Mathematicians and Their Times: History of Mathematics and Mathematics of History

MATHEMATICIANS AND THEIR TIMES

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NORTH-HOLLAND MATHEMATICS STUDIES

48

Notasde Matematica (76) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Mathematicians and Their Times Histow of Mathematics and Mathematics of History

LAURENCEYOUNG Distinguished Research Professor of Mathematics (Emeritus) at the University of Wisconsin-Madison, Wisconsin, U.S.A. Past Fellow of Trinity College, Cambridge, U.K.

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Library of Congress Cataloging in Publication Data

Young, Laurence Chisholm. Mathematicians and t h e i r times. (North-Holland mathematics s t u d i e s ; 48) Bibliography : p. I n c l u d e s index. i. Mathematics--History. I. T i t l e . QA2 1.Y68 510I.9 80 -274 13 ISBN 0-444-86135-1 ( U . S . )

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PREFACE

These lectures were given at the request of colleagues, in view of my long acquaintance with distinguished mathematicians. From early childhood, I have known almost legendary figures, known them better than people today who affect my daily life: the mailman who brings my letters to the door, the garage man who attends to my aged car, even the piano-tuner (bless him) who cares for my prized Ritmiiller from old Goettingen. As a student, I learnt a great deal from some mathematicians in this book, particularly Hardy and Littlewood, and before them the great Caratheodory -- as I mention in the text, I used to spend whole afternoons at his Munich apartment across from the English Garden. Mathematics became for me, not a store of past knowledge, but creative activity of the highest form, directed towards the future. Yet I learned that to study the newest developments is not enough: we have to go back to the sources, with a professional XX-th century outlook, to find what manner of men were able to plant the seed, and what difficulties and prejudices they had to overcome.

I learnt to associate mathematics, whether of yesterday or today, not just with definitions and theorems, algorithms and proofs, still less with masses of formulae, but with the creative minds of real people. The great ones, the pioneers, are much more than names attached to.some discovery: I was taught to ask, not just what they achieved, but what they tried, and in the case of mathematicians, the way they thought, even the I can relay only a small part of this, and errors they made. naturally I say most about mathematicians I know best. I have been at pains to talk, not just of greatness, but of the things that make up our reality: we know them only too well as errors, trials, tribulations and sometimes tragedies. This may help readers generally, not only those mathematically inclined. Everyone, sooner of later, has to make a crucial decision, and not depend on others. Some decide like automata, by rule of To them I can only say with Cromwell: "I beseech you thumb. think it possible you may be mistaken." However, those willing to reflect might consider reading, as they may in this book, how others did decide, and what came of it.

...,

The past is a laboratory of human experience, from which to learn. Each age can look at its lengthy past, judging each event by what came after. This is by no means the usual historical procedure: historians have enough trouble studying V

vi

Preface

each separate period, as if they were observing the motion of some inanimate object. Humanity, living things generally, are affected at each instant by the memory of earlier times far in the past, and by the anticipation of what is yet to come. This implies a more complex relationship to time, and without it much that occurs cannot be properly understood. In this way, history becomes anachronistic and mathematical: we need a mathematics of history. The sudden influence of centuries far back can be disturbingly destructive: I devote some time to explaining this phenomenon which amounts to a partial return to barbarism. However, the influence can also be dramatically beneficial, as we know from the Renaissance. This last occurs when we choose to learn from the more civilised aspects of certain times gone by. This is the way in which modern mathematics was almost totally changed by the introduction of Set Theory. The past, like everything else,needs to be studied and understood, if it is to be beneficial. In this respect, the history of mathematics deserves very special study. The ideas of mathematicians, after many centuries, can be incomparably more useful than the laws laid down by the most successful of rulers, backed by overwhelming force. Past ideas, past efforts are useful, much more than past achievements, now long out of date. This makes the lives of I am not over-fond of great mathematicians so instructive. "great": each reader should decide for himself, and not be told by me, whether a particular figure of the past, or present, I deserves to be considered great. I prefer the word find people with useful ideas real. It is in this sense that some mathematicians, whom I knew extremely well, are so real to I am delighted that other people now call them great, but me. for me, I see no gain in their losing their reality and becoming mere objects of admiration.

&.

Ours is an age of much re-appraisal and searching criticism, even disruption. I do not like disruption; others have not liked the re-appraisal. Poincard wrote: "Previously, when someone invented a new function, it was for some practical purpose, now people invent them solely to find fault with the way our fathers reasoned." Our likes and dislikes are neither here nor there: reality, truth, understanding -- these matter. I sympathise with those who do not like aspersions cast, as much as with those who do not like being told what to praise, and with those who want to forget whatever is crude or cruel. I once agreed, a little rashly, to oblige a now deceased Modern Languages colleague, who needed help in presenting, at a humanities conference, scenes from his translation of a heart-rending As narrator I had to acclaim the literary old Spanish play. merit; and then as actor, to make credible an off-stage suicide. I think it proper to revise appraisals: some of the honour that tradition assigned to great men can now go to neglected pioneers, who saw further ahead. We must learn from past errors, and they are many. Our world is still very young: enormous changes take place in as little as a dozen generations or less, a genetic time that many life forms go through in a single of

Preface

vi i

our days. Much is still crude: it is all the more encouraging that in our crude world greatness is still possible. I am reminded of a tiny tree, no longer than my little finger, that had somehow forced its way through a cement crack in my driveway. I rescued it, and planted it elsewhere: it is now twice as high as my house. We do not learn from the crudeness of our world, nor by pretending that it is not there. We learn by seeing how it can be overcome. Galois' fame rests on the pages he was able to write hastily overnight, before being killed at dawn in a duel, as he knew he would be. Newton overcame, by fantastic hard work, his Weierstrass apparently total lack of mathematical ability. made up for four wasted years by some fifteen years of teacher training and school teaching, during which he somehow managed to develop mathematically. Plato had to travel for eleven years, be sold as a slave, and be fortunately freed by friends, before he could bring himself to return to Athens, to found his academy in the city that had condemned his teacher, Socrates. I need not multiply the examples; many more will be found in this book. We learn from them that the greatest hardships can be overcome: we must be prepared for them. We learn, too, that And there are things in life that make hardship worthwhile. what of those who failed to survive their ordeals? The potential great ones, the saviours, perhaps, of humanity, slaughtered or stifled before they could help? The mute mathematical Miltons and Hampdens left to rot on village greens? The very fact that so many great men had to overcome obstacles of that magnitude, suggests that many more were wasted, whom humanity could not afford to lose. This is why, in our critical times, we must do what we can for the coming generation. For my part, I am only passing on what I was privileged to receive. In the preparation and collection of material, and in the tedious checking and proof-reading, many friends have assisted me: it is impossible to give adequate credit to them all. They include the late professor Behnke; the late professor Littlewood, together with other friends from Trinity College, Cambridge; also, my sister, Dr. Tanner and members of the Harriot Seminar; Dr. Burkill and Dame Mary Cartwriqht in Cambridge; Professor Sharma in Edmonton; Professors d'?mbrosio and Lintz at Unicamp in Brazil; and of course the members of the Mathematics Department and the Research Center, at Wisconsin, particularly Professors I also appreciate the secretarial help in Nohel and Miles. these places, especially here at Wisconsin, in regard to the camera-ready copy. L. c. Y.

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CONTENTS

Preface

V

INTRODUCTION

Part I, Generalities 1. The n o n - m a t h e m a t i c a l framework 2. The t r a i n e d o b s e r v e r 3. The e m b e l l i s h m e n t o f t r u t h 4. The i n t r u s i o n o f p e r s o n a l f e e l i n g s 5. Some r u l e s o f c r i t i c a l a n a l y s i s Part 6. 7. 8. 9. 10. 11.

11, E a r l y H i s t o r y

The The The The The The

h i s t o r i c a l s i g n i f i c a n c e of mathematics meaning o f m a t h e m a t i c s distant past Greek s c h o o l o f m a t h e m a t i c s d e s t r u c t i v e bends miraculous p r e s e r v a t i o n

P a r t 111, The Slow R e n a i s s a n c e 12. Dismal p r o s p e c t s i n W e s t e r n E u r o p e 13. The p i o n e e r s who came t o o s o o n 14. The l e a d e r s h i p t h a t c o u l d n o t l a s t P a r t I V , The New B e g i n n i n g 15. The b e g i n n i n g s o f a n a l y s i s 16. Newton 17. Preparing f o r bankruptcy 18. Euler 19. The r i s e o f m a t h e m a t i c s i n P a r i s

CHAPTER I , The Romantic P e r i o d 20. The f l o u t i n g o f a r i t h m e t i c 21. The f o u n d i n g o f t h e E c o l e P o l y t e c h n i q u e 22. Gauss 23. The f i r s t r o m a n t i c s 24. Riemann a n d Weierstrass 25. Between two w o r l d s

ix

13 16 19 22 32 37 39 41 49 56 72 89 103

114

124 130 142 152 187 204

Con t e n t s

X

CHAPTER 11, The D r e a m of a C o n c e p t u a l Age, and t h e Hard R e a l i t y of S t r u g g l e f o r E x i s t e n c e The u n p r i z e d h e r i t a g e 26. The s i g n i f i c a n c e o f P l a t o f o r m a t h e m a t i c s 27. 28. C a n t o r and S e t s 29. Goettingen

216 222 226 233

CHAPTER 111, The Age of C o n t r a d i c t i o n s and o f t h e G r e a t Q u e s t i o n Marks 30. The a r t of t h e p o s s i b l e 31. The e x p a n s i o n o f m a t h e m a t i c s 32. Cambridge 33. The Hardy-Littlewood era 34. The b a t t l e g r o u n d o f i d e a s 35. The u n c e r t a i n f u r t h e r blooming 36. The o t h e r s i d e of c a t a s t r o p h e

249 255 266 275 29 5 313 322

INDEX

339

Introduction I , Generalities; 11, Early History; 111, The Slow Renaissance; IV, The New Beginning Part I

1. The Non-mathematical Framework In a course in mathematics, we would start with axioms: the beginners that in a sense we all are may not always like it. This is a course in history (of mathematics): therefore we merely try to introduce some semblance of method and logic, but even this requires certain essential preliminaries. I believe in method: this may be a mathematician's illusion, but I shall try to communicate it. Of course, when we meet absurdity, I shall not ignore it in the hope that it will go away. I hope non-mathematicians will bear with me: I understand mathematician in a wide sense, but I have not devised these lectures for the exclusive benefit of fellow mathematicians, past, present or future, nor even for the many persons I consider to be, unknowingly perhaps, mathematicians at heart. History of mathematics is for all 3 us largely new territory, otherwise we would not be studying it. Before entering it, we should survey its broad lines and its pitfalls. Otherwise we might be as misled as some beloved Head of State, or some popular footballer, come to visit an integrated or comprehensive school whose pupils have suddenly become angels. I refer here, not to some places I could name, but to a school in say a reasonable London suburb, or in the States in Illinois, but the confusion would not be lessened if this were to suggest to someone that London might be a suburb of Illinois - - worse confusions are not totally unknown in our remarkable era, not even in diplomacy. To avoid any similar misunderstanding, let me repeat that this is not a course in mathematics,but one of history. However history itself may require some mathematics to help understand it, and this is what I call the mathematics of history. Similarly, we speak of the mathematics of econom i c s as meaning any mathematics with which to understand economics - - if anybody can.

I am not going to present, under the cloak of history, mathematical theories long past: they can be exciting, but less so at second hand. The place to study them is in specialist works, or in the original texts. What I shall present, in connection with the mathematicians we shall come to, is rather some picture of the background in which they worked, the times in which they lived and the personal problems they encountered: times often as cruel, and problems as acute, as our own. These 1

2

Norman L. Alling

function field (in one variable) over the reals of genus 1, whose field of constants is IR. x

and

y

such that

over IR(x).

As such it contains elements

E =IR(x,y).

Necessarily

An algebraic equation for

called a defining equation for

E.

is algebraic

over IR(x)

y

If

y

s =

2 or 1

Weierstrass p -function and its derivative are in E = IR(9,n') :

If

s = 0

will be

then the E.

Further,

and, of course, the following holds:

then this is not the case.

An elliptic function Q

can be defined using, for example, the Weierstrass zeta function (14.32:2), such that

E(Y

Ort

) =

IR(Q,Q') and, for which the

following holds: (2)

(Q')2

=

-(Q 2 + a 2) (Q2 + b 2 ) ,

To calculate

a

and

b

with

in terms of

a > 0

and

b > 0.

t, Jacobie's theory of

elliptic functions proves very useful.

These results can be

found in Chapter 14. Let (3)

P(x) E Ax4 + 4Bx3 + 6Cx2

be of degree E :: IR(x,y)r

n

=

where

field over IR exist unique

+

4Dx + E

E

IR[x]

3 or 4, and have distinct complex roots.

y2

=

P(x) : then

E

is an algebraic function

of genus 1, whose constant field is IR. s

E

{2,1,0) and

E(Yslt) are IR-isomorphic.

t > 0

Let

s

such that and

t

tively the species and geometriE modulus of

E

and

t

from the coefficients of

P(x).

There

and

be called respecE.

The problem

addressed and solved in Chapter 17 is the following: s

Let

compute

On developing ideas

that go back at least to Euler and Legendre one can see (53.2)

I. Generalities

3

are no exceptions. I need not remind you of the laughable little lapses that older persons like myself can be prone to, nor of the queer spellings that survive proof-reading. We know too, how greatly our senses can be deceived by hypnosis or by a magician's sleight of hand, and perhaps by unusual experiences that account for a vast literature on UFO's. Few people, however, realise how unreliable our senses can be under quite ordinary circumstances, and how important it is for the trained observer to be constantly on his guard. In the first place, contrary to common opinion, hallucinations are not at all uncommon. They can even arise from deliberate, but unconscious, self-deception. In mathematics, they can be theorems dreamt up overnight, and, like some desert mirages, too beautiful to be true. In other cases the deception is far from deliberate: one of the most common is the interchange of left and right. This eerie experience I have had myself: back in the States after a year in Britain, I imagined I had guarded against driving on the wrong side by noting that the steering wheel was in the half of the car nearest to the middle of the road. But when the other motorists started blinking their lights and sounding their horns, something clicked in my brain. What I then actually saw, while hearing a perfectly audible click, was my car suddenly reflecting itself about its central line. We ''see,"not what is there, but the picture our brain chooses to provide. If left and right are occasionally confused, this is not too extraordinary, considering that the actual image transmitted from the retina is upside down. Even so, I was surprised to learn that there is a whole science of Physiological Optics, due to the great Helmholtz. A few pages in the Introduction to Sommerfeld's Optics make clear that there are at all times quite systematic differences between what we "see" and what "is." This accounts for the old "Goethe versus Newton'' controversy: the tragedy of Goethe's life was his failure to recognise it. The differences normally affect colour, or in a black and white picture, intensity. To an untrained observer, they seem most erratic, and they are so great that, if he examines a true X-ray picture of a section of a patient's brain, he may be totally deceived as to the position of a tumour requiring surgical removal. A t one time astronomers were similarly deceived by what they took to be interference lines in X-ray plates of an eclipse of the Moon; and before that, there were the famous ''canals1'seen on the planet Mars, but now known to be purely subjective. Fortunately, thanks to Helmholtz, the trained observer, with even only the rudiments of mathematical analysis, can see his way through these apparently erratic distortions of our vision. Broadly speaking, we do not see white as white, nor black as black, nor do we see the intermediate shades of grey in their proper intensity. The message from our brain, like the news in our daily paper, emphasizes and exaggerates contrasts, as if in mathematics we were to replace a function by its derivative. The brain aims at providing "news": uniformity is not newsworthy. Similarly in the U . S . , when the Press was giving

Introduction

4

prominence t o American c a s u a l t i e s i n t h e Far E a s t , t h e unchanging, b u t e q u a l l y l a r g e , numbers of murders and o t h e r crimes w i t h i n t h e U . S . were i g n o r e d . I t was n o t a c a s e of dec e p t i o n , b u t of newsworthiness. The phenomenon thus a f f e c t s t h e h i s t o r i a n j u s t as much as h i s b r o t h e r s c i e n t i s t s : t h e " f a c t s " they must argue from, a r e t h o s e t h a t happened t o be "newsworthy. ' I O f c o u r s e what i s news i n one c o u n t r y , need n o t b e s o i n a n o t h e r : h i s t o r i c a l e v e n t s thus have a m u l t i p l i c i t y of v e r s i o n s . One such e v e n t , c i t e d i n a c i r c u l a r by my c o l l e a g u e Creighton Buck, w a s t h e War of 1 8 1 2 . For a U . S . c i t i z e n i t w a s won by t h e U . S . a g a i n s t B r i t a i n ; f o r a Canadian, by Canada a g a i n s t t h e U . S . ; f o r a B r i t o n , Russian, German, o r S p a n i a r d , by h i s country i n each c a s e a g a i n s t France; f o r a Frenchman, i t was a C i v i l War -- i n which France covered h e r s e l f w i t h g l o r y . P a r a d o x i c a l l y , t h i s l a s t view seems t o me n e a r e s t t o t h e t r u t h : but t h e g l o r y I speak of was n o t won on b a t t l e f i e l d s .

3.

The Embellishment

of Truth

The human mind, a f t e r a n i n i t i a l e a g e r n e s s t o l e a r n , a c q u i r e s a decided a v e r s i o n t o f u r t h e r s t o r a g e of mere f a c t s . However i t u s u a l l y c o n t i n u e s t o a c c e p t them embellished and s u g a r c o a t e d , j u s t as h o t e l s , which a r e f u l l , somehow always manage t o f i t i n s u f f i c i e n t l y d i s t i n g u i s h e d p e r s o n s . This need t o embellish i s a t the- r o o t of t h e d i s t o r t i o n of f a c t s , d e s c r i b e d i n t h e preceding s e c t i o n . Of c o u r s e i t does n o t s t o p t h e r e . Embellishment i s n o t n e c e s s a r i l y a bad t h i n g : adding c o l o u r t o a p i c t u r e i n b l a c k and w h i t e . provide i n s i g h t , i t may even i n s p i r e . In t h i s a p o e t s e e s what t o t h e r e s t of us i s f a r from c a s e , much should perhaps be f o r g i v e n t o t h o s e from i n t o l e r a b l e d u l l n e s s :

it is like I t may h e l p t o

way, p e r h a p s , c l e a r . I n any who r e s c u e us

There i s a q u a l i t y i n c e r t a i n people which i s above a l l a d v i c e , exposure, o r c o r r e c t i o n . Only l e t a man o r woman have DULLNESS s u f f i c i e n t , and they need bow t o no e x t a n t a u t h o r i t y . A d u l l a r d recognizes no b e t t e r s ; a d u l l a r d c a n ' t s e e t h a t he i s wrong; a d u l l a r d has no s c r u p l e s of c o n s c i e n c e , no doubts of p l e a s i n g , o r succeeding, o r doing r i g h t , no qualms f o r o t h e r p e o p l e ' s f e e l i n g s , no r e s p e c t b u t f o r t h e f o o l h i m s e l f . How can you make such a f o o l p e r c e i v e t h a t he i s a f o o l ? Such a person can no more s e e h i s own f o l l y t h a n he can s e e h i s own e a r s . What myriads of s o u l s a r e t h e r e of t h i s admirable s o r t ! This w a s w r i t t e n , of c o u r s e , long b e f o r e t h e admirable s o r t l e t l o o s e on t h e r e s t of us t h e d r e a d f u l r e p e t i t i o n s o f t h e i r mast e r p i e c e s , t h e a d v e r t i s i n g j i n g l e s . By comparison, t h e d u l l e s t of books and l e c t u r e s can be a r e l i e f : i n d e e d , t h e r e i s even a chance t h a t something worthwhile may t u r n up t h e r e . Thackeray may have guessed t h i s , f o r he w r i t e s elsewhere:

Introduction from about 1970 to the present.

5

The results presented in Part

I11 depend heavily on the work of Euler, Legendre, Gauss, Abel, Jacobi, Cauchy, Liouville, Weierstrass, Klein and others, which apply to the real case.

For example, Abel's defining equation

was = (1

(1)

-

c 2@ 2) (1 + e 2@ 2) ,

with

c

which can be used to define a Mobius strip

and

YlIt;

e > 0, where as

Jacobi' s was (2)

(sn')2 = (1

-

2 2 2 sn ) (1 - k sn 1 ,

with

which can be used to define an annulus

0 < k < 1,

YZlt.

(Hence, although

equations of type (1) and type (2) are equivalent over rc are not equivalent over IR.)

they

The initial inversion of real

elliptic integrals of the first kind to form elliptic functions, by Gauss, Abel, and Jacobi, was later superceded by using theta functions, having general complex parameters, to define elliptic functions; thus the case of complex defining equations become dominant, leaving the real case to languish, even though it is in the real case in which many of the applications to physics and engineering lie.

Thus the research in this monograph is

highly dependent on the results obtained on the real case between 1750 and late in the 19'th century.

Since there seems to be no

unified treatment of this subject, which prepares the way for the real case whenever possible, the author has written one in Part I and I1 of this monograph, as an introduction to Part 111. In addition,the author has written many historical and bibliographic notes throughout this volume.

These are included

in an effort to show what the founders of the subject actually wrote.

Further, those notes could serve a s a guide to possible

6

Introduction

kind arises from t h e v e r y t h i n g t h a t causes a p e r s o n t o devote h i s l i f e t o t h e t e d i o u s s t u d y o f documents of long a g o : a m e d i e v a l i s t may f e e l s o i n t u n e w i t h h i s p e r i o d , t h a t he would sooner d i e t h a n p r i n t about i t a s i n g l e derogatory word. We may w e l l a s k : When s h a l l we have a r e a l account of t h o s e times and heroes - - no good-humoured p a g e a n t , l i k e t h o s e of S c o t t romances - - b u t a r e a l a u t h e n t i c s t o r y t o i n s t r u c t and f r i g h t e n honest people of t h e p r e s e n t day, and make them t h a n k f u l t h a t t h e baker governs t h e world today i n p l a c e of t h e baron? I quote Thackeray a g a i n , because a s you s e e he i s v e r y modern i n o u t l o o k ; more p r e c i s e l y , he h a s helped form our views. I wish I could s a y t h a t h i s t o r i a n s of mathematics a r e l e s s i n c l i n e d t o pander t o t h e g u l l i b i l i t y of man than o t h e r h i s t o r i a n s - - l e s s i n c l i n e d t o g i v e way t o t h e e x t r a o r d i n a r y f a c u l t y w e have of r e c o n s t r u c t i n g a m o r e p a l a t a b l e p a s t . We a r e , a l l of u s , most g u l l i b l e , when we h e a r what we want t o h e a r . ' The f i r s t purpose of h i s t o r y i s s u r e l y t o develop a c r i t i c a l s e n s e . I t i s a t h a n k l e s s t a s k . E r n e s t Renan, w r i t i n g about t h e French Revolution, i n c u r r e d t h e h a t r e d and v i o l e n t p r o t e s t s of many of h i s countrymen; he could only s a d l y commen t :

The obvious i s powerless a g a i n s t t h e chimera t h a t a n a t i o n embraces w i t h a l l t h e f e r v o u r of i t s h e a r t . '1 might quote in this connection the provocative title 'IHistory remembered, recovered and invented," by Bernard Lewis, Yeshiva University.

4.

The I n t r u s i o n o f Personal F e e l i n g s

I have no wish t o f l o g an a i l i n g h o r s e : p a t r i o t i s m , r e l i g i o n , t r a d i t i o n , t h e o l d and c h e r i s h e d v a l u e s d i s t o r t today t h e views of r e l a t i v e l y few - - and t h e s e few i n c l u d e perhaps t h e most ded i c a t e d among u s , t h e l a s t t o r e s i s t t h e g r e a t wave o f s e l f i s h and c y n i c a l d i s i l l u s i o n m e n t t h a t w a s bound t o r e s u l t from cons t a n t and shameless r e p e t i t i o n s of commercials t h a t p r e s e r v e us from some supposed g i g a n t i c economic d i s a s t e r . But i n h i s t o r y , p a r t i c u l a r l y , p a s t d i s t o r t i o n s a r e no l e s s important t h a n t h o s e w e i n t r o d u c e o u r s e l v e s . Renan himself w a s lucky t o be w r i t i n g when he d i d , and n o t a t a time when a Holy Revolutiona r y Fury w a s e x e c u t i n g some of t h e b e s t c i t i z e n s of France. Even s o , - - and now I am t a l k i n g of a l l f i e l d s of s c i e n c e , -i t i s t h e d e d i c a t e d few, r a t h e r than t h e s e l f i s h many, who b r i n g p r o g r e s s ; and they need most t o be on t h e i r guard. I must be c a r e f u l how I say t h e s e t h i n g s , o r I s h a l l f i n d mys e l f accused o f b e i n g h y s t e r i c a l . Today t h e s l i g h t e s t h i n t of emphasis o r e x a g g e r a t i o n i s taken a s a s i g n of poor judge-

I.

Generalities

7

ment, u n l e s s one s a y s what p e o p l e want t o h e a r . This i s c e r t a i n l y a s u p e r f i c i a l way of e v a l u a t i n g what someone s a y s , b u t i t i s t h e beginning of a very n e c e s s a r y independent c r i t i c a l a n a l y s i s . Must we a l l o w o u r s e l v e s t o be taken i n by t h e mere power of a p e r s o n ' s words? I f s o , we have l e a r n t l i t t l e from r e c e n t h i s t o r y . Exaggeration and emphasis were t h e vogue i n V i c t o r i a n d a y s , p a r t i c u l a r l y on t h e s t a g e . This a l l changed w i t h o u r c e n t u r y . One person who understood t h e change extremely w e l l was t h e mathematician G H Hard . I f he wis-hed t o g i v e h i g h p r a i s e , he was most c a d g i n w i t h something s l i g h t l y derogat o r y : i n t h i s way people would be l e s s i n c l i n e d t o q u e s t i o n h i s judgement. I n one c a s e , t h i s b a c k f i r e d . L i t t l e w o o d mentions i t i n h i s r e m i n i s c e n c e s . Hardy was commenting a t d i n n e r i n H a l l on a d e l i g h t f u l i n a u g u r a l l e c t u r e by t h e p o e t A . E . Housman, a s Kennedy P r o f e s s o r of L a t i n : "Of c o u r s e a b i t on t h e cheap s i d e . . . ' I A r e f l e x nudge by L i t t l e w o o d reduced Hardy t o s i l e n c e a t t h e w o r s t p o s s i b l e moment. In t h e l e c t u r e , Housman had maintained t h a t a s c h o l a r ' s j o b has n o t h i n g t o do w i t h l i t e r a r y v a l u e s o r l i t e r a r y c r i t i c i s m . A s a happy parody p u t i t : "Does t h e chemist like s u l p h u r e t t e d hydrogen?" The q u e s t i o n of our l i k e s and d i s l i k e s , and whether w e s h o u l d a l l o w them t o i n f l u e n c e our s e a r c h f o r t r u t h , i s t h e s u b j e c t of a n o t h e r anecdote quoted by L i t t l e w o o d . The p h i l o s o p h e r s R u s s e l l and Moore w e r e having one of t h e i r many p h i l o s o p h i c a l t a l k s . Moore, i n c i d e n t a l l y , had a superb enl i g h t e n e d s i m p l i c i t y , Littlewood t e l l s u s , and was t o t a l l y i n c a p a b l e of p r e v a r i c a t i o n . Suddenly R u s s e l l s a i d : "Moore, you d o n ' t l i k e m e , do you?" "No," s a i d Moore. The d i s c u s s i o n continued w i t h o u t a f u r t h e r word on t h e p o i n t from e i t h e r s i d e . The q u e s t i o n of our l i k e s and d i s l i k e s i s perhaps n o t q u i t e so s i m p l e . I t i s d i f f i c u l t f o r some of u s t o i g n o r e them, and I am n o t a t a l l s u r e t h a t we s h o u l d make t h i s a f a s t r u l e . If we a r e s e r i o u s i n o u r p u r s u i t of t h e t r u t h , of c o u r s e our emotions become involved. The most extreme c a s e I know w a s t h a t of W i t t g e n s t e i n , a man t o whom I owe a g r e a t d e a l . L i t t l e w o o d , who knew him s t i l l b e t t e r , has t h i s t o s a y about him (among o t h e r t h i n g s ) : Having i n h e r i t e d a p r o f i t a b l e e n g i n e e r i n g b u s i n e s s , W i t t g e n s t e i n wished t o l e a r n t h e more fundamental s c i e n c e behind i t , and was a r e s e a r c h s t u d e n t a t ManChester under Rutherford a t t h e t i m e I w a s t h e r e . Everyone thought him a c r a n k . He came t o T r i n i t y i n 1 9 1 1 , where R u s s e l l had j u s t r e t u r n e d a s a L e c t u r e r . H e w a s s t i l l r e g a r d e d as a c r a n k . A f t e r a y e a r o r s o , h e asked R u s s e l l whether he thought him c a p a b l e of doing philosophy. R u s s e l l t o l d him t o w r i t e down h i s i d e a s , e x p e c t i n g , I am p r e t t y s u r e , t h a t t h e r e w a s n o t h i n g d o i n g , b u t he w a s i n s t a n t l y converted . . . I n 1 9 3 0 t h e T r i n i t y Council gave m e t h e q u e e r e s t j o b of my l i f e : t o d e c i d e whether t o make W i t t g e n s t e i n a T i t l e B Fellow. H e was a p e r s o n a l f r i e n d - - suppose

8

Introduction t h e answer had t o be no! I s t r u g g l e d through 5 o r 6 s e s s i o n s f o r him t o expound h i s i d e a s . He almost never f i n i s h e d a s e n t e n c e : e x p l a n a t i o n was p r a c t i c a l l y i m p o s s i b l e , because t o understand A , you had t o unders t a n d B , C , e t c . and then A . There w a s h i s c h a r a c t e r i s t i c o b l i g a t o of "I am an a s s . " However, i n t h e end I f e l t t h a t I could honourably say y e s . . . W i t t g e n s t e i n was much t h e s t r a n g e s t man I e v e r knew, f u l l of p a s s i o n a t e c o n v i c t i o n s I had no c l u e t o , though I w a s one of t h e few dons he d i d n o t damn f o r something - - i n c u r a b l e f r i v o l i t y , l a c k of s e r i o u s purpose, I d o n ' t know. (He had no contempt f o r o r d i n a r y p e o p l e . ) He was himself always i n t e l l e c t u a l l y on duty a t f u l l s t r e t c h , and probably d i d n ' t understand t h e o r d i n a r y human need t o r e l a x (though he d i d go t o t h e cinema). He dined j u s t once a t H a l l d i n n e r . I s a t o p p o s i t e , and e v e r y t h i n g seemed t o have gone b e a u t i f u l l y : he had even put on a t i e and exchanged some p l e a s a n t c h a f f . But according t o Hardy, something I s a i d a t t h e v e r y end shocked him f o r f r i v o l i t y , and he never dined a g a i n , though he forgave me p e r s o n a l l y . . . We f r e q u e n t l y m e t walking round t h e T r i n i t y g a r d e n , and once g o t on t h e s u b j e c t of semi-popular a r t i c l e s on s c i e n c e . I s a i d I g o t somethin worthwhile o u t of them, though w i t h no i l l u s i o n s about e p t h of u n d e r s t a n d i n g . He d i d n o t d i s s e n t , b u t when I p r a i s e d Eddington, he exploded. H e d i d n o t s a y what was wrong, b u t h i s e x a c t words were: " I f you want t o know what I f e e l about Eddington, i t i s t h a t i f I had a choice of Eddington and W i t t g e n s t e i n i n Heaven, and Eddington and W i t t g e n s t e i n i n H e l l , I would choose H e l l . "

+

That someone s u p e r f i c i a l l y l a b e l l e d a crank should f e e l i n t h i s way about anyone w i t h a t r a c e of s u p e r f i c i a l i t y , i s unde'rstandable. But t h a t i s n o t t h e p o i n t : h a t r e d of superf i c i a l i t y was t h e g r e a t f o r c e behind a l l t h a t W i t t g e n s t e i n t r i e d t o do. It i s a h a t r e d t h a t I must admit I l a r g e l y s h a r e . When I was very v e r y young, e a g e r t o l e a r n and quick t o jump t o conclus i o n s , I g o t my hands on a s o - c a l l e d "Calculus" book of about 1850. I s o l v e d t h e problems, and supposed I had understood something, e v e r y t h i n g . Years l a t e r , when my e l d e s t s i s t e r w a s r e a d i n g Mgray, I r e a l i s e d t h a t I had been "had." I am n o t o f t e n a n g r y , b u t my anger became a d r i v i n g f o r c e : sound mathematics was n o t l a c k i n g a t home, and I was determined t o g e t on t o t h e most modern works. Even now, i n popular w r i t i n g s i n t h e g e n e r a l a r e a of mathematics o r i t s h i s t o r y , I g e t p u t o f f by a Chapter Heading such a s : From Hypatia t o Hamilton," which sounds to'me l i k e a s l o g a n . Nor can I s t a n d a t i t l e such as "The Mysterious U n i v e r s e , ' ' which a d v e r t i s e s a s much as t h e b e e r o r t h e smoke, t h a t may o r may n o t r e f r e s h . P l e a s e do n o t t h i n k me c y n i c a l : t h i s l a s t t i t l e d a t e s from 50 years ago, long b e f o r e a d v e r t i s i n g became what i t i s . A t t h e t i m e , I o b j e c t e d more t o t h e s u g g e s t i o n of J e a n s , t h e a u t h o r , t h a t t h e C r e a t o r i s a mathematician: t o me t h i s would

I.

Generalities

9

have seemed i n b e t t e r t a s t e , i f t h e a u t h o r were n o t c l a i m i n g t o be a mathematician t o o . A good book should s t a n d on i t s own, n o t by some s p e c i a l d i s p e n s a t i o n o f Providence. This resembles t o o much t h e s p e c i a l p l e a d i n g by a Master of my o l d College a t t h e l a s t Judgement, a s k i n g t h e Judge t o remember t h a t H e t o o , i n a s e n s e , i s a T r i n i r y man. But my poor o p i n i o n of Jeans does n o t extend t o Eddington, my s u p e r v i s o r when I w a s I s a a c Newton S t u d e n t . Littlewood l i k e w i s e thought h i g h l y o f Eddington. He r e c a l l s t h a t he was on h o l i d a y w i t h R u s s e l l when E d d i n g t o n ' s b r i l l i a n t r e p o r t on R e l a t i v i t y had appeared, and t h a t h e was expounding t h e r e p o r t . Suddenly R u s s e l l b u r s t o u t : "TO t h i n k t h a t I have s p e n t my l i f e on m s ! 'I A s you s e e , g r e a t men can have s t r o n g f e e l i n g s t o o . I s h a l l n o t enumerate a l l t h e ways i n which t h e s e may d i s t o r t what we s e e , and what w e r e a d o r w r i t e . I want t o mention s p e c i a l l y w i s h f u l t h i n k i n g , and h e r e t h e d i s t o r t i o n may sometimes be a l l t o t h e good. Imagine two young s c i e n t i s t s d i s c u s s i n g t h e r e s u l t s of an experiment. One s a y s : "Wouldn't i t have been n i c e i f s o and s o had happened!" - - "Oh," s a y s t h e o t h e r , "I never even thought of checking t h a t ! " And s u r e enough, i t t u r n s o u t t o have been t h e c a s e . This i s n o t t o o i n f r e q u e n t i n r e s e a r c h . Here I a m concerned more w i t h r e a d i n g and s t u d y i n g , b u t t h e d i f f e r e n c e i s n o t as g r e a t as p e o p l e t h i n k . In r e a d i n g a d i f f i c u l t p a s s a g e , t h e r e can be t h e same i n t e n s e p l e a s u r e as i n r e s e a r c h , when f i n a l l y t h e r e a l s i g n i f i c a n c e f l a s h e s i n t o o u r b r a i n . Tibor Rado used t o c a l l i t t h e l u c i d moment: i t comes, u n f o r t u n a t e l y , r a t h e r r a r e l y , b u t i t i s the r e a s o n why I b e l i e v e i n day-dreaming.

5.

%me Rules o f C r i t i c a l A n a l y s i s

I s a i d e a r l i e r t h a t h i s t o r y should h e l p develop a c r i t i c a l s e n s e : t h e same a p p l i e s even more t o t h e h i s t o r y of mathem a t i c s . And c e r t a i n l y , i n a l l s u b j e c t s o u t s i d e of mathematics, t h i s c r i t i c a l s e n s e i s b a d l y needed. S t e n d h a l went s o f a r a s t o say t h a t much s o - c a l l e d s c i e n c e o t h e r t h a n mathematics, was i n h i s day t h e parade ground of simple-minded clumsy enthus i a s t s and c l e v e r h y p o c r i t e s -- nigauds e n t h o u s i a s t e s e t h y p o c r i t e s a d r o i t s . The need f o r c o n s t a n t checking i s t h e t r a d i t i o n o f t h e U n i v e r s i t y of Wisconsin, as e x p r e s s e d by a famous p l a q u e , a f u l l - s i z e d copy of which my f a t h e r proudly brought home t o Europe i n 1 9 1 7 , a f t e r one of h i s many voyages around t h e w o r l d , t o hang above h i s g r e a t r o l l - t o p d e s k . I t speaks of t h a t f e a r l e s s s i f t i n g and winnowing, by which a l o n e t h e t r u t h can be found. However t h e development of a p r o p e r c r i t i c a l s e n s e i s n o t always f o s t e r e d i n u n i v e r s i t i e s , n o t even perhaps a t Wisconsin i t s e l f . One d i f f i c u l t y i s t h e examination system: t h e importance of a t o p i c and t h e e x t e n t t o which i t should be t r e a t e d more t h a n s u p e r f i c i a l l y , i s determined, i n many s t u d e n t s ' minds, e n t i r e l y by t h e n a t u r e of t h e q u e s t i o n s t o be s e t on i t i n an examination. I n t h e c a s e of h i s t o r y , t h e r e s u l t i s paroded i n "1066 and a l l t h a t . " What a s t u d e n t r e t a i n s from a c o u r s e has been compared t o t h e r e f l e x i o n i n a s i l v e r t e a p o t . I n c i d e n t a l l y , a t Wisconsin

Introduction

10

t h e r e a r e p r i z e s f o r t e a c h i n g : I have wondered about them, and whether they encourage embellishment r a t h e r than f e a r l e s s s i f t i n g . Wisconsin now a l s o has e v a l u a t i o n s by s t u d e n t s of the courses taken - - e x c e l l e n t , no d o u b t , i f i t means t h a t t h e s t u d e n t s l e a r n t o e v a l u a t e and c r i t i c i s e . U n f o r t u n a t e l y , I have i t on good a u t h o r i t y t h a t i n one s t u d e n t dormitory a t l e a s t , l o t s were drawn t o decide how t h e e v a l u a t i o n forms should be f i l l e d o u t . This i s why I do n o t b e l i e v e t h e s t u d e n t s , o r anyone e l s e , l e a r n much from f i l l i n g o u t forms. I c e r t a i n l y do n o t w i s h t o advocate courses on t h e c r i t i c a l e v a l u a t i o n of c o u r s e s : they could only l e a d t o a r e g i m e n t a t i o n c o n t r a r y t o t h e whole s p i r i t of academic freedom. What i s i m p o r t a n t i s t h a t t h e s t u d e n t i n each c o u r s e should l e a r n i n t h a t course n o t t o t a k e on t r u s t what he i s t o l d , b u t t o form h i s own independent and r e l i a b l e judgement. A t f i r s t h i s judgement may be v e r y s u p e r f i c i a l , f o r i n s t a n c e a s I s a i d e a r l i e r , he may be s o l e l y i n t e r e s t e d i n what w i l l h e l p i n exa m i n a t i o n s , o r e l s e he may completely d i s b e l i e v e what he does n o t l i k e . I n t i m e , however, he may judge i n a more mature manner. E x a c t l y t h e same a p p l i e s t o t h e r e a d i n g of books, as t o t h e a t t e n d i n g of c o u r s e s .

I n a number of p l a c e s i n t h i s book, I s h a l l t r y t o show t h a t t h e b e n e f i t a s t u d e n t r e c e i v e s from a book o r a c o u r s e , does n o t n e c e s s a r i l y depend on t h e e x c e l l e n c e of t h e p r e s e n t a t i o n and such m a t t e r s . For every such book o r c o u r s e , t h e r e i s a book t h a t you w r i t e i n your-own mind: i t i s t h e . q u a l i t y oft h e l a t t e r t h a t you should be concerned w i t h . There a r e , f o r i n s t a n c e , i n mathematics, s t i l l books t h a t a r e no b e t t e r t h a n t h e Calculus of 1850 t h a t I mentioned e a r l i e r . From such a book, you can l e a r n t o c r i t i c i s e i t : R . H . Bing used t o enjoy t e a r i n g such a book t o p i e c e s , by p o i n t i n g o u t t h e e r r o r s . I t i s almost a p i t y t h a t bad mathematical books a r e becoming s c a r c e r ! F o r t u n a t e l y i n o t h e r s u b j e c t s one has t o be c r i t i c a l of any book, t h e p r e s e n t one i n c l u d e d . My f a t h e r once wanted me t o t a k e up economics. He had given t h e same a d v i c e t o Keynes, when he s u p e r v i s e d him i n Cambridge; l a t e r he gave i t a l s o t o a young f r i e n d of mine, who ended up a s a Very Import a n t Person a t t h e I n t e r n a t i o n a l Labour O f f i c e . I n my c a s e , I was t o r e a d S m i t h ' s Wealth of Nations - - I found i t a wond e r f u l book. Then I was p u t on t o M a r s h a l l ' s P r i n c i p l e s of Economics - - wonderful t o o . A f t e r t h a t i t w a s Gide P r i n c i p e s d'Economie P o l i t i q u e -- t h i s t h i r d I found as wonderful a s t h e two p r e c e d i n g . The t h r e e books a r r i v e d a t t o t a l l y d i f f e r e n t c o n c l u s i o n s . I decided t o s t i c k t o mathematics: I w a s t o o g u l l i b l e f o r an economist. But when t h e g r e a t d e p r e s s i o n came, I began t o wonder whether g u l l i b i l i t y was what economists a r e p a i d f o r . Of course today t h i n g s a r e d i f f e r e n t ,

. .

It i s easy t o be wise a f t e r t h e e v e n t . My m i s t a k e w a s a common o n e , and I was s t i l l r a t h e r young. I became s i m i l a r l y d i s s a t i s f i e d w i t h Atomic P h y s i c s , b e f o r e t h e days of Heisen, Schroedin e r and D i r a c . I d i d n o t r e a l i s e t h a t i t i s o r us t o t u r n a isadvantage i n t o an opportunity. As i t happens, b o t h economics and Atomic Physics have become very

9

-ga

I. Generalities

11

mathematical, and the type of mathematics they use is quite close to my own work. I also did not realise that a course or a book is not a finished product, simply to be photographed as accurately as possible in our brains: this would make human beings no better than glorified parrots or gramophones. It was more than 10 years later, on the boat returning to Newcastle from the O s l o International Congress, that I learnt, in conversations with the mathematician L. P. Eisenhart (1881-1965), of a better conception of the purpose of a University or Graduate School, that of be innin an education; the talks more than made up, in retr-or much tossing about on the rough North Sea. Clearly a book or a course of lectures is only a small part of such a beginning: the important part, the reader's or the student's own contribution, is yet to come. To him, at least, it is the important part. The proper way to study a book or a course, perhaps the worst to pass an examination, differs only in degree from research concerning it. The most important is reading between the lines: noting not just what is there, but what has been omitted. Why was this said, and not that? If we were following a game of chess, we would try out moves that did not occur there. This is critical analysis: from it we learn most. In particular we learn to recognise real mistakes, rather than apparent ones. When I originally gave these lectures at Wisconsin, the audience was very mixed, ranging from beginning undergraduates to graduate students in a variety of fields, and to some of my own colleagues in mathematics or in sister departments. A s far as my colleagues were concerned, I did not evaluate their performance -- the boot was on the other foot! It was a pleasure and a challenge. But I had to assign grades to the students. I solved this by asking each student to select for himself or in collaboration with others, some topic connected with the course, and to study in detail some material bearing on it. I recommend a similar study to the reader. There is, I think, no better way of acquiring judgement and self-reliance,.thanby studying in detail material beyond one's normal studies. In these lectures, I sometimes go out of my way to draw attention to the many errors and discouragements that great men have had to struggle with. Many people feel that such matters should be passed over, and that we should let bygones be. However it is from such signs of apparent weakness that we learn most. I have known some of these great men; others were, as I say, only a handshake away. I am sure that many would not have taken it amiss. They were not then legendary figures, such as inhabit palaces or the like. It is mainly after their deaths that they became much more than acquaintances from just round the block. I would not wish to suggest that the progress since their time makes us as good as them; nor do I imply that mathematicians are intrinsically better than other scientists, or even than journalists and lawyers. I am thinking only of the standards of truth: I believe that future generations

12

Introduction

w i l l be no more s a t i s f i e d than I am w i t h p r e s e n t o r p a s t s t a n dards of many o t h e r w i s e e x c e l l e n t p e r s o n s . I n any branch of h i s t o r y we have t o t a k e account of t h e s t a n d a r d s of persons i n many walks of l i f e , and we have t o make a p p r o p r i a t e c o r r e c t i o n s .

One of t h e main t r o u b l e s t o be guarded a g a i n s t i s t h e widespread f e e l i n g , c l e v e r l y used a g a i n and a g a i n i n Court c a s e s , of informat h a t something can be concluded from a t o t a l &k t i o n . I n my y o u t h , t h e r e was n o t a shred of eviaence t o support t h e h y p o t h e s i s of d r i f t i n g c o n t i n e n t s : t h i s h y p o t h e s i s was t h e r e f o r e s h e e r nonsense -- today i t i s t h e a c c e p t e d t r u t h . Some people f e e l such c o n t r a d i c t i o n s a r e a s i g n of p r o g r e s s , o t h e r s t h a t p r o g r e s s h a s f a r t o go. I am n o t t o o impressed w i t h Law Court t r u t h . I was once a w i t n e s s i n a f a t a l a c c i dent c a s e : my e v i d e n c e , and t h a t of t h e tyre-marks n o t e d by the p o l i c e , c l e a r l y i n d i c a t e d t h a t t h e defendent was t o blame. This evidence was simply c o n t r a d i c t e d by t h e o t h e r a l l e g e d ' " w i t n e s s e s , " none of whom had been v e r y a p p a r e n t a t t h e t i m e . The Court r u l e d t h a t t h e r e was c o n f l i c t of evidence. Vae v i c t i s : t h e dead cannot t e s t i f y . This i s a l s o one o f t 6 e things t h a t makes h i s t o r y prone t o e r r o r . A l l t h i s w h i l e , I have been speaking a s ' i f t h e one important t h i n g i n h i s t o r y , o r i n s c i e n c e i n g e n e r a l , were f a c t u a l a c c u r a c y : t h i s i s a most p r i m i t i v e and c h i l d i s h view. I n t h a t c a s e , we would r e c o r d , o u t of t h e g r e a t mass of everyday f a c t s , t h o s e t h a t seem s p e c i a l l y s t r i k i n g o r s t r a n g e : m i r a c l e s , genuine o r e m b e l l i s h e d , would f i l l t h e h i s t o r y books. However some space would be saved f o r p r i n c e s s e s born w i t h s i x t o e s on one o r b o t h f e e t , and f o r t h e remarkable e x p l o i t s and loves of some k n i g h t , a worthy p r e d e c e s s o r o f our sportsmen and f i l m - s t a r s . Much of t h i s , a more s o p h i s t i c a t e d g e n e r a t i o n would throw o u t as t r a s h , and t h e r e would be l i t t l e l e f t t o t e a c h i n s c h o o l s , o t h e r t h a n what w e f i n d i n "1066 and a l l t h a t . It

We would have a s i m i l a r s t a t e of a f f a i r s i n geometry, i f t h e p o i n t s of t h e l i n e , o r p l a n e , were spoken of a s ' ' f a c t s , " and we were o n l y i n t e r e s t e d i n a c c u r a t e l y measuring t h e i r p o s i t i o n s . I cannot say how we would become i n t e r e s t e d i n t h e number I T , s i n c e t h e r e would be n o t h i n g s p e c i a l about i t ; b u t once we had decided t o pay a t t e n t i o n t o i t , our e f f o r t s would be d i r e c t e d t o o b t a i n i n g and checking more and more of i t s decimals -- a l l of which would be f u t i l e , s i n c e t h e r e would be n o t h i n g s p e c i a l about t h e number i n t h e f i r s t p l a c e . A c t u a l l y , i n geometry, we a r e i n t e r e s t e d , n o t i n i n d i v i d u a l p o i n t s , b u t i n t h e r e l a t i o n s and f i g u r e s t h a t connect them. I n h i s t o r y , and g e n e r a l l y i n s c i e n c e , our main i n t e r e s t l i e s i n a supposed c a u s a l i t y . I t i s only by v i r t u e of t h i s r e l a t i o n t h a t t h e g r e a t mass of f a c t s a c q u i r e s an i n t e l l i g i b l e s t r u c t u r e . I n some s c i e n c e s , c a u s a l i t y can be checked t o some e x t e n t by experiment, a l t h o u g h we can never be q u i t e s u r e t h a t t h e experiment h a s n o t i t s e l f a l t e r e d t h e whole s e t up. H i s t o r y i s one of t h e s c i e n c e s where t h i s checking i s n o t v e r y f e a s i b l e : t h e r e supposed c a u s a l i t y i s a l l we can a i m f o r .

11. Early History

13

Newton speaks of gravitation, not of actual attraction. Critical analysis has thus to be for or against causality between events. This has not always been seen: we read about the strangeness, for instance, the near miracle by which Fourier came to escape execution. Let me add to this section, that no advice that I could give, can apply to all students, especially not to the best: each person ultimately forms his methods of study. Mistakes and clumsiness are a failing of great men. In a famous laboratory, when the great man was on his round, stark terror preceded him, and the delicate experiments would hurriedly be hidden away.

PART I1

6. The Historical Significance of Mathematics As a mathematician, I should be the last to speak of this: it sounds like pleading ro domo, or self-advertisement. However it is the only way to intro uce causality. In no uncertain terms, I intend to point to the paramount importance of mathematics. After what I have said about exaggeration, you should specially heed my warning to check and criticize. Between mathematics and history, a presumed causality can be expected in several ways. One of these I have mentioned: it concerns mathematical laws, or something like them possibly controlable by us, that affect history; I call it the mathematics of history. Sufficiently developed, it would constitute a separate subject, but here I shall merely draw attention to it when the occasion arises. In this section, I shall limit myself to the way in which the history of mathematics can affect the rest of history. We shall see many instances of this, but there is one particularly timely. When I gave these lectures, Americans were celebrating the second centenary of Independence, something that I was fully in favour of, like everyone in the British Isles. Even in 1776, there was support for the c o l o nists, or at least little support for the idiocy of the British government of the day, an idiocy unparallelled even in recent times; people on the other side of the Ocean are not necessarily without friends where they came from. But American Independence was bound to come. All the war did, was to ensure a particular date: in history, this is of minor importance. On the other hand, without mathematics there would have been no American Independence at all: no United States as we know it, largely Protestant, Anglo-Saxon--atmost, perhaps, a few trading posts in a Northern replica of South America, if not in some vast Spanish colonial Empire, administered from

%

14

Introduction

Havana, o r Mexico C i t y . The d i s c o v e r y of America by Columbus, and t h a t of Newfoundland by Cabot, were h i t o r m i s s a f f a i r s . The d i s c o v e r i e s were i n e v i t a b l e , and t h e s u c c e s s o r f a i l u r e of a p a r t i c u l a r e x p l o r a t i o n merely a f f e c t e d t h e t i m e - t a b l e . What w a s more t o t h e p o i n t was t h a t t h e r e were s h i p s capable of making t h e t r i p , and compasses t o g u i d e them. Of c o u r s e t h e r e had a l s o t o be men brave enough t o r i s k t h e i r l i v e s . However t o s e t t l e l a r g e numbers o f c o l o n i s t s a c r o s s t h e Oceans r e q u i r e s more than a half-chance of e v e r g e t t i n g t h e r e . A s l a t e a s t h e beginning of t h i s c e n t u r y , I understand t h a t a s h i p r a n aground on t h e P a c i f i c c o a s t of South America. The weather had been cloudy, b u t f o r a b r i e f moment they had t h e chance t o t a k e t h e i r b e a r i n g on a s t a r , and they f a i l e d t o do s o . They had no i d e a t h a t they were being c a r r i e d along by a s t r o n g s e a - c u r r e n t -they were s t e e r i n g by compass, and by t h e i r r e c o r d of t h e d i s t a n c e covered r e l a t i v e t o t h e water. A f t e r t h e discovery of America, and t h a t of Newfoundland, t h e r e were f u r t h e r voyages from Spain and P o r t u g a l , and from o t h e r c o u n t r i e s on t h e s e a - c o a s t of t h e European c o n t i n e n t ; however t h e r e were v i r t u a l l y none from England f o r .another s i x t y y e a r s . Some voyages from Plymouth t o t h e Guinea c o a s t and B r a z i l , by W. Hawkins a p p a r e n t l y e x c i t e d l i t t l e i n t e r e s t . (See f o r i n s t a n c e 'The Expansion of E l i z a b e t h a n England, by A. L . Rowse.) You may wonder how an England, ravaged and i m poverished by t h e Wars of t h e Roses, and r u l e d by t h e p i t i l e s s Tudor t y r a n n y , could suddenly become, under Queen E l i z a b e t h , a prosperous ocean-going n a t i o n , c a p a b l e , a f t e r a miraculous d e f e a t of t h e Spanish Armada, of i n i t i a t i n g t h e c o l o n i s a t i o n of North America. This has been r e c e n t l y s t u d i e d by D r . John Roche, of Oxford U n i v e r s i t y . It was a s t r a n g e combination o f c i r c u m s t a n c e s , t h a t j u s t e x a c t l y came a t t h e r i g h t t i m e . A n important f a c t o r w a s t h e m a r r i a g e of P h i l i p of Spain t o Queen Mary of England; however what w a s c r u c i a l (according t o D . W. Waters, The A r t of Navigation i n England, London 1 9 5 8 ) , was t h e f a c t t h a t w h i l e England and Spain were t h u s a l l i e d f o r a s h o r t t i m e , someone named Stephen Borough brought t o England a 1 5 5 1 book i n Spanish by Martin CortGs on t h e A r t of Naviga(An e a r l i e r 1545 copy i s i n t h e Bodleian L i b r a r y a t tion. Oxford. I n c i d e n t a l l y , t h e a u t h o r seems u n r e l a t e d t o t h e I t was t h e book used i n t h e famous school "Conquistador.") f o r P i l o t s and Navigators i n S e v i l l e . It w a s t r a n s l a t e d i n t o English and p u b l i s h e d i n London i n 1561.

