further, the desideratum of utmost symmetry and generality in the whole body of theorems. The latter means that the applicability of a theorem shall be as wide as possible. This has relation to the arrangement of the assumptions, and can be attained by using in the proof of each theorem a minimum of assumptions. 8 Symmetry can frequently be obtained by a judicious choice of terminology. This is well illustrated by the concept of "points at infinity" which is fundamental in any treatment of projective geometry. Let us note first the reciprocal character of the relation expressed by the two statements: '7 This is obviously necessary for the precise distinction between an assumption and a theorem. S If the set of assumptions used in the proof of a theorem is not categorical, the applicability of the theorem is evidently wider than In the contrary case. Cf. example of preceding section.
1702
A point lies on a line.
Oswald Veblen and John Wesley Youn,
A line passes through a point.
To exhibit clearly this reciprocal character, we agree to use the phrases
.
A pOint is on a line;
A line is on a point
to express this relation. Let us now consider the following two tions:
1. Any two distinct points oj a plane are on one and only one line. 9
proposi~
1'. Any two distinct lines oj a plane are on one and only one pOint.
Either of these propositions is obtained from the other by simply interchanging the words point and line. The first of these propositions we recognize as true without exception in the ordinary Euclidean geometry. The second, however, has an exception When the two lines are parallel. In view of the symmetry of these two propositions it would clearly add much to the symmetry and generality of all propositions derivable from these two, if we could regard them both as true without exception. This can be accomplished by attributing to two parallel lines a point oj intersection. Such a point is not, of course, a point in the ordinary sense; it is to be regarded as an ideal point, which we suppose two parallel lines to have in common. Its introduction amounts merely to a change in the ordinary terminology. Such an ideal point we call a point at infinity; and we suppose one such point to exist on every line. 10 The use of this new term leads to a change in the statement, though not in the meaning, of many familiar propositions, and makes us modify the way in which we think of points, lines, etc. Two non-parallel lines cannot have in common a point at infinity without doing violence to propositions 1 and 1'; and since each of them has a point at infinity, there must be at least two such points. Proposition 1, then, requires that we attach a meaning to the notion of a line on two points at infinity. Such a line we call a line at infinity, and think of it as consisting of aU the points at infinity in a plane. In like manner, if we do not confine ourselves to the points of a single plane, it is found desirable to introduce the notion of a plane through three points at infinity which are not all on the same line at infinity. Such a plane we call a plane at infinity, and we think of it as consisting of all the points at infinity in space. Every ordinary plane is supposed to contain just one line at infinity; every system of parallel planes in space is supposed to have a line at infinity in common with the plane at infinity, etc. By lme throughout we mean straight line should be noted that (since we are taking the point of view of Euclid) we do not thin'!' of a hne as containing more than one point at infinity; for the supposition that a hne contains two such points would imply either that two parallels can be drawn through a given point to a given line, or that two distinct lines can have more than one point in common. 9
10 It
A Mathematical Science
1703
The fact that we have difficulty in presenting to our imagination the notions of a point at infinity on a line, the line at infinity in a plane, and the plane at infinity in space, need not disturb us in this connection, provided we can satisfy ourselves that the new terminology is self-consistent and cannot lead to contradictions. The latter condition amounts, in the treatment that follows, simply to the condition that the assumptions on which we build the subsequent theory be consistent. That they are consistent will be shown at the time they are introduced. The use of the new terminology may, however, be justified on the basis of ordinary analytic geometry. This we do in the next section, the developments of which will, moreover, be used frequently in the sequel for proving the consistency of the assumptions there made.
.
4.
CONSISTENCY OF THE NOTION OF POINTS, LINES, AND PLANE AT INFINITY
We will now reduce the question of the consistency of our new terminology to that of the consistency of an algebraic system. For this purpose we presuppose a knowledge of the elements of analytic geometry of three dimensions. In this geometry a point is equivalent to a set of three numbers (x, y, z). The totality of ,all such sets of numbers constitute the analytic space of three dimensions. If the numbers are all real numbers, we are dealing with the ordinary "real" space; if they are any complex numbers, we are dealing with the ordinary "complex" space of three dimensions. The following discussion applies primarily to the real case. A plane is the set of all points (number triads) which satisfy a single linear equation
ax + by + cz + d
=o.
A line is the set of all points which satisfy two linear equations, alx a2x
+ b1y + CIZ + d1 = 0, + b2y + C2Z + d 2 = 0,
provided the relations
do not hold. l1 Now the points (x, y, z), with the exception of (0, 0, 0), may also be 11 It should be noted that we are not yet, in this section, supposing anything known regarding points, lines, etc., at infinity, but are placing ourselves on the basis of elementary geometry.
Oswald Veblen and lohn Wesley Youne
1704
denoted by the direction cosines of the line joining the point to the origin of coordinates and the distance of the point from the origin~ say by
x
where d ::;; Vx':!.
z
y
+ y:! + z:!, and 1::;; -, m ::;; -, d
n
= -. The origin itself may
d
d
be denoted by (0,0,0, k), where k is arbitrary. Moreover, any four numbers (Xl' X2, X3, X4) (X.j, =F 0), proportional respectively to
(I,
n,~}
In,
will serve equally well to represent the point (x, y, z), provided we agree that (Xl' X2, XII, X4) and (cxt> CX!"!! cXs, CX4) represent the same point for all values of e different from O. For a point (x, y, z) determines Xl
=
Xa
=
'CX
yx2 + y2 + Z2 cz
yx2 + y2 + Z2
cy
=el,
X2=
== cn,
X4::;;
=cm,
c
where e is arbitrary (c =F 0), and (1)
yx2 + y2 + :;2
C
y'X2 + y2 + Z2
(Xl' X2, Xg, X4)
--,
d
determines
Xz
y=-, X4
provided X 4 O. We have not assigned a meaning to
(Xl' X2, xs, X4)
is evident that if the point ( d, em, en,
when
Xi
== 0, but it
~) moves away from the origin
an unlimited distance on the line whose direction cosines are 1, m, n, its coordinates approach (el, cm, en, 0). A little consideration will show that as a point moves on any other line with direction cosines I, m, n, so that its distance from the origin increases indefinitely, its coordinates also approach (el, cm, en, 0). Furthermore, these values are approached, no matter in which of the two opposite directions the point moves away from the origin. We now define (X11 X2, Xg, 0) as a point at infinity or an ideal point. We have thus associated with every set of four numbers (Xl' X2' X3, X 4 ) a point, ordinary or ideal, with the exception of the set (0, O. 0, 0), which we exclude entirely from the oiscussion. The ordinary points are those for which X.j, is not zero; their ordinary Cartesian coordi-
1705
A MathematIcal SCience
nates are given by the equations (1). The ideal points are those for which X4 O. The numbers (Xl> X2, xs, X4) we call the homogeneous coordinates of the point. We now define a plane to be the set of all points (Xl, X2, xs, X4) which satisfy a linear homogeneous equation:
=
axl
+ bX2 + CXs + dX4 = O.
It is at once clear from the preceding discussion that as far as all ordi-
nary points are concerned, this definition is equivalent to the one given at the beginning of this section. However, according to this definition all the ideal points constitute a plane X4 0. This plane we call the plane at infinity. In like manner, we define a line to consist of all points (Xl. X2. Xa, X4) which satisfy t~o distinct linear homogeneous equations:
=
a1Xl a2 x l
+ b1X 2 + CIXa + dlX4 = 0, + b2X 2 + C2Xa + d2X4 = O.
Since these expressions are to be distinct, the corresponding coefficients throughout must not be proportional. According to this definition the points common to any plane (not the plane at infinity) and the plane X4 = 0 constitute a line. Such a line we call a line at infinity, and there is one such in every ordinary plane. Finally, the line defined above by two equations contains one and only one point with coordinates (Xl' X2, X~, 0); that is, an ordinary line contains one and only one point at infinity. It is readily seen, moreover, that with the above definitions two parallel lines have their points at infinity in common. Our discussion has now led us to an analytic definition of what may be called, for the present, an analytic projective space of three dimensions. It may be defined, in a way which allows it to be either real or complex, as consisting of: Points: All sets of four numbers (Xl, X2, Xa, X4), except the set (0, 0, 0, 0), where (exl, eX2. exs, CX4) is regarded as identical with (Xl' X2. Xa, X4). provided C is not zero. Planes: All sets of points satisfying one linear homogeneous equation. Lines: All sets of pajnts satisfying two distinct linear homogeneous equations. Such a projective space cannot involve contradictions unless our ordinary system of real or complex algebra is inconsistent. The definitions here made of points. lines, and the p1ane at infinity are, however, precisely equivalent to the corresponding notions of the preceding section. We may therefore use these notions precisely in the same way that we consider ordinary points, lines, and planes. Indeed, the fact that no exceptional properties attach to our ideal elements follows at once from the symmetry of the analytic formulation; the coordinate X4- whose vanishing gives rise
Oswald Veblen and John Wesley Young
1706
to the ideal points~ occupies no exceptional position in the algebra of the homogeneous equations. The ideal points~ then, are not to be regarded as different from the ordinary points. All the assumptions we shall make in our treatment of projective geometry will be found to be satisfied by the above analytic creation, which therefore constitutes a proof of the consistency of the assumptions in question. • . . 5.
PROJECTIVE AND METRIC GEOMETRY
In projective geometry no distinction is made between ordinary points and points at infinity, and it is evident by a reference forward that our assumptions provide for no such distinction. We proceed to explain this a little more fully, and will at the same time indicate in a general way the difference between projective and the ordinary Euclidean metric geometry. Confining ourselves first to the plane, let m and m' be two distinct lines, and P a point not on either of the two lines. Then the points of m may be made to correspond to the points of m' as follows: To every point A on m let correspond that point A' on m' in which m' meets the line joining A to P (Figure 2). In this way every point on either line is assigned a unique corresponding pOint on the other line. This type of correspondence is called perspective, and the pOints on one line are said to be transformed into the points of the other by a perspective transformation with center P. If the points of a line m be transformed into the points of a line m' by a perspective transformation with center P, and then the points of m' be transformed into the points of a third line m" by a perspective transformation with a new center Q; and if this be continued any finite number of times~ ultimately the points of the line m will have been brought into correspondence with the points of a line m(n), say, in such
... .-.
..... ,-.---_ .... _---" " , , " , , ,,
, ," , ,
,"
, FIGURE 2
A Mathematical SCIence
1707
a way that every point of m corresponds to a unique point of m (n). A correspondence obtained in this way is called projective, and the points of m are said to have been transformed into the points of m (n) by a projective transformation. Similarly, in three-dimensional space, if lines are drawn joining every point of a plane figure to a fixed point P not in the plane 1r of the figure, then the points in which this totality of lines meets another plane 1r~ will form a new figure, such that to every point of 1r will correspond a unique point of 1r', and to every line of 7f will correspond a unique line of 1r'. We say that the figure in 1r has been transformed into the figure in 1r' by a perspective transformation with center P. If a plane figure be subjected to a succession of such perspective transformations with different centers, the final figure will still be such that its points and lines correspond uniquely to the points and lines of the original figure. Such a transformation is again called a projective transformation. In projective geometry two figures that may be made to correspond to each other by means of a prOjective transformation are not regarded as different. In other words, projective geometry is concerned with those properties of figures that are left unchanged when the figures are subjected to a projective transformation. It is evident that no properties that involve essentially the notion of measurement can have any place in projective geometry as such; 12 hence the term projective, to distinguish it from the ordinary geometry, which is almost exclusively concerned with properties involving the idea of measurement. In case of a plane figure, a perspective transformation is clearly equivalent to the change brought about in the aspect of a figure by looking at it from a different angle, the observer's eye being the center of the perspective transformation. The properties of the aspect of a figure that remain unaltered when the observer changes his position will then be properties with which projective geometry concerns itself. For this reason von Staudt called this science Geometrie der Lage. In regard to the points and lines at infinity, we can now see why they cannot be treated as in any way different from the ordinary points and lines of a figure. For, in the example given of a perspective transformation between lines, it is clear that to the point at infinity on m corresponds in general ,an ordinary point on m', and-conversely. And in the example given of a perspective transformation between planes we see that to the line at infinity in one plane corresponds in general an ordinary line in the other. In projective geometry, then, there can be no distinction between the ordinary and the ideal elements of space. 12 The theorems of metric geometry may however be regarded as special cases of projective theorems.
The observation of phenomena cannot tell us anythmg more than that the mathematical equations are correct: the same equatIOns might equally well represent the behavior of some other material system. For example, the VibratIOns of a membrane whlch has the shape uj an ellipse can be calculated by means oj a differential equation known as Mathieu's equation: but this same equation is also arrived at when we study the dynamics of a circus performer who holds an assistant balanced on a pole while he himself stands on a spherical ball rolling on the ground. If now we Imagine an observer who discovers that the future course of a certain phenomenon can be predicted by Mathieu's equation, but who is unable for some reason to perceive the system which generated the phenomenon, then evidently he would be unable to tell whether the system in question zs an eliiptlc membrane or a variety artist. -SIR EDMUND
6
T.
WHI1TAKER
Mathematics and the World By DOUGLAS GASKING
MY OBJECT is to try to elucidate the nature of mathematical propositions, and to explain their relation to the everyday world of counting and meas· urement-of clocks, and yards of material, and income~tax forms. I should like to be able to summarize my views in a few short phrases, and then go on to defend them. Unfortunately I cannot do this, for, as I shall try to demonstrate, I do not think any short statement will do to express the truth of the matter with any precision. So I shall proceed by approxi· mations-I shall discuss several different views in the hope that in showing what is right and what is wrong with them, clarification will come about. The opinions of philosophers about the nature of mathematical propositions can be divided, as can their opinions about so many things, into two main classes. There are those who try to analyse mathematical propositions away-who say that they are really something else (like those writers on ethics who say that goodness is really only pleasure, or those metaphysicians who say that chairs and tables are really groups of sensations, or colonies of souls). I shall call such 'analysing-away' theories 'radical' theories. On the other hand there are those who insist that mathematical propositions are sui generis, that they cannot be analysed into anything else, that they give information about an aspect of reality totally different from any other (compare those philosophers who maintain, e.g., that goodness is a simple un analysable quality, or those realists who maintain that a chair is a chair, an external material substance, known, perhaps, by means of sensations, but not to be confused with those sensations). For convenience, I shall call these types of theory which 1708
Mathematics and the World
1709
oppose any analysing-away, 'conservative.' I should maintain that in general what I call 'conservative' opinions in philosophy are perfectly correct, but rather unsatisfactory and unilluminating, whereas opinions of the 'radical' type are untrue, but interesting and illuminating. I shall start by considering the 'radical' theories about the nature of mathematics. Those I know of fall into two main types. (1) Some people maintain that a proposition of mathematics is really a particularly wellfounded empirical generalization of a certain type, or that it is logically on the same footing as a very well-established scientific law. Mill's theory was of this type, and many scientists I have talked to have tended to have similar opinions. Let us call these 'empirical' theories about mathematics. (2) Then, on the other hand, there is a great variety of theories usually called 'conventionalist,' which analyse away mathematical propositions into propositions about the use of symbols. Examples: 'By a mathematical proposition the speaker or writer merely expresses his intention of manipulating symbols in a certain way, and recommends or commands that others should do likewise.' 'A mathematical proposition is really an empirical proposition describing how educated people commonly use certain symbols.' 'A mathematical proposition is really a rule for the manipUlation of symbols.' (Ayer, for example, and C. 1. Lewis have expressed opinions of this general type.) First for the 'empirical' theories. According to these a mathematical proposition just expresses a particularly well-founded empirical generalization or law about the properties and behaviour of objects, obtained by examining a large number of instances and seeing that they conform without exception to a single general pattern. The proposition '7 + 5 = 12,' for instance, just expresses (on one version of this theory) the fact of experience that if we count up seven objects of any sort, and then five more objects, and then count up the whole lot, we always ,get the number twelve. Or again. it might be maintained that the geometrical proposition 'Equilateral triangles are equiangular' just expresses the fact that wherever, by measurement, we find the sides of a triangle to be equal, we will find, on measuring the angles with the same degree of accuracy, that the angles are equal too. It is contended that such propositions are essentially like, for example, Boyle's Law of gases, only much better founded. But '7 + 5 = 12' does not mean the same as the proposition about what you get on counting groups. For it is true that 7 + 5 does equal 12, but it is not true that on counting seven objects and then five others, and then counting the whole, you will always get twelve. People sometimes miscount, and sometimes the objects counted melt away (if they are wax) or coalesce (if they are globules of mercury). Similarly the geometrical proposition that equilateral triangles are equiangular does not mean the same as the proposition that any triangle which is equilateral by measure-
Douglas Gasklllg
1710
ment will be found to be equiangular when measured. The former is true; the ,.latter false. We sometImes make mistakes with our rulers and protractors. To thIS it might be objected that this shows that the empIrical proposition offered as a translation of the mathematIcal one is not a correct translation, but that It has not been demonstrated that It is impossible to find an empirical proposition about counting and measurement, which is a correct translation. Let us try some alternatives, then. It mIght be suggested that '7 + 5 12' means 'If you count carefully and with attention, you will get such and such a result.' But, even with the greatest care in countmg, mistakes sometImes happen at any rate with large numbers. Shall we then say. '7 + 5 12' means 'If you count correctly YOtI will get such and such results'? But, In the first place, even If you count objects correctly, you do not always get a group of seven objects and a group of five addmg up to twelve. It sometimes happens that a person correctly counts seven objects, then correctly counts five, and then correctly counts the total and gets eleven. Sometimes one of the objects does disappear in the course of counting, or coalesces with another. And even If this were not so, the suggested translation would not gIve you a simple empirical propositIOn about what happened when people counted, as a translation of 7 + 5 = 12, but would gIve you a mere tautology. For what ... is the criterion of correctness m counting? Surely that when you add seven and five you should get twelve. 'Correctness' has no meaning, in thIS context, independent of the mathematIcal propositIOn. So our suggested analysIs of the meaning of '7 + 5 = 12' runs, when suitably expanded: '7 + 5 = 12' means 'If you count objects correctly (Le., In such a way as to get 12 on adding 7 and 5) you WIll, on adding 7 to 5, get 12.' No doubt there are Important connections between mathematical propoSItIons, and propOSItIons about what results people WIll usually get on countIng and measuring But It will not do to say that a mathematical proposition means the same as, or IS equivalent to, any such empirical proposition, for this reason: A mathematical propositIon is 'IncorrigIble: whereas an empmcal propOSItIOn is 'corrigIble.' The difference between 'corrIgible' and 'mcorrigible' propositions can best be explaIned by examples. Most everyday assertions that we make, such as that 'Mr. Smith has gone away for the day,' are corrigible. By this I mean simply that, whenever we make such an assertion, however strong our grounds for makmg it, we should always freely withdraw it and admit it to have been false, if certain things were to happen. Thus my assertion, that Smith is away for the day, is corrigible, because (although I may have the excellent grounds for making it that when I met hIm in the street this morning he SaId he was on hIS way to the railway-station) if, for example, I were to go to his room now and find him sitting there, I
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MathematIcs and the World
1711
should withdraw my assertion that he was away and admit it to have been false. I should take certain events as proving, if they happened, that my assertion was untrue. A mathematical proposition such as '7 + 5 = 12: on the other hand, is incorrigible, because no future happenings whatsoever would ever prove the proposition false, or cause anyone to withdraw it. You can imagine any sort of fantastic chain of events you like, but nothing you can think of would ever, if it happened, disprove '7 + 5 12: Thus, if I coun~ed out 7 matches, and then 5 more, and then on counting the whole lot, got 11, this would not have the slightest tendency to make anyone withdraw the proposition that 7 + 5 = 12 and say it was untrue. And even if this constantly happened, both to me and to everyone else, and not only with matches, but with books, umbrellas and every sort of object -surely even this would not make us withdraw the proposition. Surely in such a case we should not say: 'the proposition "7 + 5 = 12" has been empirically disproved; it has been found that 7 + 5 really equals 11.' There are plenty of alternative explanations to choose from. We ~ight try a psychological hypothesis, such as this: we might say that it had been discovered by experiment that everyone had a curious psychological kink, which led him, whenever he performed counting operations of a certain sort, always to miss out one of the objects in his final count (like the subject in some experiments on hypnosis who, under suggestion, fails to see any 't's on a printed page). Or we might prefer a physical hypothesis and say: a curious physical law of the universe has been experimentally established, namely, that whenever 5 objects are added to 7 objects, this process of addition causes one of them to disappear, or to coalesce with another object. The one thing we should never say, whatever happened, would be that the proposition that 7 + 5 = 12 had been experimentally disproved. If curious things happened, we should alter our physics, but not our mathematics. This rather sweeping assertion that mathematical propositions are completely incorrigible is, I think, an over-simplification, and needs qualifying. I shall mention the qualifications later, rather than now, for simplicity of exposition. So if you will accept it for the moment as very nearly true, I should like to draw your attention to certain of its conse. quences. A corrigible proposition gives you some information about the world-a completely incorrigible proposition tells you nothing. A corrigible proposition is one that you would withdraw and admit to be false if certain things happened in the world. It therefore gives you the information that those things (Le., those things which would make you withdraw your proposition if they happened) will 'not happen. An incorrigible proposition is one which you would never admit to be false whatever happens: it therefore does not tell you what happens. The truth, for example, of the
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Douglas Gasking
1712
corrigible proposition that Smith is away for the day, is compatible with certain things happening (e.g. t your going to his room and finding it empty) and is not compatible with certain other happenings (e.g., your going to his room and finding him there). It therefore tells you what sort of thing will happen (you will find his room empty) and what sort of thing will not happen (you will not find him in). The truth of an incorrigible proposition, on the other hand, is compatible with any and every conceivable state of affairs. (For example: whatever is your experience on counting, it is still true that 7 + 5 12.) It therefore does not tell you which events will take place and which will not. That is: the proposition '7 + 5 12' tells you nothing about the world. If such a proposition tells you nothing about the world, what, then, is the point of it-what does it do? I think that in a sense it is true to say that it prescribes what you are to say-it tells you how to describe certain happenings. Thus the proposition '7 + 5 12' does not tell you that on counting 7 + 5 you will not get 11. (This, as we have seen, is false, for you sometimes do get 11.) But it does lay it down, so to speak, that if on counting 7 + 5 you' do get 11. you are to describe what has happened in some such way as this: Either 'I have made a mistake in my counting' or 'Someone has played a practical joke and abstracted one of the objects when I was not looking' or 'Two of the objects have coalesced' or 'One of the objects has disappeared; etc. This, I think, is the truth that is in the various 'conventionalist' theories of mathematics. Only, unfortunately, the formulae expressing such theories are usually misleading and incorrect. For example, to say that: 'a mathematical proposition merely expresses the speaker's or writer's determination to use symbols in a certain way,' is obviously untrue, For if it were true, and if I decided to use the symbol '+' in such a way that 5 + 7 = 35. I would then be speaking truly if I said '5 + 7 = 35.' But this proposition is not true. The truth of any mathematical proposition does not depend on my decision or determination. It is independent of my will. This formula neglects the 'public' or 'over-individual' character of mathe... matics. Or. consider the formula: 'A mathematical proposition is really an empirical statement describing the way people commonly use certain symbols.' This) I think, is nearer. But it is open to the following obvious objection: If '7 + 5 12' were really an assertion about the common usage of symbols, then it would follow that 7 + 5 would not equal 12 if people had a different symbolic convention. But even if people did use symbols in a way quite different from the present one, the fact which we now express by '7 + 5 12' would still be true. No change in our language-habits would ever make this false. This objection is, I think, sufficient to show that the suggested formula
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Mathemailcs and the World
1713
is untrue, as it stands. But we should be blind to its merits if we did not see why it is that no change in our language-habits would make the proposition '7 + 5 = 12' untrue. The reason is this: As we use symbols at present, this proposition is incorrigible--one which we maintain to be true whatever happens in the world, and never admit to be false under any circumstances. Imagine a world where the symbolic conventions are totally different-say on Mars. How shall we translate our incorrigible proposition into the Martian symbols? If our translation is to be correct-if the proposition in the Martian language is to mean the same as our '7 + 5 = 12,' it too must be incorrigible-otherwise we should not call it a correct translation. Thus a correct Martian translation of our '7 + 5 = 12' must be a proposition which the Martians maintain to be true whatever happens. Thus '7 + 5 = 12,' and any correct translation into any other symbolic convention will be incorrigible, i.e., true whatever happens. So its truth does, in a sense, depend on the empirical fact that people use symbols in certain ways. But it is an inaccurate way of stating this fact to say that it describes how people use symbols. . A better formulation is: 'A mathematical proposition really expresses a rule for the manipulation of symbols.' But this, too, is unsatisfactory, and for the following reason: To say that it is a 'rule for the manipulation of symbols' suggests that it is entirely arbitrary. A symbolic rule is something which we can decide to use or not, just as we wish. (We could easily use 'hice' as the plural of 'house,' and get on as well as we do now.) But, it seems, we cannot just change our mathematical propositions at will, without getting into diofficulties. An engineer, building a bridge, has to use the standard multiplication tables and no others, or else the bridge will collapse. Thus which mathematical system we use does not seem to be entirely arbitrary-one system works in practice, and another does not. Which system we are to use seems to depend in some way not on our decision, but on the nature of the world. To say that '7 + 5 = 12' really expresses a rule for the use of symbols, suggests that this proposition is just like' "house" forms its plural by adding "s": But there is a difference between the two, and so the formula is misleading. I want to conclude this paper by considering in some detail the objection that you cannot build bridges with any mathematics, and that therefore mathematics does depend on the nature of reality. Before doing so, however, I should like to mention the type of theory I called 'conservative.' We saw that the (radical) theory, that mathematical propositions are 'really' empirical propositions about the results of counting, is untrue. But there is a close connection between the two sorts of proposition, and therefore the 'empirical' theory, although untrue, has a point. It emphasizes the connection between mathematical propositions and our everyday practice of counting and calculation; thus it serves as a useful corrective
Douglas Gasklng
1714
to that type of theory which would make mathematics too abstract and pure-a matter of pure intellect and Platonic 'Forms,' far from the mundane counting of change. Similarly the various 'conventionalist' theories are also, strictly speaking, untrue, but they too have their point. Mathematical propositions in certain respects are like rules for the use of symbols, like empirical propositions about how symbols are used, like statements of intention to use symbols in certain ways. But conventionalist formulae are untrue because mathematical propositions are not identical with any of these. They are what they are; they function in the way they do, and not exactly like any other sort of proposition. And this it is which makes that sort of theories I have called 'conservative' perfectly correct. Mathematical propositions are sui generis. But merely to say: 'They are what they are' is not very helpful. Nor is it any better if this is dressed up in learned language: e.g., 'Mathematical propositions state vel''';;' ~eneral facts about the structure of reality; about the necessary and synthetic relations between the universals number, shape, size, and so on.' If you are inclined to think that such answers as this, to the question 'What are mathematical propositions about?', are informa~ tive and illuminating, ask yourself: 'How does any hearer come to understand the meaning of such phrases as "structure of reality," "necessary relations between universals," and so on? How were these phrases explained to him in the first place?' Surely he was told what was meant by 'necessary relation between universals,' by being told, for example, that colour, shape, size, number, etc., are universals, and that an example of a necessary relation between universals would be 'everything that has shape has size/ '2 + 2 = 'two angles of an isosceles triangle are equal; and so on. These phrases, such as 'necessary relation between universals,' are introduced into his language via or by means of such phrases as '2 + 2 = 4'; they are introduced via mathematical propositions, among others. To use an expression of John Wisdom's,l they are 'made to measure.' So to tell someone that mathematical propositions are 'so-and~so' does not help, if, in explaining what is meant by 'so-and-so" you have to introduce mathematical propositions, among others, as illustrative examples. Compare giving a 'conservative' answer to the question 'What are mathematical propositions?' with the following example: A child learns the meaning of the words 'see,' 'can't see,' 'blindfolded' etc., before he learns the meaning of the word 'blind: The latter word is then introduced into his vocabulary by the explanation; 'A blind man is one who can't see in broad daylight even when not blindfolded.' If the child then asks of a blind man 'Why can't he see in broad daylight even when not blindfolded?', it is not much use answering 'Because he is blind.' Like the 'con-
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i My debt to the lectures of Wisdom and Wittgenstein, in writing this paper, is very great.
Mathematics and the World
1715
servative' answer in philosophy, it may serve to stop any further questions, but it usually leaves a feeling of dissatisfaction. Then what sort of answer can be given to one who is puzzled about the nature of mathematics? Mathematical propositions are what they are, so any radical answer equating them with something else, such as symbolic rules, or statements of the results of counting and measurement, or of common symbolic usage, will be untrue. Such answers will be untrue, because the two sides of the equation will have different meanings. Simi8 lady conservative answers will be unhelpful, because the two sides of the equation will have the same meaning. The definiens will be useless, because it will contain terms which are introduced into the language via the definiendum, and can only be explained in terms of it. It is 'made to measure.' No simple fonnula will do. The only way of removing the puzzle is to describe the use and function of mathematical propositions in detail and with examples. I shall now try to do this, to some extent, in considering the natural objection to the strictly untrue but illuminating theory: 'Mathematical propositions express rules for the manipulation of symbols.' The objection is that symbolic rules are essentially arbitrary, whereas mathematics does, to some extent at least, depend not on our choice of symbolic conventions, but on the nature of reality, because only our present system gives useful results when applied to the practical tasks of the world. Against this, I shall maintain that we could use any mathematical rules we liked, and still get on perfectly well in the business of life. Example 1. 6 X 4, according to our current multiplication table, equals 24. You might argue: this cannot be merely a conventional rule for our use of symbols, for if it were we could use any other rule we liked, e.g., 6 X 4:;:: 12, and still get satisfactory results. But if you tried this alternative rule, you would, in fact, find your practical affairs going all wrong. A builder, for example, having measured a room to be paved with tiles, each one yard square, and having found the length of the sides to be 6 yards and 4 yards, could not use the alternative table. He could not say to himself: 'The room is 6 by 4; now 6 X 4 12, so I shall have to get 12 tiles for this job.' For, if he did, he would find he had not enough tiles - to cover his floor. But the builder could quite easily have used an arithmetic in which 6 X 4 12, and by measuring and counting could have paved his room perfectly well, with exactly the right number of tiles to cover the fioor. How does he do it? Well, he: (1) Measures the sides, and writes down '4' and '6.' (2) Multiplies 4 by 6 according to a 'queer' multiplication table which gives 4 X 6 12. (3) Counts out 12 tiles, lays them on the fioor. And they fit perfectly.
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1716
The 'queer' multiplication table he uses gives 2 X 2 :::: 4, 2 X 4 :::: 6, 2 X 8 :::: 10, 4 X 4 :::: 9, 4 X 6:::: 12, etc. The number found by multiplying a by b according to his table, is that which in our arithmetic we should get by the formula: (a + 2)(b + 2)/4
And he could pave any other size of fioor, using the queer multiplication table described, and still always get the right number of tiles to cover it. How is this possible? He measures the sides 'of the room with a yardstick as follows: He lays his yardstick along the longer side, with the '0' mark of the yardstick in the corner, and the other end of the stick, marked '36 inches: some distance along the stick. As he does this, he counts 'one.' He then pivots the yardstick on the 36 inches mark, and swings it round through two right angles, till it is once more lying along the side of the room-this time with the '36 inches' mark nearer to the corner from which he started, and the '0' mark further along the side. As he does this, he counts 'two.' But now the direction of the stick has been reversed, and it is the convention for measuring that lengths should always be measured in the same direction. So he pivots the stick about its middle and swings it round so that the '36' mark is now where the '0' mark was, and vice versa. As he does this, he counts 'three.' He then swings the stick round through two right angles, pivoting on the '36' mark, counting 'four.' He then reverses its direction, as before, counting 'five.' He swings it over again, counting 'six.' It now lies with its end in the corner, so he writes down the length of the side as 'six yards.' (If we had measured it in our way, we should have written its length down as four yards.) He then measures the shorter side in the same way, and finds the length (using his measuring technique) to be four yards. (We should have made it three.) He then multiplies 4 by 6, according to his table, making it 12, counts out 12 tiles, and lays them down. So long as he uses the technique described for measuring lengths, he will always get the right number of tiles for any room with his 'queer' multiplication table. This example shows you that we use the method we do for multiplying lengths to get areas, because we use a certain method of measuring lengths. Our technique of calculating areas is relative to our technique of measuring lengths. Here you might say: Admitting that this is true, it is still the case that mathematics is not arbitrary, for you could not use the method 'of measuring we do, and a different multiplication table, and still get the right number of tiles for our room. Can't we? Let's see. Example 2. Suppose our 'queer' multiplication table gave 3 X 4 24. The builder measures the sides of a room exactly as we do. and finds that
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Mathematics and the World
1717
they are 3 yards and 4 yards, respectively. He then 'multiplies' 3 by 4. and gets 24. He counts out 24 tiles, places them on the fioor. and they fit perfectly, with none over. How does he do it? He measures the sides as we do, and writes down '3' and '4.' He 'multiplies' and gets 24. He then counts out 24 tiles as follows: He picks up a tile from his store, and counts ·one.' He puts the tile on to his truck and counts 'two.' He picks up another tile and counts 'three.' He puts it on his truck and counts 'four.' He goes on in this way until he reaches a count of 'twenty-four.' He then takes his 'twenty-four' tiles and paves the room, and they fit perfectly. This example shows that our technique of calculating areas is relative both to a certain technique of measurement, and to a certain technique of counting. At this stage you might make a further objection. You might say: Mathematics does tell you something about the world, and is not an arbitrary rule of symbolic usage. It tells you that if you both count and measure lengths in the way we do, you will not get the right number of tiles for a room unless you multiply the lengths according to our present table. It is not arbitrary, because if, for example, you measure the sides of a room as we do, and find them to be 4 and 3, and if you count tiles as we do, you would get the wrong number of tiles to pave your room if you used some other multiplication table-say one in which 3 X 4 = 24. I maintain, on the contrary. that we could quite well use such a 'queer' table, and count and measure as at present, and still get the right number of tiles. To help us to see what is involved here, let us consider a rather analogous case. Example 3. Imagine that the following extraordinary thing happened. You measure a room normally, and find the sides to be 6 and 4. You multiply normally and get 24. You then count out 24 tiles in the normal way. (Each tile is 1 X 1.) But when you come to try and lay the tiles in the room, you find that you can only get 12 such tiles on to the fioor of the room, and there are 12 tiles over. What should we say if this happened? The first thing we should say would be: 'You must have made a mistake in your measuring' or 'You must have made a slip in mUltiplying' or 'You must have counted your tiles wrongly, somehow: And we should immediately check again the measurements, calculations and counting. But suppose that t after the most careful checking and re-checking, by large numbers of highly qualified persons, no mistake at all of this sort can be found anywhere. Suppose, morever, that this happened to everyone con~ standy, with all sorts of rooms and tiles. What should we say then? There are still a number of ways in which we might explain this curious phenomenon. I shall mention two conceivable hypotheses:
1718
Douglas Gaskmg
( 1) Measuring rods do not, as we supposed, stay a constant length wherever they are put. They stay the same size when put against things the same length as themselves, and also when put against things larger than themselves running from north to south. But when put against things larger than themselves running east-west, they always contract to half their previous length (and this contraction happens so smoothly that we never notice it). Thus the room is in fact 6 by 2 yards, i.e., 12 square yards, and twelve tiles are needed. When the measuring rod is put along the north-south wall of six yards' length, it stays a yard long, and so we get a measurement of 6. When, however, it is put along the shorter east-west wall it contracts to half a yard in length, and can be put four times along the two-yard wall. If you now say the dimensions are 6 and 4, and mUltiply to get 24, you are overestimating the real area. (2) An alternative hypothesis: When we measure the room our yardstick always stays a constant length, and thus the area of the room is really 24 square yards. But since we can only get 12 tiles in it, each tile being 1 yard square, it follows that the tiles must expand, on being put into the room, to double their area. It is just a curious physical law that objects put into a room double their area instantaneously. We do not see this expansion because it IS instantaneous. And we can never measure it, by measuring the tiles, first out of the room and then inside, because our yardstick itself expands proportionately on being taken into the room. This example (which might easily have been put in much more detail with ad hoc hypotheses to cover every discrepancy) shows that, however much the practical predictions of builders' requirements are upset when we use our present mUltiplication table, this need never cause us to alter our present rules for multiplication. Anomalies are accounted for by saying our knowledge of the relevant physical laws is defective, not by saying that the multiplication table is 'untrue.' If, when working things out in the usual way, we found that we had constantly 12 tiles too many, we should not say that we had been wrong in thinking that 6 X 4 24. We should rather say that we had been wrong in thinking that physical objects did not expand and contract in certain ways. If things go wrong, we always change our physics rather than our mathematics. If we see, from Example 3, what we should do if things went wrong when we used our present arithmetic, we can now answer the objection it was intended to throw light on. The objection was this: 'It is wrong to say that we could use any arithmetic we liked and still get on perfectly well in our practical affairs. Mathematics is not a collection of arbitrary symbolic rules, therefore, and does tell us something about, and does depend on, the nature of reality. For if you both count and measure as we do, and use a Hqueer" multiplication table, you won't get the right number of tiles to pave a room. Thus the proposition "3 X 4
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MathematIcs and the World
1719
= 12" tells you that for a room 3 yards by 4, measured nonnalIy, you need neither more nor less than 12 tiles, counted normally. Its truth de· pends on this fact about the world.' But I deny this. I say we could have ( 1) used our present technique of counting and measurement, (2) multiplied according to the rule 3 X 4 = 24 (for example), (3) and still have got exactly the right number of tiles to pave our room.
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I therefore say that 3 X 4 12 depends on no fact about the world, other than some fact about the usage of symbols. Example 4. Imagine that we did use a 'queer' arithmetic, in which 3 X 4 24. If this was our universally accepted and standard arithmetic, we should treat the proposition '3 X 4 = 24' exactly as we now treat the proposition '3 X 4 - 12' of our present standard arithmetic. That is to say, if we did use this queer system, we should stick to the proposition '3 X 4 = 24' no matter what happened, and ascribe any failure of predic~ tion of builders' requirements, and so on, always to a mistaken view of the physical laws that apply to the world, and never to the untruth of the fonnula '3 X 4 24.' This latter proposition, if it were part of our mathematical system, would be incorrigible, exactly as '3 X 4 = 12' is to us now. In Example 3 we saw what would be done and said if things went wrong in using '3 X 4 = 12.' Now if 3 X 4 24 were our rule, and incorrigible, and if in using it we found ourselves getting the wr~ng practical results, we should do and say exactly the same sort of thing as we did in Example 3. Thus, assuming that our rule is 3 X 4 24, a builder measures his floor normally and writes down '3' and '4.' He multiplies according to his table and gets 24. He counts out 24 tiles normally and tries to put them in the room. He finds that he can only get 12 tiles in. What does he say? He does not say 'I have proved by experiment that 3 X 4 does not equal 24: for his proposition '3 X 4 24' is incorrigible, and no event in the world, however extraordinary, will ever lead him to deny it, or be counted as relevant to its truth or falsity. What he does say is something like this: 'The area of the room is really 24 square yards. Since I can only get 12 yard square tiles into it, it follows that the tiles must expand to double their area on being put into the room.' (As we have seen, he might use other hypotheses, e.g., about the behaviour of measuring rods. But this is probably the most convenient.) Thus we could easily have counted and measured as at present, and used an arithmetic in which 3 X 4 = 24, and have got perfectly satisfactory results. Only. of course, to get satisfactory practical results, we should use a physics different in some respects from our present one. Thus a builder having found the area of a room to be 24 square yards would
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1720
never attempt to put 24 tiles in it, for he would have learnt in his physics lessons at school that tiles put in a room double in area. He would therefore argue: 'Since the tiles double in area, I must put half of 24 tiles, or 12 tiles, in the room.' He would count out 12 tiles and pave the room perfectly with them. But even here an obstinate objector might admit all this, and still maintain that mathematics was not an arbitrary convention; that it did depend on certain facts about the world. He might say' "3 X 4 = 12" is true, and it is true because of this fact about the world, namely that if tiles and rulers do not expand and contract (except slightly with changes in temperature), and if we measure and count normally, we need exactly 12 tiles, no more and no less, to pave a rOom that is 3 by 4. And "3 X 4 = 24" is false, because of the "brute fact" that if tiles, etc., don't expand, and if you measure and count normally, 24 tiles are too many to pave a room that is 3 by 4.' The point that is, I think, missed by this objection could be brought out by asking: 'How do we find out whether a tile or a yardstick has or has not expanded or contracted?' We normally have two ways of doing so. We can watch it growing bigger or smaller. Or we can measure it before and after. Now in the case described in Example 4, where our queer arithmetic gives 3 X 4 24, and things double in area on being put in a room, how do we find out that the things do expand? Not by watching them growex hypothesi we do not observe this. Nor by measuring them before and after. For, since we assume that a measuring rod also expands on being taken into the room, the dimensions of the tile as measured by a yardstick outside the room are the same as its dimensions as measured by the same (now expanded) yardstick inside the room. In this case we find out that the tiles expand by measuring, counting and calculating in a certain way -by finding that the tiles each measure 1 Xl, that the room measures 3 X 4, or 24 square yards, and that we can only get 12 tiles in it. This is our sole criterion for saying that the tiles expand. That the tiles expand follows from our queer arithmetic. Similarly, as we do things at present, our criterion for saying that tiles do not expand, is that when 12 tiles measuring 1 X 1 are put into a room 3 X 4, or 12 square yards, they fit exactly. From our present arithmetic, it follows that tiles do not expand. In Example 4, where we have a 'queer' arithmetic in which 3 X 4 = 24, and a 'queer' physics, it is a 'law of nature' that tiles expand on being put into a room. But it is not a 'law of nature' which describes what happens in the world. Rather is it a law 'by convention,' analogous to that law of our present physics which says that when a body falls on the floor with a certain downward force, the floor itself exerts an equal force in the 0ppo-
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Mathematics and the World
1721
site direction. It is just put into the system to balance our calculations, not to describe anything that happens. This last objection might have been put in a slightly different form. It might have been said: '''3 X 4 12" does describe and depend on the nature of reality, because it entails a certain purely empirical proposition about what does and does not happen, namely the complex proposition. "It is not the case both that tiles do not expand and that we need less than 12 tiles to pave a floor measuring 3 by 4".' But I should maintain that this complex proposition (of the form 'Not both p and q') is not empirical; that it does not describe anything that happens in the world, because it is incorrigible. Nothing whatsoever that we could imagine happening would ever lead us, if it happened, to deny this complex proposition. Therefore it does not tell us what happens in the world. The simple propositions which are elements in this complex one-the propositions 'Tiles do not expand' and 'We need less than 12 tiles to pave a 3 by 4 floor'-are both of them corrigible, and both describe the world (one of them falsely). But the complex proposition that they are not both true is incorrigible, and therefore, for the reasons given earlier, does not describe or depend on the nature of the world. There is nothing out of the ordinary about this. The propositions 'My curtains are now red over their whole surface,' and 'My curtains are now green all over' are both of them corrigible propositions, descriptive of the world. (One is true, the other false, as a matter of fact.) But the complex proposition 'My curtains are not both red and green over their whole surface' is incorrigible, because nothing would ever make me deny it, and it is therefore not descriptive of the world. I have talked, throughout the paper, as if mathematical propositions were completely incorrigible, in the sense that whatever queer things happened, we should never alter our mathematics, and always prefer to change our physics. This was a convenient over-simplification that must now be qualified. I maintain that we need never alter our mathematics. But it might happen that we found our physical laws getting very campti· cated indeed, and might discover that, by changing our mathematical system, we could effect a very great simplification in our physIcs. In such a case we might decide to use a different mathematical system. (So far as I can understand, this seems to be what has actually happened in certain branches of contemporary physics.) And mathematics does depend on and reflect the nature of the world at least to this extent, that we would find certain systems enormously inconvenient and difficult to use, and certain others relatively simple and handy_ Using one sort of arithmetic or geometry, for example, we might find that our physics could be reduced to a, logically neat and simple system, which is intellectually satisfying,
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whereas using different arithmetics and geometries, we should find our physics full of very complicated ad hoc hypotheses. But what we find neat, simple, easy, and intellectually satisfying surely depends rather on our psychological make-up. than on the behaviour of measuring rods, solids and fluids, electrical charges-the 'external world.'
If we take In our hand any volume; of dil'mity or school metaphysics, for instance; let us ask, "Does It contain any abstract reasonmg concerning quantity or number?" No. "Does it con tam any experImental reasomng concernmg matter of fact and existence?" No. Commit it then to the flames: jor it can contam nothing but sophIstry and illusion. -DAVID HUME
7
Mathematical Postulates and Human Understanding By RICHARD VON MISES AXlOMATICS
1. High-School Axiomatics. Early in our high-school education all of us had occasion to learn certain "axioms" of geometry and arithmetic. They were presented as irrefutable truths and soon came to haunt our memories like nightmares. Here are familiar examples of such propositions: every quantity is equal to itself; the whole is bigger than any of its parts; all right angles are equal to each other; and so on. Two things were asserted of these axioms: in the first place, that they are clearly selfevident, and in the second place, that all mathematical theorems follow from them in a strictly logical way. The student, normally, does not feel any apprehension toward the assertion of obviousness. For how, indeed, could he imagine that a quantity is not equal to itself? The situation is not quite the same concerning the claim that all mathematical propositions are derived from the axioms, and exclusively from them. Here, the intelligent student soon senses that in the customary derivations many other "self-evident" concepts besides the explicitly stated axioms have been used. From the modem investigations of the foundations of geometry we know that no geometry can be built up from the few basic propositions that are listed as axioms in the school books. Hilbert's axiomatics, for example, comprises five groups of axioms, among which are propositions such as the so-called axiom of continuity, which is certainly a far cry from the simple statements of the high-school textbook. Nevertheless~ it may be argued that the simple high-school axiomatics is useful for training the student in the method of 10gical deduction. But this practice has serious deficiencies too. For in such a sentence as: the whole is larger than any of its parts, even the meanings of the words themselves are rather obscure. The statement presupposes that the student to whom it is addressed is, from his everyday language experience, ac1723
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quainted with the two relations, part to whole and larger to smaller. The axiom asserts that these two relatIons, m a certain sense, are in each instance simultaneously present or not present, and hence that one can discuss mdependently the presence of the one as well as of the other in every case. In some examples this is evidently more or less true. For if one says that France IS part of Europe, it is generally admitted that France must be smaller than Europe, although, even m this instance, some acquamtance wlth the concept of area has to be presupposed However, consider the following. accordmg to common usage It is qUlte legItlmate to say that sound sleep IS a part of one's well-bemg; here, If we really consider it as an mdependent property and not sImply as one aspect of the relation part to whole, the questlOn of bIgger and smaller breaks down. One mIght object that such cases are not meant by the aXlOm; but this would mean that the given propositlon needs a precedmg explanatlOn and thus cannot stand at the base of a logically constructed system From an ambiguous premise one cannot draw unambIguous conclusions. No propositIOn that presupposes comphcated experiences and appeals to a necessarily vague use of colloquial language can be fit to serve as the starting pomt of a rIgorously systematized branch of SCIence There have been frequent objections to the so-called axiomatIc method of instructing the begmner. However, most of them were made from a point of VIew qUite dIfferent from ours In general, such a procedure has been cntIcIzed as "too formal," appealing too lIttle to mtUltIOn and thus "apart from life" These are clearly considerations of a pedagoglc nature, with whIch we need not concern ourselves here. Our critIque is directed from the purely logical point of View. The formulatlOn of axioms found in high-school textbooks, being based on uncertain and impreCIse cllstoms of language and therefore unsuited for drawmg unambzguous conclusions, lS a failure. 2. ClassIcal AXlOmatlcs. HistOrIcally, we find the origin ofaxiomabcs 10 the tradttion of Euclid's Elements of geometry. However, the presently existing Greek text begIns with a number of definitions, whtch are followed by five postulates and nme common notlOns In the older Latm translatIOns. the latter (among which is. for example, the additIon of equals results 10 equals) are named "communes notiones sive axiomata." Accordmg to our contemporary terminology, the "postulates," e.g., the parallel postulate, would also count among the axioms The remainder of Euclid's first book conStsts of theorems and solutions of problems for which the proofs are always explicitly derived from the preceding definitIOns, postulates, and common notions. The subsequent books contain further definItIOns (of the circle and the tangent to the circle, etc.), from which addItional theorems are derived by means of the original axioms. Here the relatlOn of the various propositions is clear.
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at first definitions are given, then unproved statements are made about the defined things, and finally other statements are derived from these by ordinary deductive methods. We find the same schema in the work which about two thousand years later became the foundation of mathematical physics-Newton's mechanics in the Philosophiae naturalis principia mathematica. It begins with a series of eight definitions for the concepts of mass, force, and so on. Next, we find the assertion that words such as time, space, location, and motion do not need any definition--only certain refinements. After these follow the "axiomata sive leges motus," while the consequences of the laws of motion form the bulk of the work. We shall not discuss the question here of how far it is really possible to derive rigorously the laws of motion of a rigid body (for example, the physical pendulum) without any premises but those expressed in Newton's three laws of motion. However, the Newtonian mechanics is above all useful in offering us instructive insight into the construction of an axiomatic system, and the relation of definitions and axioms. Without a doubt the principal part of the Newtonian system is contained in the Second Law of Motion (which includes the first one, the so-called Law of Inertia): "Mutationem motus proportionalem esse vi motrici impressae ... "; "the change of motion (of momentum, as we should say today) is proportional to the impressed force . . . " Hence, if no impressed force is present, the motion remains unchanged (which is the Law of Inertia). One may compare this with the preceding definition (IV): "Vis impressa est actio in corpus exercita, ad mutandum eius statum vel quiescendi vel movendi uniformiter in directum"; "the impressed force is the action on a body that changes its state of rest or of uniform rectilinear motion." As one can see, the axiom is anticipated by the definition. For, if an impressed force is defined as that which changes the state of motion, it follows that in the absence of impressed forces the uniform rectilinear motion remains unchanged. Thus we see that definitions and axioms are not independent of each other at all, and one recognizes the naivete of the notion that the axioms by themselves state anything "about" the defined concepts. The relation between definitions and postulates in Euclid is quite the same. though not so immediately apparent. The classical axiomatics of Euclid and Newton, which for a long time were taken as the model for the construction of every branch of the exact sciences, are characterized by a subtle confusion of apparent definitions and explicit postulates, which in fact cannot be regarded as independent of each other. ~ 3. Mach's Reform. In the area of Newtonian mechanics the confusion arising from an insufficient differentiation between definitions and axioms was remedied by Ernst Mach. It was in Mach's Mechanics, published in
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1883, that for the first time the princIple was established which today is so generally accepted and WhICh forms the essence of modern axiomatics. It can be briefly stated by sayIng that the fundamental concepts are defined by the axioms, that IS, that, apart from the IntroductIon of new terms, there are no definItions In addltion to the axioms at the basIs of a deductIve sCIence. We shall elucidate thIS by the example of mechanics. There are two new basic concepts that enter mto the construction of Newtonian mechanics-those of force and of mass Newton explains mass in his first definitIon as "quantity of matter." One notIces immediately that this definitIon is completely empty and In no way helps us to gam an understandIng of the phenomena of motion All one has to do IS to reflect that It would be possIble to SubstItute the words "quantIty of matter" for the word "mass" wherever it appears m a contemporary text, i.e., not to have the sequence of letters m-a-s~s occur at all, and then one would be able to dispense with the first definitIOn completely without anythmg in mechanics bemg changed. If we dIsregard the fact that perhaps in the age of Newton one phrasing was more colloquial than the other, the definition serves, in fact, only as the equation of two expressions, both of Which are equally in need of an explanation On the other hand, we have already shown that Newton's defimtion of force largely antICIpates the content of the first two laws of motion. the force neither changes the locatIon of a body, nor determInes Its velocity, but rather changes its ve/o( lty Thus force is first defined as somethmg that changes the velOCIty, and then the law IS stated that velOCIties are changed by forces. ThIS manner of mference has been well put by Mohere; the poppy seed is soporiferous; why? because It has the power of soporiferousness. But scoffing here IS tll-advised, for Newton's Principia expresses one of the most far-reaching and onginal discoveries ever made in physics. One can convemently descnbe It by two statements. first, the circumstances m whIch a body IS at a given time (Its position WIth respect to other bodies, and other observable properties) determine the instantaneous change of velOCIty (or the acceleration) but not its velocity; and second, for different bodIes under the same Clfcumstances the observed acceleratIons dIffer by a numerical factor which is proper to the body m questIon, and hence is a constant for each body conszdered. These are the two dIScoveries of Newton. Once they are found, It is easy enough to add that the constant assocIated WIth each body shall be called "mass" (or If one wishes, "quantIty of matter") and the circumstances determining the acceleratIOn, "Impressed force." Thus definitions are reduced to explanatIons of words, to the (in principle, dIspensable) introductIOn of abbreVIations. Everything essential is contained in the axioms themselves. They delineate the concepts, for
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which verbal denotations can then be chosen as the need arises. We can summarize briefly: As Mach showed, the Newtonian foundations of mechanics can be remodeled in such a way that one does not begin with definitions, but with assertions (axioms), which also suffice to define the fundamental concepts of force and mass. Then, all one need add to the axioms is explanations of words serving as verbal abbreviations. 4. Hilbert's Geometry. It was David Hilbert who for the first time, in the year 1899, created the new form of axiomatics for geometry. This was the starting point for the extension of the axiomatic method to many branches of the exact sciences. As had become apparent from the works of Euclid and several of Hilbert's predecessors (Pasch, Veronese, and others), geometry is particularly well suited for such a treatment. The properties of space, described by geometry, are the simplest physical phenomena known. They do not refer back to experiences of a different kind as, for instance, mechanic.$ has to refer back to geometry even in its fundamental concepts of motion. Only one realm of general experience is, used in geometry from the start-that of counting. In the construction of geometry the foundations of arithmetic are taken for granted. Hilbert's system, as remarked before, differs from Euclid's in that it does not start with definitions of space elements, but considers these as defined by the axioms: We consider three different systems of things: the things of the first system we call points . . . ; those of the second, straight lines . . .; those of the third, planes . . . We think of the points, straight lines, planes in certain relations to each other and denote these relations by words as "lying," 'between,' 'parallel,' 'congruent,' 'continuous.' The precise and, for mpthematical purposes, complete description of these relations is contained in the axioms of geometry. Hilbert's first group of axioms contains the axioms of connection, examples of which are: two noncoincident points determine a straight line. . .; there are at least four points that do not lie in one plane. (Notice here that the knowledge of counting is taken for granted.) The second group, the axioms of order, mainly delineate the concept "between"; for instance: among any three points of a straight line there is always one and only one that lies between the other two. These axioms are followed by the third group, the axioms of congruence, which explain the concept of equality. Here is the legitimate place for the proposition, every straight line is equal to itself. For this serves, together with other similar propositions, to determine precisely and completely the way in which the word "congruent" shall be used thereafter. The main interest of geometric axiomatics is concentrated upon the parallel axiom, which forms Hilbert's fourth group. From the axioms of connection one can conclude only that two straight lines in a plane have
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either one point in common or none. Now the "parallel axiom" is added: through a point outside a given straight line there is at most one straight line that lies in the same plane and does not intersect the first one. It is a result (already found earlier) of far-reaching consequences that such an explicit assumption is necessary in order to draw the conclusions that form the contents of ordinary geometry. If one gives up this assumption, or replaces it by a different one, one can arrive at new geometric theories which contradict certain theorems of Euclidean geometry. Quite apart from the practical importance which these theories have acquired in modem times for the description of certain physical phenomena, they are highly instructive as regards the position of geometry, and hence also of the other exact sciences, in the total realm of our experience. We shall return to the contents of the so-called non-Euclidean geometry in the next section. The last group of Hilbert's axioms contains the axioms of continuity which are missing in Euclid and whose necessity follows only from a more profound analysis of geometric propositions. Hilbert states the essential hypothesis here in the form of a "completeness" theorem: it is impossible to add to the system of points, straight lines, and planes another system of things such that in the resulting total system all the previously stated axioms are still valid. Hilbert shows how one can construct a manifold of points, straight lines, and planes in which all the remaining axioms are valid, but which does not satisfy the demand of completeness. This manifold does not yield a useful picture of the well-known geometric phenomena. Hilbert's axiomatics of geometry rigorously carries through the principle that the elementary concepts are to be defined by the axioms themselves; and, in particular, it demonstrates the role ot the parallel axiom and the axiom ot continuity. 5. Non-Euclidean Geometry. The best-known and in many respects most important result of axiomatic investigations is the invention of the so-called non-Euclidean geometries. We know today that Gauss had much of the essential knowledge (some of it since the year 1792), but refrained from publishing it because he was afraid of the "Geschrei der Bootier." This goes to show how strong the influence of traditional scholastic opinions may become and how necessary an education toward a free outlook and unprejudiced judgment is for the progress of science. The history of non-Euclidean geometry starts with the publication of a work by the Russian mathematician Lobachevski about 1840. The essential point is to establish the following matter of fact: Dropping Euclid's parallel axiom (previously quoted in Hilbert's form.) and substituting for it another suitable hypothesis but retaining all the others, it is possible, by the customary rules of deduction, to derive from this new
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set of axioms a new geometry. Such a geometric system does not contain in itself any contradictions, although it contradicts certain theorems of Euclidean geometry. This fact refutes the allegation that the theorems of geometry taught in OUT schools are imposed by the laws of thought, and absolutely assured truths, independent of all experience. In the philosophical system of Kant's Critique of Pure Reason this assertIon plays a decisive role. Kant says, For geometric principles are always apodictlC. i.e., united with the consciousness of their necessity . . .; theorems of this kind cannot be empirical judgments or conclusions from them. There is no doubt that Kant wanted to establish with these words a fundamental and profound difference between the theorems of geometry and those of other natural sciences, such as optics or mechanics, for instance. One often tries to make clear how a non-Euclidean geometry is possible by pointing out spatial phenomena that are governed by laws similar in structure to a non-Euclidean geometry. Imagine, for instance, the spatial situation on the surface of a sphere of very large diameter with creatures whose entire existence is limited to this two-dimensional surface As long as they know only a limited portlOn of then world, ie, only a piece of the spherical surface small compared to the diameter, they would have a geometry different in nothing from our Euclidean plane geometry Their straight hnes would be, in fact, arcs of great circles, these being the lines of shortest dlstance between two points (and the shapes of stretched strings) on the surface of the sphere. As soon, however, as the surface inhabitants extend their expenence beyond the immedtate neighborhood, they will be forced to change their hypotheses. There are no "parallel" lines on the spherical surface; all great circles intersect each other in two points. Now, it is indeed possible-by means which cannot be explained here in any detaIl-to describe the SItuation on the surface of a sphere in such a way that all statements become equivalent to those of a plane geometry in which the parallel axiom does not hold. But the same relation as that between a plane and a very large spherical surface exists also between the three-dlIDensional space described by Euclidean geometry and a "curved" space not satisfying the parallel postulate. Assume that the space we live in has an extraordinarily large "radius." i.e. an extremely slight "curvature"; then our measuring instruments would not be sufficiently accurate to determme practically whether the Euclidean or a modified geometry is in better agreement with the facts. But the deCIsion would be of no practical interest, anyway, as long as the differences are not noticeable in some way. We may leave open the question whether today there is enough evidence for such a decision. The notion of "apodictic certainty" of the geometric theorems, however, has to be abandoned definitely.
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The study oj axiomatics oj geometry has proved that a consistent system oj geometry not obeying the parallel axiom and hence not agreeing with the Euclidean geometry can be constructed,' thus the assertion that the customary geometry is logically inevitable, apodictically certain, and independent of any experience, is disproved. 6. Applications of the Axiomatic Method. Contemporary axiomaticists are mainly concerned with two general problems: the questions of consistency and of independence of the axioms. By a "consistency proof' is meant the demonstration that from an assumed set of axioms, using the recognized customary rules of inference, one can never deduce both the statements A and non-A (the contradictory opposite). Such a proof has so far, strictly speaking, never been given for any axiomatic system of general interest. All one has succeeded in doing is to reduce the consistency of, e.g., geometry (Euclidean as well as non-Euclidean) to the consistency of arithmetic. In other words, it has been shown that an inconsistency in the set of axioms of the particular geometry would imply an inconsistency in the structure of elementary arithmetic. It is only because one credits arithmetic with consistency that one believes this to be proved in the other cases. We shall return later to the problem in arithmetic itself.l The question of independence is somehow related to that of consistency. The (consistent) group of axioms A is said to be independent of the (consistent) group of axioms B, if the group B can be exchanged for a different group B' such that A and B' together again form a consistent system. It is. for instance, proved that the parallel postulate is independent of the axioms of connection, etc. In each case the consistency is measured by that of arithmetic. If an axiom proves to be dependent upon others1 one can try to restrict it in such a way that the remaining weaker axiom becomes independent of them. The usual theoretical discussion of axiomatic systems suggests a question which belongs to epistemology and which seems of importance to us. Are consistency, we ask, and perhaps independence, all one can demand of a set of axioms? Is every system that satisfies these conditions permissible, and are all these systems equally justifiable objects of scientific research? As long as this question is not formulated in another way it cannot be answered reasonably, for the meaning of what is "permissible" or a "justifiable object of scientific research" is not clear. Human activities, of which research is but an example, can only be described, Le., studied and classified as to their relations to each other and to other facts. A "Valuation" would be possible only after an arbitrary standard of values had first been adopted. If we limit ourselves to an objective description of the situation, we find that there are two extreme cases. On one end we find axiomatic systems more or less similar to the rules of the game of chess. They satisfy 1
[See p. 1744 (section on Foundations of Mathematics). ED:]
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all logical postulates but are far from being applicable to more vital phenomena. On the opposite end there are investigations that explicitly aim to influence practical action in certain areas of life, by, e.g., answering directly or indirectly questions arising from technology. All axiomatic systems fall somewhere between these two extremes. In all cases, however, even in the case of the axioms of chess, there is but one way to establish a connection between the system of axioms and their consequences on the one hand and observable facts on the other: The words and phrases used for the basic or derived concepts of the axiomatic system must be given a more or less definite interpretation by reducing them to protocol sentences. 2 We may agree with Hilbert that the words "point," "straight line," "plane" within the axiomatic system are nothing but arbitrary signs. Yet, if one substitutes for them the words "shoes" and "ships" and "sealing wax," attaching to these words their customary (incidental) meanings, one will find that the relations expressed in the axioms and theorems either are not reducible to protocol sentences or contradict certain protocol sentences. Two points determine a straight line: this is not only an axiom, but with the usual meaning of the words it is also the approximate expression of a matter of fact. That analogy breaks down if one asserts as an axiom that two shoes determine a ship. This situation can be described by saying that the axiomatic system itself remained intact, but it could not be applied. One may also adopt the formulation that an axiomatic system that with no choice of names (Le., with no coordination of the arbitrary symbols to protocol statements) leads to useful applications in the indicated sense is "worthless." All systems of axioms considered in science are formulated in such a way that they are paralleled by some interpretation in terms of observable facts. As far as the non-Euclidean geometries are concerned, we have mentioned already that their application to, observable space phenomena, with the assumption of a slight space curvature, seems possible. In modern relativity theory this is an indispensable tool for the description of physical facts. The connection of a system of axioms and their consequent theorems with reality, i.e., the meaning of its statements, has to be determined in the same way as that of all other statements, by coordinating the words and idioms used in them to elementary experiences by means of reduction 2 [In Von Mises' terminology the expression "protocol sentence" is meant to indicate that "one is dealing with statements having immediate, present events as their subject and which are instantly written down (or otherwise recorded), as is the case with protocols in the juristic sense. According to the conception of Carnap, one might imagine schematically that all pertinent experiences, sensations, feelings and thoughts are recorded in the form of a written (or mentally fixed) protocol and then become the basis of further study. . . . If we say that any statement can be reduced to protocol sentences we mean that the statements can be supplemented by further (explanatory) sentences in such a way that for those who understand the protocol language a correspondence between these statements and experienceable events is established." ED.]
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to protocol statements. On this basis one can differentiate between axiomatic systems for which consistency and independence have already been proved, according to whether or not they can be used for the description of observable facts. 7. Axiomatization in General. The axiomatic method, having had so great a success in geometry, has more recently been extended to various other parts of the natural sciences. We have mentioned previously the clarification of the foundations of mechanics due to Mach, an investigation that must be considered as a forerunner of modern axiomatics. Later, the model of geometry was used in a more formal manner to set up, among others, axioms of mechanics by G. Hamel, axioms of set theory by E. Zermelo, axioms of thermodynamics and axioms of the special theory of relativity by C. Caratheodory, etc. Attempts at extending the axiomatic method to many other branches of the natural sciences are in various stages of development. The followers of modern axiomatics see in that method the highest form of scientific theory, the form toward which all scientific endeavor strives. Thus Hilbert said, . I think that everything that can be an object of scientific thought at all, as soon as it is ripe for the formation of a theory, falls into the lap of the axiomatic method and thereby indirectly of mathematics. Under the flag of the axiomatic method mathematics seems to be destined for a leading role in science. If we disregard the reference to the place of mathematics (it is not at all obvious why an axiomatics of biology should subordinate biology to mathematics), there still remains the claim that every theory in a certain stage of its development takes the form ofaxiomatics. This form was defined by Hermann Weyl as "the complete collection of the basic concepts and facts from which all concepts and theorems of a science can be derived by definitions or deductions." In the last analysis, everybody who wants to exhibit a branch of science systematically intends to proceed according to this scheme, i.e., to collect all the essential facts in the fundamental theorems; it is only the manner in which this is done, its completeness and precision, that will not always be as Hilbert meant it. Nothing can be said against Hilbert's claim as long as the following two points are kept in mind. In the first place, the axiomatic method is only a form of description, hence something that can be taken into consideration only after the essential content of what is to be described is known. It was the discovery of Newton, prepared by Galileo, that external circumstances (to a body) determine its momentary change of velocity (acceleration). and not the velocity itself. Once this discovery (or, if one prefers, the "invention") was made, it became another problem to put its content into a set of clean
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formulas that do not say too much or too little. To take another example, a long time ag) it was found that in the mathematical concept of probability one deals with the limit of the relative frequency in an unlimited number of trials, where a certain irregularity in the sequence of the results is essential. Anyone who accepts this basic idea is faced with the problem of expressing the facts in a set of precisely formulated axioms. In geometry we find the same situation, only there the essential discoveries date so far back that we are not conscious of them any more. The axiomatization is always a secondary activity which follows the actual discovery of the pertinent relations and puts them in a precise form. The second reservation one has to keep in mind against the exaggerated claims of the axiomaticists concerns the concept of absolute rigor. All followers of the axiomatic method and most mathematicians think that there is some such thing as an absolute "mathematical rigor" which has to be satisfied by any deduction if it is to be valid. The history of mathematics shows that this is not the case, that, on the contrary, every generation is surpassed in rigor again and again by its successors. All classical mathematicians, Gauss included, as any student today can show, have been guilty, on some occasion or other, of a faulty lack of rigor. Certain new developments of mathematics, of which we shall speak,s will show that some things that were taken as quite rigorous thirty or forty years ago have to be doubted today. There is by no means an eternally valid agreement about the admissible methods of logical deduction. Thus we have to regard the task ofaxiomatization of a science as a relative one, subject to change with time. One tries to give to the basic theorems of a theory such a form that they satisfy all present requirements of logical rigor. The axiomatic formulation of a discipline may be regarded in each period as the highest level of scientific presentation, if 'one keeps in mind, first, that it is merely a form, which can be taken into consideration only after the essenl,ial relations are known; and second, that the logical requirements which the axioms have to satisfy are themselves subject to evolution. LOGISTIC
1. Tautologies and Factual Statements. "Mathematical logic" or "10gisticu is a modem name for the fundamental system of logical rules, presented in the form of concise formulas, and the study of them. Before we discuss this matter, we have to find out what is meant here by ·'ogical." The present stage in the development of positivism is governed by an idea first enunciated by L. Wittgenstein. though in a slightly different form, and advanced with particular emphasis by the so-called Vienna Circle. It is the conception according to which all meaningful statements 8 [See p. 1744 (section on Foundations of Mathematics). ED.]
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-or, as we prefer to say, all connectible statements -!:-have to be divided into two groups: those expressing a state of fact which can be tested by experience, and those which, independently of all experience, are true or false by virtue of their wording. Statements of the second kind are called "tautological" ,. . in the first instance and "contradictory" in the second .. Tautological sentences form the content of logic, of pure mathematics, and of all other axiomatically formulated scientific theories. At first sight it might seem that here Kant's a priori is revived, including the "synthetic judgments a priori" which are supposed to form the content of mathematics. However, the difference is quite considerable. A sentence is tautological if it is true independently of all experience, because it does not say anything about reality at all and is nothing but a reformulation or recasting of arbitrarily fixed linguistic rules (definitions). Consider for instance, the statement, the sum of two natural numbers is. itself a natural number; everybody can see that this is a statement of a different kind from ~this: alcohol boils at a temperature of 78° C. For if we try to explain what we mean by the sum of two natural numbers, i.e., if we want to define the word "sum:~ or to give the rules according to which it is to be used, then necessarily these rules include the result that the sum is itself a natural number. The tautology is a little more difficult to recognize in the statement, the sum of two prime numbers, neither of which is 2, is an even number. Here one needs, in order to demonstrate the tautological and noncontradictory character of the statement, a "proof." But according to our fundamental conception every mathematical proof is the tautological transformation of definitions and other linguistic rules and in any case has nothing to do with observation and experience. Finally, as a third example we might take the assertion, every even number is the sum of two prime numbers. Here we have to remark that no living person today could say whether this is tautological or contradictory (true or false). But we know that this decision does not depend upon some sort of physical experiment or observation of facts, but can be made solely on the grounds of a study of definitions and a carrying out of calculations and similar transformations. Thus, the theorems of logic and mathematics are in the framework of this conception "tautological," but neither synthetic nor a priori. They are not synthetic, because they say nothing about reality, and not a priori, because they do not come from a superempirical "source" but are the result of arbitrary definitions introduced by us. Furthermore, tautological .,\ [Von Mises defines a "connectible statement" as follows: "We shall call a group of words (sentences and sequences of sentences) 'connectible' if they are compatible with a system of statements which, it is assumed, regulate the use of language--connectible, that is, with respect to this system. Strictly speaking, one would always have to say 'conr:ectible with .. :; this addition can be omitted only if the system of reference is supposed to be known." ED.}
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character is by no means confined to mathematics; it is possessed by any system of sentences that is constructed according to the axiomatic method and that serves as the theoretical exposition of an area of facts, e.g., ~ chapter of biology. Theorems of logic and pure mathematics and of any axiomatically formulated theory are, according to the conception of L. Wittgenstein, "tautological," i.e., they do not say anything about reality and are nothing but transformations of arbitrarily agreed upon linguistic rules. The notion of tautology by no means coincides with that of Kant's a priori synthetic judgments. 2. Tautological Systems. Against the thesis according to which logical, mathematical, and similar theorems are "tautologies," various objections have been raised. Some of them-and these can most easily be settledoriginate in the rather unhappy choice of the name Htautology." Obviously Wittgenstein, and the members of the Vienna' Circle (who like to call themselves today "logical empiricists"), have created a new concept by making the distinction between two kinds of statements, and for its denotation-even though in analogy with linguistic usage-they arbitrarily drew upon a word belonging to the existing language. But previously the word "tautology" had been used almost entirely in a derogatory sense, to designate empty or superfluous talk; by "tautology" one meant something like a sentence which, without any loss to the reader or listener, could just as well be omitted. The new use of the word is burdened with this old meaning and even people who are used to abstract thinking and who are quite familiar with the process of nomenclature in the exact sciences can· not always free themselves from this influence. Thus, a beautiful book about Numbers and Figures, having two well-known contemporary mathematicians as authors, ends with a remark which shows how much the authors feel hurt by the thesis that "mathematics is fundamentally but a chain of tautologies." This is a psychologically understandable resistance to a new terminology. It might be suggested that we replace the word "tautological" by "analytic," but that would result in much worse misunderstandings, since it would be reminiscent of Kant's concept of analytic judgments. We have shown above that tautologies in our sense are by no means Kant's a priori synthetic judgments. But neither are they his analytic judgments. According to Kant's theory and that of idealistic philosophy in general, there exist concepts, independently of all human influence and all linguistic conventions, whose delineation we can more or less precisely discover through pure thought. A statement which expresses the result of such acts of thinking is called an analytic judgment. We, however, mean by a tautology a sentence that is derived by arbitrarily fixed transformations of arbitrarily chosen basic assumptions. In this chapter we shall have to show
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first of all what such a system of tautological statements looks like and this will bring us back to the discussion of the axiomatic method. On the other hand, since the aim of all language, in everyday life as well as in science, is in our opinion to assert something about observable reality, i.e., to find statements verifiable in experience, we shall have to explain further the role of tautological systems in this respect. By no means can we accept a view that would assign to tautological sentences something like dogmatic or absolute validity. At most one could call them "apodictic," as long as one keeps in mind the frame of reference that determines their range of validity. The starting theorems (definitions) as well as the rules of transformation (methods of deduction) are, as previously mentioned, to a certain degree arbitrary conventions. Even though at the present time among logicians there is, on most counts, agreement about the basic assumptions, there are still differences of opinion upon certain specific points, e.g., the axiom of the excluded middle. 1i A decision about whether one or the other of the logical systems is "right" is impossible. It is only the usefulness of a specific system for the representation of observable phenomena which can prove it to be more or less expedient. What contemporary logical empiricism calls tautologies must not be confused with the customary meaning oj the word in ordinary language, nor with Kant's "analytic judgments." Tautologies have no absolute validity but are valid only within a specific system ot basic logical concepts. The application ot such a system as a means of representation of a part ot reality determines its usefulness. 3. Basic Logical Relations. The system of logistic (or, as it is often called today, of mathematical or theoretical logic) starts with the setting of the basic relations that can exist between statements. These fundamenal connectives, of which there are four, are denoted in ordinary language by the expressions "and," "or," "not," "if . . . then." We call them, in this order, conjunction (also logical product), disjunction, negation, and implication (or conditional). The sign "and," for which we write simply a comma in the formulas, means) if put between two statements, that both statements are posited or assumed. The sign "or," whose abbreviation is "v" (vel), means that of the two statements between which it stands at least one (but not one and only one) is assumed. In order to indicate negation, i.e., the contradictory opposite, we use the sign "-," put before or above the statement sign to which it applies. Implication may be expressed by an arrow, "-:~:'; it signifies that if the left-hand statement is posited, the right-hand one is assumed to be posited too: the first one implies the second~ the second is implied by the first. To these basic concepts one can add others, e.g., that of equivalence; but one recognizes at once that this is dispensable, for by the equivalence :i
[See p. 1748 (section on Foundations of Mathematics). ED.]
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of two statements one means that the first follows from the second and
the second from the first. Hence equivalence is expressible by the two relations of implication and conjunction. If we want to use the formula way of writing, we must say that "A equ Bn can be replaced by "A~Bt B~A". These remarks suggest the question whether all four of these relations are independent of one another or whether some of them can be derived from the others. The latter iS t indeed t the case. It suffices, for instance t as one easily sees, to introduce the signs "v" and ".-" as undefinable basic signs, or, as Brentano has shown, only the concepts "and" and "not." We shall show how implication and disjunction can be expressed by the two signs "," and "-"'. That B follows from A means the same as that A and non-B cannot exist simultaneously. Hence, in order to express the implication one has only to signify that the conjunction of A and non-B is not true. The sign ..A....:,.B" can therefore be replaced by "--[At-B)", which has to be read: A does not exist together with non-B. It is equally simple to express disjunction by means of conjunction and negation. That of the two statements A and B at least one is true can be expressed by stating that the statements non-A and non-B do not hold simultaneously. The compound of signs "AvB" is thus replaceable by "-[-'A,-B]", which is to be read: non-A and non-B do not exist together. A similar consideration shows that conjunction and implication are reducible to disjunction and negation. On the other hand, it is impossible to construct the other relations by means of negation and conjunction. Of course, it cannot be the purpose of such investigations to express all logical deductions, whenever they occur, by means of the two basic relations previously chosen. On the contrary, once we have seen that signs like 'lequ", "....:,.", "v" can be replaced by suitable combinations of "," and "_". we know that the new signs can be used as abbreviations. In the same way one seeks to form larger and more efficient units and by their means to reach more complicated deductive patterns. The whole of mathematics is a construction of this kind. The first steps at the basis of mathematical logic consist of shOWing how the various simplest relations--conjunction, negation, disjunction, and implication-are connected with one another and in a certain way can be reduced to each other. 4. Further Formalization. The foregoing argument can be further formalized and then yields definite rules for the derivation of new tautologies. That is the essential aim of theoretical logic since Leibnitz, who must be considered the founder of the entire discipline. We start with two symbols, "F" and "T" (which stand for the words "false" and "true"), without giving any definition for them. To every statement we assign one of these two letters; that is, only such statements are admitted as premises as are t
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accompanied by one of the symbols lOT" or uP". Then one defines operations on statements (truth functions) by directing in what way the assignment of T or P to the objects of the operation transfers to the result. The only operation acting on a single statement is the negation. It is defined by the rule that, if the statement A has the truth value F, --A has the truth value T, and if A is assigned T, -A is to be assigned F. For every truth function that connects two independent statements, e.g., the implication I'A ~B", there are four possible combinations of the assignments of T and P to the two parts. The implication "A ~B" is defined thus: the sign "A-!J-B" has the truth value P, if A has the value T and B the value P; in the other three cases "A~B" has the value t. On the other hand, the definition of the disjunction "AvB" is this: the sign "AvB" has the value F if F is the value of both A and B; otherwise its truth value is T. Once a number of symbols have been defined in this manner, one also knows the distribution of the T and F values in formulas that result from combinations. For instance, the combination "A~,....,B" has the value F whenever both A and B have the value T; in all other cases, its value is T. Of special interest are those formulas which have under aU circumstances the value T. These are the actual propositions of mathematical logic. Por example, according to the above conventions about the operations " . . . . " and "v" (negation and disjunction), the combination "Av--A" has always the value T. For A and .-A cannot, according to the definition of " ......,t', both have the value F, and according to the definition of 'V' the combination could have the value F only if both parts, the one before as well as the one after the sign "v", have the value F. The formula "Av-A" can be called the "theorem of the excluded middle" (in the simplest case). Other "always true" formulas or theorems in truth-function theory (propositional calculus) are: "[(A~B), (B-!J-C)]~(A~C)" and U(A~ . . . . A)-!J-.-A". The first one is, in words, the fact that A implies C, or more briefly (but not exactly correctly), if B follows from A and C follows from B, then it follows that C follows from A. The second formula is the basis of the s().oCalled indirect proof; for it says in words (approximately), if a statement implies its own contradiction, it follows that the statement is false. If one writes down any formula consisting of complexes of the four signs we have introduced above and any letters A,B,C• .. " then one can determine purely mechanically the truth conditions of the formula, and in this fashion it is possible to derive theorems. Further pursuit along these lines forms the first chapter of any textbook of mathematical logic, the so-called truth-function theory or propositional calculus, which has a certain analogy to elementary algebra.
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As in algebra, where according to fixed rules algebraic formulas and theorems are derived, truth-function theory arrives at formal laws determining the connection between the basic operations. S. Difficulty of Deduction. From the foregoing examples of theorems of truth-function theory, which of course can only give a faint idea of the first beginnings of mathematical logic, there are many interesting things to be learned. In the first place.. they show how vague and unreliable ordinary linguistic usage is in such simple cases. For the sign "v", introduced above, corresponds to the English word "or" only when the latter is used in the sense of the Latin ''vel'', but not in the case of the exclusive "or" (aut). If "or" in this latter sense is to be expressed one has to write "{AvB),,-,(A,B)", Le., "A or B and not at the same time A and B". The sign "~ .. of implication, too, does not always coincide with that which is customarily expressed by the words "it follows". According to our definition, "A~B" is always true when A is false, whatever B may be, i.e., a false statement implies any arbitrary true or false statement. If, e.g., to the correct theorems of arithmetic the theorem "2 X 2 == S" is added, then any true or false result can be derived from the premises, in particular, of course, the theorem "2 X 2 == 4". In the customary way of speaking it is not so; there the conception-which is not exactly definable -is used that from false antecedents only some conclusions, namely false ones, can be drawnr On the other hand, our examples show how logical theorems can be constructed by applying arbitrarily adopted rules, as in a game. Such theorems represent "eternal truths" only as long as one takes the rules for granted. They do not become statements about reality, verifiable in experience, before the language of formulas is translated by a more or less vague correspondence into the language of everyday life. Analogously, in chess it is "unshakably true" that a player has lost the game when his king is open and without defense against the attack of his opponent. If one takes this theorem and gives the word "player," "king," "attack," "opponent." etc., the meanings customary in everyday languaget it becomes vague and doubtful. There is here another analogy with the relation between logic and reality in that the theorem does not become meaningless, but must be considered false or true, according to experiential circumstances. We shall later return to this pOint. 6 The comparison with the game of chess can also serve to illustrate another important point. Thinking of certain situations in practical life where decisions are to be made, one might be inclined to believe that the difficulties arising here are due exclusively to vagueness, indefinite and flexible conditions, and the impossibility of exact description by language. If 6
[See p. 1742.]
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everything were as clear and unambiguous as in logic, one may think, then decisions would be easy, and almost obvious. The chess~game analogy refutes this idea. Taking, say, an advanced point in the development of the game as a start, in which, say, black has the next move, the set of all possible continuations of the game is completely given by the rules of chess. Thus it is uniquely "determined" which next moves of black will lead to certain loss, which will lead to certain victory, and finally, which leave open both possibilities. (Incidentally, one or another of these three classes of next moves may be empty; also, certain well-defined assumptions about the manner of playing of the opponent may be included in the premises.) But something that is determined theoretically or in principle is by no means actually known. The complete enumeration of all possible continuations of the game goes beyond our capabilities. Under certain conditions such an enumeration could be replaced by "theorems," i.e., by tautological transformations of the chess rules, perhaps of this form: For certain starting positions (x) and next moves (y), only a checkmate of black can result. To find such theorems, i.e., to derive them from the original rules, seems in the case of the chess game extraordinarily difficult. A type of game in which the players have too much knowledge of this kind at their disposal is of no interest for them; the game of nim is appreciated only by children. Theoretical logic, including mathematics, is based upon a system of rules which is immensely more abundant, diversified, and complicated than that of chess; moreover, it is subject to change in time. Of all the theorems derivable in the system of rules, only an extremely small part has been found so far and further progress often presents extraordinary difficulty. However highly one may rate the difficulties of practical deci~ sions that cannot be reduced to theoretical questions, there is no reason to underestimate the intricacies of decisions in logic and mathematics. The theorems of truth-function theory are additions to language indispensable jor the purpose oj higher precision. They do not contain anything that is not implied somehow in the basic assumptions (the rules of the game). Nevertheless, it remains a very hard problem, and one that can never be exhausted, to find explicitly all theorems that can be produced in this way. 6. Russell's Theory oj Types. We want to mention here another important chapter of logistic which shows more clearly than truth~function theory how much present-day logistic theory differs from the classical logic, of which Kant said that since Aristotle it had made neither a step forward nor a step back. We mean the so-called theory of types of B. Russell, which proved to be indispensable in getting rid of certain otherwise unsolvable contradictions. In introducing the concept of connectibility we spoke about what is
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meant by a sentential function or propositional function. 7 An important part of all linguistic rules has the purpose of delineating the area of applicability of predicates. When we say "x bears green fruit," this sentence is meaningful only if we put for "x" the name of a plant, but not if we put for it, e.g., "subway" or "circle sector." The totality of the objects x for which the individual statement is meaningful-no matter whether it is true or false-forms a "class," namely, the class belonging to the particular predicate. In our example, the class belonging to the predicate "bears green fruit" consists of the totality of plants, at least as long as one gives the words only their customary, nonsymbolic meaning, corresponding to usual parlance. Each single plant is then called an individual member of the class. Now, undoubtedly statements can be made whose subjects are classes themselves; or better, that which appears as a class in one case may in another case be an individual of another class. When we say, e.g., "x possesses 23 vertebrae," this is a sentential function defined for a vertebrate, Le., the corresponding class is formed by the totality of vertebrates. But this totality is an individual member of the class "classes of animals," of which one could, for example, make the statement "x contains more than 200 species." Russell was the first to point out clearly that the two classes, "classes of animals" and "vertebrates" stand in a certain relation of subordination; they belong to different types or rank. No statement can be meaningful for classes of different types at the same time. This concept of types must not be confused with a distinction of smaller and larger classes. The sentential function "x has four legs" is defined for a bigger class of animals than that of the vertebrates, perhaps for the totality of all animals. But this totality is not of a higher type than that of the vertebrates; it is only bigger, more comprehensive, the vertebrates being a subset of the set of aU animals. On the other hand, the concept of the class of animals, i.e., the class of which one could say "x contains many species," is from the point of view of the theory of types a higher concept than that of a vertebrate, an insect, or any individual animal, etc. The theory of types, or the doctrine of the "hierarchy of types," served originally to resolve certain contradictions that appeared in the theory of sets, a branch of pure mathematics. No satisfactory solution had existed previously_ In a work of great scope, Principia Mathematica, Whitehead and Russell showed how to reduce all mathematical concepts to the simplest logical operations. But far beyond its original purpose, the theory of types serves as an important reference for the constitution of a general conceptual system and for the logical construction of a scientific language comprehending broad areas of experience. Today we are still a long way '1 [For the meaning of sentential or propositional function, see the selections logic by Ernest Nagel and Alfred Tarskl. ED.]
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from being able to show decisive practical results. But considering that for centuries philosophical discussions were governed by the controversy between so-called universalists and nominalists, i.e., the question whether classes are as real as individuals or, on the contrary, of higher reality, or are merely abstract names, one may harbor a faint hope that today's mathematical logic, one of whose pillars is Russell's theory of types, constitutes the first step toward a useful conceptual structure which does not admit of such absurd questions. Russell's theory of types forms an important part of theoretical logic which increases its efficiency as compared to classical forms oj logic. It constitutes the first step toward a general rational conceptual structure and discards old pseudo problems, like the controversy between the universalists and the nominalists. 7. Universal Physics. In the framework of the present efforts, going on in different countries along different paths, to arrive at a "philosophie scientifique," at a conception of the world free of metaph),sics, theoretical logic certainly plays an important role. But we cannot object too strenuously to the opinion occasionally voiced by representatives of the Viennese and the Polish schools (Lukasiewicz, Ajdukiewicz, Kotarbinski, Tarski, etc.) that all one need do is to develop mathematical logic further. Certainly mathematical logic, Le., the formal construction of symbolic systems and their tautological transformations, is an indispensable means for a useful description of reality, at least as indispensable as elementary and nonelementary mathematics is for the correct presentation of physical phenomena, say the propagation of light. Occasionally one finds also mathematicians who are of the opinion that physics is reducible to mathematics; they hold, for instance, that electrodynamics has become a "part of geometry" through the theory of relativity. Such utterances are logical misconceptions and go ill with the critical subtlety which the mathematician otherwise often exhibits. By the mere manipulation of signs according to chosen rules one can indeed learn nothing about the external world. All the knowledge we gain through mathematics about reality depends upon the fact that the signs as well as the rules of transformation are in some wise made to correspond to certain observable phenomena. This correspondence is not a part of logic or mathematics and has no place withm its tautological construction. To explore how the correspondence works is a big problem. For in order to describe it in any concrete case, one cannot use the filtered, sharp, and more or less formularized language of logistic; rather one has to resort to ordinary speech which is imperfectly built up and not sufficiently reducible to element statements. Between the exact theories and reality there lies an area of vagueness and "un speak ability" in the literal sense. The question whether logic rests upon experience or originates in the
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human mind independently of all experience is wrongly posed. We observe that there are things which are black and things which are not black. But this distinction is not strict, the disjunction is incomplete, and we cannot even circumscribe precisely the range of objects to which it applies. But it is this observation, duplicated again and again, that led us to create the logical relation of A to non-A, which after suitable agreement about the use of the signs is free from uncertainty and vagueness. The same is true for the other relations of logic-the disjunction, conjunction, etc. They are abstracted from experiences in which they are realized only approximately. The beginning of this schematization (using a more general but not very expedient expression, one could say rationalization) goes back, as we saw in the first chapter, to the creation of ordinary language, hence to the distant past of the origins of human history. Its latest stage is the systematic synthesis of a logical language of formulas. Of course, one may ask the question, which qualities enable man-man alone or also some higher animals-to carryon such intelligent activity. But even this question would not lead us out of the domain of experience and into the realm of the transcendental or of metaphysics; it merely opens up another discipline of (biological) research, which is to be attacked by the usual means of empirical science. However, it is not this historical question of the genesis of the basic logical concepts that is essential for us here; rather it is the present relation between logical insights and reality. Logic does not fioat freely in midair, without connection to the world of observation. What we have said above about axiomatics also holds for logistic. The words and idioms appearing in it-that they are presented in other forms than by customary lettors does not make any difference in this respect--correspond in an unexact and never precisely determinable way to elements of observable matters of fact. It is only because logical formulas, after having been interpreted in ordinary language, according to this correspondence, yield a useful description of experienceable relations, that the study of theoretical logic is regarded as a part of scientific activity. We agree with the mathematician F. Gonseth-in a certain deviation from the "logical empiricism" of the Vienna Circle-who in a theory which he calls "idoneism" conceives of logic as of a "physique de l'objet quelconque" (physics of aU things) and says that the intuitive rules of logic and common sense are nothing else than an abstract schema drawn from the world of concrete objects. Remembering the explanation of the concept ofaxiomatization given in the previous section, we may say that theoretical logic is in axiomatic form the doctrine of the most general and the simplest typical relations that are observed among objects of any kind.
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Theoretical logic-like the axiomatic of a specific discipline-is a tautological symbolic system developed according to accepted rules; its importance for science lies merely in the fact that if the single signs and basic formulas are made to correspond to certain very general factual situations, any derived formulas correspond to certain observable matters of fact. It is, therefore, just as false to say that the logical inferences rest upon experience, as that logic has nothing whatever to do with experience. THE FOUNDATIONS OF MATHEMATICS
1. Tautological Part. When anyone, on any occasion, wants to give an example of an absolutely certain and indubitable truth, he does not hesitate to cite some mathematical theorem known to him, perhaps the Pythagorean theorem or even the formula of multiplication, "2 times 2 equals 4." If thus the results of mathematical rules are taken as completely and unshakably true, one should think that this is even more the case with the general foundations from which such results are derived. Nothing is more astonishing for the layman than to hear that among mathematicians there are differences of opinion as to the basic principles of mathematicstheir meaning, their applicability, and their content. If, moreover, an eminent contemporary mathematician declares that the present uncertainties by no means concern merely questions on the frontier of mathematical k':1owledge but go directly to its core, then the general naive faith in the "most absolute" of all sciences must be completely shaken. After all that has been said in the previous chapters it is not difficult for us to understand what one has to think of the certainty or uncertainty of mathematical theories. "Pure" mathematics in the sense of customary parlance is only a system of tautologies, i.e., of conventions about signs and transformations according to accepted rules. or, we could say, a system of deductions. From the point of view of the extreme formalists (which we do not share), all assumptions about symbols and rules are completely arbitrary, such that without any further justification one may construct different mathematical systems which are not in agreement with one another. From our point of view this is true only with the essential reservation that all tautological systems that play any role within science rest in the last analysis upon a certain coordination of the symbols with other observable experiences. Only if one disregards these coordinations completelYt which is, e.g., possible by excluding all use of colloquial expressions, can one regard different systems that are complete and consistent within themselves as equivalent to each other. This relation of mathematics to reality. which is so often neglected, is particularly well pointed out by F. Gonseth in his interesting book Les Mathematiques et la Realite. Mter alit it is not a factual question but rather one of nomenclature
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whether one wants to define the concept "pure mathematics" in such a way that it is sharply distinguished from all other natural sciences. The theory of each domain of experience has its tautological part which appears in its most complete form as an axiomatic structure. Now, one may single out a specific realm of experience and call its treatment "mathematics"; then mathematics is in principle not different from other areas of knowledge and consists of a tautological and a nontautological part. This point of view was taken by M. Pasch, who for half a century tried to clarify the basic concepts of mathematics in a number of valuable works. However, it corresponds better to present usage to reserve the name "mathematics" for the purely tautological parts within the different branches. As far as geometry, which is usually considered as a part of mathematics, is concerned, the interplay of tautological and empirical questions was clearly described earlier by various scholars like Helmholtz and Mach. It would be a grave error to think that the situation in arithmetic is fundamentally different. If we say that the number 17 is a prime number (which is a tautological proposition) we have also in mind a simple empirical fact, which can be observed with 17 apples or coins. that a division into "equal" groups is impossible. A result which is not so trivial, e.g., that the number 681,199 does not admit of any factorization but 727 X 937 can be verified by trying to divide a corresponding number of objects into equal parts. A shepherd who has to care for 30 sheep can check experimentally that % plus 1ft.; is more than a half. All these examples belong to the domain of experience the theory of which has arithmetic as its tautological part. It is only because the transition between the facts and the axiomatic concepts seems so extremely simple, familiar, and clear here that one usually pays no attention to it and forms from the tautological considerations alone the concept of a science called arithmetic. As far as geometry is concerned, the customary linguistic usage is a little vague in this respect. In mechanics or other parts of physics, it occurs to nobody any more to call their theorems, as far as they refer to real facts, independent of experience and absolutely certain truths, i.e., to confuse the empirical and the tautological aspects. There is no difference in principle between the disciplines of arithmetic, geometry, mechanics, thermodynamics, optics, electricity, etc. It is merely a habit (suggested by the actual situation) in arithmetic, and sometimes in geometry, to reserve the name "arithmetic," or "geometry," for the purely tautological part of the studies. Thus the foundations and basic assumptions of arithmetic are debatable in the same sense as those of any part of physics, i.e., on the one hand, as to the internal questions of tautological structure, and on the other hand, as to the relations with the world of experience. 2. Mathematical Evidence. In all discussions of the foundations of
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mathematics, without distinction between tautological and factual questions, the word "evidence" appears at an early stage. The simplest mathematical statements, so it is asserted, are "self-evident" (manifest) and their absolute certainty is derived from the evidence for them. In fact, even the contemporary controversy between the so-called formalists (Hilbert) and the so-called intuitionists (Brouwer) is for a large part based upon questions of "mathematical evidence." We can assume that everybody, or at least everybody beyond a certain stage of education, is familiar with an experience for which he uses the word "evidence" or "self-evident truth." If anyone claims that to him the truth of a statement is evident or immediately apparent, he is hardly inclined to resolve this impression of his into simpler parts, i.e., to analyze it and to exhibit its components. This could lead us to accept the words "This is evident to me" as a kind of element statement or protocol sentence. As in the case of the statement "This is red," or "Here I have the sensation red," the words should, then, not be analyzed further. but should be regarded as a piece of raw material that has to be used along with other material of the same kind. But physical optics, whose significance is derived from the fact that there are primitive impressions like red and blue, light and dark, or element statements of the form "Here I see red," by no means is as a scientific theory based upon such concepts as blue and red, or light and dark. Surely no course in optics starts with an attempt to make clear what the meanings of blue and red are, or derives anything from the assumption that these meanings are a priori known. The connection between the simple protocol sentences and the physical theory is not so immediate and in any case not so simple. For the moment we do not have to go deeper into the question what is the connection in other areas between element statements and the theories. The point here is that, even granted the existence of a not further analyzable elementary experience of "evidence," it does not follow that mathematics (or any theory) should be built upon the assumption of evidence as a self-explanatory and obvious basic concept. For this to be possible, one condition above all would have to be satisfied: there must be a certain agreement about what knowledge is evident, that is, not only among the mathematicians themselves, but also among those who are only starting to study mathematics and who therefore do not yet draw conclusions from the theory. It does not require much speCUlation to convince oneself of how divided, in all domains, the opinions are as to which statements have the property of evidence. We may even exclude such matters as the spherical shape of the earth, the existence of antipodes, or the rotation of the earth about its axis. Let us limit ourselves to purely mathematical concepts.
Mathematical Postulates and Human Understandzng
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To almost everybody the properties of the positive integers, as they are taught in school, are evident. Most educated people claim, moreover, that the so~called existence of an infinite series of numbers is evident to them. More serious discussions arise if one asks whether the continuum of points on a line segment or the continuum of numbers between 0 and 1 is immediately evident or not. Some people even assert that it is immediately evident that all properties of geometric figures remain unaltered if one changes the dimensions of the figure. The latter lend to Euclidean geometry the character of evidence, while the former, who take the continuum for granted, jump over a wide area of mathematical difficulties. There is no doubt about the lack of agreement as to what is evident "to us." For each person it seems to depend upon education and incidental experience what appears evident to him, not to speak of the vagueness of linguistic formulation of allegedly "evident" propositions. The famous dictum of the mathematician C. G. J. Jacobi, "Mathematics is the science of what is clear by itself" can in no way be maintained today. It is impossible to accept as the basis of mathematics merely statements that seem self-evident, if only because there is no agreement as to what statements actually belong to this class. 3. Intuitionism. Among the various controversial schools of thought in the field of the foundations of mathematics, it is the intuitionists who-as their name indicates-place the greatest emphasis upon intuition, evidence, and immediate apprehension or immediate insight. According to L. E. J. Brouwer, the founder of the intuitionist school, the simplest mathematical ideas are implied in the customary lines of thought of everyday life and all sciences make use of them; the mathematician is distinguished by the fact that he is conscious of these ideas, points them out clearly, and completes them. The only source of mathematical knowledge is, in Brouwer's opinion, the intuition that makes us recognize certain concepts and conclusions as absolutely evident, clear, and indubitable. However, he does not assume that it is possible to list in a precise and complete way all basic fundamental concepts and elementary methods of deduction, which in this sense are to serve as a basis of mathematical derivations. It should always be possible to supplement the once fixed set of assumptions by accepting new ones, if a further intuition leads that way. A first intuition yields us the concept of "two" from which the concept of multiplicity is inferred. Originally Brouwer regarded the continuum too as immediately given by intuition; later he tried to comprehend it by a new concept, the "sequences of free formation," Le., to reduce the continuum in some way, after all, to a sequence of numbers. Disregarding certain rather mystic formulations that Brouwer gave to his doctrine, one recognizes his point of view as very close to a radical empiricism. The thesis that the fundamental assumptions of mathematics
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Richard von Mises
cannot be formulated in a definitely :fixed and complete form, but are subject to continued examination and possible supplementation by intuition (we should prefer to say, by experience which changes the stock of what appears as "evident"), corresponds exactly to our conception. The opposite point of view is that of Kantian a-priorism, according to which the basic mathematical concepts are once and for all impressed upon the human race by the properties of its reasoning power. Such a prediction that a specific chapter of science will never change--and the "a priori" means just that-has no place in the concept of science that this book 8 represents. On the other hand, we see no reason why at any point of time the set of assumptions momentarily appearing as necessary should not be assembled and, in the form of axioms and fundamental rules, be made the basis of the derivations. Today's mathematics, as we know, can be derived only to a small extent from assumptions regarded as intuitive or evident by Brouwer. There is nothing to do but consider the remaining premises for the time being as further hypotheses which perhaps later will prove to be dispensable or replaceable by "intuitive" ones. This is without importance for the inner working of mathematics; witness the fact that the content of the textbooks in the various mathematical fields did not change appreciably in recent years, in spite of the immense influence intuitionism exerted, and justifiably so, upon mathematical thinking. In agreement with the empiristic conception of science. intuitionism holds that the source of mathematics is the insight which We intuitively comprehend trom experience of the external world, but which cannot once and for all be collected in a closed system of axioms. 4. The Excluded Middle. Not only did intuitionism bring new life into the discussions on the foundations of mathematics, which seemed to have reached somehow a dead end, but it also, for the first time in centuries, opened up again problems in elementary logic. Some rumOrs have spread to nonmathematicians-and were accepted with justified suspicion-that the intuitionists deny the validity of the simple rule of the excluded middle. That is, besides the two statements "today is Tuesday" and "today is not Tuesday" this revolutionary theory is supposed to admit a tertium quid. The situation here is as follows. In no case of extramathematical application is the validity of the "tertium exc1usum" questioned. Within mathematics, too, it remains absolutely unshaken as long as one deals with finite sets. If, for example, "An signifies any precisely defined property of a natural number such that it is possible to determine unambiguously whether a given natural number possesses property A or not, then the following alternative holds: among the natural numbers from 1 to ten million there exists a number which has 8
[Positivism: A Study in Human Understanding.]