-

We s e e h e r e t h e s t r a n g e n e s s - - on which I commented b e f o r e - of much recorded h i s t o r y . The t i m e was r i p e : England's c o a s t a l t r a d e had j u s t been h a r d h i t by t h e l o s s of C a l a i s ; n a v i g a t i o n on t h e High Seas was an i n v i t i n g a l t e r n a t i v e . The T r i n i t y Houses, founded by Henry V I I I f o r P i l o t T r a i n i n g , expanded t h e i r program t o an Ocean-going o n e , and i n c l u d e d p r i v a t e t e a c h i n g : Martin Cort6s became t h e s t a n d a r d t e x t . I want you t o imagine t h e tremendous i m p a c t of t h a t book. changed t h e h i s t o r y o f t h e w o r l d .

It

11.

Early History

15

P r e i o u s t o M a r t i n C o r t & , o n l y a few m a r i n e r s who knew L a t i n couyd s a i l t h e seven Seas w i t h a r e a s o n a b l e chance of r e a c h i n g t h e i r d e s t i n a t i o n : t h e y used t h e o l d a s t r o n o m i c a l t a b l e s p r i n t e d i n Venice i n 1483, which had been c i r c u l a t e d i n manus c r i p t f o r 200 y e a r s , and which had o r i g i n a l l y been commissioned by Alphonso X , King of Leon and C a s t i l e . P r e v i o u s t o t h e s e A l p h o n t i n e T a b l e s , a s t i l l o l d e r s e t of t a b l e s had been used throughout Europe: t h e y were t h e Toledan T a b l e s , t h e L a t i n v e r s i o n of t h e t a b l e s due t o t h e Arab i n v e n t o r of t h e a s t r o l a b e , A r z a c h n e l . A s you must know, mathematics i n t h e Middle Ages came t o Europe v i a t h e Arabs. C o r t 6 s ' s book, "Breve compendio d e l l a s p h e r a y de l a a r t e d i n a v i g a r , " i s b r i e f mainly by comparison w i t h a 1537 T r e a t i s e of t h e Sphere by Pedro Nunes. T h i s does n o t make i t j u s t a b r i e f s y n o p s i s of t a b l e s and formulae - - one of t h o s e r e p e l l e n t t h i n g s t h a t some p e o p l e i n s i s t on c a l l i n g p r a c t i c a l . Cort4s t a k e s g r e a t p a i n s t o e x p l a i n e v e r y t h i n g simply and i n words, w i t h t h e h e l p of diagrams h a v i n g moving p a r t s , and he puts i t a l l i n i t s h i s t o r i c a l c o n t e x t . In a prologue, address e d t o t h e Emperor C h a r l e s V , he s t r e s s e s t h e d a n g e r s of t h e s e a , and t h e added h a z a r d of t r u s t i n g t o p i l o t s who can s c a r c e l y r e a d , and a r e s c a r c e l y of c a p a c i t y t o l e a r n . He c o n t i n u e s w i t h a l e s s o n from which o u r age might p r o f i t : "And whereas i n t h e f i f t h Chapter of t h i s book, I have made m e n t i o n , t h a t t h e g o v e r n a l l o r s t e e r a g e ought t o be committed t o e x p e r t men and o f good u n d e r s t a n d i n g , h e s h o u l d s e e t h a t nowadays t h e i g n o r a n t presume t o govern o t h e r , which were n e v e r a b l e t o r u l e o r govern t h e m s e l v e s . " It w a s a good book, i t went t o s e v e r a l e d i t i o n s , and i n t h e c o u r s e of t i m e , a b o u t 1580, i t w a s succeeded by o t h e r s . One of i t s r e a d e r s , i n c i d e n t a l l y , was Thomas H a r r i o t (1560? - 1 6 2 1 ) , a m a t h e m a t i c i a n of whom I s h a l l s a y more l a t e r , and who d i d a g r e a t d e a l h i m s e l f t o i m prove t h e a c c u r a c y of n a v i g a t i o n a l a s t r o n o m y . 2 What i s h i s t o r i c a l l y s o i m p o r t a n t i s t h a t s u d d e n l y , t h a n k s t o t h e t r a n s l a t i o n of t h e book of M a r t i n CortGs, t h e a r t of n a v i g a t i o n r e q u i r e d no L a t i n . T h i s i s what r a i s e d up England and made p o s s i b l e t h e P i l g r i m s e t t l e m e n t s i n North America. Witho u t i t , w i t h o u t t h i s key t o freedom, n o t h i n g i n Europe o r America c o u l d have checked t h e power of t h e I b e r i a n p e n i n s u l a a t s e a . The United S t a t e s p r i d e s i t s e l f on i t s freedom: t h e key t o i t was m a t h e m a t i c s . I n t h e XVI-th c e n t u r y , mathematics and t h e a r t of n a v i g a t i o n meant t h e same t h i n g : t h i s i n v o l v e d t h e u s e of s i m p l e n a v i g a t i o n a l i n s t r u m e n t s , and some astronomy and s p h e r i c a l t r i g o n o m e t r y . I s p e a k of n a v i g a t i o n i n open w a t e r s , on t h e High S e a s . This made of mathematics a r e s p e c t e d and u s e f u l s c i e n c e . A t o t h e r p e r i o d s , t h e p u b l i c h a s t a k e n o t h e r views of mathem a t i c i a n s . Popular s t o r i e s suggest they a r e l i k e a b l e , absentminded, innocuous i n d i v i d u a l s , whose minds a r e i n t h e c l o u d s , and who can r a t h e r e a s i l y be t r i p p e d up and t a k e n advantage o f . T h e r e ' s t h e astronomer who g a z e s a t a s t a r , and f a l l s i n t o a w e l l . T h e r e ' s t h e geometer who's l o o k i n g f o r a lemma, t h a t ' s i n a n o t h e r lemma, t h a t i s n ' t r e a l l y t h e r e . . . A t t h e

16

Introduction

opposite extreme, we see also the hatred of the common man for what he cannot grasp. That's when a King James I (common enough by our standards) writes against "White Magic," to brighten the lives of any lrgoodrl witches, sorcerers and demons -- much as Mein Kam f brightened those of lrgoodll Jews and their w i c k e h d We can be thankful that those days are over . . . Ah, but are they? ' O r Trevelyan llIllustratedEnglish Social History", where we read also of the bankrupt state inherited by Elizabeth, and of the exclusion, after the loss of Calais, o f English cloth merchants, first from the Netherlands in 1567, then from Hamburg in 1577.

*Harriet is so important to modern mathematics that I shall defer speaking about him until Chapter 111, section 34, except f o r incidental references. 7.

The Meaning

of Mathematics

It is perhaps desirable that you should be able to describe what you are to study the history of. This, as I shall make clear in a moment, is not as easy as it sounds, even if you happen to be professional mathematicians! Moreover, failure to produce a suitable description may have unpleasant consequences, at a time when even a paper that was once respected throughout the world threatens "sacred cows'' and "aristocrats of intellect." Of course it is not the same paper at all: otherwise there would have been no incidental insult to one of the major religions of India. I can only say that when these lectures were given, I appreciated the fact that a fairly substantial audience was willing to risk being turned into those dastardly sacred cows, or whatever other words are used for thinkin man. Personally I believe with the geometer and h i s t E T Z T E h h e m a t i c s Loria that civilisation cannot exist without mathematics, but I sometimes wonder whether the dawn of civilisation will ever come? Let me make clear, first of all, that it is time to raise the level of what we are prepared to call mathematics. In the words of Edmond Rostand, in Cyrano de Bergerac, "I will not burn incense at the shrine of mediocrity." As I understaa history of mathematics, it is no longer relevant that, according to Plato, some form of arithmetic existed prior to its supposed founder: Plato asks, "Perhaps then, without Palamides, Agamemnon would have been ignorant of the number of his feet?" Far be it from me to cast aspersions on the excellent persons who study primitive arithmetic 'and the like. Their work has little to do with what I call mathematics, but if they called themselves anthropologists, it would certainly give tone to what is still full of popularisations. I do not object to popularisations per se, but once the habit is formed, there is no knowing where it will lead. At present, half the young

11. Early History

17

teachers of America are out of work, because of popular lectures by anthropologists and others on the population explosion in the United States, a country in which the population density is rather small, but in which, they assure us, that if the o ulation keeps doubling there will eventual1 be no room for gE athrooms.' If I were writing a history o the misuse of arithmetic, I would have much to say on this, but there is a perfectly respectable branch of mathematics which deals with population genetics, and the mathematicians who work in it are not scaremongers. If I mention these things, it is certainly not out of any personal animosity towards the brave people willing to forego personal comfort to study at first hand the primitive tribes of man in our time, but because I am anxious to draw attention to instances of causality: the evils of our time, according to the great Hyppolite Taine, are "products," just like sulfuric acid, or sugar.

d

However, wherever we start what we choose to call mathematics, it is clear that the subject has changed a great deal over the ages. For instance, it has changed since the XVI-th century: astronomy and navigation are no longer included - - navigation has become highly technical and mechanical, and astronomy has powerful telescopes and computers. Even s o , I personally prefer not to distinguish astronomy from mathematics: we make too many distinctions, contrary to the whole spirit of mathematics. At a meeting, my father once went up to a young man and said, 'You must be an Englishman." It was van der Corput. He replied, "No, I am a Dutchman." My father said, "Same thing." Van der Corput was delighted: he told me the story himself. Some people distinguish rigidly between "Pure'' and "Applied." Anyone who passes from Pure to Applied, or vice-versa, is looked upon as some sort of traitor: I have never heard anything so sanctimonious; it is like having the wars of religion all over again. In my Cambridge days, Burkill and I ran for two years a joint Seminar with Stan Ulam. He then counted as "Pure;" now he counts as a physicist. He's the same Ulam to me . . . You may think it a matter of taste, but I hope to convince you as we go along. Distinctions are for botanists and lawyers: they put each thing in its special pigeon-hole. Mathematicians look for what matters. People ask, "Isn't mathematics very abstract?" -- as if that made mathematics impossibly difficult. The truth is that law and botany are difficult: we have to keep in mind all manner of ridiculous legal or botanical distinctions. This is why law is costly, and why books on botany look like dictionaries. Abstraction is what makes mathematics easy: it's the freedom to forget what doesn't really matter, to remember only what's important; it's the way to escape the compartmentalisation mania that keeps our minds in little cages. Mathematics has changed, and is changing all the time, but it has one aspect that does not change: it deals with what's important. That makes it an art, even more so than I said

18

Introduction

t h a t t h e o b s e r v a t i o n and r e c o r d i n g of f a c t s i s one: indeed i t i s r a t h e r l i k e Chinese p a i n t i n g . The essence of a r t i s t o choose w h a t ' s i m p o r t a n t , but i n Chinese p a i n t i n g t h e a r t i s t d e l i b e r a t e l y i g n o r e s t h e r e s t , This i s p r e c i s e l y what t h e mathematician does. Remember t h e Cheshire Cat i n A l i c e i n Wonderland? And how i t f a d e s u n t i l only t h e g r i n remains, l a s t ; ; f a m H E G R I N , t h a t ' s w h a t ' s i m p o r t a n t : i s i t v e r y hard t o t h i n k of j u s t t h e g r i n ? I n t h e same way, we might imagine a doughnut, of t h e u s u a l ring-shaped k i n d , and l e t i t slowly fade u n t i l t h e r e remains j u s t THE HOLE. I n mathematics, i t ' s t h e h o l e t h a t m a t t e r s , we speak of i t a s t h e t o p o l o g i c a l c h a r a c t e r i s t i c . Everything e l s e about t h e doughnut w e choose t o f o r g e t . I hope t h a t those of you who a r e n o t mathematicians w i l l n o t f i n d t h i s f o r g e t t i n g too h a r d . I f w e ' r e t o t a l k about h i s t o r y of mathematics, we must l e a r n t o r e c o g n i s e what t h e r e i s i n t h i s c o n s t a n t l y changing mathematics t h a t remains, and allows u s t o keep g i v i n g i t t h e same name. This reminds m e of a most awkward predicament i n which I once found m y s e l f , and i f I t e l l you about i t , you may b e saved a s i m i l a r embarrassment. It was s h o r t l y b e f o r e World War 11, i n Cape Town, South A f r i c a . I had gone o u t t h e r e a s P r o f e s s o r and Head of Department. On a r r i v i n g , I r e c e i v e d an i n v i t a t i o n t o t h e "Owls' Club" - - RSVP, 8 pm, evening d r e s s . I a c c e p t e d , and on t h e appointed day, i n o r d e r t o do j u s t i c e t o t h e expected meal, I decided a s a r e s u l t of previous experience w i t h i n v i t a t i o n s , t o do without lunch o r t e a . I came t o t h e Owls' Club q u i t e hungry - - n o r was my a p p e t i t e diminished when I had been shown i n t o an ante-room and had c h a t t e d f o r some 20 minu t e s w i t h club o f f i c i a l s . I was then e s c o r t e d t o t h e main assembly room: t h a t was when I discovered t h a t t h e r e was no d i n n e r , and t h a t I was t o a d d r e s s t h e meeting. I had never made a speech i n my l i f e - - n o t even a t my wedding. What could I t a l k about? People had come i n f u l l evening d r e s s t o h e a r me. With a g r e a t v o i d i n s i d e , I t r i e d t o p r a i s e t h e Cape's h o s p i t a l i t y -- i t was l i k e c l u t c h i n g a t s t r a w s . C l e a r l y I was expected t o s a y more t h a n t h a t ! There was only mathematics l e f t : what does one s a y about such an a b s t r u s e t h i n g t o people i n evening d r e s s ? I t r i e d : i t was n o t my b e s t effort.

I s a i d I had thought mathematics was one t h i n g when I was a small c h i l d , and I had found i t w a s n ' t t h a t . I had thought i t was a n o t h e r t h i n g when I was p r o g r e s s i n g through High School, and i t w a s n ' t t h a t . I went on and on, I spared them t h e techn i c a l d e t a i l s -- how could anyone even t h i n k of g i v i n g such an audience anything s e r i o u s , l i k e a p r o o f ? So I s a i d n o t h i n g about mathematics a t al1;except t h a t , each time, i t w a s n ' t t h a t . I d i d n o t add t h a t p r i o r t o v i s i t i n g t h a t c l u b I had thought a mathematician's j o b was one t h i n g , and once again i t wasn't t h a t . It was t h e lamest speech imaginable, b u t b a s i c a l l y I was r i g h t : mathematics i s changing a l l t h e t i m e . And now I s h a l l t a l k o f some of t h e changes. I hope t h a t some of t h e t h i n g s I s h a l l

11. Early History

19

say will help if you find yourselves in my predicament. 'We s h o u l d r e f l e c t t h a t , f a r from c o n s t a n t l y doubling and r e d o u b l i n g , t h e p o p u l a t i o n o f t h e Roman Empire d ecr eas ed u n t i l i t was t o o weak t o w i t h s t a n d t h e a t t a c k s by b a r b a r i a n s .

8. The Distant Past There are excellent and detailed works on the history of mathematics up to about 1800. They deal with what I prefer to call the infancy of mathematics. The standard work is that of E i t z Cantor, Geschichte der Mathematik; I have a slight preference for Loria, Storia delle Matematiche, because its author is a mathematician - - a geometer reasonably well-known in his time. There are a number of other works, most of which I shall not even cite, since I intend to give of this whole period of infancy only the briefest of outlines. We can learn more, I think, from the past 200 years, with perhaps an occasional flashback.

I should mention first the fascinating story that is being studied by modern archeological research, promising mathematical developments of which we inherited hardly a trace. They occurred in Ancient Babylon, and are described in Neu ebauer's Vorgriechische Mathematik (Springer 1934) and in B k ' s Das mathematische Denken der Antiquen (Goettingen 1957). I should mention that Pliny the Elder reported that Assyrian astronomical observations went back 100,000 years, a figure that Loria dismisses as meaningless. However it is fairly clear that in Babylon Pythagoras's theorem was familiar long before Pythagoras, but presumably without any proof. The archeological work has been full of surprises: there was certainly in Babylon a remarkable understanding of algebra, that went beyond anything of its kind elsewhere in antiquity, as far as solving the cubic. One surprise, not of a mathematical or historical nature, is specially reported .by Loria (p. 28) : the discovery in 1904, in the archives of the Society of Natural Sciences of Goettingen, of a memoir by F. G. Grotenfeld, presented in 1802 and rejected on the grounds that its value could not be estimated. In this memoir, the extraordinary difficulty of deciphering Babylonian inscriptions was for the first time successfully overcome. (Other instances of rejection, or lengthy delay, of works of fundamental importance, will be met in the course of these lectures. Referees sometimes prefer a "safe mediocrity" and yesterday's problems, but one good editor is better than a thousand referees.) At the time of Neugebauer's book, a single Babylonian text - in two pieces, one in Berlin, the other in the British Museum - - was available to make intelligible the Babylonian solution of the cubic. One wonders how much more of Babylonian mathematics may have been lost in the destruction of Babylon. Much

20

Introduction

the same applies, in the Middle Ages, to Arab mathematics and to the destruction of Bagdad by the Turks. On such matters, there is a vast amount of archeological research to be done. There was of course also some mathematics in Ancient China and India. The dates are unreliable, but there is reason to believe that, as early as the VIII-th century BC, the statement of Pythagoras's theorem was known, together with various approximations to IT. In particular the triangle of sides 3 , 4 , 5 was used. Such things occur in India in altar-construction in the Shulva-Sutra, a geometrical work using the Shulva, or measuring tape. Moreover early astronomical achievements in China find a champion in Montucla Histoire des Mathgmatiques; he wrote before it was reZEEed,Loria tells u s , that the earliest eclipse claimed to have been predicted by the Chinese, was not in fact visible in China. As to the mathematics of India, opinions are extremely divergent: some assert that the Hindoos were far more advanced than I suggest, others that they copied from the works they are alleged to have anticipated. Loria comments that the East has a great capacity for inventing a more exciting past. He was writing for Western readers! But he wrote in 1929: in the West it was then not yet fashionable to see flying saucers in the Bible, and in Greek, Egyptian and Hindoo mythology. Neither had the freedom of the U . S . press, and the capacity for invention by American newsmen, reached their present prestigious heights. The balance has been restored: the assumption that the West is more free from wishful thinking, is wishful thinking. We are back in the old paradox that all Cretans are liars. It is remarkable that Loria mentions the palatable past phenomenon at all; and he certainly leans over backwards at times, Got to glorify some countrymen. Here it is my turn to lean over backwards as a Westerner. The main thing is, however, that.all these early developments had little ultimate effect on mathematics as we know it. The Babylonian period may have affected mathematics rather indirectly, via the much inferior mathematics of Egypt, just as a little of the Bagdad period trickled through to the West during the crusades or via Spain. It could hardly have done so very much. Egyptian mathematics is one more enigma to add to the riddles of the Sphinx: it is full of, to u s , absurd prohibitions. Apart from the fraction 2 1 3 , the only fractions allowed were those with numerator unity, except in the oldest inscriptions, where 314 still appears: isn't that progress in reverse? But there are many mysteries in Ancient Egypt. The first main document to throw light on the mathematics of Egypt was the Rhind papyrus, now in the British Museum except for a fragment in New York. There is still relatively little to go on, and much of it opens up endless conjectures. It is clear that priests were highly educated and powerful: they might have limited the amount of knowledge that laymen were allowed. Their amazing influence seems to have extended far outside their country: a sacrificial cloth has been found in Egypt,

TI.

Early History

21

on which a r e l i g i o u s ceremony i s d e s c r i b e d on one s i d e i n E g y p t i a n , and on t h e o t h e r i n E t r u s c a n . This i s r e m i n i s c e n t of t h e famous R o s e t t a s t o n e , by which E g y p t o l o g i s t s were f i n a l l y a b l e t o u n d e r s t a n d Egyptian h i e r o g l y p h i c s . There i s a l s o some q u e s t i o n of a r e l a t i o n s h i p between Egyptian p r i e s t s and t h e Delphic O r a c l e : how e l s e should one e x p l a i n t h e c u r i ous r e f u s a l by Delphi of a p r o t e c t i v e Greek m i l i t a r y f o r c e during t h e P e r s i a n i n v a s i o n , and a l s o t h e Delphic prophecy of a d e f e a t f o r Greece, and of t h e need t o s u r r e n d e r , a prophecy somehow changed i n r e t r o s p e c t t o p r e t e n d t o have f o r e t o l d " d i v i n e Salamis?" A s you s e e , t h e c o n j e c t u r e s a r e e n d l e s s , n o t only i n t h e mathematical c o n t e x t ; and t h e r e t h e main problem i s o f t e n t o i n t e r p r e t what w e f i n d . I t i s e s s e n t i a l t o abandon our i n g r a i n e d preconceived n o t i o n s , based on even t h e most elementary a r i t h m e t i c of our t i m e s . We have t o look a t t h i n g s p u r e l y i n t h e i r h i s t o r i c a l development, absurd though i t may seem t o u s . The p a r a l l e l w i t h Law may be h e l p f u l : many t h i n g s i n Law may seem, t o a layman, absurd and t o t a l l y a t v a r i a n c e w i t h t h e most elementary p r i n c i p l e s of j u s t i c e and common s e n s e . To t h e s t u d e n t of Law, and s i m i l a r l y t o t h e s t u d e n t of E g y p t i a n , and a l s o of Babylonian, mathematics, i t g r a d u a l l y a p p e a r s t h a t t h e s u b j e c t h a s a l o g i c of i t s own, deeply dependent on i t s h i s t o r i c a l development, and a l s o on t h e o l d - f a s h i o n e d terminology i n which i t i s couched. Of c o u r s e people complain t h a t t h e s t r u c t u r e of Law i s designed t o maximise l e g a l f e e s : s i m i l a r complaints were d o u b t l e s s made i n r e g a r d t o Babylonian, and e s p e c i a l l y E g y p t i a n , mathematics. We a l s o complain of t h e design of c a r s : t h e i r absurd i m p e r f e c t i o n s , which b e n e f i t t h e bank b a l a n c e s o f a whole c l a s s of p e r s o n s , have e q u a l l y been a r r i v e d a t by a p e r f e c t l y l o g i c a l and u n d e r s t a n d a b l e h i s t o r i c a l p r o c e s s : i t i s impossible t o e f f e c t c e r t a i n q u i t e t r i v i a l r e p a i r s o r a d j u s t m e n t s , w i t h o u t c r a w l i n g under t h e c a r , o r l i f t i n g i t b o d i l y up. I n Law, t h e complicated s t r u c t u r e a r i s e s from t h e requirement t h a t each l e g a l q u e s t i o n s h a l l have an answer es o r no ( g u i l t y o r n o t g u i l t y ) , d e r i v e d by one d e f i n i t e a n h r u c i z p r o c e d u r e , even i f t h e y a l l happen t o l e a d t o t h e same r e s u l t . H i s t o r i c a l l y t h i s goes back t o t h e w a s t e f u l wars of r i v a l r u l e r s , o r r i v a l i d e o l o g i e s , t h a t Law h a s y e t t o p r e v e n t . But i n any c a s e , t o a l l o w s e v e r a l e q u i v a l e n t procedures a t any s t a g e , would r e q u i r e e q y i v a l e n c e theorems: t h i s t h e lawyers could n o t stomach, o r e l s e t h e c o s t t o t h e i r c l i e n t s would become p r o h i b i t i v e . T h e r e f o r e t h e a d m i s s i b l e procedures a r e t o a l a r g e e x t e n t t a b u l a t e d i n Law books, and from e a r l i e r Court c a s e s . I must h e r e i n t e r p o s e a remark t h a t i s , I t h i n k , i m p o r t a n t i n t h e t e a c h i n g o f mathematics. Indeed t h e whole s u b j e c t of t h e h i s t o r y of mathematics i s f u l l of l e s s o n s i n t h i s r e s p e c t . When I speak of t h e a p p a r e n t a b s u r d i t i e s o f mathematics of v a r i o u s p a s t a g e s , o r e q u a l l y of t h o s e of Law o r of t h e d e s i g n of c a r s , and of t h e d i f f i c u l t y t h a t laymen have i n u n d e r s t a n ding them, I f e e l s u r e t h a t many r e a d e r s w i l l o b j e c t , and w i t h r e a s o n , t h a t t h e r e a r e j u s t as many a a r e n t a b s u r d i t i e s i n T e mathematician t h e mathematics t h a t i s t a u g h t today

+

22

Introduction

-+

Lebes u e , i n h i s charming l i t t l e book “ I n t h e margin o f t h e Calcu us of v a r i a t i o n s , ” mentions a proof t h a t p u r p o r t s t o show t h a t i n every t r i a n g l e t h e sum of two s i d e s e q u a l s t h e t h i r d s i d e . For h i s f r i e n d s a t s c h o o l , who t o l d him of i t , t h e whole t h i n g w a s a j o k e . But he was u n a b l e t o s e e t h e d i f f e r e n c e between i t and t h e p r o o f s he was g e t t i n g every day i n c l a s s . The b e s t s t u d e n t s , t h o s e who s e e f u r t h e s t , may be t h e ones most conscious o f apparent a b s u r d i t i e s ; and they may need most h e l p . Like Law today, Babylonian and Egyptian mathematics seem t o have been t a b u l a t e d : they w e r e based on t h e u s e of mathematic a l t a b l e s . The Babylonian t a b l e s appear t o d a t e from Mesopotamia‘s extremely o l d Sumerian a g e . P a r a d o x i c a l l y , t h i s v e r y a n t i q u i t y l e d t o a more f l e x i b l e numeration, by powers of 60 (Neugebauer p p . 151-2), w h i l e i n Egypt m u l t i p l i c a t i o n and d i v i s i o n by powers of 2 had a l r e a d y a c q u i r e d a p r i v i l e g e d p o s i t i o n , along w i t h t h e u s e of t h e f i r s t elementary f r a c t i o n s 1 1 2 , 1 1 3 , 2 1 3 and of t h e number 1 0 f o r t h e two hands. The g e n e r a l impression i s t h a t o f being i n a t o t a l l y d i f f e r e n t w o r l d , t h a t could have disappeared w i t h o u t a d e c i p h e r a b l e t r a c e : a l o s t world, i n t e r e s t i n g f o r i t s Museum v a l u e , i n which w e s e e how mathematical i d e a s came t o grow i n p r i m i t i v e human m i n d s , ’ a n d y e t a dead world i n which c o n s t r u c t i v e thought and c r e a t i v i t y have l e f t only t h e t r a c e s of t h e i r f i n i s h e d r e s u l t s , i n c o l d reminder of i t s a n c i e n t d e s t r u c t i o n .

9.

The Greek School of Mathematics

The h i s t o r i a n who s e r i o u s l y t r i e s t o i n t r o d u c e c a u s a l i t y , i s c o n s t a n t l y f o r c e d t o admit t o t h e e x i s t e n c e of n e a r - m i r a c l e s . The e x t e n t t o which t h e world of Babylonian s c i e n c e i n f l u e n c e d Ancient Greece i s f a r from c l e a r : i t would have been d i f f e r e n t i f t h a t s c i e n c e had s u r v i v e d . However i t i s only by what has been c a l l e d a m i r a c l e , t h a t t h e t r e a s u r e s of Ancient Greece, and of anything p r i o r t o i t t h a t may have helped i t t o develop, d i d n o t l i k e w i s e become t h o s e of a l o s t dead w o r l d , w i t h o u t i n f l u e n c e on what came a f t e r . I say t r e a s u r e s : I was t a u g h t t o v a l u e Greek thought. We have of c o u r s e only a f r a c t i o n of t h e g r e a t works o f Ancient Greece; t h e m i r a c l e of t h e i r p r e s e r v a t i o n i s wasted on a g e n e r a t i o n t h a t h a r d l y r e a d s any g r e a t works a t a l l . “Why w a s t e time on Greek?” Why indeed t r o u b l e t o r e a d a t a l l ? O r f o r t h a t m a t t e r , t o t h i n k ? The m i r a c l e , t o my mind, i s t h a t t h e r e w a s ever a time when i t w a s n o t s o : o r t h a t man ever managed t o evolve from t h e a p e , o r the ape from t h e dead p r i m i t i v e d u s t . W e d e a l , i n such m a t t e r s , w i t h i n c r e d i b l y s m a l l p r o b a b i l i t i e s , and t h e r e f o r e n e a r m i r a c l e s . C a u s a l i t y , as i t a f f e c t s us a l l , i s of t h e same n a t u r e a s t h a t by which, of t h e m i l l i o n s of eggs l a i d by a f i s h , a v e r y few s u r v i v e t o c o n t i n u e t h e s p e c i e s . Without some n e a r - m i r a c l e , perhaps t h e f u t u r e i s what c e r t a i n s c i e n c e f i c t i o n w r i t e r s p r e d i c t - - t h i n g s s o s i m p l i f i e d f o r u s , as f o r t h e hens i n o u r egg f a c t o r i e s , t h a t each of u s i s k e p t p e r f e c t l y happy i n h i s own l i t t l e compartment, w i t h t h e b a r e s t minimum of room t o e x e r c i z e h i s mind, i f a t a l l . However, p r o b a b i l i t y t e a c h e s u s t h a t i n a l a r g e p o p u l a t i o n of p o s s i b i l i t i e s ,

11. Early History

23

even a small probability becomes a virtual certainty. That some Napoleon should appear is what matters, not that he should happen to come from Corsica. This last is perhaps merely a curious historical fact. Miracle or not, in Greek mathematics we find for the first time the logical structure of definitions, axioms and proofs, now considered essential to mathematics. Whether any of this existed before, we do not know. This is where Martin Cort6s found the material of his Art of Navigation: for Martin Cortgs, however successful as an expositor, was no creative mathematician. He was passing on what had been preserved through the centuries by the Arabs, and had escaped destruction by the Turks. His original source, or that of his teachers, was the ilmagest, the Arabic translation of the Great Syntax, or great setting together" in 13 volumes, by the mathematician and astronomer Ptolemy of Alexandria, who lived in the 11-nd century AD. Translations of the Almagest, from the Arabic or the original Greek, were made into Latin in the XII-th century; today it is available in the principal European languages. What is important to us is the tremendous effect it had through Martin Cort&s, on Elizabethan England and ultimately on today's world. Kings and generals come and go, and cover themselves with a passing glory or shame. Within a century or so, all they have achieved proves no more decisive than the smoke that the wind blows this way and that way. But a mathematician's work - that some scoff at and term so useless, and that historians, for the most part, know so little of - - may, after a thousand years or more, still totally ,alter the course of history.

I mustn't give the impression that Ptolemy overshadowed his predecessors. The real teacher of Ptolemy was Euclid, 500 years earlier, in the IV-th century BC, just as Ptolemy was the real teacher of Martin Cort6s: teaching is not done by word of mouth only, nor by setting examinations, but by planting the good seed. This is the gist of the conversations I mentioned earlier with Eisenhart. The mathematician Felix Klein put it differently: a teacher is not there to instruct, so much as to inspire. The geometry of Euclid, as set out in his Elements - - which were still taught in English schools when my father was a boy -- , deals mainly with circles and triangles: these were the tools that Ptolemy used in his Almagest. This is why Ptolemy works with cycles and epicycles and so on, although he justifies them by a rather naive discussion, in which he says that the Earth, being so much heavier than the tiny stars, must naturally be the c e n t r e h e Universe! Euclid should perhaps share with Ptolemy the claim of bringing power and prosperity to poverty-stricken Tudor England. (We may wonder that a country so heavily imbued with tradition should have been capable of rapid change; this happened again in the industrial revolution, after the poverty and misery resulting from the Napoleonic wars. Perhaps there is nothing like intense poverty

24

Introduction

and misery f o r sharpening a n a t i o n ' s w i t s ; i t i s a b l e a k cons o l a t i o n f o r what c e r t a i n p r o g r e s s i v e governments may be prog r e s s i n g t o , n o t w i t h o u t h e l p from u n s e l f i s h and p a t r i o t i c trade-unions and l a b o u r u n i o n s , a s w e l l as from t h e i r b i t t e r e s t opponents.) B e t h a t as i t may, perhaps a mere mathematician, an o l d one now r e t i r e d , might s u g g e s t t h a t t h e way o u t of a n a t i o n ' s problems i s n o t n e c e s s a r i l y t o s t u d y and t o t e a c h less than b e f o r e ? What I f i n d e x t r a o r d i n a r y about Ptolemy, who a f t e r a l l was a most p r o d u c t i v e and o r i g i n a l mathematician of g r e a t power, i s t h a t he was so l i t t l e i n f l u e n c e d by t h e g r e a t geometers of c l a s s i c a l Greece, a f t e r E u c l i d . S u r e l y he should have conside r e d more s e r i o u s l y t h a t h i s a s t r o n o m i c a l d a t a might f i t e l l i p t i c o r b i t s b e t t e r than c i r c u l a r ones? E l l i p s e s and c o n i c s g e n e r a l l y had a l r e a d y been c o n s i d e r e d i n t h e 111-rd c e n t u r y BC by Archimedes, and were much s t u d i e d by A o l l o n i u s . Ptolemy's f a i G r e t o u s e them meant t h a t , f o r a s e b a r n i c s of c e l e s t i a l b o d i e s , mankind had t o w a i t u n t i l Modern T i m e s . S t i l l , w i t h i n t h e l i m i t s o f accuracy of h i s day, Ptolemy's theory worked. H i s e p i c y c l e s amount b a s i c a l l y t o a s u p e r p o s i t i o n of uniform c i r c u l a r motions, and s o t o t h e f i r s t few terms of what we c a l l a F o u r i e r s e r i e s . The f i r s t term r e p r e s e n t s a p a r t i c l e r o t a t i n g uniformly around a c i r c l e ; t h e f i r s t , t w o terms s i m i l a r l y correspond t o a p a r t i c l e r o t a t i n g uniformly around a s m a l l e r c i r c l e , which i s i t s e l f r o t a t i n g , l i k e t h e o r i g i n a l p a r t i c l e p r e v i o u s l y , around t h e o r i g i n a l b i g g e r c i r c l e . We could continue i n d e f i n i t e l y r e p l a c i n g p a r t i c l e s s u c c e s s i v e l y by s m a l l e r and s m a l l e r r o t a t i n g c i r c l e s . The s u c c e s s i v e mot i o n s then amount t o t a k i n g more and more F o u r i e r terms, which i s i n p r i n c i p l e how astronomers s t i l l approximate a n a c t u a l c e l e s t i a l motion w i t h i n d e s i r e d l i m i t s o f accuracy. They d i d s o , a t any r a t e , b e f o r e computer technology made i t h a r d t o f o l l o w t h e d e t a i l s ; b u t f o r i n s t a n c e i n P o i n c a r g ' s M6canique C h l e s t e , one o f t h e g r e a t modern works on astronomy, F o u r i e r s e r i e s a r e used. Why d i d Ptolemy s t i c k r i g i d l y t o c i r c l e s ? Why i g n o r e c o n i c s , which had been s t u d i e d 300 y e a r s ? S e v e r a l r e a s o n s may be g i v e n , but what r e a l l y s t a n d s o u t , i s t h a t t h e mathematics of c l a s s i c a l Greece was t h a t of E u c l i d ' s Elements. It a l o n e was considered v i t a l t o t h e whole c u l t u r e : numbers were a s s o c i a t e d w i t h musical n o t e s and harmonies, g e o m e t r i c a l f i g u r e s w i t h a r c h i t e c t u r a l d e s i g n s , c i r c l e s w i t h p r o p o r t i o n s of t h e human body as d e p i c t e d i n s t a t u e s . These t h i n g s were a p p l i c a b l e mathematics, i n a c u l t u r e t h a t had become c r y s t a l l i z e d round them. Conics, t h e t h i n g s t h a t r e a l l y e x i s t e d i n n a t u r e , were r e l e g a t e d t o t h e r o l e o f toys f o r mathematicians t o p l a y w i t h : t o o "Pure" f o r p r a c t i c a l u s e . The u t t e r a b s u r d i t y of t h e d i s t i n c t i o n between Pure and Applied has n e v e r been more c l e a r l y demons t r a t e d . Yet a s i m i l a r s t a t e of a f f a i r s r e c u r r e d a t t h e s t a r t of t h e XX-th c e n t u r y , r e g a r d i n g r i g o u r , when we had t h e e x t r a o r d i n a r y Phenomenon of a Poincar6 a t t a c k i n g S e t Theory. And now f o r a s h o r t summary o f Greek mathematics. The o l d e s t known Greek mathematician w a s Thales of M i l e t ; h i s l i f e over-

25

11. Early History

lapped the VII-th and VI-th centuries BC, and we have none of his writings. The next important name is Pythagoras of Samos; he lived in the VI-th century BC, we have none of his writings either. In this absence of first-hand information, some historians assure us that Thales and Pythagoras took nothing of substance from the East! A s van der Waerden points out, this would mean that the founders of Greek mathematics were so limited in outlook, as to be blind to the value of foreign culture. On the contrary, it is because of their contact with the East, that they were able to become founders. It was no accident that these founders happened to be born in territories bordering oriental Kingdoms and for a while subject to them. Moreover, Herodotus tells u s that Thales predicted the eclipse 585 BC, which ended a war by stopping a battle between the Medes and the Lydians. This prediction would hardly have been feasible, without a knowledge of Babylonian observations indicating approximate periodicities, and of course it would be made rather easy by them. Next, in the V-th century BC, comes Zeno, known for the paradox of Achilles and the tortoise, and forthe geometric series 1 + k + 4, + . . . = 2. (A more interesting paradox is Besicovitch's solution of the pursuit problem "Lion and Man'' in the London Mathematical Society Journal.) No one would today dispute that Zeno's paradox and Pythagoras's theorem, things that once occupied famous men, belong to the infancy of mathematics: we teach them to children. However the importance of such matters can be totally out of proportion with their elementary character. It is worth illustrating this. Consider for this purpose the more general geometric series 1 + x + x 2 + . . . , I shall speak of it simply as the geometric series. Tremendous importance is attached to it, and to its partial sums -- obtained by stopping after a finite number of terms. This is so in all parts of mathematics. Already in Egyptian and Babylonian times, an equation such as 7 = 1 + 2 + 4 , expressing 7 as such a partial sum for x = 2 , was used routinely for multiplication by 7, i.e. for reducing the latter to multiplication by powers of 2. Again in elementary trigonometry, the finite geometric series identity can be combined with a formula of de Moivre, or of Euler, that I shall have more to say about later, to derive most simply many basic trigonometric identities. By the finite geometric series identity, one normally means

(b

"

- a ")/(b-a)

=

b

+

ab"-

*+

+

a"-',

but I was referring to the case where a and b are replaced by 1 and x. The more general form was used before Newton and his Binomial Theorem, to establish more simply than he did, the formula for the derivative at x = a of the function x : this is done by choosing b to be a + h, and making h tend to 0. In the theory of infinite series, the convergence of the

26

Introduction

geometric series to the value l/(l-x) when x is numerically less than 1, is the basis of elementary convergence tests; it is also, since Euler, the basis of much of the theory of divergent series; in the same context the finite geometric series identity is the basis for finite difference identities used in summation by arithmetic means. Perhaps most important, in the theory of analytic functions, the mathematician Cauchy obtained a formula, expressing a complex analytic function f(z) as a contour integral with a denominator 2 - 2 , where 2 is the variable on the contour; thus by expanding the factor l / ( Z - z ) in a geometric series we have again many applications. In modern mathematics, the importance of the geometric series is illustrated in the most recent developments of the theory of Fourier Series, and in Analysis and Number Theory generally, as well as in many deep geometrical problems. In Fourier Series, the problem is how the sum to n terms behaves as n tends to infinity: it turned out rather simple to establish this behaviour when n is restricted to powers of 2 -- a rather surprising fact, certainly! Thus the terms of a geometric series, the powers of 2, play a part in the modern theory of Fourier Series, just as in Egyptian mathematics. Not unnatural ly, an experienced modern analyst is always looking for further strange ways of using powers of 2. I once had occasion'to ask Littlewood for assistance in proving a conjecture. For a year and a half, he would send an occasional postcard, "he hadn't forgotten my problem;" once he wrote "that he was too tired even to resist thinking about it." In the end he came up with an argument: need I add that it depended on a geometrical use of powers of 2 ? The combinatorial topologist Max Dehn (18781 9 5 2 ) , with whom I shared my office when I first came to Wisconsin, told me, as he came in from an elementary lecture on the geometric series, that one could lecture on this series to advanced graduate students for years without exhausting the subject . A geometric series trick was used by Gustin to prove a conjec-

ture of Wendell Flemin . Before the p x a p p e a r e d , two members of my Real Variab e class had found proofs for themselves, after I hinted that the geometric series could be used. Fleming's problem was the following: A number of equal cubes are put together to make up a number

of solids, in such a way that whenever two cubes have a point in common, they meet either along a whole face, or along a whole edge, or at a vertex. By the surface area of these solids, I shall mean the sum of the areas of the "free faces," i.e. of the faces of the constituent equal cubes which are not common to two such cubes. I shall denote the total area thus obtained for all the solids by a. We may imagine that the solids have been constructed by a small child, and that it is bedtime. A number of cubic boxes of various sizes are now produced, for which the total surface area, i.e. the sum of the areas of the faces of these new cubic boxes, is A . The solids constructed by the child are now put into these boxes without being broken up, except that a same solid may be shared

11. Early History

27

by several boxes, and this may require slightly separating parts of the solid by the relevant faces of the boxes (keeping of course the original small cubes of which the solid is made, intact). It is required to prove that the boxes can always be so chosen that all this is possible, provided that A is not less than 25a. The solution is by no means trivial. A first try might be to put the solids in a single huge box: if there were N original equal cubes, and they formed the solid obtained by placing them end to end in a line, the surface area of the huge box would have to be at least 6N2 times the area of a face of one small cube, and therefore roughly 6N/4 times the quantity a. By choice of N this would far exceed 25a. A second try might be to have the boxes all equal to the original cubes: if our solid consisted of a single big cube made up of N 3 original cubes, the ratio A:a would be 6N3/6N2 = N , again as large as we please. Some of you might like to think up more satisfactory attempts: if you fail in this, you will be in good company - - it is a problem that baffled some excellent mathematicians: just as a similar problem baffles many a parent at a child's bedtime. But I was speaking of the infancy of mathematics, rather than that of our children - - bless them! The problem with Thales and Pythagoras, or indeed with anyone whose writings are totally lost, is to decide exactly what they did, since subsequent mathematicians were rather fond of naming their own contributions "Theorem of Thales , " or else "of Pythagoras.'' A similar situation exists nowadays in music: when a composer produces anything in the least harmonious and musical, it is invariably presented as some lost piece by Mozart, that he had the good fortune to discover in some attic. Of course the converse can also occur, and no doubt did in Ancient Greece. Publishers of music sometimes have a hard time explaining to would be composers that Mozart wrote it first, and without mistakes . . . Nowadays, even gifted youngsters soon lose interest in the work of Thales and Pythagoras or Zeno, unless they go in for schoolteaching. They are anxious to get on to more exciting things. If I were teaching elementary mathematics, it would be from a higher point of view. Felix Klein used to give a course with such a title: the "higher point of view" involved concepts, that were, in his time, new. .It was a way of taking his audience into the heart of the mathematics of his day. I have in mind a rather different approach: a course that I shall not give here, but that you may write for yourselves, in your own minds; it would concern the uses to which elementary results can be put, as tools for finding and proving deep theorems. They can be, of course, used in solving elementary problems. I remember my old teacher's wizened face lighting up whenever he put Pythagoras's theorem to use: I'P tha ore," he would gleefully exclaim, "n'est pas mort." In t e course I am thinking of, I would not limit myself to Ancient Mathematics: many equally elementary, and equally useful, things are of recent

+

28

Introduction

date I Littlewood was fond, in this connection, of stressing an elementary inequality, due to Cauchy or to Buniatowsky, but generally referred to as Schwarz's. He often repeated the story of how Marcel Riesz finally found the proof of a result on which a whole beautiful theory hinged. Such results are not uncommon in the dreams of mathematicians, but this one happened to be true. The story will be found in Littlewood's lecture on "A mathematician's art of work," to be published, I understand, in his collected works. At the crucial point, Marcel Riesz uses Schwarz's inequality: using this at every opportunity, Littlewood tells us, was to Marcel Riesz second nature. Lessons of this kind seem to me among the most important that the history of mathematics can teach. In this way, the progress of mathematics can be affected, not just by its present, or immediate past, but by the whole development that began in its infancy. The same comment applies to the whole history of mankind -- and here I address myself to the non-mathematicians for whom I mainly write - - , against the great forces of history that overwhelmed us and crushed us in times gone by, what hope have we, if not in the lessons of much more than our present or immediate past? Even in mathematics, where I admit that we learn most from the last 200 years, we must not neglect what came before. But it is in the parallel with medicine, that we have the plainest case: the human body defends itself against ordinary everyday illnesses - - against them it needs to know only the immediate past; but against a virulent disease, it needs to remember an earlier inoculation. Personally, I do not like to be at the mercy of forces that we cannot control

. . .

In Greece, in the V-th century BC, there was also the geometer Hi ocrates, famous in his own right, and not to be confused Z h u n d e r of medicine who had the same name. His Elements are lost: they are believed to anticipate part o f Euclid's. The problem of the quadrature of the circle is attributed to him. Aristotle tells us that Hippocrates once let himself be cheated by customs officials: he was more competent in geometry than in ordinary affairs. It was, no doubt, a wicked weakness in an aristocrat of intellect, and the alert customs officials deserved to decorated, if I read today's papers aright? The next names I come to are Archytas, Eudoxus, Plato and Theaetetus, in the IV-th century BC. They are t h e e a t names 3 Greek mathematics. By all accounts, their contributions to what was still the infancy of mathematics were substantial, but are mostly lost. These are the contributions that most historians of mathematics limit themselves to, by definition. They non-Euclidean are important today mainly by what they omit and non-Archimedian geometries. I am more interested, as a XX-th century mathematician, in what the above mathematicians did for what was not yet mathematics in their day, but has become so for us: to this I shall return at the appropriate time. Archytas had a school in Southern Italy, and was famous

--

11. Early History

29

enough to be mentioned 2% centuries later by Horace. Eudoxus and Theaetetus were members of Plato's academy, and are mainly known to us by Plato's dialogues. Travel was easy: there was much contact between Plato's academy in Athens and the school of Archytas, and also with Egypt and other places. Since most of the work they did is lost, we can only judge of it through commentators such as Proclus, some 800 years later. What seems certain is that these mathematicians should be credited with much of Euclid's elements. Theaetetus is the subject of a dialogue of Plato: he proved the irrationality of numbers such as the square root of 2. The existence of irrationals was one difficulty of the whole scheme, and this was why it paradox of Achilles and was not so easy to dispose of the the tortoise. Eudoxus is presumably responsible for the "axiom of Archimedes:" magnitudes multiplied by suitable integers are capable of exceeding one another. He may also be responsible for the axiom of parallels, expressed by the 5-th postulate of Euclid: one and only one line can be drawn through a point parallel to a given line. Eudoxus is known for his theory of proportion and for his method of exhaustion - - a method for calculating areas and volumes. This means that he was conversant with inequalities and with the rudiments of XIX-th century concepts of real numbers, more so, certainly, than XVIII-th century mathematicians. He also occupied himself with astronomy, and devised the system of concentric circular orbits that Ptolemy later perfected, and that the readers and pupils of Martin Cortgs used successfully as pilots in the XVI-th and early XVII-th centuries. This could account for the observed motions of the Sun and the distant stars, and for those of Jupiter, Saturn and Mercury, and to some extent the Moon, but not for those of Venus and Mars. The question is how these earlier developments affected Euclid, who flourished around 300 BC at Alexandria. For this we-ly rely on Proclus, who tells, incidentally, two stories about Euclid. When a student asked what'good it would do him to study geometry, Euclid told his slave to "Give him three-pence, since he must needs gain by what he learns." And when the king of Egypt asked about a shorter way to geometry than the Elements, he replied: "There is no royal road to geometry." Proclus assures us that Euclid incorporated into his Elements whole books, which were mainly the work of Hippocrates, Archytas, Eudoxus and Theaetetus. If we ask who, across the ages, really inspired the mariners who brought the Pilgrim Fathers to the North American Continent, and consequently who was really responsible for the existence of the United States, we should perhaps go back to the circle of mathematicians closely associated with Plato's academy. Without them, we would have a different world. In the Britannica, the algebraist van der Waerden suggests that the books of Euclid not due to others, book VIII in particular, are full of repetitions, cumbersome enunciations, even fallacies. At least this means that we have some of the earlier work virtually at first hand, and not as a hateful "rehash." I believe in reading the

30

Introduction

In the s o u r c e , and I s h a l l n o t r e p r o d u c e any o f E u c l i d h e r e . o r i g i n a l s o u r c e c r e a t i v e i d e a s a r e t o be f o u n d : i n l a t e r vers i o n s , " t h e lemon i s squeezed d r y . " N e v e r t h e l e s s , t o u s e exc l u s i v e l y E u c l i d f o r c e n t u r i e s i n s c h o o l s , was no b e t t e r t h a n t o depend on A r i s t o t l e i n t h e Middle Ages: f o r geometry i t w a s deadening. A f t e r E u c l i d comes E r a t o s t h e n e s ( 1 1 1 - r d c e n t u r y BC), remembered m a i n l y now f o r h i s " s i e v e , " a p r o c e d u r e f o r o b t a i n i n g t h e primes by s u c c e s s i v e l y s t r i k i n g o u t , from t h e p o s i t i v e i n t e g e r s , t h e m u l t i p l e s of 2 , t h e n of 3 , and so o n . However, h e a l s o i n t r o d u c e d a c a l e n d a r w i t h l e a p y e a r s , compiled a c a t a l o g of s t a r s , c a l c u l a t e d t h e c i r c u m f e r e n c e o f t h e E a r t h . H e was a p o e t t o o , and a w r i t e r . I n h i s o l d age h e i s s a i d to have become b l i n d , and t o have s t a r v e d h i m s e l f t o d e a t h - - t h i s was before people believed s t a r v a t i o n prolongs l i f e . (Plato quotes from t h e Odyssey t h a t " t h e s a d d e s t o f f a t e s i s t o d i e and meet d e s t i n y from h u n g e r , " a l t h o u g h h e makes c l e a r , by q u o t i n g from t h e I l i a d " 0 heavy w i t h w i n e , who h a s t t h e e y e s o f a dog and t h e h e a r t o f a s t a g , " t h a t he does n o t f a v o u r o v e r - i n d u l g e n c e i n drink. ) E u c l i d and E r a t o s t h e n e s b e l o n g e d t o t h e s c h o o l o f A l e x a n d r i a . That i s a l s o where Archimedes ( a b o u t 287 BC - 2 1 2 BC) s t u d i e d , though he l i v e d m o s s y a t S y r a c u s e . H e w r o t e more t h a n a dozen b o o k s , a number o f which have s u r v i v e d , m o s t l y i n g e o m e t r y , b u t t h e r e i s a t r e a t i s e on c e n t r e s o f g r a v i t y and one on f l o a t i n g b o d i e s . H e was a l s o an i n v e n t o r , a n d even h i s t o r i a n s know him f o r h i s machines and many d e v i c e s , some u s e d w i t h e f f e c t i n t h e war w i t h Rome: p a r a b o l i c m i r r o r s t o c o n c e n t r a t e t h e S u n ' s r a y s and s e t f i r e t o Roman s h i p s , a r e a t t r i b u t e d t o him. Van d e r Waerden, i n a book on Greek m a t h e m a t i c s t h a t i s a p l e a s u r e t o r e a d , q u o t e s Roman w r i t e r s a s s a y i n g t h a t i f t h e S y r a c u s a n s happened t o l e a v e a h a r m l e s s b i t o f wood p r o t r u d i n g above t h e i r w a l l s , t h e Roman s o l d i e r s would f l e e i n t e r r o r , imagining i t t o b e a new weapon of d e s t r u c t i o n . According t o t h e B r i t a n n i c a , o n e o f Archimedes' i n v e n t i o n s , a water-screw, i s s t i l l u s e d t o i r r i g a t e f i e l d s i n E g y p t . The s t o r y o f h i s r u s h i n g home naked from t h e b a t h s , s h o u t i n g "Eureka, e u r e k a , " r e l a t e s to h i s d i s c o v e r i n g t h e p r i n c i p l e of buoyancy, by which he d e t e r m i n e d w h e t h e r t h e crown of King H i e r o n was made o f p u r e g o l d , o r had been a d u l t e r a t e d by s i l v e r . 1 s h a l l c o n t i n u e t o r e p r o d u c e such s t o r i e s - - h i s t o r y i s f u l l o f them, and w e do n o t have t o b e l i e v e them a l l . . . A l i t t l e l a t e r ( a b o u t 262 BC - a b o u t 1 9 0 BC), w e have t h e g r e a t geometer A o l l o n i u s . H e t o o s t u d i e d a t A l e x a n d r i a , where h e s e t t l e d a n t a u g h t . He became t h e g r e a t a u t h o r i t y on c o n i c s , which had been a p p a r e n t l y f i r s t c o n s i d e r e d by Menaechmus, a p u p i l of P l a t o , a c c o r d i n g t o P r p c l u s . E u c l i d t o o , had w r i t t e n on c o n i c s , b u t t h e book i s l o s t . F o r t u n a t e l y , f o u r o f Apoll o n i u s ' s books on c o n i c s s u r v i v e i n Greek, 3 o f t h e 4 o t h e r s i n A r a b i c t r a n s l a t i o n . Some o f t h i s work was e v e n t u a l l y s u p e r s e d e d by P a s c a l i n t h e XVII-th c e n t u r y AD, a n d t h e r e were g r e a t devz-nts i n geometry i n t h e XIX-th c e n t u r y , b u t much of A p o l l o n i u s s t i l l r e a d s v e r y w e l l . I n p a r t i c u l a r , i n Book

3

11. Early History

31

111, Apollonius introduces the harmonic property of poles and polars. His proofs are sound, unlike much that was done in the XIX-th century, at otherwise excellent universities! All of which we shall come to in due course.