Mathematical Postulates and Human Understanding
1749
the property A, or there is no such number (but there is no third possibility) . Brouwer, however, noticed that the problem is not so simple once one treats an "infinite" sequence of numbers (such as all even numbers) instead of a finite set (such as the numbers between 1 and ten million). The proposition, "there is a number with the property A" still signifies in this case the same as before, namely, running through the sequence of even numbers, one will find a number possessing the property A. But if We say there is no such number, that does not mean that running through the sequence of all even numbers we never hit upon a number of the property A, because it makes no sense to speak of an examination of infinitely many numbers. The negative statement is rather an abbreviation for the following much more complicated assertion: by means of the axioms and deductive methods of mathematics it can be proved that between the prop· erty of being an even number and the property A there is a contradiction. This is, we think, the only meaning one can reasonably assign to the statement that there is no number having the property A in the infinite set. After this explanation it should be clear that between the assertions, there is a number . . . , and, there is no number . . . , in the case of an infinite set of numbers, the relation of contradictory opposites no longer exists. There is no difficulty in imagining that besides the finding of a number and the provability of a contradiction there is still a third possibility, namely, that neither does one find, on running through the numbers. one that has the property A, nor is a contradiction between the definition of the numbers and the property A derivable by means of mathematics. It is only when one makes the additional assumption that every mathematical problem is solvable that the extension of the theorem of the excluded middle to all problems dealing with infinite sets becomes justified. The nonmathematician will hardly be inclined to regard such an assump· tion as "logically necessary" and thus all that seems so objectionable in the intuitionist thesis disappears. Brouwer, the founder oj intuitionist mathematics, has shown that in certain mathematical problems dealing with infinite sets of numbers the elementary rule of the excluded middle is not admissible, without an additional arbitrary assumption. Statements like: there is a number • . • and: there is no number . . ., in this case only seemingly, by virtue ot their abbreviated linguistic formulations, have the form oj contradictory opposites. 5. New Logic. We saw in the discussion of logistic that the theorem of the tertium exclusum is an immediate consequence of the basic formulas of logic if one assumes these according to the conception of classical logic. Therefore, if one wants to have a logic that satisfies the requirements of
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Richard von Mzses
intuitionist mathematics and that does not break down on application to infinite sets, it is advisable, instead of admitting "exceptions'" to formulate the basic axioms a little differently. This problem was solved for the first time by A. Heyting, but we shall follow the later exposition by A. Kolmogoroff, which points out more clearly the fundamental idea. Let the letters A, B, C . . . signify, instead of statements, as before, problems to be solved. We may think of mathematical problems, e.g., to construct a triangle under given stipulations, or to calculate a number defined in a particular way, say the root of an equation. We shall use the sign for negation in order to signify that the solution of a problem is impossible, i.e., leads to a contradiction. Letters connected by means of a comma-"A,B"-will be taken to mean both problems; those connected by "v"-"AvB"-to mean at least one of the two problems, A,B. Finally, the sign for implication, "~", will be used in the sense that "A~B" means to reduce the solution of B to that of A. Now, for the use of the four signs "v n , "--", "4", ",'. one can prescribe those and only those rules which correspond to the situations in solving problems. For example, the formula "[A,(A~B)]~B" is always true; in words it means that the solution of the problem B is reducible to the solution of the two problems, A and the reduction of B to A. Similarly "[(A~B),(B~C)]~(A~C)"; in words, if B is reducible to A, and C to B, then the reduction of C to A is reducible to these two problems, namely, to the carrying out of the two reductions. In the ordinary propositional calculus the first of these two formulas would state that if A holds and B follows from A, then B also holds; and the second, if B follows from A, and C from B, then it follows that C follows from A. Thus in these two examples there is no essential difference between the propositional calculus and the new problem calculus. But while in the former the formula "--AvA" is valid, i.e., always, either non-A or A holds, we have no reason to admit the generality of the theorem: one of the two properties of problem A must be true, either A is solvable or A is recognizable as contradictory. One can see here that with a suitable agreement about symbols there will result an algorithm that agrees on the whole with ordinary truth-function theory, but that does not contain the formula which expresses the theorem of the excluded middle. This argument has the same significance as the acceptance of the logical independence of Euclid's parallel axiom, which led to the construction of the non-Euclidean geometries. We have arrived here at a special form of "non-Aristotelian logic." a form which is also called by the misleading name (since it is reminiscent of "intuitive") "intuitionist logic." The main application of the intuitionist contribution consists in supplying us with a method of rejecting from all previous mathematical results those in whose derivation the tertium exclusum was used, including. in
Mathematical Postulates and Human Understandmg
1751
particular, all those theorems which rest upon so-called "indirect proofs." If, for instance, 'we are looking for an unknown quantity x and we can prove that the assumption that there is no such x leads to a contradiction, then, according to the new conception, the existence of x is not proved. Brouwer demands a "constructive" proof, i.e., the establishing of a method of calculating the number x. This way of looking at things proves to be fruitful in the study of the basic elements of mathematics, even though it leaves the actual work of mathematicians in most special branches almost unchanged. Mathematicians gsuaUy were satisfied with an indirect proof only if a constructive one could not be found. It makes no sense to argue about whether a quantity "really exists" if only an indirect proof can be given for its existence; the word "exist" cannot be defined independently of what one wants to admit as proofs. The construction of a "problem calculus" in the sense of Heyting and Kolmogoroff yields a model of logic in which the theorem of the excluded middle does not appear among the basic formulas. The study of such a logic widens our insight into the basic elements of mathematics and, in particular, points out the special position of the so-called indirect proofs within mathematics. 6. Formalism. It is mainly Hilbert and his followers who have objected to the thesis of the intuitionists since it first became known. According to the ideas of Hilbert and the "formalistic" school led by him, mathematics in the narrower sense is replaceable by a purely mechanical method for deriving formulas, a method which has nothing to do with the meaning or interpretation of the symbols used. Certain aggregates of symbols are assumed as premises; these are the axioms, and from them further groups of signs are derived according to fixed rules and in a purely mechanical manner, i.e., without the use of conclusions drawn from their interpretation; the new groups are then the provable theorems. Thus, the entire content of mathematics is, according to Hilbert, transformable, in principle, into a system of symbolic formulas. Besides this formal system, however, there is, as Hilbert states, still something else which serves as justification of the system of formulas and is called "metamathematics." It is not clearly stated whether the rules that govern the use of signs in the formal system and describe the methods of deriving new formulas are also considered as part of metamathematics. At any rate, it comprises all arguments that are supposed to lead to the proof of the consistency of the formal system. By a consistency proof is meant the proof that a certain "false" formula, e.g., the formula "1 = 2," cannot be derived; for, on the one hand, the appearance of this formula would make the system useless, and on the other hand, from any other "false" formula existing in the system the proposition "1 2" would follow. Everything then d.:pends upon how and with what means metamathe-
=
1752
Richard von Mlse1l
matics works. It uses meaningful deductions, i.e., it operates with words and idioms whose meaning is somehow abstracted from linguistic usage. According to Hilbert's original thesis, metamathematics should apply only the most elementary and immediately evident logical premises and deductive methods, at any rate only the simplest inferences appearing in the formal system of symbols under consideration, and should use those only in a finite number of repetitions. Therefore, the theorem of the excluded middle too can only be applied to a finite set. The essential idea was that metamathematics, by using only finite mean~ of elementary logic, should be in a position to construct and support the structure of formal mathematics, which deals with infinite sets. That is exactly the point where the questions posed by intuitionism start infiltrating into metamathematics. One cannot say that the end which Hilbert posited for metamathematics has been attained even in a single partial area or that there is any hope of reaching it in the near future. On the contrary, the mathematician Godel recently showed that, in principle, in order to furnish a consistency proof of a formal system one needs means that go beyond what is formalized in the system. That does not necessarily mean a failure of Hilbert's efforts, but it shows that metamathematics does not get around the questions thrown into the discussion by the intuitionists. The opposition between the formalists and the intuitionists, which was originally so violent and apparently irreconcilable, seems gradually to reduce to this: on the one hand, Hilbert's formal mathematics comprises more than a formalization of Brouwer's mathematics could yield, but on the other hand, the metamathematics, which is indispensable in Hilbert's total structure, has to incorporate essential ideas of Brouwer's intuitionism. There is one point in which the opponents still seem irreconcilable. While the formalists-in this they follow mainly Poincare-regard a consistency proof, once it is given, as a complete and total justification of a deductive system, the intuitionists, according to their rejection of the general principle of the excluded middle, do not consider consistency as something positive. According to our repeatedly stated view, formal (tautological) systems appear in science only because they can be coordinated with certain sets of experiences or groups of phenomena. and from this coordination they derive, in the last analysis, their "justification." Inner consistency is certainly a necessary criterion of the usefulness of a system, but for its applicability, i.e., for the possibility of coordination with the world of experience, a sufficient "scientific proof' cannot be given at all. Dividing mathematics into a formal system, which progresses according to mechanical rules, and a metamathematics, which is supposed to lead to the justification of the formal system, does not exclude the difficulties that intuitionism has pointed out. The coordination between mathematics
Mathematical Postulates and Human Understanding
1753
(its tautological side) and reality cannot be reached by a mathematicized doctrine and certainly cannot be settled by a consistency proof. 7. Logicism. With the formalists and the intuitionists one frequently mentions the "logicists" as the third party in the controversy about the foundations of mathematics. Bertrand Russell attempted in his book Principia Mathematica (together with A. N. Whitehead), based upon essential preliminary studies by Frege and Peano, to construct completely the basic concepts used in mathematics, starting from their simplest and most plausible elements. The solution of this logical problem, to which the authors apply admirable ingenuity, will never be able to command universal acceptance. For it remains undecided, and depends on each individual, whether such concepts as sequence of numbers, cardinal numbers, etc., are less simple and immediate than successor, one-to-one coordination of elements, etc. The choice of the preferred starting point will always depend upon the experience of the individual. But about the relation to reality, to the world of experience, logicists do not want to say anything; they see their goal in the complete exhibition of the tautological relations, i.e., relations that are fixed by definitions and other linguistic ru1es. In this respect, their merits are indisputable; we mention here only one example. In almost all branches of mathematics there appears a type of argument that always has been regarded as a "principle," characteristic of mathematics, and at times considered an inscrutable mystery: the process of so-called complete induction. If we divided 7 by 3, then the first digit after the decimal point of the quotient is a 3, and the remainder is 1. Furthermore it is easy to prove that in division by 3 a decimal which yields the remainder 1 must always be followed by another 3 with the remainder 1. From this one customarily "concludes" that all the "infinitely many" decimal digits of the quotient are 3's. This is the method of mathematical induction, and with respect to it the question was raised (e.g., by H. Poincare), how is it possible to draw from such a small and in any case finite number of inferences an infinite number of conclusions (namely about all infinitely many decimals). The fact is that nothing at all is concluded here. The sentence "All decimal digits are 3's" is only a different linguistic expression for "The first decimal digit is 3, and as often as there appears a 3, it is followed by another 3." The word "all," in this context, i.e., applied to the infinite sequence of digits in the decimal fraction, has no other meaning than that determined by the concept of succession (by which the sequence of natural numbers is defined). There are other problems in mathematics in which the word "all," referring to infinite sets of a different kind, has another meaning and where the method of induction is not applicable at all. This situation is fu11y cleared up in Whitehead and Russell's foundation of mathematics. While the ordinary textbooks of mathematics have the purpose of developing chains of mathematical de-
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Richard von Mises
ductions ascending from simpler to more and more complicated forms, Russell drives the examination of the tautologies in the opposite direction, toward the origin. We return now to what was said at the start of this chapter, that mathematics as a whole, like any other science, has a tautological and an empirical side; it differs from other sciences in that here the formal side is much more essential and decisive than anywhere else. Thus it becomes understandable that mathematics has often been identified completely with its tautological part, as witnessed by Goethe's well-known utterance, Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at all. And thus it goes on and on in its conclusions, except that in the higher formulas the identity fades out of sight. To this we have to say that four is not only two times two, but also three plus one, and fifteen minus eleven, and the cube root of 64, and so on. It is the task of mathematics in the narrower sense to describe the relations of these various "identities." But there is also an empirical side to mathematical doctrines, where there is no more talk about the "infallibility of conclusions," and which must not be neglected in an epistemological investigation. The logicistic foundation of mathematics deals with an analysis of its tautological or deductive part, by attempting to reduce the basic mathematical concepts and methods to the simplest and most plausible elements. There is no reference to the relation with reality; the connection with the world of experience is hidden in the choice of the basic elements. Our final conclusion is this: None of the three forms of the foundation of mathematics, the intuitionist, the formalistic. or the logicistic, is capable of completely rationalizing the relation between tautological systems and (extramathematical) experiences. which is its very purpose, i.e., to make this relation a part of the mathematical system itself.
PART XII
The Mathenlatical Way of Thinking 1. The Study That Knows Nothing of Observation
2. 3. 4. 5.
6.
by JAMES JOSEPH SYLVESTER The Essence of Mathematics by CHARLES SANDERS PEIRCE The Economy of Science by ERNST MACH Measurement by NORMAN CAMPBELL Numerical Laws and the Use of Mathematics in Science by NORMAN CAMPBELL The Mathematical Way of Thinking by HERMANN WEYL
COMMENTARY ON
An Eristic Controversy HE famous address of J. J. Sylvester to the British Association in 1869 on the nature of mathematics was a genteel polemic fired at Thomas Henry Huxley. Huxley, himself a formidable controversialist, had contributed to Macmillan's Magazine an article on the sad shortcomings of the scientific training offered by contemporary British education. and to the celebrated Fortnightly Review a decapitating assault on the positivistic philosophy of Auguste Comte. 1 The Macmillan piece, enlivened by one of Huxley's stinging attacks on the c1ergy,2 stressed the importance of scientific training-"in virtue of which it cannot be replaced by any other discipline whatsoever"-for bringing the mind "directly into contact with fact, and practicing the intellect in the completest form of induction." Mathematics, on the other hand (said Huxley), is almost purely deductive and cannot discipline the mind in this way. "The mathematician starts with a few propositions, the proof of which is so obvious that they are called self-evident, and the rest of his work consists of subtle deductions from them." In the Fortnightly Review, Huxley quoted Comte: "C'est done par l'etude des mathematiques, et seulement par elle, que ron peut se faire une idee juste et approfondie de ce que c'est qu'une science." This placing of mathematics at the summit of knowledge, pointing to its method as the • model and goal of all scientific enquiry, was the central principle of Cornte's philosophy. To Huxley the principle implied an outrageous affront to the experimental method. He felt obliged to make this outrageous reply: "That is to say, the only study which can confer 'a just and comprehensive idea of what is meant by science,' and at the same time, furnish an exact conception of the general method of scientific investigation, is that which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation." Having thus disposed of mathematics, a subject which irritated him because he knew so little about it, he moved briskly to another front. The statement, however, created some excitement, and no one was better equipped than Sylvester "to speak for mathematics" and demonstrate the silliness of Huxley's eloquent tirade. The address is typical of Sylvester. 3 It is learned, allusive, flowery, dis.
T
1 "Scientific Education: Notes of an After-dinner Speech," Macmillan's Magazine, Vol. XX, 1869; Fortnightly Review, Vol. II, N.S. 5. 2 For example: The clergy are "divisible in three sections: an immense body who are ignorant and speak out; a small proportion who know and are silent' and a minute minOrity who know and speak according to their knowledge." ' 3 For a biographical essay On this eminent mathematician see p. 341.
1756
An EnsUe Controversy
1757
cursive, amusing and long-winded. It is as richly stocked with excellent things as with irrelevancies. The crowning irrelevancy was the pUblication of the lecture, together with profuse annotations and ensuing correspondence provoked by an incidental reference to Kant's doctrine of space and time, as an appendix to Sylvester's Laws of Verse. The connection between the principles of sound versification and Huxley's, not to mention Kant's, critique is not obvious. Sylvester was never deterred by such considerations. It was his practice to give the public the benefit of his views on the subject uppermost in his mind, regardless of what topic was announced on the lecture program or title page. The story is told that he was scheduled one evening, while he was living in Baltimore, to give a reading of a 400-line poem he had written, all the lines rhyming with the name Rosalind. A large audience gathered to witness this improbable feat. "Professor Sylvester, as usual, had a number of footnotes appended to his production; and he announced that in order to save interruption in reading the poem itself, he would first read the footnotes. The reading of the footnotes suggested various digressions to his imagination; an hour had passed, still no poem; an hour and a half passed and the striking of the clock or the unrest of his audience reminded him of the promised poem. He was astonished to find how time had passed, excused all who had engagements, and proceeded to read the Rosalind poem." 4 Huxley never replied to the British Association lecture. Yet it is doubtful that Sylvester disposed of the questions it raised as completely as he imagined. The extent to which abstract mathematical concepts are suggested by experience remains a topic of lively debate. Sylvester touches upon the problem, but in less than a paragraph rushes off in another direction. The effectiveness of the lecture lies not so much in its rebuttal of Huxley. as in its kaleidoscopic survey of nineteenth-~entury mathematical thought, and its display of Sylvester's amazing erudition and imagination. 4 Alexander Macfarlane, Lectures on Ten British Mathematicians of the Nineteenth Century; New York, 1916, pp. 117-18. E. T. Bel] tells the same story in his essay on Cayley and Sylvester: see p. 341.
Having laid down fundamental principles of the wisdom of the Latins so far as they are found In language, mathematics and optics, I now wish to unfold the principles of experimental science, since without experience nothing can be sufficiently known. For there are two modes 01 acquiring knowledge, namely, by reasoning and experience. Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the inituition of truth, unless the mind discovers it by the path of experience; since many have the arguments relating to what can be known, but because they lack experience they neglect the arguments, and neither avoid what is harmful nor follow what is good. For if a man who has never seen a fire should prove by adequate reasoning that fire burns and injures things and destroys them, his mmd would not be satisfied thereby, nor would he avoid fire, until he placed his hand or some combustIble substance in the fire, so that he might prove by experience that which reasonmg taught. But when he has actual experience of combustIon his mind is made certain and rests in the full lzght of truth. Therefore, reasoning does not suffice, but experi-ROGER BACON ence does.
1
The Study That Knows Nothing of Observation By JAMES JOSEPH SYLVESTER EXCERPT OF ADDRESS TO BRITISH ASSOCIATION,
1869
IT IS said of a great party leader and orator in the House of Lords that, when lately requested to make a speech at some religious or chari~ table (at all events a non-political) meeting, he declined to do so on the ground that he could not speak unless he saw an adversary before himsomebody to attack or reply to. In obedience to a somewhat similar combative instinct, I set to myself the task of considering certain recent utterances of a most distinguished member of this Assocation, one whom I no less respect for his honesty and public spirit than I admire for his genius and eloquence,1 but from whose opinions on a subject which he has not studied I feel constrained to differ. Goethe has said"VersHindige Leute kannst du irren sehn In Sachen, namlich, die sie nieht verstehn." Understanding people you may see erring-in those things, to wit, which they do not understand. I have no doubt that had my distinguished friend, the probable President-elect of the next Meeting of the Association, applied his uncommon powers of reasoning, induction, comparison, observation, and invention to 1 Although no great lecture·goer, I have heard three lectures in my life which have left a lasting impression as masterpIeces on my memory---Clifford on Mind, Huxley
on Chalk, Dumas on Faraday. 1758
The Study That Knows Nothing of Observatzon
1759
the study of mathematical science, he would have become as great a mathematician as he is now a biologist; indeed he has given public evidence of his ability to grapple with the practical side of certain mathematical questions; but he has not made a study of mathematical science as such, and the eminence of his position and the weight justly attaching to his name render it only the more imperative that any assertions proceeding from such a quarter, which may appear to me erroneous, or so expressed as to be conducive to error, should not remain unchallenged or be passed over in silence)~ He says "mathematical training is almost purely deductive. The mathematician starts with a few simple propositions, the proof of which is so obvious that they are called self-evident, and the rest of his work consists of subtle deductions from them. The teaching of languages, at any rate as ordinarily practised, is of the same general nature-authority and tradition furnish the data, and the mental operations are deductive." It would seem from the above somewhat sin~ularly juxtaposed paragraphs that, according to Prof. Huxley, the business of the mathematical student is from a limited number of propositions (bottled up and labelled ready for future use) to deduce any required result by a process of the same general nature as a student of language employs in declining and conjugating his nouns and verbs-that to make out a mathematical proposition and to construe or parse a sentence are equivalent or identical mental operations. Such an opinion scarcely seems to need serious refutation. The passage is taken from an article in Macmillan's Magazine for June last, entitled "Scientific Education-Notes of an Mter-dinner Speech," and I cannot but think would have been couched in more guarded terms by my distinguished friend had his speech been made be/ore dinner instead of after. The notion that mathematical truth rests on the narrow basis of a limited number of elementary propositions from which all others are to be derived by a process of logical inference and verbal deduction, has been stated still more strongly and explicitly by the same eminent writer in an article of even date with the preceding in the Fortnightly Review, where we are told that "Mathematics is that study which knows nothing of observatioil, nothing of experiment, nothing of induction, nothing of causation." I think no statement could have been made more opposite to the undoubted facts of the case, that mathematical analysis is constantly invoking the aid of new principles, new ideas, and new methods, not capable of being defined by any form of words, but springing direct from the inherent powers and activity of the human mind, and from continually renewed introspection of that inner world of thought of which the phenomena are as varied and require as close attention to discern as those of the outer 2 In his eioge of Daubenton, Cuvier remarks, "Les savants jugent toujours comme le vulgaire les ouvrages qui ne sont pas de leur genre."
1760
lamtJs Joseph Sylvester
physical world (to which the inner one in each individual man may, I think, be conceived to stand in somewhat the same general relation of correspondence as a shadow to the object from which it is projected, or as the hollow palm of one hand to the closed fist which it grasps of the other), that it is unceasingly calling forth the faculties of observation and comparison, that one of its principal weapons is induction, that it has frequent recourse to experimental trial and verification, and that it affords a boundless scope for the exercise of the highest efforts of imagination and invention. Lagrange, than whom no greater authority could be quoted, has expressed emphatically his belief in the importance to the mathematician of the faculty of observation; Gauss has called mathematics a science of the eye, and in conformity with this view always paid the most punctilious attention to preserve his text free from typographical errors; the ever to be lamented Riemann has written a thesis to show that the basis of our conception of space is purely empirical, and our knowledge of its laws the result of observation, that other kinds of space might be conceived to exist subject to laws different from those which govern the actual space in which we are immersed, and that there is no evidence of these laws extending to the ultimate infinitesimal elements of which space is composed. Like his master Gauss, Riemann refuses to accept Kant's doctrine of space and time being forms of intuition, and regards them as possessed of physical and objective reality. I may mention that Baron Sartorius von Waltershausen (a member of this Association) in his biography of Gauss ("Gauss zu Gedachtniss"), published shortly after his death, relates that this great man used to say that he had laid aside several questions which he had treated analytically, and hoped to apply to them geometrical methods in a future state of existence, when his conceptions of space should have become amplified and extended, for as we can conceive beings (like infinitely attenuated bookworms 3 in an infinitely thin sheet of paper) which possess only the notion of space of two dimensions, so we may imagine beings capable of realising space of four or a greater number of dimensions. 4 Our 3 I have read or been told that eye of observer has never lighted on these depredators, living or dead. Nature has gifted me with eyes of exceptional microscopic power, and I can speak with some assurance of having repeatedly seen the creature wriggling on the learned page. On approaching it with breath or finger-nail it stiffens out into the semblance of a streak of dirt, and so eludes detection. 4 It is well known to those who have gone into these views that the laws of motion accepted as a fact suffice to prove in a general way that the space we live in is a flat or level space (a Uhoma}old"), our existence therein being assimilable to the life of the bookworm in an un rumpled page: but what if the page should be undergoing a process of gradual bending mto a curved form? Mr. W. K. Clifford has indulged in some remarkable speculauons as to the possibility of our being able to infer, from certain unexplained phenomena of light and magnetism, the fact of our level space of thr~e dime~sions being in the act of undergoing In space of four dimensions (space as inconceIvable to us as our space to the supposititious bookworm) a distortion analogous to the rumpling of the page to which that creature's powers of direct perception have been postulated to be limited.
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Cayley, the central luminary, the Darwin of the English school of mathematicians, started and elaborated at an early age, and with happy consequences, the same bold hypothesis. Most, if not all, of the great ideas of modem mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all the efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blowpipe flame of Gauss's transcendent genius; or the doctrine of double periodicity, which resulted from the observation by Jacobi of a purely analytical fact of transformation; on Legendre's law of reciprocity; or Sturm's theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected with the motion of compound pendulums; or Huyghens' method of continued fractions, characterized by Lagrange as one of the principal discoveries of "that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton"; or the New Algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this Chair, "that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has almost become new through it, algebraic geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics" (he might, if speaking at the present moment, go on to add the theory of elasticity and the highest developments of the integral calculus) "have all felt its influence." Now this gigantic outcome of modern analytical thought, itself, too, only the precursor and progenitor of a future still more heaven-reaching theory, which will comprise a complete study of the interoperation, the actions and reactions, of algebraic forms (Analytical Morphology in its absolute sense), how did this originate? In the accidental observation by Eisenstein, some score or more years ago, of a single invariant (the Quadrinvariant of a Binary Quartic) which he met with in the course of certain researches just as accidentally and unexpectedly as M. Du Chaillu might meet a Gorilla in the country of the Fantees, or anyone of us in London a \Vhite Polar Bear escaped from the Zoological Gardens. Fortunately he pounced down upon his prey and preserved it for the contemplation and study of future mathematicians. It occupies only part of a page in his collected posthumous works. This single result of observation (as well entitled to be so called as the discovery of Globigerinre in chalk or of the Confoco-ellipsoidal structure of the shells of the Foraminifera), which remained unproductive in the hands of its distinguished author, has served to set in motion a train of thought and to propagate an impulse which
1162
James Joseph Sylvest"
have led to a complete revolution in the whole aspect of modem analysis, and whose consequences will continue to be felt until Mathematics are forgotten and British Associations meet no more. I might go on, were it necessary I piling instance upon instance to prove the paramount importance of the faculty of observation to the process of mathematical discovery.5 Were it not unbecoming to dilate on one's personal experience, I could tell a story of almost romantic interest about my own latest researches in a field where Geometry, Algebra, and the Theory of Numbers melt in a surprising manner into one another, like sunset tints or the colours of the dying dolphin, "the last still loveliest" (a sketch of which has just appeared in the Proceedings of the London Mathematical Society 6), which would very strikingly illustrate how much observation, divination, induction, experimental trial, and verification, causation, too (if that means, as I suppose it must, mounting from phenomena to their reasons or causes of being), have to do with the work of the mathematician. In the face of these facts, which every analyst in this room or out of it can vouch for out of its own knowledge and personal experience, how can it be maintained, in the words of Professor Huxley, who, in this instance, is speaking of the sciences as they are in themselves and without any reference to scholastic discipline, that Mathematics "is that study which knows nothing of observation, nothing of induction, nothing of experiment, nothing of causation"? I, of course, am not so absurd as to maintain that the habit of observation of external nature will be best or in any degree cultivated by the study of mathematics, at all events as that study is at present conducted; and no one can desire more earnestly than myself to see natural and experimental science introduced into our schools as a primary and indispensable branch of education: I think that that study and mathematical culture should go on hand in hand together, and that they would greatly influence each other for their mutual good. I should rejoice to see mathematics taught with that life and animation which the presence and example of her young and buoyant sister could not fail to impart, short roads preferred to long ones, Euclid honourably shelved or buried "deeper than did ever plummet sound" out of the schoolboy's reach, morphology introduced into the elements of Algebra-projection, correlation, and motion accepted as aids to geometry-the mind of the student quickened and elevated and his faith awakened by early initiation into the ruling ideas of polarity, continuity, I> Newton's Rule was to all appearance, and according to the more received opinion, obtained inductively by its author My own reduction of Euler'S problem of the Virgins (or rather one slightly m.:re general than this) to the form of a question (or, to speak ,more exactly, a set of questions) in simple partitions was, strange to say, first obtamed by myself inductively, the result communicated to Prof. Cayley, and proved subsequently by each of us independently, and by perfectly distinct methods. 6 Under the title of "Outline Trace of the Theory of Reducible Cyclodes:'
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1763
infinity, and familiarization with the doctrine of the imaginary and inconceivable. It is this living interest in the subject which is so wanting in our traditional and mediaeval modes of teaching. In France, Germany, and Italy, everywhere where I have been on the Continent, mind acts direct on mind in a manner unknown to the frozen formality of our academic institutions; schools of thought and centres of real intellectual cooperation exist; the relation of master and pupil is acknowledged as a spiritual and a lifelong tie, connecting successive generations of great thinkers with each other in an unbroken chain, just in the same way as we read, in the catalogue of our French Exhibition, or of the Salon at Paris, of this man or that being the pupil of one great painter or sculptor and the master of another. When followed out in this spirit. there is no study in the world which brings into more harmonious action all the faculties of the mind than the one of which I stand here as the humble representative, there is none other which prepares so many agreeable surprises for its followers, more wonderful than the changes in the transformation-scene of a pantomime, or, like this, seems to raise them, by successive steps of initiation, to higher and higher states of conscious intellectual being. This accounts, I believe, for the extraordinary longevity of all the greatest masters of the Analytical art, the Du Majores of the mathematical Pantheon. Leibnitz lived to the age of 70; Euler to 76; Lagrange to 77; Laplace to 78; Gauss to 78; Plato, the supposed inventor of the conic sections, who made mathematics his study and delight, who called them the handles or aids to philosophy, the medicine of the soul, and is said never to have let a day go by without inventing some new theorems, lived to 82; Newton, the crown and glory of his race, to 85; Archimedes, the nearest akin, probably, to Newton in genius, was 75, and might have lived on to be 100, for aught we can guess to the contrary, when he was slain by the impatient and ill-mannered sergeant, sent to bring him before the Roman general, in the full vigour of his faculties, and in the very act of working out a problem; Pythagoras, in whose school, I believe, the word mathematician (used, however, in a somewhat wider than its present sense) Originated, the second founder of geometry, the inventor of the matchless theorem which goes by his name, the precognizer of the undoubtedly miscalled Copernican theory, the discoverer of the regular solids and the musical canon, who stands at the very apex of this pyramid of fame, (if we may credit the tradition) after spending 22 years studying in Egypt, and 12 in Babylon, opened school when 56 or 57 years old in Magna Grrecia, married a young wife when past 60, and died, carrying on his work with energy unspent to the last, 'at the age of 99. The mathematician lives long and lives young; the wings of his soul do not early drop off, nor
1764
James Joseph S,lVl!lter
do its pores become clogged with the earthy particles blown from the dusty highways of vulgar life. Some people have been found to regard all mathematics, after the 47th proposition of Euclid, as a sort of morbid secretion, to be compared only with the pearl said to be generated in the diseased oyster, Of, as I have heard it described, "une excroissance maladive de resprit humain." Others find its justification, its "raison d'etre," in its being either the torch-bearer leading the way, or the handmaiden holding up the train of Physical Science; and a very clever writer in a recent magazine article, expresses his doubts whether it is, in itself, a more serious pursuit, or more worthy of interesting an intellectual human being, than the study of chess problems or Chinese puzzles. What is it to us, they say, if the three angles of a triangle are equal to two right angles, or if every even number is, or may be, the sum of two primes, or if every equation of an odd degree must have a real root. How dull, stale, flat, and unprofitable are such and such like announcements! Much more interesting to read an account of a marriage in high l~fe, or the details of an international boat-race. But this is like judging of architecture from being shown some of the brick and mortar, or even a quarried stone of a public building, or of painting from the colours mixed on the palette, or of music by listening to the thin and screechy sounds produced by a bow passed haphazard over the strings of a violin. The world of ideas which it discloses or illuminates, the can" templation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance. In conformity with general usage, I have used the word mathematics in the plural; but I think it would be desirable that this form of word should be reserved for the applications of the science, and that we should use mathematic in the singular number to denote the science itself, in the same way as we speak of logic, rhetoric, or (own sister to algebra 7) music. Time was when all the parts of the subject were dissevered, when algebra, geometry, and arithmetic either lived apart or kept up cold relations of acquaintance confined to occasional calls upon one another; but that is now at an end; they are drawn together and are constantly becoming more . and more intimately related and connected by a thousand fresh ties, and I have elscv.here (in my "Tnlogy" published in the Philosophical Transactions) the close connexion between these two cultures, not merely as having Anthmetlc for theIr common parent, but as similar in th(!ir habits and affections. I ha"'e caned "Music the Algebra of '!ense, Algebra, the Music of the reason; Music the dream, A1gebra the \vaking life,-the soul of each the same!" 7
ref~rred ~o
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1765
we may confidently look forward to a time when they shall form but one body with one soul. Geometry formerly was the chief borrower from arithmetic and algebra, but it has since repaid its obligations with abundant usury; and if I were asked to name, in one word, the pole-star round which the mathematical firmament revolves, the central idea which pervades as a hidden spirit the whole corpus of mathematical doctrine, I should point to Continuity as contained in our notions of space, and say, it is this, it is this! Space is the Grand Continuum from which, as from an inexhaustible reservoir, all the fertilizing ideas of modern analysis are derived; and as Brindley, the engineer, once allowed before a parliamentary committee that, in his opinion, rivers were made to feed navigable canals, I feel almost tempted to say that one principal reason for the existence of space, or at least one principal function which it discharges, is that of feeding mathematical invention. Everybody knows what a wonderful influence geometry has exercised in the hands of Cauchy, Puiseux, Riemann, and his followers Clebsch, Gordan, and others, over the very form and presentment of the modern calculus, and how it has come to pass that the tracing of curves, which was once to be regarded as a puerile amusement, or at best useful only to the architect or decorator, is now entitled to take rank as a high philosophical exercise, inasmuch as every new curve or surface, or other circumscription of space is capable of being regarded as the embodiment of some specific organized system of continuity.s The early study of Euclid made me a hater of Geometry, which I hope may plead my excuse if I have shocked the opinions of any in this room (and I know there are some who rank Euclid as second in sacredness to the Bible alone, and as one of the advanced outposts of the British Constitution) by the tone in which I have previously alluded to it as a schoolbook; and yet, in spite of this repugnance, which had become a second nature in me, whenever I went far enough into any mathematical question, I found I touched, at last, a geometrical bottom: so it was, I may instance, in the purely arithmetical theory of partitions; so, again, in one of my more recent studies, the purely algebraical question of the invariantive criteria of the nature of the roots of an equation of the fifth degree: the first inquiry landed me in a new theory of polyhedra; the latter found its perfect and only possible complete solution in the construction of a surface of the ninth order and the subdivision of its infinite content into three distinct natural regions. Having thus expressed myself at much greater length than I originally 8 M. Camille Jordan's application of Dr. Salmon's Eikosi-heptagram to Abelian functions is one of the most recent instances of this reverse action of geometry on analysis. Mr. Crofton's admirable apparatus of a reticulation with infinitely :fine meshes rotated successively through indefinitely small angles, which he applies to ob· taming whole families of definite integrals, is another equally striking example of the same phenomenon.
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lames loseph Sylvester
intended on the subject, which, as standing first on the muster-roll of the Assocation, and as having been so recently and repeatedly arraigned before the bar of public opinion, is entitled to be heard in its defence (if anywhere) in this place,-having endeavoured to show what it is not, what it is, and what it is probably destined to become, I feel that I must enough and more than enough have trespassed on your forbearance, and shall proceed with the regular business of the Meeting.
COMMENTARY ON
CHARLES SANDERS PEIRCE HARLES SANDERS PEIRCE (1839-1914), the founder of pragmatism, was an able scientist, a creative philosopher, a great logician. Today he is recognized as one of the most gifted and influential thinkers this country has produced-an esteem in marked contrast to the neglect he suffered during his life. Peirce's failure to win recognition was due partly to outer circumstance, partly to his own prickly personality, partly to a brilliance and breadth of vision that confused and frightened lesser men. The present tendency is perhaps to overpraise him, to read into him anticipations of the central ideas of some of the best-known philosophers and logicians of the last quarter century. 1 But there can be no doubt that he achieved much of lasting value. Whitehead judged him shrewdly: "The essence of his thought was originality in every subject that he taught." Peirce was born in Cambridge, Massachusetts, the second son of Benjamin Peirce, a Harvard professor and the foremost American mathematician of his time. 2 The elder Peirce, a forceful man, was bent on turning his son into a thinking machine. He closely supervised every step of the boy's education, bearing down hard on his evident mathematical talents, but also putting him through somewhat unconventional exercises in the "art of concentration." "From time to time they would play rapid games of double dummy together, from ten in the evening until sunrise, the father sharply criticizing every error." The results of this bair-raising regimen were mixed. Charles was a rather poor student at Harvard; and it is clear that by driving him, by making excessive demands, his father aggravated, if indeed he did not instill, the unfortunate traits which contributed to Peirce's later misfortunes. S On the other hand Benjamin Peirce
C
1 See Ernest Nagel, "Charles Peirce's Guesses at the Riddle," The Journal of Philosophy, Vols XXX (July, 1933), XXI (1934), XXXIII (1936). 2 The main source of the data for this sketch is the authoritative article on Peirce by Paul Weiss in the Dictionary of Amerzcan Biography, New York, 1928-1944. Many of the unkeyed quotations are from this article. Other sources include Thomas A. Goudge, The Thought 0/ c. S. Peirce, Toronto, 1950; Studies in the. PhIlosoph}' 0/ Charles Sanders Peirce. edited by Philip P. Wiener and Frederic H. Young, Cambridge (Mass.), 1952; The Philosophy of Peirce, Selected Writings, edited by Justus Buchler, New York, 1950; Chance, Love and Logic, edited by Moms R. Cohen, New York, 1949; W. B. Gallie, Peirce and Pragmatism, Penguin Books, Baltimore, 1952; Ernest Nagel, "Charles Peirce's Guesses at the Riddle," Journal of Philosophy, July 16, 1933, pp. 365-386 (continued in Vols. XXXI and XXXUI); Ernest Nagel, "Charles S. Peirce, Pioneer of Modem Empiricism," Philosophy of Science, VoL 7. no. 1, January, 1940, pp. 69-80. The most important collection of Peirce's writings, a work which has contributed significantly to the revival of interest in his ideas, is The Collected Papers 0/ Charles Sanders Peirce. Vols. I-VI, edited by Charles Hartshorne and Paul Weiss, Cambridge (Mass.), 1931-35. 3 Gallie, op. cit., p. 35.
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Editor's Comment
was a talented man of wide interests, and the training he gave his son in experimental science, mathematics, logic and philosophy was invaluable. "He educated me," Charles could justly say, "and if I do anything it will be his work." It was his father's wish that he become a scientist. Peirce was more inM elined to pursue philosophy and found a way of following both professions. In 1861 he joined the United States Coast Survey, where he remained in various posts for thirty years. He served as a computer for the nautical almanac, made pendulum investigations, was put in charge of gravity research and wrote a number of solid scientific papers. Some of these appeared in Photometric Researches (1878), the only book of his published in his lifetime, and one that earned him "international recognition among contemporary astrophysicists." 4 The government post, though obviously not a sinecure, left him with enough time to teach and to engage in private researches in science, philosophy and logic. During the 1860s he gave lecture courses at Harvard in the philosophy of science and logic; for five years he taught logic at lohns Hopkins. In 1867 Peirce read before the American Academy of Arts and Sciences a short paper on the work of George Boole. This marked the beginning of a series of writings which established him as "the greatest formal logician of his time, and the most important single force in the period from Boole to Ernst Schroder." 5 His technical logical papers are today regarded as "primarily of historical interest," but the weight of his contribution to the advancement of this science is beyond dispute. o His main mathematical studies, published and unpublished, dealt with foundation problems, associative algebra, the theory of aggregates, transfinite arithmetic (in which he "anticipated or ran parallel with" the work of Richard Dedekind and Georg Cantor), analysis situs, and related topics.7 He was one of the first proponents of the frequency interpretation of probability. Though his writings in pure mathematics were not extensive, they were of characteristic originality, always suggestive, often prophetic; and his treatment of the logical and philosophical aspects of mathematics was of high quality. 4 Frederic Harold Young, "Charles Sanders Peirce: 1839-1914," in Wiener and Young, op. cit., p. 272. :; "He radically modified, extended and transformed the Boolean algebra, making it applicable to propositions, relations, probability and arithmetic. Practically single.. handed. following De Morgan, Peirce laid the foundations of the logic of relations, the instrument for the logical analysis of mathematics. He invented the copula of inclusion, the most important symbol in the logic of classes, two new logical algebras, two new systems of logical graphs, discovered the link between the logic of classes and the logic of propositions, was the first to gIve the fundamental principle for the loglca! development of mathematics, and made exceedingly important contributions to probability theory, induction, and the logIC of scientific methodology." Dictionary of American Biography. loco Cit. e Ernest Nagel. Philosophy of Science, op. cit., p. 72. 7 Benjamin Peirce, attracted to associative algebra by his son's work in that field, wrote the pioneer text, Lmear Associative Algebra, which opens with the famous sentence: "Mathematics is the science which draws necessary conclusions."
Charles Sanders Peirce
1769
Pragmatism is said to have had its origin in the discussions of a fortnightly "Metaphysical Club" (a name chosen, according to Peirce, "to alienate all whom it would alienate"») founded in Cambridge in the seventies, whose members included Oliver Wendell Holmes (the jurist), John Fiske and Francis E. Abbott. Others who played a significant part in the evolution of the concept were the mathematician and philosopher Chauncey Wright and. of course, William James, Peirce's "'lifelong friend and benefactor." James was unusually persuasive in publicizing the pragmatic view but he gave it a "characteristic twist" which split it away from Peirce's doctrines. S Peirce offered the first definition of pragmatism in an article published in Popular Science Monthly (January 1878) under the title "How to Make Our Ideas Clear." To achieve this laudable goal he suggested that we "consider what effects, which might conceivably have practical bearings, we conceive the object of our conception to have. Then our conception of these effects is the whole of our conception of the object." The definition itself is not a promising beginning to the task of intellectual clarification, but while philosophers grumbled over and even ridiculed its awkwardness they obviously understood what it meant. It was a maxim proposed, in Ernest Nagel's words, as a "guiding principle of analysis. It was offered to philosophers in order to bring to an end disputes which no observation of facts could settle because they involved terms with no definite meaning." It was intended to "eliminate specious problems, and unmask mystification and obscurantism hiding under the cloak of apparent profundity. . . . Above all it pointed to the fact that the 'meaning' of terms and statements relevant in inquiry consist in their being used in determinate and overt ways." 9 Peirce's pragmatism is closely related to his "critical common-sensism" and "contrite fallibilism," two of his often-used expressions. 10 By "common-sensism" he meant that on a great many matters we have no sensible alternative to adopting vague but "indubitable beliefs" which "rest on the everyday experience of many generations of multitudinous populations." Examples of such beliefs are that :fire bums, that incest is undesirable, that there is an element of order in the universe. l l To be sure, these "instinctive beliefs" may change in time, or may in certain instances be proved false; but in the main it is insincere to pretend we can disregard them, start with a Cartesian clean 8 James urged that pragmatism justified, in areas where proof was impossible, moral and religious issues, for example, the embracing of "falth" or the adoption of "unreasoned" beliefs if they were conducive to inward happiness or "beneficial" in other ways. Peirce descnoed this doctrine of the "Will-to-Believe" as "suicidal." Gallie, op. cit., pp. 25 et seq. 9 Ernest Nagel, Pluiosophr oj Science, op. cit., p. 73. 10 See, for example, Nagel. op. elf., pp. 77-79; GaIlie, op. cit •• pp. 106 er seq.; Roderick M. Chisholm. "Fallibilism and Belief," in Wiener and Young, op. cit., pp. 93-110; Arthur F. S. MulIy::n. "Some Implications of ComIl1on~Sensism:' in Wiener and Young. op. cit .. pp. 111-120. 11 See The-mas A. Goudge. op. CIt., pp. 16-17.
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slate and make any real advances in knowledge. 12 Fallibilism is the twin to the tenet of common-sensism. All beliefs and all conclusions, however arrived at, are subject to error. The methods of science are more useful than old wives' gossip for achieving "stable beliefs and reliable conclusions," but even science offers no access to "perfect certitude or exactitude. We can never be absolutely sure of anything, nor can we with any probability ascertain the exact value of any measure or general ratio. This :is my conclusion after many years study of the logic of science." 13 I can do no more than mention Peirce's other labors in philosophy, which include a fairly explicit formulation of a cosmology, social theories of reality and logic, papers on epistemology and numerous writings on his difficult and obscure but valuable "theory of signs." Peirce's interest in logic, it should be observed, was a consequence of his initial concern with philo~ophical problems. But it soon came about that "he saw philosophy and other subjects almost entirely from a logic perspective." His importance in the history of thought, says Ernest Nagel, is due not only to his contributions to logic and mathematics, but to the stimulating effect on the study of scientific method of a mind at once philosophical and dis~ ciplined by first-hand knowledge of the sciences. In ordinary social relations Peirce must have been hard to stomach. He was emotional, quarrelsome, vain, arrogant and snobbish. He was careless in money matters, gullible, impractical and slovenly in appearance. "I yield to no one," said James, "in admiration of his genius, but he is paradoxical and unsociable of intellect, and hates to make connec~ tion with anyone he is with." 14 He was erratic in the usual tiresome fashion-forgetting appointments, misplacing things, and so on; and a "queer being," as James described him, in more interesting ways: for example, he was not merely ambidextrous but could "write a question with one hand and the answer simultaneously with the other." It is not surprising that in spite of strenuous efforts on his part and the enthusiastic support of influential admirers he could never get a permanent teaching job. His reputation as a heavy drinker (now known to be 12 Peirce wrote: "A man may say 'I will content myself with common sense.' I, for one, am with him there, in the main. I shall show why I do not think there can be any direct profit in going behind common sense-meaning by common sense those ideas and beliefs that man's situation absolutely forces upon him. . . . I agree, for example, that it is better to recognize that some things are red and some others blue, in the teeth of what optical philosophers say, that it is merely that some things are resonant to shorter ether waves and some to longer ones." Hartshorne and Weiss, op. cit., Vol. I, paragraph 129. 13 Peirce's writings, Hartshorne and Weiss, op. cit., Vol. I, paragraph 147. To the objection that the proposition "There is no absolute certainty" is itself inconsistent Peirce answered "If I must make any exceptIon, let it be that the assertIon that every assertion but this is fallible, is the only one that is absolutely infallible." Hartshorne and Weiss, op. cit., Vol. 2, paragraph 75. 14 These and other quotations from James are from Ralph Barton Perry, The Thought and Character of William James, 2 vols., Boston 1935 and are quoted in Gallie, op. cit. ' ,
C1uules Sandi!rs Pezrce
1771
exaggerated) and as a "loose-living man"-mainly because he divorced his first wife-undoubtedly handicapped him in university circles; but his inability to get along with others, the independence and "violence" of his thought (James' word), the "sententiousness of his manner," were far greater obstacles to academic advancement. 15 Yet Peirce was a much more attractive man than these reports of his personality (some of which are self-descriptions) would indicate. He could be charming and witty and gracious. He was "singularly free from academic Jealousy." He was a fair-minded, indeed a most "chivalrous" opponent in controversy. There is no doubt he inspired love in the few men who knew him well, who recognized that he was a difficult but not an intractable child. James' formula for treating Peirce was to grasp him "after the famous 'nettle' receipt . . . contradict, push hard, make fun of him, and he is as pleasant as anyone." One of Peirce's warming attributes was his capacity for self-criticism. He knew himself and was brilliantly apt in hIS self-portrayals: "I insensibly put on a sort of swagger here. which is designed to say: 'You are a very good fellow in your way; who you are I don't know and I don't care, but I, you know, am Mr. Peirce, distinguished for my various scientific acquirements, but above all for my extreme modesty in which respect I challenge the world.''' 16 The selfcritical faculty was relentlessly applied to his writings. It was not unusual for him to redraft a paper a dozen times "until it was as accurate and precisely worded as he could make it." Nevertheless his works are uneven in quality, not a few papers being obscure and fragmentary, and it cannot be said that he achieved anything approaching a systematic, unified expression of his conception of philosophy. In 1887, having received a small legacy, Peirce retired to a house on the outskirts of Milford, Pennsylvania, though he continued to engage in researches for the Coast Survey until 1891. The legacy was insufficient for his needs and he hoped to eke it out with writing. He was a prodigious worker and regularly turned out, as he records, 2,000 words a day. Like Trollope, when he finished one paper he started with undiminished energy on the next. Since many of his manuscripts were not published, they piled up in his study and fell into disorder. The bulk of this work was on logic and philosophy, but he also wrote on mathematics, geodesy, religion, astronomy, chemistry, psychology, early English and classical Greek pronunciation, psychical research, criminology, the history of science, ancient history, Egyptology, and Napoleon. For the Century Dictionary he wrote 15 A most interesting account of Peirce's brief career at Johns Hopkins, Ius relations with the president, Daniel Coit Gilman, with J. J. Sylvester and other members of the faculty, Ius lectures, his library, his quarrels, and other personal items is unfolded In Max H. Fisch and Jackson I. Cope, "Peirce at the Johns Hopkins University" in Wiener and Young, op. cit., pp. 277-311. 16 Ralph Barton Perry, op. clf., Vol. I, p. 538.
Editor's Comment
1172
all the definitions on more than half a dozen subjects; for the Nation he wrote many book reviews; he did translations, prepared a thesaurus and an editor's manual. Nonetheless he could earn barely enough to keep alive. "In his home he built an attic where he could work undisturbed. or, by pulling up the ladder. escape from his creditors." His last years were darkened by illness and the constant struggle to stave off poverty. In 1909, at the age of seventy, he was still working furiously, though he needed a grain of morphine a day to deaden the pain of cancer. For five more years he held on; on April 19, 1914, he died, "a frustrated. isolated man, still working on his logic, without a publisher, with scarcely a disciple, unknown to the public at large." I should like to conclude this sketch with two quotations. The first is a poignant. remarkably revealing self-appraisal, a brief passage in which Peirce compares his own intellectual character with that of William James: "Who, for example, could be of a nature so different from his as 11 He so concrete, so living; I a mere table of contents, so abstract, a very snarl of twine." 11 The second is an illuminating appraisal by Justus Buchler: "Even to the most unsympathetic, Peirce's thought cannot fail to convey something of lasting value. It has a peculiar property. like that of the Lernean hydra: discover a weak point, and two strong ones spring up beside it. Despite the elaborate architectonic planning of its creator, it is everywhere uncompleted, often distressingly so. There are many who have small regard for things uncompleted, and no doubt what they value is much to be valued. In his quest for magnificent array, in his design for a mighty temple that should house his ideas, Peirce failed. He succeeded only in advancing philosophy." 18 17
Hartshorne and Weiss, op. cit., Vol. 6, paragraph 184, quoted in Gallie, op. CIt.,
pp. 57-58. 18
XVI.
Justus Buchler, The PhIlosophy oj Peirce, Selected Writings, New York, 1940,
Certain characteristics ot the subject are clear. To begin with, we do not, in this subject, deal with particular things or particular properties: we deal formally with what can be said about "any" thing or "any" property. We are prepared to say that one and one are two, but not that Socrates and Plato are two, because, in our capacity of logicians Or pitre mathematicians, we have ne\'er heard of Socrates or Plato. A world in which there were no such individuals would still be a world in which one and one are two. It zs not open to us, as pure mathematicians or logicians, to mention anythmg at all, because, if we do so we i11troduce somethmg irrelevant and not formal. -BERTRAND RUSSELL
2
The Essence of Mathematics By CHARLES SANDERS PEIRCE
IT DOES not seem to me that mathematics depends in any way upon logic. It reasons, of course. But if the mathematician ever hesitates or errs in his reasoning, logic cannot come to his aid. He would be far more liable to commit similar as well as other errors there. On the contrary, I am persuaded that logic cannot possibly attain the solution of its problems without great use of mathematics. Indeed all formal logic is merely mathematics applied to logic. It was Benjamin Peirce,l whose son I boast myself, that in 1870 first defined mathematics as "the science which draws necessary conclusions." This was a hard saying at the time; but today, students of the philosophy of mathematics generally acknowledge its substantial correctness. The common definition, among such people as ordinary schoolmasters, still is that mathematics is the science of quantity. As this is inevitably understood in English, it seems to be a misunderstanding of a definition which may be very 01d, 2 the original meaning being that mathematics is the science of quantities. that is, forms possessing quantity. We perceive that Euclid was aware that a large branch of geometry had nothing to do with measurement (unless as an aid"in demonstrating); and. therefore, a Greek geometer of his age (early in the third century B.C.) or later could not define mathematics as the science of that which the abstract noun quantity expresses. A line, however t was classed as a quantity. or quantum, by Aristotle and his followers; so that even perspective (which deals wholly with intersections and projections, not at all with lengths) could be said to be a science of quantities, "quantity" being taken in the conI "Linear Associative Algebra" (1870), sec. 1; see American Journal of Mathematics, vol. 4 (1881). 2 From what is said by Proclus Diadochus, A.D. 485 (Commentarii in Primum Euclidis Elementorum Librum, Prologi pars prior, c. 12), it would seem that the Pythagoreans understood mathematics to be the answer to the two questions "how many?" and "how much?"
1773
1774
Charles Sanders PeIrce
crete sense. That this was what was originally meant by the definition "Mathematics is the science of quantity," is sufficiently shown by the circumstance that those writers who first enunciate it, about A.D. 500, that is Ammonius Hermire and Boethius, make astronomy and music branches of mathematics; and it is confirmed by the reasons they give for doing SO,3 Even Philo of Alexandria (100 B.C.), who defines mathematics as the science of ideas furnished by sensation and reflection in respect to their necessary consequences, since he includes under mathematics, besides its more essential parts, the theory of numbers and geometry, also the practical arithmetic of the Greeks, geodesy, mechanics, optics (or projective geometry), music, and astronomy, must be said to take the word 'mathematics' in a different sense from ours. That Aristotle did not regard mathematics as the science of quantity, in the modern abstract sense, is evidenced in various ways. The subjects of mathematics are, according to him, the how much and the continuous. He referred the continuous to his category of quantum; and therefore he did make quantum, in a broad sense, the one object of mathematics. Plato, in the Sixth book of the Republic,4 holds that the essential characteristic of mathematics lies in the peculiar kind and degree of its abstraction, greater than that of physics but less than that of what we now call philosophy; and Aristotle follows his master in this definition. It has ever sincc been the habit of metaphysicians to extol their own reasonings and conclusions as vastly more abstract and scientific than those of mathematics. It certainly would seem that problems about God, Freedom, and Immortality are more exalted than, for example, the question how many hours, minutes, and seconds would elapse before two couriers travelling under assumed conditions will come together; although I do not know that this has been proved. But that the methods of thought of the metaphysicians are, as a matter of historical fact, in any aspect, not far inferior to those of mathematics is simply an infatuation. One singular consequence of the notion which prevailed during the greater part of the history of philosophy, that metaphysical reasoning ought to be similar to that of mathematics, only more so, has been that sundry mathematicians have thought themselves, as mathematicians, qualified to discuss philosophy; and no worse metaphysics than theirs is to be found. Kant regarded mathematical propositions as synthetical judgments a priori; wherein there is this much truth, that they are not, for the most part, what he called analytical judgments; that is, the predicate is not, in the sense he intended. contained in the definition of the subject. But if the propositions of arithmetic. for example. are true cognitions, or even forms 3 I regret I have not noted the passage of Ammonlus to whIch I refer. It is probably one of the excerpts given by Brandis My MS. note states that he gives reasons 'showing this to be his meaning. 4 SlOe to the end; but in the Laws his notion is improved.
Tile Essence of Mathematics
1775
of cognition, this circumstance is quite aside from their mathematical truth. For all modem mathematicians agree with Plato and Aristotle that mathematics deals exclusively with hypothetical states of things, and asserts no matter of fact whatever; and further, that it is thus alone that the necessity of its conclusions is to be explained. This is the true essence of mathematics; and my father's definition is in so far correct that it is impossible to reason necessarily concerning anything else than a pure hypothesis. Of course, I do not mean that if such pure hypothesis happened to be true of an actual state of things, the reasoning would thereby cease to be necessary. Only, it never would be known apodictically to be true of an actual state of things. Suppose a state of things of a perfectly definite, general description. That is, there must be no ro' For example, we prefix the descriptional operator '(,X),' which is short for 'the one and only one x such that,' to the statement-form 'x discovered argon,' so as to obtain (7X) (x discovered argon).
This expression is to be read as: 'The one and only one x such that x discovered argon,' or more briefly as 'The discoverer of argon.' The descriptional operator is itself constructed by enclosing in parentheses the inverted Greek letter iota followed by a variable.
Symbolic Notation. Haddocks' Eyes and the Dog-Walkmg Ordmance
1883
And finally, descriptional operators and definite descriptions cannot be defined explicitly in terms of notions that have already been introduced in this exposition. However, if the notion of strict identity is taken for granted and is represented by the customary sign '=', it is possible to eliminate definite descriptions from any statement in which they occur, in favor of locutions that have already been presented. For example, the statement 'The discoverer of argon was Lord Rayleigh,' which can also be rendered with the help of the descriptional operator as: ('1x)(x discovered argon) was Lord Rayleigh,
can be transformed into the statement that no longer contains a definite description: ('3: x ) (y) «y discovered argon
argon) . (x
==
x discovered
= Lord Rayleigh)
Read literally, this statement says: There is an individual x such that for any individual y, y discovered argon if and only if x discovered argon, and x is identical with Lord Rayleigh. When translated into ordinary English, it says that there is just one individual who discovered argon, that individual being Lord Rayleigh. 5.
THE USE AND MENTION OF SIGNS
One does not ordinarily confuse a name (or other linguistic expression) with what the name designates. Thus, probably no one will have difficulty in recognizing that in the following two statements: George Eliot was a woman George Eliot was a pseudonym the subject-term of the first refers to a human being, while the subjectterm of the second refers to a linguistic expression. (In terms of a locu· tion that has gained wide currency, the expression 'George Eliot' in the first statement is being used but is not being mentioned, for it is a certain human being who is being mentioned; on the other hand, in the second statement the expression 'George Eliot' is being mentioned but is not being used, for it is not being used to designate the human being whom the expression normally names.) Nevertheless, there are many contexts in which this distinction is not so readily seen, so that serious confusions sometimes arise in consequence. A mildly amusing example of the confusions that may arise, is contained in the following anecdote. A teacher of English composition, rebuking his students for their use of cheap language, remarked that there were two words in particular he wished them to avoid in the future. "These words are 'lousy' and 'awful,'" he added. The brief silence that
1884
Ernest Nagel
followed this chiding was interrupted by one student's question: "But what are the words, Professor?" The point of the story is of course that while the instructor was mentioning but not using the words~ the student understood him to be using rather than mentioning them. A more serious confusion is illustrated by the following fallacious argument. Since % = 1~1 and 7 is greater than 1%1, it follows (substituting equals for equals) that 7 is greater than %. Similarly, so it is argued, since % 1%1 and 7 is a divisor of the numerator of 1;21, it follows (substituting equals for equals) that 7 is a divisor of the numerator of %. The mistake here arises from confounding numbers with their names; and in particular from identifying ratios with fractions (which are commonly used as names or designations of ratios). Thus, in the equation '% == 1~1' we are using but not mentioning the fractional expression '1%1'; but in the statement '7 is a divisor of the numerator of I%!' we are mentioning but not using the fraction 'l¥.n; though we are using the phrase 'the numerator of 1~1' to refer to a certain numeral (namely, '14') which in turn is the name of a certain number (namely, 14). Although % =: 1%1, it is clearly not the case that the fraction ';s' is identical with the fraction '1%1'; and the error in the argument consists in confusing these two statements. Much more serious errors arising from the neglect of the distinction under discussion, but too involved for brief mention. occur in the literature on the foundations of logic and mathematics, as well as in the writings of philosophers. In. order to prevent such confusions, it is useful to formulate a general principle underlying the distinction between use and mention; and it is also convenient to introduce a notational device which can prevent us from violating the principle. In a correctly formed English sentence about anything whatsoever, the things talked about never appear in the sentence, but must instead be represented by their names or other designatory expressions. Thus, the individual who was a writer of novels, who lived in England, and so on, is obviously not a constituent of the statement 'George Eliot was a woman,' though her pen-name is such a constituent. This general rule must be observed whether we are talking about non-linguistic things and events, or whether we are constructing statements about linguistic expressions. Accordingly, since in the statement 'George Eliot was a pseudonym' we are asserting something about a linguistic expression, the statement should contain as a constituent not that expression but a name for the expression. Now there is a widely used current device for manufacturing names for written and printed expressions: it consists in placing an expression within single quotation marks. and using the complex made up out of the expression and its enclosing quotation marks as the name for the expression itself. In agreement with this convention, the statement last mentioned must be formulated as:
=
Symbolic Notation. Haddocks' Eyes and the Dog-Walking Ordinance
1885
'George Eliot' was a pseudonym. If this convention as to the use of single quotation marks is adopted,
it is evident that of the following three statements only the third is correct: George Eliot contains quotation marks 'George Eliot' contains quotation marks I 'George Eliot' , contains quotation marks. The first of these is obviously false, if not nonsensical. The second is false, because the subject-term refers to the name of a British authoress, and this name contains no quotation marks. But the third statement is true, since its subject-term refers to the name of an expression, and this name does contain quotation marks. The following locutions and notation, which in part embody the distinctions just explained, will be useful in the immediate sequel. We shall employ the descriptive function 'the name of x' as an abbreviation for •(, y) (y names x)', and shall construe the sentential form 'y names x' to mean that any expression which is substituted for 'y' is the conventional name for the "object" designated by the expression which is substituted for 'x.' Thus, it will be correct to say that: 'Napoleon' names the victor of Austerlitz or alternatively: The name of the victor of Austerlitz = 'Napoleon'; but it will be incorrect to say that: Napoleon names the victor of Austerlitz or: The name of the victor of Austerlitz = Napoleon. Moreover, we shall employ the descriptive function 'the call-name of x' as an abbreviation for '(, y)(y call-names x)', and shall understand the statement form 'y call-names x' to mean that any expression which is substituted for 'y' is not the conventional name for, but is what one calls the "object," designated by the expression substituted for 'x.' Thus, it will be correct to say that: The call-name of George Washington = 'The father of his country.' but not correct to say either that: The call-name of George Washington The father of his country or that: The call-name of George Washington = 'George Washington.'
=
Errust Nagel
1886
FROM "THROUGH THE LOOKING GLASS" BY LEWIS CARROLL
"You are sad," the Knight said in an anxious tone: "let me sing you a song to comfort you." "Is it very long?" Alice asked, for she had heard a good deal of poetry that day. "It's long," said the Knight, "but it's very, very beautiful. Everybody that hears me sing it--either it brings the tears into their eyes, or else-" "Or else what?" said Alice, for the Knight had made a sudden pause. "Or else it doesn't, you know. The name of the song is called 'Haddocks' Eyes.' " "Oh, that's the name of the song, is it?" Alice said, trying to feel interested. "No, you don't understand," the Knight said, looking a little vexed. "That's what the name is called. The name really is 'The Aged Aged Man.' " "Then I ought to have said 'That's what the song is called'?" Alice corrected herself. "No, you oughtn't: that's quite another thing! The song is called 'Ways and Means'; but that's only what it's called, you know!" "Well, what is the song, then?" said Alice, who was by this time completely bewildered. "I was coming to that," the Knight said. "The song really is 'A-sitting On A Gate': and the tune's my own invention." So saying, he stopped his horse and let the reins fall on its neck: then, slowly beating time with one hand, and with a faint smile lighting up his gentle foolish face, as if he enjoyed the music of his song, he began. II. HADDOCKS' EYES The selection above and that on p. 1890 illustrate in an amusing way two types of difficulty that are frequently encountered in every-day speech, though doubtless rarely in such exaggerated form. It is generally possible to overcome these difficulties without serious trouble, but ordinary language possesses no explicit, systematic technique for doing so. Modem logical theory does have such a technique, and it is instructive to see how it can serve for resolving the difficulties. It must be emphasized, however, that the power and significance of current logical techniques are not fully revealed in these quite elementary examples of their application. The objectives for whose realization every·day speech is an admirable instrument can normally be achieved without employing the elaborate and cumbersome machinery of modern formal logic. No one in his right mind would recommend the systematic use of the interferometer for measuring lumber required in making a table. On the other hand, it would be equally absurd to maintain that since the physicist's techniques of high precision measurement are pointless in the carpenter's shop. those techniques are completely without a raison d' 2tre. The selection from Lewis Carroll illustrates the type of misunderstanding that may arise from the failure to distinguish between fragments of
Symbolic NotatIon, Haddocks' Eyes and the Dog-Walking Ordinanctl
1881
discourse (such as names) and what linguistic expressions are about or designate. Perhaps the grossest example of this type of error is the textbook fallacy: Tigers eat meat, Meat is a word, therefore Tigers eat words. Such misunderstandings can be eliminated by strict adherence to the distinction between the use and mention of symbols, and to the notational convention of employing quotation marks for manufacturing names for linguistic expressions, both of which have' been explained above. The selection from Robert Graves makes evident the lack in every-day speech of a standard procedure for expressing unambiguously the necessary or sufficient conditions for occurrences, and for stating explicitly the time-dependence of events. For example, it is not clear in the statement "The price of corn will be high and the rainfall will increase if the prevailing winds continue" whether what is being asserted is that a consequence of the prevailing winds will be both higher com prices and increased precipitation, or only the latter. Again, it would usually be assumed that in the statement "He took off his clothes and went to bed," the action reported in the first clause preceded the one reported in the second. But the order in which clauses are written does not always suffice to fix the time order of events, as is evident from the statement "His house was destroyed and he disappeared." Such ambiguities can be obviated through the use of devices that are standard in modem logic: precise rules of punctuation (i.e., rules for grouping expressions with the help of pareni theses or some other notation), rigid prescriptions for employing statemental connectives, and the introduction of variables and quantifying operators. In the interchange between Alice and the White Knight, the latter offers to sing her a song whose tune, as is eventually revealed, is claimed by him to be his own invention. Since the statement form: 'x is a song whose tune is claimed by the Knight to be his own invention' has a length too un.. wieldy for comfort, let us abbreviate it into the shorter form:
'x is a KS.' Apparently there is only one such song, so that the definite description '(1X) (x is a KS)'
uniquely identifies it. It will be convenient occasionally to abbreviate this mode of writing the definite description into the quite brief phrase: • p (This is called the "principle of tautology" and is referred to by the abbreviated title "Taut.") *1.3 1-: q. :::> • p v q (This is called the "principle of addition," and is referred to by the abbreviation "Add.") '" 1.4 1-: p v q • :::> • q v p (This is called the "principle of permuta. tion," and is referred to as "Perm.") '" 1.5 1-: p V (q v r). :::> • q v (p V r) (This is called the "associative principle" and is referred to as "Assoc.") *1.6 1-:. q :::> r. :::> : p v q . :::> • p v r (This is called the "principle of summation" and is referred to as "Sum,") It is of some general interest to note that primitive proposition * 1.5 is now known to be unnecessary, since it can be derived from the remaining "symbolical" primitives. But it can also be shown that the remaining four of these primitives are independent of one another: no one of them is derivable from the other three. However, when Principia was first published, methods for showing the independence of logical primitives were not available. We can now tum to the theorems and their demonstration. *2.05
Demo
1-:. q :::> r . :::> : p :::> q • :J • P :J r
[Sum
7]
1-:
[(1) . (*1.01)]
0
q ::J
I- : .
T
q :J
0
::J : p v q ::J
T.
0
0
PV
T
(1)
:J : p:::> q. :::> • p:::> r
This theorem is one form of the principle of the syllogism. It asserts the following as true: If q implies T, then if p implies q then p implies r. The demonstration is straightforward. The first line asserts what is obtained by substitution from the primitive statement Sum, when ''''''p' is substituted for 'p' uniformly (that is, for every occurrence of this letter) in that primitive. The final line, which is the theorem to be established, is obtained from
Symbolic Notatlon. Haddocks' Eyes and the Dog-Walkmg Ordmance
1897
line (1) with the help of the definition .;. 1.01 (which permits the replacement of a defining expression by the defined expression, and vice versa). *2.07
f-:
p • :J • P v P [ * 1.3 ; ]
p Here we put nothing beyond H* 1.3 -," because the proposition to be
q proved is what *1.3 becomes when p is written in place of q. This theorem asserts the following as true: If p, then p or p. No comment beyond that supplied by Whitehead and Russell is needed. *2.08
1-. P :J P
Dem. [
*2.05 -p_v_p,-pJ 1-: :pvp.:J .p::J :.p.:J .pvp: :J.p:Jp
q,
(1)
r
[Taut]
(2) (3) (4)
'r-:pvp.:J.p [( 1) . (2) . *1.11] 'r- :.p.:J.pvp: :J.p:Jp
[*2.07] [(3) . (4) *1.11]
'r-:p.:J.pvp 'r-.p:Jp
This theorem simply asserts that: If p then p. It is one form of what is sometimes called "the principle of identity." The demonstration begins with making a substitution in theorem *2.05. The substitution consists in putting 'p v p' for every occurrrence of 'q' in that theorem, and putting 'po for every occurrence of 'r.' Line (1) is the result of that substitution. The second line is simply the assertion of the primitive proposition Taut. The third line of the demonstration is obtained from the first two lines with the help of primitive proposition =I< 1.11. This primitive is a slight generalization of the rule of the modus ponens. It will be noted that the second line of the proof is identical with that part of the expression in the first line which is to the left of the main implication sign-Le., to the left of the horse-shoe sign which is flanked by the largest number of dots. In accordance with the modus ponens it is therefore possible to assert what is to the right of the main-implication line-and this is line (3). The fourth line of the demonstration is simply theorem *2.07. The final line of the demonstration is obtained from lines (3) and (4) with the he]p of the primitive proposition =I< 1. II-in exactly the same manner as line (3) is obtained from lines (1) and (2).
,. 54.43
'r-:. a ,
f3 E I . :J
:
/l;
n f3 = A • == . a
U f3
E
2
1898
Dem. *54.26 . :J I-- : • a ::::; , ' X • f3 = , ' y . ::> : a U f3 E 2 . == . X oF Y • [*51.231] =.L'xnII/L'y=11. [*13.12] =. anlll{3 = A (1) 1--. (1) . *11.11.35.:J I-- : • ( ax, y) . a L' X . f3 y . :J : aU f3 E 2 . == • anlll{3 A (2) I-- • (2) . *11.54 . *52.1 . :J I-- • Prop I-- •
=
= ,'
=
From this proposition it will follow, when arithmetical addition has been defined, that 1 +. 1 2. To understand what is being said here, some new notation must first be explained. The Greek letter "E" signifies the relation of class-membership, so that in general "A E B" is to be read as "A is a member of (the class) B," and in particular "Frege E Men" is to be read as "Frege is a member of the class of men." If A and B are classes, "A n B" represents their logical product, i.e., A nil B is the class whose members belong to both A and B; thus, Male nil Parent is the class of those individuals who are both males and parents, i.e., the class of fathers. If A and B are classes, "A U III, B" represents their logical sum, i.e., AU III B is the class whose members belong either to A or to B or to both; thus, Male U:III Parent is the class of those individuals who are either male or parents or both. "A" signifies the null-class, that is, the class which contains no members; thus, since the equation "x2 + 1 0" has no real root, the phrase "the real roots of the equation 'x 2 + 1 0' " signifies the null-class. ", 'x" is the name of a unit-class whose sole member is x; thus, , 'Plato is the unit-class whose only member is the individual P·lato. "1" symbolizes the class of all unitclasses; and "2" symbolizes the class of all classes each of which is a couple, that is, each of which has just two elements as members. Theorem *54.43 accordingly asserts the following: If the classes ct and f3 are members of the class of unit-classes, then the logical product of ct and f3 is the null-class if, and only if, the logical-sum of ct and f3 is a member of the class of couples. The demonstration, translated into English, says the following. Since *54.26 is a theorem and can be asserted, it follows (and hence can also be asserted) that if ct is identical with the unit-class whose only member is x and f3 is identical with the unit-class whose sole member is y, then the logical-sum of a and f3 is a member of the class of couples if, and only if, x and yare not identical. But by *51.231, x is not identical with y if, and only if, the logicalproduct of the unit-classes whose sole members are x and y respectively is the null·class. Accordingly, under the hypothesis, the logical-sum of ct and f1 is a member of the class of couples if, and only if, the logical-
=
=
=
Symbolic Notation, Haddocks' Eyes and the Dog-Walking Ordmance
1899
product of the unit-classes whose sole members are x and y respectively is the null-class. And by *13.12, the logical-product of the unit-classes whose sole members are x and y respectively is the null-class, if and only if the logicalproduct of a and f3 is the null-class. Hence under the hypothesis, the logical-sum of a and f3 is a member of the class of couples, if and only if the logical product of a and f3 is the null-class. (1) But since (1) and *11.11 and * 11.35 are theorems and can be asserted, it follows (and hence can also be asserted) that If there is an x and a y, such that a is the unit-class whose sole member is x and f3 is the unit-class whose sole member is y, then the logical-sum of a and f3 is a member of the class of couples if, and only if, the logical(2) product of a and f3 is the null-class. But since (2) and *11.54 and *52.1 are theorems and can be asserted, *54.43 (which is required to be proved) can be asserted. *110.643.
1-.1
+c 1 =2
Dem. *110.632. ,...*101.21.28.=> I- • 1 + c 1 - f {( g' y) ,y E f . f - , ' Y E 1} [*54.3] 2 . :J I- • Prop I- •
=
Some further new notation must now be explained. The expression "+c" signifies arithmetical (or cardinal) addition. A letter used as a variable with a "cap" over it (e.g., ~) serves to specify a class whose members must satisfy the condition indicated by the expression following the A capped-letter. For example, x {(x E Prime Number) . (x> 25) . (x < 75)} is the class of prime numbers greater than 25 and less than 75. If A is a class, -A is the negative (or the negation) of A, and is the class whose members are all elements which are not members of A. Thus, -,,' Plato is the class of all individuals who are not members of the unitclass whose sole member is Plato. If A and B are classes, B - A is the logical product of B and the negation of A; thus, Men - Parents is the class of all those individuals who are men but are not parents. Theorem *110.643 accordingly asserts the following: The arithmetical sum of the cardinal number 1 and the cardinal number 1 is the cardinal number 2. The demonstration, translated into English, says the fo1l6wing. Since *110.632 and *101.21 and *101.28 are theorems and can be asserted, it follows (and can hence be asserted) that The arithmetical sum of the cardinals 1 and 1 is identical with the class of classes, such that for some y or other, y is a member of E, and the
1900
Ernest Nagel
class which is the logical-product of E and the negation of the unit-class whose sole member is y, is a member of the class of unit-classes. But since * 54.3 has already been demonstrated, it follows that this class of classes g is identical with the c1ass of couples (that is. with the cardinal number 2). so that the proposition which requires demonstration follows.
Blake wrote: "I have heard many People say, 'Give me the Ideas. It is no matter what Words you put them into.''' To this he replies, "Ideas cannot be Given but in their minutely Appropriate Words." -WILLIAM BLAKE (Quoted by Agnes Arber, "The Mmd and the Eye")
4
Symbolic Logic By ALFRED TARSKI ON THE USE OF VARIABLES CONSTANTS AND VARIABLES
EVERY scientific theory is a system of sentences which are accepted as true and which may be called LAWS or ASSERTED STATEMENTS or, for short, simply STATEMENTS. In mathematics, these statements follow one another in a definite order according to certain principles, and they are, as a rule, accompanied by considerations intended to establish their validity. Considerations of this kind are referred to as PROOFS, and the statements established by them are called THEOREMS. Among the terms and symbols occurring in mathematical theorems and proofs we distinguish CONSTANTS and VARIABLES. In arithmetic, for instance, we encounter such constants as "number/' "zero" ("0"), "one" ("I"), "sum" ("+"), and many others.l Each of these terms has a well-determined meaning which remains unchanged throughout the course of tbe considt'rations. As variables we employ, as a rule, single letters, e.g., in arithmetic the small letters of the English alpbabet: "a," "b," "e," .. " "x," "y/' "z," As opposed to the constants, the variables do not possess any meaning by themselves. Thus, the question:
does zero have such and such a property? e.g.:
is zero an integer? can be answered in the affirmative or in the negative; the answer may be 1 By "arithmetic" we shall here understand that part of mathematics which is concerned with the investigation of the general properties of numbers, relations between numbers and operations on numbers. In place of the word "arithmetic" the term "algebra" is frequently used, particularly in high-school mathematics. We have given preference to the term "arithmetic" because, in higher mathematics, the term "algebra" is reserved for the much more special theory of algebraic equations. (In recent years the term "algebra" has obtained a wider meaning,. which is, however, still different from that of "arithmetic!')-The term "number" will here always be used with that meaning which is normally attached to the term "real number" in mathematics; that is to say. it will cover integers and fractions, rational and irrational, positive and negative numbers, but not imaginary or complex numbers.
1901
Alfred Tarsld
1902
true or false, but at any rate it is meaningful. A question concerning x, on the other hand, for example the question: is x an integer?
cannot be answered meaningfully. In some textbooks of elementary mathematics, particularly the less recent ones, one does occasionally come across formulations which convey the impression that it is possible to attribute an independent meaning to variables. Thus it is said that the symbols "x," "y," ... also denote cer· tain numbers or quantities, not "constant numbers" however (which are denoted by constants like "0," "1;' ... ), but the so-called "variable num· bets" or rather "variable quantities." Statements of this kind have their source in a gross misunderstanding. The "variable number" x could not possibly have any specified property, for instance, it could be neither positive nor negative nor equal to zero; or rather, the properties of such a number would change from case to case, that is to say, the number would sometimes be positive, sometimes negative, and sometimes equal to zero. But entities of such a kind we do not find in our world at aU; their exist· ence would contradict the fundamental laws of thought. The classification of the symbols into constants and variables, therefore, does not have any analogue in the form of a similar classification of the numbers. EXPRESSIONS CONTAINING VAlUABLES-SENTENTIAL AND DESIGNATORY FUNCTIONS
In view of the fact that variables do not have a meaning by themselves, such phrases as: x is an integer
are not sentences, although they have the grammatical form of sentences; they do not express a definite assertion and can be neither confirmed nor refuted. From the expression: x is an integer
we only obtain a sentence when we replace ttx" in it by a constant denoting a definite number; thus, for instance, if "x" is replaced by the symbol "1," the result is a true sentence, whereas a false sentence arises on replacing "xl> by "%." An expression of this kind, which contains variables and, on replacement of these variables by constants, becomes a sentence, is called a SENTENTIAL FUNCTION. But mathematicians, by the way, are not very fond of this expression, because they use the term "function" with a different meaning. More often the word "CONDITION" is employed in this sense; and sentential functions and sentences which are composed
SymboliC Logic
1903
entirely of mathematical symbols (and not of words of everyday language), such as: x
+y
=5,
are usually referred to by mathematicians as FORMULAS. In place of "sentential function" we shall sometimes simply say "sentence"-but only in cases where there is no danger of any misunderstanding. The role of the variables in a sentential function has sometimes been compared very adequately with that of the blanks left in a questionnaire; just as the questionnaire acquires a definite content only after the blanks have been filled in, a sentential function becomes a sentence only after constants have been inserted in place of the variables. The result of the replacement of the variables in a sentential function by constants--equal constants taking the place of equal variables-may lead to a true sentence; in that case, the things denoted by those constants are said to SATISFY the given sentential function. For example, the numbers 1, 2 and 2% satisfy the sentential function:
x
< 3,
but the numbers 3, 4 and 4% do not. Besides the sentential functions there are some further expressions containing variables that merit our attention, namely, the so-called DESIGNATORY OR DESCRIPTIVE FUNCTIONS. They are expressions which, on replacement of the variables by constants, tum into designations {"descriptions"} of things. For example, the expression:
2x+ 1 is a designatory function, because we obtain the designatjon of a certain number (e.g., the number 5), if in it we replace the variable "x" by an arbitrary numerical constant, that is, by a constant denoting a number (e.g., "2"). Among the designatory functions occurring in arithmetic, we have, in particular, all the so-called algebraic expressions which are composed of variables, numerical constants and symbols of the four fundamental arithmetical operations, such as:
x-y,
x+l
,
2.(x+y-z).
y+2 Algebraic equations, on the other hand, that is to say, formulas consisting of two algebraic expressions connected by the symbol "=", are sentential functions. As far as equations are concerned, a special terminology has become customary in mathematics; thus, the variables occurring in an
A.lfred Tal'ski
1904
equation are referred to as the unknowns, and the numbers satisfying the equation are called the roots of the equation. E.g., in the equation: x 2 + 6= 5x the variable fiX" is the unknown, while the numbers 2 and 3 are roots of the equation. Of the variables "x," "y," •.. employed in arithmetic it is said that they STAND FOR DESIGNATIONS OF NUMBERS or tha:t numbers are VALUES of these variables. Thereby approximately the following is meant: a sentential function containing the symbols "x," "y," ... becomes a sentence, if these symbols are replaced by such constants as designate numbers (and not by expressions designating operations on numbers, relations between numbers or even things outside the field of arithmetic like geometrical configurations, animals, plants, etc.). Likewise, the variables occurring in geometry stand for designations of points and geometrical figures. The designatory funo~ which we meet in arithmetic can also be said to stand for designations of numbers. Sometimes it is simply said that the symbols "x," "y," .•• themselves, as well as the designatory functions made up out of them, denote numbers or are designations of numbers, but this is then a merely abbreviative terminology. FORMATION OF SENTENCES BY MEANS OF VARIABLES-UNIVERSAL AND EXISTENTIAL SENTENCES
Apart from the replacement of variables by constants there is still another way in which sentences can be obtained from sentential functions. Let us consider the formula:
x +y=y +x. It is a sentential function containing the two variables "x" and ''y'' that is satisfied by any arbitrary pair of numbers; if we put any numerical constants in place of y;
it may be said, therefore, that it permits the transformation of the formula "x < y" into an equivalent expression which no longer contains the sym-
Alfred Tarski
1920
bol " 5;
since the latter is a true assertion, so is the former, Similarly, the formula:
4 s'2, s'3, . . . s'n' which together will make up the pea. Then the proposition goes on to say that if the sun and the pea have been cut up in a suitable manner, so that the little portion 81 of the sun is congruent to the little portion s\ of the pea, S2 congruent to s'2, Sa congruent to s'!'I' up to Sn congruent to s"I> this process will exhaust not only all the little portions of the pea, but all the tiny portions of the sun as well. In other words, the sun and the pea may both be divided into a finite number of disjoint parts so that every single part of one is congruent to a unique part of the other, and so that after each small portion of the pea has been matched with a small portion of the sun, no portion of the sun will be left over.5 To express this giant bombshell in terms of a small firecracker: There is a way of dividing a sphere as large as the sun into separate parts, so that no two parts will have any points in common, and yet without compressing or distorting any part, the whole sun may at one time be fitted snugly into one's vest pocket. Furthermore the pea may have its component parts so rearranged that without expansion or distortion. no two parts having any points in common, they will fill the entire universe solidly, no vacant space remaining either in the interior of the pea, or in the universe. Surely no fairy tale, no fantasy of the Arabian nights, no fevered dream can match this theorem of hard, mathematical logic. Although the theorems of Hausdorff, Banach, and Tarski cannot, at the present time, be put to any practical use, not even by those who hope to learn how to pack their overflowing belongings into a week-end grip, they stand as a magnificent challenge to imagination and as a tribute to mathematical conception. s
*
*
*
•
As distinguished from the paradoxes just considered, there are those which are more properly referred to as mathematical fallacies. They arise in both arithmetic and geometry and are to be found sometimes, although not often, even in the higher branches of mathematics as, for instance, in the calculus or in infinite series. Most mathematical fallacies are too trivial to deserve attention; nevertheless, the subject is entitled to some consideration because, apart from its amusing aspect, it shows how a S We recognize this, of course, to be a simple one-to-one correspondence between the elements of one set which make up the sun, and the elements of another set which make up the pea. The paradox lies in the fact that each element is matched with one which is completely congruent to it (at the risk of repeating, congruent means identical in size and shape) and that there are enough elements in the set making up the pea to match exactly the elements which make up the SUD. 6 In the version given of the theorems of Hausdorff, Banach, and Tarski, we have made liberal use of the lucid explanation given by Karl Menger in his lecture: "Is the Squaring of the Circle Solvable?" in Alte Probleme-Neue Losungen, Vienna: Deuticke, 1934.
Edward Kamer and James R. Newman
1946
chain of mathematical reasoning may be entirely vitiated by one fallacious step. ARITHMETIC FALLACIES
I. A proof that 1 is equal to 2 is familiar to most of us. Such a proof may be extended to show that any two numbers or expressions are equal.
The error common to all such frauds lies in dividing by zero, an operation strictly forbidden. For the fundamental rules of arithmetic demand that every arithmetic process (addition, subtraction, multiplication, division, evolution, involution) should yield a unique result. Obviously, this requirement is essential, for the operations of arithmetic would have little value, or meaning, if the results were ambiguous. If 1 + 1 were equal to 2 or 3; if 4 X 7 were equal to 28 or 82; if 7 -+ 2 were equal to 3 or 3lh, mathematics would be the Mad Hatter of the sciences. Like fortunetelling or phrenology, it would be a suitable subject to exploit at a boardwalk concession at Coney Island. Since the results of the operation of division are to be unique, division by 0 must be excluded, for the result of this operation is anything that you may desire. In general, division is so defined that if a, b, and care three numbers, a -+ b c, only when c X b a. From this definition, what is the result of 5 -:- O? It cannot be any number from zero to infinity, for no number when mUltiplied by 0 will be equal to 5. Thus 5 -:- 0 is mean" ingless. And even 5 -:- 0 = 5 -:- 0 is a meaningless expression. Of course, fallacies resulting from division by 0 are rarely presented in so simple a form that they may be detected at a glance. The following example illustrates how paradoxes arise whenever we divide by an expression, the value of which is 0:
=
Assume A
+ B = C,
=
and assume A
= 3 and B = 2.
Multiply both sides of the equation A We obtain A 2 + 2AB + B2
+ B = C by
(A
+ B).
=C(A + B).
Rearranging the terms, we have A2 +AB-AC= -AB _B2+BC.
+ B - C), we have A (A + B - C) = -B( + A B - C). Dividing both sides by (A + B - C" that is. dividing A = -B, or A + B = O. which is evidently absurd. Factoring out (A
n.
by zero, we get
In extracting square roots, it is necessary to remember the algebraic rule that the square root of a positive number is equal to both a negative and a positive number. Thus, the square root of 4 is -2 as well as +2 (which may be written V4 +2), and the square root of 100 is equal
=
Parado~
Lost and Paradox Regained
1947
to +10 and -10 (or, yroo = ±10). Failure to observe this rule may generate the following contradiction: 7
=
(a) (n + 1)2 n 2 + 2n + 1 (b) (n+l)2-.(2n+l)=n 2 (c) Subtracting n(2n + 1) from both sides and factoring, we have (d) (n + 1) '}. - (n + 1) (2n + 1) = n 2 - n (2n + 1) ( e) Adding '4 (2n + 1) 2 to both sides of (d) yields (n + 1)2 - (n + 1)(2n + 1) + 14(2n + 1)2 n2 - n(2n + 1) + lA,(2n + 1)2.
=
This may be written: (f) [en + 1) - 'h(2n + 1)]2
=[n -
lh(2n + 1)]2.
Taking square roots of both sides, (g) n + 1 - %(2n + 1)
=n -7'2(2n + 1)
and, therefore, (h) n = n
+ 1.
III. The following arithmetic fallacy the reader may disentangle for himself: 8 (1) (2)
yaxy'b=yaXb ~ X Y -1 yl'"":"""(~1:-:-)-x......(--."....l)
(3) Therefore, (y'=i)2
........ true ........ true
=VI; Le., -1 = 1
'" ............ ?
IV. A paradox which cannot be solved by the use of elementary mathematics is the following; Assume that log (-1) x. Then, by the law of logs,
=
log (_1)2 = 2 X log (-1) = 2x.
=
But, on the other hand, log (-1) 2 log (1), which is equal to O. Therefore, 2· x = 0. Therefore, log (-1) = 0, which is obviously not the case. The explanation lies in the fact that the function that represents the log of a negative, or complex, number is not single-valued, but is manyvalued. That is to say, if we were to make the usual functional table for the logarithm of negative and complex numbers, there would be an infinitude of values corresponding to each number. 9 V. The infinite in mathematics is always unruly unless it is properly treated. Instances of this were found in the development of the theory of 7 Lietzmann, Lustiges und Merkwurdzges von Zahlen und Formen, Breslau: Feed. Hirt, 1930. & Ball, W. W. Ro, Mathematical Recreations and Essays, Eleventh Edition, N. Y., 1939. 9 Weismann, Ein:fiJhrung in d4s mathematische Denken, Vienna, 1937.
Edward Kasner and James R Newman
1948
aggregates and further examples will be seen in the logical paradoxes. One instance is appropriate here. Just as transfinite arithmetic has its own laws differing from those of finite arithmetic, special rules are required for operating with infinite series. Ignorance of these rules, or failure to observe them brings about inconsistencies. For instance, consider the series equivalent to the natural logarithm of 2: Log 2 = 1 - 112
+ %-
l;4
+ lAi -
% . . .
If we rearrange these terms as we would be prompted to do in finite arithmetic, we obtain:
+ % + 75 + 17
+ l;4 + % +
* . ..)
Log 2
=
Log 2
= {(l + % + %+"* . . .) + (112 + l;4 + % + lk) }
(l
. . .) -
(lt2
Thus,
= {l
*
* ...)
- 2(112 + + % + + 112 + % + + % + ... } - {I + 112 + % + + 75 + ... }
*
=0
*
Therefore, log 2 = O. On the other hand, log 2 1 -1;2
=
+ *-
* + %- % ... =0.69315,
an answer that can be obtained from any logarithmic table. Rearranging the terms in a slightly different way:
= 1 + % - 112 + ;i + 17 - * + *+ ;11 = % X 0.69315 or, in other words, log 2 = 72 X log 2.
log 2
%
A famous series which had troubled Leibniz is the beguiling simple: + 1 - 1 + 1 - 1 + 1 - 1 + 1 . . . By pairing the terms differently, a variety of results is obtained; for example: (1 - 1)
+ (1 -
1)
+ (1 -
1)
+. . . = 0,
but 1 - (l - 1)
+ (l -
1) . . .
= 1.
GEOMETRIC FALLACIES
Optical illusions 10 concerning geometric figures account for many deceptions. We confine our attention to fallacies which do not arise from physiological limitations, but from errors In mathematical argument. A well-known geometric "proof" is that every triangle is isosceles. It assumes 10 For additional illustrative material see Plates I-Von pp. 1954-1955. [ED.]
Paradox Lost and. Paradox Regained
1949
that the line bisecting an angle of the triangle and the line which is the perpendicular bisector of the side opposite this angle intersect at a point inside the triangle. The fol1owmg is a similarly fallacious proof, namely, that a right angle is equal to an angle greater than a right angle. l1
o FIGURE 8
In Figure 8, ABCD is a rectangle. If H is the midpoint of CB, through H draw a line at right angles to CB, which will bisect DA at I and be perpendicular to it. From A draw the line AE outside of the rectangle and equal to AB and DC. Connect C and E, and let K be the midpoint of this line. Through K construct a perpendicular to CEo CB and CE not being parallel, the lines through Hand K will meet at a point O. Join OA, OE, OB, OD and ~C. It will be made clear that the triangle ODe and DAE are equal in all respects. Since KO is the perpendicular bisector of CE and thus any point on KO is equidistant from C and E, DC is equal to OE. Similarly, since HD is the perpendicular bisector of CB and DA, OD equals 0 A. As AE was constructed to equal DC, the three sides of the triangle ODC are equal respectively to the three sides of the triangle OAE. Hence, the two triangles are equal, and therefore, the angle ODC is equal to the angle OAE. But angle ODA is equal to angle OAD, because side AD is equal to side OD in the triangle OAD and the base angles of the isosceles triangle are equal. Therefore, the angle IDe, which is equal to the difference of ODe and ODJ, equals IAE, which is the difference between OAE and ~Al. But the angle IDC is a right angle, whereas the angle lAE is greater than a right angle, and hence the result is contradictory. Can you find the flaw? Hint: Try drawing the figure exactly. LOGICAL PARADOXES
Like folk tales and leg~nds, the logical paradoxes had their forerunners in ancient times. Having occupied themselves with philosophy and with the foundations of logic, the Greeks formulated some of the logical conundrums which, in recent times, have returned to plague mathematicians and 11
Ball, op. cit.
195()
Edward Kasner and James R. Newman
philosophers. The Sophists made a specialty of posers to bewilder and confuse their opponents in debate, but most of them rested on sloppy thinking and dialectical tricks. Aristotle demolished them when he laid down the foundations of classical logic-a science which has outworn and outlasted all the philosophical systems of antiquity, and which, for the most part, is perfectly valid today. But there were troublesome riddles that stubbornly resisted unraveling.1.2 Most of them are caused by what is known as "the vicious circle fallacy," which is "due to neglecting the fundamental principle that what involves the whole of a given totality cannot itself be a member of the totality." 18 Simple instances of this are those pontifical phrases, familiar to everyone. which seem to have a great deal of meaning, but actually have none, such as "never say never," or "every rule has exceptions," or, "every generality is false." We shall consider a few of the more advanced logical paradoxes involving the same basic fallacy, and then discuss their importance from the mathematician's point of, view. (A) Poaching on the hunting preserves of a powerful prince was punishable by death, but the prince further decreed that anyone caught poaching was to be given the privilege of deciding whether he should be hanged or beheaded. The culprit was permitted to make a statement-if it were false, he was to be hanged; if it were true, he was to be beheaded. One logical rogue availed himself of this dubious prerogative-to be hanged if he didn't and to be beheaded if he did-by stating: "I shall be hanged." Here was a dilemma not anticipated. For, as the poacher put it, "If you now hang me, you break the laws made by the prince, for my statement is true, and I ought to be beheaded; but if you behead me, you are also breaking the laws, for then what I said was false and I should, therefore, be hanged." As in Frank Stockton's story of the lady and the tiger, the ending is up to you. However, the poacher probably fared no worse at the hands of the executioner than he would have at the hands of a philosopher, for until this century philosophers had little time to waste on such childish riddles--especially those they could not solve. (B) The village barber shaves everyone in the vil1age who does not shave himself. But this principle soon involves him in a dialectical plight analogous to that of the executioner. Shall he shave himself? If he does, then he is shaving someone who shaves himself and breaks his own rule. If he does not, besides remaining unshaven, he also breaks his rule by failing to shave a person in the village who does not shave himself. (C) Consider the fact that every integer may be expressed in the English language without the use of symbols. Thus, (a) 1400 may be written 12 For jnstance, the riddle of the Epimenides concerning the Cretan who says that all Cretans are liars. 13 Ramsay, Frank Plumpton. Articles on "Mathematics," and "Logic," Encyc1opaedio. Britannica, 13th edition.
Paradox Lost and Paradox Regained
1951
as one thousand, four hundred, or (b) 1769823 as one million, seven hun. dred and sixty-nine thousand, eight hundred and twenty-three. It is evident that certain numbers require more syllables than others; in general, the larger the integer, the more syllables needed to express it. Thus, (a) requires 6 syllables, and (b) 21. Now, it may be established that certain numbers will require 19 syllables or less, while others will require more than 19 syllables. Furthermore, it is not difficult to show that among those integers requiring exactly 19 syllables to be expressed in the English language, there must be a smallest one. Now, "it is easy to see" 14 that "The least integer not nameable in fewer than nineteen syllables" is a phrase which must denote the specific number, 111777. But the italicized expression above is itself an unambiguous means of denoting the smallest integer expressible in nineteen syllables in the English language. Yet, the italicized statement has only eighteen syllables! Thus, we have a contradiction, for the least integer expressible in nineteen syllables can be expressed in eighteen syllables. (D) The simplest form of the logical paradox which arises from the indiscriminate use of the word all may be seen in Figure 9. What is to be said about the statement numbered 37 1 and 2 are false, but 3 is both a wolf dressed like a sheep and a sheep dressed like a wolf. It is neither the one thing nor the other: It is neither false nor true. An elaboration appears in the famous paradox of Russell about the
1. This Book has 597 Pages 2 The Author of this Book is ConfucliJs
3 The Statements Numbered 1,2, and 3 are all Folse
FIGUllE 9
class of all classes not members of themselves. The thread of the argument is somewhat elusive and will repay careful attention: (E) Using the word class in the customary sense, we can say that there are classes made up of tables, books, peoples, numbers, functions, ideas, etc. The class, for instance, of an the Presidents of the United 14 This expression may, perhaps, be taken in the sense in which Laplace employed it. When he wrote his monumental Mecanique Celeste, he made abundant use of the expression, "It is easy to see" often prefixing it to a mathematical formula which he had arrived at only after months of labor. The result was that scientists who read his work almost invariably recognized the expression as a danger signal that there was very rough going ahead.
1952
Edward Kasner and lames R. Newman
States has for its members every person, living or dead, who was ever President of the United States. Everything in the world other than a person who was or is a President of the United States, including the concept of the class itself is not a member of this class. This then, is an example of a class which is not a member of itself. Likewise, the class of all members of the Gestapo, or German secret police, which contains some, but not all, of the scoundrels in Germany; or the class of all geometric figures in a plane bounded by straight lines; or the class of all integers from one to four thousand inclusive, have for members, the things described, but the classes are not members of themselves. Now, if we consider a class as a concept, then the class of all concepts in the world is itself a concept, and thus is a class which is a member of itself. Again, the class of all ideas brought to the attention of the reader in this book is a class which contains itself as a member, since in mentioning this class, it is an idea which we bring to the attention of the reader. Bearing this distinction in mind, we may divide all classes into two types. Those which are members of themselves and those which are not members of themselves. Indeed, we may form a class which is composed of all those classes which are not members of themselves (note the dangerous use of the word "all"). The question is presented: Is this class (composed of those classes which are not members of themselves) a member of itself, or not? Either an affirmative or a negative answer involves us in a hopeless contradiction. If the class in question is a member of itself, it ought not be by definition, for it should contain only those classes which are not members of themselves. But if it is not a member of itself, it ought to be a member of itself, for the same reason. It cannot be too strongly emphasized that the logical paradoxes are not idle or foolish tricks. They were not included in this volume to make the reader laugh, unless it be at the limitations of logic. The paradoxes are like the fables of La Fontaine which were dressed up to look like innocent stories about fox and grapes, pebbles and frogs. For just as all ethical and moral concepts were skillfully woven into their fabric, so all of logic and mathematics, of philosophy and speculative thought, is interwoven with the fate of these little jokes. Modem mathematics, in attempting to avoid the paradoxes of the theory of aggregates, was squarely faced with the alternatives of adopting annihilating skepticism in regard to all mathematical reasoning, or of reconsidering and reconstructing the foundations of mathematics as well as logic. It should be clear that if paradoxes can arise from apparently legitimate reasoning about the theory of aggregates, they may arise anywhere in mathematics. Thus, even if mathematics could be reduced to logic, as Frege and Russell had hoped, what purpose would be served if logic itself were insecure? In proposing their "Theory of Types" Whitehead and
Paradox Lost IllId Paradox Regained
1953
Russell, in the Principia Mathematica, succeeded in avoiding the contradictions by a formal device. Propositions which were grammatically correct but contradictory, were branded as meaningless. Furthermore, a principle was formulated which specifically states what form a proposition must take to be meaningful; but this solved only half the difficulty, for although the contradictions could be recognized, the arguments leading to the contradictions could not be invalidated without affecting certain accepted portions of mathematics. To overcome this difficulty, Whitehead and Russell postulated the axiom of reducibility which, however, is too technical to be considered here. But the fact remains that the axiom is not acceptable to the great majority of mathematicians and that the logical paradoxes, having divided mathematicians into factions unalterably opposed to each other, have still to be disposed Of.15
'"
'"
*
*
'"
It has been emphasized throughout that the mathematician strives always to put his theorems in the most general form. In thil respect, the
aims of the mathematician and the logician are identical-to formulate propositions and theorems of the form: if A is true, B is true, where A and B embrace much more than merely cabbages and kings. But if this is a high aim, it is also dangerous, in the same way that the concept of the infinite is dangerous. When the mathematician says that such and such a proposition is true of one thing, it may be interesting, and it is surely safe. But when he tries to extend his proposition to everything, though it is much more interesting, it is also much more dangerous. In the transition from one to ali, from the specific to the general, mathematics has made its greatest progress, and suffered its most serious setbacks, of which the logical paradoxes constitute the most important part. For, if mathematics is to advance securely and confidently it must first set its affairs in order at home. 1S As was pointed out in discussing the googol, * there are the followers of Russell who are satisfied with the theory of types and the axiom of reducibility; there are the Intuitionists, led by Brouwer and Weyl, who reject the axiom and whose skepticism about the infinite in mathematics has carried them to the point where they would reject large portions of modem mathematics as meaningless, because they are interwoven with the infinite; and there are the Formalists, led by Hilbert, who, while opposed to the beliefs of the Intuitionists, differ considerably from Russell and the Logistic school. It is Hilbert who considers mathematics a meaningless game, comparable to chess, and he has created a subject of metamathematics which has for its program the discussion of this meanirtgless game and its axioms. '" [For the meaning of "googol" See selection by Kasner and Newm~ "New Names f6r Old," p. 2007.]
1954
Edward Kasner and James R. Newman
THE FOLLOWING OPTIC.AL ILLUSIONS, WHILE NOT PROPERLY PART OF A BOOK ON MATHEMATICS, MAY BE OF SOME INTEREST-
AT LEAST TO THE IMAGINATION
~
~~~ ~~~ PLATE I-Are the three honwntal lmes parallel?
PLATE II-The whIte square
PLATE
1$
of course larger than the black. Or is It smaller?
m-The two shaded regions ha'l'e the. same area.
Paradox Lost and Paradox Regained
PLATE IV-Wlnch of the two penclls
1955
IS
longer? Measure them and lind out.
PLATE V-What do you see? Now look agam.
Truth can never be told so as to be understood, and not be believ'd. -WILLIAM BLAKE
Earthly minds, like mud walls, restst the strongest batteries; and though, perhaps, sometimes the force of a clear argument may make some impression, yet they nevertheless stand firm, keep out the enemy, truth, that would captivate or disturb them. -JOHN LOCKE How often have I said to you that when you have elimmated the impossible, whatever remams, however improbable, must be the truth. -SIR ARTHUR CONAN DOYLE (The Szgn oj Four)
2
The Crisis in Intuition By HANS HAHN
OF ALL the leading philosophers Immanuel Kant was undoubtedly the one who assigned the greatest importance to the part played by intuition in what we call knowledge. He observed that two opposite factors are basic to our knowledge: a passive factor of simple receptivity and an active factor of spontaneity. In his "Critique of Pure Reason," at the beginning of the section entitled "Transcendental theory of elements; Part two: Transcendental logic," we find the following: "Our knowledge comes from two basic sources in the mind, of which the first is the faculty of receiving sensations (receptivity to impressions), the second the ability to recognize an object by these perceptions (spontaneity in forming concepts). Through the first an object is given to us, through the second this object is thought in relation to these perceptions, as a simple determination of the mind. Thus, intuition and concepts constitute the elements of all our knowledge . . ." That is to say, we conduct ourselves passively when through intuition we receive impressions, and actively when we deal with them in our thought. Further, according to Kant, we must distinguish between two ingredients of intuition. One of these, the empirical, a posteriori part, arises from experience and forms the content of intuition, such as colors, sounds, smells, and sensations of touch (hardness, softness, roughness, etc.). The other is a pure, a priori part, independent of all experience; it constitutes the form of intuition. We possess two such pure intuitional forms: space, the intuitional form of our external sense by means of which we "picture things as outside ourselves"; and time, the intuitional form of our inner sense "by means of which the mind observes itself or its inner state." In Kant's system, as I have said, this pure intuition plays an extremely important role. He believed that mathematics is founded on pure intuition, not on thought. Geometry, as it has been taught since ancient times, deals 1956
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with the propertIes of the space that is fully and exactly presented to us by pure intuition, arIthmetIc (the study of the real numbers) rests on our pure and fully exact intuItion of time. The 1OtUltional forms of space and time constitute the a priorl frame mto which we fit all physical happenings that experIence presents to us Every physical event has its precise and exactly determined place 10 space and tIme. However plausible these ideas may at first seem, and however well they corresponded to the state of science in Kanfs day, their foundations have been shaken by the course that science has taken since then. The physIcal side of the questIon has already been treated in the first two lectures, so I can here confine myself to mentioning It briefly. Kant's ideas about the place of space and time in physics correspond with New" tonian phYSICS, which was supreme in Kant's day and which remained so down to very recent times. ThIS conception received its first serious jolt from Einstein's theory of relativity. According to Kant, space and time have nothing to do WIth each other, for they stem from quite different sources. Space is the mtuitional form of our outer sense, time of our lOner sense. We have an absolutely stationary space and an absolute time that flows independent of it. The theory of relativity holds, on the contrary, that there is no absolute space and no absolute time; it is only a combination of space and time-the "universe"-that has absolute physical meaning. A much worse blow was struck at Kant's conception of space and time as a priori intUltlonal forms by the most recent developments of physics. We have already noted that, according to Kant's conception, every physical event has its precisely fixed location in space and time. But there has always been a certain difficulty about this. We know physical events only through experience; but all experience is inexact and every observation involves observational errors. Thus the earher conception embodies the inconsistency that, while every physical event has its exact place in space and time, we can never precIsely determine those places. Let us take, for example, a circular piece of chalk; once a unit of length is chosen, the distance between any two points on this piece of chalk is measured by an exact real number. Imagine that the distance has been determined between every pair of points on this piece of chalk and call the greatest of these distances its Hdiameter:~ Assuming that the chalk 0ccupies an exact fixed portion of the space given to us by precise intuition, it would be reasonable to ask, "Is the diameter of the chalk disc expressed by a rational or an irratIonal number?" But the question could never be answered, for the difference between rational and irrational is much too fine ever to be determined by observation. Thus what mIght be called the classical conception raises questions that are fundamentally unanswerable; which is to say that the conception is metaphysical.