I should mention next two astronomers. Aristarchus (about 217 - about 145 BC), who lived in Samos, maintained that the Earth revolves around the Sun - - for which "impiety" the Stoic

Cleanthes said that he should be indicted: so that we have, in the 111-rd century BC, a partial re-enactment of the confrontation of Galileo with Pope Urban VIII. The heliocentric hypothesis found few adherents: "there was no shred of evidence to prove it true, therefore it was false." Astronomical observations were still unsystematic and unreliable. This changed only with the astronomer Hi archus, who lived in the 11-nd century BC, mainly in R h o d e h o partly in Alexandria. Hipparchus was satisfied to fit his observations into the generally accepted geocentric theory, modified by such devices as epicycles. This is no doubt why Ptolemy, who had a great admiration for Hipparchus, hardly gave a thought to anything but geocentrism a couple of centuries later. Indeed the astronomical and mathematical apparatus that Ptolemy inherited from Hipparchus and Apollonius, was considerable. He had in effect, partly thanks to them spherical trigonometry, a topic on which my mother still wrote a doctorate as late as 1893. He had also what amount to discussions of inequalities and stationary points of trigonometric expressions: these were implied (as noted in the Introduction to Great Books, vol 16, University of Chicago), by his study in Book 12 of the Almagest of conditions under which a star will appear to regress. After Ptolemy, there were still a few Greek mathematicians of note. Sophantus, at Alexandria in the 111-rd century AD, was a number-theorist, but only half of the 13 books of his Arithmetica survive. He is known for his so-called Diophantine equations, i.e. equations whose solutions are required to be positive integers, for instance the equation n = x2 + y 2 , which expresses the integer n as the sum of two squares. The most famous Diophantine equation remains unsolved: it is

Fermat's last theorem conjectures that, for integers p greater than 2, this equation has no positive integer solutions x , y , z. On the other hand, it seems to have been known to Plato that the simultaneous Diophantine equations x2

+ y2

= 22,

x3

+ y3 +

2 3

=

t3,

have exactly one system of positive integer solutions, namely, x, y , z , t equal, apart from a permutation of the first triple, to 3 , 4 , 5 , 6. The corresponding value 216 of the cube of t is known as Plato's "nuptial number." The Greeks had their numerical superstitions too.

32

Introduction

Finally in the IV-th century AD, there was the geometer Pappus, also of Alexandria, known to geometers by Pappus's theorem, a fundamental one of plane projective geometry: if A, B , C are collinear, and X, Y, 2 are collinear, then a third triple of collinear points consists of the intersections P , Q, R of the pairs of lines BZ, CY and CX, A2 and AY, BX. This is a special case of the same conclusion, when A, B , C , X, Y, Z are six points of a conic, and thus of a famous theorem of Pascal. To non-geometers, other theorems are known as Pappus's, for instance those also attributed to Guldini about centroids of figures. For historians of mathematics, Pappus is mainly a commentator, without whom more of Greek mathematics would have been lost. He was the last Ancient Greek mathematician. After him, in the IV-th and V-th centuries AD, there are only the further commentators Theon and Theon's daughter Hypatia. Conceivably, Hypatia might have become the first known mathematically creative woman; unfortunately she was murdered. Greek mathematics ended in mere scholasticism: it lacked the simplifying and revitalizing drive of a new generation determined to throw out the cobwebs. The past is there to teach and inspire - - not to be venerated, but also not to be totally thrown out and lost as almost happened in Greece. Therefore let me speak of the historical context and explain its trends mathematically.

10. The Destructive Bends Let us go back to the V-th century BC. For Persia, the invasion of Greece had ended in disaster at Salamis in 480 BC. Athens had borne the brunt of the war, and had survived. The following year the demoralised Persians were defeated on land at Platea, and finally left. It was total victory for Greece: but not every nation can stand the strain of victory. Evolution through the ages has taught man to withstand the pressure of difficult times. It seems like a miracle when a peaceful countrylike England in 1914 survives even the economic dislocation resulting from a sudden war. People had predicted that no country -- least of all one totally unprepared -- could survive the inevitable stock exchange panic. Memoirs of those times show that survival was no miracle, but the result of much difficult and dangerous hard work. However it was work that human beings understand: the sudden great pressures were balanced. The biologist J.B. S. Haldane once asked me to look into the mathematics of a similar phenomenon in deep-sea diving. Three Haldanes are well-known -- and if all free spirits were no worse the world would be a better place. Littlewood said that each of them broke some commandment. J. B. S. broke the seventh, and the inevitable reaction made him a rebel; his father, famous for prescribing salt to miners, broke the second law of thermodynamics; the uncle, of whom we shall hear more, was a Hegelian, and so broke the law of contradiction. It seems a miracle that human tissues stand up to the terrific

11.

Early History

33

underwater p r e s s u r e : they f i l l themselves w i t h n i t r o g e n , o r a t g r e a t e r depths w i t h helium, a t a r a t e governed by a d e f i n i t e mathematical law. I n d i g n a n t r e l a t i v e s , and an i g n o r a n t v o c i f e r o u s p r e s s , could w e l l make a p u b l i c o u t c r y , demanding an immediate end t o such p r e s s u r e s and i n s i s t i n g t h a t t h e d i v e r be brought s t r a i g h t back t o t h e s u r f a c e . This i s p r e c i s e l y what would k i l l him: he would s u f f e r t h e ''bends." When t h e o u t s i d e p r e s s u r e i s suddenly reduced, t h e i n s i d e p r e s s u r e r e mains: t h e poor f e l l o w v i r t u a l l y explodes. How s h a r k s go up and down w i t h o u t e x p e r i e n c i n g t h i s , I have no i d e a : human beings cannot s t a n d i t . This i s a l s o t h e danger a n a t i o n f a c e s , a f t e r a sudden outbreak of peace. The U.S. s t i l l exp e r i e n c e s i t , a f t e r a minor w a r i n Vietnam, Laos, Cambodia. The world has experienced i t many times : h i s t o r i a n s s u g g e s t t h a t , f o r t h i n g s t o g e t b e t t e r , t h e y have t o g e t worse f i r s t ( a s i f w e can do n o t h i n g about i t ) . However, we know t h a t a d i v e r can be brought s a f e l y t o t h e s u r f a c e , i f we t a k e account of t h e r a t e a t which h i s t i s s u e s can s a f e l y a l l o w t h e n i t r o g e n , o r t h e helium, t o be r e l e a s e d . The bends account f o r what happened t o Greece a f t e r i t s escape from conquest by P e r s i a . A s a mathematical phenomenon, a f f e c t i n g a most important p e r i o d i n t h e development of mathem a t i c s , I s h a l l d e s c r i b e i t i n some d e t a i l . Human b e i n g s , I s a i d , a r e accustomed t o l i v i n g under s t r e s s , when they must suppress u r g e s and d e s i r e s : t h e y have done s o f o r thousands of y e a r s . But when t h e p r e s s u r e s a r e r e l e a s e d , even t h e most p r i m i t i v e u r g e s , from f a r back i n t i m e , a r e no l o n g e r h e l d i n check. Suddenly, people have, from every p e r i o d of human evol u t i o n , every c o n c e i v a b l e d e s i r e , however i n c o m p a t i b l e w i t h a l l t h e o t h e r s , o r w i t h l i f e a s i t now e x i s t s . Throughout t h e Greek w o r l d , Thucydides t e l l s u s , t h e r e w a s a d e t e r i o r a t i o n o f c h a r a c t e r . Of t h i s we s h a l l soon b e i n a p o s i t i o n t o judge f o r o u r s e l v e s , b u t f o r t h e moment l e t m e q u o t e f u r t h e r : "The simple way of looking a t t h i n g s , which i s S O much t h e mark of a n o b l e n a t u r e , w a s regarded a s a r i d i c u l o u s q u a l i t y and soon ceased t o e x i s t . " From o t h e r w r i t e r s ' w e i n f e r t h a t a t t h e same time t h e whole meaning of r e l i g i o n changed. The c u l t of Bacchus became an excuse f o r o r g i e s i n which i t w a s impious n o t t o t a k e p a r t , o r g i e s n o t u n l i k e some t h a t we r e a d about today: t h e y a r e d e s c r i b e d i n t h e l'Bacchantesl' of E u r i p i d e s . A l o o s e n i n g of t h e moral code i s unashamedly proclaimed by S o p h i s t s i n P l a t o ' s d i a l o g u e s , l i k e what much l a t e r took p l a c e i n Nero's Rome - Nero w a s a v e r y p o p u l a r Emperor. I n Athens t h e r e w a s a l s o a demand f o r changing t h e a n c i e n t and w e l l - t r i e d l a w s of Solon, by g i v i n g "more power t o t h e p e o p l e : " t h i s w a s n o t n e c e s s a r i l y a bad t h i n g , and a t f i r s t i t meant power t o t h e g r e a t P e r i c l e s ; but a f t e r h i s banishment i n 431 B C , and h i s d e a t h 2 y e a r s l a t e r , i t r e s u l t e d i n m i l i t a r y and o t h e r d e c i s i o n s amounting t o s h e e r f o l l y , and was d i r e c t l y r e s p o n s i b l e f o r l o s i n g t h e Peloponnesian War w i t h S p a r t a . That long c i v i l w a r w a s i t s e l f a m a n i f e s t a t i o n of t h e bends, and of what I can o n l y c a l l t h e madness t h a t came a f t e r t h e p r e s s u r e o f t h e w a r w i t h P e r s i a had been removed.

34

Introduction

There was a t l e a s t a t e r r i b l e r e v e r s i o n t o barbarism: i t spread t o t h e whole o f Greece, Thucydides s a y s , and f i r s t appeared i n Corcyra. "When t h e Corcyrians r e a l i s e d t h a t t h e Athenian f l e e t was approaching, . . , t h e y s e i z e d upon a l l t h e i r enemies t h a t they could f i n d and p u t them t o d e a t h . They d e a l t w i t h t h o s e t h e y had persuaded t o go on board t h e s h i p s , k i l l i n g them a s they landed. They went t o t h e temple of Hera and persuaded about 50 of t h e s u p p l i a n t s t o submit t o a t r i a l : t h e y condemned every one of them t o d e a t h . Most of t h e o t h e r s u p p l i a n t s , who had r e f u s e d t o be t r i e d , k i l l e d each o t h e r i n t h e temple; some hanged themselves on t r e e s . . . During seven days . . . t h e Corcyrians continued t o massacre any o f t h e i r own c i t i z e n s whom they considered enemies. The v i c t i m s w e r e accused o f c o n s p i r a c y . . . b u t were o f t e n k i l l e d on grounds of ,personal h a t r e d . . . There was d e a t h i n every shape and form . . . There were f a t h e r s who k i l l e d t h e i r s o n s ; men were dragged from t h e temples o r b u t c h e r e d on t h e very a l t a r s ; some were a c t u a l l y w a l l e d up i n t h e temple of Dionysus and d i e d t h e r e , ' I The Peloponnesian War was n o t o n l y a c i v i l w a r between Greeks: i t was a l s o a c i v i l w a r w i t h i n each c i t y , a w a r fought w i t h t h e utmost f e r o c i t y and savagery, and consequently w i t h i n c r e d i b l e s t u p i d i t y . Thucydides t e l l s us t h a t , a s a r u l e , t h o s e l e a s t remarkable f o r i n t e l l i g e n c e showed t h e g r e a t e r powers o f s u r v i v a l : such people launch r u t h l e s s l y i n t o a c t i o n , as t h e i r s o l e hope of a c h i e v i n g a n y t h i n g , and they have no s e n s e of honour o r humanity t o s t o p them. This p a r t l y e x p l a i n s why Athens, t h e more c u l t u r e d s i d e , ended by l o s i n g ; b u t t h e f o l l y of Athenian d e c i s i o n s was more d i r e c t l y r e s p o n s i b l e . H i s t o r i a n s speak of t h e " d e c l i n e " o f Athens; l a t e r , when i t i s Rome's t u r n , we have " d e c l i n e and f a l l . " Such a g e n t l e p a s s ing away! Like some dear o l d s o u l , i n t h e r i p e n e s s of y e a r s , h a p p i l y l e a n i n g back and s a y i n g good-bye f o r e v e r ! I n p o i n t o f f a c t , i t was sudden, i t was d r a m a t i c , and i t was, as I s a i d , i n c r e d i b l y s t u p i d -- n o t t h a t t h e same may n o t happen i n our t i m e , on a v a s t l y g r e a t e r s c a l e . The 30 y e a r Peloponesian War, i n which i t o c c u r r e d , i s d e s c r i b e d by Thucydides, and i s t h e r e f o r e b e t t e r known t h a n almost any o t h e r p e r i o d of H i s t o r y of comparable l e n g t h . Most of t h e w a r was a s t a l e m a t e , w i t h i n Lervals of s o - c a l l e d t r u c e t h a t we would c a l l "cold w a r . " m e r e was a l s o what w e would c a l l a l i t t l e "Vietnam War," waged by Athens on t h e s i d e f a r away i n S i c i l y , mainly a g a i n s t Syracuse, and n o t u n s u c c e s s f u l l y u n t i l , as we s h a l l s e e , t h i s l i t t l e w a r was e s c a l a t e d . This l a s t d i d n o t happen i n t h e f i r s t 15 y e a r s , and during t h a t t i m e , on t h e whole, t h e Peloponnesian War proceeded s l i g h t l y i n Athens' f a v o u r , The 16-th y e a r began w i t h an a t t a c k by Athens on t h e i s l a n d of Melos, whose crime was t o have remained n e u t r a l w h i l e t h e o t h e r i s l a n d s had j o i n e d t h e Athenian empire. The Melians s u r r e n d e r e d , and t h e " c u l t u r e d and c i v i l i s e d " Athenians p u t t o d e a t h t h e

11.

35

Early History

men of m i l i t a r y a g e , ar.d s o l d t h e women and c h i l d r e n as s l a v e s , thus e n s u r i n g t h a t a l l f u r t h e r enemies would f i g h t back w i t h t h e courage of d e s p e r a t i o n . This was Soon t o be p u t t o t h e t e s t , a s t h e l i t t l e side-war i n S i c i l y was e s c a l a t e d t o huge p r o p o r t i o n s . The wicked S y r a c u s i a n s were t o be t h e n e x t t o be crushed f o r t h e i n s u l t of n o t s u b m i t t i n g t o t h e might of Athens. To S i c i l y , Athens s e n t over a huge army and a huge f l e e t . Never b e f o r e o r a f t e r , was an e x p e d i t i o n so mismanaged. Of t h e t h r e e commanders, one, t h e prime mover, w a s a t once r e c a l l e d t o f a c e t r i a l f o r a l l e g e d l y d e f a c i n g s t a t u e s of Hermes, and chose t o d e f e c t t o S p a r t a ; t h e second commander wanted q u i c k a c t i o n b u t was t a l k e d o u t of i t ; and t h e t h i r d , N i c i a s , had a l l along opposed t h e e x p e d i t i o n . Two y e a r s p a s s e d : n o t h i n g had been achieved and N i c i a s wrote u r g i n g w i t h d r a w a l . I n s t e a d , Athens s e n t y e t more s h i p s and men, and a l s o some f u r t h e r commanders, one of whom, named Demosthenes l i k e t h e subsequent g r e a t o r a t o r , had p r e v i o u s l y , by a n i g h t a t t a c k , i n f l i c t e d a d e f e a t on S p a r t a on l a n d , a t h i n g no one e l s e had been a b l e t o do. H e t r i e d t h e same gamble a t Syracuse w i t h v a s t l y g r e a t e r f o r c e s : t h e confusion was u n b e l i e v a b l e , i t was a complete f i a s c o . Things were now s o bad t h a t he urged immediate w i t h drawal, which N i c i a s t h i s time opposed b u t f i n a l l y agreed t o . The Athenians were a l l s e t t o l e a v e , b u t now t h e r e w a s a n e c l i p s e of t h e Moon: t h e t r o o p s were t e r r i f i e d , N i c i a s i n s i s t e d on d e l a y i n g s a i l i n g f o r 2 7 days. I t was a f a t a l d e l a y : t h e demoralised e x p e d i t i o n l o s t a l l i t s s h i p s and was t o t a l l y des t r o y e d on l a n d . The i n c o n c e i v a b l e had t h u s happened: Athens was now suddenly deprived of h a l f h e r f o r c e s and was v i r t u a l l y a t t h e mercy of S p a r t a . By t h e m e r e s t l u c k t h e w a r dragged on 8 y e a r s more: t h e crowning f o l l y was y e t t o come. I t was when t h e Athenian f l e e t h a d , i n s p i t e of e v e r y t h i n g , once a g a i n won a g r e a t v i c t o r y a t s e a : on t h e i r r e t u r n t o Athens, t h e s i x v i c t o r i o u s commanders were condemned t o d e a t h f o r f a i l i n g t o r e c o v e r t h e b o d i e s of Athenian seamen, l o s t i n t h e heavy s e a s . There were no f u r t h e r Athenian v i c t o r i e s a t s e a , anymore t h a n on l a n d : i n t h i s v e r y p e r i o d of g r e a t e s t e x t e r n a l d a n g e r , Athens w a s t o r n by p o l i t i c a l u p h e a v a l s , t h a t followed one a n o t h e r i n q u i c k s u c c e s s i o n . Athens was blockaded and s t a r v e d i n t o submission. People remembered t h e f e r o c i t y towards t h e i s l a n d of Melos, and how t h i s had been good f o r a laugh i n t h e " B i r d s , " when A r i s tophanes had a l l u d e d t o t h e "Melian famine." By comparison, t h e t r e a t m e n t Athens r e c e i v e d from v i c t o r i o u s S p a r t a was almost 30 Athenians humane: a d i c t a t o r s h i p of 30 w a s p u t i n power who r e i g n e d only 8 months and i n t h a t time p u t 1500 c i t i z e n s t o death.

--

The v i r t u a l s e l f - d e s t r u c t i o n of Athens was no i s o l a t e d madness, b u t a p r e l u d e t o t h a t of a l l Greece. For S p a r t a t o o , t h e r e followed a " d e c l i n e : " t h e r e t h e madness w a s of a more common k i n d , t h a t t h r i v e s on ignorance and s u p e r s t i t i o n . I n educated and c u l t u r e d c l a s s i c a l Greece, S p a r t a remained t h e anachronism of a legendary e n c l a v e of muscle-bound Homeric h e r o e s , and i t s continued e x i s t e n c e depended on t h e myth of i t s i n v i n c i b i l i t y .

36

I n t r o duc t i o n

The p e r p e t u a t i o n of t h e myth demanded t h a t Spartan t r o o p s p a r t i c i p a t e only i n v i c t o r i e s , The a l l i e s o f S p a r t a were worse o f f than enemies: when h a r d p r e s s e d , t h e y r e c e i v e d no h e l p . Thucydides f i n d s p l e a s u r e i n d e s c r i b i n g how t h e Spartans s e t o u t i n f u l l f o r c e under t h e i r King - - only t o r e t u r n home because t h e omens f o r c r o s s i n g t h e f r o n t i e r were unfavourable. Already a g a i n s t t h e P e r s i a n s , t h e Athenians a t Marathon had been l e f t t o f i g h t a l o n e : t h e Spartans had waited f o r f u l l Moon, and a r r i v e d t h e day a f t e r t h e b a t t l e . The automatic i n v i n c i b i l i t y o f heroes i s t h e myth of t h e Gorgon's Head, a mere cabbage, p e r h a p s , s u i t a b l y d r e s s e d up. The t e r r o r i n s p i r e d o r i g i n a t e s indeed i n v e r y r e a l , though long p a s t , a c t s of heroism. Leonidas and h i s 300 w e r e h e r o e s , as much as Perseus h i m s e l f , when they d i e d a t Thermopilae t o g a i n f o r Greece a v e r y l i t t l e time. For a w h i l e , p a s t heroes i n s p i r e s u c c e s s o r s t o emulate them. Then g r a d u a l l y , t h e new s o - c a l l e d heroes r e l y more and more on t h e t e r r o r t h a t t h e i r myth -- t h e i r Gorgon's Head, as I might c a l l i t - s t r i k e s i n t o t h e i r enemies. One moment of t e r r o r - s t r i c k e n p a r a l y s i s may b e enough, and a f t e r w a r d s dead men t e l l no t a l e s . But f o r t h i s t h e myth must be p e r p e t u a t e d a t a l l c o s t s . Hence, as t h e y e a r s went by, S p a r t a r e l i e d more and more on f i g h t i n g b a t t l e s i n which, through t r e a c h e r y o r o t h e r w i s e , t h e r e w a s every chance of v i c t o r y and o f s i l e n c i n g whoever might p e r c e i v e t h a t S p a r t a n s had been l e s s t h a n h e r o i c . I n t h e long r u n , t r e a c h e r y , and t h e f a i l u r e t o h e l p a l l i e s i n t h e i r need, breed h a t r e d . S p a r t a d i d n o t even s c r u p l e t o j o i n w i t h P e r s i a i n o r d e r t o r u l e a l l Greece. F i n a l l y h a t r e d of S p a r t a d i d i t s work: S p a r t a w a s d e f e a t e d by her long-time a l l i e s , t h e Thebans, l e d by one of t h e g r e a t g e n e r a l s of a l l t i m e . For a few y e a r s , Thebes w a s t h e r u l i n g c i t y of Greece: t h e n she had h e r s e l f t o f a c e a c o a l i t i o n of t h e r e s t of t h e H e l l e n e s -- a g r e a t and i n d e c i s i v e b a t t l e l e f t so many dead, t h a t t h e w a r ended 'by sheer e x h a u s t i o n . Now Greece w a s s o weak, t h a t she was unable t o r e s i s t t h e Macedonian b a r b a r i a n s , P h i l i p , who was a s s a s s i n a t e d i n 336 B C , and Alexander, who d i e d i n 323 B C . A f t e r t h a t , t h e Romans g r a d u a l l y conquered t h e whole Medit e r r a n e a n b a s i n -- Greek p o l i t i c a l freedom ceased i n 146 BC, and had been somewhat p a r t i a l w e l l b e f o r e . These h i s t o r i c a l e v e n t s , t h e l o g i c a l consequences of t h e f r e e and unchecked a c t i o n of t h e mathematical f o r c e s unleashed i n t h e ' b e n d s ' , e x p l a i n how t h e r e could be a Greek school of mathematics a t Alexandria, t o which E u c l i d belonged about 300 B C , and which continued long a f t e r ; and how i t could t a k e t h e p l a c e of t h e school of P l a t o , which f l o u r i s h e d i n Athens a t a time of s o c i a l and moral upheaval, v e r y similar t o our age. A t such a t i m e , and i n such an atmosphere, anyone who b e l i e v e s t h a t human beings have a b e t t e r purpose i n l i f e , has n o t much c h o i c e b u t t o seek i t a s we do, i n a U n i v e r s i t y , o r i n P l a t o ' s t i m e , i n h i s academy. Perhaps what we do h e r e w i l l a g a i n s t a r t humanity on an upward p a t h , even i f , as w i t h P l a t o ' s s c h o o l , t h i s does n o t t a k e e f f e c t f o r a n o t h e r 2000 y e a r s - - and t h e world l a s t s u n t i l t h e n .

11. Early History

37

11. The Miraculous Preservation. After the fall of the Roman empire, mathematics in its former territories dropped for 700 years to an abysmal level. In that darkest of ages, to save, to restore to Europe even the rudimentary things that we teach young children, a miracle was needed -- the Arab miracle, as Loria calls it. Some blame the Arabs for the destruction of the Alexandria Library, that Caesar had burnt once before, - - the Arabs are a convenient scapegoat, ignorant fanatic desert tribes swooping down on the wonderfully civilised Christian world in which Hypatia had been murdered. I shall not argue the point: the miracle is all the greater that those same tribes used the pitiless scimitar of their religion, within a century, as perhaps the greatest civilising force the world has known. This is why I am by no means dismayed by today's sudden surge of Arab power -- a possible safety valve to the West's grave unbalanced internal pressures. There is, perhaps, more safety in diversity of nations, than in their uniformity. The Caliphs a1 Mansur (the Victorious, who reigned 754 - 775), Harun ar Rashid (the well-guided, 786 - 809), and a1 Mamun (the Blessed with Trust, 813 - 833), made of Bagdad the merging of the cultures of Greece and India. Just to determine the direction of Mecca and the hours set by for religious purification, the Coran itself requires a minimum of astronomy already beyond the capacity of the few Western scholars at the time. In Bagdad mathematics flourished, and under a1 Mamun a meridian arc was measured for the first time since Eratosthenes. Much work, even by the best mathematicians, was translation into Arabic: this is what preserved Greek texts. Thus ibn Luca, a Christian, was only an assiduous translator; he died about 912. How far others may have been original, is not clear: we know only a fraction of their work, and by poor Latin translations. A vast amount of material in Arabic awaits more serious study, even just in Spain. We may not even have the names of the most important mathematicians; and in many cases it is difficult to identify a mathematician, since he may be given many names. Two valuable sources are biographies, one by an Nadim in 987 (German by Suter, 1892), another much later by ibn Kahldun (1332 - 1406), the latter published by Woepcke in researches on Leonard0 of Pisa (Fibonacci) in 1856. I shall list only names that became known in Europe. One of these was ibn Musa, known as a1 Chwarizmi, from whose name the word algorithm was coined. He was a Court mathematician under a1 Mamun: a 1342 copy of his Algebra is in the Bodleian at Oxford, and a Latin translation of his Arithmetic, entitled Algoritmi de numero Indorum, was published in 1857 by Boncompagni. A little later than a1 Chwarizmi - around 900 - - the Egyptian Abu Kamil introduced rather s i m p l e procedures for linear equations, including the so-called 'regula falsoruml later used by the Spanish Jew Rabbi ben Esdra (who died in 1167, aged more than 70). Two very well known astronomers were a1 Battani (approx. 878 - 919) and Abu'l Wafa (940 - 997): tE w o r k s f the former, translated into Latin about 1120, led to modern trigonometric notation.

38

Introduction

A f t e r t h e e a r l y C a l i p h s , t h e r e were upheavals i n Bagdad, and s i g n s t h a t t h e Arab empire was s p l i t t i n g up. E v e n t u a l l y t h e r i v a l r y between E a s t and West became such t h a t s c i e n t i f i c c o n t a c t was d i f f i c u l t . I n S p a i n , t h e Arab c a p i t a l Cordova was s o out of touch w i t h Bagdad t h a t t h e mathematical n o t a t i o n was n o t i c e a b l y d i f f e r e n t . The symbol 0 used i n t h e Western p a r t meant 4 i n t h e E a s t , and t h e s i g n s f o r 7 , 8 were 3 3 i n t h e West, and v A i n t h e E a s t . Even without t h e s e v a r i a t i o n s , Arab symbolism i s hard t o make o u t ! Between E a s t and West, Egypt acquired an importance of i t s own. The mathematical p h y s i c i s t ibn Haitam, known a s a 1 Hazen (965 - 1 0 3 9 ) , came t h e r e w i t h t h e i d e a of b u i l d i n g an Azu-m, b u t gave i t up when he saw t h e magnitude of t h e work involved, and t h e g r e a t Egyptian monuments t h a t would be a f f e c t e d ; s o t h a t , a f t e r being welcomed, h e suddenly found himself i n danger of e x e c u t i o n , and only escaped by pretending i n s a n i t y . U n t i l t h e r u l e r a 1 Hakim d i e d , he d i d n o t d a r e show h i s f a c e . Some 1 3 0 of h i s works a r e unpublished, b u t h i s O p t i c s i n t h e L a t i n t r a n s l a t i o n Opticae Thesaurus (an e d i t i o n of which was much l a t e r Drinted i n Base1 i n 1622) had a g r e a t i n f l u e n c e on Roger Bacon and even on Kepler. I n t h e E a s t , mathematics continued during t h e upheavals. I n t h e same period a s a1 Hazen, t h e r e was t h e g r e a t s c h o l a r and s c i e n t i s t i b n S i n a , mentioned by Dante a s Avicenna (980 - 1037) and famous a l s o a s t h e g r e a t e s t of Arab p h y s i c i a n s . H e wrote an encyclopedic work on Geometry, Astronomy, A r i t h m e t i c and Music, and c a l l e d i t 'The l i b e r a t i o n ' of t h e s o u l from ignorance -- s u r e l y t h e t r u e s t of l i b e r a t i o n s . Next, from about 1036 t o 1123, t h e r e was t h e poet-mathematician, philosopher and astronomer, a 1 Kayami, g e n e r a l l y known a s Omar Kha am, from Persia. The E n g l i s h poet Edward F i t z g e r a d a f r e e t r a n s l a t i o n of h i s Rubaiyat i n t h e l a s t c e n t u r y , and t h i s again was t r a n s l a t e d i n t o almost every known language. H i s most quoted l i n e s a r e Here w i t h a Loaf of Bread beneath t h e bough, A F l a s k of Wine, a Book of Verse and Thou Beside me s i n g i n g i n t h e Wilderness -And Wilderness i s P a r a d i s e enow!

--

Omar Khayyam i s known t o have reformed t h e calendar and t o have w r i t t e n on many s u b j e c t s (Law, medicine, h i s t o r y among them), b u t l i t t l e of a l l t h i s has come t o u s , a p a r t from h i s Algebra (French by Woepcke 1851), considered one of t h e b e s t books of t h e e n t i r e Arab l i t e r a t u r e i n mathematics. F i n a l l y I should mention Nasir ed Din (1201 - 1274), and Kemal ed Din born some 50 y e a r s e a r l i e r ; OF t h e l a t t e r , I can only say t h a t a d e l e g a t i o n was s e n t by t h e emperor F r e d e r i c I1 a l l t h e way t o Mosul t o c o n s u l t him; however t h e works of Nasir ed Din i n c l u d e Schakl a 1 k a t t a , published by Alexander Pascha Karathgodory i n 1892, and we know more about him. (The p u b l i s h e r was an u n c l e of t h e mathematician Carathgodory.) N a s i r was i n t r o u b l e w i t h t h e r u l e r s of Bagdad, and s i d e d w i t h t h e Mongols, who took Bagdad i n 1258 and rewarded him by b u i l d i n g f o r him a magnific e n t observatory and a l i b r a r y w i t h 4 0 0 , 0 0 0 books. H i s name means 'defender of t h e f a i t h ' . H i s work on E u c l i d ' s axiom of

111. The Slow Renaissance

39

parallels was important enough for Wallis to publish with his own works; he was also the forerunner of trigonometry, anticipating Levi ben Gerson and Regiomontanus. I have said nothing of the Western Arab empire, but in mathematics Cordova fancied itself as a rival to Bagdad, and Granada took its place in 1248, when the Moors had lost Cordova and Seville. The great Alphonso X, an astronomer as well as King of Castile, had many Moorish texts translated into Latin; but much remains untranslated. Just from looking at Granada's buildings and monuments, with their highly mathematical decorations, one can see the relations of geometry to the decorative arts. They had learnt design symmetries: they were on the way to group-theory. A s to what they actually did, there is much deciphering to be done. One problem, we do not have: they were honest - - they neither falsely glorified themselves, nor the memory of their teachers.

PART I11 12.

Dismal Prospects in Western Europe

In Rome and its West-European territories, mathematics was slow in coming. Rome had done precious little to supplement or even preserve Greek thought. From the Greeks, the Romans learnt something, certainly; and I have no wish to minimize it. But it is a poor pupil who does mot somewhere surpass his master, and I have yet to find where - - hardly in plumbing, if lead plumbing poisoned Rome; nor, I would say, in Law. Not everyone cares for Roman Law's ruthless rule-of-thumb,supported by overwhelming force. A deeper purpose of Law is described in the trilogy of Aeschylus, when Athenae casts her deciding vote for the acquital of Orestes, and with infinite patience wins over, without show of force, his accusers, the Furies. This i s also the concept of Law that has hope of succeeding between nations. The best Romans - - and I admire them - - were admirers of Greece. They would have liked to subscribe to Horace's wishful thinking "Graecia capta ferum victorem capit, et artes intulit agresti Latio", and in mathematics they might regret with Cicero that "In summo honore apud Graecos geometria fuit; itaque nihil mathematicis illustrius; at nos ratiocinandi metiendique utilitate hujus artis terminamus modum." But mostly the Romans absorbed the very things that had destroyed Greece, and that ended by destroying Rome. I take these, and the next few, quotations from Loria. There is another interesting one. It comes from the Saturnali of Volpis: "Nemo christianorum presbyter non-mathematicus." Would you not imagine this to imply that there were rather m a n y mathematicians in Imperial Rome: Who were these mathe-

40

Introduction

matical Christian priests? How is it we haven't heard of them? But in fact, mathematics had become so foreign, that mathematicus had come to mean magician or sorcerer, and this is what the line accuses Christian priests of being. They were none of them mathematicians: the Christian Church had absolutely nothing to do with mathematics. However, this changed after the fall of Rome. For the next thousand years, in the territories of the former Roman empire of the West, the Church did most for mathematics. In the terrible first half-dozen centuries, the half-dozen or s o scholars from its territories who attempted to rescue some little mathematics along with other fragments of scholarship, were Christian theologians. It was characteristic of those times, that the first and most famous, Boetius, was tortured to death. Of him Dante writes

Lo corpo ond'ella fu cacciata giace Giuso in Ciel dauro, ed essa da martiro E da esilio venne a questa pace. He was buried, Dante tells us, in the church of the Golden Sky; and from martyrdom and exile he came to this peace. It is surely worth something to be mentioned with affection by Dante? Mathematics and mathematicians, Loria tells us, appear in several places in Dante, with special appreciation. The Divina Commedia itself is divided into 1 , 3 3 , 3 3 , 3 3 cantos, and even elementary geometry is given a poetic appeal in the lines. Qua1 e '1 geometra che tutto s'afflige Per misurar lo cerchio, e non ritrova Pensando, quel principio ond'egli indige. One cannot quote Dante anymore: Italian news-stands defile his quotations with 'allusions'. What times we live in! Boetius (about 480 - 5 2 4 ) , a great advocate of a return to Ancient Greek sources in science, mathematics and the humanities, was reputed to have a deep understanding of Greek culture. Writers of the Middle Ages credit him with the translation into Latin of 15 Greek mathematical works, including those of Pythagoras, Euclid and Ptolemy. Much of this Loria considers probably spurious, and the same applies to his supposed geometry, an inferior half-baked presentation. In the rest, the translation is not good, and there is a total lack of any original contribution, however modest. It is most disappointing. Should one expect better in those terrible times, in which the aristocracy of anti-intellect had been pushed to its logical conclusion? Nor did conditions improve until the reign of Charlemagne, and then only during that reign. After the Roman Boetius, Dante mentions another mathematician and theological scholar, the venerable Bede (about 673 - 7 3 5 ) , from just South of the Scottish B o r d e r . 7 was not a matter of chance that such a person s h o u l d come from what had been a distant outpost of the Roman empire: from similar outposts in nearby Ireland, most of Europe was reconverted to a semblance of Christian religion . . . However Bede's scholarly works,

111. The Slow Renaissance

41

including his mathematics, are less known than his system for describing numbers by gestures, visible at a distance and intelligible to deaf-mutes. Did Bede deal, I cannot help wondering, with bee-dealers? Bees have a way of describing the exact location of any find of theirs. I suspect that at the time the bees were the better mathematicians . . , I come next to the Yorkshireman Alcwyn (735 - 8 0 4 1 , who settled in the France of Charlemagne, and was made abbot of Tours; he is credited with a mathematical text of sorts, not free from error, which includes the familiar conundrum of how to take across a river, at most one at a time, a wolf, a lamb, and a cabbage. The title is "Propositiones acuendos juvenes". The next name of note is that of the Frenchman Gerbertus, who became Pope Sylvester I1 in 999 and died in 1003. He spent much time in that dark period, trying to persuade people that the world would not end at the year 1000. Today, as we approach another number divisible by the cube of 10, there are still people who imagine that the Creator is standing by with a stopwatch . . . Such superstitions about numbers suggest that we have not progressed very far from the X-th century. That was also the period known until recently, on account of matters I prefer not to discuss, as that of the lowest degradation of human beings. To be Pope at such a time is no pleasure. We cannot wonder that, in mathematics, Gerbertus ranks no higher than Boetius. He is credited with a book on geometry: from whole chapters apparently copied by subsequent writers, it does not seem to have been a good book. Even someone cultured was incapable of proper proofs. But the book was no worse than others. Loria cites a booklet, written between 1036 and 1056 by the director Francone of the Cathedral school of St Lambert at LiGge under the emperor Otto 111: from the inaccuracy that the ratio of a circumference to its diameter is exactly 22/7, it equates the area of a circle of diameter 14 to that of a rectangle of sides 14 and 11, and after a number of pages concludes that to construct a square of equal area is arithmetically impossible! One wonders how.mathematics could ever recover from such a state? There were universities in the X-th century: they existed from about the year 800 unofficially. Officially, of course, Bologna was founded in 1100, Paris in 1150, Cambridge prior to 1226 (when it possessed a Chancellor). But what is the use of universities and schools, if the teachers are lamentably deficient? Or if they are chosen f o r any reason other than competence and creativity? In the 14 centuries after the age of Pericles, there had been progress of a sort - - people had done their best, under their metal or coarse cloth carapace, to evolve in the only manner they understood. Perhaps this is how nature produces its locust or rhinoceros . . . However mathematics had progressed only towards extinction. 13. The Pioneers Who Came Too Soon Thewhole atmosphere of Western Europe was changed by the crusades. I cannot say how the enlightened Churchmen felt about

42

Introduction

C h r i s t i a n s being s e n t t o f i g h t f o r s a c r e d s t o n e s of a r e l i g i o n they understood s o l i t t l e . However t h e s p i r i t o f d e d i c a t i o n enhanced t h e l e a d e r s h i p of t h e Pope, t h e Church, Rome; and t h e r e f o r e t h e i n f l u e n c e of I t a l y , i t s r e l a t i v e a f f l u e n c e , i t s t r a d e , i t s a r c h i t e c t u r e . Mathematics, which i s c l o s e l y l i n k e d w i t h t r a d e a n d a l s o w i t h t h e d e c o r a t i v e a r t s , came t o I t a l y , which was becoming t h e s p i r i t u a l and i n t e l l e c t u a l l e a d e r o f Europe. I know I t a l y w e l l , as a country long l i n k e d t o my own E r i t i s h I s l e s by i n t e l l e c t u a l and a r t i s t i c t r a d i t i o n s . Perhaps t h e o r i g i n a l l i n k was w i t h I r e l a n d - - a s a l r e a d y mentioned, t h e I r i s h Monks had s e t o u t t o r e c o n v e r t Euaope i n t h e d a r k e s t pagan days. Much l a t e r , a s we know, Shakespeare wrote some of h i s p l a y s f o r an I t a l i a n s e t t i n g . I n England e s p e c i a l l y , t r a d i t i o n l i v e s on: i t i s a p i t y t h a t a r t and t h e i n t e l l e c t a r e a t times l e s s r o b u s t . Byron, S n e l l e y , t h e Brownings s t i l l l i k e d I t a l y , but i t was because they f e l t t h e r e more a t home. S p i r i t u a l l e a d e r s h i p can_ h e l p p r o g r e s s : i t produced a new I t a l y , n o t h e l d back by t h e s h o r t c o m i n g s of Ancient Rome and t h e Dark Ages. The Renaissance of mathematics followed t h a t of Church a r c h i t e c t u r e : i t s f i r s t g r e a t r e p r e s e n t a t i v e , Leonardo of P i s a , known a s F i b o n a c c i (about 1173 t o sometime a f t e r 1 2 4 0 ) , came from a ci-eady famous f o r i t s Duomo, d a t t i s t e r o arid Leaning Tower--a c i t y t h e r e f o r e t h a t d i d n o t r e j e c t a r t i s t i c and i n t e l l e c t u a l v a l u e s a s e l i t i s m ( i t was moving away from b a r b a r i s m ) , n o r e l i m i n a t e i n consequence a l l v a l u e s except money. Money was needed t o o : competition was f i e r c e between t h e r i v a l p o r t s of P i s a , Genoa and Venice--the l i v e l i h o o d o f c i t i z e n s depended on i t . F i b o n a c c i ' s f a t h e r , a s c r i b e , was thus s e n t on a permanent m i s s i o n t o t h e Customs' o f f i c e of t h e Algerian c i t y of Bugia. The son j o i n e d him, and w a s encouraged t o study Arab mathematics: f o r t h i s , and a l s o t o b e u s e f u l t o h i s f a t h e r , he t r a v e l l e d around t h e Mediterranean. Fibonacci i s an a b b r e v i a t i o n f o r f i l i u s Bonnaci, which i s how he i s d e s c r i b e d on t h e t i t l e page of t h r e e X I I I - t h century manus c r i p t s of h i s f i r s t and main work. There i s a l s o t h e p r o p e r name Leonardo P i s a n o , and i n l a t e r works ' B i g o l l o , ' a p o s s i b l e v a r i a n t of vagabond, reminiscent of Sindbad t h e S a i l o r . Matter of f a c t h i s t o r i a n s have nad d i f f i c u l t y w i t h t h i s c u r i o u s e p i t h e t ; b u t i t was a time when some Westerners were c i v i l i s e d enough t o f e e l t h e romantic appeal of t h e Arabian N i g h t s , and of a philosophy i n which what m a t t e r s i s n o t what you have, b u t what you a r e . This may account f o r B i g o l l o , b u t t h e r e i s a problem too about ' F i b o n a c c i : ' t h e f a t h e r ' s name was Guglielmo. I t has been suggested t h a t t h e f a t h e r became known a s "good f e l l o w , " f o r h i s s e r v i c e s i n g r e a s i n g t h e wheels of Arab customs, and t h a t t h e son thus became t h e I t a l i a n equival e n t of " f i l s du bonace." Fibonacci r e t u r n e d t o P i s a a b o u t b h e y e a r 1 2 0 0 , aged 30 o r s o , and i n 1 2 3 2 he published h i s L i b e r Abaci. The term abbacus, o r abacus, had come t o mean a r i t h m e t i c . The book had an enormous and immediate impact, which i s most s u r p r i s i n g when one considers t h a t i t s innovations were comparable i n magnitude

111. The Slow Renaissance

43

to the introduction of Set Theory into Analysis a hundred years ago, or to that of the 'New Math' in our schools, both of which raised storms of protests. It seems that the book of Fibonacci had suddenly become a factor in the cut-throat competition between Pisa and its rivals, and it is amazing how quickly people accept radical changes when their purses benefit.

In its most elementary aspects, which have today only historical interest, the object of the book was nothing less than to change the whole system of numerotation of Europe to the IndoArabic one. (By comparison, decimalisation, now being gradually introduced in the U.K. and the U . S . , is a trivial affair!) The advantages are made clear by such comparisons as MMMMCCCXXI and 4 3 2 1 . There follow tables for addition, multiplication and prime-factor decomposition, and two chapters on fractions - - here the persistence, even then, of old Egyptian calculations appears in identities such as 9 8 - 1 + 1 + 1 + 1 + 4 5 '50

m-2

1

m

Four further chapters treat problems of commerce, to appeal to a city of merchants, while the final four chapters are for amateur mathematicians of the time: we find there arithmetic and geometric progressions, square and cube roots, perfect numbers, friendly numbers, Fibonacci numbers, Diophantine equations. What has mainly remained of interest are the Fibonacci numbers, introduced in a finite difference problem of breeding rabbits, and actually expressible in the closed form

where a,b

=

%(1+6).

Fibonacci wrote a.second book, Practica geometriae (about 1 2 2 5 ) , and various, perhaps more original, smaller writings. His Liber quadratorum includes enough number theory to have given a possible impetus to that subject long before Fermat, if the book had not, unfortunately, remained buried for 6 centuries. It contains, incidentally, the oft rediscovered rittle identity' 2 2 2 2 2 (a +b ) (C +d ) = (ad+bc)2 + (bd-ac) . Both in this writing and in a paper entitled Flos Leonardi Bigollo Pisani super solutionibus quarundam quaestionum etc, reference is made to a discussion with Johannes of Palermo, held at the Palace at Pisa in the presence of Emperor Frederic 11. (Note the epithet Bigollo.) This seems to confirm that Fibonacci was honoured by the Emperor; he was honoured also by the city of Pisa, which eventually decided,

44

Introduction

by a d e c r e e of 1240 i n s c r i b e d i n a marble t a b l e t , t o pay f o r h i s h i t h e r t o f r e e s e r v i c e s a s accountant. A l l t h i s would be f i n e , except f o r t h e annoying l i t t l e i m p r o b a b i l i t i e s of which h i s t o r y i s so f u l l . To t h e doubts we may e n t e r t a i n r e g a r d i n g Bonaccio and B i g o l l o , we must now add o t h e r s concerning t h e a l l e g e d d a t e of F i b o n a c c i ' s i n t r o d u c t i o n t o t h e Emperor, a much t r a v e l l e d man whose movements a r e n o t unrecorded! Nowadays, someone who speaks of F r e d e r i c I1 u s u a l l y means t h e s o - c a l l e d ' g r e a t ' F r e d e r i c , t h e l i t t l e Hohenzollern XVIII-th c e n t u r y King of P r u s s i a , of whom I s h a l l have occasion t o speak more f r e e l y than i s u s u a l . When a few more c e n t u r i e s have p a s s e d , i t w i l l be c l e a r e r which F r e d e r i c I1 was r e a l l y ' g r e a t ' . The one I speak of h e r e , t h e grandson of Barbarossa, was t h e l a s t of t h e g r e a t Hohenstaufen r u l e r s : f o r 30 y e a r s (1220 - 1250), he was Holy Roman Emperor. T h i s e a r l i e r F r e d e r i c I1 (1194 - 1250) was l e f t a s a c h i l d i n charge of Pope Innocent 111, a s a r e s u l t of which he became probably t h e most c u l t u r e d and i n t e l l i g e n t sovereign known t o h i s t o r y . I mentioned e a r l i e r t h a t he s e n t a d e l e g a t i o n t o c o n s u l t t h e Arab mathematician Kemal ed Din. H e a c t u a l l y g o t on extremely w e l l w i t h t h e Arabs: he could a p p r e c i a t e t h a t they were t h e most c i v i l i s e d peoples of t h e t i m e . H e was himself brought up i n S i c i l y , of which he was King from t h e age of 5 w i t h t h e Pope a s Regent: t h e r e he was introduced t o a mixed c u l t u r e t h a t uniquely combined elements of a n t i q u i t y , Arab and Jewish wisdom, t h e Occidental s p i r i t of t h e Middle Ages, and Norman r e a l i s m . He i s b e l i e v e d t o have been a f r e e - t h i n k e r : however he d i d embark on two c r u s a d e s . The f i r s t i n 1 2 2 7 he had t o g i v e up on account of i l l n e s s -- which l e d t o h i s excommunic a t i o n by a new and impatient Pope. The second, c a r r i e d o u t i n s p i t e of excommunication, was t h e most s u c c e s s f u l i n h i s t o r y -and e n t i r e l y b l o o d l e s s . S u l t a n al-Kamil of Egypt was so s u r p r i s e d t o meet a Western c u l t u r e d s o v e r e i g n , t h a t he handed over Jerusalem, Bethlehem and Nazareth i n 1 2 2 9 : t h e r e was peace i n t h e Holy land f o r 1 0 y e a r s . However, a g a i n s t h i s own s u b j e c t s i n Western Europe, F r e d e r i c I1 understood v e r y w e l l t h a t only t h e crude and r u t h l e s s methods of b a r b a r i a n s could succeed i n winning v i c t o r i e s . H e d i d what he could t o promote c i v i l i s a t i o n : f o r i n s t a n c e he founded t h e U n i v e r s i t y of Naples. A s f o r h i s v i s i t s t o P i s a , t h e r e was a v e r y b r i e f one i n J u l y 1226, then n o t u n t i l December 1239, August 1244, May 1245, A p r i l 1247, May 1249. However Liber quadratorum i s c i t e d a l r e a d y i n t h e second e d i t i o n (1228) of L i b e r a b a c i , and i t s p r i n t e d v e r s i o n s t a r t s o f f " I n c i p i t l i b e r quadratorum compoThe t i t l e has managed s i t u s a Leonard0 Pisano. Anni MCCXXV." t o be e a r l i e r than t h e f i r s t v i s i t of t h e Emperor . . . B e t h a t was helped by t h e e n l i g h t e n e d a s i t may, F i b o n a c c i ' s =SS monarch on t h e t h r o n e . Indeed t h e new c a b a l i s t i c s i g n s t h a t went by t h e name of numerals were n o t unopposed: f o r a t i m e a f t e r F i b o n a c c i ' s d e a t h , they w e r e even p r o s c r i b e d by Law i n Florence, i n an a r t i c l e of 1299 on exchange, quoted by L o r i a . I n t h e end, s e n s i b l e a r i t h m e t i c won, b u t F i b o n a c c i , i t s p i o n e e r , was f o r g o t t e n i n I t a l y . Dante mentions only h i s ast r o l o g e r f r i e n d "Michele S c o t t o . . . che veramente d e l l e magiche f r o d e seppe il g i o c o . " S t i l l I t a l y ' s Church helped produce

111.

The Slow Renaissance

45

F i b o n a c c i ' s pioneer s u c c e s s o r s . I now speak of r e a l mathematicians, f o r whose work t h e r e i s no need t o a p o l o g i s e by d e s c r i b i n g t h e u n p l e a s a n t times t h e y l i v e d i n and t h e abysmal ignorance a l l around. I n England t h e r e was Thomas Bradwardine (1290 - 1 3 4 9 ) , Archbishop of Canterbury; i n France, Levi ben Gerson (1288 - 1347), supported by t h e Pope a t Avignon, and Nicholas Oresme (1323 - 1382), Bishop of L i s i e u x ; i n Germany, Cardinal N i c h o l a s of Cusa (1401 - 1464), and Regiomontanus (1436 - 1476), whom Pope S i x t u s I V c a l l e d t o Rome i n 415 t o reform t h e c a l e n d a r . Such men could h a r d l y appear suddenly i n a t o t a l i n t e l l e c t u a l d e s e r t : t h e r e were o t h e r s who would have seemed g i a n t s a couple of c e n t u r i e s b e f o r e . Roger Bacon (1214 - 1294) was one of t h e s e : he was n o t o u t s t a n d i n g a s a mathematician - - he t y p i f i e d r a t h e r t h e no-nonsense mathem a t i c s t h a t some p r a c t i c a l people claim t o p r e f e r . H e d e c l a r e d t h a t e could by himself l e a r n t h e whole of geometry i n a week, and he demonstrated t h i s by proving f a l s e theorems: he might a t l e a s t have t e s t e d them by h i s famous experimental method! H e must have been s t i l l i n f l u e n c e d by some a t r o c i o u s t h i n g s t h a t had gone by t h e name of mathematics b e f o r e him: no wonder he o b j e c t e d on p r i n c i p l e t o r e a d i n g ! A c t u a l l y t h i s i s n o t such a bad a t t i t u d e i n one who wants t o do h i s own t h i n k i n g , but i t seems a p i t y t o have l e a r n t Arabic, L a t i n and Greek t o such l i t t l e purpose: t h e t r o u b l e , L o r i a s u g g e s t s , was h i s j o i n i n g t h e Franciscan o rd e r, r a t h e r than t h e highly i n t e l l e c t u a l Dominicans.

Among t h e lesser mathematicians of t h e p e r i o d , s e v e r a l wrote anonymously: one b e f o r e , and one a f t e r , Levi ben Gerson, wrote on Trigonometry; and a t h i r d one, who used t h e pseudonym Jordanus Nemorarius, wrote a number of works - - a fragment ent i t l e d 'De Ponderibus' on Mechanics, a t r a c t e n t i t l e d P l a n i s p h a e r i u s on s t e r e o g r a p h i c p r o j e c t i o n , and f o u r t r e a t i s e s e n t i t l e d A r i t h m e t i c a , Algorithmus demonstratus, D e numeris d a t i s , D e t r i a n g u l i s . I n t h e l a s t of t h e s e , by a ' c o i n c i d e n c e ' a s when s t u d e n t s copy from one a n o t h e r , t h e s i d e Ln of t h e r e g u l a r n-gon i n s c r i b e d i n a c i r c l e i s given by a formula eq u i v a l e n t t o t h e erroneous e q u a t i o n . 3/sini

=

Jn(n-1)

+

6

,

which happens t o be t r u e f o r n = 3 , 4 , 6 , b u t f a l s e f o r n=7, i n which c a s e i t amounts t o Nemorarius's ' I n d i a n r u l e ' L7 = 4L3 , a f a l s e r u l e t o be found i n Abu'l Wafa. (See L o r i a . ) Of t h e ' r e a l m a t h e m a t i c i a n s ' , Regiomontanus became b e s t k n o w , b u t o n l y because of t h e r e v e r s a l e f f e c t t h a t t h e i n v e n t i o n of p r i n t i n g had on what came b e f o r e , t h e l a t e r works being p r i n t e d f i r s t . H e and Levi ben Gerson d i d s i m i l a r work, and count among t h e founders of our trigonometry. The work of Regiomontanus on t h i s , D e t r i a n g u l i s omnimodis' , appeared i n 1464 and was p r i n t e d i n 1533: he never c i t e s Levi ben Gerson, and n e i t h e r of them c i t e a1 B a t t a n i and N a s i r ed Din, a l t h o u g h t h e r e was much o v e r l a p . Levi ben Gerson's work on trigonometry

46

I n t r o duc t ion

was more than a c e n t u r y e a r l i e r : i t appeared i n Hebrew i n 1321, and i n L a t i n t r a n s l a t i o n i n 1342. He a l s o published Lunar Tables -- b o t h he and Regiomontanus were a l s o known a s astronomers. On t h e whole, Levi ben Gerson seems t o have been t h e b e t t e r man, e s p e c i a l l y s i n c e he came 100 y e a r s e a r l i e r : i n c i d e n t a l l y he i s considered a f o r e r u n n e r of non-Euclidean geometry. However Regiomontanus had t h e g r e a t e r impact because of t h e r e v e r s a l e f f e c t , and Moritz Cantor devotes t o him a whole c h a p t e r . Regiomontanus means 'from Koenigsberg', n o t t h e famous c i t y of seven b r i d g e s , but t h e l i t t l e mountain town i n t h e duchy of Coburg. The proper name was Johannes M u e l l e r , and was no doubt s h a r e d , even a t t h a t t i m e , w i t h thousands. H e deserved e a r l y t o be d i s t i n g u i s h e d : a t t h e age of 1 2 he a t t e n d e d t h e U n i v e r s i t y of L e i p z i g , and two o r t h r e e y e a r s l a t e r t h a t of Vienna. I n t h o s e days, t o be a mathematician, i t was n e c e s s a r y a l s o t o be a L a t i n , Greek and Arabic s c h o l a r . Regiomontanus, who d i e d a t 4 0 , devoted much of h i s s h o r t l i f e t o t r a n s l a t i n g , i n a d d i t i o n t o h i s p a r t l y o r i g i n a l contribut ions. About Bradwardine and O r e s m e I can be e n t h u s i a s t i c . Bradward i n e had been p r o c t o r a t Oxford and had l e c t u r e d t h e r e f o r 1 2 y e a r s a s ' d o c t o r p r o f u n d u s ' , b e f o r e being named Chancellor of S t P a u l ' s i n London, and he t h e n accompanied Edward 111 i n h i s French wars a s Confessor. Edward found him so i n v a l u a b l e t h a t he held up t h e appointment a s Archbishop. I t was Bradwardine, f o r i n s t a n c e , who preached t h e v i c t o r y sermon a f t e r t h e b a t t l e of Crecy: h i s sermons were remarkable, and make him a f o r e runner of XIX-th c e n t u r y determinism. H i s theology i s summar i z e d i n Dean M i l t o n ' s Church H i s t o r y ( i v . 7 9 - 1 0 6 ) : i t was f i r s t published a t t h e r e q u e s t of t h e f e l l o w s of Merton College, and l a t e r expanded i n t o a 3 volume t r e a t i s e . H e maint a i n e d t h a t human n a t u r e i s a b s o l u t e l y i n c a p a b l e of conquering a s i n g l e temptation without a supply of d i v i n e g r a c e : I t o t a l l y a g r e e w i t h him. Of c o u r s e I i n t e r p r e t d i v i n e g r a c e i n a more modern way i n terms of c i v i l i s i n g f o r c e s such a s moral laws, proper e d u c a t i o n , o r i n some c a s e s sheer n e c e s s i t y . They a r e t h e f o r c e s by which a ' d i v i n e g r a c e ' could be made e f f e c t i v e , and they can a c t a s a brake. I s h a l l say more about them i n a moment: we s h a l l s e e i n t h e c o u r s e of t h e s e l e c t u r e s t h a t such m a t t e r s h e l p t o make h i s t o r i c a l events understanda b l e , and t h a t t h e y a r e r e l a t e d t o an important mathematical contribution, The c a r e e r of Oresme was v e r y s i m i l a r , b u t s l i g h t l y l a t e r and i n another country. H e e n t e r e d t h e College de Navarre i n P a r i s , and remained t h e r e , a s s t u d e n t , a s t e a c h e r , and f i n a l l y a s Grand Master, f o r a t o t a l of 13 y e a r s , i n which time he a l s o became ' d o c t e u r de P a r i s ' . Then he found favour w i t h Charles t h e Wise, and he ended up a s Bishop: he was a s i n v a l u a b l e t o h i s King a s Bradwardine had been t o Edward 111. He was a g r e a t writer and p r e a c h e r , p e r f e c t i n both French and L a t i n . For h i s King he t r a n s l a t e d i n t o b e a u t i f u l French some works of A r i s t o t l e , from t h e i r a v a i l a b l e L a t i n v e r s i o n s : he included comments, by no means orthodox i n h i s day, t h a t show him, L o r i a t e l l s u s , t o have been a f o r e r u n n e r of Copernicus.