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For a long time this difficulty was not taken very seriously. It was answered somewhat as follows: "Even if every single observation is inexact and subject to observational errors, yet our methods of observation are becoming more and more accurate. Let us rrow imagine a specific physical quantity measured over and over again with more and more precise observational methods. The results thus obtained, though each one is inexact, will nevertheless approach without limit a definite limiting value, and this limit is the exact value of the physical quantity in question." This argument is scarcely satisfactory from a philosophical standpoint, and the recent advances of physics seem to prove that it is also untenable on purely physical grounds. For it now seems that on purely physical grounds the location of an event in space and time cannot be determined with unlimited precision. If, then, measurements cannot be pushed beyond certain limits of exactness we are left with this result: The doctrine of the exact location of physical events in space and time is metaphysical, and therefore meaningless. This most recent and revolutionary development of physics will necessarily come as a shock to most persons-including most physicists-grounded as they are in dogmatic and metaphysical theories; but for the thinker trained in empirical philosophy it contains nothing paradoxical. He will recognize it at once as something familiar and will welcome it as a major step forward along the road toward the "physicalization" of physics, toward cleansing physics of metaphysical elements. After this very brief reference to the physical side of the question we turn to the field of mathematics where the opposition to Kant's doctrine of pure intuition manifested itself considerably earlier than in physics'. From here on I shall deal exclusively with the subject "mathematics and intuition"; moreover, even within this sphere I shall pass over a whole group of questions, as important as they are difficult, which Menger will deal with in the final lecture of this series. I shall not discuss the vehement and successful opposition to Kant's thesis that arithmetic, the study of numbers, also rests on pure intuition-an opposition inextricably bound up with the name of Bertrand Russell, and which has set out to prove that, in complete contradiction to Kant's thesis, arithmetic belongs exclusively to the domains of the intellect and of logic. Thus I have narrowed my subject to "geometry and intuition," and I shall attempt to show how it came about that, even in the branch of mathematics which would seem to be its original domain, intuition gradually fell into disrepute and at last was completely banished. One of the outstanding events in this development was the discovery that, in apparent contradiction to what had previously been accepted as intuitively certain, there are curves that possess no tangent at any point, or (and we shall see that this amounts to the same thing) that it is pos-
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sible to imagine a point moving in such a manner that at no instant does it have a definite velocity. Mathematicians were tremendously impressed when their great Berlin colleague K. Weierstrass made this discovery known in the year 1861. But manuscripts preserved in the Vienna National Library show that the fact was recognized considerably earlier by the Austrian philosopher, theologian and mathematician, Bernhard Bolzano. Since some of the questions involved here directly affect the foundations of the differential calculus as developed by Newton and Leibniz, I shall first say a few words about the basic concepts of that discipline. Newton started with the concept of velocity. Imagine a point moving along a straight line, as shown in Figure 1. At time t the moving point
• FIGURE 1
will be, say, at q. What is to be understood by the expression "the velocity of the moving point at the instant t"1 If we determine the position of the point at a second instant f (at this second instant think of the point as being at q), then we can ascertain the distance qq that it has traversed in the time that has elapsed between the instants t and 1'. We now divide the distance qq that the point has traversed. by the time that has elapsed between the instants t and 1', and get the so-called "mean velocity" of the moving point between t and f. This "mean" velocity is in no sense the velocity at time t itself. The mean velocity may, for instance, tum out to be very great, even though the velocity at instant t is quite small, if the point moved very rapidly during the greater part of the time interval in question. But if the second instant f is chosen sufficiently close to the :first instant t, then the mean velocity between t and I' will provide a good approximation to the velocity at time t itself, and this approximation will be closer, the closer I'is to t. Newton's reasoning about this matter ran somewhat as fallows: Think of the instant f chosen closer and closer to t; then the average velocity between t and ( will approach closer and closer to a certain definite value; it will-to use the language of mathematics-tend toward a definite limit, which limit is called the "velocity of the moving point at the instant t." In other words, the velocity at t is the limiting
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value approached by the average velocity between t and t', as t' approaches t without limit. Leibniz started from the so-called tangent problem. Consider the curve shown in Figure 2; what is its slope (relative to the horizontal) at point p?
FIGURE 2
Choose a second point, p', on the curve and construct the "average slope" of the curve between P and p', This is obtained by dividing the height (represented in Figure 2 by the line pI! p') gained in ascending the section of the curve from p to p' by the horizontal projection of the distance passed over (represented in Figure 2 by the line pp", which indicates how far one moves in the horizontal direction by following along the section of the curve from p to p'). The average slope of the curve between p and p' is not, of course, identical with its slope at the point p itself (in Figure 2 the slope at p is obviously greater than the average slope between p and p~). However, it will give a good approximation to the slope at p, if only p' is chosen sufficiently close to Pi and the approximation will be more accurate, the closer p' is to p. Now again as in the Newtonian example: If p' is permitted to approach p without limit, the average slope of the curve between p and p' will tend toward a definite limit, which limit is called the "slope of the curve at the point p." That is to say, the slope at p is the limiting value approached by the average slope between p and p' as p' approaches p without limit. One designates as the "tangent of the curve at the point p" the straight line passing through p which (throughout its entire length) has the same slope as the curve at p. There is thus a striking resemblance between the procedure for obtaining the slope of a curve and the procedure for determining the velocity of a moving point. In fact, the problem of determining the velocity of a moving point at a given instant becomes identical with the problem of determining the slope of a curve at a given point if we employ a simple device. familiar from its use in [German] railroad timetables. [Hahn apparently refers to a graphic type of timetable with which Americans are not acquainted. ED.] Along a horizontal straight line (a "time axis") mark off time intervals so that every point on the line represents a definite
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point of time; and on the straight line in Figure 1, along the path of the moving point in question select an arbitrary point o. If at instant t the moving point is at q, erect at right angles to the time axis at t the line segment oq. (See Figure 2.) It can be seen that the point p thus obtained will represent as in Figure 2 the position of the moving point at instant t. If this process were carried out for every single instant of time, one would obtain a smooth curve portraying the path of the moving point, namely its "time-distance curve." From this curve one can derive all the particulars of the motion of the point, just as one can work out a train schedule from the graphic type of timetable referred to above. Now it is evident that the average slope of the time-distance curve between p and p' is identical with the average velocity of the moving point between t and t', and thus the slope of the time-distance curve at p is identical with the velocity of the moving point at the instant t. This is the simple connection between the velocity problem and the tangent problem; the two are the same in principle. The fundamental problem of the differential calculus is this: Let the path of a moving point be known; from this data its velocity at any instant is to be calculated; or, let a curve be given-for each of its points the slope is to be calculated (at every point the tangent is to be found) We shall now examine the tangent problem, bearing in mind that everything we say about this problem can, on the basis of the foregoing, be carried over directly to the velocity problem. We noted that if the point p' on the curve in question approaches the point p without limit, the average slope between p and p' will approach more and more closely a definite limiting value, which will represent the slope of the curve at the point p itself. It may now be asked whether this is true for every curve. The principle holds for the standard curves that have been studied since early times: circles, ellipses, hyperbolas, parabolas, cycloids, etc. But a relatively simple example will show that it is not true of every curve. Take the curve shown in Figure 3; it is a wave curve, and in the neighborhood of the point p it has infinitely many waves. The wave
FIGURE 3
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length as well as the amplitude of the separate waves decrease without limit as they approach p. Using the method described above we shall attempt to ascertain the slope of the curve at the point p. We take a second point p' on the curve and obtain the average slope between p and p'. If we take PI as the point pI (Figure 3) the average slope between p and PI turns out to be equal to 1. If p' is permitted to approach p along the curve, it can be seen that the average slope at once decreases; when pI reaches P2 the average slope (between p and P2) becomes O. If p' moves farther along the curve toward p, the average slope between p and p' decreases further, becoming negative, and drops to -1 when p' reaches Ps. 1£ p' moves still closer to p the average slope now begins to increase: it becomes 0 again when p' reaches P4; then keeps increasing and attains the value 1 when p' reaches P5' And if p' moves farther along the curve toward p the same cycle is repeated: As p' approaches p and traverses a complete wave of the wave curve, the average slope between p and p' drops from 1 to -1, only to rise again from -1 to 1. Observe that if p' approaches p without limit, it must travel through infinitely many waves, since the curve as we have defined it generates this pattern. That is, as p' approaches p without limit the average slope between p and p' keeps oscillating between the values 1 and -1. Thus, as regards this slope, there can be no question of its limit nor of a definite slope of the curve at the point p. In other words the curve we have been considering has no tangent at p. This relatively simple intuitable illustration demonstrates that a curve does not have to have a tangent at every point. It used to be thought, however, that intuition forced us to acknowledge that such a deficiency could occur only at isolated and exceptional points of a curve~ never at all points. It was believed that a curve must possess an exact slope, or tangent, if not at every point, at least at an overwhelming majority of them. The mathematician and physicist Ampere, whose contribution to the theory of electricity is well known, atteIQpted to prove this conclusion. His proof was false, and it was therefore a great surprise when Weierstrass announced the existence of a curve that lacked a precise slope or tangent
FIGURE 4
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at any point. Weierstrass invented the curve by an intricate and arduous calculation, which I shall not attempt to reproduce. But his result can today be achieved in a much simpler way, and this I shall attempt to explain, at least in outline. We start with the simple figure shown in Figure 4, which consists of an ascending and a descending line. The ascending· line we shall replace, as shown in Figure 5, by a broken line of six parts, which first rises to half
FIGURE S
the height of the original line, then drops all the way down, then again rises to half height, continues on to full height, drops back again to half height, and finally rises once more to full height. Similarly we replace the descending line of Figure 4 by a broken line of six parts, which drops from full height to half height, rises again to full height, then again drops to half height and continues all the way down, rises once more to half height, and finally drops all the way down. From this figure composed of 12 line segments, we evolve by an analogous method the figure of 72 line segments, shown in Figure 6; that is by replacing every line segment of
FIGURE 6
Figure 5 by a broken line of 6 parts. It is easy to see how this procedure can be repeated, and that it will lead to more and more complicated figures. There exists a rigorous proof (though I cannot give it here) that the succession of geometric objects constructed according to this rule approach without limit a definite curve possessing the desired property: namely, at no point will it have a precise slope, and hence at no point a tangent. The character of this curve entirely eludes intuition; indeed after
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a few repetitions of the segmenting process the evolving figure has grown so intricate that intuition can scarcely follow; and it forsakes us completely as regards the curve that is approached as a limit. The fact is that only thought, or logical analysis can pursue this strange object to its final form. Thus, had we relied on intuition in this instance, we should have remained in error, for intuition seems to force the conclusion that there cannot be curves lacking a tangent at any point. This first example of the failure of intuition involves the fundamental concepts of differentiation; a second example can be derived from the fundamental concepts of integration. The basic problem of differentiation is: given the path of a moving point, to calculate its velocity, or given a curve, to calculate its slope; the basic problem of integration is the inverse: given the velocity of a moving point at every instant, to calculate its path, or given the slope of a curve at each of its points, to calculate the curve. This latter problem, however, has meaning only if the path of the moving point is in fact determined by its velocity, if the curve itself is actually determined by its slope. The question facing us can be more precisely phrased as follows: If two movable points whose track is a single straight line are set in motion in the same direction, at the same instant, from the same place on the line, and at every instant have the same velocity, must they remain together or can they become separated?--or: If two curves in a plane start from a common origin and continuously have the same slope, must they coincide in their entire course or can one of them rise above the other? The dictate of intuition is that the two moving points must always remain together, and that the two curves must coincide in their entire course; yet logical analysis shows that this is not necessarily so. The intuitive answer is true of course for ordinary curves and motions, but we can conceive of certain rather complicated motions and curves for which it is not true. I am sorry not to have the space to go into this in more detail; we must content ourselves with recognizing that in this instance also the apparent certainty of intuition proves to be decei.ving. 1 The foregOing examples of the failure of intuition involve the concepts of the calculus-a subject whose difficulty is acknowledged by the customary designation of "higher mathematics." Lest it be supposed, therefore, that intuition fails only in the more complex branches of mathematics, I propose to examine an occurrence of failure in the elementary branches. At the very threshold of geometry lies the concept of the curve; everyone believes that he has an intuitively clear notion of what a curve is, and since ancient times it has been held that this idea could be expressed by the following definition: Curves are geometric figures generated by the • 1 [Hahn showed in a technical article (Monatshejte f. Mathematik und Physik, 16 (1905), p. 161) that the motions (or curves) in question assume infinite velocities (or slopes of infinite value). ED.]
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motion of a point)! But, attend! In the year 1890 the Italian mathematician Giuseppe Peano (who is also renowned for his investigations in logic) proved that the geometric figures that can be generated by a moving point also include entire plane surfaces. For instance, it is possible to imagine a point moving in such a way that in a finite time it will pass through all the points of a square-and yet no one would consider the entire area of a square as simply a curve. With the aid of a few diagrams I shall attempt to give at least a general idea of how this space-filling motion is generated. Divide a square into four small squares of equal size (as shown in
FIGURE 7
Figure 7) and join the center points of these squares by a continuous curve composed of line segments; now imagine a point moving in such a way that at uniform velocity it will traverse the curve in a finite timesay, in some particular unit of time. Next, divide (Figure 8) each of the four small squares of Figure 7 into four smaller squares of equal size and
FIGURE 8 2 [Hahn refers to continuous motion in which the point makes no "jumps" and the curve it generates therefore shows no "discontinuities" or gaps. ED.]
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connect the center points of these 16 squares by a similar line. and again imagine the point moving so that in unit time it will traverse this second curve at uniform velocity. Repeat this procedure (Figure 9) by dividing
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FIGURE 9
each of the 16 small squares of Figure 8 into four still smaller squares, connecting the center points of these 64 squares by a third curve, and imagining the point to move so that in unit time it will traverse this new system of lines at a uniform velocity. It is easy to see how this procedure is to be continued; Figure 10 shows one of the later stages, when the
FIGURE 10
original square has been divided into 4096 small squares. It is now possible to give a rigorous proof that the successive motions considered here approach without limit a definite course, or curve, that takes the moving point through aU the points of the large square in unit time. This motion cannot possibly be grasped by intuition; it can only be understood by logical analysis. While certain geometric objects which no one regards as curves (e.g .• a square) can, contrary to intuition, be generated by the motion of a point, other objects that one would not hesitate to classify as curves cannot
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be so generated. Observe, for instance, the geometric shape shown in Figure 11. It is a wave curve which in the neighborhood of the line segment ab (as shown in the figure) consists of infinitely many waves, b
FIGURE 11
whose lengths decrease without limit but whose amplitudes (in contrast to the curve of Figure 3), do not decrease. It is not difficult to prove that this figure, in spite of its linear character cannot be generated by the motion of a point; for no motion of a point is conceivable that would carry it through all the points of this wave curve in a finite time. Two important questions now suggest themselves. 1. Since the time· honored definition of a curve fails to cover the fundamental concept, what other more serviceable definition can be substituted for it? 2. Since the class of geometric objects that can be produced by the motion of a point does not coincide with the class of all curves, how shall the former class be defined? Today both questions are satisfactorily answered; I shall with· hold for a moment the answer to the first question and speak briefly about the second. This was solved with the aid of a new geometric concept, "connectivity in the small" [HZusammenhang im Kleinem"] or "local connectivity." 3 Consider certain figures that can be generated by the motion of a point, such as (Figure 12) a line, a circle, or a square. In each of
fO
FIGURE 12
a [The German expression, not easy to translate, may also be rendered as "connection . . ." and "connectedness in the small." and the function describing the property is sometimes given as a "piecewise continuous function." ED.]
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these figures we fix our attention on two points p and q that lie very close together. It is evident that in each case we can move from p to q along a path that does not leave the confines of the figure and remains through· out in close proximity to p and q. This property is called (in an appropri· ately precise formulation) "connectivity in the small." The structure shown in Figure 11 does not have this property. Take for example the neighboring points p and q; in order to move from p to q without leaving the curve it is necessary to traverse the infinitely many waves lying be· tween them. This path does not remain in close proximity to p and q, since all the intervening waves have the same amplitude. Now it is important to realize that "connectivity in the small" is the basic characteristic of figures that can be generated by the motion of a point. A line, a circle, and a square can be generated by the motion of a point, because they are connected in the small; the construction in Figure 11 cannot be generated by the motion of a point, because it is not connected in the small. We can convince ourselves by a second example of the undependability of intuition even as regards very elementary geometrical questions. Think of a map (Figure 13) showing the areas of three countries. On this map there will be boundary points at which two of the countries touch each other; but there may also be points at which all three countries come
FIGURE 13
together-so-called "three-country corners," like the points a and b in Figure 13. Intuition seems to indicate that these three-country comers can occur only at isolated points, that at the great majority of boundary points on the map only two countries will be in contact. Yet the Dutch mathematician L. E. J. Brouwer showed in 1910 how a map can be divided into three countries in such a way that at every boundary point all three countries will touch one another. 4 Let us attempt briefly to describe how this is done. We start with the map shown in Figure 14, on which there are three 4, [A, point is called a "boundary point" if in each of its neighborhoods lie points of, vanous countt;ies. Three countries meet in a boundary point If In each of its neIghborhoods pomts are to be found of each of the three countries. ED.]
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FIGURE 14
different countries, one hatched (A) > one dotted (B) and one solid (C); the unmarked remainder is unoccupied land. Country A, seeking to bring this land into its sphere of influence, decides to push through a corridor (Figure 15) which approaches within one mile of every point of the t
FIGURE 15
unoccupied territory; but-in order to avoid trouble-the corridor is not to impinge upon either of the two other countries. After this has been done country B decides that it must make a similar move, and proceeds to drive a corridor into the remaining unoccupied territory (Figure 16)a corridor that comes within one-half mile of all the unoccupied points but does not touch either of the other two countries. Thereupon country
FIGURE 16
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FIGURE 17
C, deciding that it cannot lag behind, also extends a corridor (Figure 17) into the territory as yet unoccupied, which comes to within a third of a mile of every point of this territory but does not touch the other countries. Country A now feels that it has been outwitted and proceeds to push a second corridor into the remaining unoccupied territory, which comes within a quarter of a mile of all points of this territory but does not touch the other two countries. The process continues: country B extends a corridor that comes within a fifth of a mile of every unoccupied point; country C extends one that comes within a sixth of a mile of every unoccupied point; country A starts over again, and so on and on. And since we are giving imagination free rein, let us assume further that country A required a year for the construction of its first corridor, country B the following half-year for its first corridor, country C the next quarter year for its first corridor; country A the next eighth of a year for its second, and so on; thus each succeeding extension is completed in half the time of its predecessor. It can easily be seen that after the passage of two years none of the originally unoccupied territory will remain unclaimed; moreover the entire map will then be divided among the three countries in such a fashion that at no point will only two of the countries touch each other, but instead all three countries will meet at every boundary point. Intuition cannot comprehend this pattern, although logical analysis requires us to accept it. Once more intuition has led us astray. Because intuition turned out to be deceptive in so many instances, and because propositions that had been accounted true by intuition were repeatedly proved false by logic, mathematicians became more and more sceptical of the validity of intuition. They learned that it is unsafe to accept any mathematical proposition, much less to base any mathematical discipline on intuitive convictions. Thus a demand arose for the expulsion of intuition from mathematical reasoning, and for the complete formalization of mathematics. That is to say, every new mathematical concept was to be introduced through a purely logical definition; every mathematical proof was to be carried through by strictly logical means. The pioneers of this
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program (to mention only the most famous) were Augustin Cauchy (1789-1857), Bernhard Bolzano (1781-1848), Karl Weierstrass (18151897), Georg Cantor (1845-1918) and Richard Dedekind (1831-1916). The task of completely formalizing mathematics, of reducing it entirely to logic, was arduous and difficult; it meant nothing less than a reform in root and branch. Propositions that had formerly been accepted as intuitively evident had to be painstakingly proved. To cite one example: the simple geometric proposition that "every closed polygon that does not cross itself divides the plane into two separate parts" requires a lengthy and highly complicated proof. This is true to an even greater degree of the analogous proposition of solid geometry: "every closed polyhedron that does not intersect itself divides space into two separate parts." As the prototype of an a priori synthetic judgment based on pure intuition Kant cites the proposition that space is three-dimensional. But by present~day standards even this statement calls for searching logical analysis. First it is necessary to define purely logically what is meant by the "dimensionality" of a geometric figure, or "point set," and then it must be logically proved that the space of ordinary geometry-which is also the space of Newtonian physics-as embraced in this definition, is in fact three-dimensional. This proof was not achieved till recent times, in 1922, and then simultaneously by the Viennese mathematician K. Menger and the Russian mathematician P. Urysohn-the latter having since" suc~ cumbed to a tragic accident at the height of his creative powers. I wish to give at least a sketchy explanation of how the dimensionality of a point set is defined. A point set is called null-dimensional if for each of its points there exists an arbitrarily small neighborhood whose boundary contains no point of the set: for example, every set consisting of a finite number of points is nuB-dimensional (cf. Figure 18), but there are also a great many
FIGURE 18
very complicated null-dimensional point sets that consist of infinitely many points. A point set that is not null-dimensional is called one· dimensional if for each of its points there is an arbitrarily small neighborhood whose boundary has only a null-dimensional set in common with the point set. Every straight line, every figure composed of a finite number of straight lines, every circle, every ellipse, in short all geometrical constructs that we ordinarily designate as curves are one-dimensional in this
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FIGURE 19
sense (cf. Figure 19); in fact, even the geometric object shown in Figure 11, which we saw could not be generated by the motion of a point, is onedimensional. Similarly. a point set that is neither null-dimensional nor one-dimensional is called two-dimensional if for each of its points there is an arbitrarily small neighborhood whose boundary has at the most a one-dimensional set in common with the point set. Every plane, every polygonic or circular plane area, every spherical surface, in short every geometric construct ordinarily classified as a surface is two-dimensional in this sense. A point set that is neither null-dimensional, one-dimensional, nor two-dimensional is called three-dimensional if for each of its points there is an arbitrarily small neighborhood whose boundary has at most a two-dimensional set in common with the point set. It can be proved-not at all simply, however-that the space of ordinary geometry is a threedimensional point set in the foregoing theory. This theory provides what we have been seeking, a fully satisfactory definition of the concept of a curve. The essential characteristic of a curve turns out to be its one-dimensionality. But beyond that the theory also makes possible an unusually precise and subtle analysis of the structure of curves, about which I should like to comment briefly. A point on a curve is called an end point if there are arbitrarily small neighborhoods surrounding it, each of whose boundaries has only a single point in common with the curve (cf. points a and b in Figure 20). A point on the curve
FIGURE 20
that is not an end point is called an ordinary point if it has arbitrarily small neighborhoods each of whose boundaries has exactly two points in common with the curve (cf. point c in Figure 20). A point on a curve
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is called a branch point if the boundary of any of its arbitrarily small neighborhoods has more than two points in common with the curve (cf. point d in Figure 20). Intuition seems to indicate that the end points and branch points of a curve are exceptional cases; that they can Occur only sporadically, that it is impossible for a curve to be made up of nothing but end points or of branch points. This intuitive conviction is specifically confirmed by logical analysis as far as end points are concerned; but as regards branch points it has been refuted. The Polish mathematician W. Sierpinski proved in 1915 that there are curves all oj whose points are branch points. Let us attempt to visualize how this comes about. Suppose that an equilateral triangle has been inscribed within another equilateral triangle (as shown in Figure 21) and the interior of the in-
FIGURE 21
scribed triangle erased (hatched in Figure 21); there remain three equilateral triangles with their sides. In each of these three triangles (Figure 22) inscribe an equilateral triangle and again erase its interior; there are now left nine equilateral triangles together with their sides. In each of
FIGURE 22
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these nine triangles an equilateral triangle is to be inscribed and its interior erased so that 27 equilateral triangles are left. Imagine this process continued indefinitely. (Figure 23 shows the fifth step, where 243 tric
FIGURE 23
angles remain.) The points of the original equilateral triangle that survive the infinitely numerous erasures can be shown to form a curve, and specifically a curve all of whose points, with the exception of the vertex points a, b, c, of the original triangle, are branch points. From this it is easy to obtain a curve with all its points branch points; for instance, by distorting the entire figure so that the three vertices a, b, c, of the original triangle are brought together in a single point. But enough of examples-let us now summarize what has been said. Again and again we have found that in geometric questions, and indeed even in simple and elementary geometric questions., intuition is a wholly unreliable guide. It is impossible to permit so unreliable an aid to serve as the starting point or basis of a mathematical discipline. The space of geometry is not a fonn of pure intuition, but a logical construct. The way is then open for other non-contradictory logical constructs in the form of spaces differing from the space of ordinary geometry; spaces, for instance, in which the so-called Euclidean parallel postulate is replaced by a contrary postulate (non-Euclidean spaces) spaces whose dimensionality is greater than three, non-Archimedean spaces. I shall say a few words about the last named. The possibility of measuring the length of a line segment by a real number, and the possibility that followsArom this, namely, fixing the position of a point, as is done in analytic geometry, by assigning real numbers as its "coordinates," rests on the so-called "postulate of Archimedes." 5 This postulate reads as follows: given lengths, there is always a multiple of the first that is greater than the second. As logical constructs. a [It should properly be called the postulate of Eudoxus. Eudoxus' dates are 408355 B.C.; Archimedes: 287-212 B.C. ED.]
The Crisis
In
IntuItion
1975
however, spaces can be devised in which the Archimedean postulate is replaced by its opposite, that is, in which there are lengths that are greater than any multiple of a given length. 6 Hence in these spaces infinitely large and infinitely small lengths can exist (as determined by any arbitrarily chosen unit of measure); while in the space of ordinary geometry there are no infinitely large and infinitely small lengths. In a "nonArchimedean" space, lengths can be measured, and a system of analytical geometry can be developed. Of course, the real numbers of ordinary arithmetic are of no help in this geometry, but one uses "non-Archimedean" number systems, which can be interpreted and applied in calculation as well as the real numbers of ordinary arithmetic. But what are we to say to the often heard objection that the multidimensional, non-Euclidean, non-Archimedean geometries, though consistent as logical constructs, are useless in arranging our experience because they are non-intuitional? For projecting our experience, it is said, only the conventional three-dimensional, Euclidean, Archimedean geometry is usable, for it is the only one that satisfies intuition. My first comment on this score-and this is the point of my entire lecture-is that ordinary geometry itself by no means constitutes a supreme example of the intuitive process. The fact is that every geometry-three-dimensional as well as multi-dimensional, Euclidean as well as non-Euclidean, Archimedean as well as non-Archimedean-is a logical construct. Traditional physics is responsible for the fact that until recently the logical construction of three-dimensional, Euclidean, Archimedean space has been used exclusively for the ordering of our experience. For several centuries, almost up to the present day, it served tbis purpose admirably; thus we grew used to operating with it. This habituation to the use of ordinary geometry for the ordering of our experience explains why we regard this geometry as intuitive; and every departure from it unintuitive, contrary to intuition, intuitively impossible. But as we have seen, such "intuitional impossibilities," also occur in ordinary geometry. They appear as soon as we no longer restrict ourselves to the geometrical entities with which we have long been familiar, but instead reflect upon objects that we had not thought about before. Modern physics now makes it appear appropriate to avail ourselves of the logical constructs of multi-dimensional and non-Euclidean geometries for the ordering of our experience. (Although we have as yet no indication that the inclusion of non-Archimedean geometry might prove useful, this possibility is by no means excluded.) But these advances in physics are so recent that we are not yet accustomed to the manipulation of these logical constructs; hence they still seem an affront to intuition. 6 The first to investigate thoroughly the properties of non-Archimedean spaces was the Italian mathematician G. Veronese. [H.H.]
1976
Hans Hahn
The theory that the earth is a sphere was also once an affront to intuition. This hypothesis was widely rejected on the grounds that the existence of the antipodes was contrary to intuition. However, we have got used to the idea and today it no longer occurs to anyone to pronounce it impossible because it conflicts with intuition. Physical concepts are also logical constructs and here too we can see clearly how concepts whose application is familiar to us acquire an intui· tive status which is denied to those whose application is unfamiliar. The concept "weight" is so much a part of common experience that almost everyone regards it as intuitive. The concept "moment of inertia," however, does not enter into most persons' activities and is therefore not regarded by them as intuitive. Yet among experimental physicists and engineers, who constantly work with it, "moment of inertia" possesses an intuitive status equal to that generally accorded to "weight." Similarly, the concept "potential difference" is intuitive for the electrical technician, but not . for most men . If the use of multi-dimensional and non-Euclidean geometries for the ordering of our experience continues to prove itself so that we become more and more accustomed to dealing with these logical constructs; if they penetrate into the curriculum of the schools; if we, so to speak, learn them at our mother's knee, as we now learn three-dimensional Euclidean geometry-then nobody will think of saying that these geometries are contrary to intuition. They will be considered as deserving of intuitive status as three-dimensional Euclidean geometry is today. For it is not true, as Kant urged, that intuition is a pure a priori means of knowledge, but rather that it is force of habit rooted in psychological inertia.
PAR T XV
How to Solve It 1. How to Solve It by G. POLYA
COMMENTARY ON
The Tears of Mathematics HE most painful thing about mathematics is how far away you are from being able to use it after you have learned it. It differs in this respect from golf, say, or piano playing. In these activities you learn as you go, but in mathematics it is possible to acquire an impressive amount of information as to theorems and methods and yet be totally incapable of solving the simplest problems. Entrants for tripos, the difficult competitive examinations which used to be held at Cambridge, customarily enrolled with a mathematical coach whose job it was to drill them until they were blue in the face and, more important, could work out problems quickly and almost by instinct. It is debatable whether this grind enriched the spirit but it obviously improved one's arithmetic. Besides learning how problems should be solved, can one learn how to solve problems? In 1945 Princeton University Press published a little book, How to Solve It, which promised that the art could be learned. The book jacket carried the blurb: "A System of thinking which can help you solve any problem." The claim is too large, but the book is both delightful and instructive. George Polya, its author and a professor of mathematics at Stanford University, describes his book as heuristiC, a word which means "serving to discover." To solve a problem, Polya says, is to make a discovery: a great problem means a great discovery but "there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery." How to Solve It considers the general questions that problem solvers face and proposes general methods to deal with them. It also ferrets out the small, often unspoken, questions that gnaw at the mind and leave it a ruin of confusion before the problem has even been put on paper. The book is full of valuable hints and urbane, soothing suggestions. Problems can be "decomposed" into their elements and "recombined," the new arrangement often being easier to solve; good use can be made of analogy and related methods; there are tricks to setting up equations and to paring the inessentials; it is often profitable to work backwards. Polya does not scorn the value of having a bright idea but does not assume that you will get one. "If you cannot solve the proposed problem," he says cheerfully, "do not let this failure amict you too much but try to find consolation with some easier success, try to solve first some related problem; then you may find courage to attack your original problem again." I have selected a few
T
1918
The Tears of Mathematics
1979
items from the book almost at random: Mathematical Induction-the best exposition I have ever seen; Setting up Equations; The Traditional Mathematics Professor; Working Backwards. I am sorry I cannot give more. Polya is an American born in Budapest in 1887. He was educated abroad, has taught at Princeton, Brown and Stanford, and has done important research in function theory, probability, applications of mathematics and mathematical method. JUdging from his book, he has thought deeply about improving the teaching of mathematics and he cares what is learned in his classes. His students are to be envied.
We are in the ordinary posttton oj SCIentISts oj havmg to be content with piecemeal zmprovements. we can make several thmgs clearer. but we cannot make anythmg clear. -FRANK PLUMPTON RAMSEY
1
How to Solve It By G. POLYA
INDUCTION AND MATHEMATICAL INDUCTION. Induction is the process of discovering general laws by the observation and combination of particular instances. It is used in all sciences, even in mathematics. Mathematical induction is used in mathematics alone to prove theorems of a certain kind. It is rather unfortunate that the names are connected because there is very little logical connection between the two processes. There is, however, some practical connection; we often use both methods together. We are going to illustrate both methods by the same example. 1. We may observe, by chance, that 1 + 8 + 27
+ 64 == 100
and, recognizing the cubes and the square, we may give to the fact we observed the more interesting form: 13
+ 23 + 38 + 4 3 == 102 •
How does such a thing happen? Does it often happen that such a sum of successive cubes is a square? In asking this we are like the naturalist who. impressed by a curious plant or a curious geological formation, conceives a general question. Our general question is concerned with the sum of successive cubes
13 +-28
+ 38 + ... + n8 •
We were led to it by the "particular instance" n == 4. What can we do for our question? What the naturalist would do; we can investigate other special cases. The special cases n == 2, 3 are still simpler, the case n == 5 is the next one. Let us add, for the sake of uniformity and completeness. the case n == 1. Arranging neatly all these cases, as a geologist would arrange his specimens of a certain ore, we obtain the following table: 1 1== 1+8 9 == 1 + 8 + 27 36 == 1 + 8 + 27 + 64 == 100 == 1 + 8 + 27 + 64 + 125 == 225 == 1980
12 32 6~
102 152•
H ow to Solve It
1981
It is hard to believe that all these sums of consecutive cubes are squares by mere chance. In a similar case, the naturalist would have little doubt that the general law suggested by the special cases heretofore observed is correct; the general law is almost proved by induction. The mathematician expresses himself with more reserve although fundamentally, of course, he thinks in the same fashion. He would say that the following theorem is strongly suggested by induction: The sum of the first n cubes is a square.
2. We have been led to conjecture a remarkable, somewhat mysterious law. Why should those sums of successive cubes be squares? But, apparently, they are squares. What would the naturalist do in such a situation? He would go on examining his conjecture. In so doing, he may follow various lines of investigation. The naturalist may accumulate further experimental evidence; if we wish to do the same, we have to test the next cases, n 6, 7, .... The naturalist may also reexamine the facts whose observation has led him to his conjecture; he compares them carefully, he tries to dis· entangle some deeper regularity, some further analogy. Let us follow this line of investigation. Let us reexamine the cases n 1, 2, 3, 4, 5 which we arranged in our table. Why are all these sums squares? What can we say about tl:!.ese squares? Their bases are 1, 3, 6, 10, 15. What about these bases? Is there some deeper regularity, some further analogy? At any rate, they do not seem to increase too irregularly. How do they increase? The difference between two successive terms of this sequence is itself increasing,
=
=
3 - 1 = 2, 6 - 3 = 3, 10 - 6 = 4, 15 - 10 = 5.
Now these differences are conspicuously regular. We may see here a surprising analogy between the bases of those squares, we may see a remarkable regularity in the numbers 1, 3, 6, 10, 15:
=
1 1 3=1+2 6=1+2+3 10= 1 + 2 + 3 +4 15 = 1 + 2 + 3 + 4 + 5. If this regularity is general (and the contrary is hard to believe) the
theorem we suspected takes a more precise form: It is, for n I, 2, 3, ...
=
13 + 23
+ 38 +
...
+ n8 = (1 + 2 + 3 +
... + n)2.
3. The law we just stated was found by induction, and the manner in which it was found conveys to us an idea about induction which is neces·
G. PDlya
1982
sarily one-sided and imperfect but not distorted. Induction tries to find regularity and coherence behind the observations. Its most conspicuous instruments are generalization, specialization, analogy. Tentative generalization starts from an effort to understand the observed facts; it is based on analogy, and tested by further special cases. We refrain from further remarks on the subject of induction about which there is wide disagreement among philosophers. But it should be added that many mathematical results were found by induction first and proved later. Mathematics presented with rigor is a systematic deductive science but mathematics in the making is an experimental inductive science. 4. In mathematics as in the physical sciences we may use observation and induction to discover general laws. But there is a difference. In the physical sciences, there is no higher authority than observation and induction but in mathematics there is such an authority: rigorous proof. After having worked a while experimentally it may be good to change our point of view. Let us be strict. We have discovered an interesting result but the reasoning that led to it was merely plausible, experimental. provisional, heuristic; let us try to establish it definitively by a rigorous proof. We have arrived now at a "problem to prove": to prove or to disprove the result stated before (see 2, p. 1981). There is a minor simplification. We may know that 1 +2+3+
n(n + 1) ... +n=---
2
At any rate, this is easy to verify. Take a rectangle with sides nand n + 1, and divide it in two halves by a zigzag line as in Figure 1a which shows the case n 4. Each of the halves is "staircase-shaped" and its area
=
r-r---
I a
b FIGURE 1
has the expression 1 + 2 + ... + n; for n = 4 it is 1 + 2· 3 + 4, see Figure lb. Now, the whole area of the rectangle is n(n + 1) of which the staircase-shaped area is one half; this proves the formula. We may transform the result which we found by induction into
How to Solve It
l'
+ 2' + 33 + ... + n' = (n(n:
1983
y.
1)
5. If we have no idea how to prove this result, we may at least test it. Let us test the first case we have not tested yet, the case n == 6. For this value, the formula yields 1 + 8 + 27
+ 64 + 125 + 216 =
C:
7)'
and, on computation, this turns out to be true, both sides being equal to 441.
We can test the formula more effectively. The formula is, very likely, generally true, true for all values of n. Does it remain true when we pass from any value n to the next value n + I? Along with the formula as written above we should also have 13
+ 28 + 33 + ... + n8 + (n + 1)3 == (
(n
+ l)(n + 2) 2
)2 .
Now, there is a simple check. Subtracting from this the formula written above, we obtain (n
+ 1)3 == (
n+l)(n+2))2 2
..
-
(n(n+l))2 2
.
This is, however, easy to check. The right hand side may be written as
C-+-2_1)'
[In
+
+ 2)2 -
(n 1)2 - - - (4n
4
n') =
C-;_1 Y[n2 + 4n + 4 -
n2)
+ 4) == (n + 1)2 (n + 1) == (n + 1)8,
Our experimentally found formula passed a vital test. Let us see clearly what this test means. We verified beyond doubt that (n
+ 1)8 == (
(n
+ l)(n + 2) 2
)2 -
(n(n +
2
We do not know yet whether l' + 23 + 3" +
... + n3 =( n(n :
1)
1))2.
y.
is true. But if we knew that this was true we could infer, by adding the equation which. we verified beyond doubt, that 13 + 28
+ 38 + ... + n3 + (n + 1)3 == (
(n
+ 1)( n + 2) 2
)2
G. Poly"
1984
is also true which is the same assertion for the next integer n + 1. Now, we actually know that our conjecture is true for n 1, 2, 3, 4, 5, 6. By virtue of what we have just said, the conjecture, being true for n = 6, must also be true for n == 7; being true for n == 7 it is true for n == 8; being true for n 8 it is true for n 9; and so on. It holds for all n, it is proved to be true generally. 6. The foregoing proof may serve as a pattern in many similar cases. What are the essential lines of this pattern? The assertion we have to prove must be given in advance, in precise form. The assertion must depend on an integer n. The assertion must be sufficiently "explicit" so that we have some possibility of testing whether it remains true in the passage from n to the next integer n + 1. If we succeed in testing this effectively, we may be able to use our experience, gained in the process of testing, to conclude that the assertion must be true for n + 1 provided it is true for n. When we are so far it is sufficient to know that the assertion is true for n 1; hence it follows for n = 2; hence it follows for n == 3, and so on; passing from any integer to the next, we prove the assertion generally. This process is so often used that it deserves a name. We could call it "proof from n to n + 1" or still simpler "passage to the next integer." Unfortunately. the accepted technical term is "mathematical induction." This name results from a random circumstance. The precise assertion that we have to prove may come from any source, and it is immaterial from the logical viewpoint what the source is. Now, in many cases, as in the case we discussed here in detail, the source is induction, the assertion is found experimentally, and so the proof appears as a mathematical complement to induction; this explains the name. 7. Here is another point, somewhat subtle, but important to anybody who' desires to find proofs by himself. In ,the foregoing, we found two different assertions by observation and induction, one after the other, the first under 1, the second under 2; the second was more precise than the first. Dealing with the second assertion, we found a possibility of checking the passage from n to n + I, and so we were able to find a proof by "mathematical induction." Dealing with the first assertion, and ignoring the precision added to it by the second one, we should scarcely have been able to find such a proof. In fact, the first assertion is less precise, less "explicit," less "tangible," less accessible to testing and checking than the second one. Passing from the first to the second, from the less precise to the more precise statement, was an important preparative for the fin~tl proof. This circumstance has a paradoxical aspect. The second assertion is
=
=
=
=
flow to Solve It
1985
stronger; it implies immediately the first, whereas the somewhat "hazy" first assertion can hardly imply the more "clear-cut" second one. Thus, the stronger theorem is easier to master than the weaker one; this is the inventor's paradox. is like translation from one language into another. This comparison, used by Newton in his Arithmetica Universalis, may help to clarify the nature of certain difficulties often felt both by students and by teachers. 1. To set up equations means to express in mathematical symbols a condition that is stated in words; it is translation from ordinary language into the language of mathematical formulas. The difficulties which we may have in setting up equations are difficulties of translation. In order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language. The situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression. An English sentence is relatively easy to translate into French if it can be translated word for word. But there are English idioms which cannot be translated into French word for word. If our sentence contains such idioms, the translation becomes difficult; we have to pay less attention to the separate words, and more attention to the whole meaning; before translating the sentence, we may have to rearrange it. It is very much the same in setting up equations. In easy cases, the verbal statement splits almost automatically into successive parts, each of which can be immediately written down in mathematical symbols. In more difficult cases, the condition has parts which cannot be immediately translated into mathematical symbols. If this is so, we must pay less attention to the verbal statement, and concentrate more upon the meaning. Before we start writing formulas, we may have to rearrange the condition, and we should keep an eye on the resources of mathematical notation while doing so. In all cases, easy or difficult, we have to understand the condition, to separate the various parts of the condition, and to ask: Can you write them down? In easy cases, we succeed without hesitation in dividing the condition into parts that can be written down in mathematical symbols; in difficult cases, the appropriate division of the condition is less obvious. The foregoing explanation should be read again after the study of the following examples. 2. Find two quantities whose sum is 78 and whose product is 1296. SETTING UP EQUATIONS
G. Polya
1986
We divide the page by a vertical line. On one side, we write the verbal statement split into appropriate parts. On the other side, we write algebraic signs, opposite to the corresponding part of the verbal statement. The original is on the left, the translation into symbols on the right. STATING THE PROBLEM
in English Find two quantities whose sum is 78 and whose product is 1296
in algebraic language x, y x+ y 78 xy 1296.
=
=
In this case, the verbal statement splits almost automatically into successive parts, each of which can be immediately written down in mathematical symbols. 3. Find the breadth and the height of a right prism with square base, being given the volume, 63 cu. in., and the area of the surface, 102 sq. in. What are the unknowns? The side of the base, say x, and the altitude of the prism, say y. What are the data? The volume, 63, and the area, 102. What is the condition? The prism whose base is a square with side x and whose altitude is y must have the volume 63 and the area 102. Separate the various parts of the condition. There are two parts, one concerned with the volume, the other with the area. We can scarcely hesitate in dividing the whole condition just in these two parts; but we cannot write down these parts "immediately." We must know how to calculate the volume and the various parts of the area. Yet, if we know that much geometry, we can easily restate both parts of the condition so that the translation into equations is feasible. We write on the left hand side of the page an essentially rearranged and expanded statement of the problem, ready for translation into algebraic language. Of a right prism with square base find the side of the base x and the altitude. y First. The volume is given. 63 The area of the base which is a square with side x and the altitude determine the volume which is their product. x2y 63 Second. The area of the surface is given. 102 The surface consists of two squares with side x and of four rectangles, each with base x and altitude y, 4xy whose sum is the area. 2x2 4xy = 102. 4. Being given the equation of a straight line and the coordinates of a point, find the point which is symmetrical to the given point with respect to the given straight line.
=
How to Solve 11
1987
This is a problem of plane analytic geometry. What is the unknown? A point, with coordinates, say p, q. What is given? The equation of a straight line, say y == mx + n, and a point with coordinates, say a, b. What is the condition? The points (a, b) and (p, q) are symmetrical to each other with respect to the line y == mx + n. We now reach the essential difficulty which is to divide the condition into parts each of which can be expressed in the language of analytic geometry. The nature of this difficulty must be well understood. A decomposition of the condition into parts may be logically unobjectionable and nevertheless useless. What we need here is a decomposition into parts which are fit for analytic expression. In order to find such a decomposition we must go back to the definition of symmetry, but keep an eye on the resources of analytic geometry. What is meant by symmetry with respect to a straight line? What geometric relations can we express simply in analytic geometry? We concentrate upon the first question, but we should not forget the second. Thus, eventually, we may find the decomposition which we are going to state. The given point and the point required are so related that first, the line joining them is perpendicular to the given line, and second, they lie on opposite sides of the given line but are at equal distance from it.
(a, b) (P. q) q- b
1
--=-p-a
b - ma-n
vl+m~
m q- mp-n
VI + m
2
THE TRADITIONAL MATHEMATICS PROFESSOR of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face the blackboard and to tum his back on the class. He writes a, he says b, he means c; but it should be d. Some of his sayings are handed down from generation to generation. "In order to solve this differential equation you look at it till a solution occurs to you." ''This principle is so perfectly general that no particular application of it is possible." "Geometry is the art of correct reasoning on incorrect figures." "My method to overcome a difficulty is to go round it." "What is the difference between method and device? A method is a device which you use twice," After all, you can learn something from this traditional mathematics professor. Let us hope that the mathematics teacher from whom you cannot learn anything will not become traditional.
1988
WORKING'BACKWARDS.
G. Polya
If we wish to understand human behavior we
should compare it with animal behavior. Animals also "have problems" and "solve problems." Experimental psychology has made essential progress in the last decades in exploring the "problem-solving" activities of various animals. We cannot discuss here these investigations but we shall describe sketchily just one simple and instructive experiment and our description will serve as a sort of comment upon the method of analysis, or method of "working backwards." . . .1 1. Let us try to find an answer to the following tricky question: How can you bring up from the river exactly six quarts of water when you have only two containers, a four quart pail and a nine quart pail, to measure with? Let us visualize clearly the given tools we have to work with, the two containers. (What is given?) We imagine two cylindrical containers having equal bases whose altitudes are as 9 to 4, see Figure 2. If along the lateral surface of each container there were a scale of equally spaced horizontal
FIGURE 2
lines from which we could tell the height of the waterline, our problem would be easy. Yet there is no such scale and so we are still far from the solution. We do not know yet how to measure exactly ,6 quarts; but could we measure something else? (If you cannot solve the proposed problem try to solve first some related problem. Could you derive something useful from the data?) Let us do something, let us play around a little. We could fill the larger container to full capacity and empty so much as we can into the smaller container; then we could get 5 quarts. Could we also get 6 quarts? Here are again the two empty containers. We could also . . . We are working now as most people do when confronted with this puzzle. We start with the two empty containers, we try this and that, we empty and fill, and when we do not succeed, we start again, trying something else. We are working forwards, from the given initial situation to 1 [The Greek mathematician Pappus, Polya points out, gave an important description of the method. ED. J
How to Solvl! It
1989
the desired final situation, from the data to the unknown. We may succeed, after many trials, accidentally. 2. But exceptionally able people~ or people who had the chance to learn in their mathematics classes something more than mere routine operations~ do not spend too much time in such trials but turn round, and start working backwards. What are we required to do? (What is the unkown?) Let us visualize the final situation we aim at as clearly as possible. L~t us imagine that we have here, before us, the larger container with exactly 6 quarts in it and the smaller container empty as in Figure 3. (Let us start from what is required and assume what is sought as already found, s~ys Pappus.)
u
FIGURE ;)
From what foregoing situation could we obtain the desired final situation shown in Figure 3? (Let us inquire from what antecedent the desired result could be derived, says Pappus.) We could, of course, fill the larger container to full capacity, that is, to 9 quarts. But then we should be able to pour out exactly three quarts. To do that . . . we must have just one quart in the smaller container! That's the idea. See Figure 4.
FIGURE 4
(The step that we have just completed is not easy at all. Few persons are able to take it without much foregoing hesitation. In fact. recognizing the significance of this step, we foresee an outline of the following solution.) But how can we reach the situation that we have just found and fius-
G. Polya
1990
trated by Figure 4? (Let us inquire again what could be the antecedent oj that antecedent.) Since the amount of water in the river is, for our purpose, unlimited, the situation of Figure 4 amounts to the same as the next one in Figure 5
FIGURE 5
or the following in Figure 6.
u FIGURE 6
It is easy to recognize that if anyone of the situations in Figures 4, 5, 6 is obtained, any other can be obtained just as well, but it is not so easy to hit upon Figure 6, unless we have seen it before, encountered it acci~ dentally in one of our initial trials. Playing around with the two containers, we may have done something similar and remember now, in the right moment, that the situation of Figure 6 can arise as suggested by Figure 7: We fill the large container to full capacity, and pour from it four quarts
u FIGURE 7
How to Solve It
1991
}pto the smaller container and then into the river, twice in succession. We carne eventually upon something already known (these are Pappus's words) and following the method of analysis, working backwards, we have discovered the appropriatc? sequence of operations. It is true, we have discovered the appropriate sequence in retrogressive order but all that is left to do is to reverse the process and start from the point which we reached last of all z'n the analysis (as Pappus says). First, we perform the operations suggested by Figure 7 and obtain Figure 6; then we pass to Figure 5, then to Figure 4, and finally to Figure 3. Retracing our steps, we finally succeed in deriving what was required. 3. Greek tradition attributed to Plato the discovery of the method of analysis. The tradition may not be quite reliable but, at any rate, if the method was not invented by Plato, some Greek scholar found it necessary to attribute its invention to a philosophical genius. There is certainly something in the method that is not superficial. There is a certain psychological difficulty in turning around, in going away from the goal, in working backwards, in not following the direct path to the desired end. When we discover the sequence of appropriate operations, our mind has to proceed in an order which is exactly the reverse of the actual performance. There is some sort of psychological repugnance to this reverse order which may prevent a quite able student from understanding the method if it is not presented carefully. Yet it does not take a genius to solve a concrete problem working backwards; anybody can do it with a little common sense. We concentrate upon the desired end, we visualize the final position in which we would like to be. From what foregoing position could we get there? It is natural to ask this question, and in so asking we work backwards. Quite primitive problems may lead naturally to working backwards. Working backwards is a common-sense procedure within the reach of everybody and we can hardly doubt that it was practiced by mathematicians and nonmathematicians before Plato. What some Greek scholar may have regarded as an achievement worthy of the genius of Plato is to state the procedure in general terms and to stamp it as an operation typically useful in solving mathematical and nonmathematical problems. 4. And now we tum to the psychological experiment-if the transition from Plato to dogs, hens, and chimpanzees is not too abrupt. A fence forms three sides of a rectangle but leaves open the fourth side as shown in Figure 8. Wc place a dog on one side of the fence, at the point D, and some food on the other side, at the point F. The problem is fairly easy for the dog. He may first strike a posture as if to spring directly at the food but then he quickly turns about, dashes off around the end of the fence and, running without hesistation, reaches the food in a smooth curve. Sometimes, however, especially when the points D and F are close I
1992
G. Polya
F (I
•••••••••• ••••••••••••••• • • • • • • D • • •• ••• ••• ••• FIGURE 8
to each other, the solution is not so smooth; the dog may lose some time in barking, scratching, or jumping against the fence before he "conceives the bright idea" (as we would say) of going around. It is interesting to compare the behavior of various animals put into the place of the dog. The problem is very easy for a chimpanzee or a fouryear-old child (for whom a toy may be a more attractive lUre than food). The problem, however, turns out to be surprisingly difficult for a hen who runs back and forth excitedly on her side of the fence and may spend considerable time before getting at the food if she gets there at all. But she may succeed, after much running, accidentally. 5. We should not build a big theory upon just one simple experiment which was only sketchily reported. Yet there can be no disadvantage in noticing obvious analogies provided that we are prepared to recheck and revalue them. Going around an obstacle is what we do in solving any kind of problem; the experiment has a sort of symbolic value. The hen acted like people who solve their problem muddling through, trying again and again, and succeeding eventually by some lucky accident without much insight into the reasons for their success. The dog who scratched and jumped and barked before turning around solved his problem about as well as we did ours about the two containers. Imagining a scale that shows the waterline in our containers was a sort of almost useless scratching, showing only that what we seek lies deeper under the surface. We also tried to work forwards first, and came to the idea of turning round afterwards. The dog who, after brief inspection of the situation, turned round and dashed off gives, rightly or wrongly, the impression of superior insight. No, we should not even blame the hen for her clumsiness. There is a certain difficulty in turning round, in going away from the goal, in proceeding without looking continually at the aim, in not following the direct path to the desired end. There is an obvious analogy between her difficulties and our difficulties.
PART XVI
The Vocabulary of Mathematics 1. New NamesfoI Old by EDWARD KASNER and JAMES R.
NEWMAN
COMMENTARY ON
D,ouble Infinite Rapport and Other Mathematical Jargon HIS essay, written a few years ago, offers a twenty-minute tour of modern mathematics. The intention was to convey a notion of recent developments by reviewing the words that have been attached to new concepts. It will be seen that most of them are terms in ordinary usage, but given a special meaning. This is a fairly common practice in mathematics. One might make a similar tour of medical therapy, say, by explaining the names of the latest drugs. The names, though by now familiar, are essentially difficult: streptomycin, penicillin, sulfanilamide, and so on. Adrenocorticotropic hormone is so exceedingly polysyllabic that everyone, in the trade and out, now calls it ACTH. We suggested that mathematics, on the other hand, could be defined as the science that uses easy words"group," "ring," "limit," "simple curve," "inside," and "outside"-for hard ideas. There are exceptions to this definition, depending to some extent on the concept-maker's tempera~ent. The great British mathematician J. J. Sylvester, a prolific inventor of both concepts and names, was flowery of speech and poetic. He coined strange names such as "Hessian," "Jaco.. bian," "discriminant," "umbral notation" (denoting quantities which he said were "mere shadows")-so many, in fact, that he was called the mathematical Adam: Looking through the mathematics books which have accumulated on my shelves in the last decade, I now find that the easy word definition has become less suitable. The vocabulary of mathematics is getting more abstruse, though it does not follow that the ideas are getting more profound. One of the reasons for the harder words is that mathematics has moved further into the great world of affairs and has been serving strange gods. Communication and information theory, operational research, the calculus of games, are among the subjects in which mathematics has assumed major responsibilities, requiring, in each case, an appropriate jargon. One encounters, to give a small sample from some thoroughly respectable books and pamphlets, the "maximization of 'moral expectation'" (having to do with improving businessmen's profits and not the level of ethics) '. "correlogram analysis," "isoquants," "asymptotic minimax solutions of sequential point estimation problems," "confidence regions for linear regression," "subminimax solutions of compound statistical decision problems," the "Monte Carlo method," the "homotopy groups of Hurewicz," "vacuous games," "half-spaces," "pay-off functions," "hemibel
T
1994
Double Infinite Rapport and Other Mathematical Jargon
1995
thinking," "eigen values," "hole functions," "double infinite rapport," "pheno-typical inversion." Most of these words and expressions were unknown to mathematics twenty-five years ago; some are newly coined, some newly changed. It is safe to say that the vocabulary gives a somewhat inflated impression of the recent progress in mathematical thought. Mathematics has advanced, to be sure, but neither as fast nor as far as the grandiose terminology would suggest. Edward Kasner, who was my collaborator in the writing of Mathematics and the Imagination, from which the following selection and two others in this book were taken, was a leading mathematician of the twentieth century. His specialty was higher geometry, to which he contributed a large number of original papers; he also was known for his skill as a teacher. ' Born in New York City in 1878, Kasner was educated at the College of the City of New York, Columbia and Gottingen universities. In 1900 he joined the teaching staff at Columbia, in 1910 he was appointed professor, and in 1937 he received the Adrain chair in mathematics. His exceptional capacity as a lecturer made his courses among the most popular at the university. To a fertile imagination and natural mathematical gifts he joined a sense of humor and a wide appreciation of cultural values. He could make you see intricate mathematical relationships by his verbal images, by the graceful gestures of his delicate hands, and by the spidery, badly executed, but remarkably illuminating diagrams he liked so much to scrawl on the blackboard. It was his way to tease his students, to lead them disingenuously down the garden path. Then, by the absurdity of their position, he would force them to find the way out. By such methods, tinctured with irony and gentle malice, he succeeded both in explaining his subject and in whetting the student's appetite for more. I had the good fortune to attend several of his courses as a graduate 'student, and, like many others, I owe to him a true awakening of interest in mathematics and an appreciation of its rare excellence. Some of his pupils attained distinction in mathematics; all remembered their instruction under him as an intellectual delight. And in this large assembly are included not only college and graduate school aUdiences, and those who attended his courses at the New York School for Social Research, but the children of kindergarten age to whom he would often lecture on the mathematics of infinity, topology and other recondite matters. Kasner's numerous honors included membership in the National Academy of Sciences. He never married. He died in New York, January 7, 1955, at the age of seventy-six.
For out of olde feldes, as men seith, Cometh al thz.; newe corne fro yeere to yere; And out of oide bokes, in good feith, Cometh al this newe SCIence that men lere.
1
New Names for Old By EDWARD KASNER and JAMES R. NEWMAN
EVERY once in a while there is house cleaning in mathematics. Some old names are discarded, some dusted off and refurbished; new theories, new additions to the household are assigned a place and name. So what our title really means is new words in mathematics; not new names, but new words, new terms which have in part come to represent new concepts and a reappraisal of old ones in more or less recent mathematics. There are surely plenty of words already in mathematics as well as in other SUbjects. Indeed, there are so many words that it is even easier than it used to be to speak a great deal and say nothing. It is mostly through words strung together like beads in a necklace that half the population of the world has been induced to believe mad things and to sanctify mad deeds. Frank Vizetelly, the great lexicographer~ estimated that there are 800,000 words in use in the English language. But mathematicians, generally quite modest, are not satisfied with these 800,000; let us give them a few more. We can get along without new names until, as we advance in science, we acquire new ideas and new forms. A peculiar thing about mathematics is that it does not use so many long and hard names as the other sciences. Besides, it is more conservative than the other sciences in that it clings tenaciously to old words. The terms used by Euclid in his Elements are current in geometry today. But an Ionian physicist would find the terminology of modern physics, to put it colloquially, pure Greek. In chemistry, substances no more complicated than sugar, starch, or alcohol have names like these: Methylpropenylenedihydroxycinnamenylacrylic acid~ or, O-anhydrosulfaminobenzoine, or, protocatechuicaldehydemethylene. It would be inconvenient if we had to use such terms in everyday conversation. Who could imagine even the aristocrat of science at the breakfast table asking, "Please pass the O-anhydrosulfaminobenzoic acid," when all he wanted was sugar for his coffee? Biology also has some tantalizing tongue twisters. The purpose of these long words is not to frighten the exoteric, but to describe with scientific curtness what the literary man would take half a page to express. 1996
New Names lor Old
1997
In mathematics there are many easy words like "group," "family," "ring," "simple curve," "limit," etc. But these ordinary words are sometimes given a very peculiar and technical meaning. In fact, here is a boobyprize definition of mathematics: Mathematics is the science which uses easy words for hard ideas. In this it differs from any other science. There are 500,000 known species of insects and every one has a long Latin name. In mathematics we are more modest. We talk about "fields:' "groups/' "families," "spaces," although much more meaning is attached to these words than ordinary conversation implies. As its use becomes more and more technical, nobody can guess the mathematical meaning of a word any more than one could guess that a "drug store" is a place where they sell ice-cream sodas and umbrellas. Noone could guess the meaning of the word "group" as it is used in mathematics. Yet it is so important that whole courses are given on the theory of "groups," and hundreds of books are written about it. Because mathematicians get along with common words, many amusing ambiguities arise. For instance, the word "function" probably expresses the most important idea in the whole history of mathematics. Yet, most people hearing it would think of a "function" as meaning an evening social affair, While others, less socially minded, would think of their livers. The word "function" has at least a dozen meanings, but few people suspect the mathematical one. The mathematical meaning is expressed most simply by a table. Such a table gives the relation between two variable quantities when the value of one variable quantity is determined by the value of the other. Thus, one variable quantity may express the years from 1800 to 1938, and the other, the number of men in the United States wearing handle-bar mustaches; or one variable may express in decibels the amount of noise made by a political speaker, and the other, the blood pressure units of his listeners. You could probably never guess the meaning of the word "ring" as it has been used in mathematics. It was introduced into the newer algebra within the last thirty years. The theory of rings is much more recent than the theory of groups. It is now found in most of the new books on algebra, and has nothing to do with either matrimony or bells. Other ordinary words used in mathematics in a peculiar sense are "domain," "integration," "differentiation." The uninitiated would not be able to guess what they represent; only mathematicians would know about them. The word "transcendental" in mathematics has not the meaning it has in philosophy. A mathematician would say: The number 1r, equal to 3.14159 . . . , is transcendental, because it is not the root of any algebraic equation with integer coefficients. Transcendental is a very exalted name for a small number, but it was coined when it was thought that transcendental numbers were as rare as
Edward Kasner and James R. Newman
1998
quintuplets. The work of Georg Cantor in the realm of the infinite has since proved that of all the numbers in mathematics, the transcendental ones are the most common, or, to use the word in a slightly different sense, the least transcendental. Immanuel Kant's "transcendental epistemology" is what most educated people might think of when the word transcendental is used, but in that sense it has nothing to do with mathematics. Again, take the word "evolution," used in mathematics to denote the process most of us learned in elementary school, and promptly forgot, of extracting square roots, cube roots, etc. Spencer, in his philosophy, defines evolution as "an integration of matter, and a dissipation of motion from an indefinite, incoherent homogeneity to a definite, coherent heter~ ogeneity," etc. But that, fortunately, has nothing to do with mathematical evolution either. Even in Tennessee, one may extract square roots without running afoul of the law. As we see, mathematics uses simple words for complicated ideas. An example of a simple word used in a complicated way is the word "simple." "Simple curve" and "simple group" represent important ideas in higher mathematics.
FIGURE 1
The above is not a simple curve. A simple curve is a closed curve which does not cross itself and may look like Figure 2. There are many impor~ tant theorems about such figures that make the word worth while. They are found in a queer branch of mathematics called "topology." A French
FIGURE 2
mathematician, Jordan, gave the fundamental theorem of this study: every simple curve has one inside and one outside. That is, every simple curve
New Names for Old
1999
divides the plane into two regions, one inside the curve, and one outside. There are some groups in mathematics that are "simple" groups. The definition of "simple group" is really so hard that it cannot be given here. If we wanted to get a clear idea of what a simple group was, we should probably have to spend a long time looking into a great many books, and then, without an extensive mathematical background, we should probably miss the point. First of all, we should have to define the concept "group." Then we should have to give a definition of subgroups, and then of selfconjugate subgroups, and then we should be able to tell what a simple group is. A simple group is simply a group without any self-conjugate subgroups-simple, is it not? Mathematics is often erroneously referred to as the science of common sense. Actually, it may transcend common sense and go beyond either imagination or intuition. It has become a very strange and perhaps frightening subject from the ordinary point of view, but anyone who penetrates into it will find a veritable fairyland, a fairyland which is strange, but makes sense, if not common sense. From the ordinary point of view mathematics deals with strange things. We shall show you that occasionally it does deal with strange things, but mostly it deals with familiar things in a strange way. If you look at yourself in an ordinary mirror, regardless of your physical attributes, you may find yourself amusing, but not strange; a subway ride to Coney Island, and a glance at yourself in one of the distorting mirrors will convince you that from another point of view you may be strange as well as amusing. It is largely a matter of what you are accustomed to. A Russian peasant came to Moscow for the first time and went to see the sights. He went to the zoo and saw the giraffes. You may find a moral in his reaction as plainly as in the fables of La Fontaine. "Look," he said, "at what the Bolsheviks have done to our horses." That is what modem mathematics has done to simple geometry and to simple arithmetic. There are other words and expressions, not so familiar, which have been invented even more recently. Take, for instance, the word "turbine." Of course, that is already used in engineering, but it is an entirely new word in geometry. The mathematical name applies to a certain diagram. (Geometry, whatever others may think, is the study of different shapes, many of them very beautiful, having harmony, grace and symmetry. Of course, there are also fat books written on abstract geometry, and abstract space in which neither a diagram nor a shape appears. This is a very important branch of mathematics, but it is not the geometry studied by the Egyptians and the Greeks. Most of us, if we can play chess at all, are content to play it on a board with wooden chess pieces; but there are some who play the game blindfolded and without touching the board. It
Edward Kasner and lames R. Newman
2000
--
--
and arguing from the fact that the two triangles thus created are similar to one another and to the original triangle, and that the proportions which their corresponding sides bear to one another are therefore equal, one can show in algebraical form that C2 + D~ (the squares on the other two sides) are equal to A'.!. + B2 (the squares on the two segments of the hypotenuses) + 2AB; which last, it is easy to show geometrically, is equal to (A + B)2, or the square on the hypotenuse. Guido was as much enchanted by the rudiments of algebra as he would have been if I had given him an engine worked by steam, with a methylated spirit lamp to heat the boiler; more enchanted, perhaps-for the engine would have got broken, and, remaining always itself, would in any case have lost its charm, while the rudiments of algebra continued to grow and blossom in his mind with an unfailing luxuriance. Every day he made the discovery of something which seemed to him exquisitely beautiful; the new toy was inexhaustible in its poten-
tialities. In the intervals of applying algebra to the second book of Eudid t we experimented with circles; we stuck bamboos into the parched earth, measured their shadows at different hours of the day, and drew exciting conclusions from our observations. Sometimes, for fun, we cut and folded sheets of paper so as to make cubes and pyramids. One afternoon Guido arrived carrying carefully between his small and rather grubby hands a flimsy dodecahedron. tiE tanto bello!" he said, as he showed us his paper crystal; and when I asked him how he had managed to make it, he merely smiled and said it had been so easy. I looked at Elizabeth and laughed. But it would have been more symbolically to the point, I felt, if I had gone down on all fours, wagged the spiritual outgrowth of my os coccyx, and barked my astonished admiration. It was an uncommonly hot summer. By the beginning of July our little Robin, unaccustomed to these high temperatures, began to look pale and tired; he was listless, had lost his appetite and energy. The doctor advised mountain air. 'Ve decided to spend the next ten or twelve weeks in Switzerland. My paning gift to Guido was the first six books of Euclid in Italian. He turned over the pages, looking ecstatically at the figures . . "If only I knew how to read properly," he said. "I'm so stupid. But now I shall really try to learn." From our hotel near Grindelwald we sent the child, in Robin's name, various postcards of cows, alphorns, Swiss chalets, edelweiss, and the like. We received no answers to these cards; but then we did not expect answers. Guido could not write, and there was no reason why his father or his sisters should take the trouble to write for him. No news, we took it,
Young Archimedes
2245
was good news. And then one day, early in September, there arrived at the hotel a strange letter. The manager had it stuck up on the glass-fronted notice board in the hall, so that all the guests might see it, and whoever conscientiously thought that it belonged to him might claim it. Passing the board on the way in to lunch, Elizabeth stopped to look at it. "But it must be from Guido," she said. I came and looked at the envelope over her shoulder. It was unstamped and black with postmarks. Traced out in pencil, the big uncertain capital letters sprawled across its face. In the first line was written: AL BABBO Dr ROBIN, and there followed a travestied version of the name of the hotel and the place. Round the address bewildered postal officials had scrawled suggested emendations. The letter had wandered for a fortnight at least, back and forth across the face of Europe. "AI Babbo di Robin. To Robin's father." I laughed. "Pretty smart of the postmen to have got it here at all." I went to the manager's office, set forth the justice of my claim to the letter and, having paid the fifty-centime surcharge for the missing stamp, had the case unlocked and the letter given me. We went into lunch. "The writing'S magnificent," we agreed, laughing, as we examined the address at close quarters. "Thanks to Euclid," I added. "That's what comes of pandering to the ruling passion." But when I opened the envelope and looked at its contents I no longer laughed. The letter was brief and almost telegraphic in style. SONO DALLA PADRONA, it ran, NON MI PlACE HA RUBATO IL MID LIBRO NON VOGLIO SUONARE PIU VOGLIO TORNARE A CASA VENGA SUB ITO GUIDO.
"What is it?" I handed Elizabeth the letter. "That blasted woman's got hold of him," I said. Busts of men in Homburg hats, angels bathed in marble tears extinguishing torches, statues of little girls, cherubs, veiled figures, allegories and ruthless realisms-the strangest and most diverse idols beckoned and gesticulated as we passed. Printed indelibly on tin and embedded in the living rock, the brown photographs looked out, under glass, from the humbler crosses, headstones, and broken pillars. Dead ladies in cUbistic geometrical fashions of thirty years ago-two cones of black satin meeting point to point at the waist, and the arms: a sphere to the elbow, a polished cylinder below-smiled mournfully out of their marble frames; the smiling frames; the smiling faces, the white hands. were the only recognizably human things that emerged from the solid geometry of their clothes. Men with black mustaches, men with white beards, young c1eanshaven men, stared or averted their gaze to show a Roman profile. Children in their stiff best opened wide their eyes, smiled hopefully in anticipation of
Aldous Huxley 2246
the little bird that was to issue from the camera's muzzle, smiled skepti-
cally in the knowledge that it wouldn't, smiled laboriously and obediently because they had been told to. In spiky Gothic cottages of marble the richer dead reposed; through grilled doors one caught a glimpse of pale Inconsolables weeping, of distraught Geniuses guarding the secret of the tomb. The less prosperous sections of the majority slept in communities, close-crowded but elegantly housed under smooth continuous marble floors, whose every flagstone was the mouth of a separate grave. These Continental cemeteries, I thought, as Carlo and I made our way among the dead, are more frightful than ours, because these people pay more attention to their dead than we do. That primordial cult of corpses, that tender solicitude for their material well-being, which led the ancients to house their dead in stone, while they themselves lived between wattles and under thatch. still lingers here; persists, I thought, more vigorously than with us. There are a hundred gesticulating statues here for every one in an English graveyard. There are more family vaults, more "luxuriously appointed" (as they say of liners and hotels) than one would find at home. And embedded in every tombstone there are photographs to remind the powdered bones within what form they will have to resume on the Day of Judgment; beside each are little hanging lamps to bum optimistically on All Souls' Day. To the Man who built the Pyramids they are nearer, I thought, than we. "If I had known," Carlo kept repeating, "if only I had known." His voice came to me through my reflections as though from a distance. "At the time he didn't mind at all. How should I have known that he would take it so much to heart afterwards? And she deceived me, she lied to me." I assured him yet once more that it wasn't his fault. Though, of course, it was, in part. It was mine too, in part; I ought to have thought of the possibility and somehow guarded against it. And he shouldn't have let the child go, even temporarily and on trial, even though the woman was bringing pressure to bear on him. And the pressure had been considerable. They had worked on the same holding for more than a hundred years, the men of Carlo's family~ and now she had made the old man threaten to turn him out. It would be a dreadful thing to leave the place; and besides, another place wasn't so easy to find. It was made quite plain, however, that he could stay if he let her have the child. Only for a little to begin with; just to see how he got on. There would be no compulsion whatever on him to stay if he didn't like it. And it would be all to Guido's advantage; and to his father's, too, in the end. All that the Englishman had said about his not being such a good musician as he had thought at .first was obviously untrue-mere jealousy and little-mindedness: the man wanted to take credit for Guido himself, that was all. And the boy, it
Y Dun, Archimedes
2241
was obvious, would learn nothing from him. What he needed was a real good professional master. All the energy that, if the physicists had known their business, would have been driving dynamos, went into this campaign. It began the moment we were out of the house, intensively. She would have more chance of success, the Signora doubtless thought, if we weren't there. And besides, it was essential to take the opportunity when it offered itself and get hold of the child before we could make our bid-for it was obvious to her that we wanted Guido just as much as she did. Day after day she renewed the assault. At the end of a week she sent her husband to complain about the state of the vines: they were in a shocking condition; he had decided, or very nearly decided, to give Carlo notice. Meekly, shamefacedly, in obedience to higher orders, the old gentleman uttered his threats. Next day Signora Bondi returned to the attack. The padrone, she declared, had been in a towering passion; but she'd do her best, her very best, to mollify him. And after a significant pause she went on to talk about Guido. In the end Carlo gave in. The woman was too persistent and she held too many trump cards. The child could go and stay with her for a month or two on trial. After that~ if he really expressed a desire to remain with her, she would formally adopt him. At the idea of going for a holiday to the seaside-and it was to the seaside, Signora Bondi told him, that they were going--Guido was pleased and excited. He had heard a lot about the sea from Robin. "Tanta aequa!" It had sounded almost too good to be true. And now he was actually to go and see this marvel. It was very cheerfully that he parted from his family. But after the holiday by the sea was over, and Signora Bondi had brought him back to her town house in Florence, he began to be homesick. The Signora, it was true, treated him exceedingly kindly, bought him new clothes, took him out to tea in the Via Tornabuoni and filled him up with cakes7 iced strawberry-ade, whipped cream, and chocolates. But she made him practice the piano more than he liked, and what was worse, she took away his Euclid, on the score that he wasted too much time with it. And when he said that he wanted to go home, she put him off with promises and excuses and downright lies. She told him that she couldn't take him at once, but that next week, if he were good and worked hard at his piano meanwhile, next week . . . And when the time came she told him that his father didn~t want him back. And she redoubled her petting, gave him expensive presents, and stuffed him with yet unhealthier foods. To no purpose. Guido didn't like his new life, didn't want to practice scales, pined for his book, and longed to be back with his brothers and sisters. Signora Bondi~ meanwhile, continued to hope that time and chocolates
2248
Aldous Huxlt)'
would eventually make the child hers; and to keep his family at a distance, she wrote to Carlo every few days letters which still purported to come from the seaside (she took the trouble to send them to a friend, who posted them back again to Florence), and in which she painted the most charming picture of Guido's happiness. It was then that Guido wrote his letter to me. Abandoned, as he supposed, by his family-for that they should not take the trouble to come to see him when they were so near was only to be explained on the hypothesis that they really had given him up-be must have looked to me as his last and only hope. And the letter, with its fantastic address. had been nearly a fortnight on its way. A fortnight-it must have seemed hundreds of years; and as the centuries succeeded one another, gradually, no doubt, the poor child became convinced that I too had abandoned him. There was no hope left. "Here we are," said Carlo. I looked up and found myself confronted by an enormous monument. In a kind of grotto hollowed in the flanks of a monolith of gray sandstone, Sacred Love. in bronze, was embracing a funeral urn. And in bronze letters riveted into the stone was a long legend to the effect that the inconsolable Emesto Bondi had raised this monument to the memory of his beloved wife. Anunziata, as a token of his undying love for one whom, snatched from him by a premature death, he hoped very soon to join beneath this stone. The first Signora Bondi had died in 1912. I thought of the old man leashed to his white dog; he must always, I reflected, have been a most uxorious husband. "They buried him here." We stood there for a long time in silence. I felt the tears coming into my eyes as I thought of the poor child lying there underground. I thought of those luminous grave eyes, and the curve of that beautiful forehead. the droop of the melancholy mouth, of the expression of delight which illumined his face when he learned of some new idea that pleased him, when he heard a piece of music that he liked. And this beautiful small being was dead; and the spirit that inhabited this form, the amazing spirit, that too had been destroyed almost before it had begun to exist. And the unhappiness that must have preceded the final act, the child's despair~ the conviction of his utter abandonment-those were terrible to think of, terrible. "1 think \\;: had better come away now," I said at last, and touched Carlo on the arm. He was standing there like a blind man, his eyes shut, his face slightly lifted towards the light; from between his closed eyelids the tears welled out, hung for a moment, and trickled down his cheeks. His lip:; trembled and I could see that he was making an effort to keep them I:itiI1. "Come away," I repeated.
Young Archimedes
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The face which had been still in its sorrow was suddenly convulsed; he opened his eyes, and through the tears they were bright with a violent anger. "I shall kill her," he said, "I shall kill her. When I think of him throwing himself out, falling through the air . . . " With his two hands he made a violent gesture, bringing them down from over his head and arresting them with a sudden jerk when they were on the level with his breast. "And then crash." He shuddered. "She's as much responsible as though she had pushed him down herself. I shall kill her." He clenched his teeth. To be angry is easier than to be sad, less painful. It is comforting to think of revenge. "Don't talk like that," I said. "It's no good. It's stupid. And what would be the point'?" He had had those fists before, when grief became too painful and he had tried to escape from it. Anger had been the easiest way of escape. I had had, before this, to persuade him back into the harder path of grief. "It's stupid to talk like that," I repeated, and I led him away through the ghastly labyrinth of tombs, where death seemed more terrible even than it is. By the time we had left the cemetery, and were walking down from San Miniato towards the Piazzale Michelangelo below, he had become calmer. His anger had subsided again into the sorrow from which it had derived all its strength and its bitterness. In the Piazzale we halted for a moment to look down at the city in the valley below us. It was a day of floating clouds-great shapes, white, golden, and gray; and between them patches of a thin, transparent blue. Its lantern. level, almost, with our eyes, the dome of the cathedral revealed itself in all its grandiose lightness, its vastness and aerial strength. On the innumerable brown and rosy roofs of the city the afternoon sunlight lay softly, sumptuously, and the towers were as though varnished and enameled with an old gold. I thought of aU the Men who had lived here and left the visible traces of their spirit and conceived extraordinary things. I thought of the dead child.
COMMENTARY ON
Mr. Fortune HE Reverend Timothy Fortune had been a bank clerk but his heart was set neither on riches nor advancement. When, at middle age, a small sum was left to him by an aunt, he went to a training college, was ordained a deacon and quitted England to become a missionary at St. Fabien, a port on an island of the fictional Raritongan Archipelago in the Pacific. After a time he felt the call to go to Fanua, a small, remote island, to make Christians of its peaceful, childlike natives. Mr. Fortune was a humble man and easygoing. "Even as a young man he had learnt that to jump in first doesn't make the bus start any sooner; and his favorite psalm was the one which begins: My soul truly waitest still upon God." He intended no pressure to convert the islanders. He knew they were a happy people; after he had dwelt among them, he thought, they would come to him and he would teach them "how they might be as happy in another life as they were in this." Three years Mr. Fortune spent on Fanua; he made not a single convert. At first he thought he had converted a beautiful native boy named LueH; he loved the boy and was loved by him. But one day he discovered that Lueli had only feigned to be a Christian so as not to offend Mr. Fortune; "in secret, in the reality of secretness," he continued to worship an idol. Mr. Fortune was angry, then puzzled, and finally ashamed of his failure. It was clear that he himself was unworthy and this failure his punisbment. His tormenting reflections were interrupted by a terrific earthquake that suddenly struck the island. The hut he occupied collapsed and but for Lueli's efforts, at the risk of his life, Mr. Fortune would have perished. The earthquake had awful consequences for both the boy and the priest. The fire in the hut had destroyed LueU's idol-which he might have saved had he not thought first of his friend's safety; the hideous shaking of the earth, "the flames, that had burst roaring and devouring from the mountain top" had also destroyed Mr. Fortune's belief in God. He had "departed in clouds of smoke, He had gone up and was lost in space." Lueli felt the loss of the idol more acutely than Mr. Fortune the evaporation of his faith. The boy was listless and utterly miserable. His friends teased him for having lost his God. Mr. Fortune knew he must find ways to draw Lueli from his despair. "After three years of such familiarity it would not be easy to reconstruct his first fascination as something rich and strange. But it must be done if he Were to compete successfully with his rival in Lueli's affections. It must be done because that rival was death." He tried to bring about a change in Lueli's mood by introducing him to
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games: ping-pong, spellikins, dicing, skittles. He caught a baby flying-fox and reared it for Lueli as a pet. He introduced him, with the aid of a magnifying glass, to the wonderful details of natural history. Nothing worked. Lueli got hit in the nose playing ping~pong, stoutly resisted spellikins, was bored by dicing and was regularly scratched and bitten by the fox. Then one morning Mr. Fortune remembered mathematics. The sequel to this inspiration is recounted in the excerpt below, taken from Sylvia Townsend Warner's gentle satire, Mr. Fortune's Maggot. It is a witty, enchanting episode of modern literature.
He knew what's what, and that's m high As metaphysic wit can fly.
3
SAMUEL BUTLEll (Hudibras)
Geometry in the South Pacific By SYLVIA TOWNSEND WARNER [EXCERPT FROM "MR. FORTUNE'S MAGGOT"]
AND then one morning when they had been living in the new hut for about
six weeks he [Mr. Fortune] woke up inspired. Why had he wasted so much time displaying his most trivial and uncompelling charms, opposing to the magnetism of death such fripperies and titbits of this world, such gew~ gaws of civilization as a path serpentining to a parrot-cote (a parrot-cote which hadn't even allured the parrots), or a pocket magnifying glass, while all the time he carried within him the inestimable treasures of intellectual enjoyment? Now he would pipe Lueli a tune worth dancing to, now he would open for him a new world. He would teach him mathematics. He sprang up from bed, full of enthusiasm. At the thought of all those stretches of white beach he was like a bridegroom. There they were, hard and smooth from the tread of the sea, waiting for that noble consummation of blank surfaces, to show forth a truth; waiting, in this particular instance, to show forth the elements of plane geometry. At breakfast Mr. Fortune was so glorified and gay that Lueli caught a reflection of his high spirits and began to look more life-like than he had done for weeks. On their way down to the beach they met a party of islanders who were off on a picnic. Mr. Fortune with delight heard Lueli answering their greetings with something like his former sociability, and even plucking up heart enough for a repartee. His delight gave a momentary stagger when Lueli decided to go a-picnicking too. But, after all, it didn't matter a pin. The beach would be as smooth again to-morrow, the air as sweet and nimble; Lueli would be in better trim for learning after a spree, and, now he came to think of it, he himself wouldn't teach any the worse for a little private rUbbing-up beforehand. It must be going on for forty years since he bad done any mathematics; for he had gone into the Bank the same year that his father died, leaving Rugby at seventeen because, in the state that things were then in. the Bank was too good an opening to be missed. He had once got a prize-The Poetical \Vorks of Longfellow-for Algebra, and he had scrambled along ",ell enough in other branches of mathematics; but he had not learnt with 2~52
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any particular thrill or realized that thrill there might be until he was in the Bank, and learning a thing of the past. Then, perhaps because of that never-ending entering and adding up and striking balances, and turning on to the next page to enter, add up and strike balances again, a mental occupation minute, immediate and yet, so to speak, wool-gathering, as he imagined knitting to be, the absolute quality of mathematics began to take on for him an inexpressibly romantic air. "Pure Mathematics." He used to speak of them to his fellow clerks as though he were hinting at some kind of transcendental debauchery of which he had been made free--and indeed there does seem to be a kind of unnatural vice in being so completely pure. After a spell of this holy boasting he would grow a little uneasy; and going to the Free Library he took out mathematical treatises, just to make sure that he could follow step by step as well as soar. For twenty pages perhaps, he read slowly, carefully, dutifully, with pauses for self-examination and working out the examples. Then, just as it was working up and the pauses should have been more scrupulous than ever, a kind of swoon and ecstasy would fall on him, and he read ravening on, sitting up till dawn to finish the book, as though it were a novel. After that his passion was stayed; the book went back to the Library and he was done with mathematics till the next bout. Not much remained with him after these orgies. but something remained: a sensation in the mind, a worshipping acknowledgment of something isolated and unassailable, or a remembered mental joy at the rightness of thoughts coming together to a conclusion, accurate thoughts, thoughts in just intonation. coming together like unaccompanied voices coming to a close. But often his pleasure flowered from quite simple things that any fool could grasp. For instance he would look out of the bank windows, which had green shades in their lower halves; and rising above the green shades he would see a row of triangles, equilateral, isosceles, acute-angled, rightangled, obtuse-angled. These triangles were a range of dazzling mountain peaks, eternally snowy, eternally untrodden; and he could feel the keen wind which blew from their summits. Yet they were also a row of triangles. equilateral. isosceles, acute-angled, right-angled, obtuse-angled. This was the sort of thing he designed for Lueli's comfort. Geometry would be much better than algebra, though he had not the same certificate from Longfellow for teaching it. Algebra is always dancing over the pit of the unknown, and he had no wish to direct LueJi's thoughts to that quarter. Geometry would be best to begin with, plain plane geometry~ immutably plane. Surely if anything could minister to the mind diseased it would be the steadfast contemplation of a right angle, an existence that no mist of human tears could blur. no blow of fate deflect. Walking up and down the beach, admiring the surface which to-morrow
Sylvia Townsend
Warn~r
with SO much epiphany and glory was going to reveal the first axioms of Euclid, Mr. Fortune began to think of himself as possessing an universal elixir and charm. A wave of missionary ardour swept him along and he seemed to view, not Lueli only, but all the islanders rejoicing in this new dispensation. There was beach·board enough for all and to spare. The picture grew in his mind's eye, somewhat indebted to Raphaers Cartoon of the School of Athens. Here a group bent over an equation, there they pointed out to each other with admiration that the square on the hypotenuse equalled the sum of the squares on the sides containing the right angle; here was one delighting in a rhomboid and another in conic sections, that enraptured figure had secured the twelfth root of two, while the children might be filling up the foreground with a little long division. By the morrow he had slept off most of his fervour. Calm, methodical, with a mind prepared for the onset, he guided Lueli down to the beach and with a stick prodded a small hole in it. "What is tbis?" "A hole:' "No, Lueli, it may seem like a hole, but it is a point." Perhaps he had prodded a little too emphatically. Lueli's mistake was quite natural. Anyhow, there were bound to be a few misunderstandings at the start. -He took out his pocket knife and whittled the end of the stick. Then he tried again. "What is this?" "A smaller hole." "Point," said Mr. Fortune suggestively. "Yes, I mean a smaller point." "No, not quite. It is a point, but it is not smaller. Holes may be of different sizes. but no point is larger or smaller than another point." LueU looked from the first point to the second. He seemed to be about to speak, but to think better of it. He removed his gaze to the sea. Meanwhile, Mr. Fortune had moved about, prodding more points. It was rather awkward that he should have to walk on the beach-board, for his footmarks distracted the eye from the demonstration. "Look, Lueli!" Lueli turned his gaze inland. "Where?H said he. "/(t all these. Here; and here; and here. But don't tread on them." Lueli stepped back hastily. When he was well out of the danger-zone he stood looking at Mr. Fortune with great attention and some uneasiness. "These are all points." Lueli recoiled a step further. Standing on one leg he furtively inspected the sole of his foot.
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"As you see, Lueli, these points are in different places. This one is to the west of that and consequently that one is to the east of this. Here is one to the south. Here are two close together, and there is one quite apart from all the others. Now look at them, remember what I have said, think carefully and tell me what you think." Inclining his head and screwing up his eyes Lueli inspected the demonstration with an air of painstaking connoisseurship. At length he ventured the opinion that the hole lying apart from the others was perhaps the neatest. But if Mr. Fortune would give him the knife he would whittle the stick even finer. "N ow what did I tell you? Have you forgotten that points cannot be larger or smaller? If they were holes it would be a different matter. But these are points. Will you remember that?" Lueli nodded. He parted his lips, he was about to ask a question. Mr. Fortune went on hastily. "Now suppose I were to cover the whole beach with these: what then?" A look of dismay came over Lueli's countenance. Mr. Fortune withdrew the hypothesis. "I don't intend to. I only ask you to imagine what it would be like if I did." The look of dismay deepened. "They would all be points," said Mr. Fortune, impressively. "All in different places. And none larger or smaller than another. "What I have explained to you is summed up in the axiom: a point has position but not magnitude. In other words if a given point were not in a given place it would not be there at all." Whilst allowing time for this to sink in he began to muse about those other words. Were they quite what he meant? Did they indeed mean anything? Perhaps it would have been better not to try to supplement Euclid. He turned to his pupil. The last words had sunk in at any rate, had been received without scruple and acted upon. Lueli was out of sight. Compared with his intentions actuality had been a little quelling. It became more quelling as time went on. Lueli did not again remove himself without leave; he soon discovered that Mr. Fortune was extremely in earnest, and was resigned to regular instruction every morning and a good deal of rubbing-in and evocation during the rest of the day. No one ever had a finer capacity for listening than he, or a more docile and obliging temperament. But whereas in the old days these good gifts had flowed from him spontaneously and pleasurably he now seemed to be exhibiting them by rote and in a manner almost desperate, as though he -were listening and obliging as a circus animal does its tricks. Humane visitors to circuses often point out with what alacrity the beasts run into the ring to perform their turn. They do not understand that in the choice
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Sylvia Townsend Wtml"
of two evils most animals would rather flourish round a spacious ring than be shut up in a cage. The activity and the task is a distraction from their unnatural lot, and they tear through paper hoops all the better because so much of their time is spent behind iron bars. It had been a very different affair when Lueli was learning Bible history and the Church Catechism, The King of Love my Shepherd is and The Old Hundredth. Then there had been no call for this blatant submission; lessons had been an easy-going conversation, with Lueli keeping his end up as an intelligent pupil should and Mr. Fortune feeling like a cross between wise old Chiron and good Mr. Barlow. Now they were a succession of harangues, and rather strained harangues to boot. Theology, Mr. Fortune found, is a more accommodating subject than mathematics; its technique of exposition allows greater latitude. For instance when you are gravelled for matter there is always the moral to fall back upon. Comparisons too may be drawn, leading cases cited, types and antetypes analysed and anecdotes introduced. Except for Archimedes mathematics is singularly naked of anecdotes. Not that he thought any the worse of it for this. On the contrary he compared its austere and integral beauty to theology decked out in her fiaunting charms and wielding all her bribes and spiritual bonuses; and like Dante at the rebuke of Beatrice he blushed that he should ever have followed. aught but the noblest. No, there was nothing lacking in mathematics. The deficiency was in him. He added line to line, precept to pre cept; he exhausted himself and his pupil by hours of demonstration and exposition; leagues of sand were scarred, and smoothed again by the tide, and scarred afresh: never an answering spark rewarded him. He might as well have made the sands into a rope-walk. Sometimes he thought that he was taxing LueH too heavily, and desisted. But if he desisted for pity's sake, pity soon drove him to work again. for if it were bad to see Lueli sighing over the properties of parallel lines, it was worse to see him moping and pining for his god. Teioa's words, uttered so matter-of-factly, haunted his mind. "I expect he will die soon." Mr. Fortune was thinking so too. Lueli grew steadily mOre lacklustre, his eyes were dull, his voice was fiat; he appeared to be retreating behind a film that thickened and toughened and would soon obliterate him. "If only, if only I could teach him to enjoy an abstract notion! If he could once grasp how it all hangs together, and is everlasting and harmonious, he would be saved. Nothing else can save him, nothing that I or his fellows can offer him. For it must be new to excite him and it must be true to hold him, and what else is there that is both new and true?" There were women, of course. a race of beings neither new nor true, yet much vaunted by some as a cure for melancholy and a tether for the M
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soul. Mr. Fortune would have cheerfully procured a damsel (not that they were likely to need much of that), dressed her hair, hung the whistle and the Parnell medal round her neck 1 dowered her with the nineteen counters and the tape measure and settled her in Lueli's bed if he had supposed that this would avail. But he feared that Lueli was past the comfort of women, and in any case that sort of thing is best arranged by the parties concerned. So he resorted to geometry again, and once more Lueli was hurling himself with frantic docility through the paper hoops. It was really rather astonishing, how dense he could be! Once out of twenty, perhaps, he would make the right answer. Mr. Fortune, too anxious to be lightly elated, would probe a little into his reasons for making it. Either they were the wrong reasons or he had no reasons at all. Mr. Fortune was often horribly tempted to let a mistake pass. He was not impatient: he was far more patient than in the palmiest days of theology-but he found it almost unendurable to be for ever saying with various inflexions of kindness: "No, Lueli. Try again," or: "Well, no, not exactly," or: "I fear you have not quite understood," or: "Let me try to make that clearer." He withstood the temptation. His easy acceptance (though in good faith) of a sham had brought them to this pass, and tenderness over a false currency was not likely to help them out of it. No, he would not be caught that way twice. Similarly he pruned and repressed LueH's talent for leaking away down side-issues, though this was hard too, for it involved snubbing him almost every time he spoke on his own initiative. Just as he had been so mistaken about the nature of points, confounding them with holes and agitating himself at the prospect of a beach pitted all over, Lueli contrived to apply the same sort of well·meaning misconceptions to every stage of his progress-if progress be the word to apply to one who is hauled along in a state of semiconsciousness by the scruff of his neck. When the points seemed to be tolerably well-established in his mind Mr. Fortune led him on to lines, and by joining up points he illus-trated such simple figures as the square, the triangle and the parallelogram. Lueli perked up, seemed interested, borrowed the stick and began joining up points too. At first he copied Mr. Fortune, glancing up after each stroke to see if it had been properly directed. Then growing rather more confident. and pleased-as who is not?-with the act of drawing on sand, he launched out into a more complicated design. "This is a man," he said. Mr. Fortune was compelled to reply coldly: "A man is not a geometrical figure." At length Mr. Fortune decided that he had better take in sail. Pure mathematics were obviously beyond Lue1i; perhaps applied mathematics
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Syl'tlio. Townsend Warner
would work better. Mr. Fortune, as it happened, bad never applied any, but he knew that other people did so, and though he considered it a rather lower line of business he was prepared to try it. "If 1 were to ask you to find out the height of that tree, how would you set about it?" Lueli replied with disconcerting readiness: "1 should climb up to the top and let down a string." "But suppose you couldn't climb up it?" "Then I should cut it down." "That would be very wasteful: and the other might be dangerous. I can show you a better plan than either of those." The first thing was to select a tree, an upright tree, because in all elementary demonstrations it is best to keep things as clear as possible. He would never have credited the rarity of upright trees had he not been pressed to :find one. Coco-palms, of course, were hopeless: they all had a curve or a list. At length he remembered a tree near the b~thing-pool, a perfect specimen of everything a tree should be, tall, straight as a die, growing by itself; set apart, as it were, for purposes of demonstration. He marched Lueli thither, and when he saw him rambling towards the pool he recalled him with a cough. "Now I will show you how to discover the height of that tree. Attend. You will find it very interesting. The first thing to do is to lie down." Mr. Fortune lay down on his back and Lueli followed his example. Many people find that they can think more clearly in a recumbent position. Mr. Fortune found it so too. No sooner was he on his back than he remembered that he had no measuring stick. But the sun was delicious and the grass soft; he might well spare a few minutes in exposing the theory. "It is all a question of measurements. Now my height is six foot two inches, but for the sake of argument we will assume it to be six foot exactly. The distance from my eye to. the base of the tree is so far an unknown quantity. My six feet however are already known to you." Now Lueli had sat up. and was looking him up and down with an intense and curious scrutiny, as though he were something utterly unfamiliar. This was confusing, it made him lose the thread of his explanation. He felt a little uncertain as to how it should proceed. Long ago on dark January momings t when a septic thumb (bestowed on him by a cat which he had rescued from a fierce poodle) obliged him to stay away from the Bank, he had observed young men with woollen comforters and raw-looking wind-bitten hands practising surveying under the snarling elms and whimpering poplars of Finsbury Park. They had tapes and tripods, and the girls in charge of perambulators dawdled on asphalt paths to watch their proceedings. It was odd how vividly frag-
the
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ments of his old life had been coming back to him during these last few months. He resumed: "In order to ascertain the height of the tree I must be in such a position that the top of the tree is exactly in a line with the top of a measuringstick---or any straight object would do, such as an umbrella-which I shall secure in an upright position between my feet. Knowing then that the ratio that the height of the tree bears to the length of the measuringstick must equal the ratio that the distance from my eye to the base of the tree bears to my height, and knowing (or being able to find out) my height, the length of the measuring stick and the distance from my eye to the base of the tree, I can, therefore, calculate the height of the tree." "Wbat is an umbrella?" Again the past flowed back, insurgent and actual. He was at the Oval, and out of an overcharged sky it had begun to rain again. In a moment the insignificant tapestry of lightish faces was exchanged for a noble pattern of domes, blackish, blueish and greenish domes, sprouting like a crop of miraculous and religious mushrooms. The rain fell harder and harder, presently the little white figures were gone from the field and, as with an abnegation of humanity, the green plain, so much smaller for their departure, lay empty and forsaken, ringed round with tier upon tier of blackly glistening umbrellas. He longed to describe it all to Lueli, it seemed to him at the moment that he could talk with the tongues of angels about umbrellas. But this was a lesson in mathematics: applied mathematics, moreover, a compromise, so that all further compromises must be sternly nipped. Unbending to no red herrings he replied: «An umbrella, Lueli, when in use resembles the-the shell that would be formed by rotating an arc of curve about its axis of symmetry, attached to a cylinder of small radius whose axis is the same as the axis of symmetry of the generating curve of the shell. When not in use it is properly an elongated cone, but it is more usually helicoidal in form." Lueli made no answer. He lay down again, this time face downward. Mr. Fortune continued: "An umbrella, however, is not essential. A stick will do just as well, so find me one, and we will go on to the actual meas'QIement." Lueli was very slow in finding a stick. He looked for it rather languidly and stupidly, but Mr. Fortune tried to hope that this was because bis mind was engaged on what he had. just learnt. Holding the stick between his feet, Mr. Fortune wriggled about on his back trying to get into the proper position. He knew he was making a fool of himself. The young men in Finsbury Park had never wriggled about on their backs. Obviously there must be some more dignified way of
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getting the top of the stick in line with the top of the tree and his eye, but just then it was not obvious to him. Lueli made it worse by standing about and looking miserably on. \Vhen he had placed himself properly he remembered that he had not measured the stick. It measured (he had had the forethought to bring the tape with him) three foot seven, very tiresome: those odd inches would only serve to make it seem harder to his pupil. So he broke it again, drove it into the ground, and wriggled on his stomach till his eye was in the right place, which was a slight improvement in method, at any rate. He then handed the tape to Lueli, and lay strictly motionless, admonishing and directing while LueH did the measuring of the ground. In the interests of accuracy he did it thrice, each time with a different result. A few minutes before noon the height of the tree was discovered to be fifty-seven foot, nine inches. Mr. Fortune now had leisure for compassion. He thought LueH was looking hot and fagged, so he said: "'Vhy don't you have a bathe? It will freshen you up." Lueli raised his head and looked at hini with a long dubious look, as though he bad heard the words but without understanding what they meant. Then he turned his eyes to the tree and looked at that. A sort of shadowy wrinkle, like the blurring on the surface of milk before it boils, crossed his face. "Don't worry any more about that tree. If you hate all this so much we won't do any more of it, I will never speak of geometry again. Put it all out of your head and go and bathe."
COMMENTARY ON
Statistics as a Literary Stimulus OU would not, perhaps, think statistics a subject likely to inspire the literary imagination; yet offhand I can recall at least half a dozen fables based on the theory of probability, among them Clerk Maxwell's celebrated conjecture about the demon who could r~verse the second law of thermodynamics (making heat flow the wrong way) and one or two remarkable anecdotes by Augustus de Morgan. Charles Dickens issued an interesting tribute to theoretical statistics by refusing, one day late in December, to travel by train, on the ground that the average annual quota of railroad accidents in Britain had not been filled and therefore further disasters were obviously imminent. All of us, I suppose, are a little afraid of statistics. Like Atropos, the sister who cut the thread, they are inexorable; like her too, they are not only impersonal but terribly personal. One dreams of :flouting them. A modem Prometheus would not waste his time showing up the gods by stuffing a sacrificial bull with bones; he would :flaunt his artfulness and independence by juggling the law of large numbers. Neither heaven nor earth could be straightened out thereafter. That bit of mischief is essentially what the next two fables are about. Inflexible Logic by Russell Maloney is a widely known and admired story built around a famous statistical whimsy. Eddington gave currency to it in one of his lectures but I am far from certain that he made it up. Maloney was a writer of short stories, sketches, profiles, anecdotes, many of which appeared in The New Yorker magazine between the years 1934 and 1950. He conducted for several years the magazine's popular department, "Talk of the Town," and claimed to have written for it "something like 2600 perfect anecdotes." He died in New York, September 5, 1948, at the age of thirty-eight. The Law is a fascinating, and, in a way, terrifying story of a sudden, mysterious failure of the "law of averages." To be sure there is no such law, but when it fails the consequences are much worse than if death had taken a holiday. If men cannot be depended on to behave like a herd or like the molecules of a gas the entire social order falls to ruin. Robert Coates, the author of this and other equally entrancing tales, is a writer and art critic. He too contributes frequently to The N ew Yorker. His books include Wisteria Cottage (1948), The Bitter Season (1946), The Outlaw Years (1930), The Eater of Darkness (1929).
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How often might a man, after he had Jumbled a set of letters in a bag, fling them out upon the ground before they would fall into an exact poem, yea, or so much as make a good discourse in prose. And may not a little book be as easily made by chance as this great volume of the world. -ARCHBISHOP TILLOTSON
4
Inflexible Logic By RUSSELL MALONEY
WHEN the six chimpanzees came into his life, Mr. Bainbridge was thirtyeight years old. He was a bachelor and lived comfortably in a remote part of Connecticut, in a large old house with a carriage drive, a conservatory, a tennis court, and a well-selected library. His income was derived from impeccably situated real estate in New York City, and he spent it soberly, in a manner which could give offence to nobody_ Once a year, late in April, his tennis court was resurfaced, and after that anybody in the neighborhood was welcome to use it; his monthly statement from Brentano's seldom ran below seventy-five dollars; every third year, in November, he turned in his old Cadillac coupe for a new one; he ordered his cigars, which were mild and rather moderately priced, in shipments of one thousand, from a tobacconist in Havana; because of the international situation he had cancelled arrangements to travel abroad, and after due thought had decided to spend his travelling allowance on wines, which seemed likely to get scarcer and more expensive if the war lasted. On the whole, Mr. Bainbridge's life was deliberately, and not too unsuccessfully, modelled after that of an English country gentleman of the late eighteenth century, a gentleman interested in the arts and in the expansion of science, and so sure of himself that he didn't care if some people thought him eccentric. Mr. Bainbridge had many friends in New York, and he spent several days of the month in the city, staying at his club and looking around. Sometimes be called up a girl and took her out to a theatre and a night club. Sometimes be and a couple of classmates got a little tight and went to a prizefight. Mr. Bainbridge also looked in now and then at some of the conservative art galleries, and liked occasionally to go to a concert. And he liked cocktail parties, too, because of the :fine footling conversation and the extraordinary number of pretty girls who had nothing else to do with the rest of their evening. It was at a New York cocktail party, however. that Mr. Bainbridge kept his preliminary appointment with doom. At one of the parties given by Hobie Packard, the stockbroker, he learned about the theory of the six chimpanzees. 2262
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2263
It was almost six-forty. The people who had intended to have one drink
and go had already gone, and the people who intended to stay were fortifying themselves with slightly dried canapes and talking animatedly. A group of stage and radio people had coagulated in one corner, near Packard's Capehart, and were wrangling about various methods of cheating the Collector of Internal Revenue. In another corner was a group of stockbrokers, talking about the greatest stockbroker of them all, Gauguin. Little Marcia Lupton was sitting with a young man, saying earnestly, "Do you really want to know what my greatest ambition is? I want to be myself," and Mr. Bainbridge smiled gently, thinking of the time Marcia had said that to him. Then he heard the voice of Bernard Weiss, the critic, saying, "Of course he wrote one good novel. It's not surprising. Mte,r all, we know that if six chimpanzees were set to work pounding six typewriters at random, they would, in a million years, write all the books in the British Museum." Mr. Bainbridge drifted over to Weiss and was introduced to Weiss's companion, a Mr. Noble. "What's this about a million chimpanzees, Weiss?" he asked. "Six chimpanzees," Mr. Weiss said. "It's an old cliche of the mathematicians. I thought everybody was told about it in schooL Law of averages, you know, or maybe it's permutation and combination. The six chimps, just pounding away at the typewriter keys, would be bound to copy out all the books ever written by man. There are only so many possible combinations of letters and numerals, and they'd produce all of them-see? Of course they'd also tum out a mountain of gibberish, but they'd work the books in, too. All the books in the British Museum/' Mr. Bainbridge was delighted; this was the sort of talk he liked to hear when he came to New York. "Well, but 100k here," he said, just to keep up his part in the foolish conversation, "what if one of the chimpanzees finally did duplicate a book, right down to the last period, but left that off? Would that count?" "I suppose not. Probably the chimpanzee would get around to doing the book again, and put the period in." "What nonsense!" Mr. Noble cried. "It may be nonsense, but Sir James Jeans believes it," Mr. Weiss said, huffily. "Jeans or Lancelot Hogben. I know I ran across it quite recently." Mr. Bainbridge was impressed. He read quite a bit of popular science, and both Jeans and Hogben were in his library. "Is that so?" he murmured, no longer feeling frivolous. "Wonder if it has ever actually been tried? I mean, has anybody ever put six chimpanzees in a room with six typewriters and a lot of paper?" Mr. Weiss glanced at Mr. Bainbridge's empty cocktail glass and said drily, "Probably not."