111. The Slow Renaissance

47

In Latin, his sermon on Christmas Eve in 1363 at Avignon, openly attacking the weaknesses and errors of the Pope and his Cardinals, is a classic, both in form and content; he also did not hesitate to attack astrologers and the powerful orders of mendicants. In this he was carrying out the wishes of his King, and he was most effective. In mathematics his most famous works are two treatises, entitled Tractatus de latitudine formarum and Algoritmus proportionum. In the former, his latitude and longitude are in effect the coordinate system of Descartes; while in the latter we find what we would now call the basic properties of the exponential ax for rational x , and from there it is only a step to Napier and his logarithms. Oresme also notes that near a maximum the increment of a variable quantity becomes 0 , and in this he anticipates the great Fermat himself. Finally, in his Tractatus de uniformitate et difformitate intensionum, still available only in manuscript, he has a cryptic reference to the work having suggested itself to him from the 'irnaginatio veterum' , Who were these 'old ones'? Did he mean people like Bradwardine and Levi ben Gerson, or further back, Arabs that ben Gerson fails to cite and whose works may be still buried among the manuscripts of Spain? Or did he mean the Ancient Greeks? Whichever it is, Oresme, a man clearly 300 years ahead of his time, sets the example of reading what is old. It is as if he said: "Read works that have become classic, read between the lines, looking for what is not done there." This is the exhortation of a good teacher. It i s h o w to get ideas to try out, and, in the same spirit one should also re-read one's own work. Oresme's virtual introduction of the exponential is a landmark in the History of Mathematics, a total departure from Ancient Greek limitations; it also makes it possible to describe mathematically a number of phenomena in biology and medicine, and in human history as well. The natural growth of living organisms and systems is exponential, while it remains unchecked. This makes it very difficult to control. It was, for instance, some five centuries after Oresme that Jenner was finally able to control smallpox. What is so characteristic of exponential growth, is that it is extremely slow to begin with, and extremely fast once a noticeable size is reached. The human reaction to fight off an infection is therefore late in starting, and proceeds at first very slowly by its own exponential growth, just when the infection, which is now noticeable, proceeds at a most rapid rate. Jenner's method is to maintain a low level of infection and of human reaction, as a result of which the body's defense starts 'growing at a higher position on the exponential. In human affairs generally, this is the only way to fight a disturbance with any chance of success, whether the disturbance is a revolution, a population explosion, a war, or simply an individual temptation. The alternative, for one who wishes to survive, is to run away: I personally recommend this solution in most cases -- for instance I avoid driving a car in the rush-hour - - but in the rare cases where moral obligations prevent any such escape, I adapt to the best of my ability the mathematical principle used in vaccination. In this way, thanks to Oresme, I remain

48

Introduction

master of my own f a t e , a b l e f o r a while t o hold o f f t h e f o r c e s of determinism t h a t Bradwardine d e s c r i b e s w i t h such l o g i c a l cogency and mathematical p r e c i s i o n a s i r r e s i s t i b l e . I t i s impossible n o t t o be tremendously impressed w i t h Oresme. Besides d i s c o v e r i n g what made v a c c i n a t i o n p o s s i b l e , n o t t o speak of s i m i l a r a p p l i c a t i o n s i n t h e f u t u r e t o our p o l i t i c a l and economic i l l s a t t h e v e r y l e a s t , he a n t i c i p a t e d by 300 y e a r s , n o t one b u t f o u r o f t h e g r e a t mathematicians. Even s o , he d i d n o t , i n my o p i n i o n , r e a l i s e h i s f u l l p o t e n t i a l : h i s whole way of t h i n k i n g , i n keeping w i t h h i s v e r y p r e c i s e and f i n i s h e d mode of e x p r e s s i o n , was t o go r i g h t t o t h e p o i n t , and t o l i m i t himself t o what he completely understood. This was t o become c h a r a c t e r i s t i c of French thought a s expressed by Boileau: "Qui ne s u t s e b o r n e r , ne s u t jamais k r i r e . " For t h i s r e a s o n , Oresme has l o s t some of h i s i n t e r e s t ; so have Copernicus, Napier, D e s c a r t e s , even Fermat. Remarkably t h i s i s n o t t h e c a s e of Bradwardine, nor of Nicholas of Cusa. The t i t l e ' d o c t o r profundus' o b t a i n e d by Bradwardine a t Merton College, Oxford, w a s n o t i n a p p r o p r i a t e , even though Roger Bacon was ' d o c t o r m i r a b i l i s ' and o t h e r s were d o c t o r a n g e l i c u s , d o c t o r u n i v e r s a l i s , s u b t i l i s , illumunatus, i n v i n c i b i l i s , s i n u l a r i s , solemnis, r e s o l u t u s . What was considered Bradwardine s main work managed t o g e t p r i n t e d a s e a r l y a s 1495 ( s o t h a t t h e r e was no r e v e r s a l e f f e c t ) . This c o n s i s t e d of h i s Arithmetica S p e c u l a t i v a , De p r o p o r t i o n i b u s v e l o c i t a t u m , T r a c t a t u s d e quadratura c i r c u l i , and Geometria s p e c u l a t i v a . Moritz Cantor f i n d s t h e l a s t of t h e s e works remarkable: i t i n t r o d u c e s k - t h s t a r polygons of t h e n - t h o r d e r , and g i v e s f o r t h e sum of t h e i r a n g l e s a formula proved only i n t h e XIX-th c e n t u r y . However f o r u s today t h e r e a l l y e x c i t i n g work of Bradwardine i s t h e T r a c t a t u s de c o n t i n u o , s t i l l only a v a i l a b l e i n m a n u s c r i p t : i n i t he i n t r o d u c e s concepts of i n f i n i t y , among o t h e r t h i n g s , t h a t amount t o T r a n s f i n i t e Numbers, and t h a t had t o w a i t 600 y e a r s ' t o be developed by Georg Cantor, t h e man many regarded a s t h e g r e a t e s t mathematician of h i s t i m e .

F

Georg Cantor was unaware of t h i s work of Bradwardine, but he was w e l l r e a d , and he c i t e s i n s t e a d t h e e q u a l l y i n t e r e s t i n g p e r s o n a l i t y of Cardinal Nicholas of Cusa. The Cardinal came from t h e v i l l a g e o f Cues on t h e l e f t bank of t h e Mosel; h i s family name was Krebs, and h i s f a t h e r had been a fisherman. I n Cues, t h e C a r d i n a l ' s L i b r a r y and o t h e r e f f e c t s a r e c a r e f u l l y p r e s e r v e d . Nicholas s t u d i e d i n Heidelberg and Padua, but h i s s u b j e c t was Law. He took up Theology when he l o s t h i s f i r s t c a s e . H i s f i r s t connection w i t h mathematics i n h i s w r i t i n g s , came from h i s p r o p o s a l s t o c o r r e c t t h e c a l e n d a r and improve t h e Alphontine t a b l e s . A s a member o f t h e Council of Basel, which m e t i n t h e y e a r s 1432 - 1437, and on which he r e p r e s e n t e d t h e view of t h e Pope, he brought up t h e s e p r o p o s a l s , b u t t h e Council turned them down, although he pointed o u t t h e i m portance of making c l e a r t h e f a s t days on which no meat was t o be consumed. H e became known t o mathematicians of h i s t i m e f o r some c u r i o u s approximations i n t h e q u a d r a t u r e of t h e c i r c l e . I t must be borne i n mind t h a t many people s t i l l h e l d f a s t t o t h e b e l i e f t h a t TI was e x a c t l y 2 2 / 7 - - t o change t h e c a l e n d a r

111.

The Slow Renaissance

49

was bad enough, b u t t o change T ? However, t o us and i n t h e h i s t o r y of mathematics, C a r d i n a l Nicholas o f Cusa i s mainly known f o r h i s De d o c t a i g n o r a n t i a and De B e r y l l o , p h i l o s o p h i c a l works which s t r a y i n t o v e r y o r i g i n a l ways of looking a t mathematics - - and which l a t e r a t t r a c t e d Georg Cantor. 14.

The Leadership That Could Not L a s t .

The d e d i c a t i o n of t h e c r u s a d e s made p o s s i b l e t h e emergence of t h e mathematicians I have mentioned; i t w a s a l s o t h e s p i r i t t h a t had b u i l t t h e c a t h e d r a l s of Europe. I n t h a t u n d e r t a k i n g , everybody h e l p e d : t h e p o o r e s t p e a s a n t w a s n o t t o o p o o r , nor t h e p r o u d e s t l o r d too proud, t o l e n d a hand w i t h t h e t e d i o u s and backbreaking t a s k of c a r t i n g t h e n e c e s s a r y s t o n e s . T h i s i s a l s o how i n I t a l y , where F i b o n a c c i w a s s o soon f o r g o t t e n , mathematics took on a new d i r e c t i o n , i n s p i r e d by a r c h i t e c t u r e , and soon by p a i n t i n g : i t developed a geometry of e r s e c t i v e . This w a s new o n l y i n p a r t : t h e Greeks were f a m i l i m a r t of g i v i n g t o t h e p a i n t e d s c e n e r y of a t h e a t r e t h e i l l u s i o n of r e a l i t y -- t h e y d i d s o from t h e time of Aeschylus. P e r s p e c t i v e a r i s e s from what used t o be o p t i c s and c a t o p t i c s ; we have w r i t i n g s on t h e s e a t t r i b u t e d t o E u c l i d , u s i n g t h e S o c r a t i c h y p o t h e s i s t h a t v i s i o n r e s u l t s from r a y s i s s u i n g from t h e eye, t o g e t h e r w i t h o t h e r p o s t u l a t e s b a s i c i n p e r s p e c t i v e . I should a l s o remind t h e r e a d e r of t h e O p t i c s of a1 Hazen, which was a l r e a d y - - according t o Moritz Cantor - - being t a u g h t i n Oxford a t t h e time of Roger Bacon. Textbooks on p e r s p e c t i v e , c o v e r i n g t h i s m a t e r i a l , were w r i t t e n by such people as John Peckham (1242 - 1 2 9 2 ) , archbishop of Canterbury; and i n t h e f o l l o w i n g c e n t u r y t h e P e r s p e c t i v a V i t e l l i o n i s of t h e German-Pole Witelo became well-known. However a l l t h i s w a s g i v e n a new d i r e c t i o n by t h e I t a l i a n s . The b a s i c n o t i o n s were t h o s e of t h e v i s u a l cone, determined by a n o b j e c t , and i t s i n t e r s e c t i o n w i t h t h e p l a n e of r e f e r e n c e , on one hand, and t h e n o t i o n of l i n e s of e q u a l i l l u m i n a t i o n , on t h e o t h e r . Such t h i n g s as t h e h o r i z o n , t h e enlargement of a p a i n t i n g , become m a t t e r s of p u r e geometry, w h i l e t h e l i n e s of e q u a l i l l u m i n a t i o n , which c o n s t i t u t e t h e b a s i s f o r shading and c h i a r o s c u r o , a r e more properly t o p i c s f o r d i f f e r e n t i a l geometry, The men who i n t r o d u c e d a l l t h i s had t o be mathematicians i n a r a t h e r wider s e n s e . They were c r e a t i v e , t h e y were n o t necess a r i l y q u i t e r i g h t , s i n c e t h e y were concerned w i t h p r a c t i c a l t h i n g s . S i m i l a r remarks a p p l y t o t h e men who helped develop Western Music, which l i k e p a i n t i n g h a s some r e l a t i o n t o t h e development of mathematics. I n Ancient Greece t h e r e l a t i o n of music t o mathematics w a s t a k e n f o r g r a n t e d , and t h e r e i s l i k e w i s e a w r i t i n g on music a t t r i b u t e d t o E u c l i d : i t i n c l u d e s a fragment on t h e mathematical t h e o r y of sound. I n t h e new p e r s p e c t i v e , t h e f i r s t important names a r e t h o s e of t h e Renaissance a r c h i t e c t B r u n e l l e s c i (1377 - 1446) and h i s p u p i l Uccello (1397-1475). Next we come t o Leon B a t t i s t a A l b e r t - r n 4 - 1472) and P i e r o d e l l a Francesca (1420 - 1 4 9 2 ) , writers and t h e o r e t i c i a n s as w e l l a s a r t i s t s . A l b e r t i ' s books,

50

Introduction

Della s t a t u a , Elementi d i p i t t u r a , De a r t e e d i f i c a t o r i a , became famous. I n a d d i t i o n he wrote Ludi m a t e m a t i c i , i n which, f o r i n s t a n c e , he e f f e c t s t h e q u a d r a t u r e of t h e H i p p o c r a t i c c r e s c e n t . Della Francesca wrote De p e r s p e c t i v a p i n g e n d i , which became t h e b a s i s of s i m i l a r t e x t s under o t h e r names. I n a d d i t i o n he wrote D e c o r p o r i b u s r e g u l a r i b u s a p u r e g e o m e t r i c a l work, t h e I t a l i a n v e r s i o n of which, by Luca P a c i o l i , was p r i n t e d i n 1915. The most famous I t a l i a n a r t i s t - s c i e n t i s t w a s of c o u r s e Leonardo da Vinci (1452 - 1519). The T r a t t a t o d e l l a p i t t u r a a t t r i b u t e d t o him i s s t a t e d by L o r i a t o be a hodge-podge randomly p u t t o g e t h e r by o t h e r s ; what i s c e r t a i n l y h i s i n i t , i s t h e l e s s o n he preaches t o would-be p a i n t e r s : " F i r s t s t u d y t h e s c i e n c e , t h e n only i t s practical use." This might n o t s u i t t o d a y ' s a r t i s t s , b u t Leonardo p r a c t i s e d what he preached: t h i s p r a c t i c a l u s e l e d him t o become a g r e a t i n v e n t o r of p r a c t i c a l t h i n g s , as w e l l as a g r e a t c r e a t i v e a r t i s t . I n mathematics, h i s main i n t e r e s t s , a p a r t from p e r s p e c t i v e t h e o r y , were i n mechanics, which he c a l l e d t h e ' p a r a d i s e of m a t h e m a t i c s ' . H i s f r i e n d Luca P a c i o l i (about 1445 t o sometime a f t e r 1514) w a s mainly a mathematical t e x t book w r i t e r who had b e n e f i t e d from t h e i n v e n t i o n of p r i n t i n g , and who could g e t Leonardo t o draw some of h i s diagrams. P r i n t i n g had been invented i n 1456, a n A r i t h m e t i c had been p r i n t e d a t T r e v i s o i n 1478, and E u c l i d ' s Elements had been p r i n t e d i n 1482. L o r i a says t h a t by 1500 t h e r e were i n Europe some 70 p r i n t i n g f i r m s , 50 of them i n Venice. A f l o o d of mathematical textbooks began t o a p p e a r , many of them copied from one a n o t h e r , m i s t a k e s i n c l u d e d , j u s t l i k e a t l a t e r t i m e s . E d i t i o n s of E u c l i d ' s Elements succeeded one a n o t h e r , i n which t h e a u t h o r was confused w i t h E u c l i d of Megara, a contemporary of S o c r a t e s and P l a t o . P a c i o l i produced t h e f i r s t encyclopedia of mathematics, under t h e t i t l e Summa d i A r i t h m e t i c a , Geometria, P r o p o r t i o n i e t P r o p o r t i o n i t a ; i t w a s p r i n t e d i n Venice; i t s main h i s t o r i c a l i n t e r e s t i s i n showing, according t o L o r i a , t h a t t h e r e had been no r e a l p r o g r e s s f o r t h r e e c e n t u r i e s i n t h e s e matters, s i n c e F i b o n a c c i . P a c i o l i a l s o brought o u t a work e n t i t l e d Divina p r o p o r t i o n e : i t was i n t h r e e p a r t s , t h e t h i r d of which i s t h e unacknowledged I t a l i a n t r a n s l a t i o n of P i e r o d e l l a F r a n c e s c a ' s D e c o r p o r i b u s r e g u l a r i b u s . However i n P a c i o l i ' s Summa two t h i n g s a r e worth n o t i n g : one i s t h e f a i r d i v i s i o n of pledgemoney between two p l a y e r s when a game i s i n t e r r u p t e d - - a q u e s t i o n of p r o b a b i l i t y i n which he goes badly wrong - - , the o t h e r i s h i s u s e of e f u n c t i o n s 2X , ~2~ f o r t h e approximate s o l u t i o n of 2(f7X) = l + r / 1 0 0 and of x2X = 30 which reminds u s of Oresme and a g a i n g i v e s u s a f o r e t a s t e of Napier. I should a l s o say something a t t h i s p o i n t of Albrecht Durer (1471 - 1528) and Nicholas Copernicus (1473 - 1543): i t may

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come a s a s u r p r i s e t h a t I p u t t h e i r names t o g e t h e r , b u t t h e y were b o t h amateur mathematicians and t h e y b o t h s t u d i e d i n Bologna. W r e r was t h e r e a s h o r t time t o s t u d y p e r s p e c t i v e , b u t was o t h e r w i s e mainly i n Venice d u r i n g h i s I t a l i a n s t a y (1505 - 1507). Copernicus s p e n t 1 0 y e a r s (1496 - 1505) i n I t a l y , t h e f i r s t f i v e a t Bologna, and became Doctor of Laws a t F e r r a r a i n 1503: he had p r e v i o u s l y s t u d i e d 4 y e a r s (1491 1494) a t Crakow, T i m e , i n such m a t t e r s , does n o t c o u n t , nor how much one l e a r n s : what counts i s t h e sowing of t h e s e e d , t h e awaking of t h e s p a r k .

- - everyone knows t h e epoch-making importance of h i s work: he was Doctor of Laws and canon of t h e c a t h e d r a l of Frauenburg, y e t he s p e n t t h e l a s t 40 y e a r s of h i s l i f e w r i t i n g D e r e v o l u t i o n i b u s orbium c o e l e s t i u m , t h e f i r s t p r i n t e d copy of which i s s a i d t o have been handed t o him on h i s death-bed. The mathematical achievements of Durer, t h e g r e a t p a i n t e r , a r e l e s s known, b u t t h i s i s because t h e y were even f u r t h e r ahead of h i s t i m e , and because he was w r i t i n g f o r a r t i s t s . H e ended by w r i t i n g , i n 1525, a famous g e o m e t r i c a l work, t h e L a t i n t r a n s l a t i o n of which b e a r s t h e t i t l e I n s t i t u t i o n e m geometricarum L i b r i q u a t u o r . H e i s t h e f o r e r u n n e r of Monge, and l i k e Monge he a p p l i e d h i s knowledge t o a t h e o r y of m i l i t a r y f o r t i f i c a t i o n s , t h e l a s t t o p i c one would expect a g r e a t a r t i s t t o w r i t e a b o u t ! Of c o u r s e he a p p l i e d h i s mathematics a l s o t o t h e a r t of p a i n t i n g , and he wrote a f i n a l work on t h e p r o p o r t i o n s of t h e human body, which was p r i n t e d posthumously. I do n o t have t o s a y much of Copernicus

I mentioned t h e d e d i c a t i o n t h a t came w i t h t h e c r u s a d e s . Clearl y t h i s d e d i c a t i o n , t h i s e x a l t a t i o n , was n o t much i n evidence among t h e w r i t e r s and p r i n t e r s of mathematical t e x t b o o k s , P a c i o l i among them. I t had more t h a n worn o f f i n t h e i r c a s e and f o r most of t h e p u b l i c a t l a r g e . A s i n Athens when S o c r a t e s was condemned, i n t h e Golden Age of i t s a r i s t o c r a c y of i n t e l l e c t , t h e g r e a t p h i l o s o p h e r s , p l a y w r i g h t s and s c u l p t t o r s , i t no doubt remained l o n g e s t o n l y w i t h t h e f i n e s t of men, and w i t h them f o r a w h i l e even grew s t r o n g e r as i t ebbed away from a l l around them. And s o i t was perhaps i n t h e I t a l y of Leonard0 da V i n c i : an I t a l y t o r n by t h e r i v a l r y of i t s S t a t e s , and ravaged by t h e greed of i t s neighbours.

I n t h i s I t a l y , i n mathematics, some triumphs were s t i l l t o come: one of t h e s e was t h e s o l u t i o n of t h e c u b i c -- b u t t h i s was a minor a f f a i r compared w i t h t h e i n t r o d u c t i o n of complex numbers, which came w i t h i t . For u s t h e s o l u t i o n i s e a s y , because we a r e n o t hampered by cumbrous n o t a t i o n s ; b u t of c o u r s e , we t o o must u s e complex numbers -- o t h e r w i s e we a r e faced w i t h a d r e a r y d i s t i n c t i o n of p a r t i c u l a r c a s e s , c o n t r a r y t o t h e whole s p i r i t of mathematics. However I p r e f e r t o l e a v e t h e mathematical a s p e c t s of such m a t t e r s t o t h e n e x t s e c t i o n , which d e a l s w i t h a s l i g h t l y l a t e r t i m e i n which our n o t a t i o n had come i n .

We can h a r d l y imagine t h a t such t h i n g s a s t h e s q u a r e r o o t of a n e g a t i v e number would be accepted without p r o t e s t a t a t i m e

Introduction

52

when t h e d e v i l was seen i n a l l t h i n g s i n t h e l e a s t mysterious and obscure, i n a country where t h e Church had much s a y . I t was bad enough f o r Fibonacci t o have introduced h i s pagan c a b a l i s t i c s i g n s . However I t a l i a n mathematicians employed t h e a n c i e n t defense used by Egyptian p r i e s t s and by t h e Pythago r e a n s : they kept t h e i r d i s c o v e r i e s l a r g e l y t o themselves. An a l t e r n a t i v e method i s t o p u b l i s h anonymously: we saw i n t h e l a s t s e c t i o n t h a t s e v e r a l mathematicians had done s o : i n t h e next s e c t i o n we s h a l l s e e a g a i n good r e a s o n t o do t h e same. I n our t i m e s , i n France, t h e school of Bourbaki was begun i n t h e same way, and l i k e t h a t of Pythagoras of o l d , i t acquired a c e r t a i n p o l i t i c a l power. A u t h o r i t i e s a r e agreed t h a t t h e f i r s t s o l v e r of t h e cubic was Scipione d a l F e r r o (1465 - 1 5 2 6 ) . Some i n s i s t t h a t he solved only one s p e c i a l f o r m : t h i s i s supported by t h e l a t e 0 s t e i n Ore (1900-1968), who wrote a book on Cardano and p u b l i b a r t i c l e i n t h e B r i t a n n i c a . However, i n t h e Bologna U n i v e r s i t y L i b r a r y , a manuscript of B o l o g n e t t i , p r o f e s s o r t h e r e from 1565 t o 1582, r e f e r s t o t h e method of d a l F e r r o f o r s o l v i n g t h e cubic. Dal F e r r o ' s r i v a l s i n t h i s m a t t e r , i f he can be s a i d t o have had any, a r e Jeromexardano (1501 - 1576) and Nicolo T a r t a l i a (about 1505 - 1557); i n a d d i t i o n , t h e p u p i l of Car ano Ludovic F e r r a r i (1522 1565), solved t h e q u a r t i c .

+

w,

-

Dal F e r r o ' s s o l u t i o n was seen i n 1515 by d a l F e r r o ' s Venetian according t o both Cardano and T a r t a g l i a . (The pupil l a t t e r s a y s ' 3 0 y e a r s a g o ' , and was w r i t i n g t h i s i n o r about 1545.) F i n a l l y , i n a Challenge t o T a r t a g l i a - - t h e second - - , F e r r a r i says t h a t d a l F e r r o ' s e l e g a n t complete s o l u t i o n of t h e cubic was i n h e r i t e d by d a l F e r r o ' s son-in-law, Hannibal d e l l a Nave, who showed i t t o Cardano i n 1542. The meaning of the word 'complete' i s n o t perhaps c l e a r : t h e important word i s , t o my mind, ' e l e g a n t ' . Elegance i s n o t p o s s i b l e without t h e imaginary; and i f t h e imaginary i s u s e d , t h e v a r i o u s s p e c i a l cases a r e a l l equivalent, so t h a t t h e s o l u t i o n i s n e c e s s a r i l y complete. C e r t a i n l y , i n t h e whole d i s p u t e t h a t followed, t h e r e i s nothing t o r e p r o a c h t h e f i r s t s o l v e r d a l F e r r o w i t h , except h i s understandable u n w i l l i n g n e s s t o p u b l i s h what might i n a s u p e r s t i t i o u s a g e c o s t him h i s head. Dal F e r r o was r i c h l y honoured by Bologna a s d o t t o r e eminente, matematico e c c e l e n t i s s i m o , and he had been dead 20 y e a r s b e f o r e t h e o t h e r p r o t a g o n i s t s h u r l e d t h e i r i n v e c t i v e s , and t h e i r c l a i m s and counter-claims, a t one o t h e r . The d i s p u t e i t s e l f , a s we s h a l l see, h a r d l y adds honour t o t h e s e p r o t a g o n i s t s ; however, an account of t h e i r l i v e s s u g g e s t s t h a t i t d i d not s u b t r a c t much e i t h e r -- i t merely showed what they were; and by comparison w i t h t h e people around them, t h e i r f a u l t s , t h a t may seem g l a r ing t o u s were r e l a t i v e l y minor. I n f a c t t h e whole episode reminds me of nothing so much a s of a d i s p u t e between Venice g o n d o l i e r s , h u r l i n g i n v e c t i v e s a t one another while r e c l i n i n g i n utmost indolence i n t h e i r c r a f t , o r e q u a l l y of p o l i t i c i a n s who then manage t o go o f f t o g e t h e r t o a v e r y good lunch. I would i g n o r e t h e whole s t o r y , b u t f o r t h e l i g h t i t sheds on t h e s t a t e of what I t a l y , once t h e s p i r i t u a l l e a d e r of Europe, t h e land of Fibonacci, of Dante, of Leonard0 da V i n c i , had become.

111. The Slow Renaissance

53

It is not that I blame the Italians: they were the victims. The blame was in the times. To realise this, we have only to read what was happening to Italians. Tartaglia gives some details of his life and personal views, things that some chroniclers of historical events would have preferred left out. At the infamous sac of Brescia (19-th February 1511), Tartaglia, aged about 6 - - some say as much as 12 - - , had taken refuge in the Duomo, only to be dragged out and have his face and head slashed and hacked at, by a soldier of the nephew Gaston de Foix of the King of France. The wounds, seven terrible wounds, were kept clean by licking by Tartaglia's mother, and they finally healed, but a slash right across the face healed badly, so that his teeth remained wobbly, and he was for a long time unable to speak, and then only with the stutter that gave him the nickname he retained as his own name. His father was dead, his mother penniless with three small children. He was at school only long enough to learn the letters A to K, then there was no money for school fees. He became his own teacher, studying, he says, the works of the defunct. In time, he took to teaching others, both publicly and privately, in various cities of the republic of Venice; later, from 1534 onwards, it was in Venice itself, except for a brief, and not pleasant, return to his home town of Brescia. During these years as teacher, there seems to have been no evidence of his having attained any great wealth, and in fact he was not above a little quiet swindle on the side: he published as his own a Latin translation by von Moerbecke of Archimedes' Floating Objects, and claimed in t h e preface that he had been at pains to decipher and translate a torn and barely legible Greek manuscript of the same, that had come into his hands. On the other hand, his translation of Euclid's Elements was undoubtedly his own, Loria tells us, since he did not scruple to alter any definition not to his taste! The works of Tartaglia that are relevant to the cubic, apart from his replies to the Challenges - - Cartelli di sfida - - are his Questi et inventioni diverse (1546), and his General Trattato. The latter was unfinished when he died, and the cubic part was compiled by someone else. The former ranges from ballistics and military formations to geometrical problems, and ends with cubics: its geometry is faulty, and the part on cubics shows signs, Loria says, of having benefited from Cardano's Ars Magna. I quote Loria in such matters, because he is t o t a l l y free from any tendency to favour his countrymen in his account of what they did. For instance, in the case of Fibonacci, about whom Moritz Cantor becomes quite ecstatic - "like a meteor he appeared," etc - - , Loria is restrained, and even critical in parts. In the case of Tartaglia, the picture is not flattering. Still it is not very different, perhaps, from what our own teachers may become, if, while demanding of them the utmost dedication and integrity, we not only make their task virtually impossible, but also economise on their salaries. Even s o , what I have said of Tartaglia will hardly induce the reader to give excessive credence to Tartaglia's claims in the matter of the cubic or in anything else, unless

54

Introduction

indeed he happens t o be an experienced d e t e c t i v e - s t o r y r e a d e r , who knows how d e c e p t i v e appearances can b e . The man who s t r o n g l y d i s p u t e d T a r t a g l i a ' s c l a i m , by i s s u i n g t h e C a r t e l l i d i s f i d a , was F e r r a r i : he had been taken i n and cared f o r , a s an orphan a t t h e age of 15, by Cardano i n Milan, and h e had a s s i s t e d h i s b e n e f a c t o r w i t h t h e work t h a t culminated i n A r s Magna. H e was of a v i o l e n t temperament, and a t 1 7 smashed a l l t h e f i n g e r s of h i s r i g h t hand i n a brawl. I n s p i t e of t h i s , he managed a s Cardano's co-worker, t o f i n d soon a f t e r , t e a c h i n g p o s i t i o n s i n Milan. He was l e d t o seek t h e s o l u t i o n of t h e q u a r t i c from a problem by Zuanne da Coi, and h i s s u c c e s s l e d t o a Chair a t Milan a t 2 2 . This he gave up t o t a k e charge of a census of t h e duchy of Mantua, where he stayed 8 y e a r s . He l e f t Mantua a f t e r a s e v e r e i l l n e s s t o s t u d y philosophy a t Bologna, where he was o f f e r e d a Chair 3 y e a r s l a t e r , but d i e d b e f o r e he could occupy i t : he had been poisoned by h i s s i s t e r , who wanted t o i n h e r i t h i s e s t a t e . The C a r t e l l i d i s f i d a were w r i t t e n when he was 25 o r 26 y e a r s of age. The above d e t a i l s , which a g a i n c a s t l i g h t on t h e u n b e l i e v a b l e contingencies t h a t could a f f e c t people i n t h o s e t e r r i b l e times, a r e s u p p l i e d by Cardano, i n a b i o g r a p h i c a l s k e t c h of h i s p u p i l . Cardano a l s o wrote h i s own autobiography. We r e a d t h e r e t h a t he s t u d i e d i n h i s n a t i v e P a v i a , and t h e n i n Padua where he became r e c t o r of s t u d e n t s and took a degree i n medicine. H i s g r e a t r e p u t a t i o n i n t h i s s u b j e c t , we a r e t o l d , reached o t h e r c o u n t r i e s , but i t was n o t u n t i l 1539 t h a t he was admitted t o t h e College of P h y s i c i a n s . However i n 1534 he was awarded a Chair i n t h e P a l a t i n e Academy a t Milan t o t e a c h mathematics, a p o s i t i o n t h a t he l o s t when he was d e f e a t e d i n a Challenge by Zuanne da Coi. He r e t u r n e d t o P a v i a , where he became p r o f e s s o r of medicine. There he had t h e g r i e f of s e e i n g h i s e l d e s t son brought t o j u s t i c e (25 A p r i l 1560) a s a p o i s o n e r , f o r u x o r i c i d e . H e l e f t P a v i a , and was awarded a Chair a t Bologna, and t h a t c i t y ' s honorary c i t i z e n s h i p . However he found i t necess a r y t o have h i s o t h e r son permanently e x p e l l e d from Bologna; and a f t e r 1 0 y e a r s , he himself was c a s t i n t o p r i s o n (14 October 1 5 7 0 ) , accused of magic, and h i s Chair and honorary c i t i z e n s h i p ended. He was e v e n t u a l l y r e l e a s e d s u b j e c t t o a promise n o t t o l e c t u r e i n t h e Papal S t a t e s , and he moved t o Rome, where h i s medical s k i l l earned t h e Pope's favour and a pension t o t h e end of h i s l i f e . Of h i s own c h a r a c t e r , Cardano p a i n t s an e x t r a o r d i n a r y , and by no means p l e a s i n g , p i c t u r e . I s h a l l s a y no more of i t , nor of h i s works ( q u i t e e x t e n s i v e i n o t h e r f i e l d s ) o t h e r t h a n A r s Magna, except t h e Sermo de p l u s e t minus, w r i t t e n a t t h e end of h i s l i f e , a f t e r Bombelli's Algebra: i n i t he e x p r e s s e s doubts about i m a g i n a r i e s , doubts e v i d e n t l y implied by a l l h i s e a r l i e r enumeration of c a s e s f o r c u b i c and q u a r t i c e q u a t i o n s , although he u s e s i m a g i n a r i e s himself i n Chapter XXXVII (De r e g o l a falsum ponendi) of A r s Magna. I n t h e background I have d e s c r i b e d , of poisoning of wives o r b r o t h e r s , and of t h e t o r t u r e and m u t i l a t i o n of small c h i l d r e n ,

111.

The Slow Renaissance

55

t h e a c t u a l s t o r y of t h e c u b i c i s comparatively tame. Between 1530 and 1539, T a r g a g l i a had solved a number of s p e c i a l c u b i c s , proposed t o him by Zuanne da Coi and by F i o r . Then on t h e 2-nd January 1539, he r e c e i v e s a l e t t e r from Cardano a s k i n g f o r h i s method f o r s o l v i n g , f o r p o s i t i v e p , q , t h e c u b i c e q u a t i o n x3 + px = q , and r e q u e s t i n g a l s o t h e s t a t e m e n t s of 30 problems s o l v e d . T a r t a g l i a sends t h e problems on t h e 28-th February; Cardano a g a i n a s k s f o r t h e method, swearing n o t t o d i v u l g e i t . T a r t a g l i a sends i t , f i r s t i n v e r s e , and t h e n , Cardano having f a i l e d t o u n d e r s t a n d t h i s , i n complete d e t a i l . F i n a l l y , a f t e r h e a r i n g from a t h i r d p a r t y t h a t Cardano i s w r i t i n g a book on t h e c u b i c , he writes t o accuse Cardano o f bad f a i t h . (The l e t t e r s a r e i n May and J u l y 1539.) F u r t h e r r e q u e s t s from Cardano n e v e r t h e l e s s r e a c h T a r t a g l i a (August 1539, January 1540) - - and a r e r e j e c t e d : t h e y concern two o t h e r t y p e s of c u b i c s . A l l t h i s i s more t h a n two y e a r s b e f o r e t h e p e r s u a s i v e Cardano o b t a i n s from d e l l a Nave i n 1542 t h e o r i g i n a l method of d a l F e r r o ; i t i s a l s o more t h a n f i v e y e a r s p r i o r t o A r s Magna. There seems t o be l i t t l e room f o r doubt a s t o who c h e a t e d whom. " I t was", w e a r e t o l d , " i n t h e i n t e r e s t s of s c i e n c e " . The excuse has a f a m i l i a r r i n g , except t h a t t h e word s c i e n c e sounds o u t of p l a c e . What seems more i m p o r t a n t , i s t h e p i c t u r e of I t a l y as i t had become. So l i t t l e t i m e had e l a p s e d s i n c e i t had a t t a i n e d p r o s p e r i t y , and i n t e l l e c t u a l , a r t i s t i c and s p i r i t u a l g r e a t n e s s . But t h a t , as I e x p l a i n e d , w a s p r e c i s e l y why i t w a s i n g r a v e danger: t h e p r e s s u r e s had been removed. The c o u n t r y was t o r n a p a r t by t h e bends, and t h e n ravaged by f o r e i g n t r o o p s . R e markably mathematics s u r v i v e d t h e r e 1 0 0 more y e a r s . ' 'Attributed, for instance, to Lagrange. Gauss later found all identities of this type. * I n Italy, mathematics began to bloom again, as we shall see, in the XIX-th century Risorgimento. It was a rather long time to wait: in the intervening period, mathematical life of a sort continued of course at a depressingly low level. However we should, perhaps, look on this rather as an illustration of the way in which the stimulus of a more distant past can give heart to a land, and help it overcome the handicap of a less inspiring intervening period. Thus the fact that historical events can depend on the distant as well as the immediate past, has its advantages, as we see again and again.

56

Introduction

PART I V

15.

The beginnings of a n a l y s i s .

T h e - p e r i o d we now come t o , s t r o n g l y b r i n g s t o mind t h e beginning of D a n t e ' s V i t a Nuova: "In q u e l l a p a r t e d e l l i b r o de l a mia memoria, d i n a n z i a l a q u a l e s i p o t r e b b e l e g g e r e , s i t r o v a una r u b r i c a , l a q u a l e d i c e : I n c i D i t v i t a nuova." Great s p i r i t u a l f o r c e s n o t o n l y played a part--good o r bad--in h i s t o r y , they a l s o changed mathematics and t h e l i v e s of i n d i v i d u a l mathem a t i c i a n s , such a s P a s c a l and Newton, G a l i l e o and de Moivre. Before going back i n time, we reached t h e p o i n t where Martin C o r t g s ' s A r t of Navigation had been p u b l i s h e d i n E n g l i s h i n 1561. L e t me s a y a t once t h a t , by c o n t r a s t w i t h what was happening i n I t a l y , t h e r e l a t i v e p r o s p e r i t y and t h e i n t e l l e c t u a l p r o g r e s s t h a t came t o England d i d n o t produce t h e bends. The r u l e o f Queen E l i z a b e t h w a s s t i l l too h a r s h , and t h e escape from c a l a m i t y a t t h e hands of t h e Spanish Aimada f a r too r e c e n t , f o r t h e p r e s s u r e s on i n d i v i d u a l s t o be c o n s i d e r e d removed. Moreover, a country t h a t depends o n . t h e sea, i s never wholly unaware of danger: i t i s s p a r e d t h e f u l l f o r c e of the bends when they come, and t h e s e a f u r t h e r p r o t e c t s i t from onslaught d u r i n g t i m e s of weakness. E l i z a b e t h a n iingland w a s v e r y conscious o f i t s dependence on t h e sea. With n a v i g a t i o n now i m p o r t a n t , astronomy began t o a t t r a c t much a t t e n t i o n : as a r e s u l t , t h e r e were i n Cambridge, as l a t e as t h e beginning o f t h i s c e n t u r y , no l e s s than t h r e e Chairs of Astronomy, and only one of Pure Mathematics. I n E l i z a b e t h a n and subsequent t i m e s , t h i s d i s p r o p o r t i o n a t e i n t e r e s t i n astronomy was n o t wholly c o n f i n e d t o England. However mathematics and astronomy f r e q u e n t l y went t o g e t h e r : they were t h e key t o n a v i g a t i o n , t o t h e f a b u l o u s r i c h e s of t h e New World and t h e F a r E a s t . Even i n t h e h e a r t of Germany, f a r f r o m t h e o c e a n s , a r e q u e s t was made a l i t t l e l a t e r , i n 1640, by t h e U n i v e r s i t y of Marburg t o i t s Landgrave--as r e c e n t l y found i n t h e U n i v e r s i t y a r c h i v e s by my f r i e n d t h e l a t e p r o f e s s o r Heinrich Behnke (1898-1979)-t o a u t h o r i z e a l e c t u r e r i n mathematics, ' ' i n view," i t s t a t e s , "of t n e importance o f mathematics a t t h e p r e s e n t t i m e . " The l e c t u r e r appointed was a Denis Pappin: t h e name i s a l s o t h a t of tile i n v e n t o r o f a steam-engine, who, however, was n o t born u n t i l 1647. U n i v e r s i t y r e c o r d s s p e c i f y s i x semesters of c o u r s e s i n mathematics: two of them on a r i t h m e t i c a l f o u n d a t i o n s , l o g i s t i c , s t u d y of t r i a n g l e s and t h e books of E u c l i d ; two on astronomy; one on geography and t h e d e s c r i p t i o n of t h e g l o b e ; and one on o p t i c s . Optics-- t n e t e l e s c o p e - - h a d r e v o l u t i o n i z e d astronomy, making c l e a r t h a t t h e Ptolemaic t h e o r y was n o t q u i t e a c c u r a t e , and needed r e p l a c i n g . Oresme, Copernicus, Thomas H a r r i o t , had a n t i c i p a t e d t h i s . Now i t was c l e a r t o G a l i l e o (1564-1642).

IV.

The New Beginning

57

T cho Brahe (1564-1601), Kepler (1571-1630), b u t by no means

b o s e knowledge comes d i r e c t from Heaven w i t h o u t t h e need o f t e l e s c o p e s . . .

How p l e a s a n t i t i s t o comment i n our s u p e r i o r wisdom, w i s e by h i n d s i g h t : However, as a f r i e n d , a well-known mathematician, i s fond of s a y i n g , "a c o i n h a s two s i d e s . " The t e l e s c o p e was a mechanical g a d g e t , f o r combining l e n s e s whose m a g n i f i c a t i o n p r i n c i p l e had long been f a m i l i a r . Crude l e n s e s , t h e B r i t a n n i c a t e l l s u s , have been u n e a r t h e d i n C r e t e and Asia Minor b e l i e v e d t o d a t e from 2000 B . C . Roger Bacon had l e a r n e d about l e n s e s from A 1 ilazen, and commented t h a t t h e Sun, Moon and s t a r s may be made t o descend h i t h e r i n appearance. Must o u r l i v e s , o u r r e l i g i o n , o u r whole o u t l o o k , depend on l i t t l e g a d g e t s ? Up t o t h a t time, t h e Renaissance had merely r e s t o r e d t h e l o s t t r e a s u r e o f Ancient Knowledge. Now t h e s u p e r i o r s c i e n c e of a n t i q u i t y - - a s i t s t i l l seemed--was b e i n g c o n t r a d i c t e d . To s u p p o r t t h i s r e v o l u t i o n i n t h o u g h t , would have c a l l e d f o r courage, and r i s k e d y e t a n o t h e r s p l i t of t h e C a t h o l i c F a i t h . B e s i d e s , one should perhaps be wery of new claims by s c i e n t i s t s , o r anyone e l s e . . . Mathematics may be common s e n s e , b u t t h i s i s e a s y t o confuse w i t h some p l a u s i b l e t h i n g we t h i n k t o s e e a t a g l a n c e , some obvious t h i n g t h a t t u r n s o u t t o be f a l s e . M i s c a l l e d common s e n s e i s a t r a p : t h i s i s how w e g e t p r o o f s t h a t e v e r y t r i a n g l e i s e q u i l a t e r a l , and s i m i l a r nonsense. People who f i n d mathematics h a r d , may be once b i t t e n , twice shy: p r e v i o u s l y caught i n t h e confidence t r a p . They l e a r n how n u m i l i a t i n g i t i s t o have f a i l e d t o d i s t i n g u i s h common s e n s e from t h e a r t o f jumping t o f a l s e c o n c l u s i o n s . Mathematicians a r e c a r e f u l t o t e a c h t h a t t h e most obvious t h i n g s r e q u i r e p r o o f . (When a non-mathematician claims something t o b e obvious, he may j u s t be p l a i n wrong; i f s o , he i s i n good company. The p h i l o s o p h e r Kant claimed i n t h i s way t o have ' p r o v e d ' t h e axiom o f p a r a l l e l s . There i s no l i m i t t o what a p e r s o n can do, once he confuses h i m s e l f w i t h God. Kant a l s o wrote a whole book of n e b u l a r nonsense, i n which a c o l d p r i m i t i v e g a s was t o have g e n e r a t e d r o t a t i o n through c o l l i s i o n s , c o n t r a r y t o t h e c o n s e r v a t i o n o f a n g u l a r momentum. The chemist Arrhenius commented t h a t Kant showed a d e f i c i e n c y of c r i t i c a l power which i s u n f o r t u n a t e l y n o t r a r e . ) I n the experience o f p a s t ages, t h e t r a d i t i o n a l p r o t e c t i o n a g a i n s t t h i s t r a p w a s t h e p r i e s t : he w a s t r a i n e d t o r e c o g n i z e human weakness. I t w a s a p r o t e c t i o n v i t a l l y needed: no one would go t o war w i t h o u t f i r s t c o n s u l t i n g t h e a u g u r i e s . The p r i e s t might s a y "NO, you must n o t go t o war today: a b l a c k crow f l e w p a s t t h e r e . " The p r i e s t s o f o l d knew t h e i r j o b ; s o d i d t h e Delphic O r a c l e . B e s i d e s , i n t h e Middle Ages, Kings had t h e i r J e s t e r - - a most i n t e l l i g e n t man; and they had t h e i r a s t r o l o g e r s . D i s a s t e r s due t o o f f i c i a l s t u p i d i t y and ignorance could b e delayed f o r a few c r u c i a l d a y s . We have l o s t t h i s s a f e g u a r d : World War I and a l l t h e m i s e r i e s s i n c e might have been avoided i f i t could have been delayed a few days. Today our only p r o t e c t i o n i s the soundness of o u r

58

Introduction

r e a s o n i n g , of our l o g i c , of o u r mathematics. This i s why mathematics i s s o i m p o r t a n t : i t i s where we l e a r n t o t h i n k , We need sound t h i n k i n g : w i t h o u t b r a k e s , we can no l o n g e r a f f o r d t n e d i c t a t o r s h i p of t h e s e c o n d - r a t e . The tragedy of t h e c o n f l i c t between G a l i l e o and Pope Urban V I I I i s n o t t h a t i t drove s c i e n c e i n t o P r o t e s t a n t c o u n t r i e s , b u t t h a t i t d e s t r o y e d humanity's brakes p r e m a t u r e l y . A f t e r t h a t , t h e Popes and t h e P r o t e s t a n t c l e r g y could s t i l l s t a r t a war, b u t they c o u l d n ' t s t o p one. Not anymore. Such t h i n g s pass unnoticed. What t h e world n o t i c e d , what h i s o o r i a n s emphasize so much, i s t h a t t h e s p i r i t u a l and i n t e l l e c t u a l l e a d e r s h i p of I t a l y was a t an end. The Pope w a s only p a r t i a l l y t o blame: t h e l a n d of Dante was going through d i f f i c u l t times: even a g r e a t goldsmith and s c u l p t o r , Benvenuto C e l l i n i , was n o t above a q u i e t murder on t h e s i d e . Astronomy, mathematics, moved North. I n I t a l y two names s t i l l s t o o d o u t , b u t n o t f o r v e r y l o n g : they were t h o s e of G a l i l e o ' s p u p i l s E v a n g e l i s t a T o r r i c e l l i (1608-1647), and Bonaventura C a v a l i e r i (1598-164/) . I n mathematics, t h e move had an a d d i t i o n a l c a u s e : t h e r e v o l u t i o n w i t h i n mathematics, r e s u l t i n g from changes i n n o t a t i o n . This s t a r t e d i n Vienna and e l s e w h e r e , b u t i t k s i n England t h a t t h e changes f i r s t began t o b e adopted. The sudden importance o f mathematics i n England had something t o do w i t h t h i s . Spain and P o r t u g a l had a c o n s i d e r a b l e s t a r t i n Ocean n a v i g a t i o n . I t was only 7 months a f t e r t h e Moors had f i n a l l y been t u r n e d o u t of Granada (2nd January 1492), t h a t Columbus s a i l e d i n t h e s e r v i c e of S p a i n . P o r t u g a l t o o undertook g r e a t voyages under Don d e n r i q u e t h e S e a f a r e r , Martin Behaim t h e p u p i l of Regiomontanus, and Vasco da Gama who went round t h e t i p of A f r i c a . Such voyages presuppose a knowledge of n a v i g a t i o n , and t h e r e f o r e o f mathematics and astronomy. This w a s what t h e I b e r i a n p e n i n s u l a had s o f o r t u n a t e l y gained by i t s long s t r u g g l e w i t h t h e I s l a m i c world. Yet t h e very manner i n which i t had been a c q u i r e d t i e d i t permanently t o a m i l i t a n t C h r i s t i a n i t y , r e a c t i o n a r y i n t h e extreme: i t remained a s t knowledge, t h e m a t e r i a l of t e x t b o o k - w r i t e r s r a t h e r than h e r s . I t p u t t h e i r c o u n t r i e s f o r a t i m e w e l l ahead i n t h e e a r l y long s e a voyages, b u t England, a late-comer, could h a r d l y a f f o r d t o do no b e t t e r i n i t s mathematical p r e p a r a t i o n , Of c o u r s e , from what I s a i d of John Peckham and Thomas Bradwardine, we can b e s u r e t h a t , p r i o r t o 1561, mathematics i n England had n o t e x a c t l y been n e g l e c t e d - - a s i t happens, i t had been promoted, i f a n y t h i n g , too e n e r g e t i c a l l y , by FLOGGING. No doubt t h e Normans used f l o g g i n g a f t e r t h e Conquest, on t h e d e f e a t e d B r i t o n s , b u t t h i s w a s g i v e n up i n time--except i n t h e s c h o o l s , where i t i s d e s c r i b e d , f o r i n s t a n c e , by Roger Bacon ( s e e Moritz Cantor 11, p . 9 6 , and f o o t n o t e ) , The t e r r o r i n s p i r e d by t h e s u c c e s s i v e p r o p o s i t i o n s of E u c l i d by t h i s method can be imagined: p r o p o s i t i o n V w a s t h e " f l i g h t o f t h e u n f o r t u n a t e . " The method does n o t seem t o have been too s u c c e s s f u l . Robert Recorde

IV.

The New Beginning

59

(1510-1558) complains (Moritz Cantor 11, p , 477) t h a t t h e E n g l i s h , a l t h o u g h of n a t u r a l i n t e l l i g e n c e i n f e r i o r t o v e r y few o t h e r r a c e s , are t e r r i b l y i g n o r a n t . L e t m e h a s t e n t o add t h a t I do n o t mean t o imply t h a t f l o g g i n g ,

as an a i d t o l e a r n i n g , d i d n o t come i n b e f o r e t h e Normans: c e r t a i n l y t h e y had no monopoly o f i t . Moreover mathematics came long b e f o r e t o t h e A r i t i s h I s l e s , as w e know from Alcwyn and t h e v e n e r a b l e Bede. An unknown v e r s i f i e r , c i t e d by Moritz Cantor from H a l l i w e l l , Rara Mathematica, t e l l s us Thys c r a f t com ynto England, a s y ghow s a y , Yn tyme of good k i n g Adelstones day,

although I would p u t t h e d a t e a couple of c e n t u r i e s e a r l i e r than t h a t . I n any c a s e , t h e d a t e i s immaterial compared w i t h t h e q u a l i t y , and I might mention i n t h i s c o n n e c t i o n Richard o f W a l l i n g f o r d , about whom p r o f e s s o r John North h a s r e c e n t l y w r i t t e n a book. Moritz Cantor mentions only t h a c W a l l i n g f a r d w a s p r o f e s s o r of philosophy and t h e a r t s a t Oxford about 1 3 2 6 , and t h a t he w r o t e , n o t i n a p p r o p r i a t e l y , about a n i n s t r u m e n t i t s name ( a l l by one) makes Albyon, which w a s a timepiece: a f i n e motto f o r B r i t a i n ( l i k e t h e S w i s s "Un pour t o u s , t o u s pour un"), and symbolizes t h e t r a d i t i o n a l B r i t i s h s u p p o r t of the individual. A s an a i d t o t e a c h i n g mathematics, f l o g g i n g may be about on a p a r w i t h t h e s t r i n g e n t examination system t h a t e v e n t u a l l y r e p l a c e d i t : probably some s t u d e n t s would have p r e f e r r e d i t . Of c o u r s e no system f u n c t i o n s u n l e s s i t can b e used i n t e l l i g e n t l y and i n moderation. I s u s p e c t t h a t t h i s only

t a k e s e f f e c t as a r e s u l t o f t h e mellowing o f c e n t u r i e s , such as a t l a s t allowed i n t e l l i g e n c e and moderation t o f i n d a p l a c e i n B r i t i s h Education f o r a w h i l e - - b u t t h a t w a s b e f o r e e v e r y t h i n g w a s t u r n e d u p s i d e down i n t h e b e l i e f , i t would s e e m , t h a t ignorance i s b l i s s , . . ( o r s h o u l d I say t h a t i t ' s f o l l y t o be w i s e , and s o t o b e a n a r i s t o c r a t of i n t e l l e c t , a t a time when t h e L a w of Conservation o f Ignorance i s s o c l e a r l y a s s e r t i n g i t s e l f ) . A t any r a t e , good o r b a d , - - 1 w i l l n o t argue t h e point--we can a l l a g r e e t h a t f l o g g i n g and examinations [administered w i t h t h e admirable enthusiasm f o r c a s t i n g o u t o f o t h e r s than o n e s e l f t h e d e v i l and a l l h i s works - - o r a t l e a s t t h e c a r d i n a l s i n o f l a z i n e s s ] can have a ' v e r y marked e f f e c t . ' "Les A n g l a i s , " t h e French s a y , " s o n t t r e s pratiques

."