.2264
Russell Malon',
Nine weeks later, on a winter evening, Mr. Bainbridge was sitting in his study with his friend James Mallard, an assistant professor of mathematics at New Haven. He was plainly nervous as be poured himself a drink and said, "Mallard, I've asked you to come here-Brandy? Cigar?-for a particular reason. You remember that I wrote you some time ago, asking your opinion of . . . of a certain mathematical hypothesis or supposition." "Yes," Professor Mallard said, briskly. "I remember perfectly. About the six chimpanzees and the British Museum. And I told you it was a perfectly sound popularization of a principle known to every schoolboy who had studied the science of probabilities." "Precisely," Mr. Bainbridge said. "Well, Mallard, I made up my mind . . . . It was not difficult for me, because I have, in spite of that fellow in the White House, been able to give something every year to the Museum of Natural History, and they were naturally glad to oblige me. . . . And after all, the only contribution a layman can make to the progress of science is to assist with the drudgery of experiment. . . . In short, I-If "I suppose you're trying to tell me that you have procured six chimpanzees and set them to work at typewriters in order to see whether they will eventually write all the books in the British Museum. Is that it?" "Yes, that's it," Mr. Bainbridge said. "What a mind you have, Mallard. Six fine young males, in perfect condition. I had a-I suppose you'd call it a dormitory-built out in back of the stable. The typewriters are in the conservatory. It's light and airy in there, and I moved most of the plants out. Mr. North, the man who owns the circus, very obligingly let me engage one of his best animal men. Really, it was no trouble at all." Professor Mallard smiled indulgently. "After all, such a thing is not unheard of," he said. "I seem to remember that a man at some university put his graduate students to work flipping coins, to see if heads and tails came up an equal number of times. Of course they did." Mr. Bainbridge looked at his friend very queerly. "Then you believe that any such principle of the science of probabilities will stand up under an actual test?" "Certainly. " "You had better see for yourself." Mr. Bainbridge led Professor Mallard downstairs, along a corridor, through a disused music room, and into a large conservatory. The middle of the floor had been cleared of plants and was occupied by a row of six. typewriter tables, each one supporting a hooded machine. At the left of each typewriter was a neat stack of yellow copy paper. Empty wastebaskets were under each table. The chairs were the unpadded, spring-backed kind favored by experienced stenographers. A large bunch of ripe bananas was hanging in one comer, and in another stood a Great Bear water-cooler and a rack of Lily cups. Six piles of
Inflexible Logic
2265
typescript, each about a foot high, were ranged along the wall on an improvised shelf. Mr. Bainbridge picked up one of the piles. which he could just conveniently lift, and set it on a table before Professor Mallard. "The output to date of Chimpanzee A, known as Bill," he said simply. " , "Oliver Twist," by Charles Dickens,' " Professor Mallard read out. Re read the first and second pages of the manuscript, then feverishly leafed through to the end. "You mean to tell me," he said, "that this chimpanzee has written-" "Word for word and comma for comma," said Mr. Bainbridge. "Young, my butler, and I took turns comparing it with the edition I own. Having finished 'Oliver Twist,' Bil1 is, as you see, starting the sociological works of Vilfredo ParetQ, in Italian. At the rate he has been going, it should keep him busy for the rest of the month." "And all the chimpanzees"-Professor Mallard was pale, and enunciated with '~ifficulty-"they aren't a11-" "Oh, yes, all writing books which I have every reason to believe are in the British Museum. The prose of John Donne, some Anatole France. Conan Doyle, Galen, the conected plays of Somerset Maugham, !vJarcel Proust, the memoirs of the late Ivfarie of Rumania, and a monograph by a Dr. Wiley on the marsh grasses of Maine and Massachusetts. I can sum it up for you, Mallard, by telling you that since I started this experiment, four weeks and some days ago, none of the chimpanzees has spoiled a single sheet of paper." Professor Manard straightened up, passed his handkerchief across his brow, and took a deep breath. '"I apologize for my weakness," he said. "It was simply the sudden shock. No, looking at the thing scientificallyand I hope I am at least as capable of that as the next man-there is nothing marvellous about the situation. These chimpanzees, or a succession of similar teams of chimpanzees, would in a million years write all the books in the British Museum. I told you some time ago that I believed that statement. Why should my belief be altered by the fact that they produced some of the books at the very outset? After all. I should not be very much surprised if I tossed a coin a hundred times and it came up heads every time. I know that if I kept at it long enough. the ratio would reduce itself to an exact fifty per cent. Rest assured, these chimpanzees will begin to compose gibberish quite soon. It is bound to happen. Science tells us so. Meanwhile, I advise you to keep this experiment secret. Uninfonned people might create a sensation if they knew." "I will, indeed," ~fr. Bainbridge said. "And I'm very grateful for your rational analysis. It reassures me. And now, before you go, you must hear the new Schnabel records that arrived today." During the succeeding three months, Professor ,Mallard got into the habit of telephoning Mr. Bainbridge every Friday afternoon at five-thirty,
Russell Maloney
immediately after leaving his seminar room. The Professor would say, HWell?," and Mr. Bainbridge would reply, "They're still at it, Mallard. Haventt spoiled a sheet of paper yet." If Mr. Bainbridge had to go out on Friday afternoon, he would leave a written message with his butler, who would read it to Professor Mallard: "Mr. Bainbridge says we now have Trevelyan's 'Life of Macaulay,' the Confessions of St. Augustine, 'Vanity Fair/ part of Irving's 'Life of George Washington,' the Book of the Dead, and some speeches delivered in Parliament in opposition to the Com Laws. sir." Professor Mallard would reply, with a hint of a snarl in his voice, "Tell him to remember what I predicted," and hang up with
a clash. The eleventh Friday that Professor Mallard telephoned, Mr. Bainbridge said, "No change. I have had to store the bulk of the manuscript in the cellar. I would have burned it, except that it probably has some scientific value." "How dare you talk of scientific value?" The voice from New Haven roared faintly in the receiver. "Scientific value! You-you--chimpanzee!U There were further inarticulate sputterings, and Mr. Bainbridge hung up with a disturbed expression. "I am afraid Mallard is overtaxing himself," he munnured. Next day, however, he was pleasantly surprised. He was leafing through a manuscript that had been completed the previous day by Chimpanzee D, Corky. It was the complete diary of Samuel Pepys, and Mr. Bainbridge was chuckling over the naughty passages, which were omitted in his own edition, when Professor Mallard was shown into the room. "I have come to apologize for my outrageous conduct on the telephone yesterday," the Professor said. "Please don't think of it any more. I know you have many things on your mind," Mr. Bainbridge said. "Would you like a drink?" "A large whiskey, straight, please," Professor Mallard said. "I got rather cold driving down. No change, I presume?" "No, none. Chimpanzee F, Dinty, is just finishing John Florio's translation of Montaigne's essays, but there is no other news of interest." Professor Mallard squared his shoulders and tossed off his drink in one astonishing gUlp. "I should like to see them at work," he said. ''Would I disturb them, do you think?" "Not at all. As a matter of fact, I usually look in on them around this time of day. Dinty may have finished his Montaigne by now, and it is always interesting to see them start a new work. I would have thought that they would continue on the same sheet of paper, but they don't, you know. Always a fresh sheet, and the title in capitals." Professor Mallard, yvithout apology, poured another drink and slugged it down. "Lead on," he said.
Infi4xible Legic
2267
It was dusk in the conservatory, and the chimpanzees were typing by the light of student lamps clamped to their desks. The keeper lounged in a corner, eating a banana and reading Billboard. "You might as well take an hour or so off," Mr. Bainbridge said. The man left. Professor Mallard, who had not taken off his overcoat, stood with his hands in his pockets, looking at the busy chimpanzees. "1 wonder if you know, Bainbridge, that the science of probabilities takes everything into account," he said, in a queer, tight voice. "It is certainly almost beyond the bounds of credibility that these chimpanzees should write books without a single error, but that abnormality may be corrected by-these!" He took his hands from his pockets, and each one held a .38 revolver. "Stand back out of harm's way!" he shouted. "Mallard! Stop itt" The revolvers barked, first the right hand. then the left, then the right. Two chimpanzees fell, and a third reeled into a corner. Mr. Bainbridge seized his friend's arm and wrested one of the weapons from him. "Now I am armed, too, Mallard, and I advise you to stop 1" he cried. Professor Mallard's answer was to draw a bead on Chimpanzee E and shoot him dead. Mr. Bainbridge made a rush, and Professor Mallard fired at him. Mr. Bainbridge, in his quick death agony, tightened his finger on the trigger of his revolver. It went off, and Professor Mallard went down. On his hands and knees he fired at the two chimpanzees which were still unhurt, and then collapsed. There Was nobody to hear his last words. "The human equation . . . always the enemy of science . . ." he panted. "This time . . . vice versa . . . I, a mere mortal . . . savior of science . . . deserve a Nobel . . ." When the old butler carne running into the conservatory to investigate the noises, his eyes were met by a truly appalling sight. The student lamps were shattered, but a newly risen moon shone in through the conservatory windows on the corpses of the two gentlemen, each clutching a smoking revolver. Five of the chimpanze,es were dead. The sixth was Chimpanzee F. His right arm disabled, obviously bleeding to death, he was slumped before his typewriter. Painfully, with his left hand, he took from the machine the completed last page of Florio's Montaigne. Groping for a fresh sheet, he inserted it, and typed with one finger, "UNCLE TOM'S CABIN, by Harriet Beecher Stowe. Chapte . . ." Then he, too, was dead.
Chaos umpire sits And by decision more embroils the fray Bi' which he reigns: next him. high arbiter Chance governs all.
-MILTON
Lo! th;r' dread empire, Chaos! is restord.
"II the law
SIIPPOSi!S
--ALEXANDER POPE
'..J' t" that," said Mr. Bum ble,... "h t e l aw'IS a ass--a '",10 • -DICKENS (Oliver Twist)
Stand not upon the order 01 )'our going, SHAKESPEARE
But go at once.
5
(Macbeth)
The Law By ROBERT M. COATES
THE first intimation that things were getting out of hand came one earlyfall evening in the late nineteen-forties. What happened, simply, was that between seven and nine o'clock on that evening the Triborough Bridge had the heaviest concentration of outbound traffic in its entire history. This was odd, for it was a weekday evening (to be precise, a Wednesday), and though the weather was agreeably mild and clear, with a moon that was close enough to being full to lure a certain number of motorists out of the city, these facts alone were not enough to explain the phenomenon. No other bridge or main highway was affected, and though the two preceding nights had been equally balmy and moonlit, on both of these the bridge traffic had run close to normal. The bridge personnel, at any rate, was caught entirely unprepared. A main artery of traffic, like the Triborough, operates under fairly predictable conditions. Motor travel, like most ot1)er large·scale human activities. obeys the Law of Averages-that great, ancient rule that states that the actions of people in the mass will always follow consistent patterns-and on the basis of past experience it had always been possible to foretell, almost to the last digit, the number of cars that would cross the bridge at any given hour of the day or night. In this case, though, all rules were broken. The hours from seven till nearly midnight are normally quiet ones on the bridge. But on that night it was as if all the motorists in the city, or at any rate a staggering proportion of them, had conspired together to upset tradition. Beginning almost exactly at seven o'clock, cars poured onro the bridge in ')uch numbers and with such rapidity that the staff at the t\111 booths \\'as overwhelmed almost from the start. It was soon 2::68
The Law
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apparent that this was no momentary congestion, and as it became more and more obvious that the traffic jam promised to be one of truly monumental proportions, added details of police were rushed to the scene to help handle it. Cars streamed in from all directions-from the Bronx approach and the Manhattan one, from 125th Street and the East River Drive. (At the peak of the crush, about eight-fifteen, observers on the bridge reported that the drive was a solid line of car headlights as far south as the bend at Eighty-ninth Street, while the congestion crosstown in Manhattan disrupted traffic as far west as Amsterdam Avenue.) And perhaps the most confusing thing about the whole manifestation was that there seemed to be no reason for it. Now and then, as the harried toll-booth attendants made change for the seemingly endless stream of cars, they would question the occupantst and it soon became clear that the very participants in the monstrous tieup were as ignorant of its cause as anyone else was. A report made by Sergeant Alfonse O'Toole, who commanded the detail in charge of the Bronx approach, is typical. "I kept askin' them," he said, n'Is there night football somewhere that we don't know about? Is it the races you're goin~ to?' But the funny thing was half the time they'd be askin' me. 'What's the crowd for, Mac?' they would say. And I'd just look at them. There was one guy I mind, in a Ford convertib1e with a girl in the seat beside him, and when he asked me, I said to him, 'Hell, you're in the crowd. ain't you?' I said. 'What brings you here?' And the dummy just looked at me. 'Me?' he says. 'I just come out for a drive in the moonlight. But if I'd known there'd be a crowd like this .. .' he says. And then he asks me, 'Is there any place I can turn around and get out of this?'" As the Herald Tribune summed things up in its story next morning, it ~~just looked as if everybody in Manhattan who owned a motorcar had decided to drive out on Long Island that evening." The incident was unusual enough to make all the front pages next morning, and because of this, many similar events, which might otherwise have gone unnoticed. received attention. The proprietor of the Aramis Theatre, on Eighth Avenue, reported that on several nights in the recent past his auditorium had been practically empty, while on others it had been jammed to suffocation. Luncheon owners noted that increasingly their patrons were developing a habit of making runs on specific items; one day it would be the roast shoulder of veal with pan gravy that was ordered almost exclusively, while the next everyone would be taking the Vienna loaf, and the roast veal went begging. A man who ran a small notions store in Bayside revealed that over a period of four days two hundred and seventy-four successive customers had entered bis shop and asked for a spool of pink thread.
2270
Robert M. Coates
These were news items that would ordinarily have gone into the papers as fillers or in the sections reserved for oddities. Now, however, they seemed to have a more serious significance. It was apparent at last that something decidedly strange was happening to people's habits, and it was as unsettling as those occasional moments on excursion boats when the passengers are moved, all at once, to rush to one side or the other of the vessel. It was not till one day in December when, almost incredibly, the Twentieth Century Limited left New York for Chicago with just three passengers aboard that business leaders discovered how disastrous the new trend could be, too. Until then, the New York Central, for instance, could operate confidently on the assumption that although there might be several thousand men in New York who had business relations in Chicago, on any single day no more-and no less-than some hundreds of them would have occasion to go there. The play producer could be sure that his patronage would sort itself out and that roughly as many persons would want to see the performance on Thursday as there had been on Tuesday or Wednesday. Now they couldn't be sure of anything. The Law of Averages had gone by the board, and if the effect on business promised to be catastrophic, it was also singularly unnerving for the general customer. The lady starting downtown for a day of shopping, for example, could never be sure whether she would find Macy's department store a seething mob of other shoppers or a wilderness of empty, echoing aisles and unoccupied salesgirls. And the uncertainty produced a strange sort of jitteriness in the individual when faced with any impulse to action. "Sball we do it or shan't we?" people kept asking themselves, knowing that if they did it, it might tum out that thousands of other individuals had decided similarly; knowing, too, that if they didn't, they might miss the one glorious chance of all chances to have Jones Beach, say, practically to themselves. Business languished. and a sort of desperate uncertainty rode everyone. At this juncture, it was inevitable that Congress should be called on for action. In fact, Congress called on itself, and it must be said that it rose nobly to the occasion. A committee was appointed, drawn from both Houses and headed by Senator J. Wing Slooper (R.), of Indiana, and though after considerable investigation the committee was forced reluctantly to conclude that there was no evidence of Communist instigation, the unconscious subversiveness of the people's present conduct was obvious at a glance. The problem was what to do about it. You can't indict a whole nation, particularly on such vague grounds as these were. But, as Senator Slooper boldly pointed out, "You can control it," and in the end a system of reeducation and reform was decided upon, designed to lead people
The Uzw
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back to--again we quote Senator Slooper-"the basic regularities, the homely averageness of the American way of life." In the course of the committee's investigations, it had been discovered, to everyone's dismay, that the Law of Averages had never been incorporated into the body of federal jurisprudence, and though the upholders of States' Rights rebelled violently, the oversight was at once corrected, both by Constitutional amendment and by a law-the Hills-Slooper Act -implementing it. According to the Act, people were required to be average, and, as the simplest way of assuring it, they were divided alphabetically and their permissible activities catalogued accordingly. Thus, by the plan, a person whose name began with "G," "N," or "U," for examplet could attend the theatre only on Tuesdays, and he could go to baseball games only on Thursdays, whereas his visits to a haberdashery were confined to the hours between ten o'clock and noon on Mondays. The law, of course, had its disadvantages. It had a crippling effect on theatre parties, among other social functions, and the cost of enforcing it was unbelievably heavy. In the end, too, so many amendments had to be added to it-such as the one permitting gentlemen to take their fiancees (if accredited) along with them to various events and functions no matter what letter the said fiancees' names began with-that the courts were frequently at a loss to interpret it when confronted with violations. In its way, though, the law did serve its purpose, for it did inducerather mechanically, it is true, but still adequately-a return to that average existence that Senator Slooper desired. All, indeed, would have been well if a year or so later disquieting reports had not begun to seep in from the backwoods. It seemed that there, in what bad hitherto been considered to be marginal areas, a strange wave of prosperity was making itself felt. Tennessee mountaineers were buying Packard convertibles, and Sears, Roebuck reported that in the Ozarks their sales of lUXUry items had gone up nine hundred per cent. In the scrub sections of Vermont, men who formerly had barely been able to scratch a living from their rock-strewn acres were now sending their daughters to Europe and ordering expensive cigars from New York. It appeared that the Law of Diminishing Returns was going haywire, too.
PART XXIV
Mathematics and Music 1. Mathematics of Music by
SIR JAMES JEANS
COMMENTARY ON
SIR JAMES JEANS IR JAMES .:'EANS was a mathematical physicist whose writi~gs were much admired by Tallulah Bankhead. I do not know that MISS Bankhead ~as especially moved by Jeans' contributions to the theory of gases or to the study of the equilibrium of rotating fluid masses, but it is recorded that she described the best known of his works, The Mysterious Universe, as a book every girl should read.! Jeans had a productive and varied career which divides more or less into two periods. He was born in 1877 in Ormskirk, Lancashire, to parents in comfortable circumstances. His father was a journalist attached to the press gallery of the House of Commons but with interests a good deal broader than the chicaneries and trivia of daily politics. He published two popular books on science,2 which reflected not only his admiration for scientific knowledge but his conviction that students of the subject have a duty to follow the example of men like Tyndall, Huxley and Clifford in kindling "a love of science among the masses. n 3 His outlook, a strange compound of strict Victorian religious orthodoxy and free thinking, must have been confusing to his son. Jeans was a precocious child, inc1ined to melancholy. He amused himself by memorizing seven-place logarithms and by dissecting and studying the mechanism of clocks; he Was also trained at an early age to read the first leader of the Times each morning to his parents. One of his biographers; J. O. Crowther, remarks that "there are cases of the balance of infants' minds having been disturbed by this practice." 4 At nineteen Jeans entered Trinity College, Cambridge, where he read mathematics and soon gave evidence of exceptional powers. One envies him his undergraduate days in a college whose faculty included 1. W. L. Glaisher, W. W. Rouse Ball, Alfred North Whitehead and Edmund T. Whittaker. He finished second in the stiff competitive examination known as the tripos, two places above his classmate G. H. HardY, who later became the foremost British mathematician of his generation. Jeans first applied his imagination and formidable mathematical technique to problems concerned with the distribution of energy among the molecules in a gas. Clerk Maxwell and Ludwig Boltzmann had invented theories which treated gas molecules as if they were tiny billiard balls-
S
J. G. Crowther, British Scientists of the Twentieth Century, London, 1952, p. 95. The Creators of the Age oj Steel (1884); Lives oj the Electricians (1887). 3 Crowther, op. cit., p. 96. 4 Crowther, op. cit., p. 97. 1
2
2274
SII' lam~$ Jeans
2275
"rigid and geometrically perfect spheres." While these theories fitted the observed facts fairly well, they produced serious dilemmas connected with the law of the conservation of energy and the second law of thermodynamics. In Jeans' first treatise, Dyn~mical Theory 0/ Gases (1904), he refined the older theories and overcame some of the principal difficulties to which they gave rise. What is generally regarded as Jeans' masterwork, his Problems 0/ Cosmogony and Stellar Dynamics (1917), also has to do with the behavior of gases. This book discusses the cosmogonic problems involving "incompressible masses acted on by their own gravitation." 5 What happens to a mass of liquid "spinning about an axis and isolated in space"? What are the changes of form assumed by masses of gas (like the sun) under rotation, and as the masses shrink? What is the bearing of these matters on the evolution of planets and of our solar system? Questions of this kind had attracted the greatest scientists from Newton and Laplace to Poincare and Sir George Darwin. Jeans' treatise, while claiming no finality for its conclusions, is acknowledged to be a landmark in the history of astronomy, a contribution to the solution of the underlying mathematical problems which must be ranked a "permanent achievement, come what may in the future development of cosmogony." 6 Some men turn to religion as they grow old; Jeans turned to a mixture of religion and popular science. At the age of fifty-two, when he was knighted, he could look back upon his research and teaching career with unmixed satisfaction. He had been elected to a fellowship in Trinity in 1901, had served (1905-1910) as professor of applied mathematics at Princeton, as Stokes Lecturer at Cambridge, and as one of the joint secretaries of the Royal Society (to which he had been elected when he was only twenty-eight) from 1919 to 1929. Marriage to a wealthy American girl-it was a very happy union-had given him financial independence; Jeans was not compelled to assume teaching duties Which might interfere with his research. By 1928 he had published seven books and seventy-six original papers and had earned a reputation as an outstanding mathematician. At this point, perhaps because he felt, as Milne reports, that his powers as a mathematician were declining, he abandoned pure science for popularization. 7 The conversion was a tremendous success. His first book in a nontechnical vein was The Universe Around Us (1929); it was followed by The Mysterious Universe (1930) and by haIf a dozen similar volumes, the last appearing in 1947. Jeans' style was "transparent, trenchant, and dignified. His scientific narrative flowed like a grand river under complete E. A. Milne, Sir James Jeans, Cambridge, 1952, p. 110. E. A. Milne, op. cit., p. 114 passim. 7 Milne, op. cit., p. 73. Milne gives this on the authority of Jeans' second wife, whom he married after the death of the first Lady Jeans in 1934. S
S
2276
Editor's Comment
control." 8 His images were vivid, often breathtaking. Their object was to make the reader goggle at immensities and smallnesses, excellent items for conversational gambits. Because he was primarily a mathematician, Jeans expatiated less on physical ideas than on startling numerical contrasts. The scale of matter, he noted, ranges "from electrons of a fraction of a millionth of a millionth of an inch in diameter, to nebulae whose diameters are measured in hundreds of thousands of millions of miles"; a model of the universe in which the sun was represented "by a speck of dust 1/3400 of an inch in diameter" would have to extend four million , miles in every direction to encompass a few of our island-universe neighbors; "Empty Waterloo Station of everything except six specks of dust, and it is still far more crowded with dust than space is with stars"; the molecules in a pint of water "placed end to end . . . would form a chain capable of encircling the earth over 200 million times"; the energy in a thimble of water would drive a large vessel back and forth across the ocean twenty times; a pinhead heated to a temperature equal to that at the center of the sun would "emit enough heat to kill anyone who ventured within a thousand miles of it"; the sun loses daily by radiation 360,000 million tons of its weight, the earth only ninety pounds. Most of us were brought up on these images and have never shaken off their emotional impact, though it is true that the facts of the atomic age make some of Jeans' best flesh-creepers sound commonplace. The public took Jeans' excitements to its heart, but scientists and philosophers were able to restrain their enthusiasm. As, more and more, a religious, emotional and mystical note crept into his books, they came under sharp attack. In her biting Philosophy and the Physicists, Susan Stebbing let fly at both Jeans and Eddington for their philosophical interpretation of physicaJ theories. 9 "Both of these writers approach their task through an emotional fog; they present their views with an amount of personification and metaphor that reduces them to the level of revivalist preachers." Many of Jeans' "devices," wrote Miss Stebbing, "are u~ed apparently for no other purpose than to reduce the reader to a state of abject terror." Other critics were kinder (and less witty) but raised essentially the same objections. It is hard to escape the feeling that Jeans capitalized increasingly on his charm and virtuosity as a popular expositor, that he yielded to an inner need to write more even as his ideas were petering out,lo Jeans died of a heart attack on September 16, 1946. He was reading Cro", ther, op. cit., p. 93. Penguin Books, Harmondsworth, 1944; Chapters 1-3 passim. 10 "The .\f,nteriolls ['nil erse received fierce criticism when it was published. Ru.therford was he~rd to say that Jeans had told him that 'that fellow Eddington has wntten a book which has sold 50,000 copies; I will write one that will sell 100,000.' And Rutherford added: 'He did.' " Crowther, op. cit., p. 136, 8
[l
Sir James Jea1ls
2277
proofs of his last book. The Growth of Physical Science, only a few days before his death. In this biographical sketch, brief though it is, I have said a good deal more about Jeans' work as a popularizer than as an original investigator. This seemed desirable because of his real contribution, now disparaged, to scientific education. He was a first~rate mathematical physicist and for that he will be remembered. It would be unjust, however, to overlook the impetus he gave to scientific understanding in much broader circles, even though some of his ideas were muddled and grossly misleading; even though, in later years, he came close to being a hack. On the whole, I think more persons got a glimpse of the meaning of science, a taste of its excitement, integrity and beauty from Jeans' vivid primers than were led astray by his misty philosophy. The selection below is from Jeans' Science and Music. This is an excellent account of the science of sound, written in a lucid and straightforward style, without theological dissonances; it is exper~ in discussions of both the experimental and mathematical sides of the subject. I have chosen excerpts which illustrate the remarkable contribution that mathematicians have made to the analysis of musical structure, from the profound discovery made by Pythagoras through the magnificent labors of Helmholtz and his successors. Jeans deeply loved music all his life; he often played the organ for three or four hours a day, and thought in musical images even in his scientific work. l l His second wife was a brilliant organist and their mutual interest in music led to the writing of this book. 11
Crowther, op. cit., p. 93.
Discoursed with Mr. Hooke about the nature of sounds, and he did make me understand the nature of musicall sounds made by strings, mighty prettily; Dnd he told me that having come to a certain number oj vibrations proper to make any tone, he is able to tell how many strokes a fly makes with her wings (those flies that hum in their flying) by the note that it answers to in musiqlle, during their flying. That 1 suppose is a little too much refined; but his discourse in general ot sound was mighty fine. -SAMUEL PEPYS (Diary, Aug. 8, 1666) Through and through the world is injested with quantity: To talk sense is to talk quantities. It is nc use sa}'ing the nation is large-How large? It is no use saying that radium is scarce-How scarce? You cannot evade quantity. You may fly to poetry and music, and quantity and number will face you in your rhythms and your octaves. -ALFRED NORTH WHITEHEAD
1
Mathematics of Music By SIR JAMES JEANS TUNING-FORKS AND PURE TONES
WE have seen that every sound, and every succession of sounds, can be represented by a curve, and our first problem must obviously be to find the relation between such a curve and the sound or sequence of sounds it represents-in brief, we must learn to interpret a sound-curve. PURE TONES
Let us start by taking an ordinary tuning-fork as our source of sound. We begin with this rather than, let us say, a violin or an organ-pipe, because it gives a perfectly pure musical note, as we shall shortly see. If we strike its prongs on something hard, or draw a violin-bow across them, they are set into vibration. We can see that they are in vibration from their fuzzy outline. Or we can feel that they are in vibration by touching them with our fingers, when we shall experience a trembling or a buzzing sensation. Or, without trusting our senses at all, we may gently touch one prong with a light pith ball suspended from a thread, and shall find that the ball is knocked away with some violence. When the prongs of the fork vibrate, they communicate their vibrations to the air surrounding them, and this in turn transmits the agitation to our ear-drums, with the result that we hear a sound. We can verify that the air is necessary to the hearing of the sound by standing the vibrating fork inside an air-pump and extracting the air. The fuzzy appearance of the prongs shews that the fork is still in vibration, but we can no longer hear the sound, because the air no longer provides a path by which the vibrations can travel to our ears. 2278
2279
MalhemoJics of Music
FIGURE I-The vibrations of a tuDing-fork give a fuzzy appearance to the prongs and cause them to repel a llgbt pith baU with some violence.
.. FIGURE 2-The trace of a vibrating fOIl: can be obtained by drawing a piece of paper or smoked glass under it.
To study these vibrations in detail, we may attach a stiff bristle or a light gramophone needle to the end of one prong of the fork, and while the fork is in vibration, run a piece of smoked glass under it as shewn in Figure 2, taking care that it moves in a perfectly straight line and at a perfectly steady speed. If the fork were not vibrating, the point of the needle would naturally cut a straight furrow through the smoky deposit on the glass; if we held the glass up to the light, it would look like Figure 3. In actual fact, we shall find it looks like Figure 4, which is a copy of an actual photograph; the vibrations have left their record in the smoke, 80 that the needle has not cut a straight but a wavy furrow. Each complete wave obviously corresponds to a single to-and-fro motion of the needle point, and so to a complete vibration of the prong of the tuning-fork.
Sir James Jeans
FIGURE 3-The trace of a non-vibrating fork.
FIGURE 4-Tbe trace of a vibrating fork. The waves are produced by the vibrations of the fork, one complete wave by one complete vibration.
This wavy curve must clearly be the sound-curve of the sound emitted by the vibrating fork. For if we reverse the motion and compel the needle to follow the furrow, the sideways motions of the needle will set up similar motions in the prong to which it is attached, and these will produce exactly the same sound as was produced when the fork vibrated freely of itself. In fact, the whole process is like that of listening to a gramophone record, except that the tuning-fork, instead of a mica diaphragm, transmits the sound-vibrations to the air. This simple experiment has disclosed the relation between the musical sound produced by a tuning-fork and its curve, which we now find to consist of a succession of similar waves. The extreme regularity of these waves is striking; they are all of precisely the same shape. so that their lengths are aU exactly the same, and they recur at perfectly regular intervals. Indeed, it is this regularity which distinguishes music from mere noise. So long as a gramophone needle is moving regularly to-and-fro in its groove we hear music; the moment it comes upon an accidental scratch on the record, so that its motion experiences a sudden irregular jerk, we hear mere noise. In such ways as this, we discover that regularity is the essential of a musical sound-curve.
2281
Mathematics of Music
Yet the regularity can be overdone, and absolute unending regularity produces mere unpleasing monotony. The problem of designing a curve which shall give pleasure to the ear is not altogether unlike that of designing a building which shall give pleasure to the eye. A mere collection of random oddments thrown together anyhow is not satisfying; our aesthetic sense calls for a certain amount of regUlarity, rhythm and balance. Yet these qualities carried to excess produce monotony and lifelessness-the barracks in architecture and the dull fiat hum of the tuning-fork in music. PERIOD, FREQUENCY AND PITCH
When a tuning-fork is first set into vibration, we hear a fairly loud note, but this gradually weakens in intensity as the vibrations transfer their energy to the surrounding air. Unless the fork was struck very violently in the first instance, we notice that the pitch of this note remains the same throughout; if the fork sounded middle C when it was first struck, it will continue to sound this same note until its sound dies away into silence. l On taking a trace of the whole motion, in the manner shewn in Figure 2, we find that the waves slowly decrease in height as the sound diminishes in strength, but they remain always of the same length. If we measure the speed at which the fork is drawn over the smoked glass in taking this trace, we can easily calculate the amount of time the needle takes to make each wave. This is, of course, the time of a single vibration of the fork, and is only a minute fraction of a second; we call it the "period" of the vibration. The number of complete vibrations which occur in a second is called the "frequency" of the vibration. Actual experiment shews that a tuning-fork which is tuned to middle C of the pianoforte executes 261 vibrations in a second. regardless of whether the sound is loud or soft. This frequency of 261 is associated with the pitch of middle C not only for the sound of a tuning-fork, but also for all musical sounds, no matter how they are produced. For instance, a siren which runs at such a rate that 261 blasts of air escape in a second will sound middle C. Or we may hold the edge of a card against a rotating toothed wheel; if 261 teeth strike the card every second we again bear middle C. If a steam-saw runs at such a rate that 261 teeth cut into the wood every second, it is again middle C that we hear. The hum of a dynamo is middle C when the current alternates at the rate of 261 cycles a second, and this is true of all electric machinery. There are electric organs on the market in which the sound of a middle C pipe is copied, sometimes very faithfully, by an 1
If the fork was struck very violently in the first instance, there may be a very
slight sharpening of pitch as the vibrations become of more usual intensity.
Sir James
228Z
J~Q1IS
electric current which is made to alternate at the rate of 261 cycles a second. Again, when a motor-car is running at such a rate that the pistons make 261 strokes a second, a vibration of frequency 261 is set up, and we hear a note of pitch middle C in the noise of the engine. AU this shews that the pitch of a sound depends only on the frequency of the vibration by which it is produced. It does not depend on the nature of the vibration. Thus we may say that it is the frequency of vibration that determines the pitch of a sound. If there is no clearly defined frequency, there is nO clearly defined pitch, because the sound is no longer musical. When a siren or steam-saw or dynamo is increasing its speed, the sound we hear rises in pitch, and conversely. Thus we learn to associate high pitch with high frequency, and vice versa. If we experiment with a series of forks tuned to all the notes in the middle octave of the piano, we shall find the following frequencies:
438-9 a 348·4 f 261·0 465·0 369·1 a# 276-5 f# 492·7 b 391·1 g 293·0 c' 522·0 310·4 g# 414·3 328·8 . . . These frequencies might at first sight be thought to be a mere random collection of numbers, but a little study shews that they are not. We notice at once that the first number 261 is just half of the last number 522. Thus our experiments have shewn that in this particular case the interval of an octave corresponds to a 2 to 1 ratio of frequencies, and other experiments shew that this is universally true-doubling the frequency invariably raises the pitch by an octave. The octave interval is fundamental in the music of all ages and of all countries; we now see its physical significance. We may further notice that the interval from c to c# represents a rise in frequency of just about 6 per cent., and a little arithmetic will shew that the same is true for every other interval of a semi tone. The rise cannot be precisely 6 per cent. for each semitone, since if it were, the rise in the whole octave, consisting of twelve such intervals, would be equal to 1 . 06 Xl· 06 xl· 06 X .. _ etc., there being twelve factors in all, each equal to 1·06. This is the quantity which the mathematician describes as (1·06) 12. and it is equal to 2· 0122, and not to exactly 2. In an instrument such as the piano or organ, which is tuned to "equal temperament" the exact interval of 2 is spread equally over the twelve semi tone intervals which make the octave. Each step accordingly represents a frequency ratio of 1· 05946. since this is the exact twelfth root of 2. . . . c c# d d# e
2283
MathemJJtics of Music
SIMPLE HARMONIC CURVES
Having learned all we can from the regUlarity and length of the waves in Figure 4, let us next examine their form. The extreme simplicity of their shape is very noticeable, although it must be said at once that this is not a property of all sound-curves; these particular curves are simple because they are produced by the simplest of all musical instrumentsthe tuning-fork. Exact measurement shews that the curve has a shape with which the mathematician is very well acquainted. It is called a "sine" curve, or a "simple harmonic" curve, while the motion of the needle which produces it is described as "simple harmonic motion." These simple harmonic curves and the simple harmonic motion by which they are produced are of fundamental importance in all departments of mechanics and physics, as well as in many other branches of science. They are particularly important in the theory of vibrations, and this makes them of especial interest in the study of music, since musical sound is almost invariably produced by the vibrations of some mechanical structure-a stretched string, a column of air, a drum-skin, or some metallic object such as a cymbal, triangle, tube or bell. For this reason, we shall discuss vibrations in some detail. GENERAL THEORY OF VIBRATIONS
Generally speaking, every material structure can find at least one position in which it can remain at rest--otherwise it would be a perpetual motion machine. Such a position is called a "position of equilibrium." When a structure is in such a position, the forces on each particle of itas for instance the weight of the particle, and the pushes and pulls from neighbouring particles-are exactly balanced. Any slight disturbance, such as a push, pull or knock from outside, will cause the structure to move out of this position of equilibrium to some new position, in which the forces on a particle are no longer evenly- balanced; each particle then experiences a "restoring force" which tends to pull it back to the position it originally occupied. This force starts by dragging the particle back towards its original position of equilibrium. In time it regains this position, but as it is now moving with a certain amount of speed, it overshoots the position and travels a certain distance on the other side before coming to rest. Here it experiences a new force tending to pull it back; again it yields to this force, gets up speed, overshoots the mark, and so on, the motion repeating itself time after time. Clearly the trace of the motion of any particle will be a succession of waves, like those we have already obtained from the tuning-fork in Figure 4 (p. 2280).
SIr James Jeans
2284
Motion of this kind is described by the general term "oscillation." In the special case in which each particle only moves through a very small distance, the motion is called a "vibration." Thus a vibration is a special kind of oscillation, and, as it happens, possesses certain very simple properties which are not possessed by oscillations in general. It is usually true of oscillations that the farther a particle moves from its position of equilibrium, the greater is the restoring force pulling it back. But in a vibration the restoring force is exactly proportional to the distance the particle has moved from its position of equilibrium; draw it twice as far from this position, and we double the force pulling it back. A simple mathematical investigation shews that when this relation holds. the motion of every particle will be of the same kind, whatever the structure to which it belongs. Motion of this kind is defined to be "simple harmonic motion." We have alreadY found a concrete instance of this kind of motion in the tuning·fork. Another is provided by what is perhaps the simplest mechanical structure we can irnagine-a weight suspended by a fine thread. The position of equilibrium is one in which the weight lies at a
FIGURE 5-A poSItion of equillbrium. The weight can rest in equilibrium at C but nowhere else. If we pull It aside to 8, it tends to return to C.
point C exactly under the point of suspension. When the weight is drawn a. short distance aside to an adjacent position B, there is no longer equilibrIum, . and the weight tends to fall back to C. In technical language, a restormg force acts on the weight. tending to draw it back to its position of equilibrium C, and it is a simple problem in dynamics to find its amount. So long as the displacement of the weight is not too large, we find that the restoring force is exactly proportional to the extent of the displacement BC, so that the condition for simple harmonic motion is
Mathematics 01 Music
2285
fulfilled. Indeed, if we take a trace by attaching a needle to the weight and running a piece of paper horizontally under it, as in Figure 6, we shall find that this trace is a simple harmonic curve exactly like that made by our tuning-fork. If we set our suspended weight swinging more violently, and again take a trace of its motion, we shall again obtain a simple harmonic curve. The waves will, of course, be greater in size, but their period will be exactly the same as before. We find that the swinging weight makes just as many swings per second, no matter what the extent of these swings may be, provided always that they are small enough to qualify as vibrations. This illustrates the well-known fact that the Deriod of vibration of
FIGURE 6--Taking the trace of a swinging pendulum. The trace is found to be a simple harmonic curve, exactly similar to that given by a vibrating tuning-fork (Figure 2).
a pendulum depends only on its length, and not on the extent of its swing; it is because of this that our pendulum clocks keep time. We found a similar property in the tuning.fork, the period of its vibra~ tions being the same whether we struck it fairly hard or only very softly. And all true vibrations possess the same property-the period is independent of the extent and energy of the swing. This is a most important fact for the musician. It means that every musical instrument in which the sound is produced by vibrations will "keep time" like a pendulum clock, and so will give a note of the same frequency, and therefore of the same pitch? whether it is played soft or loud. Without this property it may almost be said that music, as we know it, would be impossible. We can hardly imagine an orchestra acquitting'itself with credit if every note was out of tune unless it was played with exactly the right degree of force. Crescendos and diminuendos could only be produced by adding and sub-
Sir lamel letIIU
tracting instruments. As the note of a piano or any percussion instrument decreased in strength it would also change in pitch, and every piece would inevitably begin with a howl and end with a wail. At the same time, every musician is familiar with cases in which the pitch of an instrument is changed appreciably by playing it softer or louder. The flautist can always pull his instrument a bit out of tUDe by blowing strong or weak. while the organist knows only too well the dismal wail of flattened notes which is heard when his wind gives out. We shan discuss the theory of such sounds as these later, and shall find that they are not produced by absolutely simple vibrations like those of the tuningfork or pendulum. SIMULTANEOUS VIBRATIONS
Many structures are capable of vibrating in more than one way, and so may often be performing several different vibrations at the same time. There is a very general principle in mechanics, which asserts that when any structure whatever is set into vibration-provided only that the displacement of each particle is sman-the motion of every particle is either a simple harmonic motion or else is a more complicated motion which results from superposing a number of simple harmonic motions, one for each vibration which is in progress. A simple illustration will shew how this can be. Let us suppose that while our tuning-fork is in vibration we hit it on the top of one of the prongs with a hammer. We shall hear a sharp metallic click, which is
(b) FIGURE 7-The superposition of two vibrations. The two wavy curves in
(a)
have periods which
stand in the ratio of 6~ to 1. On superposing them we obtain the curve
(b),
which
represents very closely the sound-curve of a tun.iD~fork which is sounding its clang toile.
2287
Mathematics of Music
known as the "clang tone" of the fork. A good musical ear may perhaps recognise that its pitch lies about 2;6 octaves above the ordinary note of the fork. Clearly the blow of the hammer has started new vibrations in the fork, of much higher frequency than the original vibration. If we had taken a trace of the motion when the original vibration was acting alone we should have obtained a curve like that shewn in Figure 8. This is reproduced as the long~waved curve in Figure 7(a). If we take a trace of the clang tone alone, it will be like the short-waved curve in Figure 7 (a), this representing a simple harmonic motion having 6% times the frequency of the main vibration. N ow suppose we take a trace when the two vibrations are going on together. At the instant of time represented at the point P, the particle under consideration is displaced through a distance PQ by the main vibration, and through a distance PR by the vibration which produces the clang tone. Thus the operation of the two vibrations together displaces it through a distance PQ + PR, and this is equal to PS if we make QS equal to PRo By adding together displacements in this way all along the curve, we obtain the curve shewn in Figure 7(b) as the trace to be expected when both vibrations are in action together. The photograph of an actual trace is shewn in Figure 9. In addition to the clang tone just mentioned, we may often hear a second clang tone about four octaves higher than the fundamental note of the fork. Indeed, it is difficult to start the fork sounding in such a way that the pure tone of the fork is heard without any admixture of these higher tones. We more usually obtain a mixture of all three tones, but this does not interfere with the utility of the tuning-fork as a source of pure musical tone, since the sounds of higher frequency die away quite rapidly, and the ear soon hears nothing but the fundamental tone of the fork.
FIGURE 8-The sound-curve of the simple tone from a tunini-forlt. The note Is of frequency 2S6 (middle C), and the dots indicate intervals ot
Moo second.
Sir James JI!Il1U
A second example of simultaneous vibrations can be made to teach us something new. If we return to our weight suspended by a string and knock it sideways, it will swing from side to side pendulum-wise through
FIGURE 9-The souna-cun'e of the note from a tuning-fork when the clang tone is sounding. The dang tone superposes small waves onto the longer waves. shewn in Figure S above.
which represent !he mam tone of !he fork.
SOUND-CURVES OF A TUNING-FORK.
some such path as AB in Figure 5 (p. 2284), and its motion, as we have already seen, will be simple harmonic motion. Suppose, however, that when the weight is at B, we give it another slight knock in the direction at right angles to AB, i.e., through the paper of our page in Figure 5. This sets up a new vibration in a direction at right angles to AB, and the motion in this direction also must be simple harmonic motion. As we have seen that the period of a pendulum depends only on its length.
FIGURE 10
FIGURE 11
FIGURE 12
FIGURES 10, 11 an4 12-Three'different types Qf motion which can be executed by the bob of a conical pendulum.
the new motion will have the same period as the original motion. The whole motion is accordingly obtained from the superposition of two simple harmonic motions whose periods are equal.
2289
Mathematics oj Music
If we watch the weight from a point directly above it, we shall see it moving in a curved path round its central position C. If the second knock was violent, its path will be an elongated ellipse such as AA'BB' in Figure 10. If the knock was gentle, its path will be an ellipse elongated in the other direction such as AA'BB' in Figure 11. But if the knock was of precisely the same strength as that which originally set the pendulum in motion along AB, then the weight will move in the circle AA'BB' in Figure 12, forming the arrangement which is generally described as a conical pendulum. It must move with the same speed at each point of its journey, for it is moving in a perfectly level path, so that there is no reason why it should move faster at anyone point than at any other. Thus we learn that each of the motions illustrated in Figures 10, 11 and 12 can be regarded as the superposition of two simple harmonic motions of equal periods. The last of the three is by far the most interest,A'
FIGURE 13-A geometrical interpretation of simple harmonic motion. As the pOint P moves steadily around the circle, the point N moves backwards and forwards along AB. and its motion is simple harmonic: motion.
ing, because it shews us that a simple circular motion performed at uni, form speed can be regarded as made up of two simple harmonic motions in directions at right angles to one another. To put this more definitely, let us imagine that the point P in Figure 13 moves round the circle AA'BB' with uniform speed, like the hand of a clock. Wherever P is, let us draw perpendiculars PN, PM on to the lines AB, A'B'. Then, as P moves steadily round the circle, N moves backwards and forwards along AB, while M moves backwards and forwards along A'S'. We have learnt that the motion of each of these points will be simple harmonic motion. This gives us a simple geometrical explanation of simple harmonic motion-as P moves steadily round in a circle, the point N moves in simple harmonic motion. It is easy to see from this definition that the motion of the piston in the cylinder of a locomotive or a motor-car must be approximately simple harmonic motion. Or we may look at the problem from the other end, and see that as the point N moves to-and-fro in simple harmonic motion along AB, the point
Sir Janus Jeans
P moves steadily round the circle AA'BB". This circle is called the "circle of reference" of the simple harmonic motion. Its diameter AB is called the "extent" of the motion, while its radius CA or CB is called the "ampli. tude" of the motion. ENERGY
The amplitude of a vibration gives an indication of its energy, for it is a general law that the energy of a vibration is proportional to the square of the amplitude. For ~nstance, a vibration which has twice the amplitude of another has four times the energy of the other; in other words, the vibrating structure to which it belongs has four times as much capacity for doing work stored up within itself, and it must get rid of this in some way or other before it can come to rest. The energy stored up in a musical instrument is usually expended in setting the air around it into vibration; indeed it is only through its steady outpouring of energy into the surround* ing air that we hear the instrument at all. It follows that if we want to maintain a vibration at the same level of energy we must continually supply energy to it-as we do with an organ~ pipe or a violin-string. If energy is not supplied the vibration will die away -as with a piano-string or a bell or a cymbal. The amplitude of the vibration then slowly decreases, and the circle of reference shrinks in size. When a structure is performing several vibrations at the same time, energy does not usually pass from one vibration to another. The vibrations are independent, each possessing its own private store of energy which it preserves intact, except for what it may pass on to other outside structures-as for instance, the air around it. Thus the energy of a number of simultaneous vibrations may be thought of as the sum of the energies of the separate vibrations. SIMULTANEOUS SOUNDS
When a tuning-fork is sounding, every particle of its substance moves in simple harmonic motion, and those particles which form its surface trans~ mit their motion to the surrounding air. The final result is that every particle of air which is at all near to the tuning-fork is set into motion and moves with a simple harmonic motion, which will naturally have the same period as the tuning-fork. This period is still preserved when the vibration is passed on to the ear-drum of a listener-that is why the note heard by the ear has the same pitch as the fork. A more complicated situation arises when two tuning-forks are standing side by side. Each then imposes a simple harmonic motion on to the particle of air, so that this has a motion which is obtained by superposing the two motions. We must study motions of this kind in some detail, because they are of
2291
Mathematics of Music
great importance in the practical problems of music. We begin with the simplest problem of all-the superposition of two motions which have the same period. The resulting motion is that which would be forced on a particle of air by the simultaneous vibrations of two forks of the same pitch standing side by side. SUPERPOSING VIBRATIONS OF THE SAME PERIOD
The two simple harmonic motions can be represented by two simple harmonic curves, such as those which pass through X and Y in Figure 14. These particular curves have been drawn with their amplitudes in the ratio of 5 to 2, so that YN %XN, and the same relation holds all along
=
FIGURE 14-The superposition of two simple harmonic motions of equal period. Here the vibratory motions (represented by the thin curves) are "in the same phase"--aest ovet crest and trough over trough. The vibrations now reinforce one another. and their resultant (represented by the thick curve) has an amplitude which is equal to the sum of the amplitudes of the two constituents.
the curves. At the instant of time represented at the point N. the first harmonic motion produces a displacement through a distance XN, while the second produces a displacement through a distance YN which is % times XN. Thus the combined effect of the two motions is a displacement through a distance equal to 1% times XN. This is represented by ZN in Figure 14. The thick curve through Z is drawn so that its distance above or below the central line is everywhere exactly 1% times that of the thin curve through X. This curve must then represent the motion of which we are in search. It is simply the thin curve through X magnified 1% times vertically, while its horizontal dimensions remain unchanged. Thus the new motion is a simple harmonic motion having an amplitude equal to the sum of the amplitudes of the constituent motions. and the same period as both. The foregoing instance is only a very special case of the general problem, for the thin curves in Figure 14 are drawn in a very special way. The crests of the waves of the two curves occur at the same instants. as also
Sir James Jeans
2292
FIGURE IS-The supelJlOSition of two simple harmonic motions of equal period. Here the vibratory motIons (represented by the thin curves) are "m opposite phase "--crest over trough and trough over crest. The constituent vibrations now pull in opposite directions. and so partially neutralise one another, the amplitude of their resultant (repre-
sented by the thIck curve) being equal to the difference of the amplItudes of the two constituents.
the troughs; in the diagram, crest lies directly over crest and trough over trough. Vibrations in which this relation holds are said to be "in the same phase." The curves might equally well have been drawn as in Figure 15 t the crests of one set of waves occurring at the same instants as the troughs of the other set. Vibrations in which this relation holds are said to be "in opposite phase." Crest lies over trough and vice versa, so that the two constituents produce displacements in opposite directions. The resultant motion is again that shewn in the thick curve, but its amplitude is no longer (l +~) times the amplitude of the larger constituent, but only (1 - %) times. We must not, however, expect as a matter of course that two motions which occur simultaneously will be either in the same, or in opposite, phase. Such simplicity is unusual, and it is far more likely that the crests
FIGURE 16-The superposition of two Simple harmonic motions of equal period. Here there is no simple phase relation between the two constituent vibratory motions (represented by the thin c;urves), but their resultant is still a simple harmonic motion (represente4 by the thick curve).
2293
Mathtmatics of Music
of one set of waves will be neither over the crests nor over the troughs of the other set, but somewhere in between, as shewn in Figure 16. If we add together the displacements represented by the two thin curves here, using the method illustrated in Figure 14 (i.e., making ZN XN + YN, and so on), we shall find that the resultant motion is represented by the thick curve shewn in the figure. We may judge by eye that this is yet another simple harmonic curve, as in actual fact it is, but we can only prove this by a new method of attack on the problem, to which we now turn. We have seen that any simple harmonic motion can be derived from the steady motion of a point round a circle. For instance, as the point P moves round the circle in Figure 13, the point N moves backwards and forwards along the line AB in simple harmonic motion. The two simple harmonic motions which we now want to superpose can of course be derived from the motions of two points, each moving steadily round a circle of its own. Let the two points be P and Q in Figure 17, so that the
=
R
FIGURE 17-The superposition of two simple harmonic motions. As P and Q move round their respective circles, N and 0 execute simple harmonic motions. The resultant motion is that executed by S, because CO
+
CN
=
CS.
points N, 0 immediately beneath them execute the simple harmonic motions with which we are concerned. At the instant to which Figure 17 refers, the motion of P has produced a displacement CN, while that of Q has produced a displacement CO, so that the total displacement, being the sum of the two, is equal to CO+CN.
Sir Jamu Jeans
To represent this in Figure 17, we start from Q, and draw the line QR in a direction parallel to CP and of length equal to CPo Then, because QR and CP are parallel and equal, the length OS which lies directly under QR must be exactly equal to the length CN which lies directly under CPo Hence the sum we need, namely CO + CN, must be equal to CO + OS, and so to CS. Thus as P and Q move round their respective circles, the points Nand o execute the two constituent simple harmonic motions, and the point S executes the motion which results from their superposition. We are at present supposing the two simple harmonic motions performed by Nand 0 to be of the same frequency, so that the radii CP and CQ rotate at exactly the same rate and the angle PCQ remains always the same. Indeed, we can visualise the whole motion by imagining that we cut the parallelogram CPRQ out of cardboard, and then make it rotate round C at the same rate as P and Q. We see that R will move in a circle at uniform speed, so that S will move backwards and forwards along AB in simple harmonic motion. This shews that when two simple harmonic motions have the same frequency, the result of superposing them is a third simple harmonic motion of the same frequency as both. In terms of music, the simultaneous sounding of two pure tones of the same pitch produces a pure tone which is still of the same pitch. . . .
THE VIBRATIONS OF STRINGS AND HARMONICS We began our study of sound-curves by examining the curve produced by a tuning-fork. A tuning-fork was chosen, because it emits a perfectly pure tone. But, as every musician knows, its sound is not only perfectly pure, but is also perfectly uninteresting to a musical ear-just because it is so pure. The artistic eye does not find pleasure in the simple figures of the geometer-the straight line, the triangle or the circle--but rather in a subtle blend of these in which the separate ingredients can hardly be distinguished. In the same way, the painter finds but little interest in the pure colours of his paint-box; his real interest lies in creating SUbtle, rich or delicate blends of these. It is the same in music; our ears do not find pleasure in the simple tones we have so far been studying but in intricate blends of these. The various musical instruments provide us with readymade blends. which we can combine still further at our discretion. In the present chapter we shall consider the sounds which are emitted by stretched strings-such as, for instance, are employed in the piano, violin, harp. zither and guitar-and we shall find how to interpret these as blends of the pure tones we have already had under consider~ ation.
229S
Mathematics of Music
EXPERIMENTS WITH THE MONOCHORD
Our source of sound will no longer be a tuning-fork but an instrument which was known to the ancient Greek mathematicians, Pythagoras in particular. and is still to be found in every acoustical laboratory-the monochord. Its essentials are shewn in Figure 18. A wire, with one end A fastened rigidly to a solid framework of wood, passes over a fixed bridge Band
DeB
~----------------------------~A
FIGURE IS-The monochord. The string is kept in a state of tension by the suspended weight W. while "bridges" like those of a violin limit the vibration to a range
Be.
The instru-
ment is arranged so that both range and tension are under control.
a movable bridge C, after which it passes over a freely turning wheel D, its other end supporting a weight W. This weight of course keeps the wire in a state of tension, and we can make the tension as large or small as we please by altering the weight. Only the piece BC of the string is set into vibration, and as the bridge C can be moved backwards and forwards, this can be made of any length we please. It can be set in vibration in a variety of ways-by striking it, as in the piano; by stroking it with a bow, as in the violin; by plucking it, as in the harp; possibly even by blowing over it as in the Aeolian harp. or as the wind makes the telegraph wires whistle on a cold windy day. On setting the string vibrating in any of these ways, we hear a musical note of definite pitch. While this is still sounding, let US press with our hand on the weight W. We shall find that the note rises in pitch, and the harder we press on the weight, the greater the rise will be. The pressure of our hand has of course increased the tension in the string, so that we learn that increasing the tension of a string raises the pitch of the note it emits. This is the way in which the violinist and piano~tuner tune their strings and wires; when one of these is too low in pitch, they screw up the tuning-key. A series of experiments will disclose the exact relation between the pitch and the tension of a string. Suppose that the string originally sounds c! (middle C). the tension being 10 lb. To raise the note an octave, to c",
81,. lamtJs IIHIIU
we shan find we must increase the tension to 40 lb.; to raise it yet another octave to em, we need a total tension of 160 lb., and so on. In each case a fourfold increase in the tension is needed to double the frequency of the note sounded, and we shall find that this is always the case. It is a general law that the frequency is proportional to the square root of the tension. We can also experiment on the effect of changing the length of our string, repeating experiments such as were performed by Pythagoras some 2500 years ago. Sliding the bridge C in Figure 18 to the right shortens the effective length BC of the string, but leaves the tension the same-that necessary to support the weigbt W. When we shorten the string, we find that the pitch of the sound rises. If we halve its length, the pitch rises exactly an octave, shewing that the period of vibration has also been halved. By experimenting with the bridge C in all sorts of positions, we discover tbe general law that the period is exactly proportional to the length of the string, so that the frequency of vibration varies inversely as the length of the string. This law is exemplified in all stringed instruments. In the violin, the same string is made to give out different notes by altering its effective length by touching it with the finger. In the pianoforte different notes are obtained from wires of different lengths. We may experiment in the same wayan the effect of changing the thickness or the material of aUf wire.
FIGUR.E 19
FIGURE 20
FIGURE 2J FIGURES 19. 20 and 21-Characteristic vibrations of a stretched string. '!be string vibrates in one, two and three equal parts respectively, and emits its fundamental tone, the octave and the twelfth of this in
liO
doing.
MER.SENNE'S LAWS
The knowledge gained from all these experiments can be summed up in the following laws, which were :first formulated by the French mathematician Mersenne (Harmonie Universelle, 1636):
Mathematics of Music
2297
1. When a string and its tension remain unaltered, but the length is varied, the period of vibration is proportional to the length. (The law of Pythagoras. ) II. When a string and its length remain unaltered, but the tension is varied, the frequency of vibration is proportional to the square root of the tension. III. For different strings of the same length and tension, the period of vibration is proportional to the square root of the weight of the string. The operation of all these laws is illustrated in the ordinary pianoforte. The piano-maker could obtain any range of frequencies he wanted by using strings of different lengths but similar structure, the material and tension being the same in all. But the 71A. octaves range of the modem pianoforte contains notes whose frequencies range from 27 to 4096. If the piano-maker relied on the law of Pythagoras alone, his longest string would have to be more than 150 times the length of his shortest, so that either the former would be inconveniently long, or the latter inconveniently short. He accordingly avails himself of the two other laws of Mersenne. He avoids undue length of his bass strings by increasing their weight-usually by twisting thinner copper wire spirally round them. He avoids inconvenient shortness of his treble strings by increasing their tension. This had to be done with caution in the old wooden-frame piano, since the combined tension of more than 200 stretched strings imposed a great strain on a wooden structure. The modern steel frame can, however, support a total tension of about 30 tons with safety, so that piano-wires can now be screwed up to tensions which were formerly quite impracticable. . . . HARMONIC ANALYSIS
Several times already we have superposed two simple harmonic curves, and studied the new curves resulting from the superposition. The essence of the process of superposition has already been illustrated in Figure 7(a) on p. 2286, and Figure 14 on p. 2291. In each of these cases the number of superposed curves is only two; when a greater number of such curves is superposed, the resultant curve may be of a highly complicated form. There is a branch of mathematics known as "harmonic analysis" which deals with the converse problem of sorting out the resultant curve into its constituents. Superposing a number of curves is as simple as mixing chemi· cals in a test-tube; anyone can do it. But to take the final mixture and discover what ingredients have gone into its composition may require great skill. Fortunately the problem is easier for the mathematician than for the analytical chemist. There is a very simple technique for analysing any curve, no matter how complicated it may be. into its constituent simple
22.98
Sir James Jeans
harmonic curves. It is based on a mathematical theorem known as Fourier's theorem, after its discoverer, the famous French mathematician J. B. 1. Fourier (1768-1830). The theorem tells us that every curve, no matter what its nature may be, or in what way it was originally obtained, can be exactly reproduced by superposing a sufficient number of simple harmonic curves-in brief, every curve can be built up by piling up waves. The theorem further tells us that we need only use waves of certain specified lengths. If, for instance, the original curve repeats itself regularly at intervals of one foot, we need only employ curves which repeat themselves regularly 1) 2t 3, 4. etc. times every foot-i.e., waves of lengths 12, 6,4, 3, etc. inchQs. This is almost obvious, for waves of other lengths, such as 18 or 5 inches, would prevent the composite curve repeating regularly every foot. If the original curve does not repeat regularly, we treat its whole length as the first half-period 2 of a curve which does repeat, and obtain the theorem in its more usual form. It tells us that the original curve can be built up out of simple harmonic constituents such that the first has one complete half-wave within the range of the original curve, the second has two complete half-waves, the third has three. and so on; constituents which contain fractional parts of half-waves need not be employed at all. There is a fairly simple rule for calculating the amplitudes of the various constituents, but this lies beyond the scope of the present book. We obtain a first glimpse into the way of using this theorem if we suppose our original curve to be the curve assumed by a stretched string at any instant of its vibration. Figures 19, 20 and 21 on p. 2296 shew groups of simple harmonic curves which contain one, two and three complete half-waves respectively within the range of the string. Let us imagine this series of diagrams extended indefinitely sO as to exhibit further simple harmonic curves containing 4, 5, 6, 7 and all other numbers of complete half-waves. Then the series of curves obtained in this way is precisely the series of constituent curves required by the theorem. We take one curve out of each diagram, and superpose them all; the theorem tells us that by a suitable choice of these curves, the final resultant curve can be made to agree with any curve we happen to have before us. Or, to state it the other way round, any curve we please can be analysed into constituent curves, one of which will be taken from Figure 19, one from Figure 20, one from Figure 21, and so on. This is not, of course, the only way in which a cui've can be decomposed into a number of other curves. Indeed, the number of ways is infinite, just as there is an infinite number of ways in which a piece of 2 It might seem simpler to treat the original curve as a whole period of a repeating curve, but there are mathematical reasons against this.
2299
Mathe1TUltics of Music
paper can be torn into smaller pieces. But the way just mentioned is unique in one respect, and this makes it of the utmost importance in the theory of music. For when we decompose the curve of a vibrating string into simple harmonic curves in this particular way, we are in effect decomposing the motion of the string into its separate free vibrations, and these represent the constituent tones in the note sounded by the vibration. ~ the vibratory motion proceeds, each of these free vibrations persists without any change of strength, apart from the gradual dying away already explained. If, on the other hand, we had decomposed the vibration in any other way, the strength of the constituent vibrations would be continually changing-probably hundreds of times a second-and so would have no reference to the musical quality of the sound produced by the main vibration. So general a theory as this may well seem confused and highly complicated, but a single detailed illustration will bring it into sharp focus and shew its importance. STRING PLUCKED AT ITS MIDDLE POINT
Let us displace the middle point of a stretched string AB to C, so that the string forms a :flat triangle ACB as in Figure 22. The shape of the string ACB may still be regarded as a curve, although a somewhat unusual one, and our theorem tells us that this "curve" can be obtained from the superposition of a number of simple harmonic curves. In actual fact, Figure 23 shews how the curve ACB can be resolved into its constituent curves; if we superpose all the curves shewn in this latter figure, we shall find we have restored our original broken line ACE, except for a difference in scale; the vertical scale in Figure 23 has been made ten times the horizontal in order that the fluctuations of the higher harmonics may be the more clearly seen. Suppose we now let go of the point C, and allow the natural motion of the string to proceed. We may imagine each of the curves shewn in Figure 23 to decrease and increase rhythmically in its own proper period, and the superposition of the curves at any instant will give us the shape of the string at that instant. These curves correspond to the various harmonics that are sounded on plucking a string at its middle point. We notice that the second, fourth and sixth harmonics are absent. This is not a general property of harmonics, but is peculiar to the special case we have chosen. We have plucked the string in such a way...that its two halves are bound to move in similar fashion, and as a consequence the second, fourth and sixth harmonics, which necessarily imply dissimilarity in the two halves, cannot possibly appear. If we had plucked it anywhere else than at its middle point, some at least of these harmonics would have been present.
Sfr
2300
lam~$ }~ans
c A
R FlGURE 22
First hannonic Second harmonic
Fourth harmonic
--
,.
Fifth harmonic Sixth harmonic
FIGURE 23 FIGURES 22 and 23-The string is displaced to form the triangle ACB. This "curve" can be analysed into the simple harmonic curves shewn in Figure 20. On superposing these we restore the "curve" ACB of Figure 19. (The vertical seales in
Figure 23 are all magnified ten-fold.)
ANALYSIS OF A SOUND-CURVE
Let US next apply Fourier's theorem to a piece of a sound-curve. The theorem tells us that any sound-curve whatever can be reproduced by the superposition of suitably chosen simple harmonic waves. Consequently any sound~ no matter how complex-whether the voice of a singer or a motor-bus changing gear--can be analysed into pure tones and reproduced exactly by a battery of tuning-forks, or other sources of pure tone. Professor Dayton Miller has built up groups of organ-pipes, which produce the various vowels when sounded in unison; other groups say papa and mama.
The sound-curve of a musical sound is periodic; it recurs at perfectly regular intervals. Indeed, we have seen that this is the quality which distinguishes music from noise. Fourier's theorem tells us that such a sound-
2301
MathemlJtic$ of Music
curve can be made up by the superposition of simple harmonic curves such that 1, 2, 3, or some other integral number of complete waves occur within each period of the original curve. If, for instance, the sound-curve has a frequency of 100, it can be reproduced by the superposition of simple harmonic curves of frequencies 100, 200, 300, etc. Each of these curves represents a pure tone, whence we see that any musical sound of frequency 100 is made up of pure tones having respectively 1, 2, 3, etc. times the frequency of the original sound. These tones are called the "natural harmonics" of the note in question. NATURAL HARMONICS AND RESONANCE
Vibrations are often set up in a vibrating structure by a force or disturbance which continually varies in strength; such a force may be periodic in the sense that the variations repeat themselves at regular intervals. Fourier's theorem now tells us that a variable force of this kind can be resolved into a number of constituent forces each of which varies in a simple harmonic manner, and that the frequencies of these forces will be 1, 2, 3 ... times that of the total force. For instance, if the force repeats itself 100 times a second, the simple harmonic constituents of the force will repeat themselves 100, 200, 300, etc., times a second. If the structure has free vibrations of frequencies 100, 200, 300, etc., these will be set vibrating strongly by resonance, while any vibrations of other frequencies that the structure may possess will not be set going in any appreciable strength. In other words, a disturbing force only excites by resonance the "natural hannonics" of a tone of the same period as itself. This is a result of great importance to music in general. Amongst other things, it explains why the stretched string has such outstanding musical qualities; the reason is simply that its free vibrations coincide exactly in frequency with the natural harmonics of its fundamental tone, so that when the fundamental tone is set going, the harmonics are set going as well. . . . HEARING
We have now considered the generation of sound and its transmission through the air to the ear; we must finally consider its reception by the ear, and transmission to the brain. When the air is being traversed by sound-waves, we have seen that the pressure at every point changes rhythmically, being now above and now below the average steady pressure of the atmosphere-just as, when ripples pass over the surface of a pond, the height of water in the pond changes rhythmically at every point, being now above and now below the average steady height when the water is at rest. The same is of course
Sir lames leans
true of the small layer of air which lies in contact with the ear-drum, and it is changes of pressure in this layer which cause the sensation of hearing. The greater the changes of pressure, the more intense the sound, for we have seen that the energy of a sound-wave is proportional to the square of the range through which the pressure varies. The pressure changes with which we are most familiar are those shewn on our barometers-half an inch of mercury, for instance. The pressure changes which enter into the propagation of sound are far smaller; indeed they are so much smaller that a new unit is needed for measuring themthe "bar." For exact scientific purposes, this is defined as a pressure of a dyne per square centimetre, but for our present purpose it is enough to know that a bar is very approximately a millionth part of the whole pressure of the atmosphere. When we change the height of our ears above the earth's surface by about a third of an inch, the pressure on our ear-drums changes by a bar; when we hear a fairly loud musical sound, the pressure on our ear-drums again changes about a bar.
THE THRESHOLD OF HEARING
Suppose that we gradually walk away from a spot where a musical note is being continuously sounded. The amount of energy received by our ears gradually diminishes, and we might perhaps expect that the intensity of the sound heard by our brains would diminish in the same proportion. We shall, however, find that this is not so; the sound diminishes for a time, and then quite suddenly becomes inaudible. This shews that the loudness of the sound we hear is not proportional to the energy which falls on our ears; if the energy is below a certain amount we hear nothing at all. The smallest intensity of sound which we can hear is said to be at "the threshold of hearing." We obtain direct evidence that such a threshold exists if we strike a tuning-fork and let its vibrations gradually die away. A point is soon reached at which we hear nothing. Yet the fork is still vibrating, and emitting sound, as can be proved by pressing its handle against any large hard surface, such as a table-top. This, acting as a sound-board, amplifies the sound so much that we can hear it again. Without this amplification the sound lay below the threshold of hearing; the amplification has raised it above the threshold. In possessing a threshold of this kind, hearing is exactly in line with all the other senses; with each our brains are conscious of nothing at all until the stimulus reaches a certain ''thresholdu degree of intensity. The threshold of seeing. for instance, is of special importance in astronomy; our eyes see stars down to a certain limit of faintness, roughly about 6·5 magnitudes, and beyond this see nothing at all. Just as a sound-board may
Math~matic$ of Music
2303
raise the sound of a tuning-fork above the threshold of hearing, so a telescope raises the light of a faint star above the threshold of seeing. We naturally enquire what is the smallest amount of energy that must fall on our ears in order to make an impression on our brains? In other words, how much energy do our ears receive at their threshold of hearing? The answer depends enormously on the pitch of the sound we are trying to hear. Somewhere in the top octave of the pianoforte there is a pitch at which the sensitivity of the ear is a maximum, and here a very small amount of sound energy can make itself heard, but when we pass to tones of either higher or lower pitch, the ear is less sensitive, so that more energy is needed to produce the s!,-me impression of hearing. Beyond these tones we come to others of very high and very low pitch, which we cannot hear at all unless a large amount of energy falls on our ears, and finally, still beyond these, tones which no amount of energy can make us hear, because they lie beyond the limits of hearing. The following table contains results which have been obtained by Fletcher and Munson. 3 The first two columns give the pitch and frequency of the tone under discussion, the next column gives the pressure variation at which the tone first becomes audible, while the last column gives the amount of energy needed at this pitch in terms of that needed at fiT, at which the energy required is least:
Tone
ecce
(32-ft. pipe of organ; close to lower limit of hearing) AM (bottom note of piano) CCC (lowest C on piano) CC C e' (middle C) c" em elv flv (maximum sensitivity) eV (top of piano) cvt evil
Close to upper limit of hearing
Frequency
16 27
Pressure variation at which note is first heard
Energy required in terms of minimum
100 bars
1,500,000,000,000
1 bar
150,000,000
32
% "
64 128 256 512 1,024 2,048 2,734
" lhoo ,."
100,000
lh1500
"
2S
~oooo
..
4,096 8,192 16.384 20,000
*0
lhooo
25,000.000
3,800
ISO
%000 " "
6
'Aoooo " lhooo "
1.5 38
~2500
~oo
..
500 bars
I.S
1.0
IS,OOO
38,000,000,000,000
3 Many other investigators have worked at the problem, their results generally agreeing fairly closely, although not always exactly, with those stated in the table. Investigations, by Andrade and Parker (1937), yield results which are in very close agreement with those of Fletcher and Munson.
Sir JlJmes Jeans
2304
We see that the ear can respond to a very small variation of pressure when the tone is of suitable pitch. Throughout the top octave of the piano, less than a ten-thousand-millionth part of an atmosphere suffices; as already mentioned, this is produced by an air-displacement of less than a ten-thousand-millionth part of an inch, which again is only about a hundredth part of the diameter of a molecule. We also notice the immense range of figures in the last column. Our ears are acutely sensitive to sound within the top two octaves of the piano, and quite deaf, at least by comparison, to tones which are far below or above this range; to make a pure tone of pitch ecce audible needs a million million times more energy than is needed for one seven octaves higher. The structure of an ordinary organ provides visual confirmation of this. The pipe of pitch ecce is a huge 32-foot monster, with a foot opening which absorbs an enormous amount of wind, and yet it hardly sounds louder than a tiny metal pipe perhaps three inches long taken from the treble. A child can blow the latter pipe quite easily from its mouth, but the whole force of a man's lungs will not make the 32-foot pipe sound audibly. . . .
THE SCALE OF SOUND INTENSITY
The change in the intensity of a sound which results from a tenfold increase in the energy causing this sound is called a "bel." The word has nothing to do with beauty or charm, but is merely three-quarters of the surname of Graham Bell, the inventor of the telephone. We have already thought of this tenfold increase as produced by ten equal steps of approximately 25 per cent. each. More exactly, each of these must represent an increase by a factor of y( lO)f of which the value is 1 ·2589. Each of these steps of a tenth of a bel is known as a "deciber'; as we have seen, it represents just about the smallest change in sound intensity which our ears notice under ordinary conditions. The intensity at the threshold of hearing is usually taken as zero point, so that, if we take the smallest amount of energy we can hear as unit: I unit of energy gives a sound intensity of o decibels I . 26 " " " U" H " l' 1 decibel 1 . 58 " u " """ " " 2 decibels 2 """ "'," u " 3 " 4 """ "" u " " 6 "
8 10
100 1000
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Mathematics of Music
2305
THE SCALE OF LOUDNESS
The scale of sound intensity had its zero fixed at the threshold of hearing, but as the position of this depends enormously on the pitch of the sound under discussion, this scale is only useful in comparing the relative loudnesses of two sounds of the same pitch. It is of no use for the comparison of two sounds of different pitches. For this latter purpose we must introduce a new scale, the scale of loudness. The zero point of this scale is taken to be the loudness, as heard by the average normal hearer, of a sound-wave in air, which has a frequency of 1000 and a pressure range of %000 bar--or. more precisely, O· 0002 dynes -at the ear of the listener. This, as we have already seen, is just about the threshold of bearing for a sound of this particular frequency. The unit on this scale is called a "phon." 50 long as we limit ourselves to sounds of frequency 1000, the phon is taken to be the same thing as the decibel, both as regards its amount and its zero point. Thus if a sound of frequency 1000 has an intensity of x decibels on the scale of sound intensity, it has a loudness of x phons on the new scale of loudness. But the phon and decibel diverge when the frequency of the sound is different from 1000. Two sounds of different pitch are said to have the same number of phons of loudness when they sound equally loud to the ear. Thus we say that a sound has a loudness of x phons when it sounds as loud to the ear as a sound of frequency 1000 and an intensity of x decibels. Such a sound lies at x decibels above the threshold of hearing for a sound of frequency 1000, not above that for a sound of its own pitch.4 THE THRESHOLD OF PAIN
We have already considered what is the smallest amount of sound we can hear; we consider next what is the largest amount. This is not a meaningless problem. For, if we continually supply more and more energy to a source of sound-as for instance by beating a gong harder and harderthe sound will get louder and louder and, in time, we shall find it becoming too loud for pleasure. At first it is merely disagreeable, but from being disagreeable it soon passes to being uncomfortable. Finally the vibrations set up in our ear..d,rums and inner ear may become so violent as to give us acute pain, and possibly injure our ears. If we note the number of bels our ears can endure without discomfort, we shall find that this again, like the position of the threshold of hearing, 4 This defines the British standard phon. The Americans use the same phon as the British, but frequently describe it as a decibel. The Gennans use a different zero point. 0·0003 dyne in place of 0·0002.
Sir lames leans
2306
depends on the pitch of the sound. At the bass end of the pianoforte it is about six bels; it has risen to eleven bels by middle C; it rises further to twelve bels in the top octave of the pianoforte, after which it probably falls rapidly. The intensity of sound at the threshold of hearing, and also the range above the threshold which we can endure without undue discomfort, both vary greatly with the pitch of the sound, but their sum, which fixes a sort of threshold of pain, varies much less. Throughout the greater part of the range used in music, the intensity at this threshold is given by a pressure
t
~
;
~
(H
0-01 0-001
0-0001 O-ooxl1.
6
32
FIGURE 24-The limits of the area of hearing. as determined by Fletcher and Munson. Each
point in
thiS
diagram represents a sound of a certain specified frequency (as shewn
on the scale at the bottom.) and of a certain specified intensity (as shewn by the scale
on the left). If the point lies within the shaded area, the sound can be heard with comfort. If the pomt lies above the shaded area. the hearing of the sound is painful. If the point lies below the shaded area, the sound lies below the threshold of hear-
Ing. and so cannot be heard at all.
variation of about 600 bars, except that it falls to about 200 bars in the region of maximum sensitivity. We can represent this in a diagram as in Figure 24, and the shaded area which is the area of hearing can be divided up further by curves of equivalent loudness as shewn in Figure 25. Both the limits of the area of hearing and the curves of equal loudness have been determined by Fletcher and Munson. We see at a glance how the ear is both most sensitive to faint sound, and also least tolerant to excessive sound. in the range of the upper half of the piano. To be heard at a moderate comfortable loudness of say 50
Mathematics 01 Music
2307
120
120
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1024 . . . 4096 .192 16. .
~in,:pIum.. FIGURE 2S-Tbe loudness of sounds which lie within lhe area of hearing, as determined by Fletcher and Munson. As in Figure 24. each point of the diagram represents a sound
of specified frequency (as shewn on the scale at the bottom) and of specified intensity in decibels (as shewn on the scale: on the left)" the zero point being the faintest sound of frequency t 000 which can be heard. at all. The loudness of the sound in phons is the number written on the curved line which passes through the point; thus these curves are curves of equal loudness.
or 60 phons, treble music needs but little energy, while bass music needs a great deal. This is confirmed by exact measurements of the energy em· ployed in playing various instruments. The following table gives the results of experiments made at the Bell Telephone Laboratories: ORIGIN OF SOUND
Orchestra of seventy-five performers. at loudest .. Bass drum at loudest ..................... . Pipe organ at loudest ...........•.......... Trombone at loudest ..................... . Piano at loudest ........•................. Trumpet at loudest .•................ _..... Orchestra of seventy-five performers, at average. Piccolo at loudest . . . . . . . . . . . . _. . . . . . .. _.. . Clarinet at loudest ....................... . Bass singing D•............... Human voice { Alto singing pp ...•.... _... _ . Average speaking voice _ ..... _. Violin at softest used in a concert ........... .
ENERGY
Watts 70 25 13 6 0·4 0-3 0-09 0-08 0-05 0·03 0-001 0-000024 0·0000038
Sir James Jeans
2308
We may notice in passing how very small is the energy of even a loud sound. A fair·sized pipe organ may need a 10,000-watt motor to blow it; of this energy only 13 watts reappears as sound, while the other 9987 watts is wasted in friction and heat. A strong man soon tires of playing a piano at its loudest, his energy output being perhaps 200 watts; of this only 0·4 watts goes into sound. A thousand basses singing fortissimo only give out enough energy to keep one 30-watt lamp alight; if they turned dynamos with equal vigour, 6000 such lamps could be kept alight. The first and last entries in the table above represent the extreme range of sounds heard in a concert room, and we notice that the former is more than eighteen million times the latter. Yet this range. large though it is, is only one of 7* bels, and so is not much more than half of the range of 12 bels which the ear can tolerate in treble sounds. For a person well away from the instruments, we may perhaps estimate the violin at its softest as being about 1 bel above the threshold of hearing for the note it is playing, so that the full orchestra is about 8·3 bels, or 83 decibels. This may be compared with the intensities of various other sounds, as shewn in the following table: Threshold of hearing •...................... o decibels Gentle rustle of leaves ..................... . 10 " Quiet London garden ..................... . 20 " Whisper at 4 feet ..........•............... 20 " Quiet suburban streett London .............. . 30 " Quietest time at night, Central New York ...... . 40 " Conversation at 12 feet .................... . 50 " u Busy traffic. London. . . . . . . . . . . . . . . . . . ..... . 60 Busy traffic, New York ...............•.•..• 68 Very heavy traffic, New York ............... . 82 " Lion roaring at 18 feet .................... . .88 " Subway station with express passing, New York .. 95 " Boiler factory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 " u Steel plate hammered by four men, 2 feet away .. 112
.
Owing to the different thresholds of hearing, the sounds in the above tables are not strictly comparable, unless they happen to be of the same pitch. The following table shews the differences of subjective loudness for a few common sounds: Threshold of hearing ......•......... Ticking of a watch at 3 feet .......... . Sounds in a quiet residential street .... . Quiet conversation ................. . Sounds in a busy main street ........ . Sounds in a tube train .............. , Soun,ds .in a busy machine shop ...... . PrOXImIty of aeroplane engine ....... .
o phons
20 40 60 75 90 100 120
" " " " " " "
Experiments shew that a faint sound will not be heard at all through a louder sound of the same pitch, if the difference in intensity is more than
Mathematics of Music
2309
about 1 . 2 beis, but the difference in loudness may be greater if the sounds are of very different pitch. Conversation at 12 feet should just be heard against busy traffic in London, because the difference in intensity only amounts to 1·0 bels; it will not, however, be heard against busy traffic in New York, because the difference here is 1·8 bels. In the same way a roaring lion would only jUst be heard in a boiler factory, although he might hope to attract considerable attention in a New York subway station. . . .
PART XXV
Mathematics as a Culture Clue 1. Meaning of Numbers by OSWALD SPENGLER 2. The Locus of Mathematical Reality: An Anthropological Footnote by LESLIE A. WHITE
COMMENTARY ON
OSWALD SPENGLER SWALD SPENGLER (1880-1936) was a sickly German high-school teacher of apocalyptic inclination who at the age of thirty-one retired from his post to write an immense and sensational book on the philosophy of history. The book which brought fame to this obscure scholar was The Decline of the West; it was conceived in 1911 and completed in 1917. Exempt from military service because of a weak heart and defective eyesight, Spengler had ample time during the war years to elaborate his theme; but throughout he was harassed by poverty and other adversities, and only a sense of mission based on the conviction that he had discovered a great truth about "history and the philosophy of destiny" sustained him'! In 1918 a reluctant Viennese publisher was persuaded to take the book. It appeared in a small edition but in a few weeks began to sell. In Germany, where its gloomy tone suited the post-war mood, the book provoked vehement controversy; abroad "it won the admiration of the halfeducated and the scorn of the judicious." 2 Now in another post-war period, the issues raised by Spengler are again at the focus of attention. His theory has found few adherents, yet it has impelled many thoughtful men to sober reflection. "It is easy," as one critic has observed, "to criticize Spengler, but not SO easy to get rid of him." 3 Spengler's main thesis is that the patterns of history are cyclical, not linear. Man does not improve. He experiences the inexorable biological progression of "birth, growth, maturity and decay"; he accommodates himself to circumstance, changing his ways in order to survive; but his basic attitudes remain unchanged. The same principle applies to the several cultures man has produced over the centuries. Like living organisms cultures flourish, decline and die. They do not progress; they merely recur. Their course is as predestined as the course of their creators. In the history of every culture there is discernible a "master pattern" 4_"a characteristic cast of the human spirit working itself out." This master pattern shapes each of the activities which compose the culture. While the master patterns differ, and thus distinguish one culture from another, they pass inevitably
O
1 The words in quotation marks are Spengler's and are taken from the preface to the English translation of his book, by Charles Francis Atkinson: The Decline oj the Wes:: Form and Actuality, New York, 1926, p. XIV. For biographical and other detafls I have drawn on H. Stuart Hughes, Oswald Spengler, A Critical Estimate, New York, 1952. 2 Hughes, op. cit., p. L 3 The Times Literary Supplement, October 3, 1952, p. 637. 4 This is A. L. Kroeber's expression (Configurations of Culture Growth, Berkeley and Los Angeles, 1944, p. 826) quoted by Hughes, op. cit., p. 10.
2312
Oswald Spengler
2313
through the same "morphological" stages. Spengler has epitomized his am~ bitious program in these words: "I hope to show that without ex.ception all great creations and forms in religion, art, politics, social life, economy and science appear, fulfill themselves and die down contemporaneously in all the Cultures; that the inner structure of one corresponds strictly with that of all the others; that there is not a single phenomenon of deep physiognomic importance in the record of one for which we could not find a counterpart in the record of every other; and that this counterpart is to be found under a characteristic form and in a perfectly definite chronological position." 5 Even in two massive volumes Spengler was unable to persuade men of insight and dispassionate judgment that he had fulfilled his promise. Both his arguments and his presentation were vulnerable. Scholars hacked at his blunders, scientists at his pseudo-scientific reasoning, philosophers at his conclusions, literary critics at his swollen, unlovely style. It was pointed out that the cyclical view of history was a "hoary commonplace"; that Spengler had borrowed his main ideas from his betters; that he was antirational, pompously prophetic, crude and melodramatic. None of these charges were altogether baseless; some, indeed, were painfully true. Yet The Decline of the West contains elements of great originality, flashes of extraordinary insight. Spengler exaggerated but he also brilliantly illuminated comers of history which less passionate philosophers had overlooked; his ex.pression was pretentious but it was powerful. Above all, as H. Stuart Hughes said in his recent excellent study: .~ . . . The Decline of the West offers the nearest thing we have to a key to our times. It formulates more comprehensively than any other single book the modem malaise that so many feel and so few can express." 6
•
..
•
*
Spengler was convinced that mathematics is no exception to his principle of cultural parallelism. There are no. eternal verities even in this most abstract, seemingly disembodied intellectual activity. "There is not, and cannot be. number as such. There are several number-worlds as there are several Cultures." 7 Mathematics, like art or religion or politics, expresses man's basic attitudes~ his conception of himself; like the other elements in a culture it exemplifies "the way in which a soul seeks to actualitize itself in the picture of its outer world." S The first selection below is taken from the chapter "Meaning of Numbers," one of the most remarkable discussions in The Decline of the West. It is unnecessary to agree with Spengler's thesis to be stimulated by this performance. No one else has Spengler, op. cit., p. 112. Hughes, op. cit., p. 165. '1 Spengler, OPe cit., p. 59. 8 Ihid., p. 56. 5 6
1114
Editor's CQ17l11U!!1It
made even a comparable attempt to cast a synoptic eye over the evolving concept of number. A good deal of what Spengler has to sayan this subject strikes one as far-fetched and misty. But he was a capable mathematician; his ideas cannot be dismissed as hollow; and I think you will find this a disturbing and exciting essay. The second selection is less disturbing but no less exciting. It attacks the question "Do mathematical truths reside in the external world, or are they man-made inventions?" There are very few sensible discussions of this problem. Leslie White's approach is that of a cultural anthropologist. What he has to say is balanced and persuasive. It is worth comparing both with Spengler's vehement opinions and with the moderate, lucidly reasonable views of Richard von Mises (see pp. 1723-1754). Dr. White is chairman of the department of anthropology at the University of Michigan. He has taught at the University of Chicago, Yale, Columbia and Yenching University, Peiping, China. He has had extensive experience as a field investigator among the Pueblo Indians. His best known book The Science of Culture, was published in 1948.
In the study of ideas, it is necessary to remember that insistence on hardheaded clarity issues from sentimental feeling, as it were a mist, cloaking the perplexities of fact. Insistence on clarity at a11 costs is based on sheer superstition as to the mode in which human intelligence functions. Our reasonings grasp at straws for premises and /loat on gossamers for deductions. -ALFRED NORTH WHITEHEAD (Adventures in Ideas)
1
Meaning of Numbers By OSWALD SPENGLER
IN order to exemplify the way in which a soul seeks to actualize itself in the picture of its outer world-to show, that is, in how far Culture in the "become" state can express or portray an idea of human existence-I have chosen number, the primary element on which all mathematics rests. I have done so because mathematics, accessible in its full depth only to the very few, holds a quite peculiar position amongst the creations of the mind. It is a science of the most rigorous kind, like logic but more comprehensive and very much fuller; it is a true art, along with sculpture and music, as needing the guidance of inspiration and as developing under great conventions of form; it is, lastly, a metaphysic of the highest rank, as Plato and above all Leibniz show us. Every philosophy has hitherto grown up in conjunction with a mathematic belonging to it. Number is the symbol of causal necessity. Like the conception of God, it contains the ultimate meaning of the world-as-nature. The existence of numbers may therefore be called a mystery, and the religious thought of every Culture has felt their impress. Just as all becoming possesses the original property of direction (irreversibility), all things-become possess the property of extension. But these two words seem unsatisfactory in that only·an artificial distinction can be made between them. The real secret of all things-become, which are ipso facto things extended (spatially and materially), is embodied in mathe~ matical number as contrasted with chronological number. Mathematical number contains in its very essence the notion of a mechanical demarcation, number being in that respect akin to word, which, in the very fact of its comprising and denoting, fences off world-impressions. The deepest d~pths, it is true, are here both incomprehensible and inexpressible. But the actual number with which the mathematician works, the figure, formula, sign, diagram, in short the number-sign which he thinks. speaks or writes exactly, is (like the exactly-used word) from the first a symbol of these depths, something imaginable, communicable, comprehensible to the inner 'and the outer eye, which can be accepted as representing the demarcation. The origin of numbers resembles that of the myth. Primitive 2315
2316
Oswald Spengler
man elevates indefinable nature-impressions (the "alien," in our terminology) into deities, numina, at the same time capturing and impounding them by a name which limits them. So also numbers are something that marks off and captures nature-impressions, and it is by means of names and numbers that the human understanding obtains power over the world. In the last analysis, the number-language of a mathematic and the grammar of a tongue are structurally alike. Logic is always a kind of mathematic and vice versa. Consequently, in all acts of the intellect germane to mathematical number-measuring, counting, drawing, weighing, arranging and dividing I-men strive to delimit the extended in words as well, i.e., to set it forth in the form of proofs. conclusions, theorems and systems; and it is only through acts of this kind (which may be more or less unintentioned) that waking man begins to be able to use numbers, normatively, to specify objects and properties, relations and differentire, unities and pluralities-briefly, that structure of the world-picture which he feels as necessary and unshakable, calls "Nature" and "cognizes." Nature is the numerable, while History. on the other hand, is the aggregate of that which has no relation to mathematics-hence the mathematical certainty of the laws of Nature, the astounding rightness of Galileo's saying that Nature is "written in mathematical language," and the fact, emphasized by Kant. that exact natural science reaches just as far as the possibilities of applied mathematics anow it to reach. In number, then, as the sign oj completed demarcation. lies the essence of everything actual, which is cognized, is delimited, and has become all at once-as Pythagoras and certain others have been able to see with complete inward certitude by a mighty and truly religious intuition. Nevertheless, mathematics-meaning thereby the capacity to think practically in figures-must not be confused with the far narrower scientific mathematics, that is, the theory of numbers as developed in lecture and treatise. The mathematical vision and thought that a Culture possesses. within itself is as inadequately represented by its written mathematic as its philosophical vision and thought by its philosophical treatises. Number springs from a source that has also quite other outlets. Thus at the beginning of every Culture we find an archaic style, which might fairly have been called geome,trical in other cases as well as the Early Hellenic. There is a common factor which is expressly mathematical in this early Classical style of the 10th Century B.C .• in the temple style of the Egyptian Fourth Dynasty with its absolutism of straight line and right angle, in the Early Christian sarcophagus-relief, and in Romanesque construction and ornament. Here every line. every deliberatelY non-imitative figure of man and beast, reveals a mystic number-thought in direct connexion with the mystery of death (the hardset). 1
Also "thinking in money."
Meaning oj Numbers
2317
Gothic cathedrals and Doric temples are mathematics in stone. Doubtless Pythagoras was the first in the Classical Culture to conceive number scientifically as the principle of a world-order of comprehensible thingsas standard and as magnitude-but even before him it had found expression, as a noble arraying of sensuous-material units, in the strict canon of the statue and the Doric order of columns. The great arts are, one and all, modes of interpretation by means of limits based on number (consider, for example, the problem of space-representation in oil painting). A high mathematical endowment may, without any mathematical science whatsoever, come to fruition and full self-knowledge in technical spheres. In the presence of so powerful a number-sense as that evidenced, even in the Old Kingdom,2 in the dimensioning of pyramid temples and in the technique of building, water-control and public administration (not to mention the calendar), no one surely would maintain that the valueless arithmetic of Ahmes belonging to the New Empire represents the level of Egyptian mathematics. The Australian natives, who rank intellectually as thorough primitives, possess a mathematical instinct (or, what comes to the same thing, a power of thinking in numbers which is not yet communicable by signs or words) that as regards the interpretation of pure space is far superior to that of the Greeks. Their discovery of the boomerang can only be attributed to their having a sure feeling for numbers of a class that we should refer to the higher geometry. Accordingly-we shall justify the adverb later-they possess an extraordinarily complicated ceremonial and, for expressing degrees of affinity, such fine shades of language as not even the higher Cultures themselves can show. There is analogy, again, between the Euclidean mathematic and the absence, in the Greek of the mature Periclean age, of any feeling either for ceremonial public life or for loneliness, while the Baroque, differing sharply from the Classical, presents us with a mathematic of spatial analysis, a court of Versailles and a state system resting on dynastic relations. It is the style of a Soul that comes out in the world of numbers, and the world of numbers includes something more than the science thereof.
..
..
..
..
..
From this there follows a fact of decisive importance which has hitherto been hidden from the mathematicians themselves. There is not, and cannot be, number as such. There are several numberworlds as there are several Cultures. We find an Indian, an Arabian, a Classicalt a Western type of mathematical thought and, corresponding with each, a type of number-each type fundamentally peculiar and ~ Dynasties I-VIII, or, effectively, I-VI. The Pyramid period coincides with Dynasties IV-VI. Cheops, Chephren and Mycerinus belong to the IV dynasty, under which also great water-control works were carried out between Abydos and the Fayum.-Tr.
2318
Oswald Spengler
unique, an expression of a specific world-feeling, a symbol having a specific validity which is even capable of scientific definition, a principle of ordering the Become which 'reflects the central essence of one and only one soul, viz., the soul of that particular Culture. Consequently, there are more mathematics than one. For indubitably the inner structure of the Euclidean geometry is something quite different from that of the Cartesian, the analysis of Archimedes is something other than the analysis of Gauss, and not merely in matters of form, intuition and method but above all in essence~ in the intrinsic and obligatory meaning of number which they respectively develop and set forth. This number, the horizon within which it has been able to make phenomena self-explanatory, and therefore the whole of the "nature" or world-extended that is confined in the given limits and amenable to its particular sort of mathematic, are not common to all mankind, but specific in each case to one definite sort of mankind. The style of any mathematic which comes into being, then, depends wholly on the Culture in which it is rooted, the sort of mankind it is that ponders it. The soul can bring its inherent possibilities to scientific development, can manage them practically, can attain the highest levels in its treatment of them-but is quite impotent to alter them. The idea of the Euclidean geometry is actualized in the earliest forms of Classical ornament, and that of the Infinitestimal Calculus in the earliest forms of Gothic architecture, centuries before the first learned mathematicians of the respective Cultures were born. A deep inward experience, the genuine awakening of the ego, which turns the child into the higher man and initiates him into community of his Culture, marks the beginning of number-sense as it does that of language-sense. It is only after this that objects come to exist for the waking consciousness as things limitable and distinguishable as to number and kind; only after this that properties, concepts, causal necessity, system in the world-around, a form 0/ the world. and world laws (for that which is set and settled is ipso facto boWlded, hardened, number-governed) are susceptible of exact definition. And therewith comes too a sudden, almost metaphysical, feeling of anxiety and awe regarding the deeper meaning of measuring and counting, drawing and form. Now, Kant has classified the sum of human knowledge according to syntheses a priori (necessary and universally valid) and a posteriori (experiential and variable from case to case) and in the former class has included mathematical knowledge. Thereby, doubtless, he was enabled to reduce a strong inward feeling to abstract fonn. But, quite apart from the tact (amply evidenced in modem mathematics and mechanics) that there is no such sharp distinction between the two as is originally and unconditionally implied in the prinCiple, the a priori itself, though certainly one
Meaning of Numbers
2319
of the most inspired conceptions of philosophy, is a notion that seems to involve enormous difficulties. With it Kant postulates-without attempting to prove what is quite incapable of proof-both unalterableness of form in all intellectual activity and identity of form for all men in the same. And, in consequence, a factor of incalculable importance is-thanks to the intellectual prepossessions of his period, not to mention his ownsimply ignored. This factor is the varying degree of this alleged "universal validity." There are doubtless certain characters of very wide-ranging validity which are (seemingly at any rate) independent of the Culture and century to which the cognizing individual may belong, but along with these there is a quite particular necessity of form which underlies all his thought as axiomatic and to which he is subject by virtue of belonging to his own Culture and no other. Here, then, we have two very different kinds of a priori thought-content, and the definition of a frontier between them, or even the demonstration that such exists, is a problem that lies beyond all possibilities of knowing and will never be solved. So far, no one has dared to assume that the supposed constant structure of the intellect is an illusion and that the history spread out before us contains more than one style of knowing. But we must not forget that unanimity about things that have not yet become problems may just as well imply universal error as universal truth. True, there has always been a certain sense of doubt and obscurity--so much so, that the correct guess might have been made from that non-agreement of the philosophers which every glance at the history of philosophy shows us. But that this non-agreement is not due to imperfections of the human intellect or present gaps in a perfectible knowledge, in a word, is not due to defect, but to destiny and historical necessity-this is a discovery. Conclusions on the deep and final things are to be reached not by predicating constants but by studying differentil:C and developing the organic logic of differences. The comparative morphology of knowledge forms is a domain which Western thought has still to attack.
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If mathematics were a mere science like astronomy or mineralogy, it would be possible to define their object. This man is not and never has been able to do. We West-Europeans may put our own scientific notion of number to perform the same tasks as those with which the mathematicians of Athens and Baghdad busied themselves, but the fact remains that the theme, the intention and the methods of the like-named science in Athens and in Baghdad were quite different from those of our own. There is no mathematic but only mathematics. What we call "the history of mathematics"-implying merely the pro,ressive actualizing of a single
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cIIS of Mathematical Reality: A.n Anthropological Footnote
2349
dolle, and who was herself a scholar of distinction, l expressed a view widely held when she said: 2: "Nothing has afforded me so convincing a proof of the unity of the Deity as these purely mental conceptions of numerical and mathematical science which have been by slow degrees vouchsafed to man, and are still granted in these latter times by the Differential Calculus, now superseded by the Higher Algebra, all of which must have existed in that sublimely omniscient Mind from eternity." Lest it be thought that Mrs. Somerville was more theological than scientific in her outlook, let it be noted that she was denounced, by name and in public from the pulpit by Dean Cockburn of York Cathedral for her support of science. 3 In America, Edward Everett (1794-1865), a distinguished scholar (the first American to win a doctorate at Gottingen), reflected the enlightened view of his day when he declared: " "In the pure mathematics we contemplate ,absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven." In our own day, a prominent British mathematician, G. H. Hardy, has expressed the same view with, however, more technicality than rhetorical flourish: 5 "I believe that mathematical reality lies outside us, and that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations' are simply our notes of our observations." 6 Taking the opposite view we find the distinguished physicist, P. W. Bridgman, asserting that "it is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention." 7 Edward Kasner and James Newman state that "we have overcome the notion 1 She wrote the following works, some of which went into several editions: The Mechanism of the Heavens. 1831 (which was, it seems, a popularization of the Mlcanique Celeste of Laplace); The Connection of the Physical Sciences, 18S8; Molecular and Microscopic Science, 1869; Physical Geography, 1870. 2 Personal Recollections oj Mary Somerville, edited by her daughter, Martha Somerville, pp. 140-141 (Boston, 1874). . a ibid., p. 375. See, also, A. D. White, The History oj the Warfare of Science with Theology &:c, Vol. I, p. 225, ftn.* (New York, 1930 printing). 4 Quoted by E. T. Bell in The Queen of the Sciences, p. 20 (Baltimore, 1931). :5 G. H. Hardy, A Mathematician's Apology, pp. 63-64 (Cambridge, England; 1941). 6 The mathematician is not, of course, the only one who is inclined to believe that his creations are discoveries of things in the external world. The theoretical physicist, too, entenains this belief. "To him who is a discoverer in this field:' Einstein observes, "the products of his imagination appear so necessary and natural that he regards them, and would like to have them regarded by others, not as creations of thought but as given realities," ("On the Method of Theoretical Physics," in The World as 1 See It, p. 30; New York, 1934). 7 P. W. Bridgman, The Logic of Modern Physics, -po 60 (New York, 1927).
Ltslie A. White
that mathematical truths have an existence independent and apart from our own minds. It is even strange to us that such a notion could ever have
existed." 8 From a psychological and anthropological point of view, this latter conception is the only one that is scientifically sound and valid. There is no more reason to believe that mathematical realities have an existence independent of the human mind than to believe that mythological realities can have their being apart from man. The square root of minus one is real. So were Wotan and Osiris. So are the gods and spirits that primitive peoples believe in today. The question at issue, however, is not, Are these things real?, but Where is the locus of their reality? It is a mistake to identify reality with the external world only. Nothing is more real than an hallucination. Our concern here, however, is not to establish one view of mathematical reality as sound, the other illusory. What we propose to do is to present the phenomenon of mathematical behavior in such a way as to make clear, on the one hand, why the belief in the independent existence of mathematical truths has seemed so plausible and convincing for so many centuries, and, on the other, to show that all of mathematics is nothing more than a particular kind of primate behavior. Many persons would unhesitatingly subscribe to the proposition that "mathematical reality must lie either within us, or outside us." Are these not the only possibilities? As Descartes once reasoned in discussing the existence of God, "it is impossible we can have the idea or representation of anything whatever, unless there be somewhere, either in us or out of us, an original which comprises, in reality . . .u 9 ( emphasis ours) . Yet, irresistible though this reasoning may appear to be, it is, in our present problem, fallacious or at least treacherously misleading. The following propositions, though apparently precisely opposed to each other, are equally valid; one is as true as the other: 1. "Mathematical truths have an existence and a validity independent of the human mind," and 2. "Mathematical truths have no existence or validity apart from the human mind." Actually, these propositions, phrased as they are, are misleading because the term "the human mind" is used in two different senses. In the first statement, "the human mind" refers to the individual organism; in the second, to the human species. Thus both propositions can be, and actually are, true. Mathematical truths exist in the cultural tradition into which the individual is bom, and so enter his mind from the outside. But apart from cultural tradition, mathematical concepts have neither existence nor meaning. and of course, cultural tradition has no existence apart S Edward Kasner and James Newman, Mathematics and the Imagination, p. 359 (New York, 1940). II Principles 0/ Philosophy, Pt. I, Sec. XVIII, p. 308, edited by J. Veitch (New York, 1901).
The Locus of Mathe11llltical Reality: A.n Anthropological Footnote
2351
from the human species. Mathematical realities thus have an existence independent of the individual mind, but are wholly dependent upon the mind of the species. Or, to put the matter in anthropological terminology: mathematics in its entirety, its "truths" and its "realities," is a part of human culture, nothing more. Every individual is born into a culture which already existed and which is independent of him. Culture traits have an existence outside of the individual mind and independent of it. The individual obtains his culture by learning the customs, beliefs, techniques of his group. But culture itself has, and can have, no existence apart from the human species. Mathematics, therefore--like language, institutions, tools, the arts, etc.-is the cumulative product of ages of endeavor of the human species. The great French savant :emile Durkheim (1858-1917) was one of the first to make this clear. He discussed it in the early pages of The Elemen tary Forms of the Religious Life. Io And in The Rules of Sociological Method 11 especially he set forth the nature of culture 12 and its relationship to the human mind. Others, too, have of course discussed the relationship to the human mind. Others, too, have of course discussed the relationship between man and culture,lS but Durkheim's formulations are especially appropriate for our present discussion and we shall call upon him to speak for us from time to time. Culture is the anthropologist's technical term for the mode of life of any people, no matter how primitive or advanced. It is the generic term of which civilization is a specific term. The mode of life, or culture, of the human species is distinguished from that of all other species by the use of symbols. Man is the only living being that can freely and arbitrarily impose value or meaning upon any thing, which is what we mean by "using symbols." The most important and characteristic form of symbol behavior is articulate speech. All cultures~ all of civilization, have come into being, have grown and developed, as a consequence of the symbolic faculty, unique in the human species.14 w
Les Formes eUmentaires de La Vie Religieuse (Paris, 1912) translated by I. W. Swain (London, 1915). Nathan Altslnller-Court refers to Durkheim's treatment of this point in "Geometry and Experience," (Scientific Monthly, Vol. LX, No. I, pp. 63-66, Jan., 1945). 11 Les Regles de la Mithode SocioLogique (Paris, 1895; translated by Sarah A. Solovay and John H. Mueller, edited by George E. G. Catlin; Chicago, 1938). 12 Durkheim did not use the term culture. Instead he spoke of the "collective consciousness," "collective representations," etc. Because of his unfortunate phraseology Durkheim has been misunderstood and even branded mystical. But it is obvious to one who understands both Durkheim and such anthropologists as R. H. Lowie, A. L. Km.eber and Clark Wissler that they are all talking about the same thing: culture. Iliiiiiiiiii