This p r a c t i c a l i t y h a s n o t been w i t h o u t i t s own 'marked e f f e c t ' on B r i t i s h mathematics. To t h e l a t t e r i t h a s g i v e n , i n t h e c o u r s e of c e n t u r i e s , a p e c u l i a r t w i s t , t h a t I have no i n t e n t i o n of c r i t i c i z i n g , s i n c e I a m a g r e a t b e l i e v e r i n d i v e r s i t y . A t t h e t i m e w e are d i s c u s s i n g , t h i s p r a c t i c a l i t y proved, as w e s h a l l s e e , most i m p o r t a n t and b e n e f i c i a l . I t i s t h e r e f o r e no more t h a n f a i r t o show a l s o some l e s s d e s i r a b l e r e s u l t s , t o which i t h a s l e d . By a l l means, l e t us have people who a r e p r a c t i c a l ; I see no r e a s o n t o u s e o n e ' s head as a n a t t i c , f o r s t o r i n g u s e l e s s i n f o r m a t i o n . However i n t h i s

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connection I wish t o g i v e an example of what p r a c t i c a l i t y can lead t o . I n n i s p u b l i c l e c t u r e a t Wisconsin on t h e mathematical l i f e a t Cambridge around 1900 and b e f o r e , Littlewood mentions an examination i n Greek t h a t p r o s p e c t i v e e n t r a n t s t o t h e Univers i t y were r e q u i r e d t o p a s s . Candidates l i k e Eddington and Chapman, who came from u n i v e r s i t i e s where t h e r e was no such r e q u i r e m e n t , found t h i s irksome. The most c o n c e n t r a t e d cramming needed t o p a s s , took a t l e a s t s i x weeks. A contemporary of Littlewood's--we a r e n o t t o l d who, b u t i t could h a r d l y have been Eddington--managed t o do i t i n t h r e e : h e memorized a l l t h e E n g l i s h , and t r a n s l a t e d a passage o f t h e Greek testament he had n o t reached, counting t h e numerous Bamt C E L v ' s t o s t o p a t t h e r i g h t p l a c e . G e n e r a l l y , i n examinations, i t i s n e c e s s a r y t o know a number of r a t h e r elementary t h i n g s , and t o know them extremely w e l l , even a t t h e expense of any f u r t h e r knowledge whatsoever. The s u p e r f i c i a l i t y thus i n b r e d i n t o t h e s y s tem i s compensated nowadays, when examiners go o u t of t h e i r way t o r e c o g n i s e c r e a t i v i t y , o r depth of u n d e r s t a n d i n g . I n t h e XVI-th c e n t u r y , i t does n o t seem t o have been o f f s e t i n t h i s way: t h i s can be s e e n , f o r i n s t a n c e , by t h e s e t t i n g o f many r u l e s o f mathematics i n t o v e r s e (Moritz Cantor 11, p . 9 0 ) ) and by Bishop Tons t a l p u b l i s h i n g De a r t e supputandi (1522) , WITHOUT ANY PROOFS. Verse i s one o f t h e a i d s t o memory--a Greek d i c t i o n a r y ( r a t h e r s m a l l , and i n t o French) e x i s t s i n v e r s e . Robert Recorde, an i m a g i n a t i v e Welshman, thought of a p r a c t i c a l way of speeding up l e a r n i n g i n England: t h i s was t o become a n a t i o n a l c h a r a c t e r i s t i c , and i t c o n s i s t e d i n t h e widespread use of a b b r e v i a t i o n s . He was himself t h e f i r s t t o u s e o u r e q u a l i t y s i g n , and t o use i n England our s i g n s f o r p l u s and miilus. Thomas H a r r i o t added o u r i n e q u a l i t y s i g n s and s e v e r a l o t h e r a b b r e v i a t i o n s , Then W i l l i a m Ou h t r e d (1574-1660) i n t r o d u c e d i n h i s C l a v i s m a t h e m a t i c h e s s t h a n 150 mathematical symbols: i t was England's answer t o i t s system of t e r r o r . I mentioned t h a t changes i n n o t a t i o n came e a r l i e r i n Vienna: t h a t was a l r e a d y i n o r about 1510, and some of t h e a b b r e v i a t i o n s a r e f a i t h f u l l y reproduced i n L o r i a . However they were n o t very s u c c e s s f u l , Many more occur i n t h e work of the French mathematician F r g o i s V i e t e (1540-1603), who became famous i n decoding. (Spain accused France of having used w i t c h c r a f t t o break i t s code!) U n f o r t u n a t e l y , decoding w a s a l s o what V i S t e ' s own works r e q u i r e d . These works, p r i n t e d a t h i s own c o s t and l i b e r a l l y s e n t a l l over Europe, were s o l i t t l e understood t h a t a would-be t r a n s l a t o r , L o r i a t e l l s u s , exclaimed "I1 f a u d r a i t un second ViSte pour t r a d u i r e l e p r e m i e r . " This i s a l s o why t h e works are now s c a r c e , although t h e y were r e p r i n t e d i n 1646 by t h e Dutch mathematician F r a n c i s van Schooten: u n i n t e l l i g i b i l i t y had n o t y e t become, a s i n m o d e r n i s t i c t h i n g s , t h e supreme q u a l i t y i n a c o l l e c t o r ' s i t e m . A c t u a l l y Vibte was by no means t h e only mathematician of h i s t i m e t o l a c k c l a r i t y : L o r i a terms C a v a l i e r i ' s famous book, Geometria d e g l i i n d i v i s i b l i ,

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one 06 t h e most proKound and obscure i n t h e whole mathematical l i t e r a t u r e ; and h e adds t h a t many lesser w r i t i n g s , t h a t claimed t o be s c i e n t i f i c , w e r e so f u l l of f a s h i o n a b l e mythological a l l u s i o n s t o g i v e them a veneer of c u l t u r e , t h a t i t was h a r d t o s e e what they were d r i v i n g a t .

I t was thus i n England t h a t t h e s i m p l i f i e d n o t a t i o n s took r o o t . T h i s , and t h e g e n e r a l atmosphere favouring mathematics i n t h e B r i t i s h I s l e s , helped p r e p a r e t h e way f o r e x c i t i n g developments. I n t h i s s e c t i o n , I must s t i l l speak o f t h e mathemat i c i a n s who came b e f o r e t h a t . The f i r s t name I now wish t o mention i s John Na i e r (15501617), t h e i n v e n t o r of l o g a r i t h m s . This h e d&his spare t i m e , a s h e was mainly a very v o c a l P r o t e s t a n t Statesman and Defender of t h e P r o t e s t a n t f a i t h i n S c o t l a n d . Thanks t o h i s I f we l o g a r i t h m s , banks have made m i l l i o n s s i n c e h i s t i m e . could t a x them a t a s m a l l p e r c e n t a g e of what they have thus s a v e d , our new P h . D . ' s might n o t b e wondering where t h e n e x t meal i s coming from. U n f o r t u n a t e l y , mathematicians b e n e f i t t h e f u t u r e and a r e p a i d i n t h e p r e s e n t . A r t i s t s a r e much i n t h e same b o a t : t h e i r p a i n t i n g s may e v e n t u a l l y b e worth a f o r t u n e , b u t during t h e i r l i f e t i m e may go f o r a song o r a meal. I n mathematics, logarithms a r e important f o r a d i f f e r e n t I t i s no a c c i d e n t t h a t l o g a r i t h m i c t a b l e s a r e reason. p r i n t e d t o g e t h e r w i t h t a b l e s of t r i g o n o m e t r i c f u n c t i o n s . The l a t t e r f u n c t i o n s were d i s c o v e r e d much e a r l i e r and were t a b u l a t e d by Ptolemy. But t h e r e i s a remarkable connection between t h e two s e t s o f t a b l e s , and i t s d i s c o v e r y was one of t h e most remarkable of modern times i n mathematics, w i t h f a r reaching consequences i n a l l manner of f i e l d s , even i n t h e theory of numbers. There i t was used i n a s e n s a t i o n a l way by H i l b e r t t a g i v e a r a t h e r s h o r t proof of t h e transcendence of T I . S h o r t p r o o f s of transcendence e x i s t e d p r e v i o u s l y f o r t h e number e , t h e b a s e of Naperian l o g a r i t h m s , which i s d e f i n e d by s e t t i n g l o g e equal t o 1. S i m i l a r l y t h e number x = ~ / i4s d e f i n e d by s e t t i n g t a n x = 1. Before H i l b e r t , no one had found a way of using t h i s s i m i l a r i t y i n a proof of transcendence, and indeed t h e transcendence of T was only proved by Lindemann, w i t h g r e a t e f f o r t and a t g r e a t l e n g t h , snortly betore.' Suddenly, H i l b e r t produced h i s t h r e e page p r o o f ! Imagine t h e f e e l i n g s of Lindemann; I spoke e a r l i e r of p r a c t i c a l i t y : w e s e e how i t can a f f e c t mathematicians. We a p p r e c i a t e i t , a s w e do t h e t r a f f i c p o l i c e b u t only s o long a s i t i n t e r f e r e s w i t h ' o t h e r d r i v e r s . ' The paper o f H i l b e r t , c o n t a i n i n g h i s p r o o f , I recommend t o any g r a d u a t e s t u d e n t wishing t o l e a r n German: i t w i l l a l s o t e a c h him how a paper should b e w r i t t e n - - e v e n t h e s e n t e n c e s a r e s h o r t . The s t r a n g e connection between logarithms and t r i g o n o m e t r i c f u n c t i o n s i n v o l v e s i m a g i n a r i e s . I mentioned them i n connection w i t h c u b i c s . I would l i k e t o spend t i m e on them, p a r t l y because, a s I s a i d , we a r e d e a l i n g w i t h one of t h e g r e a t d i s c o v e r i e s , and p a r t l y because of deep p h i l o s o p h i c a l

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q u e s t i o n s i n v o l v e d , which throw l i g h t on t h e whole n a t u r e of mathematics. You need f e e l no alarm: I s h a l l t a l k about very simple t h i n g s . My f a t h e r had a s t u d e n t come t o him from theology t o switch t o mathematics. My f a t h e r began t a l k i n g t o him about i m a g i n a r i e s : t h e s t u d e n t decided t h a t h e would r a t h e r s t a y i n t h e o l o g y - - i t made more s e n s e t o him than t h e square r o o t of -1. The very word 'imaginary' f r i g h t e n s p e o p l e , as i f i t conjured up v i s i o n s of g h o s t s and s p i r i t s from a n o t h e r world. Can we wonder t h a t Cardano w a s imprisoned f o r w i t c h c r a f t ? A s l a t e a s 1 8 2 0 , t h e P a r i s e n g i n e e r i n g s t u d e n t s s t a g e d a r i o t a g a i n s t i m a g i n a r i e s : they complained t h a t i m a g i n a r i e s were u s e l e s s and do n o t r e a l l y e x i s t a t a l l . They were dead wrong! I n t h a t very y e a r , as Henri Poincar6 remarks i n one of h i s popular b u t h i g h l y i n s t r u c t i v e l i t t l e books, t h e r e occurred t h e worst and most t e r r i b l e of shipwrecks i n t h e memory o f man, t h e wreck o f t h e French s h i p " l a Mgduse. I ' Such shipwrecks, Poincar6 goes o n , a r e now f a r l e s s t o be f e a r e d , thanks t o w i r e l e s s t e l e g r a p h y . . And guess what w i r e l e s s t e l e g r a p h y depends on?

.

I s a i d b e f o r e t h a t one of t h e c h a r a c t e r i s t i c s of mathematics i s t h a t i t d e a l s w i t h a b s t r a c t t h i n g s . This i s p r a c t i c a l i t y a t work. I f we h a d n ' t invented numbers, which a r e a b s t r a c t , a c a l c u l a t i o n concerning a number of a l l e g e d a p p l e s ( n i c e l y wrapped up i n s p e c i a l p a p e r , as t h e b e s t ones a r e ) , would have t o .be done a l l over a g a i n , i f on unwrapping them, we d i s c o v e r e d t h a t t h e a p p l e s were r e a l l y p e a r s ! I experienced something o f t h e k i n d , b e l i e v e i t o r n o t , i n Lisbon i n t h e summer of 1 9 7 5 . I had bought a R a i l p a s s , v a l i d 15 days, t o s t a r t on J u l y 1 - s t . The lady a t t h e c o u n t e r was by no means uneducated--she could speak s e v e r a l languages--but s h e wrote t h e e x p i r y d a t e J u l y 1 4 - t h . When I p r o t e s t e d , s h e began t o c o u n t - - J u l y 1, f i r s t day; J u l y 2 , second day; a l l t h e way t o J u l y 1 5 , f i f t e e n t h day. "Oh, y e s : y o u ' r e r i g h t . " I hope s h e r e a l i s e d a t t h e end t h a t i n any such c a l c u l a t i o n i t does n o t m a t t e r whether you count days of t h e month, o r numbers, o r f o r t h a t m a t t e r , a p p l e s o r p e a r s o r any o t h e r items: S i m i l a r l y we might be counting squares o r c i r c l e s , o r some o t h e r kinds o f f i g u r e s , s o t h a t t h e process o f a b s t r a c t i o n a p p l i e s j u s t a s w e l l w i t h i n mathematics i t s e l f . I n t h i s way, a c a l c u l a t i o n , a theorem, o r a c o n c e p t , can be made more a b s t r a c t t h a n i t w a s t o b e g i n w i t h : we s a y i t can be g e n e r a l i s e d . For i n s t a n c e , a p l a n e geometrical r e s u l t may be a s p e c i a l c a s e of one i n 3-space. Indeed a problem i n mathematics may seem d i f f i c u l t s i m p l y because t h e r e a r e some i r r e l e v a n t d a t a : i t then becomes much e a s i e r , i f we f o r g e t them, and c o n c e n t r a t e on t h e r e a l l y important f a c t s ,

Nhen a PhD s t u d e n t has done a n i c e p i e c e o f work, one of t h e f i r s t q u e s t i o n s t h e a d v i s e r a s k s i s "Can you g e n e r a l i s e i t ? " . That means, can you remove some unnecessary axioms o r assumpt i o n s , i n o t h e r words, can you make i t more a b s t r a c t . This may n o t be t h e most e x c i t i n g t h i n g i n t h e w o r l d , b u t i t i s good honest work f o r a b e g i n n e r . The i n t r o d u c t i o n o f t h e imaginary i s a more e x c i t i n g g e n e r a l i s a t i o n : i t s o b j e c t i s n o t j u s t t o make t h i n g s work t h a t d i d s o b e f o r e , b u t t o make t h e m work when

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formerly they d i d n o t . We must b e a r i n mind t h a t MATHEMATICS IS NOT GRAMZUR: i t s r u l e s a r e n o t designed t o have a long l i s t of e x c e p t i o n s . I remember a s m a l l French boy s a y i n g t o h i s f a t h e r "Regarde, papa, l e s chevals!" Kany a f o r e i g n e r would have sympathised: however, s i n c e t h i s was a m a t t e r o f grammar, r u l e d upon by t h e a u g u s t Acadgmie F r a n c a i s e , t h e c h i l d w a s reminded of h i s e r r o r . Now t a k e t h e e q u a t i o n x 2 = a . The r u l e f o r solvLng i t i s n o t beyond t h e c a p a c i t y of a s m a l l c h i l d ; i t i s x = Ja, o r more c o r r e c t l y x = ? Ja. "Mais mon p e t i t , " s a y s t h e f a t h e r , "what i f a i s -1?" And t h e c h i l d , b e i n g a French c h i l d , and t h e r e f o r e l o i c a l and u n a f r a i d , r e p l i e s t h a t i n t h a t c a s e , p a p a , x i s J- , o r e l s e - R . Wlio a r e we t o s a y t h e c h i l d i s wrong? A c h i l d could have i n t r o d u c e d , i n t h i s i n some ways t h o s e simple-minded way, t h e imaginary J q : e a r l y a n a l y s t s were c h i l d r e n . But i t w a s some c e n t u r i e s b e f o r e tile new symbol could b e g i v e n a s a t i s f a c t o r y i n t e r p r e t a t i o n .

g-r

For two c e n t u r i e s , mathematicians were c o n t e n t t o u s e a p e r f e c t l y meaningless symbol! Imagine humanity, chained l i k e Prometneus t o t h e rock t h a t we c a l l r e a l i t y . A small c h i l d b r i n g s t h e key w i t h which t o unlock t h e c h a i n s : and l o and behold, w e cannot a c c e p t t h e key, because i t i s n o t chained l i k e us t o t h e g r e a t rock! Do we a c c e p t i t ? Of c o u r s e w e do! This i n v o l v e s us i n a l o g i c a l p o i n t t h a t a f f e c t s t h e v e r y e s s e n c e o f mathematics. I once l i s t e n e d f o r a whole y e a r t o t h e mathematical p h i l o s o p h e r Ludwig W i t t g e n s t e i n (1889-1951) . H e had suddenly, t o h i s d i s g u s t , become p o p u l a r . H i s Seminar, o r Conversation c l a s s a s such t h i n g s were c a l l e d i n Cambridge, was t o have been h e l d i n h i s College rooms, i n a s m a l l s i t t i n g room k e p t completely b a r e , e x c e p t f o r a t r e s t l e b l a c k board and a dozen s m a l l Woolworth c h a i r s - - t h e k i n d bought i n two p a r t s , s i x p e n c e f o r t h e s e a t , s i x p e n c e f o r t h e r e s t : t h e buyer was expected t o n a i l t h e s e a t onto t h e r e s t . A t t h e beginning of t h e academic y e a r , no l e s s t h a n 100 p e o p l e t u r n e d up: I w a s lucky t o have a r r i v e d f i r s t , and t o s e c u r e a s e a t . The o t h e r s crammed themselves i n t o t h e room, and a l l t h e way down t h e s t a i r s . . We agreed t h a t t h e n e x t meeting would be i n my own p a l a t i a l F e l l o w ' s rooms i n N e v i l e ' s Court. However a f t e r a few m e e t i n g s , W i t t g e n s t e i n i n s i s t e d on a r e t u r n t o h i s own rooms, and i n t h i s way he succeeded i n r e d u c i n g h i s audience t o about 4 0 . I n t h e c o u r s e o f t h e y e a r , t h e audience dwindled f u r t h e r , u n t i l a t t h e end only my f r i e n d Ursell and I were l e f t .

.

W i t t g e n s t e i n ' s w a s an amazing performance. I sometimes wonder wnat people would have thought o f h i s t e a c h i n g a t Wisconsin, where t h e r e are p r i z e s f o r s o - c a l l e d e x c e l l e n c e . I probably l e a r n t as much from him as from any o f my t e a c h e r s . But l e t me d e s c r i b e what a c t u a l l y took p l a c e . W i t t g e n s t e i n a d d r e s s e d himself almost e n t i r e l y t o a s i n g l e paragraph i n Hardy's Pure Mathematics, a book s u i t a b l e f o r a good E n g l i s h P u b l i c Schoolboy. Of t h i s book, L i t t l e w o o d s a y s i n h i s l e c t u r e on Cambridge, t h a t i t had a g r e a t i n f l u e n c e i n b r i n g i n g mathematical r i g o u r The paragraph W i t t g e n s t e i n s e l e c t e d t o England--belatedly. nad t o do w i t h i r r a t i o n a l numbers. I knew Hardy w e l l : he had

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Introduction

excellent qualities, quite apart from his mathematics and his great reputation, but he was a confirmed nominalist -- he felt entitled to make up any definition he pleased. He was an easy target for Wittgenstein. The odd thing is that Wittgenstein never attacked him! On the contrary, he tried very hard to understand that paragraph - - he tried for a whole year. Much of the time he stood in front of his little trestle blackboard, with his mouth open, At other times he would talk quite volubly. I can only say that it kept me thinking hard, with my brain racing furiously, trying to disentangle all manner of difficulties that I had never before dreamt of, or more often to imagine what difficulty could possibly be stopping him in mid-sentence at this particular point. One such difficulty concerned the equation a=b . If a and b are equal, why not use the same symbol for both? However in that case the equation becomes a=a , and this cannot be denied, and is consequently meaningless. To have a meaning, it must have a contrary which is thinkable; if it has no meaning, it should be eliminated. The insistence on using only one symbol for a given quantity takes us back to what I said earlier about Law. It also explains the Egyptian exclusion of fractions with numerator other than unity, the point being that as long as the numerator is unity, we automatically disallow the equal representations obtained by multiplying numerator and denominator by a same integer 2 , 3 , 4 , . . . Today, any such restriction sounds ridiculous. Does mathematics have to have meaning? Can we treat it as a game with symbols, that need no meaning as long as the rules are consistent? This is the whole question behind those two centuries' use of the imaginary: it is a question nobody asked. There is a more practical question: how is it possible for a meaningless thing to be so extraordinarily useful? The usefulness of the ima inary was soon most apparent, If we write i , we can solve by definition the for short for equation x2=-1 . However we can solve also an arbitrary quadratic equation, by the usual formula. Better still, it turns out that in terms of the complex numbers a+ib thus introduced every n-th degree equation, with real or complex coefficients, has exactly n roots. This so-called Fundamental Theorem of Algebra was not proved for some centuries, but it was at least conjectured rather early - - Felix Klein (Entwicklung der Mathematik) mentions that a certain Albert Girard asserted it in 1629. What more can one ask of a concept half-born, to which no meaning has yet been assigned? Nevertheless, a sensational discovery was yet to come -- the linkage I announced earlier, in terms of the imaginary, between the trigonometric functions and Napier's logarithm, or more precisely, between them and the exponential, the inverse function of the logarithm. This was Euler's formula I cos x + i sin x = eix

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a formula already implicit in (cos x + i sin x)" = cos nx + i sin nx, which was proved already by Abraham de Moivre (16671754), a Huguenot who escaped to England after being imprisoned in France, and who then had to make a living as best he could, and did mathematics in spare time. Euler's formula is crucial for elementary differential equations with constant coefficients, equations that engineers use all the time. But even the older form, due to de Moivre, is remarkable: it reduced all manner of complicated trigonometric identities to the old and familiar geometric series identity of Zeno's paradox of Achilles and the tortoise. What is the meaning of meaning? Should all this have been pruned away? Must we throw out a concept, however useful or promising, unless it is born perfect like Pallas Athenae, fully dressed, fully armed, and at the age of 21? And what if the imaginary did arise, perhaps, from some remaining trace of the naive mysticism of the Middle Ages: why should this matter? Clearly it served a need: this would not appear from solving quadratics - - as long as we are interested only in real solutions, we would be satisfied to say that the formula becomes meaningless when there are no real solutions. However I said I would leave to the present section the need for imaginaries in solving the cubic simply. Our modern notation, which was not available to dal Ferro and his successors, nor of course to the Babylonian solvers, greatly simplifies the work: in fact anyone can now solve the cubic without too much effort. We observe that by replacing the variable x by x+a, for a suitable a , we can remove the term in the second power of x ; alternatively, by next replacing x by l/x, we derive a form from which the term in the first power is gone. The problem is to get rid of both powers at once: for this we have only to try substituting x+a+b/x, for suitable a,b. We then obtain, after multiplying by x 3 , a quadratic in x 3 ; however, when we have solved for x 3 , we still have to solve a cubic of the form x 3 = A , with at least two roots involving imaginaries. Since these correspond to two roots of the original equation which may be real, we see that imaginaries are involved in an essential way. An equivalent method, based on substituting (ax+b)/ (cx+d), was much later found by Lagrange, in a memoir of the Academy of Berlin that I shall have occasion to refer to.

In the days of Napier, and those of de Moivre, some people did not ask for Euclid's threepence; nor was it unpatriotic to do mathematics, rather than watch the latest sporting injuries on television, nor even to find a substitute in a game, so as to prepare for an important mathematical examination in London (a crime for which a pupil at an English Public School was recently expelled). At any rate, the next three famous names I come to were, like Napier and de Moivre, those of amateurs: the Frenchmen Girard D6sar ues (1591-1661), Pierre de Fermat (1601-1665), Blaise P&23-1662), Ren6 Descartes (1596-1650). Of these, Pascal and Descartes were also great

66

Introduction

writers, and many a young man in France or elsewhere has learnt to love, or at least to respect, mathematics, by trying to emulate them. As mathematicians, however, Desargues and Fermat are the more esteemed today. With mathematicians, as with other kinds of people, some are in high esteem during their lifetime, others long after their death. This is another reason, why I do not subscribe to the mania of putting people in order of so-called excellence. DQsargues was a consultant engineer to Cardinal Richelieu. His 'Brouillon projet' on conic sections was written in a jargon of his own, according to Fermat, and it was not well received - - most copies remained with the publisher, who eventually destroyed them. It had two appreciative readers: one was Pascal, who wrote back to Fermat that he was perfectly able to understand DQsargues' language, and who elsewhere acknowledged his d e ! t to D4sargues and called him 'un des grands esprits de ce temps . The other was Chasles two centuries later, who helped found a whole s c h o o l o m e t r y on DQsargues' projective methods. When DQsargues wrote, books V to VII of Apollonius had not yet been discovered: they were found about the middle of the XVII-th century, in an Arabic translation. Pascal's own famous theorem on conics was a l s o prior to the finding of those books. However perhaps the most important part of DBsargues concerned the geometry of poles and polars, which occur already in Apollonius' book 111. The duality theory which arises from this played an immense part both in Analysis and in Geometry in the XIX-th century: it has continued to do so in today's Functional Analysis, with Minkowski's theory of dual convex figures and its extensions to general linear spaces by Hahn, Banach and, in a still more general form, by DieudonnC and Schwartz. Descartes, in complete contrast, was an excellent writer, and had an enormous impact on his time. He came from a good provincial family of lawyers and magistrates of the Touraine, the region of France that includes the city of Tours. He was able to live comfortably on the interest of the proceeds of the sale of a family estate, inherited from his mother; and although he was trained in his father's profession o f lawyer, he preferred to become, like many a young gentleman of his time, an amateur military officer without pay. He joined various armies in succession, the first being that of the Protestant leader, William of Orange. Descartes was an ardent Catholic, brought up in a Jesuit College, but it was customary to ignore personal feeling in such matters; he also had no overwhelming interest in fighting, and was no great military asset, he merely wanted to travel. When stationed in a city, he would meet any bright local intellects, and have great discussions, no doubt over a glass or two of Cognac. After some eight years of this pleasantly stimulating life, he met a papal nuncio, who suggested that his ideas deserved to be printed. He left his military occupations, and settled down for five years to write an ambitious work, entitled the World. However, when he heard what had happened to Galileo, he rushed to destroy most of what he had written: the rest he recast

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as a much shorter 'Discours de la Mgthode', which he published anonymously four years later. His analytic geometry, which consists of what mathematical beginners study today, was an appendix, intended merely as an illustration. From the 'Discours', people still quote the one sentence 'Cogito, ergo sum' - - I think, therefore I am. It made him one of the great philosophers of all time. It all seems depressingly easy: I detest slogans, and I have no interest in elementary analytic geometry. I keep preaching the need to go deeply into things. On the top of that, Descartes was not without his faults: he did not need to think correctly - - few great men do! He was also less original than he supposed: he was accused by Wallis, Loria tells us, of copying from Thomas Harriot's Artis analyticae praxis, published posthumously in 1631, six years before the appearance of the 'Discours'. Fermat too, may have anticipated Descartes' coordinates. Actually, as we saw, they were all anticipated by Oresme, whose works in manuscript were available: those works, Loria suggests, had readers, and influenced Kepler, and indirectly Newton. Such matters, and my feelings, and prejudices, are irrelevant here. Nor does it count in history that Descartes had an illegitimate daughter, who died before he could bring her to France. Descartes was leading a great movement to free thought from the older conceptions: in man's thought, true freedom and a true way of life are to be looked for. Spelt out, this may sound sanctimonious, and somewhat superfluous. A German song says 'My thoughts are free' - - Meine Gedanken sind frei; this is exactly what our thoughts are not, and never can be, as long as we fail to clear our minds of the prejudices and self-deceptions mentioned at the beginning of this introduction. I am not about to deliver a sermon on the arrogance of creatures who imagine they can think - - with the push-button thinking whose terrible implications menace us all. Descartes' movement was not concerned with thought of that sort: it was an inspiring surge forward, a great force, we realize in retrospect, that has made us what we are and continues to shape our future. If we look for causality, this is it. But I am concerned here with mathematics: if we wonder at its progress since that time, we must give credit to Descartes. The movement towards greater mathematical freedom was not started by Descartes: it is as old as time. We have long thrown out the rather absurd restrictions of Egyptian fractions. The movement was also manifest in the observations and experiments of Galileo, not only in his astronomical work, which goes beyond that of Copernicus, Kepler, Tycho Brahe, by introducting non-uniform mtion of a planet in an elliptic orbit, but also in the motion of a pendulum and in that of a body in free fall or along an inclined plane, -- all of which totally contradicts the classical assumption that in nature you have only the symmetry of straight lines and circles, and of uniform velocity along them. Or to go very slightly outside of mathematics proper, the movement could be seen some-

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Introduction

what e a r l i e r i n a r t , w i t h i t s p e r s p e c t i v e of c o n i c s , of i n f i n i t y , and i t s dynamism of c o l o u r , of Gothic a r c h e s , and o f t h e music t h a t came w i t h them, of harmonies and blending melodies. The movement was n o t unopposed--it never i s . Moritz Cantor speaks o f t h e "Law of Conservation o f I n t h e South, t h e Ignorance:" i t i s a good d e s c r i p t i o n . 2000-year o l d charges of impiety (heresy) were r e v i v e d ; and i n t h e North, f a r from t h e I n q u i s i t i o n , i t was--as we s h a l l s e e - - e a s i e r t o p r i n t a v e r s i o n of some a l l e g e d Greek mathemat i c s than any new r e s u l t s . LuckilymathematicPans were n o t blamed f o r t h e misdeeds of t h e i r b r o t h e r astnonomers--guilt by a s s o c i a t i o n was s t i l l some c e n t u r i e s ahead. With Descartes , t h e C a r t e s i a n axes provided t h e freedom of more g e n e r a l f u n c t i o n a l dependence, and h e nimself was much concerned w i t h h i g h e r a l g e b r a i c l o c i and e q u a t i o n s : t h i s i s c l e a r , f o r i n s t a n c e , from t h e folium of D e s c a r t e s , and from h i s r u l e of s i g n s i n l o c a t i n g r o o t s o f e q u a t i o n s ; t r a n s c e n d e n t a l f u n c t i o n s were being used a l s o , f o r i n s t a n c e N a p i e r ' s logarithm and e x p o n e n t i a l . Much l a t e r with E u l e r , Lagrange, and above a l l Cauchy, we come t o a n a l y t i c f u n c t i o n s , and then of course t o continuous f u n c t i o n s , and s o on. But I have s a i d a l l t h e good I can o f Descartes: he was t h e MAN OF HIS TIME. Mathematics has n o t remained s t a g n a n t : t o continue t o a t t r a c t g r e a t i n t e r e s t , h e would have had t o b e of a d i f f e r e n t stamp. The main c o n t r i b u t i o n s of Descartes are now r a t h e r obvious--he h a s n e i t h e r t h e depth and i n t e n s i t y of thought of P a s c a l , nor t h e i n g e n u i t y of Fermat. He s a i d h i m s e l f : "Fermat e s t Gascon, moi non!" And now t o come t o P a s c a l , who ranks i n France among t h e w r i t e r s and t h i n k e r s , of whom no c o u n t r y , o u t s i d e of Ancient Greece, seems a b l e t o produce more than one. With a French t e a c h e r l i k e mine, I could imagine many a youth t a k i n g up mathematics, simply because P a s c a l loved i t , i f only t h e enthusiasm and admiration passed on could s u r v i v e t h e l a t e r endless boredom of e s s a y s and examinations. A s t h i n g s a r e , t h e n e g l e c t of French a u t h o r s o u t s i d e France has i t s advantages: chere a r e people who r e a d French books f o r t h e i r own enjoyment, i n French. This i s how I recommend reading P a s c a l ' s Pensges. I t i s b e t t e r than any sermons on my p a r t , on t h e need t o go deeper i n t o t h i n g s . Moreover t o a p p r e c i a t e t h e way a mathematician t h i n k s , i t i s n o t s u f f i c i e n t t o r e a d only h i s mathematics: t h e purpose of h i s t o r y of mathematics i s t o make f u r t h e r i n f o r m a t i o n a v a i l a b l e , than can be found i n p u r e l y mathematical books. I n t h e c a s e of P a s c a l , about whom a g r e a t d e a l has been w r i t t e n , and whose w r i t i n g s a r e r e a d i l y o b t a i n a b l e , I can be b r i e f . P a s c a l ' s s i s t e r t e l l s us t h a t a s a mere c h i l d h e found t h e theorem on conics t h a t b e a r s h i s name. This i s remarkable, because i t was a theorem t h a t Apollonius missed: Apollonius has a most complicated c o n s t r u c t i o n , t o make up f o r n o t knowing t h i s s i m p l e way of deciding whether s i x given p o i n t s do o r do n o t l i e on a same c o n i c . Pappus of c o u r s e , missed i t t o o : h i s theorem, a s I mentioned, w a s t h e s p e c i a l c a s e o f

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The New Beginning

69

s i x p o i n t s s i t u a t e d on t h e d e g e n e r a t e c o n i c , c o n s i s t i n g of a p a i r of l i n e s . I s h a l l s a y no more of P a s c a l ' s e a r l y y e a r s , n o r s h a l l I e x p l a i n why a t t h e age o f 2 3 he d e c i d e d t o d e v o t e r i i s l i f e t o t h e J a n s e n i s t c o n c e p t i o n of t h e C h r i s t i a n r e l i g i o n . I n h i s c a s e , r e l i g i o u s f e r v o u r adds t o t h e depth o f h i s views on t h e r e a l aims and p u r p o s e of man. Even s o , h e n e v e r q u i t e gave up h i s mathematics, n o t even when he f i n a l l y e n t e r e d t h e community of P o r t Royal i n 1 6 5 5 . His J a n s e n i s t c o l l e a g u e s themselves a s k e d him t o h e l p w i t h a geometry t e x t b o o k f o r t h e i r s c h o o l . He a l s o belonged t o a s m a l l group of mathematicians who used t o meet i n P a r i s : i t i n c l u d e d p e o p l e l i k e Mersenne, mainly known f o r h i s correspondence w i t h Fermat and f o r a f a l s e c o n j e c t u r e a b o u t p r i m e s , and Roberval, t h e o r i g i n a t o r , w i t h T o r r i c e l l i , of a method, r e m i n i s c e n t of Oresme, f o r f i n d i n g maxima and minima, and more g e n e r a l l y t a n g e n t s . P a s c a l was g i v e n a s p e c i a l d i s p e n s a t i o n t o a t t e n d . The group corresponded w i t h Fermat, and a meeting between Fermat and P a s c a l h a d a t one time been a l l a r r a n g e d , b u t had t o b e g i v e n u p , as b o t h P a s c a l and Fermat were i n v e r y poor h e a l t h . I n s p i t e o f t h i s , Fermat and P a s c a l c o l l a b o r a t e d , b u t by c o r r e s p o n d e n c e . They d i d n o t w r i t e a n y t h i n g j o i n t l y , as Fermat would have l i k e d , b u t t h e y c a n b e c o n s i d e r e d , f o r i n s t a n c e , j o i n t founders o f t h e theory of P r o b a b i l i t y , a s u b j e c t i n which they became i n t e r e s t e d b e c a u s e Fermat was q u i t e a gambler. They were a l s o among t h e f i r s t t o p o s s e s s , b e f o r e Newton, some s o r t of C a l c u l u s , and t o d i s p r o v e by examples one of t h o s e f a l s e theorems t h a t a number o f p e o p l e k e p t r e p e a t i n g , by v i r t u e of t h e 'Law of C o n s e r v a t i o n of I g n o r a n c e ' - - t h i s one concerned t h e a l l e g e d i m p o s s i b i l i t y o f r e c t i f y i n g r a t i o n a l l y c e r t a i n types o f a r c s . F e r m a t w a s a c o u n c i l l o r i n t h e P a r l i a m e n t of Toulouse. Of n i s e a r l y l i f e l i t t l e i s known, e x c e p t t h a t he w a s Basque and s t u d i e d law. A t t h e age o f 30 h e became Bachelor of Laws o f t h e U n i v e r s i t y o f O r l e a n s . I t i s n o t even known whether h i s name w a s o r i g i n a l l y de Fermat, o r whether i t became s o ; t h e r e c o r d s m e r e l y i n d i c a t e t h a t he w a s s o named when h e w a s a p p o i n t e d t o t h e C r i m i n a l Court o f T o u l o u s e , a t t h e age of 37. Fermat i s one o f t h e p e o p l e b e l i e v e d t o have a n t i c i p a t e d D e s c a r t e s ' i n t r o d u c t i o n o f c o o r d i n a t e s i n geometry (by some h a l f - d o z e n y e a r s ) , and a l s o t o have a n t i c i p a t e d p a r t s o f t h e C a l c u l u s by u s i n g d e r i v a t i v e s t o d e f i n e t a n g e n t s . Lagrange champions F e r m a t ' s p r i o r i t y i n t h i s . The main q u e s t i o n i s what w a s t h e r e a l l y c r u c i a l s t e p . F e r m a t ' s c o n t r i b u t i o n s on t h e t o p i c s I mentioned a p p e a r i n h i s I n t r o d u c t i o n t o L o c i , d a t i n g from h i s s t u d e n t d a y s , b u t n o t p u b l i s h e d u n t i l 1 7 y e a r s a f t e r h i s d e a t h : t h i s w a s p l a n n e d a s an imagined l o s t work o f A p o l l o n i u s - - i t w a s t h e f a s h i o n t o compose such h y p o t h e t i c a l r e c o n s t r u c t i o n s . Genuine a n t i q u e works were b e i n g a v i d l y r e a d : Fermat h i m s e l f p o s s e s s e d a 1 6 2 1 e d i t i o n o f t h e A r i t h m e t i c o f Diophantus, i n t h e margin o f which he w r o t e h i s famous ' l a s t theorem'. P u b l i s h e r s w e r e w i l l i n g enough t o p r i n t a n t i q u e works; however F e r m a t ' s own works remained u n p u b l i s h e d u n t i l w e l l a f t e r h i s d e a t h , e x c e p t f o r a s i n g l e p a p e r p r i n t e d two years b e f o r e h e d i e d .

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Introduction

One t h i n g Fermat i s famous f o r i s h i s P r i n c i p l e of Least Time i n O p t i c s , I t was the f i r s t o f v a r i o u s g r e a t P r i n c i p l e s of Mathematical P h y s i c s , t h a t came i n t h e n e x t t h r e e c e n t u r i e s , and t h a t w i l l d o u b t l e s s continue t o come; i t p l a y s a v i t a l r o l e i n R e l a t i v i t y , where l i g h t - r a y s a r e geodesics. I t was i n complete disagreement w i t h Descartes o p t i c a l t h e o r y , one of t h e appendices t o t h e d i s c o u r s e on method. I n France, Descartes i s made t o appear a s one of t h e founders of t h e corpuscular theory of O p t i c s : i n p a r t i c u l a r t h e laws of r e f l e x i o n and r e f r a c t i o n , elsewhere c a l l e d S n e l l ' s Laws, a r e a t t r i b u t e d t o him.' They are c o r r e c t l y formulated i n t h e appendix I spoke o f , except f o r one a p p a r e n t l y unimportant det a i l . I once a s s i s t e d a s an examiner i n elementary a r i t h m e t i c f o r eleven-year o l d c h i l d r e n ; t h e Chief Examiner o b j e c t e d t o my g i v i n g 0 i n one q u e s t i o n , t o a g i r l who had found f o r t h e d i s t a n c e from the E a r t h t o t h e Sun a small f r a c t i o n of a m i l l i m e t e r : t h e r e was only one mistake, a l l t h e f r a c t i o n s were upside down! Descartes made t h e same m i s t a k e . . .But he made up f o r i t by supposing t h a t l i g h t t r a v e l s f a s t e s t i n very dense m a t e r i a l , and s i n c e t h e r e were no experiments t o show now f a s t l i g h t t r a v e l s , f o r i n s t a n c e , r i g h t through t h e E a r t h , t h i s was accepted even by Newton on D e s c a r t e s ' a u t h o r i t y . fIowever Newton had t h e g o o d e t o a l l o w f o r c e r t a i n c o r r e c t i v e f a c t o r s , which he c a l l e d " f i t s of easy r e f l e x i o n and r e f r a c t i o n , " and which a r e a c t u a l l y very modern i n o u t l o o k , b u t of c o u r s e , i n those t i m e s , r a t h e r hazy. A much b e t t e r theory of l i g n t , a wave t h e o r y , was given by Hu ens ( 1 6 2 9 - 1 6 9 5 ) i n h i s T r a i t 6 de l a LumiSre: he i s b e s t remem e r e d f o r Hu ens s P r i n c i p l e i n O p t i c s , a p r i n c i p l e t h a t c a r r i e s over i n t o t e Calculus of V a r i a t i o n s , b u t I s h a l l have occasion t o mention him a g a i n . The problem of r e c o n c i l i n g a corpuscular theory of l i g h t w i t h a wave theory takes us r i g h t i n t o Modern Quantum Theory,

+-+

We must n o t be s u r p r i s e d a t i n a c c u r a c i e s i n t h e work of preNewtonian mathematicians, nor even i n Newton h i m s e l f . Mathem a t i c a l l y , they were s t i l l c h i l d r e n , p a r t i c u l a r l y D e s c a r t e s , who must have found i t d i f f i c u l t t o avoid looking a t t h i n g s witn t h e breezy assurance of an amateur m i l i t a r y o f f i c e r , t o whom a l l questions have simple answers. Descartes produced, f o r i n s t a n c e , a ' v o r t e x motion of p l a n e t s ' ; he a l s o r e p e a t e d , a s d i d Newton a f t e r him i n a r a r e c a r e l e s s n e s s , t h e r e c t i f y i n g i m p o s s i b i l i t y I mentioned e a r l i e r . However, even P a s c a l , t h e model of French w r i t i n g , the champion of t r u t h , becomes badly i n a c c u r a t e i n h i s H i s t o i r e de l a r o u l e t t e , as L o r i a has p o i n t e d o u t . A l s o , even t h i n g s t h a t were b a s i c a l l y c o r r e c t , were a t t h a t time d i f f i c u l t t o e x p r e s s . Fermat, i n what Lagrange considered t o be h i s approach t o t h e C a l c u l u s , begs tile q u e s t i o n of l i m i t s , by i g n o r i n g t h e 010 indeterminacy; while Roberval, i n t h e d e l i c a t e q u e s t i o n s of maxima and minima, and of t a n g e n t s , t h a t I mentioned e a r l i e r , solved t h e problem of p r e s e n t i n g the m a t e r i a l , by having a p u p i l do i t f o r him! I n a d d i t i o n , much subsequent work i s r e p l e t e w i t h j u s t i f i c a t i o n s such as ' p a t e t ' , o r t h e French e q u i v a l e n t ' i l e s t f a c i l de v o i r e . ' The l a t t e r i s i n f u r i a t i n g much l a t e r i n L a p l a c e ' s M6canique C g l e s t e , L o r i a s a y s , where i t simply means

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a t o t a l absence o f r i g o u r ; w h i l e ' p a t e t ' i s t h e s o l e j u s t i f i c a t i o n g i v e n by Wallis i n h i s A r i t h m e t i c a i n f i n i t o r u m f o r t h e v a l u e x n + l / ( n + l y m h e area under t h e curve y = xn, between t h e o r i g i n and t h e v e r t i c a l through x . John Wallis (1616-1703) was regarded a f t e r Newton a s the g r e a t B r i t i s h mathematician of t h e t i m e : h e had s t u d i e d a t Emmanuel College, Cambridge, and became Fellow of Queens' u n t i l he m a r r i e d , and was made S a v i l i a n P r o f e s s o r a t Oxford i n 1649 and Court Chaplain i n 1 6 6 1 . H i s s u c c e s s o r a t Oxford w a s h i s f r i e n d , t h e famous a r c h i t e c t S i r C h r i s t o p h e r Wren (1632-1723), who had a l s o worked--in t h e same s l i p s h o d s p i r i t - - o n t h e r e c t i f i c a t i o n , and t h e t a n g e n t s , of p l a n e c u r v e s . I s h o u l d say something a l s o oE I s a a c Barrow (1630-1677), who had a g r e a t i n f l u e n c e on Newton's c a r e e r . A f t e r an M . A . degree i n Cambridge, he t r a v e l l e d 4 y e a r s on t h e C o n t i n e n t , a s he w a s uniiappy under Cromwell; h e r e t u r n e d (1659) t o t e a c h Greek i n Cambridge, and then mathematics a t Gresham College i n London. A t t h e a g e o f 33 he was appointed t h e f i r s t Lucasian p r o f e s s o r of mathematics a t Cambridge; t h i s he r e s i g n e d i n 1669, t o be Chaplain t o C h a r l e s 11. Barrow's main connection with t h e Calculus appears i n t h e second of t h r e e l e c t u r e There h e determines t a n g e n t s s e r i e s as Lucasian p r o f e s s o r . t o a p l a n e curve f ( x , y ) = 0 , by expanding i n powers o f h , k t h e d i f f e r e n c e f(x+h,y+k) - f ( x , y ) . This w a s no more r i g o r o u s than t h e work of t h e o t h e r s . I t i s n o t only i n t h e matter o f r i g o u r , b u t i n t h e v a r i o u s

i n a c c u r a c i e s g e n e r a l l y , and i n t h e r a t h e r c h i l d i s h approach used, t h a t i t h a s p a i d us t o go e v e r so much deeper t h a n those e a r l y p i o n e e r s . The r e a l s u r p r i s e i s n o t what they f a i l e d t o do, b u t what t h e y , i n e f f e c t , s t a r t e d . L e t me go back, i n t h i s c o n n e c t i o n , t o t h e d i s p u t e on O p t i c s between Fermat and D e s c a r t e s , and l e t u s t r y t o f o r g e t , f o r a moment, t h a t s i n c e t h a t time, experiments w i t h l i g h t , and a l s o R e l a t i v i t y and t h e e l e c t r o m a g n e t i c t h e o r y of l i g h t , appear t o s u p p o r t Fermat. Both Fermat and D e s c a r t e s a s s e r t t h a t a c o r p u s c l e of l i g h t changes i t s speed when i t p a s s e s i n t o a denser medium, and resumes t h e o r i g i n a l speed a s i t emerges a g a i n . I t i s easy t o understand t h e change, whichever i t i s , on e n t e r i n g , s i n c e t h e m a t t e r around can produce t h e n e c e s s a r y f o r c e s ; b u t how do we e x p l a i n t h e p r e c i s e r e v e r s a l on emerging? For i n s t a n c e , p u t t i n g on t h e b r a k e s w i l l s l o w a c a r , b u t r e l e a s i n g them w i l l merely l e t i t c o n t i n u e a t t h e reduced speed. This s u g g e s t s t h a t n o t only D e s c a r t e s , b u t Fermat a l s o , i s wrong. To a v o i d t h e paradox, we must suppose t h a t i n t h e d e n s e r medium, l i g h t t r a v e l s a t t h e same speed as b e f o r e . But how do w e t h e n e x p l a i n t h e e x p e r i m e n t a l and o t h e r e v i d e n c e , which s u p p o r t s Fermat? The e x p l a n a t i o n i s r a t h e r s i m p l e , f o r anyone who h a s s e e n t h e t u b e s i n which l i g h t i s made t o f o l l o w curved p a t h s o f a l l manner of shapes. A l i g h t r a y e n t e r i n g t h e tube i s r e f l e c t e d by t h e t a n g e n t p l a n e back i n t o t h e t u b e , wherever i t i s about t o emerge; i t t h e r e f o r e s t a y s i n s i d e u n t i l i t r e a c h e s t h e o t h e r end. A hundred y e a r s ago, such tubes were a c u r i o s i t y ; now t h e y a r e used i n s u r g e r y . I n p a r t i c u l a r , i f t h e t u b e i s

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s t r a i g h t and t h e r a y comes i n a t an a n g l e , t h e path i n s i d e the tube w i l l c o n s i s t of a s e r i e s of back and f o r t h r e f l e c t i o n s , and have t h e shape of a zigzag. I f t h e tube i s made f i n e r and f i n e r , and approximates more and more t o a s t r a i g h t l i n e , t h e angles of bhe zigzag can remain u n a l t e r e d . For a very f i n e t u b e , t h e p a t h of t h e l i g h t r a y w i l l look a s i f i t were s t r a i g h t , b u t i t i s r e a l l y a very f i n e zigzag, w i t h t h e same angles a s b e f o r e , s o t h a t i t s l e n g t h exceeds t h a t of t h e s t r a i g h t p a t h by a c o n s t a n t f a c t o r , and t h e t i m e taken by l i g h t t o pass along i s i n c r e a s e d by t h e same f a c t o r . For us t h e p a t h seems s t r a i g h t , s o t h a t , i n appearance, l i g h t seems t o have slowed down. I n r e a l i t y , however, t h e p a t h i s a kind of i n f i n i t e s i m a l z i g z a g , and along t h i s longer p a t h t h e r e i s no such slowing down. S i m i l a r l y , when l i g h t passes i n t o a medium, i t i s f o r c e d by t h e s t r u c t u r e of atoms and molecules of t h e medium t o follow an i n f i n i t e s i m a l z i g z a g , and t h i s causes t h e same apparent l o s s of speed. Zigzagging can of course occur along a l i n e which i s n o t s t r a i g h t , j u s t as o u r tube of l i g h t could curve round i n a l l manner of p r e t t y shapes. The corresponding i n f i n i t e s i m a l zigzag i s j u s t as easy t o imagine: i t c o n s t i t u t e s what I c a l l a ' g e n e r a l i s e d c u r v e . ' Such concepts, which e n l a r g e our freedom i n A n a l y s i s , help t o provide b e t t e r and more f l e x i b l e models o f what goes on i n t h e r e a l world. Another very convenient concept, which g r e a t l y i n c r e a s e s t h i s freedom, i s t h a t of a Schwartz d i s t r i b u t i o n : t h e l a t t e r i s based on d i f f e r e n t i d e a s , and I s h a l l n o t g i v e an i n t u i t i v e p i c t u r e . I might add t h a t t h e n o t i o n of an i n f i n i t e s i m a l zigzag along a given curve i s c l o s e t o conceptions of n a t u r e i n Modern P h y s i c s . I t i s almost e v i d e n t t h a t p a r t i c l e s moving i n such a manner w i l l resemble waves. A c t u a l l y g e n e r a l i s e d curves have been used q u i t e r e c e n t l y by Pedro Nowosad i n a promising approach t o Modern Quantum Theory. 'Even s o , Lindemann used the relation ei' case of Euler's formula below.

= -1, which is a special

2Although really due to Thomas Harriot (see Chapter 111, section 30).

16.

Newton.

I explained why mathematics could suddenly f l o u r i s h i n E l i z a b e t h a n England. This continued f o r 150 y e a r s : t h e B r i t i s h I s l e s h e l d t h e i r own a g a i n s t t h e r e s t of Europe, n o t only a t Sea, b u t i n mathematics. There i s a f e e l i n g t h a t Newton was somehow t o blame f o r t h e subsequent d e c l i n e , t h a t came with h i s d e a t h : t h i s i s t o t a l l y u n c a l l e d f o r , and I am anxious t o d i s p o s e of i t b e f o r e w e t a l k of Newton h i m s e l f , a s otherwise w e might tend t o downgrade a l l he d i d , We might f i n d o u r s e l v e s t h i n k i n g "Yes, b u t a r e a l l y g r e a t man would have had p u p i l s g r e a t e r than h i m s e l f , " o r e l s e "The t e s t o f g r e a t n e s s comes from what i t l e a d s t o . " I n Newton's

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c a s e , t h i s would be most u n f a i r . The d e c l i n e t h a t f o l l o w e d him s h o u l d b e blamed on anyone r a t h e r t h a n on Newton, A s I i n d i c a t e d e a r l i e r , growth and decay a r e a s t r u g g l e o f e x p o n e n t i a l s : each t e r m grows w i t h m i s e r a b l e slowness u n t i l w o r t h n o t i c i n g a t a l l , and t h e n w i t h e x t r a o r d i n a r y speed i f unchecked. Thus p r o g r e s s can b e r a p i d , o r i t can be o v e r t a k e n and overwhelmed, by t h e sudden growth of something p r e v i o u s l y i n s i g n i f i c a n t . What came a f t e r Newton w a s a l l t h e more d i s a p p o i n t i n g : i t was a decay a f f e c t i n g much more than m a t h e m a t i c s , a v i r u s , a m e n t a l a b e r r a t i o n o r f a s h i o n o f t h e day, m u l t i p l i e d t o n a t i o n w i d e p r o p o r t i o n s . Mathematics d i d n o t d i e o u t : t h e B r i t i s h a d m i r a l t y even r e t a i n e d i n i t enough i n t e r e s t t o award l a t e r b 300 t o E u l e r . A l e t t e r i n h i s c o l l e c t e d works i n d i c a t e s t h a t i t was f o r theorems t h a t had h e l p e d t h e i r Lunar T a b l e s : W . L . S h a p i r o s u g g e s t s t h a t t h e Tables were l a t e r v i t a l t o m a i n t a i n i n g B r i t i s h s u p e r i o r i t y a t Sea i n t h e Napoleonic w a r s . However mathematics i n B r i t a i n c e r t a i n l y d i d n o t f l o u r i s h . What s t o p p e d i t ? What w a s t h i s v i r u s , t h i s d e s t r u c t i v e f a s h i o n o f t h e d a y ? - - t h e same t h i n g t h a t l e d t o t h e South Sea Bubble w i t h d i s a s t r o u s e f f e c t on t h e p r o s p e r i t y of t h e l a n d , t h e atmosphere under t h e Georges : gambling, h u n t i n g , s w e a r i n g and t h e r e s t . "Most o f t h e i r d i s c o u r s e , ' ' Pepys t e l l s u s , ''was a b o u t h u n t i n g , i n a d i a l e c t I u n d e r s t o o d v e r y l i t t l e . " Pope s a y s , " E x p l e t i v e s t h e i r f e e b l e a i d do j o i n , and t e n low words o f t c r e e p i n one d u l l l i n e . " I t w a s b e t t e r than k i l l i n g one a n o t h e r f o r t h e n o b l e s t of r e a s o n s i n Cromwell's d a y , b u t i t w a s n o t conducive t o m a t h e m a t i c s , Not t h a t i f you swear, you cannot do mathematics: one g r a i n o f sand i s no mountain. However once t h e whole atmosphere i s r i b a l d and a n t i - i n t e l l e c t u a l , you and I have b u t a s m a l l chance of e m u l a t i n g Gauss. I had t h e good f o r t u n e myself o f growing up i n a n i d e a l atmosphere, b u t I a l s o know how l i t t l e i t t a k e s t o keep p e o p l e from mathematics. One of my d a u g h t e r s w a s t o l d by a female a d v i s o r , r i g h t h e r e i n Wisconsin, t h a t o f -c_o u r s e mathematics i s u n s u i t a b l e f o r women! Today, twenty y e a r s l a t e r , s h e h a s gone back t o t h e U n i v e r s i t y t o l e a r n some of t h e mathematics t h a t s h e now f i n d s s h e n e e d s . There i s a myth a b o u t m a t h e m a t i c i a n s b e i n g b o r n and n o t made-- t h e y a r e b o r n , b u t unmade: d i s c o u r a g e d by b u s y b o d i e s . Turn a deaf e a r when someone t r i e s t o d i s c o u r a g e you from what i s i m p o r t a n t t o you: You may be wrong, b u t i t i s f o r you t o find out.

I t i s p r e c i s e l y i n t h i s , and n o t merely as a g r e a t mathemat i c i a n , t h a t I s a a c Newton (1643-1727) i s r e m a r k a b l e . He w a s b o r n i n t h e middle o f B r i t a i n ' s f l o u r i s h i n g m a t h e m a t i c a l p e r i o d . He overshadowed h i s p r e d e c e s s o r s ; b u t i n B r i t a i n , as I e x p l a i n e d , h e u n f o r t u n a t e l y a l s o overshadowed h i s s u c c e s s o r s . I n t h e i g n o r a n c e o f my y o u t h , and p a r t l y because I w a s then educated abroad, I used t o b e p r e j u d i c e d a g a i n s t him. What I r e a l l y c o u l d n o t s t a n d , w a s t h e way i n which f e l l o w countrymen, come abroad as t o u r i s t s and t o t a l l y

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Introduction

i g n o r a n t of mathematics themselves, would grandly inform me i n t h e XX-th c e n t u r y , t h a t England was then and f o r e v e r supreme i n mathematics, because i t had once had Newton.. . I knew, of c o u r s e , how wrong they were, b u t a s I have e x p l a i n e d , t h e f a u l t was E n g l a n d ' s , n o t Newton's, I t was what had a l s o forced America i n t o Revolution, a whole atmosphere, s t i f l i n g i n i t s incomprehension. The most t h a t could be p u t t o Newton's account, was t h a t , thanks t o him, t h e whole development of B r i t i s h mathematics was reduced t o t h e s t a t u s of e x e r c i z e s such as we now ask s t u d e n t s t o s o l v e , o r e l s e t o laws so fundamental, as i n g r a v i t a t i o n o r t h e d i f f r a c t i o n of l i g h t , t h a t no f u r t h e r improvement seemed f e a s i b l e , I f i t had been France i n 1810, i n s t e a d of England about a hundred years e a r l i e r , some ' s e c r e t a i r e p e r p g t u e l ' might w e l l have r e p o r t e d t h a t i n mathematics t h e r e were only minor p o i n t s t o be c l e a r e d up, a m a t t e r about which I s h a l l be speaking a t a l a t e r stage. To have seen beyond t h e s e t h i n g s would have r e q u i r e d a mind comparable t o Newton's own. H e a l o n e could r e a l i s e how much was s t i l l t o b e done. He says I do n o t know what I may appear t o t h e world, b u t t o myself I seem t o have been only l i k e a boy p l a y i n g on t h e s e a s h o r e , and d i v e r t i n g myself now and then f i n d i n g a smoother pebble o r a p r e t t i e r s h e l l t h a n o r d i n a r y , w h i l s t t h e g r e a t ocean of t r u t h l a y a l l undiscovered b e f o r e me.

There i s something v e r y l i k e a b l e about a man who can say such t h i n g s when h i s g r e a t powers and achievements a r e c l e a r . Small minds a r e t h e ones t o imagine t h a t l i t t l e remains t o be done, j u s t a s b r a g g a r t s b o a s t t h a t l i t t l e was done b e f o r e them, whereas Newton goes on t o s a y t h a t i f he has seen f u r t h , e r than o t h e r s , i t i s because h e h a s s t o o d on t h e shoulders of g i a n t s . L e t o t h e r s comment a t once t h a t t h i s had become a c l i c h 6 a t t h e t i m e : my f i r s t thought i s r a t h e r t h a t t h i s i s how I f e e l myself. I am no hero-worshipper, b u t I can o c c a s i o n a l l y b e e n t h u s i a s t i c about someone who i s modest and r e s p e c t s h i s t e a c h e r s , a s I do mine. I n t h e s e l e c t u r e s , t h e company t h a t we s h a l l mix w i t h w i l l be of t h e h i g h e s t l e v e l : Newton and o t h e r s t h a t w e s h a l l come t o a r e u n i v e r s a l l y considered g r e a t . I am a l l t h e more anxious t o make c l e a r t h a t I do n o t f e e l even t h e g r e a t e s t mathematicians o r i n t e l l e c t u a l s - - a n d I have known some p e r s o n a l l y - - t o b e b a s i c a l l y d i f f e r e n t from t h e r e s t of us : n e i t h e r i s t h e b e s t of v i o l i n s n e c e s s a r i l y made of d i f f e r e n t wood from a l l t h e o t h e r s . A v i o l i n i s fashioned bv a c r a f t s man, b u t a human being f a s h i o n s h i m s e l f . This l a s i seems most c l e a r i n t h e c a s e of Newton. H e must have been r a t h e r l i k e a b l e : t h i s i s borne o u t by t h e s t o r i e s about him, spurious o r not--they show what people thought of him. We a l l know t h e s t o r y of Newton and t h e apple t h a t woke him by f a l l i n g : t h e moral i s what m a t t e r s - -

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a g r e a t man t u r n s a minor mishap, such as b e i n g h i t , n o t i n t o an excuse f o r p r o f a n i t y , b u t i n t o an o p p o r t u n i t y . Then t h e r e ' s t h e s t o r y of t h e c a t and t h e k i t t e n , and how Newton had h o l e s made i n h i s door t o l e t them i n o r o u t - - a b i g h o l e f o r t h e c a t , a s m a l l one f o r t h e k i t t e n . Would t h e b i g h o l e have done f o r b o t h ? Not if t h e c a t remained i n t h e way: F i n a l l y t h e r e ' s t h e s t o r y o f t h e v i s i t o r who w a i t e d i n t h e d i n i n g room. A f t e r an hour a s e r v a n t brought i n a covered d i s h . Another hour went by: t h e v i s i t o r began t o f e e l uncomfortably hungry. E v e n t u a l l y h i s c u r i o s i t y g o t t h e b e t t e r of him and he removed t n e c o v e r : i t w a s r o a s t chicken. He p u t t h e cover back, l i f t e d i t o f f a g a i n , and b e f o r e he r e a l i s e d what he w a s doing, only bones remained. He had h a r d l y p u t t h e cover back o n , wnen Newton rushed i n , s n a t c h e d t h e cover o f f , s a w t h e b o n e s , and t u r n e d t o h i s v i s i t o r , s a y i n g , " I ' m so s o r r y , I had q u i t e f o r g o t t e n I had had l u n c h . " Has a p r o f e s s o r ' s f o r g e t f u l n e s s e v e r been more endearing--and more t a c t f u l ? Not a l l g r e a t men are l i k e d , b u t mathematicians, i n s p i t e of e v e r y t h i n g , t e n d t o be remembered w i t h a f f e c t i o n . X i l b e r t w a s a h a r d t a s k m a s t e r , P o i n c a r 6 , when a v i s i t o r from Sweden had w a i t e d a c o n s i d e r a b l e t i m e , p u t h i s head round t h e c o r n e r and c a l l e d o u t "Plonsieur, vous m e d6rangez beaucoup." S t i l l , t h e r e a r e innumerable more p l e a s a n t s t o r i e s about H i l b e r t and P o i n c a r 6 . Once, when Frau H i l b e r t had g u e s t s , she asked h e r husband t o p u t on a c l e a n s h i r t : he went up t o h i s room, took o f f h i s s h i r t , and mechanically c o n t i n u e d t o go t o bed. And h e r e i s what Poincar6 s a i d t o a l i t t l e boy o f 8 , who asked him how t o ' t h i n k mathematically': You look f o r a n i n c l i n e d sandy p a t h , you walk u p , you walk down, t h e n up a g a i n , and down. Mathematical t h i n k i n g i s g e n e r a t e d by t h e f r i c t i o n of t h e s o l e s o f t h e f e e t . " The c h i l d t r i e d t h i s i n t h e Luxembourg, b u t t h e p a t h was n o t i n c l i n e d : h e e v e n t u a l l y became p r o f e s s o r a t T r i n i t y C o l l e g e , Cambridge, b u t n o t of mathematics. Newton w a s n o t always p l e a s a n t , n o t even t o h i s f r i e n d s . Over q u e s t i o n s of p r i o r i t y , he could perhaps have a f f o r d e d t o be generous: i n s t e a d h e w a s v i o l e n t and unscrupulous. IIe i s a l s o r e p u t e d t o have s u f f e r e d breakdowns: i f s o t h e y damaged n e i t h e r h i s f r i e n d s h i p s - - w h i c h must have been s o r e l y t r i e d - n o r h i s mathematical d r i v e . H e probably s u f f e r e d a t l e a s t t h e s t r a i n s t h a t are an o c c u p a t i o n a l h a z a r d o f mathematicians, j u s t as muscle s t r a i n i s a h a z a r d i n s p o r t ; h i s experiments t o o , may have involved risks--mercury p o i s o n i n g h a s been suggested--but I am n o t i n a p o s i t i o n t o d i s c u s s t h i s . I f i n d i t a p p a l l i n g t h a t f u t u r e mathematicians a r e n o t warned and p r e p a r e d , as a t h l e t e s a r e , of t h e s t r a i n s o f t h e i r p r o f e s s i o n . The b r a i n i s more d e l i c a t e and v a l u a b l e t h a n mere muscle: do people s t i l l imagine t h a t i t can be d i s p e n s e d w i t h , as w e l l a s b a t t e r e d w i t h impunity i n v a r i o u s s p o r t s ? I n mathematics, no f o o t b a l l helmet p r o t e c t s i t .

. .

Mental s t r a i n s a r e Ear from unknown i n t h e h i s b o r y of mathema4 t i c s , and i t i s h i g h t i m e t h e l e s s o n s t o b e d e r i v e d were b e t t e r known. G e n e r a l l y , a mathematician a d j u s t s t o h i s s t r a i n s , as long as t h e y do come j u s t from h i s mathematical work, What can b r e a k him, as may b e s e e n t i m e and a g a i n , i s

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some added s t r a i n from o u t s i d e . I s a i d t h a t Newton, i n p e r i o d s of s t r a i n , went on doing mathematics. To do t h e t h i n g s we r e a l l y w a n t t o do, however e x h a u s t i n g , n a t u r e provides us w i t h a r e s e r v e o f s t r e n g t h , and even h a p p i n e s s . Madame Curie writes : C ' e s t dans ce m i s e r a b l e v i e u x hangar que s ' 6 c o u l 3 r e n t l e s m e i l l e u r e s e t l e s p l u s heureuses annges de n o t r e v i e . J e p a s s a i s p a r f o i s l a journge e n t i e r e 3 remuer une masse en g b u l i t i o n , avec une t i g e de f e r p r e s q u ' a u s s i grande que moi. Le s o i r j ' 6 t a i s b r i s 6 e de fatigue. Persons whose p r o f e s s i o n i s t o t r e a t t h o s e who a r e under a s t r a i n , do n o t always understand what s c i e n t i s t s l i v e f o r . I knew a young refugee mathematician of g r e a t promise, one of many welcomed t o f r e e B r i t a i n by a government t h a t denied them a l i v i n g . I t d i d n o t m a t t e r t h a t he l i t e r a l l y s t a r v e d : what he could n o t s t a n d was t h a t s o d i d h i s devoted w i f e . H i s manner had f o r some t i m e become u n p l e a s a n t , and t h i s h e r e t a i n e d i n h i s breakdown and a f t e r i t s ' c u r e ' ; b u t of h i s once b r i l l i a n t b r a i n , h e was l e f t only enough t o become a somewhat u n s a t i s f a c t o r y a g r i c u l t u r a l worker. Somewhere i n s i d e him, t h e r e s t o f him, h i s r e a l s e l f , remained imprisoned f o r another 33 y e a r s , u n t i l r e l e a s e d by d e a t h . The only p l a c e I know where t h e s e q u e s t i o n s a r e r e a l i s t i c a l l y c o n s i d e r e d , i s L i t t l e w o o d ' s l e c t u r e "A mathematician's a r t of work," one of t h r e e p u b l i c l e c t u r e s given a t Wisconsin, and t o be published w i t h h i s c o l l e c t e d works. A few b r i e f e x c e r p t s may n o t b e out of p l a c e : E i t h e r work a l l o u t , o r r e s t completely. I t i s too easy, when r a t h e r t i r e d , t o f r i t t e r a whole day away w i t h t h e i n t e n t i o n of working, b u t never g e t t i n g p r o p e r l y down t o i t . This i s pure w a s t e , nothing i s done, 2nd you have had no r e s t o r r e l a x a t i o n . . . I want t o say something about t h e v a r i o u s symptoms of overwork, with i t s u r g e n t need of a r e s t . I have wrongly d i s r e g a r d e d them i n t h e p a s t ; s o do d o u b t l e s s o t h e r s too. One symptom can be muscular t r o u b l e . . An ominous symptom i s an o b s e s s i o n w i t h t h e importance of work, and f i l l i n g every moment t o t h a t end. The most i n f a l l i b l e symptom i s t h e a n x i e t y dream. One s t r u g g l e s t e n s e l y a l l n i g h t w i t h a pseudo-problem-p o s s i b l y with some odd r e l a t i o n t o o n e ' s c u r r e n t j o b ; and one wakes i n t h e morning q u i t e u n r e f r e s h e d .

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A s r e g a r d s h o l i d a y s , a governing p r i n c i p l e i s t h a t 3 weeks, e x a a t l y 2 1 d a y s - - t h e p e r i o d i s c u r i o u s l y p r e c i s e - - i s enough f o r recovery from t h e s e v e r e s t mental f a t i g u e , provided t h e r e i s nothing a c t u a l l y p a t h o l o g i c a l . This i s e x p e r t o p i n i o n , and my experience agrees e n t i r e l y . . . I t i s v i t a l , however,

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t h a t i t ( t h e 3 weeks h o l i d a y ) be a b s o l u t e l y unbroken, whatever t h e t e m p t a t i o n o r p r o v o c a t i o n . Of c o u r s e , i t i s n o t f a i r t o quote t h e s e t h i n g s o u t o f c o n t e x t . I should a t l e a s t mention how c a r e f u l L i t t l e w o o d i s t o b e g i n by saying how very d i f f e r e n t some people can be from o t h e r s . But h e does a l s o g i v e what h e c a l l s 'awful e x a m p l e s , ' f o r i n s t a n c e t n e c a s e of a man who s t a r t e d o f f extremely w e l l , b u t whose work became d u l l e r and d u l l e r : " i f h e had done t h e work i n t h e i n v e r s e o r d e r , he would have become Fellow of t h e Royal S o c i e t y . ' ' What d i d t h e man do wrong? I a m a f r a i d i t w a s p r e c i s e l y what most o f us do a l s o : working y e a r However, L i t t l e w o o d , i n , year o u t , w i t h n e v e r a h o l i d a y who reached t h e good age o f 9 1 , followed h i s r u l e s w i t h g r e a t s e l f - d i s c i p l i n e , and few o f us can e x p e c t t o do as w e l l . He t e l l s us f u r t h e r :

...

Mathematics i s very h a r d work, and dons t e n d t o be above average i n h e a l t h and v i g o u r . Below a c e r t a i n t h r e s h o l d a man c r a c k s u p , b u t above i t h a r d mental work K e s f o r h e a l t h and v i g o u r (and a l s o , on much h i s t o r i c a l : evidence, f o r l o n g e v i t y ) . The h i g h e r mental a c t i v i t i e s a r e p r e t t y tough and r e s i l i e n t , b u t i t i s a d e v a s t a t i n g e x p e r i e n c e i f t h e d r i v e does s t o p , and a long h o l i d a y i s t h e only hope. Some people do l o s e i t i n t h e i r f o r t i e s , and can only s t o p . I n England t h e y are a s o u r c e o f Vice-Chancellors. Minor d e p r e s s i o n s w i l l o c c u r , and most o f a mathemat i c i a n ' s l i f e i s spent i n f r u s t r a t i o n , punctuated with rare i n s p i r a t i o n s . A b e g i n n e r cannot e x p e c t quick r e s u l t s - - i f they a r e q u i c k , they are p r e t t y s u r e t o be poor.

..

To me, t h e most important l e s s o n from a l l t h i s , i s t h e need t o r e c o g n i s e overwork. My own method used t o b e t o p l a y c h e s s : i t w a s immaterial whether I won o r n o t , t h e q u e s t i o n w a s whether my p l a y was mechanical, o r t h e r e s u l t of i d e a s . Each person should make up h i s own method. I am n o t s u r e t h a t my method w a s v e r y good: i f I l a c k e d i d e a s i n t h e evening, I n e v e r seemed t o be s h o r t o f them by n e x t morning. Newton seems n e v e r t o have been too t i r e d t o do mathematics. However, l i k e t h e r e s t of u s , he must have been, as I s a i d , a t r i a l t o h i s f r i e n d s . They l e a r n t , no doubt, t o make allowances a t c e r t a i n times. A f t e r a l l , a s an American c r i t i c , C l i f t o n Fadiman, h a s s a i d : Do w e want o u r s c i e n t i s t s t o w a s t e t h e i r time on a p p r o a c h a b i l i t y ? Are t h e y n o t worth more t o u s , i f t h e y ' r e as c r u s t y as Newton, as odd as P a s c a l , as remote as W i l l a r d Gibbs? So Newton was ' c r u s t y . " Moritz Cantor goes f u r t h e r , and s a y s i n a number of p l a c e s , a f t e r s u i t a b l e ' p r o o f s , ' t h a t Newton w a s , i f n o t an o u t r i g h t l i a r , something n o t v e r y d i f f e r e n t . I t i s remarkable t h a t s o many p e o p l e who knew Newton p e r s o n a l l y , were w i l l i n g t o s t a n d by him! Minor d e p r e s s i o n s w i l l o c c u r , L i t t l e w o o d s a y s , and we can b e s u r e t h e y d i d . When a mathematician i s engaged on a h a r d problem, what k i n d o f l i f e does h e l e a d ? The c a s e of t h e

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mathematician Hamilton, t h a t we s h a l l come t o i n t h e n e x t chapter, i s r a t h e r illuminating. Imagine our mathematician a t h i s desk, day a f t e r day, r a c k i n g h i s b r a i n s . Have you n o t i c e d how exhausting i t i s t o watch f o r a s i g n , o r l i s t e n f o r a sound, t h a t never seems t o come? I t i s t h e same i n looking f o r an i d e a ; except t h a t one must keep looking, n o t j u s t f o r a few h o u r s , b u t f o r days, f o r weeks, f o r months, o f t e n f o r y e a r s . And how does t h e f i r s t day, o r month, o r y e a r , d i f f e r from t h e second, from t h e t h i r d , . . . ? Only i n t h a t , nothing having once more been achieved, only t h e weariness has i n c r e a s e d . I t seems a m i r a c l e t h a t d i s c o v e r i e s a r e e v e r made: of c o u r s e t h e r e a r e s t r a i n s ! I n a deep q u e s t i o n , you w i l l have long ago exhausted every t r i c k : a m i r a c l e i s a l l you can hope f o r . I n Newton's c a s e , i t was a l r e a d y a m i r a c l e t h a t he came t o take up mathematics a t a l l . This i s now an o l d s t o r y , and we must look f o r a r a t i o n a l e x p l a n a t i o n , b u t t h a t i s n o t so easy t o f i n d . Yet we know more now about Newton than about almost any o t h e r mathematician, thanks t o t h e p u b l i c a t i o n of h i s rough n o t e s : i t seems t h a t he d i d n o t use a wastepaper basket--we can b e thankful f o r t h a t : A s f a r a s w e can s e e , Newton showed a b s o l u t e l y no promise i n mathematics p r i o r t o h i s t h i r d undergraduate y e a r . H e had come up t o T r i n i t y C o l l e g e , Cambridge, a t the age o f 18-112. His school education had been i n t e r r u p t e d when h i s s t e p f a t h e r d i e d : h i s mother wanted him t o look a f t e r t h e farm. When he turned o u t t o be no good a s a farmer, i t was back t o school t h a t he was s e n t , and then t o Cambridge, t o g e t r i d of him a t a d i f f i c u l t a g e . His mother could a f f o r d i t , a f t e r being widowed a second time, and he was h a r d l y t h e k i n d of e l d e s t son t o be missed. His childhood had been, t o a l l i n t e n t s and purposes, t h a t of an orphan. His f a t h e r , a farmer, d i e d b e f o r e Newton was b o r n ; h i s mother then married a clergyman, and went o f f t o a n o t h e r p a r i s h , l e a v i n g Newton i n t h e c a r e o f a grandmother. How Newton f e l t about a l l t h i s can be imagined, b u t h i s f e e l i n g s l a t e r gave him some remorse. I n Cambridge i n 1 6 6 2 , he went s o f a r as t o c a t a l o g h i s s i n s i n s h o r t h a n d , i n c l u d i n g "Threatening my f a t h e r and mother Smith t o burne them and t h e i r home over them." A t s c h o o l , he showed s k i l l i n t h e s o r t of t h i n g s boys l i k e : making mechanical models of clocks and windmills. However he was a l s o good enough t o be considered f o r a s c h o l a r s h i p a t T r i n i t y - - u n t i l i t turned o u t t h a t h e was lamentably d e f i c i e n t i n mathematics: h e would f i r s t have t o s a t i s f y Barrow of h i s knowledge of E u c l i d ' s Elements,

The t r a d i t i o n a l s c h o l a s t i c sys tem i n f o r c e i n Cambridge was not i n s p i r i n g - - b e s i d e s formal l e c t u r e s and d i s p u t a t i o n s on A K i s t o t l e , something c a l l e d 'mathematics' c o n s i s t e d almost e n t i r e l y of t h e more elementary p o r t i o n s of E u c l i d , and of their 'logical analysis. This 'mathematics was what Newton was t o s a t i s f y Barrow a b o u t : h e found i t b o r i n g . This g e t t i n g nowhere, t h i s impasse, continued f o r two y e a r s , I wonder whether even we, who know something of t h e i n i t i a l slowness of

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e x p o n e n t i a l growth, could a t t h i s a l m o s t h o p e l e s s s t a g e have hazarded a most improbable l o o k i n g g u e s s . Suddenly a l l t h i s changed. The T r i n i t y L i b r a r y has Newton's copy o f Barrow's 1655 e d i t i o n o f E u c l i d , w i t h many crowded m a r g i n a l n o t e s b e l i e v e d t o b e no l a t e r than 1664--Newton's t h i r d y e a r a t Cambridge. I n t h a t y e a r he s t a r t e d h i s s e l f - e d u c a t i o n a t a f u r i o u s p a c e t o make up f o r t o t a l l a c k of p r e p a r a t i o n . Whence came t h i s sudden g r e a t e n e r g y ? H i s a p p a r e n t l y s o l e a s s e t i n t h i s , h a d shown i t s e l f i n t h e i n t e n s i t y o f h i s h a t r e d , when h e had t h r e a t e n e d t o burn down t h e house of h i s mother and s t e p f a t h e r . I t was l i k e t h e e l o q u e n t h a t r e d o f which Renan speaks i n an a r t i c l e a b o u t t h e n o t o r i o u s r e v o l u t i o n a r y de Lamenais, i n h i s E s s a i s de Morale e t de C r i t i q u e . I s i t c o n c e i v a b l e t h a t t h i s e l o q u e n t h a t r e d , a i d e d perhaps by an e q u a l l y e l o q u e n t remorse, and an i n t e n s e d i s g u s t of h i s two w a s t e d Cambridge y e a r s , w a s now d i r e c t e d t o d e s t r o y i n g h i s own i g n o r a n c e ? Have we t h e r e p e r h a p s , t h e s o l u t i o n of a w o r l d p r o b l e m - - t h e r e d i r e c t i n g and p u t t i n g t o good u s e of t o d a y ' s i n t e n s i t y o f f e e l i n g , t h a t a t p r e s e n t f i n d s a n o u t l e t o n l y i n h a t r e d , d e s t r u c t i o n and t e r r o r i s m ? C l e a r l y Newton was c a p a b l e of h e r o i c e f f o r t s : perhaps we have found t h e i r s o u r c e . I f s o , i t remains t o d i s c o v e r what c o u l d have d i r e c t e d them t o m a t h e m a t i c s , and made t h e i r s u c c e s s p o s s i b l e - - w h a t , i n o t h e r words, h e l p e d o u r n o b l e e x p o n e n t i a l t o s t a r t i t s remarkable a s c e n t ? Newton threw h i m s e l f i n t o t h e mathematics o f t h e t i m e : h e h e a r d of D e s c a r t e s , and w i t h i n a few months he had r e a d e v e r y t h i n g of D e s c a r t e s a v a i l a b l e i n p r i n t . I t w a s tough g o i n g . According t o de Moivre, who became much l a t e r a c l o s e f r i e n d , Newton was skimming through some book on a s t r o l o g y and found h e needed Trigonometry t o u n d e r s t a n d i t . When he g o t h o l d of a book on t h a t , h e c o u l d n ' t f o l l o w i t b e c a u s e h e d i d n ' t know enough o f E u c l i d ' s Elements. He t h e r e f o r e Got E u c l i d t o f i t h i m s e l f f o r Trigonometry. Read t i t l e s o f p r o p o s i t i o n s , found them e a s y t o unders t a n d , wondered how anybody would amuse themselves by w r i t i n g p r o o f s . Changed h i s mind on r e a d i n g o f t h e e q u a l i t y of P a r a l l e l o g r a m s upon a same b a s e and between t h e same p a r a l l e l s , and t h a t t h e s q u a r e of t h e hypothenuse o f a r i g h t - a n g l e d T r i a n g l e i s e q u a l t o t h e sum o f t h e s q u a r e s o f t h e o t h e r two s i d e s . Began t o go r i g h t through E u c l i d w i t h more a t t e n t i o n . Read O u g h t r e d ' s Key t o Mathematics--understood i t o n l y p a r t l y . Took D e s c a r t e s ' Geometry i n hand, began a g a i n a f t e r t e n p a g e s , went a l i t t l e f u r t h e r , went back t o t h e b e g i n n i n g , r e a d on by d e g r e e s t o t h e e n d . Understood i t b e t t e r t h a n E u c l i d . Read E u c l i d a g a i n , t h e n D e s c a r t e s a second t i m e . Read Wallis' A r i t h m e t i c a I n f i n i t o r u m .

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Introduction

Newton's abysmal ignorance o f even t h e most elementary t h i n g s , when he s t a r t e d on h i s new program, could n o t be more p l a i n . Yet while reading W a l l i s , he found t h e Binomial Theorem. By then he had many of t h e t h i n g s he l a t e r published i n 'Curvis secundi g e n e r i s . ' S t a r t i n g from s c r a t c h a t 2 1 , he had turned himself i n t o a mathematician i n a few months. What was t h e s p u r t o a l l t h i s ? What s t u n g him i n t o a c t i o n ? Some chance d i s p a r a g i n g remark overheard? A c u r t a i n l e c t u r e b y Barrow? A l e t t e r from home? O r perhaps even no more than t h e g e n e r a l atmosphere of T r i n i t y , which has produced s o many g r e a t men a f t e r him, and which has done s o i n s p i t e of t h e handicaps t h a t t h e Cambridge system--the Cambridge ' g r i n d , ' as i t i s c a l l e d - - h a s many times imposed? I n Newton's t i m e , t h e system, with t h e s y l l a b u s I d e s c r i b e d , was h a r d l y i n s p i r i n g . O r was i t perhaps something he read i n t h a t a s t r o l o g y book? Did he imagine he saw i n i t , f o r h i m s e l f , a g r e a t d e s t i n y , a g r e a t o p p o r t u n i t y ? Whatever i t was, i t induced him t o look i n t o h i m s e l f , t o r e a l i s e h i s g r e a t p o t e n t i a l d r i v e and h i s r e s e r v e s of s t r e n g t h . Here was a c h a l l e n g e , and h i s chance t o prove h i m s e l f . We s h a l l meet something s i m i l a r w i t h the mathematician Weiers t r a s s , two c e n t u r i e s l a t e r . But t h e r e i s more t o i t . Newton's g r e a t f i r s t e f f o r t s were clumsy. He needed guidance, he needed i n s p i r a t i o n . He had t h e good luck t o f i n d them i n D e s c a r t e s . I have s a i d what I thought of D e s c a r t e s , and I would p r e f e r t o p o i n t t o some o t h e r source f o r Newton's i n s p i r a t i o n . A f t e r a l l , Newton might have seen a performance of t h e Tempest, and been f i l l e d with t h e ambition t o emulate Prospero. But D e s c a r t e s i t i s , i t must b e : Descartes and h i s s l o g a n . There i s a l s o l i t t l e doubt t h a t Newton was helped by t h e r e l a t i v e l y r e c e n t improvements i n mathematical n o t a t i o n - - a n d h e r e a g a i n I wish I could minimize t h e i r importance , compared w i t h t r u l y c r e a t i v e work. Another important p o i n t i s t h a t Newton was s t u d y i n g what h e wanted t o s t u d y , n o t what h e was r e q u i r e d t o f o r some examination o r o t h e r purpose. These t h i n g s must be borne i n mind: i f some s t u d e n t , who had shown n o t t h e remotest a b i l i t y o r i n t e r e s t i n mathematics, were t o come t o m e saying t h a t h e wished t o emulate Newton, I could h a r d l y ignore t h e h i g h p r o b a b i l i t y of h i s knocking h i s head a g a i n s t a b r i c k w a l l , and y e t I would h a t e being given by h i s t o r y a p l a c e s i m i l a r t o t h a t of t h e man who t r i e d t o dissuade Sarah Bernhardt from t a k i n g up a c t i n g . I wonder how Barrow f e l t , when Newton suddenly threw himself i n t o mathematics. I f Barrow had been anything l i k e one of t h e h i g h l y s u c c e s s f u l mathematical coaches, t h a t Cambridge possessed 200 y e a r s l a t e r , h e would have a t t h e very l e a s t drummed i n t o Newton t h e 'paramount n e c e s s i t y , ' i n mathematics, 6or c o n c e n t r a t i n g on a l i m i t e d number of well-known t h i n g s . Since Newton d i d nothing of t h e k i n d , I i n c l i n e t o t h e view t h a t &arrow must have been a man of q u i t e e x c e p t i o n a l unders t a n d i n g , a s w e l l a s most praiseworthy and generous i n subsequently honouring Newton. An a d v i s o r w i t h a p u p i l l i k e Newton i s i n a d i f f i c u l t p o s i t i o n . The d i f f i c u l t y i s n o t l i m i t e d t o mathematics: I spoke of Sarah

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B e r n h a r d t - - a s a boy, I s a w h e r p e r f o r m i n Cyrano de B e r g e r a c t h e most famous of h e r p a r t s , f r a i l and o l d a s s h e t h e n was, w i t h a d e d i c a t i o n , a s p i r i t u a l q u a l i t y , f a r above any l i m i t a t i o n s o f t i m e . The m i r a c l e , as I s h a l l c a l l i t , t h a t c o u l d t u r n a d i f f i c u l t , v i n d i c t i v e , i g n o r a n t f a r m e r ' s son i n t o one of t h e g r e a t e s t of m a t h e m a t i c i a n s , i s perhaps n o t as r a r e as might be supposed. S p i r i t u a l f o r c e s , t h a t c a u s e such a p p a r e n t m i r a c l e s , a r e n o t s u b j e c t t o p r o b a b i l i t y . To many of us t h e r e comes a moment when we are no l o n g e r c o n t e n t w i t h what we have been: c o n t e n t t o c a l l i t l i v i n g , p l a c i d l y t o chew l i k e an o x ; t o b e s a i n t s on Sunday, b u t hedge and p r e v a r i c a t e s i x days a week; o r t o b e cogs i n some g i a n t machine t h a t no one unders t a n d s . The g r e a t u r g e comes, t h e s p a r k of d i v i n e f i r e , t h e h a t r e d of t h o s e t h i n g s i n us t h a t P a s c a l meant when he w r o t e t h i s i s what i s meant a l s o by t h e l i n e ' ' l e moi e s t h a h s a b l e : " -

I l o v e n o t man, b u t t h a t which d e v o u r e t h him. The g r e a t moment can come e a r l y o r l a t e , b u t perhaps o n l y o n c e , and f o r b e t t e r o r worse--not f o r w o r s e , we a l l hope, b u t merely t o hope i s h a r d l y enough. Nothing can go wrong more e a s i l y t h a n i n t e n d e d d e d i c a t i o n : i t needs supplementing by c o n s t a n t and s e a r c h i n g s e l f - c r i t i c i s m . Those who p r e s s t h e a c c e l e r a t o r , must b e r e a d y w i t h t h e b r a k e s . They must be c o n s t a n t l y on t h e l o o k o u t f o r something t o go wrong. I t i s j u s t i n t h i s r e s p e c t t h a t t h e rough n o t e s o f Newton a r e most r e v e a l i n g . He, t h e man o f whom Pope w r i t e s "Let iJewton be! And a l l was l i g h t ! " , and whose s t a t u e i n t h e T r i n i t y Chapel Wordsworth terms ' ' t h e marble image of a mind f o r e v e r voyaging through s t r a n g e seas of t h o u g h t , a l o n e , " w a s c o n s t a n t l y , i n h i s n o t e s , making e r r o r s and c o r r e c t i n g them. I c a n n o t conceive o f s u c h a man b e i n g a r r o g a n t , as some d e s c r i b e d him. There i s n o t h i n g more humbling t h a n c o r r e c t i n g o n e s e l f . I t i s a l s o a f a r more r e l i a b l e and a c c u r a t e way o f t h i n k i n g , t h a n t o b e c o m p a r a t i v e l y e r r o r - f r e e , and t o imagine t h a t a l l one does i s , by some s p e c i a l d i s p e n s a t i o n of P r o v i dence, p e r f e c t as i t s t a n d s . I r e l y on no one w h o l l y : even i n h i s p u b l i s h e d work, Newton w a s n o t i n f a l l i b l e . T h e r e , however, h i s e r r o r s a r e r a r e : he would r e w r i t e t h i n g s many t i m e s before the f i n a l version. I need h a r d l y p o i n t o u t t h a t Newton would have been most u n s u c c e s s f u l as an examination c a n d i d a t e . T h i s makes i t a l l t h e more remarkable t h a t h e , t h e g r e a t t h i n k e r , s h o u l d have grown t o h i s powers i n t h e l a n d above a l l o t h e r s , where q u i c k c l e v e r a n a l y s e s of r e l a t i v e t r i v i a l i t i e s a r e encouraged as e x a m i n a t i o n t e c h n i q u e s . This i s r a t h e r where h i s r i v a l L e i b n i z s h o u l d have f e l t a t home w i t h h i s c o n v e n i e n t l i t t l e n o t a t i o n s , as a worthy s u c c e s s o r t o Oughtred. I wonder whether o t h e r p o t e n t i a l Newtons have been l o s t t o u s , a l o n g w i t h t h e mute M i l t o n s o f o u r c o u n t r y s i d e ? I t i s i n d e e d s p l e n d i d , t h a t Newton s h o u l d have found, i n t h i s l a n d of h i s b i r t h , a l m o s t u n i v e r s a l s u p p o r t and a c c l a i m , and n o t what h e a t one time had r e a s o n t o t h i n k h e was g e t t i n g from a b r o a d , t h e p e t t y p e r s i s t e n t p i n - p r i c k s of some j e a l o u s c o n s p i r a c y o f

a2

nonentities.

Introduction

I t was i n d e e d , f o r B r i t a i n , a Golden Age.

Questions of p r i o r i t y seem unimportant a f t e r 300 y e a r s , b u t anyone can now s e e f r o m t h e r o u g h noees t h a t t h e a c c u s a t i o n s of p a r t i a l i t y , and of f a v o u r i n g a countryman and i t s p r e s i d e n t , l e v e l l e d a t t h e Royal S o c i e t y a c r o s s t h e c e n t u r i e s by no l e s s a person than t h e h i s t o r i a n o f mathematics Moritz Cantor, a r e q u i t e u n c a l l e d f o r . The work of L e i b n i z , Geometria r e c o n d i t a , appeared i n 1684. I f we s t r e t c h t h i n g s t o t h e utmost, and a c c e p t a s genuine a d a t e t h a t Moritz Cantor himself d e c l a r e s forged by L e i b n i z , t h e e a r l i e s t time when Leibniz possessed any s o r t of Calculus was 1673, more than 7 y e a r s a f t e r Newton, who devised a C a l c u l u s , a s can be seen from t h e rough n o t e s , i n t h e y e a r s 1664-1666, b e f o r e he w a s even Fellow of T r i n i t y , i n f a c t p a r t l y b e f o r e he even had a Bachelor of Science d e g r e e . T r a d i t i o n has i t t h a t Newton's b e s t year w a s 1666, annus m i r a b i l i s , and t h i s t h e n o t e s seem t o confirm. I t was a l s o , u n f o r t u n a t e l y , t h e y e a r of t h e F i r e of London; a s a r e s u l t , i t became f o r a time more d i f f i c u l t than e v e r , i n England, t o g e t anything new published--after t h e i r l o s s e s i n t h e F i r e , publishing firms were more r e l u c t a n t t h a n e v e r t o r i s k i n t h i s way f u r t h e r f i n a n c i a l l o s s . We know now t h a t Newton t r i e d h a r d , w i t h t h e support of people 1 i k e . J o h n C o l l i n s , t o p u b l i s h h i s Calculus .work i n t h e e a r l y 1 6 7 0 ' s . However, as we know, t h i n g s were n o t much b e t t e r on t h e C o n t i n e n t , s i n c e Fermat's work had t o wait u n t i l 1 6 7 9 . What i s most e v i d e n t from t h e n o t e s , i s t h a t f o r Newton mathematics was what I quoted Littlewood as s a y i n g , v e r y hard w o r k . Newton was s t i l l b a s i c a l l y t h e slow f a r m e r ' s boy t h a t he had always been, j u s t as t h e wood t h a t goes i n t o a v i o l i n does n o t change. What d i d change w i t h Newton was t h e d r i v e . I t wap r e l e n t l e s s , b u t i t s t i l l could only proceed a t t h e d e l i b e r a t e country-bred pace w i t h which he went on and on, going over t h e same ground a g a i n and a g a i n . This was how he had mastered E u c l i d and D e s c a r t e s , and t h i s i s how he continued t o work i n h i s long y e a r s of c r e a t i v i t y . He took c a l c u l a t i o n s i n h i s s t r i d e : i n t h e plague y e a r of 1665, when he was away from Cambridge a t h i s f a m i l y ' s Manor Farm a t Boothby i n Lincolnshire--where he worked i n a s p e c i a l l y c o n s t r u c t e d t i n y room w i t h i n h i s bedroom, t h a t can s t i l l be seen t h e r e - - , he d i d n ' t h e s i t a t e t o compute logarithms t o 250 p l a c e s . H i s n o t e s c o n t a i n many long c a l c u l a t i o n s , w i t h f o o t n o t e s such a s : two m i s t a k e s v i t i a t e a l l t h i s hard work . . . A s t o t h e manner i n which Newton went t o work on a problem, he d e s c r i b e s i t i n a l e t t e r : " t h e b e s t and s a f e s t way t o p h i l o s o p h i z i n g seems t o b e , f i r s t d i l i g e n t l y t o i n v e s t i g a t e t h e p r o p e r t i e s o f t h i n g s and e s t a b l i s h them by experiment, and t h e n t o seek hypotheses t o e x p l a i n them." It w a s i n t h e 'annus m i r a b i l i s ' t h a t Newton r e a l i s e d "more than a t any t i m e s i n c e " t h e import and r a p i d flow of h i s

d i s c o v e r i e s i n n a t u r a l philosophy. He became Fellow of T r i n i t y a y e a r l a t e r , i n 1667. Two y e a r s a f t e r t h a t , i n 1669, he succeeded, a s Lucasian P r o f e s s o r of Mathematics, h i s

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t e a c h e r Barrow, t h e man who had o r i g i n a l l y h e l d up h i s s c h o l a r s h i p , and who now recommended him a s a s u c c e s s o r . ( I u s e , a s always, t e a c h e r i n t h e wide s e n s e : a t e a c h e r need n o t be p h y s i c a l l y p r e s e n t , nor even a l i v e . ) Barrow had r e s i g n e d t o become Chaplain t o King Charles 11--by no means an unusual t r a n s i t i o n i n t h o s e days p r i o r t o t h e Georges. This helped t o make Newton's r i s e s o e x t r a o r d i n a r y ; however, a s I keep reminding you, a m a t h e m a t i c i a n ' s development, w h i l e i t proceeds unhindered--and t h e p h y s i c a l and mental energy s t i l l l a s t s - - , i s e x p o n e n t i a l , l i k e any o t h e r form of growth. Moreo v e r , t o make what I have s a i d more p r e c i s e , t h e r a p i d i t y of growth becomes n o t i c e a b l e when t h e e x p o n e n t i a l r e a c h e s t h e value e . A s f o r t h e g r e a t d r i v e , t h e g r e a t force t h a t takes us from t h e i n i t i a l i n t o l e r a b l e slowness of t h e e x p o n e n t i a l t o t h e m i r a c l e o f i t s m e t e o r i c r i s e , I can only d e s c r i b e i t a s " t h e p a s s i o n t h a t l e a v e s t h e ground t o l o s e i t s e l f i n t h e s k y . " To i n s t i l l t h i s d r i v e , t h i s p a s s i o n , i s s u r e l y t h e dream of every t e a c h e r , from t h e humblest i n t h e l i t t l e v i l l a g e s c h o o l , t o t h e most d i s t i n g u i s h e d i n t h e high i n s t i t u t e s o f t h e l a t e s t u p - t o - d a t e Research. And i t cannot be t a u g h t . I t can be d e s t r o y e d , s u r e enough, as I i n t i m a t e d b e f o r e : t h e e x p o n e n t i a l never h a s a chance. I t can a l s o , p e r h a p s , be r e s c u e d , when almost gone: I hope s o . But t a u g h t i t cannot b e , anymore t h a n we can t e a c h a dead t h i n g t o l i v e : i t i s , I b e l i e v e , what we a r e born w i t h . A t any r a t e we a l l e x p e r i e n c e t h e i n i t i a l slow p a r t a s f a r as t h e t u r n i n g p o i n t of t h e e x p o n e n t i a l : t h e r e we come t o t h e c r i t i c a l moment, when t h e g r e a t urge a p p e a r s of which I spoke before. We may s p e c u l a t e i n t h i s way: we s h o u l d , I t h i n k , t o unders t a n d Newton, and perhaps o u r s e l v e s . But Newton w a s f a r t o o busy t o g i v e i t a moment's t h o u g h t . He w a s a l s o t o o busy on h i s n e x t u n d e r t a k i n g t o make f u r t h e r f r u i t l e s s e f f o r t s t o p u b l i s h h i s e a r l y work. N e i t h e r d i d he t r y t o i n c o r p o r a t e t h e e a r l y work i n t h e new, s i n c e t h e l a t t e r w a s d i f f i c u l t enough t o g e t p u b l i s h e d on i t s own. Newton's new and g r e a t e s t work, t h e P r i n c i p i a , o r i n f u l l "Philosophiae N a t u r a l i s P r i n c i p i a Mathematica," appeared i n 1 6 8 7 : t h e f i r s t Book w a s p r e s e n t e d i n manuscript t o t h e Royal S o c i e t y t h e p r e c e d i n g y e a r . Newton went o u t of h i s way t o g i v e a l e n g t h y i n t u i t i v e d e s c r i p t i o n of v e l o c i t y , t o a v o i d C a l c u l u s , j u s t as Gauss much l a t e r avoided t h e imaginary i n h i s Fundamental Theorem o f Algebra. Newton's Calculus work w a s now s o well-known i n h i s whole c i r c l e , t h a t t h e r e w a s n o t t h e same inducement t o p u b l i s h i t a s when i t w a s f r e s h . I know of no one, except f o r Cayley i n t h e XIX-th c e n t u r y , who would c o n t e n t himself t o send i n f o r p u b l i c a t i o n a manuscript a l r e a d y t e n o r twenty y e a r s o l d , w i t h o u t t o t a l l y r e w r i t i n g i t . Thus Newton's Calculus became one of t h o s e many t h e o r i e s , well-known t o a whole group of p e o p l e , b u t unpublished f o r y e a r s . I f t h e o r i g i n a t o r does n o t a t l a s t p u b l i s h h i s t h e o r y , some member of t h e group may e v e n t u a l l y do s o , knowing v e r y w e l l t h a t he thus adds n o t h i n g t o h i s r e p u t a t i o n , and t h a t he r i s k s b e i n g c r i t i c i z e d f o r a l l e g e d d i s t o r t i o n s . Meanwhile, s i n c e t h e t h e o r y i s h a r d l y a dead s e c r e t , i t s ideas,

84

Introduction

n o t always understood, n o r u n d i s t o r t e d , c i r c u l a t e by word of mouth o r by t h e i m p l i c a t i o n s of r e l a t e d i n v e s t i g a t i o n s , u n t i l , as w i t h Newton's 'Methodus fluxionum e t s e r i e r u m i n f i n i t a r u m , ' someone develops h i s own t h e o r y t o cover t h e same t h i n g s . We a r e used t o t h i s t o d a y , b u t f o r some r e a s o n none of i t occurs t o Moritz Cantor, who wrote a t a time when, i n Germany, a p r o f e s s o r had a s s i s t a n t s f o r h i s donkey work. Moritz Cantor keeps a s k i n g "Why d i d n ' t Newton p u b l i s h i t ? " , and "What evidence have w e t h a t t h e r e was a n y t h i n g much worth p u b l i s h i n g a t t h a t time?" The i n f e r e n c e , as w i t h t h e h y p o t h e s i s of d r i f t i n g c o n t i n e n t s , was t h a t n o t h i n g of t h e s o r t d i d i n f a c t e x i s t . We a r e more r e a l i s t i c , and t h e evidence i s now t h e r e f o r a l l t o s e e . N e v e r t h e l e s s , I have no doubt t h a t Newton d i d become, a f t e r a w h i l e , r e l u c t a n t t o p u b l i s h Methodus fluxionum. He may have f e l t uncomfortable about i t s p r e s e n t a t i o n , w i t h i t s rudimentary n o t i o n s of l i m i t : he w a s a p e r f e c t i o n i s t , a s anyone who c r i t i c i z e s himself s o much i s bound t o b e ; he was a l s o t h e man of h i s time most l i k e l y t o s e e f u r t h e r and deeper than h i s contempora r i e s , i f only i n a vague way. Nor was he unaware of what Moritz Cantor himself c a l l s t h e Conservation of i g n o r a n c e , as a cause of h i s p u b l i c a t i o n t r o u b l e s . To f i g h t b a c k , when. 'White Magic' had once been a t t a c k e d , was no l i g h t u n d e r t a k i n g . And i n a s e n s e t h e cranks were r i g h t : a mathematics w i t h u n c l e a r concepts w a s White Magic. John C o l l i n s , who had made two e f f o r t s on Newton's behalf i n 1 6 7 1 , t o g e t Methodus fluxionum p u b l i s h e d as p a r t of o t h e r t h i n g s , t r i e d a g a i n i n 1680, t h i s time t o g e t t h e Royal S o c i e t y t o p u b l i s h i t by i t s e l f : i t took t h a t S o c i e t y two and a h a l f y e a r s f i n a l l y t o a g r e e , b u t Newton had l o s t p a t i e n c e , o r e l s e was t o t a l l y engaged w i t h t h e P r i n c i p i a . Even i f t h e Methodus fluxionum had appeared t h e n , i t would s t i l l have been 1 6 o r 1 7 years l a t e . I n t h e h i s t o r y of mathematics d e l a y s of t h i s o r d e r w i l l be met a g a i n , n o t a b l y i n t h e c a s e of F o u r i e r , and i n t h a t of Cauch . I n a c t u a l f a c t , t h e Methodus appeared f i r s t , t r a n s a t e d i n t o E n g l i s h , i n 1736, n i n e y e a r s a f t e r Newton's d e a t h ; t h i s was t r a n s l a t e d back i n t o L a t i n i n 1744, and a French t r a n s l a t i o n appeared i n 1740. The p u b l i c a t i o n d e l a y was t h u s over 70 y e a r s . Even a t t h a t l a t e d a t e , t h e book had q u i t e an impact. Many mathematicians p r e f e r r e d Newton's v e r s i o n of t h e Calculus t o t h a t of L e i b n i z , on t h e Continent a s much as i n B r i t a i n . This was r e v e r s e d l a t e r , l a r g e l y on account of Whewell i n B r i t a i n . 1736, of However t h e p u b l i c a t i o n , even a t t h e l a t e Methodus fluxionum a l s o had a less d e s i r a b l e , though p r e d i c t a b l e , r e s u l t : Bishop Berkele d e c l a r e d f l u x i o n s t o be ' d e v i l ' s w o r k , ' and duly ulminated a g a i n s t them from t h e p u l p i t and i n p r i n t . I t was a c l e a r i n s t a n c e of t h e prolonged e f f e c t of t h e Law I mentioned e a r l i e r . I t was a l s o a c o n t i n u a t i o n of a t r a d i t i o n t h a t had always e x i s t e d , a t l e a s t a s a s o r t of u n d e r - c u r r e n t , i n t h e Middle Ages, when, a s emphasized by t h e l a t e D r . Bronowski, t h e l i n e of demarcation between s c i e n c e and w i t c h c r a f t was sometimes r a t h e r tenuous: on one s i d e was t h e Church;representing t h e Good and t h e

+

+

IV.

The New Beginning

85

B e l i e f i n what I S - - t h a t i s t o say a s t a t i c conception t o t a l l y incompatible w i t h c r e a t i v i t y and s c i e n t i f i c p r o g r e s s ; on t h e o t h e r s i d e were t h e f o r c e s o f E v i l , r e p u t e d l y a l l t o o eager t o a s s i s t ! T h e r e f o r e , i f p r o g r e s s was indeed made, i t could only be what Bishop Berkeley l a t e r d e c l a r e d . . . Poor D r . F a u s t ! A s m i s r e p r e s e n t e d t h e n , as perhaps he always w i l l be . . . But w e s h a l l r e t u r n t o t h e good Bishop i n t h e n e x t s e c tion. One of t h o s e who d i d most t o e n s u r e , n o t w i t h o u t expense, t h e p u b l i c a t i o n of P r i n c i p i a was t h e astronomer Edmund Halle (1656-1742), famous f o r H a l l e y ' s comet and h i s c a t a l o g t h e s o u t e r n s t a r s , b u t well-known i n h i s time f o r b r i n g i n g o u t i n 1710 t h e f i r s t r e a l l y good L a t i n e d i t i o n of A p o l l o n i u s , i n c l u d i n g an a t t e m p t e d r e c o n s t r u c t i o n of t h e l o s t V I I I - t h book. It must have been w i t h H a l l e y ' s approval t h a t Newton avoided f l u x i o n s i n t h e main t e x t of P r i n c i p i a ; however he d i d add a s k e t c h of h i s i d e a s on t h e C a l c u l u s , which appear more i n d e t a i l i n two n o t e s a t t h e end of h i s O p t i c k s , h i s n e x t g r e a t work, i n 1704. This was followed by h i s A r i t h m e t i c a U n i v e r s a l i s i n 1 7 0 7 . These d a t e s a r e m i s l e a d i n g : most of h i s Opticks goes back t o 1666, when he experimented w i t h d i f f r a c t i o n of l i g h t i n h i s rooms i n T r i n i t y and when he f i r s t introduced t h e Calculus. Unfortunately, both t h e P r i n c i p i a , and t h e O p t i c k s , involved him i n a p r i o r i t y d i s p u t e w i t h Hooke, a p r o f e s s o r of geometry a t a l i t t l e c o l l e g e i n London, c a l l e d Gresham C o l l e g e , which had been founded by a London merchant and which continued t o e x i s t f o r a t o t a l of 160 y e a r s . Hooke claimed p r i o r i t y f o r b o t h d i f f r a c t i o n and g r a v i t a t i o n . Newton took t h e d i s p u t e s i n bad p a r t , accusat i o n s were t o s s e d back and f o r t h : Newton had some excuse f o r f e e l i n g t h a t t h e r e a l work, t h e h a r d w o r k , t h e work t h a t m a t t e r e d , had been h i s own.

6

Newton's mathematical c r e a t i v i t y was n o t reduced by h i s involvement i n p u b l i c a f f a i r s , as Member o f P a r l i a m e n t f o r t h e U n i v e r s i t y o f Cambridge i n 1689 and 1690: one can guess how he employed h i s t i m e t h e r e . H e i s r e p u t e d t o have made one s p e e c h - - h i s maiden speech and swan song, a r e q u e s t t o open a window. The rough n o t e s show t h a t he worked on relentlessly. They' show a l s o s u r p r i s i n g t h i n g s , i n p a r t i c u l a r t h e h i g h l y e m p i r i c a l way i n which he seems t o have t h o u g h t . Also i n t h e y e a r o f h i s a l l e g e d s e v e r e s t breakdown, when he might be expected t o have s u f f e r e d a t 1 e a s t . s o m e s t r a i n , t h e y e a r 1 7 9 2 , n o t h i p g of t h e k i n d shows up i n t h e n o t e s : i n s t e a d we f i n d - - a n o t h e r s u r p r i s e - - t h a t h e was working on a second e d i t i o n of P r i n c i p i a , an e d i t i o n which never appeared, and which w a s q u i t e d i f f e r e n t from t h e o f f i c i a l second e d i t i o n of 1713, e d i t e d by Roger Coates. What d i d slow Newton down, and almost t o t a l l y ended h i s o r i g i n a l mathematical c o n t r i b u t i o n s , w a s t h e work he d i d a t t h e Mint from 1696 onwards, f i r s t as S u p e r v i s o r t h e n t h r e e y e a r s l a t e r a s M a s t e r . A l s o , i n 1703, he w a s e l e c t e d P r e s i d e n t o f t h e Royal S o c i e t y , and subsequently r e - e l e c t e d each y e a r . I n 1705, Queen Anne bestowed a knighthood on him: he was now S i r I s a a c Newton.

86

Introduction

H i s appointment a t t h e Royal Mint was due t o t h e f a c t t h a t a former p u p i l , Charles Montague, had become Chancellor of t h e Exchequer i n a Whig government, and w a s anxious i n t h i s m a t t e r t o c o n c i l i a t e t h e Tory o p p o s i t i o n . A f i n a n c i a l c r i s i s loomed ahead: t h e Bank o f Amsterdam, a l l - i m p o r t a n t i n i n t e r n a t i o n a l exchange, was r e f u s i n g t o a c c e p t B r i t i s h s i l v e r c o i n s , a s they were considered t o have been debased t o 3 1 4 o f t h e i r f a c e v a l u e . The T o r i e s had proposed t h a t t h e l o s s be borne by those who owned s i l v e r c o i n s : i t . w a s a s i m p l e , b u t n o t very honorable way o u t . What a c t u a l l y happened, was t h a t t h e o l d c o i n s were withdrawn from c i r c u l a t i o n , a n d exchanged a g a i n s t new o n e s . This was what Newton had t o s u p e r v i s e , and i t meant w a r a g a i n s t c o u n t e r f e i t e r s . I have l i t t l e doubt t h a t i n Newton's mind, one c o u n t e r f e i t e r deserved hanging f a r more t h a n t h o s e he caught! This accounts f o r some of t h e a s p e c t s of t h e g r e a t p r i o r i t y f i g h t about t h e C a l c u l u s . But a t l e a s t a t t h e Mint Newton was e n t i r e l y s u c c e s s f u l : t h e B r i t i s h s y s t e m of coinage w a s good f o r s e v e r a l hundred y e a r s . Newton had worked as r e l e n t l e s s l y a t t h i s , a s formerly a t mathematics. A t t h e Royal S o c i e t y , he made up f o r i t , by o c c a s i o n a l l y dozing o f f . . .

In r e g a r d t o Newton's O p t i c k s , fundamental though i t i s t o g e t h e r w i t h Huygens' T r a i t 4 de l a Lumigre, I s h a l l n o t add t o t h e i n d i c a t i o n s t h a t I gave i n t h e p r e c e d i n g s e c t i o n . However I must say something of t h e second of t h e two n o t e s on t h e C a l c u l u s , p u b l i s h e d w i t h i t . This n o t e , e n t i t l e d 'De Quadratura Curvarum' was s i n g l e d o u t f o r a t t a c k by L e i b n i z , a f t e r John B e r n o u l l i had w r i t t e n t o Newton, c l a i m i n g t h a t t h e n o t e c o n t a i n e d an e r r o r . Moritz Cantor makes much o f t h i s , r e p e a t i n g t h e c l a i m of an e r r o r , b u t he adds t h a t t h i s does n o t mean t h a t Newton w a s unable t o f i n d t h e second d e r i v a t i v e of an n - t h power! This i s s o elementary a p o i n t , t h a t any rank beginner should be a b l e t o p u t Moritz Cantor r i g h t : i t seems t h a t t h e v a r i o u s persons who c r i t i c i z e d t h e n o t e do n o t understand t h e word p r o p o r t i o n a l . What t h e n o t e a c t u a l l y shows, t o my mind, i s t h a t Newton a l r e a d y p o s s e s s e d , and used a s d e f i n i t i o n f o r h i g h e r d e r i v a t i v e s , what i s g e n e r a l l y known a s T a y l o r ' s Theorem: f(x+h) = f ( x )

+

f'(x)h

+

+

f"(x)h2/2

+

f'"(x)h3/6

+ ...

What Newton s a y s , i n e f f e c t , i s t h a t , by d e f i n i t i o n , t h e t e r m i n hn i s r o o r t i o n a l t o t h e n - t h d e r i v a t i v e of f ( x ) . I t would be e q u a l y t r u e t o s a y t h a t i t i s p r o p o r t i o n a l t o t h i s same q u a n t i t y d i v i d e d by f a c t o r i a l n . There can be l i t t l e doubt t h a t he knew about t h e f a c t o r i a l n : i f he had s t a t e d t h e p r e c i s e v a l u e of t h e c o e f f i c i e n t of h n , he would have s t a t e d T a y l o r ' s Theorem. He simply d i d n o t choose t o do s o i n an i n t r o d u c t o r y n o t e . It i s a g e n e r a l f a c t t h a t t h e g r e a t w r i t e r s do n o t choose t o draw a l l p o s s i b l e consequences from what they p r e s e n t : t h i s i s p r e c i s e l y why t h e i r books a r e more worth r e a d i n g than many t h a t come a f t e r . This i s a p o i n t t h a t I s h a l l have o c c a s i o n t o i l l u s t r a t e many times.

IV.

The New Beginning

Passing on to Newton's Principia, I might point out that we now accept universal gravitation as more or less self-evident, except for refinements from such matters as relativity, pressure of radiation, electro-magnetic effects, and so forth. At the time it was by no means accepted by everyone, not even in the world of science. Leibniz and Huygens condemned it as a 'relapse into scholasticism,' because it involved action at a distance--what they considered 'crushing,' was that this made it 'non-mechanical.' A century later, it was, on the contrary, regarded as the prototype of the mechanistic view, according to which God created the laws by which the Universe created itself, and the term mechanistic was now the one regarded as crushing. Against such superficial judgements, one simply cannot win: the words 'mechanistic' and 'non-mechanical' had become what Littlewood terms 'parrot cries.' (See his public lecture on Cambridge.) It would have been even more damning, and irrelevant, to say that Newton dabbled in astrology and alchemy. By contrast, Newton was most careful not to let his prejLdices, whatever they were, influence his work. He never says attraction, he speaks of gravitation: to gravitate is to move as if attracted, it does not presuppose the cause of this motion. Right at the end of Principia, in the famous Scholium Generale, after summing up arguments against Descartes' vortex motion of planets, he does allow himself to depart from the narrow framework of a scientific theory. He says that the structure of the Solar System and of the Stars points to the existence of a Being who has arranged it all, and that Stars are placed at immense distances to prevent their falling into each other by gravity. Prior to Laplace, such a Being could conceivably be needed to restore stability: Newton was honest enough to admit it, though Leibniz said he had made of God a clumsy watchmaker. Actually, Laplace, as I mentioned, is not altogether reliable; neither is the subsequent stability work entirely convincing, although it now goes very deeply into modern mathematical methods. The role Newton assigns to his Superior Being, is in my opinion, not only unobjectionable, but ideal: that of interfering only when the stability of the Universe, and of the World of Man, is threatened. Similarly the best governments are those that do not constantly interfere, but only do so, unobtrusively, when things might otherwise get out of hand. The ship of State should not need constantly to change course, but only to be pushed back into its proper course, when it begins to deviate ever so slightly from it. This, however is the kind of view that one finds only in science-fiction, or in writers somewhat ahead of our turbulent times. For them the Superior Beings normally limit their action, like the traditional deus ex machina of a play, to putting right what has got out of hand. However, we ourselves habitually attribute to a Superior Being things we do not yet understand, as when we term 'acts of God' the winds and hurricanes that start from small disturbances, and dissipate themselves we know not how. These do not differ in kind from the Solar storms, that expel matter and radiation in one direction or

88

Introduction

another, making the Sun and the planets zigzag slightly on their dynamical orbits. Can these correct instability? They certainly %--I do not say they do, nor that they are needed--, only that they could theoretically act like the secondary rockets in a Moonshot, which get a capsule back on its course, and which could equally keep it on a predetermined, dynamically unstable orbit. It is indeed very likely that such unstable orbits will actually be used. Recent research by Arenstorf and Conle has shown the existence of periodic orbits in the three ody problem which make possible the sending of material cheaply from near the Earth to near the Moon, or vice-versa, in a few months. The orbit on which this is done would however be unstable, so that it would be necessary constantly to push the material back onto its proper orbit. (Ideally, this material would follow a generalised curve, zigzagging infinitesimally on the unstable orbit, in the manner described in the preceding section.) In the case of the Sun and the planets, it is more plausible that the Solar storms, if they occur randomly, would increase instability, rather than correct it, but plausibility is not science. Any proper discussion of instability leads to contemporary research, but the simplifying assumptions are still not totally warranted. All that can be said with certainty, is that no one is going to throw out gravitation, on the grounds that it makes God a clumsy watchmaker! The Law of persistence of ignorance has not gone that far as yet.

+

In the Scholium Generale, Newton goes on to discuss the properties of his Superior Being, and ends: "And s o much concerning God; to discourse of whom from the appearance of things does certainly belong to Natural Philosophy." Newton's point of view in this is perhaps borne out by the comments I have just made. It is also pure Plato: God geometrizes. Eddington believed it, Jeans gave lectures on it, astronomers have it as their motto. It has no further effect on Newton's mathematical writings. However, Newton did also write on theology, in purely theological works: it was the fashion, and in Trinity the clergy's influence was strong. It had been strong also, too strong, in his early life, and to some extent Newton remained a rebel, if only because a mathematician balks at inconsistencies and at the equation 3 = 1. Thus in his theological writings, Newton mainly tried to straighten out discrepant dates in the Gospels and to show the doctrine of the Trinity to be spurious. In the Scholium Generale, the final passage, the most quoted, speaks of the cause of gravitation, which Newton was unable to discover from phenomena. He says: "I frame no hypotheses: whatever is not deduced from phenomena is hypothesis, and hypotheses have no place in experimental philosophy." He mentions the attraction of particles, the action of electric bodies, the emission, reflection and so on of light, the transmission of sensations and commands to and from the brain through the nerves. He adds: "These things cannot be explained in a few words, nor are we furnished with a

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s u f f i c i e n c y o € e x p e r i m e n t s f o r a c c u r a t e d e t e r m i n a t i o n and d e m o n s t r a t i o n o f l a w s . . . " D e s c a r t e s , IIooke, EIuygens would n o t have s a i d t h i s : f o r them a few words would b e q u i t e enough. T h e i r s remains t h e p o p u l a r view: t h e u n i v e r s e can be p u t in a n u t s h e l l . Tile wonder i s t h a t Cngland s u p p o r t e d Newton-a l m o s t a s i f Athens had honoured S o c r a t e s . Xewton's l i f e was no bed of r o s e s , b u t h e was l u c k i e r t h a n some mathematicians we s h a l l come t o . Bulwer L y t t o n d e p i c t s a more f a m i l i a r e x p e r i e n c e : i n t n e l a s t of t h e Barons, a t t n e c l o s e of t h e Middle Ages, tlie c r e a t i v e g e n i u s i s s t o n e d , f r a u d f l o u r i s h e s , t h e u l t i m a t e triumph i s t h a t of t h e MAN OF THE AGE. I n few c o u n t r i e s , and few p e r i o d s or' h i s t o r y , i s a man ahead o f h i s honoured d u r i n g h i s l i f e t i m e . But I am s a y i n g what e v e r y body knows.

17.

Preparing f o r bankruptcy.

A once p o p u l a r Encyclopedia informs us t h a t Newton was succeeded by a t r i u m v i r a t e : C l a i r a u l t , d ' A l e m b e r t , E u l e r . 'This i s pushing e q u a l i t y too Ear: I s h a l l h a r d l y mention t h e f i r s t two. Between Newton and E u l e r , t h e m a t h e m a t i c a l developmenrs were n o t v e r y e x c i t i n g . I n B r i t a i n , Newton w a s on a p e d e s t a l ; i n F r a n c e , which w a s e n t e r i n g what i t chose t o c a l l a n 'Age 02 R e a s o n , ' V o l t a i r e went s o f a r as t o p o p u l a r i s e Newton, and h i s m i s t r e s s t h e . de C h a t e l e t t r a n s l a t e d t h e P r i n c i p i a . Fermat, P a s c a l were f o r g o t t e n : V o l t a i r e , l i k e everybody e l s e , had o t h e r f i s h t o f r y , and t h i s had been going on f o r some t i m e . Both England and France were busy making bankruptcy i n e v i t a b l e : t h e y engaged i n a s e r i e s o f wars i n which p a r t n e r s and c a u s e s were exchanged a s e a s i l y as money can change h a n d s , b u t England and F r a n c e , by a s t r a n g e c o i n c i d e n c e , always remained on o p p o s i t e s i d e s . England r a n t h e s e w a r s ' o n t h e c h e a p ' - - a s i s t h e o r e t i c a l l y p o s s i b l e from an i s l a n d f o r t r e s s - - a n d n o t w i t h o u t c a u s i n g a n x i e t y t o g e n e r a l s s u c h as Malborough: i n d e e d , by s a v i n g on a s t a n d i n g army, England became v u l n e r a b l e even t o Bonnie P r i n c e C h a r l i e and h i s Highland r a b b l e (which i s a l l h i s b r a v e folLowers e v e r w e r e , s o t h a t t h e y n e v e r r e a l l y s t o o d a c h a n c e ) . Howe v e r , any economies were wasted when t h e p u b l i c threw away much money i n t h e South Sea Bubble and i n f a s h i o n a b l e gambling and p r o f l i g a c y , i n s t e a d o f u s i n g i t f o r c o n s t r u c t i v e work: t h e t a x e s c o l l e c t e d by t h e government d i d n o t b e n e f i t . I n t h e e n d , t h e a t t e m p t w a s made t o p u t f i n a n c e s i n o r d e r a t t h e expense o f c o l o n i s t s i n America, who c o u l d h a r d l y h o l d t h e i r own a g a i n s t Mother Nature and i t s b i t t e r c o l d w i n t e r s : t h i s merely b r o u g h t a n o t h e r w a r , t h a t o f American Inde?endence. France d i d n o t r u n t h e wars ' o n t h e c h e a p ; ' a l s o i t s economy had been weakened by i t s p e r s e c u t i o n o f Huguenots; t h u s i n t h e long r u n t h i s l e d t o b a n k r u p t c y , and s o t o R e v o l u t i o n . A t any r a t e , i n England t h e Golden Age w a s a t a n e n d ; i n France i t had n o t begun. Mathematics, l i k e o t h e r i n v e s t m e n t s f o r t h e f u t u r e , i s one o f t h e f i r s t p r o f e s s i o n s t o s u f f e r , when m e d i o c r i t y i n o f f i c e i s s h o r t of c a s h . However, mathematics w a s p r e p a r i n g i t s own bankruptcy as w e l l , and we s h a l l h e a r more o f t h i s . The game w i t h meaningless symbols, t h a t

90

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mathematics had become, was t o culminate w i t h t h e formal s o l u t i o n of a l l equations expressed by l e t t e r s of t h e alphabet! This nonsense had t o be thrown o u t by Gauss and Cauchy, although some g l a r i n g howlers s u r v i v e d almost t o t h e XX-th c e n t u r y , F o r t u n a t e l y some good t h i n g s , and some n o t wholly bad, can come i n t h e w o r s t of t i m e s : a bad mathematic a l p e r i o d may do r e l a t i v e l y l i t t l e harm i f t h e r e a r e good books. I n t h i s s e n s e , i t i s n o t q u i t e t r u e t o s a y t h a t ilewton was s e l f - t a u g h t , nor t h a t h e had no p u p i l g r e a t e r than h i m s e l f : h i s t e a c h e r s were t h e books he r e a d , h i s p u p i l s were h i s r e a d e r s . People had n o t y e t f o r g o t t e n t h e g r e a t l e s s o n of t h e Renaissance: t o r e a d . This i s how, i n a s e n s e , Newton's p u p i l was t h e g r e a t E u l e r . I n between many t h i n g s happened. A remedy was found t o t h e d i f f i c u l t y of p u b l i s h i n g : E u l e r , Lagrange, Laplace had no problems of t h a t k i n d - - t h e i r c o l l e c t e d works f i l l many books h e l v e s . I n Newton's c a s e , i t had been t h e fons e t o r i g o m a l i ( f o u n t a i n and o r i g i n of e v i l ) , of h i s w o r s t t r o u b l e s and s t r a i n s , The remedy was t h e formation o f s c i e n t i f i c s o c i e t i e s , which u n d e r t o o k t h e p u b l i c a t i o n of s u i t a b l e work. I n P a r i s , t h e group t o which P a s c a l belonged f i n a l l y became t h e Acad6mie des s c i e n c e s i n 1 6 9 9 ; i t h a s remained s o t o t h i s day, except t h a t , f o r a f e w y e a r s under t h e Revolution and under. Napoleon, i t was temporarily renamed I n s t i t u t de France. L a t e r an independent I n s t i t u t de France was r e v i v e d . I should say t h a t t h e Academie des Sciences r e a l l y came i n t o being i n some f a s h i o n as e a r l y a s 1 6 6 6 , b u t i t was a l l very unsystematic and i n f o r m a l , s o t h a t - - e x c e p t f o r t h e name--it s t i l l h a r d l y d i f f e r e d from what i t had been when P a s c a l had t o g e t s p e c i a l d i s p e n s a t i o n t o a t t e n d i t s m e e t i n g s , A number of o t h e r academies were formed. That of B e r l i n , founded i n 1 7 0 0 , r a n i n t o some t r o u b l e when Leibniz p r e s i d e d : i t g o t r i d of him by r e c o n s t i t u t i n g i t s e l f and being formally opened i n 1 7 1 1 . Then t h e r e was t h e Academy of P e t e r t h e Great i n S t . P e t e r s b u r g , opened i n 1724 b y P e t e r ' s widow, Catherine I , as t h e Russian Academy of S c i e n c e s . I n f a c t t h e founding of academies and endowing them w i t h funds became f a s h i o n a b l e , and added t o t h e s t a t u s and p r e s t i g e of n a t i o n s and r u l e r s . I n t h i s r e s p e c t , however, t h e e a r l i e r academies were n a t u r a l l y , a t f i r s t , a t a disadvantage, from t h e f i n a n c i a l p o i n t of view. One of t h e e a r l i e s t of t h e s c i e n t i f i c s o c i e t i e s was t h e Royal S o c i e t y , i n London: i t was i n c o r p o r a t e d i n 1 6 6 2 . However i t d i d n o t undertake p u b l i c a t i o n u n t i l l a t e r , and f o r i n s t a n c e , when i t published Newton's P r i n c i p i a , i t s f i n a n c e s were s t i l l s o shaky t h a t Halley had t o p a y f o r t h e p r i n t i n g . The o r i g i n of t h e S o c i e t y was s i m i l a r t o t h a t of t h e P a r i s academy: informal meetings of what was known a s t h e ' I n v i s i b l e College' had taken p l a c e a s e a r l y as 1645 i n Oxford. However C i v i l War was r a g i n g , Charles I was beheaded i n 1 6 4 9 , Cromwell took o v e r ; then Cromwell d i e d i n 1658, Charles I1 was proclaimed i n 1660, and t h e f i n a l s e t t l e m e n t w i t h Parliament was only i n 1 6 6 2 , so t h a t between 1645 and 1 6 6 2 people had o t h e r t h i n g s t o t h i n k about tlian t h e formation of a Royal S o c i e t y . Royalty had l i t t l e t o do w i t h i t : i t s t r a d i t i o n s ,

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a t t h a t time, were n o t p r e c i s e l y academic, s t i l l l e s s s c i e n t i f i c . I f w e go back t o James I , who w r o t e , a s you w i l l remember, a t t a c k i n g White Magic, and who might t h e r e f o r e b e considered t o q u a l i f y r a t h e r f o r t h e o p p o s i t e camp, t h e Cambridge H i s t o r y o f English L i t e r a t u r e does s u g g e s t , i n i t s r e f e r e n c e t o t h e p e r i o d o f 1600-1660, t h a t h e would have been h a p p i e r i f , i n s t e a d of King, he had been p r o f e s s o r a t Oxford, u t t e r i n g words t h a t d i d n o t matter. ( H i t l e r t o o , no doubt, would have been h a p p i e r i n such a c a p a c i t y , poor f e l l o w . . . ) But Charles 1 ' s main i n t e r e s t , a s long as he k e p t h i s head, was i n a r t c o l l e c t i n g , which i s a t l e a s t genuinely c l o s e t o t h e i n t e l l e c t u a l l i E e of a n a t i o n , although i t was wasted on t h e Vandals who succeeded him. This i n t e r e s t i n beauty somewhat changed i t s d i r e c t i o n w i t h Charles 11. He d i d go s o f a r a s t o have a mathematician a s h i s Royal c h a p l a i n , and t h u s t o f o l l o w t h e example of h i s g r e a t a n c e s t o r King Edward 111. However, t h e a c t u a l p r e f e r e n c e s of Charles I1 may b e i n f e r r e d from h i s having no l e s s than 14 i l l e g i t i mate c h i l d r e n - - h e b e l i e v e d t h a t "God w i l l never damn a man f o r allowing himself a l i t t l e p l e a s u r e . " Presumably t h i s p a r t i c u l a r q u e s t f o r p l e a s u r e could n o t be h u r t by a r e p u t a t i o n f o r i n t e l l i g e n c e : h e l i k e d t h e i d e a of appearing t o t a k e an i n t e r e s t i n s c i e n c e , t h e more s o as he had a genuine boyhood a t t r a c t i o n t o s h i p s . S t i l l , t h e Royal S o c i e t y d i d n o t e x a c t l y t h r i v e under t h i s regime, and any semblance of i n t e r e s t t h e Crown may have had i n s c i e n t i f i c m a t t e r s , vanished when ( C u r i o u s l y , George I1 of George I came t o t h e t h r o n e . England deserves mention i n t h e H i s t o r y o f Mathematics. He seems t o have done nothing f o r Science i n England--and nothing a g a i n s t i t - - b u t he founded two g r e a t u n i v e r s i t i e s : t h a t of Goettingen i n h i s n a t i v e Hanover, and t h a t o f P r i n c e t o n i n New J e r s e y . ) Fellowship of a S o c i e t y of r e c e n t formation and w i t h somewhat shaky f i n a n c e s i s n o t normally an overwhelming honour, b u t r a t h e r a s i g n of r e a s o n a b l e r e s p e c t a b i l i t y t h a t no one has q u e s t i o n e d , and o f a c e r t a i n i n t e r e s t i n perhaps promoting t h e o b j e c t s of t h e S o c i e t y i t s e l f . I do n o t know of anyone who might have wished t o be Fellow, and who was i n f a c t turned down by t h e Royal S o c i e t y a f t e r h i s claims t o t h a t honour had been examined, i n t h e XVII-th and XVIII-the c e n t u r i e s , b u t i t i s conceivable t h a t they d i d e x i s t . The same could n o t b e s a i d of t h e P a r i s academy, and t h i s had u n f o r t u n a t e r e s u l t s i n t h e p e r i o d of T e r r o r of t h e French Revolution, a s w e s h a l l s e e . S i m i l a r l y , H i s t o r y was a f f e c t e d when Adolf H i t l e r was r e j e c t e d f o r admission by t h e U n i v e r s i t y and t h e A r t s School i n Vienna; and t h e r e a r e a few s i m i l a r c a s e s on r e c o r d . The problems of t h e Royal S o c i e t y came r a t h e r from t h e o p p o s i t e cause--the e l e c t i o n of someone who was l a t e r n o t very p o p u l a r , namely L e i b n i z . The B e r l i n academy had t h e same problem, and I have mentioned how i t s o l v e d i t . The Royal S o c i e t y began p u b l i s h i n g i t s P h i l o s o p h i c a l T r a n s a c t i o n s i n 1665. This was a l s o t h e y e a r t h e P a r i s academy began p u b l i s h i n g i t s J o u r n a l des Gcavans : t h i s had t h e s u p p o r t o f C o l b e r t , t h e r i g h t - h a n d man of Louis X I V ,

92

Introduction

although i t r a n i n t o some d i f f i c u l t i e s w i t h i t s c r i t i c a l reviews, f o r n o t seeming t o approve some a c t i o n by t h e I n q u i s i t i o n i n Rome, and f o r thus provoking a complaint by t h e p a p a l nuncio. One may w e l l a s k , "Why d i d t h e J o u r n a l des ' hy d i d t h e SCavans n o t p u b l i s h F e r m a t ' s and P a s c a l ' s p a p e r s ? W P h i l o s o p h i c a l T r a n s a c t i o n s n o t p u b l i s h Newton's Methodus The f a c t i s both s o c i e t i e s were s t i l l fluxionum i n 1670?" r a t h e r v u l n e r a b l e : new s o c i e t i e s o f t e n a r e , u n l e s s supported by some Croesus w i t h no s t r i n g s a t t a c h e d , But when t h e s o c i e t i e s have e x i s t e d f o r a c e r t a i n t i m e , i t i s w i t h i n them t h a t t h e f o r c e of t r a d i t i o n o p e r a t e s . I n e i t h e r c a s e , whatever i s ahead of t h e times i s l i a b l e t o be delayed: The Law of c o n s e r v a t i o n of e r r o r c o n t i n u e s t o o p e r a t e . Newton, Gauss understood t h i s ; b u t Leibniz was n o t t o be p u t o f f , as we s h a l l s e e . By t h e n , p u b l i s h i n g w a s n o t q u i t e s o d i f f i c u l t - Leibniz p u b l i s h e d h i s main work on t h e Calculus some 20 y e a r s a f t e r Newton had devised h i s f l u x i o n s - - i n a d d i t i o n t o which, p u b l i s h a b i l i t y w a s helped by t h e f a c t t h a t t h e Leibniz approach w a s more s u p e r f i c i a l l y a t t r a c t i v e , and r a t h e r easy t o work w i t h . I n Newton's c a s e , t h e Royal S o c i e t y was a t f i r s t somewhat lukewarm a t b e s t about p u b l i s h i n g : no doubt Hooke had f r i e n d s , and Hooke w a s accusing Newton of s t e a l i n g h i s ideas--which was p a r t l y t r u e . Hooke's c o n t r i b u t i o n s were l i m i t e d t o two c o r r e c t i o n s , made i n answer t o l e t t e r s by Newton: one c o r r e c t i o n suggested t h e i n v e r s e s q u a r e l a w a s more s a t i s f a c t o r y than what Newton had had i n mind; t h e o t h e r was an e q u a l l y fundamental one i n O p t i c s ; t h e y were s u g g e s t i o n s o n l y , and Newton d i d a t f i r s t acknowledge them as s u c h , though h e d e l e t e d any such r e f e r e n c e s when Hooke demanded t h e s o l e c r e d i t , The d e l e t i o n s were made, w e a r e t o l d , i n a f i t of r a g e : t h i s i s n o t u n l i k e l y , c o n s i d e r i n g h i s childhood f u r y i n wanting t o burn h i s m o t h e r ' s and s t e p f a t h e r ' s house, b u t h e d i d no a c t u a l harm t o anybody i n e i t h e r e p i s o d e . L a t e r on, when Newton had h i s p r i o r i t y f i g h t w i t h L e i b n i z , t h e Royal S o c i e t y , which could have s p a r e d him a l l t h i s by p u b l i s h i n g h i s Methodus fluxionum much e a r l i e r , was s o l i d l y behind him. There w a s a proper e n q u i r y , o f c o u r s e , and i t has been accused o f p a r t i a l i t y , b u t a l l t h i s now sounds r a t h e r r i d i c u l o u s and f a r - f e t c h e d . L e i b n i z ' s work i s q u i t e c l e a r l y very much l a t e r than Newton's. A s t o t h e a c c u s a t i o n s d i r e c t e d a g a i n s t L e i b n i z , of having s t o l e n t h e i d e a s o f Newton, a f t e r s t e a l i n g t h o s e of v a r i o u s o t h e r p e r s o n s , i t i s f a r more l i k e l y t h a t t h e s e i d e a s were s u g g e s t e d t o him by s c r a p s of i n f o r m a t i o n picked up unconsciously from t h e c o n v e r s a t i o n s o r p a p e r s of persons i n t h e know. This happens many times i n mathematics. I should mention a l s o Newton's q u a r r e l w i t h t h e Astronomer Royal John Flams t e e d : t h i s was when Newton was P r e s i d e n t of t h e Royal S o c i e t y . Flamsteed r e f u s e d t o be h u r r i e d i n t o making p u b l i c t h e r e s u l t s of many o b s e r v a t i o n s made a t t h e Greenwich Observatory; t h i s was n a t u r a l l y i n f u r i a t i n g t o anyone who needed t h i s o b s e r v a t i o n a l m a t e r i a l t o check t h e o r e t i c a l work. I sympathise w i t h Newton: a f t e r a l l Flamsteed was a p u b l i c s e r v a n t , n o t an amateur working a t h i s own expense. However t h e Courts decided i n F l a m s t e e d ' s f a v o u r , and I suppose t h i s might be construed a s a d e c i s i o n extending academic freedom.

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And now f o r t h e s t o r y c f L e i b n i z , and of h i s f e u d . G o t t f r i e d Wilhelm Leibniz (1646-1716) belonged t o a f a m i l y t h a t had come from Poland. The name was f i r s t germanized t o Leibniitz, and was s o used by L e i b n i z ' s f a t h e r , a n o t a r y and p r o f e s s o r of m o r a l i t y a t L e i p z i g , according t o Moritz Cantor. L e i b n i z himself s i g n e d himself a t f i r s t i n h i s w r i t i n g s Leibnuzius, and subsequently L e i b n i t i u s . That was when h e wrote i n L a t i n , otherwise h i s s i g n a t u r e was g e n e r a l l y L e i b n i z , o r l e s s f r e q u e n t l y L e i b n i t z . Both t h e s e s p e l l i n g s a r e t h e r e f o r e p e r m i s s i b l e . L e i b n i z was a b r i l l i a n t s t u d e n t : he f i n i s h e d High School a t 1 5 i n L e i p z i g , took a degree i n philosophy a t Leipzig a t 1 7 , and one i n Law two y e a r s l a t e r . The following y e a r , a t t h e age of 2 0 , he s u b m i t t e d a d i s s e r t a t i o n , b u t Leipzig r e j e c t e d him f o r t h e d o c t o r a t e on t h e grounds of extreme youth. C h a r a c t e r i s t i c a l l y , t h i s d i d n o t h o l d him up i n t h e l e a s t : he simply s u b m i t t e d i t t o t h e U n i v e r s i t y of A l t d o r f , which n o t only awarded t h e d o c t o r a t e , b u t o f f e r e d him a professorship. The l a t t e r he t u r n e d down t o e n t e r t h e d i p l o m a t i c s e r v i c e , and i n 1 6 7 2 we f i n d him i n P a r i s , a t t a c h e d t o t h e d e l e g a t i o n of t h e Archbishop E l e c t o r of Mainz. He loved P a r i s , and h e could t a l k i n t e l l i g e n t l y about almost anyt h i n g , b u t h i s hobby, f o r some r e a s o n , had become mathematics, which he s t u d i e d by h i m s e l f , and i n which he had by t h i s t , i m e a l r e a d y w r i t t e n a couple of n o t e s (1670-1671) on Motion. Mostly, a s might b e e x p e c t e d , h e was, however, r e d i s c o v e r i n g r e s u l t s and methods t h a t h e imagined new. I t i s one of t h e i n j u s t i c e s of h i s t o r y , t h a t amateurs, have on t h e whole, l e s s i n f l u e n c e than p r o f e s s i o n a l s , and t h a t r o u t i n e t a k e s precedence over c r e a t i v i t y . I n mathematics, t h e c a s e of Leibniz i s , i n t h i s , q u i t e unique. H e amused himself w i t h mathematics, and t h i s i n t h e end a f f e c t e d , n o t only mathematics b u t h i s t o r y . The way i n which i t d i d s o , was perhaps n o t a l l t o t h e good: one cannot h e l p wondering how much f u r t h e r mathematics and t h e s c i e n c e s might have p r o g r e s s e d i f Leibniz could have b e e n , l i k e Newton, a s t u d e n t a t Cambridge. Unlike Newton, he would have f i t t e d t h e system p e r f e c t l y ; and w i t h Newton a s h i s t e a c h e r , what could have been b e t t e r ?

I n P a r i s , L e i b n i z was p a r t i c u l a r l y p l e a s e d w i t h a summation method he nad found f o r s e r i e s , based on forming d i f f e r e n c e s , and much l a t e r used q u i t e e x t e n s i v e l y by E u l e r . H e showed t h e method t o Huygens, and t r i e d i t out on some s p e c i a l s e r i e s t h a t Huygens was i n t e r e s t e d i n . Encouraged by t h i s , he found an excuse t o b e s e n t t o England e a r l y i n 1673, and h e t h e r e p r e s e n t e d t h e method t o t h e Royal S o c i e t y . I t was a f i a s c o . A w e l l - r e a d minor mathematician of t h e name of P e l 1 (occasiona l l y c i t e d f o r P e l l ' s e q u a t i o n ) m i l d l y s u g g e s t e d t h a t such a w e l l - t r a v e l l e d gentleman must s u r e l y be a c q u a i n t e d w i t h . . . You can imagine t h e r e s t . L e i b n i z had been a n t i c i p a t e d by P i e t r o Mengoli of Bologna; and t h e method was exposed i n a book by Mouton who was n o t even a mathematician, b u t a t h e o l o g i a n . The S e c r e t a r y of t h e S o c i e t y k i n d l y l e n t L e i b n i z a copy of Mouton, and back i n P a r i s L e i b n i z looked up t h e n o t e of Mengoli; i t was a l l t o o t r u e . The blow was l e s s e n e d , when L e i b n i z was, n e v e r t h e l e s s , unanimously e l e c t e d Fellow of t h e

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Royal S o c i e t y a few months l a t e r . A d i p l o m a t , w i t h an i n t e r e s t i n mathematics, w a s t h e s o r t o f Fellow who might prove u s e f u l t o know. S t i l l , i t had been h u m i l i a t i n g , t h e r e had been j u s t a h i n t of a p o s s i b i l i t y t h a t h e , L e i b n i z , had t r i e d t o pass on a s h i s own, what h e had r e a d i n a r e c e n t book. However, he had a second shock: on r e t u r n i n g t o P a r i s , h e l e a r n t t h a t t h e Archbishop E l e c t o r had j u s t d i e d . L e i b n i z ' s d i p l o m a t i c c a r e e r w a s c u t s h o r t : he was w i t h o u t a j o b , Now n o t h i n g seemed t o go r i g h t f o r him. He became t u t o r t o a young nobleman: t h i s w a s n o t a s u c c e s s . Then he had t h e c h a s t e n i n g e x p e r i e n c e of showing h i s ignorance t o Huygens, by taking f o r g r a n t e d t h a t a p l a n e convex a r e a i s halved by any l i n e through i t s c e n t r e o f g r a v i t y . He w a s t o l d t o go and l e a r n some mathematics: he d i d . I n p a r t i c u l a r , h e s t u d i e d t h e unpublished papers of P a s c a l , which were shown t o him by P a s c a l ' s nephew, E t i e n n e P g r i e r , i n 1673. The n e x t y e a r Leibniz composed Quadrature a r i t h m g t i q u e , which c o n t a i n e d t h e series

H e s e n t i t t o Iluygens, who r e t u r n e d i t w i t h h i g h p r a i s e . He s e n t i t some months l a t e r t o t h e Royal S o c i e t y , b u t t h e r e he was. once a g a i n t o l d t h a t he had been a n t i c i p a t e d , t h i s t i m e by James Gregory, who had j u s t d i e d (1675) : Gregory had g i v e n i n a l e t t e r of 1671 t h e expansion of t h e f u n c t i o n a r c t a n x , and L e i b n i z ' s r e s u l t w a s t h e s p e c i a l c a s e x = 1. The Royal S o c i e t y ' s c o n t e n t i o n was i n f a c t f a l s e : q u e s t i o n s of convergence would have had t o be looked i n t o , i f t h e formula was t o b e regarded a s a s p e c i a l c a s e of G r e g o r y ' s . I n any c a s e , Gregory himself gave h i s s e r i e s w i t h o u t p r o o f . For t h e s e r e a s o n s , t h e s e r i e s f o r x = 1 i s g e n e r a l l y r e f e r r e d t o as L e i b n i z ' s series for n J 4 . T h i s i s perhaps n o t q u i t e f a i r t o Gregory, a n honest Scot who was perhaps t h e only mathematician of h i s time t o p o s s e s s s a t i s f a c t o r y i d e a s about convergence, and who u n f o r t u n a t e l y d i e d p r e m a t u r e l y . The y e a r 1675 w a s a l s o t h e one i n which Leibniz made f r i e n d s w i t h a mathematician of t h e name of Walther Tschirnhaus, who had j u s t v i s i t e d England, and brought a l e t t e r o f recommendation from t h e S e c r e t a r y o f t h e Royal S o c i e t y . I t i s n o t impossible t h a t Tschirnhaus had h e a r d o r s e e n something of Newton's f l u x i o n s , p a r t i c u l a r l y a ' l e t t e r about t a n g e n t s ' o f 1672. A t any r a t e , w i t h i n two months, on t h e 1 1 - t h of November 1675, L e i b n i z wrote h i s f i r s t Calculus p a p e r , 'Methodi tangentium i n v e r s e exempla.' (This I S THE PAPER WHOSE DATE WAS LATER CORRECTED t o 1673.) A c t u a l l y , i t a p p e a r s from L e i b n i z ' s rough n o t e s , which are d a t e d , t h a t he was working on t a n g e n t s s i n c e August 1673, i.e. w i t h i n a few months o f h i s own f i r s t v i s i t t o London, I do n o t a t t a c h much importance t o t h e s e d a t e s , n o r t o t h e

independence of L e i b n i z ' s work. Moritz Cantor does h i s h e r o l e s s t h a n j u s t i c e , i n s a y i n g t h a t t h e d a t e o f Methodi tangentium w a s ' f a l s i f i e d ; ' b u t I a l s o cannot a c c e p t h i s s a y i n g t h a t

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t h i s ' p r o v e s ' t h e d a t e 1675 t o be genuine. I am only a mathematician, n o t a lawyer f o r t h e d e f e n s e , o r f o r t h e p r o s e c u t i o n . I t i s i n my o p i n i o n p e r f e c t l y p o s s i b l e f o r a l l the d a t e s t o have been added from memory, and Leibniz may have changed h i s mind about 1675: " N O , I remember now--it was j u s t a f t e r s o and s o , - - i t must have been 1673." I n any c a s e , memory can p l a y s t r a n g e t r i c k s , even w i t h o u t any oonscious o r unconscious attempt t o r e c o n s t r u c t a more p a l a t a b l e p a s t . What i s important i s t h a t L e i b n i z ' s approach and n o t a t i o n a r e q u i t e d i f f e r e n t from Newton's, and t h a t L e i b n i z ' s work was p u b l i s h e d long b e f o r e Newton's. Here and t h e r e , L e i b n i z may borrow a word from Newton, o r a c o n s t r u c t i o n , b u t t h i s i s only n a t u r a l : t h e s e t h i n g s were being t a l k e d a b o u t . Leibniz d i d n o t p u b l i s h h i s Methodi tangentium. His f i r s t o p p o r t u n i t y t o p u b l i s h mathematical papers came i n 1682, i n t h e Acta Eruditorum, and h e began with a paper on O p t i c s . To h i s g r e a t annoyance, i t was h i s acquaintance Tschirnhaus who f i r s t p u b l i s h e d a Calculus paper t h e r e : Tschirnhaus used t h e i d e a s h e had a c q u i r e d from L e i b n i z , o r s o L e i b n i z s a i d , and i t was w i t h g r e a t d i f f i c u l t y t h a t t h e Acta Eruditorum escaped becoming a b a t t l e g r o u n d . The f i r s t paper by L e i b n i z on t h e Calculus appeared i n 1674: i t was on maxima and minima, and on t a n g e n t s . The main theory came i n 1 6 8 6 , w i t h h i s Geometria r e c o n d i t a , i n t h e same j o u r n a l . By t h a t time L e i b n i z ' s p o s i t i o n had g r e a t l y changed. The d i f f i c u l t P a r i s days, when h e was h a r d p u t t o e a r n a l i v i n g , had ended i n 1676 w i t h h i s appointment as L i b r a r i a n a t Hanover: he was even a b l e f i r s t t o spend a month o r s o i n England and Holland. Gradually h i s work a s L i b r a r i a n took on p o l i t i c a l importance, a s a f t e r some r e s e a r c h of a h i s t o r i c a l n a t u r e h e was a b l e t o s e c u r e f o r Hanover an E l e c t o r s h i p of t h e Holy Roman Empire. H e was now a most important p e r s o n , and i t was h e , who u l t i m a t e l y , had t h e t a s k of persuading B r i t a i n t o a c c e p t George I a s i t s r u l e r . He a l s o had o t h e r i r o n s i n t h e f i r e : f o r i n s t a n c e he worked a t a r e - u n i f i c a t i o n of t h e C h r i s t i a n Churches. The c a r e e r of diplomat, which h e t h u s , i n e f f e c t , r e t u r n e d t o , although he remained L i b r a r i a n , was n o t w i t h o u t i n f l u e n c e on h i s c h a r a c t e r . H e was an amateur mathematician, b u t one w i t h t h e outlook of a diplomat and lawyer. H e was a l s o i n a p o s i t i o n t o p u t c o n s i d e r a b l e p r e s s u r e on p e o p l e . Both h i s mathematics, and h i s behavior t o o t h e r mathematicians, show very c l e a r l y , i n my o p i n i o n , t r a c e s of t h i s o u t l o o k . I do n o t f e e l t h a t h e can j u s t l y be c r i t i c i z e d f o r t h i s , u n l e s s we wish t o i n d i c t t h e whole p r o f e s s i o n of diplomat, and t h a t of lawyer, a t t h e t i m e . One c a s e i n p o i n t concerns h i s r e l a t i o n s w i t h a minor mathematician of t h e name of Hermann, whom we s h a l l meet a g a i n i n t h e n e x t s e c t i o n . Hermann, a S w i s s from B a s e l , had been u s e f u l : he had w r i t t e n a Response t o an a t t a c k on t h e Leibniz Calculus by a Dutchman Nieuwenti * t ; h e had a l s o allowed L e i b n i z t o c o r r e c t an O b i E d James B e r n o u l l i . A s o r i g i n a l l y w r i t t e n by Hermann, t h e o b i t u a r y was a l r e a d y used as an excuse f o r p r a i s i n g L e i b n i z , by coupling h i s name w i t h t h a t of James B e r n o u l l i . This was n o t n e a r l y good enough for L e i b n i z ! What we r e a d i n s t e a d i n t h e c o r r e c t e d v e r s i o n i s t h a t , when t h e g r e a t discovery of

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t h e c e n t u r y , the i n f i n i t e s i m a l a n a l y s i s of L e i b n i z , had made i t s appearance, the deceased had used an e a s i l y found example obtained by t h e d i s c o v e r e r , t o shed new l i g h t on a p p l i c a t i o n s t o . . . e t c . This w a s p r e c e d z by j u s t one s e n t e n c e , saying s a t t h e numerous and b e a u t i f u l c o n t r i b u t i o n s of t h e deceased would n o t be gone i n t o . The o b i t u a r y , t h u s r e v i s e d , appeared i n January 1 7 0 6 : Hermann's reward was G a l i l e o ' s Chair a t Padua i n 1 7 0 7 . Leibniz was c e r t a i n l y i n f l u e n t i a l , Moreover, i n r e t u r n f o r h i s good o f f i c e s , which r e s u l t e d i n P r u s s i a becoming a Kingdom, and i n t h e crowning of F r e d e r i c I a t Koenigsberg t h e following y e a r , he was named P r e s i d e n t of t h e B e r l i n academy when i t was founded i n 1 7 0 0 . However, a s I mentioned e a r l i e r , L e i b n i z ' s p o p u l a r i t y i n P r u s s i a waned, and a way was found of g e t t i n g r i d of him, I t i s u s u a l , i n t h e h i s t o r y of mathematics, t o b a s e mathema-

t i c a l comments on what w e t a k e t o have been accepted views a t t h e times concerned. Whether t h i s i s e n t i r e l y f a i r t o t h e p i o n e e r , who happens t o b e ahead of h i s t i m e , i s n o t perhaps c l e a r , b u t t o anyone e l s e i t i s r a t h e r generous than o t h e r w i s e , i n i g n o r i n g e r r o r s t h a t were n o t then r e c o g n i s e d , a s i f what mattered i n h i s t o r y was de mortuis n i h i l n i s i bonum. However, i f h i s t o r y i s t o b e n e f i t t h e l i v i n g , f o r whom t h e r e may be v a l u a b l e l e s s o n s i n such e r r o r s of t h e p a s t , I t h i n k i t important f o r us t o judge a l s o , a s f a r a s we c a n , from t h e p o i n t of view of p o s t e r i t y . Therefore what I s h a l l now say w i l l b e coloured by my views as a XX-th c e n t u r y a n a l y s t ( o t h e r than non-standard o r non-Archimedean). I t i s i n my o p i n i o n no disparagement of Leibniz t o s a y t h a t h i s c o n t r i b u t i o n t o t h e Calculus ranks above a l l a s a n o t h e r d i p l o m a t i c triumph. Diplomacy o p e r a t e d under r u l e s s l i g h t l y d i f f e r e n t from t h o s e of today--honesty was n o t one of them, b u t appearances had t o b e k e p t up. Newton was no diplomat, and may even have had misgivings as t o t h e a c c e p t a b i l i t y o f f l u x i o n s : h i s Methodus fluxionum was n o t published u n t i l w e l l a f t e r h i s death, and then i n an E n g l i s h v e r s i o n w i t h verbose comments by t h e t r a n s l a t o r . Mathematics was p a r t l y White Magic: t h i s Leibniz understood, and a s a diplomat he had no qualms. For u s , an honest Calculus pnoceeds from convergence and passage t o t h e l i m i t : t h e s e a r e j u s t t h e n o t i o n s t h a t Leibniz ( u n l i k e t h e honest S c o t , James Gregory) wants t o a v o i d , I n s t e a d he i n v e n t s , a s diplomats do, meaningless conventions as a common face-saving d e v i c e . We know them a l l t o o w e l l a t peace conferences : they a r e s i m i l a r t o t h e d i s t i n c t i o n s made by t a x o f f i c i a l s - - t h e y a r e meaningless, they a r e absurd, b u t they a r e t h e ' f o r m u l a ' t h a t i s adopted. There i s no o t h e r way t o break a deadlock. Then they become r u l e s of i r o n , t h e more r u t h l e s s l y enforced because they l a c k any b a s i s i n common s e n s e . When people f i n a l l y come t o t h e i r s e n s e s , i t a l l has t o be done over a g a i n , b u t i n t h e meantime i t works, we must be t h a n k f u l f o r t h a t . This may n o t sound f l a t t e r i n g , b u t a t times diplomacy and mathematics have had much i n common. Leibniz i s i m p o r t a n t , because h e found a way of making Calculus p a l a t a b l e - - a n easy game w i t h symbols. They were meaningless symbols; and i n t h i s

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connection, t h e most d e v a s t a t i n g c r i t i c i s m i s implied by t h e q u e s t i o n i n v a r i a b l y asked i n e n g i n e e r i n g c l a s s e s by beginning s t u d e n t s , when an e n g i n e e r i n g p r o f e s s o r w r i t e s dyldx, " S i r , why d o n ' t you s i m p l i f y t h e d ' s ? " I n t h e o l d days, t h e p r o f e s s o r used t o s e e t h e l o g i c of t h e q u e s t i o n , and d i d what h e was asked: t h e answer t h e n d i s a g r e e d w i t h t h e book, which was t h e r e f o r e a s u s u a l wrong. Such t h i n g s a r e l i a b l e t o happen, when meaningless symbols a r e used. The Calculus becomes a badly designed machine, when i t was t o have been f o o l p r o o f . There a r e many such machines i n e x i s t e n c e today: they have t h e i r i d i o s y n c r a s i e s . To l e a r n t o use such machines does n o t t a k e too much e f f o r t , a s l o n g a s they work. We have t o watch o u t f o r c e r t a i n t h i n g s , b u t o t h e r w i s e i t becomes v i r t u a l l y unnecessary t o t h i n k of what i s being done. Some people had m i s g i v i n g s : mathematics w i t h o u t meaning i s too l i k e banknotes without a g o l d r e s e r v e . The c r i t i c s included N i e u w e n t i j t , a medical man and burgomaster whom I mentioned; t h e r e was a l s o t h e p h i l o s o p h e r F a t h e r Malebranche, a u t h o r of 'La Recherche de l a V 6 r i t 6 ; ' and t h e r e were s e v e r a l mathematicians, t h e geometer d e l a H i r e , a f o l l o w e r of Dgsargues, t h e a l g e b r a i s t Rolle , a u t h o r of R o l l e ' s theorem-which i s today formulated d i f f e r e n t l y - - a n d l a s t b u t n o t l e a s t t h e g r e a t Huygens h i m s e l f . They i n s i s t e d t h a t you cannot n e g l e c t what i s n o t 0 , t h e r e f o r e you cannot n e g l e c t i n f i n i t e s i m a l s : t n e r e p l y was t h a t you can and do: And t h a t t h i s was a new and e a s i e r way o f doing mathematics. They were unconvinced, b u t i t was a l l t a l k : a s long a s t h e i n f i n i t e s i m a l s had no meaning, t h e r e was n o t h i n g s p e c i f i c t o a t t a c k . Imaginaries t o o , had none, and everybody used them. Diplomats a r e m a s t e r s of t h i s type of s o p h i s t i c a t i o n : d i p l o m a t i c f i c t i o n s a r e i n c o n s t a n t u s e , and keep being i n v e n t e d : i t i s u n f o r t u n a t e l y t h e only way t o make s e n s e o u t of t h e law of t h e j u n g l e which o p e r a t e s between n a t i o n s . But t h e r e was, of c o u r s e , a l s o t h e o p p o s i t i o n of t h o s e , l i k e t h e S w i s s F a t i o , who wrote t o Huygens, who had taken a l i k i n g t o him and c o n s i d e r e d him promising, t h a t compared w i t h f l u x i o n s , L e i b n i z ' s i n f i n i t e s i m a l s were l i k e a m u t i l a t e d and v e r y i m p e r f e c t copy of a f i n i s h e d o r i g i n a l . This was p r e c i s e l y what Newton and n e a r l y every o t h e r B r i t i s h mathematician was t h i n k i n g . I t turned o u t , however, when Newton d i e d and h i s f l u x i o n s w e r e f i n a l l y p u b l i s h e d , t h a t they were much more v u l n e r a b l e : they were n o t e l u s i v e d i p l o m a t i c f u n c t i o n s - - t h e y . h a d meaning, they came from t h e world of f a c t s , from Newton's a c t u a l observat i o n s and c a l c u l a t i o n s , they were needed t o d e s c r i b e what h e could s e e . However tney were a l s o n o t based on t h e l o g i c of c l e a r d e f i n i t i o n s , any more than L e i b n i z ' s i n f i n i t e s i m a l s . F a c t s a r e n o t enough, e s p e c i a l l y i n anything remotely resembling a witchhunt. People b e l i e v e what they w i l l , w i t h a f r i g h t e n i n g i n t e n s i t y , i n t h e f a c e of a l l evidence and common s e n s e , and I a s s u r e you, I speak from e x p e r i e n c e . A t any r a t e , t h e n o t o r i o u s Bishop Berkeley r e a l l y b e l i e v e d i n s p i t e of a l l c a l c u l a t i o n s t o t h e c o n t r a r y , t h a t even f o r t h e f u n c t i o n f ( x ) given by x i t s e l f , t h e v a n i s h i n g of f(x+h) - f ( x ) and of h f o r h = 0 , n o t only makes t h e i r r a t i o t h e i n d e t e r m i n a t e 0 1 0 , b u t a l s o compels i t t o approach every v a l u e a s h becomes'

98

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s m a l l : t o a s s e r t anything d i f f e r e n t from t h i s , he m a i n t a i n e d , was t o c a l l i n t h e f a t h e r of l i e s . I f t h e r e were more t h i n g s i n Heaven and E a r t h than h e dreamt o f , h e would n o t s e e them i f they were s t a r i n g him i n t h e f a c e . I t i s only f a i r t o add t h a t of course h i s n o n s e n s i c a l a t t a c k s were based on some i n n e r c o n v i c t i o n t h a t somethin was not, q u i t e r i g h t ; and t h a t , u n l i k e what happens i n F d i t y o f outbreaks of a b s u r d i t y , h i s i n n e r c o n v i c t i o n had some j u s t i f i c a t i o n , a s l a t e r h i s t o r y showed. The same i n n e r c o n v i c t i o n was what L e i b n i z ' s c r i t i c s must have f e l t a l s o . I n both c a s e s , u n f o r t u n a t e l y , t h e r e p l i e s t o t h e c r i t i c i s m s never r o s e above t h e l e v e l of s u p e r f i c i a l i t y . L e i b n i z ' s main e f f o r t i n t h i s connection was d i r e c t e d e l s e where, namely towards e s t a b l i s h i n g h i s p r i o r i t y , perhaps even towards o b t a i n i n g some s o r t of revenge, Sentiment was s t r o n g i n B r i t a i n , t h a t h i s Calculus amounted t o a t h i r d attempt t o claim f o r himself a discovery made b y someone e l s e . Of t h i s enough has been s a i d , b u t i t had r a t h e r important consequences, both f o r h i s t o r y and f o r mathematics. Leibniz was now q u i t e a powerful f i g u r e , as w e s h a l l soon s e e , and i t was i n e v i t a b l e t h a t he should a t t e m p t t o r e t a l i a t e a s only h e knew how. H i s b e s t chance f o r t h i s was t o g e t B r i t a i n t o a c c e p t George I as i t s sovereign. I n t h i s he succeeded, i t was a triumph f o r , h i s work behind t h e s c e n e s : t h e T o r i e s had been f o r t h e e l e c t o r , b u t had been p a r t l y p u t o f f ; t h e Whigs were lukewarm--only Marlborough was a c t i v e l y i n f a v o u r . It must be admitted t h a t t h e choice was no more i n s p i r i n g than t h a t of some of t o d a y ' s e l e c t o r a l c o n t e s t s . I n B r i t i s h history-books, L e i b n i z ' s s e r v i c e s a r e h a r d l y mentioned i f a t a l l , b u t t h e i r e x t e n t can perhaps be i n f e r r e d from t h e i r r e w a r d - - v i r t u a l h o u s e - a r r e s t - while i n Marlborough's c a s e t h e p r o v e r b i a l g r a t i t u d e of p r i n c e s merely k e p t him o u t o f t h e government f o r a w h i l e . Leibniz s u f f e r e d g r e a t l y from g o u t , and d i e d two y e a r s a f t e r h i s p r o t 6 g g ' s a c c e s s i o n t o t h e B r i t i s h Throne; however, i n a s e n s e , h i s t o r y c a r r i e d out h i s revenge--not on t h e two g r a t e f u l sovereigns he had helped crown i n P r u s s i a and B r i t a i n b u t c e r t a i n l y on B r i t i s h mathematics. Of c o u r s e , h i s t o r y Prussia i t s e l f had n o t been uninfluenced b y h i s diplomacy: and England, w i t h t h e i r new Kings, had s t a r t e d on t h e i r subsequent c o u r s e s , w h i l e France, t h e t r a d i t i o n a l f r i e n d of t h e S t e w a r t s , became England's century-long f o e . Much t h a t has happened s i n c e , good o r bad, can be t r a c e d back t o t h i s : Leibniz may n o t have made i t i n e v i t a b l e , b u t he p a r t l y s e t i t i n m o t i o n . Much t h e same can b e s a i d o f what happened a f t e r him t o B r i t i s h mathematics: I l e a v e you t o judge whether i t has been good o r bad. A t t h e Royal Court, where a s e r i e s of d i s t i n g u i s h e d mathematicians had f o r some t i m e been Chaplains, mathematics now l o s t a l l s u p p o r t : t h e t r a d i t i o n o f Edward I11 meant nothing t o t h e new King, a man whom Leibniz had found d i s g u s t i n g , and who had himself l i t t l e l o v e f o r mathematics, s c h o l a r s h i p , o r any form a f c u l t u r e . More s e r i o u s s t i l l , perhaps, d r i t i s h mathematics became f o r a long p e r i o d i s o l a t e d from t h e Continent: t h i s s t a r t e d because of t h e feud between Newton and L e i b n i z , b u t o f course i t need n o t have continued s o long. I t was a l s o , perhaps, n o t i n e v i t a b l e t h a t i n t h e Royal S o c i e t y i t s e l f , t h a t g r e a t s t r o n g h o l d of mathematics

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from which L e i b n i z had s u f f e r e d some h u m i l i a t i o n , t h e s c i e n c e t h a t had made B r i t a i n Queen o f t h e Seas should have g r a d u a l l y been s u p p l a n t e d by o t h e r s , w i t h o t h e r concepts o f p r o o f , u n t i l i t had t h e r e t h e s t a t u s o f a C i n d e r e l l a , i f n o t o f a t o t a l l y u s e l e s s t o y , a s t a t u s i t has n o t f u l i y recovered from. C a u s a l i t y i n h i s t o r y o f t e n works s l o w l y : slowly enough, p e r h a p s , f o r u s t o c o u n t e r a c t i t . I g e t t h e impression t h a t B r i t i s h mathematics missed i t s g r e a t chance: t h e r a t h e r absurd c r i t i c i s m s of p e o p l e l i k e Bishop Berkeley c o u l d and should have been t h e o c c a s i o n f o r going f a r more deeply and c a r e f u l l y i n t o a p r o p e r f o u n d a t i o n f o r t h e Calculus and f o r A n a l y s i s . We have t o l e a r n , I a l w a y s s a y , t h e a r t of t u r n i n g a d i s a d v a n t a g e i n t o an o p p o r t u n i t y - - n o t an e a s y t h i n g , I must admit. England could have maintained i t s l e a d , and have a n t i c i p a t e d Cauchy: S c o t l a n d , p e r h a p s , almost d i d s o , b u t James Gregory, who had been on t h e r i g h t t r a c k , d i e d p r e m a t u r e l y , as I mentioned. I n s t e a d , B r i t i s h mathematicians c o n g r a t u l a t e d themselves on n o t u s i n g i n f i n i t e s i m a l s , w h i l e on t h e Continent i n f i n i t e s i m a l s were merely regarded as no worse t h a n imagina r i e s . That i s t h e danger o f i n f l a t i o n : i f you have n o t e s w i t h o u t a g o l d r e s e r v e - - o r symbols w i t h o u t a meaning--why n o t produce some more? I n the end t h e convenience of i n f i n i t e s i m a l s was too much f o r B r i t a i n t o r e s i s t , l i k e t h a t of i n f l a t i o n today: t h e prime mover, as I s a i d w a s Whewell, p i o n e e r of t h e i n t r i n s i c geometry of c u r v e s . This s e t B r i t a i n f u r t h e r back, s i n c e s a t i s f a c t o r y d e f i n i t i o n s f o r t h e L e i b n i z Calculus were n o t found b e f o r e S t o l t z , a f r i e n d of F e l i x K l e i n ' s e a r l y days. A s l a t e a s m r a c c o r d i n g t o one o f L i t t l e w o o d ' s p u b l i c l e c t u r e s , t h i s meant t h a t i n Cambridge maLhematica1 r i g o u r , a s we know i t , w a s almost t o t a l l y a b s e n t . France, meanwhile, had switched t o some e x t e n t t o Newton's f l u x i o n s under t h e i n f l u e n c e o f Lagrange, and t h e i r h o n e s t y and f a c t u a l c h a r a c t e r made them e a s i e r t o p u t on a f i r m logical footing. )

There w a s more t o i t than t h a t : people l i v e I N THEIR TIMES. Mathematicians a r e no e x c e p t i o n s . The d r i v e t h a t pushes them on, i s f e d by t h e whole atmosphere around, j u s t as a t i n y t r e e i s f e d by t h e s o i l i t grows i n , i f i t i s e v e r t o r e a c h deep down w i t h i t s r o o t s , and h i g h up w i t h i t s b r a n c h e s . What t h e atmosphere w a s l i k e i n England under t h e f i r s t Georges, I have d e s c r i b e d : Shakespeare might have s a i d t h a t t h e r e a l , t h e t r u e , England s l e p t . On t h e C o n t i n e n t , t h i n g s were on t n e whole no b e t t e r a t f i r s t , e x c e p t t h a t w e f i n d t h e r e t h e beginnings of a r e a s o n a b l e t h e o r y of convergence f o r s e r i e s : t h i s appears i n t h e work of d ' h l e m b e r t and o f Nicholas I B e r n o u l l i , b u t a t t h e time i t w a s n o t c a r r i e d T u r t h e r . D'Alembert's name i s l i n k e d r a t h e r t o a p r i n c i p l e of Dynamics, w h i l e t h e work of Nicholas I i s mainly, w i t h t h a t o f de Moivre, on t h e beginnings of p r o b a b i l i t y . G e n e r a l l y s p e a k i n g , e x c e p t f o r a few important names we s h a l l come t o , t h e r e w a s no g r e a t d i f f e r e n c e between t h e two s i d e s o f t h e B r i t i s h Channel: a f t e r L e i b n i z d i e d and Newton had v i r t u a l l y l e f t mathematics, t h e Calculus became a new t o y , and mathematicians merely played w i t h i t . T h i s provided t h e XIX-th c e n t u r y w i t h l i t t l e

100

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e x e r c i s e s : t h e e q u a t i o n s o f C l a i r a u t and R i c c a t i , tthe r u l e s of Simpson and L ' H a p i t a l , t h e formula o f S t i r l i n g . Even t h e importance of T a y l o r ' s theorem and t h e MacLaurin expansion w a s t o some e x t e n t missed. MacLaurin (who claimed no c r e d i t f o r h i s expansion) was somewhat b e t t e r than t h e g e n e r a l r u n : h i s account of f l u x i o n s was l a t e r p r a i s e d by Lagrange, and he i s remembered f o r h i s s h a r e i n t h e Euler-NacLaurin formula. Other i s o l a t e d e x c e p t i o n s t o t h e g e n e r a l t r i v i a l i t y o f t h e XVIII-th c e n t u r y , a r e Waring's problem and W i l s o n ' s theorem, i n t h e theory o f numbers, and t h e Lambert s e r i e s X X2 x3 l-x l-x'+ 7-x"+ +

...

F o r t u n a t e l y t h i s i s n o t a l l ; t h e r e were a few mathematicians of s t a t u r e , t o whom we s h a l l g i v e more s p a c e . I s h a l l devote t h e r e s t of t h i s s e c t i o n t o t h e B e r n o u l l i

family, Their University w a s Basel, t h e o l d e s t i n Switzerland: i t was founded i n 1460 by Pope Pius 11. Erasmus t a u g h t t h e r e i n 1521-1529.

There w a s James ( o r Jacob) B e r n o u l l i (1654-1705) and h i s nephew Nicholas I (1687-1759) ; t h e r e w a s John B e r n o u l l i (1667-1748) and h i s two sons Nicholas I1 (1695-1726) and Daniel (1700-1759) ; t h e r e was a l s o t h e t h i r d son of John, namely John I1 (17101790) and h i s two sons John I11 (1744-1807) and James I1 (1759-1789). Outside of B a s e l , i t w a s a t f i r s t , a s I made c l e a r , a l e a n time f o r mathematics. I n speaking o f i t , I d i d n o t count t h e B e r n o u l l i ' s , a s they had a v e r y d i f f e r e n t o u t l o o k : t h e r e was so much t h a t i n t e r e s t e d them o u t s i d e o f mathematics p r o p e r . They could look on mathematics, n o t a s a t o y , b u t as something t o be used i n p r a c t i c a l ways: t h e t r a d i t i o n a l S w i s s p o i n t o f view, Moreover, b o t h James and Daniel had i d e a s w e l l ahead o f t h e i r times. James B e r n o u l l i i s famous f o r h i s complete s o l u t i o n o f t h e soc a l l e d b r a c h i s t o c h r o n e problem: I g i v e i t i n my book on t h e Calculus of V a r i a c i o n s . The s o l u t i o n s given by o t h e r s a t t h e time a r e today considered u n s a t i s f a c t o r y , i n t h a t they assume an e x i s t e n c e theorem. However, Huygens, who had j u s t d i e d , had made important c o n t r i b u t i o n s long b e f o r e t h e problem was even formulated: i n h i s work on t h e pendulum, w r i t t e n i n 1673 and e n t i t l e d Horologium o s c i l l a t o r i u m , he had had o c c a s i o n to s t u d y s p e c i a l l y some p r o p e r t i e s of t h e curve which was t h e b r a c h i s t o c h r o n e s o l u t i o n ; and i n h i s famous T r a i t 6 de l a LumiSre, he had formulated t h e b a s i c p r i n c i p l e t h a t I mentioned e a r l i e r , which James unconsciously used, and which

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The New Beginning

remained o t h e r w i s e b u r i e d u n t i l t h e beginning of our c e n t u r y . The problem thus has a s p e c i a l h i s t o r i c a l i n t e r e s t ; i n c i d e n t a l l y i t has l a t e l y found i t s way even i n t o S c i e n c e - f i c t i o n . A t any r a t e I s h a l l now d e s c r i b e i t . Two p o i n t s A and B a r e i n a v e r t i c a l p l a n e , w i t h B lower t h a n A , o r a t t h e same l e v e l , b u t n o t necess a r i l y on t h e same v e r t i c a l l i n e . A bead i s allowed t o s l i d e under g r a v i t y and w i t h o u t f r i c t i o n , on a w i r e from A t o B. What i s t h e shape of t h e w i r e , €or which t h e bead reaches B i n t h e s h o r t e s t time?

James proved t h a t t h e time i s l e a s t when t h e w i r e has t h e shape o f an a r c of a c y c l o i d , t h e a r c t o be c o n t a i n e d i n a s i n g l e a r c h beginning a t A . Huygens proved t h a t i f a bead i s p l a c e d anywhere on an a r c h of a c y c l o i d , whose ends a r e a t t h e same l e v e l , w h i l e t h e r e s t of t h e a r c h i s below them, then t h e t i m e taken by t h e bead t o s l i d e from r e s t t o t h e lowest p o i n t of t h e a r c h i s t h e same f o r a l l i n i t i a l p o s i t i o n s of t h e bead h i g h e r than t h e lowest p o i n t . (This r e s u l t was a l s o found by t h e J e s u i t P a r d i e s . ) A

kAJJ B

F i g . 1: Cycloid and t h e points A,B.

OB Fig. 2 .

The c y c l o i d , w i t h i t s a r c h e s , i s i l l u s t r a t e d i n f i g . 1. I t s e q u a t i o n - - i n terms of a parameter 0 - - i s x = c cos 0 , y = c0 - c s i n 0 . The v e r t i c a l p l a n e i s taken as t h a t of x , y w i t h t h e o r i g i n a t A , and t h e x - a x i s downwards. I n t h e s l i g h t l y d i f f e r e n t S c i e n c e - f i c t i o n . v e r s i o n , A and B a r e c i t i e s on t h e E a r t h ' s s u r f a c e ( f i g . 2 ) , and t h e w i r e connecting them has become a t u n n e l , along which a t r a i n s l i d e s on f r i c t i o n l e s s r a i l s under g r a v i t y , and w i t h no a i r r e s i s t a n c e . I n e i t h e r v e r s i o n , t h e problem i s one o f t h e Calculus of Variations. I n mathematical terms, i n t h e o r i g i n a l form of t h e problem, a t any p o i n t P = ( x , y ) of t h e w i r e , t h e component of g r a v i t y along t h e t a n g e n t t o t h e w i r e a t P i s t h e only one causing a c c e l e r a t i o n , s i n c e t h e component p e r p e n d i c u l a r t o t h e w i r e i s balanced by t h e r e a c t i o n of t h e c o n s t r a i n t . I n o t h e r words, t h e t a n g e n t i a l a c c e l e r a t i o n i s g cosd, where d i s t h e a n g l e t n e t a n g e n t makes w i t h t h e v e r t i c a l . This can be w r i t t e n g d x l d s , where d s i s t h e element of l e n g t h . I f m i s t h e mass of t h e bead, t h e work done by g r a v i t y a s t h e bead s l i d e s from A i s thus mg times t h e i n t e g r a l of ( d x / d s ) d s , i . e . of dx, and s o has t h e v a l u e mgx. By e q u a t i n g t h i s t o t h e k i n e t i c energy 3;mv2, where v i s t h e v e l o c i t y d s / d t , w e f i n d t h a t

102

Introduction d t 2 = ( 2 g x ) - l * d s 2 = ( 2 g ~ ) - ' *( l + y t 2 )

*

dX2.

Omitting a c o n s t a n t f a c t o r , w e s e e t h a t t h e q u a n t i t y t o be made as s m a l l a s p o s s i b l e i s a n i n t e g r a l of t h e form

taken from t h e o r i g i n A t o t h e p o i n t B , and t h a t t h e f u n c t i o n More g e n e r a l l y , f i s h e r e t h e s q u a r e r o o t of (1 y " ) / x . f o r a g i v e n f ( x , y , y ' ) , t h e s i m p l e s t problem o f t h e Calculus of V a r i a t i o n s c o n s i s t s i n f i n d i n g a curve y ( x ) , j o i n i n g two given p o i n t s , f o r which the above i n t e g r a l (where y ' denotes t h e d e r i v a t i v e of y) i s minimal.

+

Of a l l t h e t o p i c s we have come t o s o f a r , i n t h e h i s t o r y of mathematics, t h e Calculus o f V a r i a t i o n s i s perhaps t h e d e e p e s t and most modern. I t has p l a y e d , s i n c e t h e t i m e of Huygens and t h e B e r n o u l l i ' s , a fundamental p a r t i n a number of branches of mathematics. P r i n c i p l e s , such as t h a t of L e a s t Action i n Dynamics, o r of Fermat's Least T i m e i n O p t i c s , reduce p a r t s of Physics t o problems of t h e Calculus of V a r i a t i o n s : w e r e f e r t o them as V a r i a t i o n a l P r i n c i p l e s . S i m i l a r l y t h e famous D i r i c h l e t P r i n c i p l e dominates p o t e n t i a l theory and t h e t h e o r y of harmonic f u n c t i o n s , and i t became f o r Riemann t h e b a s i s of h i s c e l e b r a t e d e x i s t e n c e theorem f o r t h e conformal mapping of a Riemann s u r f a c e . F i n a l l y , i n r e c e n t times, The Calculus of V a r i a t i o n s has reappeared i n t h e form of Control Theory, which has a p p l i c a t i o n s i n a whole range of problems of E n g i n e e r i n g , Economics, Adminis t s a t i o n , and Space Science. A s p i o n e e r s i n such a s u b j e c t , Huygens and James B e r n o u l l i count as g r e a t names s t i l l . There i s a l e s s o n i n t h i s €or s t u d e n t s about t o embark on r e s e a r c h , whether i n mathematics, o r i n some o t h e r f i e l d : choose a t o p i c t h a t w i l l grow, n o t one l i a b l e t o p e t e r o u t l e a v i n g you as the e x p e r t i n something t h a t has ceased t o e x i s t . Huygens i s t h e f i r s t g r e a t Dutch mathematician: he w a s brought t o t h e P a r i s academy by C o l b e r t , b u t r e t u r n e d t o Holland a f t e r t h e r e v o c a t i o n of t h e E d i c t of Nantes. James B e r n o u l l i i s t h e f i r s t g r e a t S w i s s mathemat i c i a n . C h r o n o l o g i c a l l y , they belong s t i l l t o t h e Newton p e r i o d , b u t t h e i r mathematical i d e a s c a r r y us t o l a t e r times. Another g r e a t member of t h e B e r n o u l l i c l a n w a s D a n i e l , whose work on t h e v i b r a t i o n o f s t r i n g s c o n t a i n s b a s i c i d e a s of t h e Theory o f F o u r i e r s e r i e s ; I s h a l l speak of i t i n a l a t e r s e c t i o n . The most p r o d u c t i v e B e r n o u l l i was John: h i s work i s n o t as deep as t h a t of James o r D a n i e l , w i t h b o t h o f whom he q u a r r e l l e d , b u t he c e r t a i n l y deserves a p l a c e i n t h e h i s t o r y of mathematics. To John, we owe t h e B e r n o u l l i d i f f e r e n t i a l e q u a t i o n , and a l s o t h e s o - c a l l e d l ' H 6 p i t a l r u l e : they are t h i n g s we l e a r n i n elementary Calculus. Then t h e r e i s t h e B e r n o u l l i s e r i e s Jydx = yx

-

X2

r y '

+ $-y" - ...,

IV.

The New Beginning

103

t h e r e a r e B e r n o u l l i numbers and polynomials, and n e a t i d e n t i t i e s such as Jolxxdx = 1 - 2 - 2 + 3 - 2 - h-2 + . . . , t h a t might almost have been produced by E u l e r . Nearly everyt h i n g t h a t b e a r s t h e name of B e r n o u l l i i s due t o John: i t i s easy t o follow. This made John a s p e c i a l l y good t e a c h e r : i n h i s c a s e t h i s turned o u t i m p o r t a n t : i t was John B e r n o u l l i , whose l e c t u r e s t h e young E u l e r was p r i v i l e g e d t o a t t e n d .

18.

Euler.

But f o r John B e r n o u l l i , E u l e r might have become simply a clergyman i n some obscure S w i s s v i l l a g e . I n s t e a d , E u l e r u l t i m a t e l y l e f t a v a s t legacy of c o l l e c t e d works, ranging o v e r an impressive number of f i e l d s and expected t o f i l l 120 v o l umes. Leonard E u l e r (1707-1783) was born i n Base1 a s t h e son of a P r o t e s t a n t clergyman, and h i s family intended h i s becoming a clergyman i n h i s turn--no doubt an e x c e l l e n t p r o f e s s i o n . S i m i l a r l y , t h e composer Schoenberg was made t o s t u d y music, and f o r h i s son h e chose t e n n i s , which he p r e f e r r e d - - e x c e l l e n t p r o f e s s i o n s , no doubt, t o o . However, today f o r a change, t h e f u t u r e of each person i s l a r g e l y i n h i s own hands. P e o p l e ' s mistakes a r e t h e i r own: they need more t h a n ever t o develop proper judgement. Luckily E u l e r was r a t h e r quick a t s c h o o l , quick enough f o r t h e following legend, t h a t w e do n o t have t o b e l i e v e s i n c e i t m i g r a t e s from E u l e r t o Gauss and o t h e r s . We a r e t o l d t h a t when h e was S o r 9 y e a r s o l d , he was i n c l i n e d t o look o u t of t h e window a t s c h o o l , while t h e r e s t of t h e c l a s s caught up w i t h what h e had grasped r i g h t away. To keep him occupied, t h e t e a c h e r asked him t o add t o g e t h e r a l l t h e numbers 1 t o 1 0 0 0 , b u t i t was h a r d l y a moment b e f o r e h e was a g a i n watching t h e snowflakes o u t s i d e : he had summed t h e a r i t h m e t i c p r o g r e s s i o n . Normally t h i s k i n d of p r e c o c i t y i n mathematics need n o t have world-shaking i m p l i c a t i o n s , o t h e r than t h e f a c t t h a t mathemat i c s i s probably n o t much more d i f f i c u l t t h a n Chinese. Prec o c i t y may b e mainly s e l f - c o n f i d e n c e and courage, n o t t o be i n t i m i d a t e d by r e p u t e d d i f f i c u l t i e s t h a t a r e i n r e a l i t y r a t h e r s u p e r f i c i a l , More l a s t i n g q u a l i t i e s a r e s e n s i t i v i t y and t h e response t o a r e a l c h a l l e n g e . Indeed most s o - c a l l e d c h i l d p r o d i g i e s do n o t impress m e : Mozart, of c o u r s e , i n a n except i o n . I n c i d e n t a l l y , legends about mathematicians almost i n v a r i a b l y m i g r a t e from one of them t o a n o t h e r , l i k e t h e one a t t r i b u t e d v a r i o u s l y t o Norbert Wiener, t o Hardy, e t c . , about w r i t i n g some r e s u l t on t h e board, s a y i n g i t ' s obvious, s t o p p i n g suddenly, s i t t i n g down a t t h e desk, a p p a r e n t l y going t o s l e e p i n f r o n t of t h e c l a s s , and waking up a t t h e end o f t h e hour t o s a y , "Yes, i t i s obvious." I normally p u t c h i l d r e n who c a l c u l a t e w i t h l i g h t n i n g

speed i n t h e same c a t e g o r y a s winners o f s p e l l i n g bees and w e l l - t r a i n e d I t h e r e f o r e consider t h e Swiss r i g h t t o b e r e l u c t a n t lap-dogs. t o allow precocious c h i l d r e n t o move up i n t o c l a s s e s f o r o l d e r c h i l d r e n . However no r u l e should b e h a r d and f a s t : I was allowed t o s k i p one c l a s s , and my e l d e r b r o t h e r t o s k i p

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t h r e e . I n E u l e r ' s c a s e , h e was allowed, a t t h e age of 13, t o a t t e n d t h e l e c t u r e s of John B e r n o u l l i along w i t h J o h n ' s sons Nicholas I1 and Daniel. Only the r e c o g n i t i o n thus accorded t o h i s p r e c o c i t y made i t p o s s i b l e f o r him t o end up a s a mathematician. He became Bachelor of A r t s a t 1 4 , Master of A r t s a t 1 6 , and he completed h i s t h e s i s a t 19. I n John B e r n o u l l i ' s house, he was t r e a t e d a s one of t h e f a m i l y . His p a r e n t s g r a d u a l l y became r e c o n c i l e d t o h i s t a k i n g up mathemat i c s , and f i n a l l y t o h i s dropping theology when h i s t h e s i s was w r i t t e n : t h e t o p i c of t h e t h e s i s was r e c i p r o c a l a l g e b r a i c t r a j e c t o r i e s . Unfortunately, n i s unusual education had i t s drawbacks: t o t h e end of h i s l i f e , he spoke German w i t h a s t r o n g B a s l e r d i a l e c t a c c e n t , s l i g h t l y r i d i c u l o u s elsewhere. Mathematically t o o , he may have been under a disadvantage: h e became, n o t s o much a t h i n k e r , as a marvellous p r o f e s s i o n a l magician of mathematics. On t h e p o s i t i v e s i d e , he acquired from t h e B e r n o u l l i ' s a wide range of i n t e r e s t s i n e v e r y t h i n g connected w i t h mathematics. This i s how, t h e y e a r a f t e r h i s t h e s i s , he came t o w r i t e a D i s s e r t a t i o n on Sound: he was hoping t o e a r n a p r o f e s s o r s h i p i n Physics a t B a s e l , b u t was considered too young t o f i l l t h e vacancy. The following y e a r , a t t h e alze of 2 0 . h e submitted t o t h e P a r i s academy a Paper on t h e best-way t o p u t masts on s h i p s : i t s m e r i t was-recbgnised by an a c c e s s i t . This i s when Daniel B e r n o u l l i , who was now i n S t . P e t e r s b u r g , suggested t h a t E u l e r j o i n him t h e r e . T r a d i t i o n a l l y , t h e more e n t e r p r i s i n g Swiss seek t h e i r l i v i n g o u t s i d e t h e i r small country, a s do a l s o t h e S c o t s . There were S o o t t i s h guards i n P a r i s , Swiss guards i n t h e Vatican. Other p r o f e s s i o n s followed s u i t . A t t h e Court of Emperor F r e d e r i c 11, w e had t h e S c o t t i s h a s t r o l o g e r Michael S c o t t , f r i e n d of F i b o n a c c i . There i s a Swiss colony today i n Wisconsin, an h o u r ' s d r i v e from Madison. The S w i s s , whatever t h e i r o f f i c i a l n a t i o n a l i t y where they r e s i d e , r e t a i n t h e i r o r i g i n a l c i t i z e n s h i p i f they wish: i t i s a s p i r i t u a l bond t h a t transcends t h e tyranny o f geography and t h e n e c e s s i t y o f e a r n i n g a l i v i n g . I n c i d e n t a l l y , Daniel was n o t tile only B e r n o u l l i who went t o S t . P e t e r s b u r g : Nicholas I1 was t h e r e i n 1725-1726, and James I1 l a t e r i n 17861789, a s academicians, b u t both d i e d i n t h e h a r d weather. Moreover, when E u l e r a r r i v e d i n 1 7 2 7 , t h e S w i s s Hermann, whom we have m e t b e f o r e , was t h e r e a s p r o f e s s o r of P h y s i c s . I might mention t o o t h a t i n Padua, which Hermann had l e f t i n 1 7 1 2 t o go t o F r a n k f u r t , Nicholas I B e r n o u l l i occupied G a l i l e o ' s Chair i n 1 7 1 6 - 1719, Unfortunately, when E u l e r a r r i v e d i n S t . P e t e r s b u r g , Catherine I had j u s t d i e d , and t h e f u t u r e of t h e Russian Academy was u n c e r t a i n . Euler was g l a d of any p o s i t i o n - - s h i p ' s o f f i c e r , a s s i s t a n t t o t h e medical s c h o o l . F i n a l l y i n 1730 h e succeeded Hermann, who went back t o Basel with an i n t e r i m p r o f e s s o r s h i p of morals and n a t u r a l law, while w a i t i n g f o r a vacancy i n mathematics which d i d n o t m a t e r i a l i s e b e f o r e h i s death. E u l e r ' s p r o f e s s o r s h i p a t S t . P e t e r s b u r g became one of mathematics, when h e succeeded Daniel i n 1733, t h e l a t t e r having r e t u r n e d t o Basel a s p r o f e s s o r of botany and anatomy!

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The New Beginning

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P r o f e s s o r s i n Base1 seem t o have been most a d a p t a b l e ! T h i s was only s h o r t l y b e f o r e D a n i e l ' s open b r e a k w i t h h i s f a t h e r John, o c c a s i o n e d by t h e i r s h a r i n g a p r i z e of t h e P a r i s academy i n 1734. Thus a t 2 6 , E u l e r was a t l a s t p r o f e s s o r of mathematics. He m a r r i e d t h e Swiss d a u g h t e r o f t h e d i r e c t o r of t h e l o c a l P a i n t i n g academy: they had 1 3 c h i l d r e n , of whom 5 s u r v i v e d . A t 70, when h i s f i r s t w i f e had d i e d , he m a r r i e d her half-sister. By t h e n he had been p r a c t i c a l l y b l i n d f o r t e n y e a r s , having l o s t t h e s i g h t o f one eye i n 1735, and t o a l l i n t e n t s and purposes t h a t o f t h e o t h e r i n 1 7 6 6 . I t was t h e p r i c e he p a i d f o r t h e r a p i d r a t e a t which h e t u r n e d o u t books and p a p e r s . During t h e e a r l y P e t e r s b u r g p e r i o d , E u l e r d i d t h e p l a n n i n g and r e p o r t - w r i t i n g , f o r a n e x p e d i t i o n t o Lapland which made measurements o f t h e E a r t h ' s s u r f a c e . H e a l s o had o f f i c i a l d u t i e s r e g a r d i n g a High School o r Gymnasium. T h i s , t o g e t h e r w i t h h i s o r i g i n a l p r o f e s s o r s h i p of P h y s i c s , h i s i n t e r e s t i n s h i p s , and h i s problem o f v i s i o n , a c c o u n t s f o r h i s w r i t i n g on such d i v e r s e s u b j e c t s a s Music and Harmony; Elementary Geometry f o r High S c h o o l s ; A r i t h m e t i c f o r High S c h o o l s ; H i s t o r y o f Elementary Methods i n Geometry; Leitters t o a German P r i n c e s s on v a r i o u s s u b j e c t s o f P h y s i c s and Philosophy ( t r a n s l a t e d i n t o 8 l a n g u a g e s ) ; O p t i c s ; Theory of V i s i o n ; C o n s t r u c t i o n o f Lenses; Geodesics; The shape o f t h e E a r t h ; A t h e o r y o f t h e Moon; Ebb and T i d e ; Naval S c i e n c e ; Hydro-machines ; B a l l i s t i c s ; Nature and p r o p e r t i e s o f F i r e (crowned by t h e P a r i s academy); Mechanics ( 2 v o l s ) ; The motion of P l a n e t s and Comets. The b a l l i s t i c s book was planned a s a t r a n s l a t i o n o f an E n g l i s h gunnery book, t h e n completely r e v i s e d w i t h t h e h e l p of t h e C a l c u l u s ; i t was i t s e l f t r a n s l a t e d i n t o E n g l i s h a n d F r e n c h , and became t h e o f f i c i a l t e x t o f French m i l i t a r y s c h o o l s . B e s i d e s a l l t h i s , a v a r i a t i o n a l p u b l i c a t i o n of 1744 i n Lausanne c o n t a i n s h i s famous ' K n i c k f o r m e l ' , w i t h o u t which no b r i d g e can be b u i l t . I n E u l e r ' s c o l l e c t e d works, t h e s e v a r i o u s t h i n g s occupy 40 volumes--enough f o r a n o r d i n a r y man. I have n o t y e t spoken of E u l e r ' s main work. When he came t o S t . P e t e r s b u r g , t h e main t o p i c o f i n t e r e s t t h e r e happened t o be summation o f i n f i n i t e s e r i e s . One o f t h e people who i n t r o d u c e d E u l e r t o t h i s , was Goldbach, a mathematician from E a s t Germany, who u l t i m a t e l y became a Russian M i n i s t e r of S t a t e : he i s famous f o r h i s l a t e r c o n j e c t u r e t h a t e v e r y even p 2 , where each of number 2n i s e x p r e s s i b l e a s a sum p l p l , p 2 i s e i t h e r a prime o r u n i t y . I n 1730-1731, E u l e r summed the s e r i e s n c n-2 = . r r 2 / 6 ,

+

t h e sum tro 1 7 decimals had been c a l c u l a t e d by S t i r l i n g , which shows how s u r p r i s i n g E u l e r ' s t r u e v a l u e must have b e e n . This was perhaps t h e f i r s t appearance, and o n l y f o r s = 2 , of t h e famous Riemann z e t a f u n c t i o n , f o r which, as i s well-known,

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E u l e r l a t e r proved t h e i d e n t i t y

where Pn i s t h e n - t h prime. I n f i n d i n g t h e precise v a l u e of t h e z e t a f u n c t i o n s e r i e s f o r s = 2 , E u l e r introduced t h e E u l e r i a n i n t e g r a l s , t h e s o - c a l l e d E u l e r Beta and Gamma functions.

E u l e r ' s name i s a l s o a t t a c h e d t o a c o n s t a n t , namely t o t h e l i m i t y a s n tends t o i n f i n i t y , of t h e e x p r e s s i o n

1 + T1 +

... +

1

n - - log n ;

t n i s c o n s t a n t p r e s e n t s a problem--we have y e t t o prove i t irrational. Goldbach a l s o induced E u l e r t o take up number t h e o r y , and t o look i n t o problems ltaised by Fermat. This was when E u l e r proved easy t h i n g s , such a s t h e i r r a t t o n a l i t y of e and i t s s q u a r e , a s a consequence of his study of continued f r a c t i o n s ; we come t o continued f r a c t i o n s , i f we continue i n d e f i n i t e l y t h e r u l e f o r p a s s i n g from each of t h e following f r a c t i o n s t o t h e next a,

al l

Non-mathematicians sometimes f i n d i t d i f f i c u l t t o understand t h a t t h e r e can be any i n t e r e s t i n mere numbers: they may even t h i n k some numbers ' u n l u c k y . ' By way of c o n t r a s t , Hardy s a i d o f Ramanujan t h a t every number was h i s - ' p e r s o n a l f r i e n d . ' If you t r e a t numbers, o r any mathematical symbols, i n t h i s way, you soon f i n d them most i n t e r e s t i n g , Take, s a y , t!ie numbers 7 and 1 5 : i s t h e r e anything s p e c i a l about t h e p a i r ? Yes, indeed! They a r e p a r t i a l sums of t h e geometric p r o g r e s s i o n 1, 2 , 4 , 8 . I f you group t o g e t h e r a few numbers, as you might acquaintances, you w i l l probably be s u r p r i s e d t o f i n d s i m i l a r l y some remarkable p r o p e r t y . I t i s l i k e f i n d i n g t h a t two f r i e n d s , who d i d n o t know each o t h e r , l i v e i n t h e same s t r e e t . Then from i n d i v i d u a l numbers, you n a t u r a l l y switch your i n t e r e s t t o remarkable p r o p e r t i e s t h a t c e r t a i n numbers can have. This i s how w e g e t i n t e r e s t e d i n such t h i n g s a s prime numbers, s q u a r e s , cubes, n- t h powers, f a c t o r i a l s , B e r n o u l l i numbers, Fibonacci numbers, Mersenne numbers , binomial c o e f f i c i e n t s , t r i a n g u l a r and polygonal numbers, p e r f e c t numbers, and many o t h e r k i n d s . The n e x t s t a g e i s t o prove something about t h e s e p r o p e r t i e s , and for t h i s w e s t a r t looking

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f o r l i t t l e t r i c k s and a l g o r i t h m s , and g r a d u a l l y f o r more g e n e r a l methods. I have s a i d enough, I t h i n k , t o make c l e a r t h a t E u l e r , who was such a wizard a t f i n d i n g l i t t l e t r i c k s and algorithms o u t of h i s h a t , would enjoy number t h e o r y , and would c e r t a i n l y go on working i n i t . This work of E u l e i ' s a c t u a l l y s t a n d s o u t , compared w i t h h i s work i n o t h e r f i e l d s , more than i t d i d a t t h e t i m e : number theory p r o g r e s s e s r a t h e r more slowly than t h e r e s t of mathematics, s o t h a t i t t a k e s longer f o r a r e s u l t i n number theory t o become o u t of d a t e . For i n s t a n c e , E u l e r considered what were c a l l e d Numeri 5donei: however i t was only proved f o r t h e f i r s t t i m e i n 1 K b y Heilbronn and Chowla, t h a t t h e r e a r e an i n f i n i t y of them. The t o p i c of i n f i n i t e s e r i e s and t h e i r summation, which was of i n t e r e s t i n S t . P e t e r s b u r g when E u l e r a r r i v e d , i s l i k e number t h e o r y , one on which he continued t o work a l l h i s l i f e . Indeed, almost a l l h i s l a t e r r e s u l t s grew o u t o f what he thought up i n t h o s e e a r l y P e t e r s b u r g y e a r s - - t h i s i s a n o t h e r i l l u s t r a t i o n of my recommendation t o choose a s u b j e c t , o r s u b j e c t s , t h a t w i l l grow. E u l e r ' s work on i n f i n i t e s e r i e s s t a n d s o u t , l i k e h i s number t h e o r y , b u t f o r a d i f f e r e n t r e a s o n : t n e r e were, i n t h o s e days, no p r o p e r d e f i n i t i o n s . Convergence and t h e l i k e w e r e n o t understood, and i n f i n i t y was a s meaningless a s t h e imaginary had been f o r 200 y e a r s . I f w e go back t o t h e geometric s e r i e s

w e may w e l l wonder how a g r e a t man l i k e E u l e r could w r i t e down such a p p a r e n t l y absurd e q u a t i o n s a s r e s u l t by s e t t i n g x e q u a l t o - 2 , 2 , o r 3 , and y e t f i n d q u i t e c o r r e c t l y t h a t l i m (1 + 1 + 1

1+ +2

... + n-1 -

. .. +

1 - l o g n)

= y

1 - -2 B1 log n - y = 2n 2n

+

, B2

2- - * +

'

where t h e B ' s a r e B e r n o u l l i numbers. There must have been rudimentary n o t i o n s o f convergence? Even d ' Alembert used them i n h i s r a t i o t e s t (Opuscules 1768). Moritz Cantor q u o t e s on t h i s E u l e r ' s l e t t e r of 7 - t h August 1745 t o Goldbach: I c h habe s e i t e i n i g e r Z e i t m i t dem Herrn Nicolao B e r n o u l l i zu Base1 e i n e k l e i n e Disput ueber d i e s e r i e s divergentes, dergleichen i s t 1 - 1 2 6 + 24 - 120 + 720 + e t c , g e h a b t , indem d e r s e l b e g e l a u g n e t , dass a l l e d e r g l e i c h e n s e r i e s e i n e d e t e r m i n i e r t e Summ haben, i c h a b e r das Gegentheil b e h a u p t e t , w e i l e n i c h g l a u b e , d a s s e i n e j e g l i c h e s e r i e s e i n e n bestimmten Werth haben muss. Um a l l e n S c h w i e r i g k e i t e n , welche dagegen gemacht werden zu begegnen, s o s o l l t e d i e s e r Werth n i c h t m i t dem IJamen d e r Summ b e l e g t werden

...

..

+

108

I n t r o duc t i on

However, because o f ' a b s u r d ' e q u a t i o n s such a s those d e s c r i b e d above, much of what E u l e r wrote on i n f i n i t e s e r i e s was discarded by h i s s u c c e s s o r s , when they introduced what we regard a s t h e 'obvious' d e f i n i t i o n s . I t was n o t u n t i l t h e b e t t e r p a r t of a century l a t e r , t h a t l e s s obvious systems o f d e f i n i t i o n s were c o n s i d e r e d , i n which some o f h i s methods could be made t o f i t . This meant t h a t suddenly E u l e r ' s work was a g a i n r e s p e c t a b l e , and i n f a c t most modern. Todaynoone uses t h e geometric s e r i e s formula t o sum t h e s e r i e s f o r x e q u a l t o 2 o r 3 ; however t h e formula i s used f o r n e g a t i v e o r complex values of x , even when t h e magnitude o f x i s l a r g e ( b u t n o t then f o r r e a l p o s i t i v e x ) . This amounts t o summing t h e s e r i e s i n a s p e c i a l way, due t o L i n d e l o e f . For v a l u e s of x , o t h e r than 1, b u t s i t u a t e d on t h e r i m of the u n i t c i r c l e of convergence of t h e s e r i e s , f o r i n s t a n c e f o r t h e value -1, t h e much simpler summation method of Cesaro can be used. These m a t t e r s a r e t r e a t e d i n e x c e l l e n t b o o k s f B o r e l , Knopp and Hardy. A g e n e r a l theory i s due t o Otto T o e p l i t z (1881-1940). T o e p l i t z i s known a l s o f o r t h e ' T o e p l i t z q u e s t i o n , ' t h a t some people s t i l l a s k : "What have w e done wrong, i n t h e e d u c a t i o n of our s t u d e n t s , t h a t h a s made p o s s i b l e f o r t h e young g e n e r a t i o n t o become what i t has?'' The answer may be t h a t s t u d e n t s must l e a r n t h e l e s s o n s o f t h e p a s t , perhaps even from t h e h i s t o r y of mathematics. I n r e g a r d t o t h e ' a b s u r d ' equations of t h e type + m equal t o a n e g a t i v e number, t h a t even t h e modern theory of d i v e r g e n t s e r i e s does n o t admit, E u l e r , i f q u e s t i o n e d , might have s a i d : "Ah, you s e e only t h e obvious i n t e r p r e t a t i o n ! " The equations mean t h a t t h e r e i s a deeper i n t e r p r e t a t i o n ; and a s t o a b s u r d i t y , even the n o n s e n s i c a l 3 = 1 is t r u e modulo 2 . This i s connected w i t h t h e q u e s t i o n o f t h e n e g m f an i n f i n i t e c o n s t a n t , i f we can e x p l a i n what t h a t means. Quantum theory h a s been plagued w i t h d i f f i c u l t i e s of t h a t s o r t . A c t u a l l y , i n our c e n t u r y , Hadamard n e g l e c t e d i n f i n i t y i n h i s F i n i t e p a r t s of d i v e r g e n t i n t e g r a l s , and h i s methods a r e c a r r i e d f u r t h e r i n Laurent Scliwartz d i s t r i b u t i o n s and van d e r Corput n e u t r i c e s . I t i s remarkable t h a t , i n connection w i t h i n f i n i t e s e r i e s , Euler n e a r l y discovered F o u r i e r s e r i e s : t h i s was when h e gave h i s second proof o f t h e value he had found f o r t h e z e t a f u n c t i o n a t s = 2 . F o u r i e r s e r i e s were a l s o almost discovered by Daniel B e r n o u l l i , but i n h i s c a s e i t was E u l e r who squashed t h i s , as I s h a l l e x p l a i n i n a l a t e r s e c t i o n .

E u l e r ' s f i r s t s t a y i n S t . P e t e r s b u r g ended i n 1741. The d e a t h of t h e Empress Anna had made t h i n g s d i f f i c u l t t h e r e s i n c e 1740. H e agreed t o go t o B e r l i n , where h e s t a y e d 25 y e a r s . The new p o s i t i d n was by no means i d e a l , and t h e reasons f o r t h i s do n o t r e f l e c t c r e d i t on t h e ',Great' F r e d e r i c o f P r u s s i a , who made h i s t o r y b u t who can now b e s e e n f o r what he was,

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i n s t e a d of by comparison with i n e p t r u l e r s of neighbouring s t a t e s . H i s t o r i a n s tend t o judge t o o much by r e s u l t s , however ephemeral: they p a s s l i g h t l y over h i s v a n i t y and t h e a b s u r d i t y of h i s r e q u i r i n g o f f i c e r s t o b e s i x f o o t t a l l . He f l a t t e r e d himself t h a t h e more than made up f o r h i s own diminutive s i z e by b e i n g e l e g a n t and w i t t y . Apart from hims e l f , he admired only P a r i s i a n s . Unfortunately t h e r e was nothing r e f i n e d o r e l e g a n t o r w i t t y about E u l e r ' s uncouth B a s l e r speech, t o make up f o r n o t being French. On t h e top o f t h a t , he had only one e y e : F r e d e r i c r e f e r r e d t o him a s ' t h e Cyclops. '

-+

E u l e r g o t on w e l l w i t h t h e P r e s i d e n t of t h e B e r l i n Academy, Mau e r t u i s , t h e l e a d e r of t h e Lapland e x p e d i t i o n t h a t E u l e r ha s o w e l l o r g a n i z e d . Maupertuis i s known f o r h i s P r i n c i p l e of L e a s t Action i n Dynamics. He was one of t h e few n o n - a t h e i s t French i n t e l l e c t u a l s . E u l e r , as a clergyman's son--or s u r p r i s i n g l y f o r one?--agreed w i t h him i n t h i s , and approved of an attempt t o prove t h e e x i s t e n c e o f God by t h e P r i n c i p l e of L e a s t Action. This was too much f o r V o l t a i r e , who proceeded t o cover them both w i t h r i d i c u l e . P l e a s e excuse me i f I now recount a s t o r y t h a t should long have been thrown o u t of t h e r e p e r t o i r e of anyone w i t h t h e l e a s t h i s t o r i c a l c o n s c i e n c e , b u t t h a t keeps b e i n g r e p e a t e d by t h e Law of Conservation of Ignorance. I t came o r i g i n a l l y from de Mor a n ' s Budget o f Paradoxes. In i t , Euler i s alleged t o l i F & - a d i s p u t a t i o n w i t h t h e French e n c y c l o p e d i s t Didgrot about t h e e x i s t e n c e of God. Didgrot, I may s a y , was no s t r a n g e r t o mathematics, b u t of course n o t i n E u l e r ' s c l a s s . The s t o r y goes t h a t E u l e r simply wrote a mathematical formula on t h e board, and s a i d "Therefore God e x i s t s ; " t h e u n f o r t u n a t e Didgrot i s s t a t e d t o have f l e d ignominiously, a f t e r excusing h i m s e l f . What t h e s t o r y does n o t s a y , and h e r e I r e c o n s t r u c t t h e most l i k e l y s e q u e l , i s t h a t DidGrot, on h i s r e t u r n t o h i s apartment, s e n t h i s l a c k e y w i t h a n o t e : "Monsieur E u l e r , j e n ' a i pas voulu vous embarrasser publiquement, mais v o t r e Dieu e s t - i l s i f a i b l e , q u ' i l vous f a u t t r i c h e r pour l e dgfendre?" I hope t h i s p u t s t h e s t o r y i n i t s r i g h t p e r s p e c t i v e . Many people do b e l i e v e t h e i r God s o powerless, t h a t h e must be helped by l y i n g and c h e a t i n g , i f n o t by murder and t o r t u r e of u n b e l i e v e r s , crimes f o r which they w i l l , according t o t h e i r own b e l i e f s , s u f f e r most u n s e l f i s h l y i n t h e n e x t world. One would l i k e t o t h i n k t h a t t h e r e l i g i o n o f a mathematician can b e f r e e o f such a b s u r d i t i e s .

I n t h i s m a t t e r o f a proof o f t h e e x i s t e n c e o f God, I must admit

t h a t I s i d e w i t h V o l t a i r e - - n o t because of any r e l i g i o u s c o n v i c t i o n s o r o t h e r w i s e , b u t simply because I f i n d t h e misuse o f mathematics by mathematicians even more i n t o l e r a b l e than t h a t by non-mathematicians. Maupertuis, i n c i d e n t a l l y , was no S a i n t : a s t h e t e a c h e r of V o l t a i r e ' s m i s t r e s s , h e may n o t have earned V o l t a i r e ' s j e a l o u s y , b u t he d i d b r i n g back from Lapland two 'lap-women' t o t h e shocked s u r p r i s e even of t h e P a r i s o f Louis XV, s o t h a t h e was somewhat under a cloud when h e l e f t f o r B e r l i n , a f t e r managing t o d i v e s t himself of them. Now, however, V o l t a i r e ' s b a r b s made F r e d e r i c of P r u s s i a f u r i o u s :

110

Introduction

Maupertuis and E u l e r were b o t h s u b j e c t e d t o a most h u m i l i a t i n g t r e a t m e n t . Maupertuis became q u i t e i l l , and f i n a l l y r e t i r e d i n 1756. E u l e r , more t h i c k - s k i n n e d , took over a d m i n i s t e r i n g t h e Academy of Sciences f o r a n o t h e r 1 0 y e a r s , u n t i l even he could s t a n d i t no l o n g e r . He was g l a d t o a c c e p t a most tempting o f f e r from Empress C a t h e r i n e I1 t o r e t u r n t o S t . P e t e r s b u r g i n 1766. F r e d e r i c l e t him go w i t h o u t a word of t h a n k s , and i n f a c t was a s p l e a s e d as a naughty schoolboy. To t h e man he had r e a l l y wanted i n p l a c e of E u l e r , t h e Frenchman d'Alembert, he w r o t e : Monsieur E u l e r , q u i aime B l a f o l i e l a grande e t l a p e t i t e o u r s e , s ' e s t approch6 du nord pour observer B son a i s e . Un v a i s s e a u q u i p o r t a i t s e s xz e t son kk a f a i t n a u f r a g e : t o u t e s t p e r d u , e t c ' e s t dommage p a r c e q u ' i l a u r a i t eu de quoi r e m p l i r 6 volumes i n f o l i o de m6moires c h i f f r 6 s d ' u n bout B l ' a u t r e , e t 1'Europe s e r a v r a i s e m b l a b l e ment p r i v 6 e de l ' a g r 6 a b l e amusement que c e t t e l e c t u r e l u i a u r a i t donn6. The B e r l i n p e r i o d (25 y e a r s , 1741-1766) i s t h e t i m e when E u l e r produced h i s b i g textbook s e r i e s . H e t u r n e d o u t t o be a remarkable e x p o s i t o r - - h e s e t o u t 00 p o p u l a r i s e t h e Calculus by i n t r o d u c i n g the r e a d e r t o t h e most d i f f i c u l t p a r t s as i f ' p l a y i n g . ' The f i r s t two volumes a r e t h e I n t r o d u c t i o n , . t h e next t h e D i f f e r e n t i a l Calculus, t h e l a s t t h r e e t h e I n t e g r a l Calculus. I t i s e x c i t i n g reading, but s t i l l a naive i n t u i t i v e Calculu9,today l o g i c a l l y u n s a t i s f a c t o r y . Such a book by a master of h i s c r a f t i s a l w a y s worth r e a d i n g . S i m i l a r l y , i n modern S e t Theory today, I recommend t h e c l a s s i c book of S i e r p i n s k i on t r a n s f i n i t e numbers (Bore1 C o l l e c t i o n 1928) : and y e t i t s t a r t s o f f "Everybody knows what i s meant by a s e t , for instance...". To keep up w i t h t h e l a t e s t w r i n k l e may be i m p o r t a n t , b u t t h e r e i s much t o be gained by reading how a r e a l l y g r e a t man looks a t t h i n g s , and how e x c i t i n g he can make them, The shipwreck r e p o r t e d by F r e d e r i c was u n f o r t u n a t e l y t r u e . However, on h i s r e t u r n t o S t . P e t e r s b u r g , E u l e r was given e v e r y f a c i l i t y . H e was o f c o u r s e no l o n g e r i n h i s prime: h i s e y e s i g h t and hBs h e a l t h gave him t r o u b l e - - h e w a s almost t o t a l l y b l i n d . Nevertheless h i s r e s e a r c h e s continued unabated a t t h e same high l e v e l . He w a s h e l p e d i n every way: i t w a s very d i f f e r e n t from t h e r e s e a r c h of h i s e a r l y days, when h i s s t u d y was by no means o u t of bounds t o t h e c h i l d r e n , then v e r y young! They a r e s a i d t o have played w i t h h i s m a n u s c r i p t s , which f r e q u e n t l y g o t p u t back i n any o r d e r , so t h a t t h e p r i n t e r , who came t o c o l l e c t them, a t t i m e s p r i n t e d t h e l a s t p a r t of a paper b e f o r e t h e f i r s t . Both i n t h e B e r l i n p e r i o d , and i n t h e subsequent second P e t e r s burg p e r i o d , E u l e r continued t o t u r n o u t a v a s t number of o r i g i n a l papers and memoirs, as w e l l a s books. Much o f t h i s w a s , a s 1 e x p l a i n e d , d i s c a r d e d i n t h e more d i s c r i m i n a t i n g p e r i o d t h a t followed, when people ceased t o b e i n t i m i d a t e d by t h e g r e a t names t h a t came b e f o r e . The t r u t h i s t h a t t h e way r e a l l y t o r e a d any mathematical o r o t h e r work, however eminent o r obscure i t s a u t h o r , i s t o ask o n e s e l f n o t s o much

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what i s t h e r e , a s what i s missing? T h i s "reading between t h e l i n e s ' ' goes much f u r t h e r than mere checking t h e c o r r e c t n e s s of what s t a n d s i n p r i n t - - a t e d i o u s b u s i n e s s t h a t we a r e s o e a s i l y tempted t o f o r e g o . I n E u l e r ' s c a s e , a g r e a t d e a l was m i s s i n g : t h i s i s one of t h e reasons why E u l e r ' s works remain v a l u a b l e . I n t h e Calculus of V a r i a t i o n s , E u l e r o b t a i n e d h i s famous "Euler e q u a t i o n " , a c o n d i t i o n t o b e s a t i s f i e d by any s o l u t i o n . What i s missing t h e r e i s an understanding of what a c o n d i t i o n means, t h e c l e a r d i s t i n c t i o n between what i s s u f f i c i e n t , and what i s n e c e s s a r y . However, i n d e r i v i n g h i s e q u a t i o n , E u l e r missed something e l s e , something almost found, and t h a t remained hidden f o r a n o t h e r 200 years--something fundamental today i n t h e theory o f Schwartz d i s t r i b u t i o n s and i n f u n c t i o n a l analysis--namely a p r i n c i p l e of d u a l i t y which t a k e s us a l l t h e way t o t h e modern theory of DieudonnC and Schwartz. The p r i n c i p l e i s i m p l i c i t i n a lemma--by no means a d i f f i c u l t one--which i s c r u c i a l f o r E u l e r ' s approach. However, i n o r d e r t o r e a l i s e t h e g r e a t p o s s i b i l i t i e s o f t h i s lemma, Euler would have needed t o a p p r e c i a t e b e t t e r t h e d u a l i t y work of De'sargues, some 200 y e a r s e a r l i e r . H i s t o r y of mathematics i s n o t w i t h o u t i t s p o i n t s . The lemma i t s e l f i s n o t u n r e l a t e d t o important p h i l o s o p h i c a l and p r a c t i c a l q u e s t i o n s . One such q u e s t i o n ( t h a t o f g h o s t s , of u n i v e r s e s u n r e l a t e d t o our own, e t c . ) a r o s e a l r e a d y when I s a i d t h a t i n my youth t h e r e was n o t a s h r e d of evidence t o s u p p o r t t h e h y p o t h e s i s o f d r i f t i n g c o n t i n e n t s : can something be regarded a s n o n - e x i s t e n t , i f no evidence can b e found of i t s e x i s t e n c e ? I n an e q u i v a l e n t form, t h e q u e s t i o n i s one o f i d e n t i t y : whbther two p e r s o n s , two o b j e c t s , two chemical s u b s t a n c e s , can be regarded a s i d e n t i c a l , i f no o b s e r v a t i o n can d e t e c t any d i f f e r e n c e s between them. These m a t t e r s a r e themselves v a r i a n t s of t h e l o g i c a l q u e s t i o n o f d e c i d a b i l i t y and u n d e c i d a b i l i t y , which has l e d t o s t a r t l i n g developments during my l i f e t i m e . Mathematiaally, t h e q u e s t i o n can b e imagined t o be t h a t of t h e e x i s t e n c e of a c e r t a i n n o n - i d e n t i a a l l y v a n i s h i n g continuous f u n c t i o n f ( x ) on t h e i n t e r v a l from 0 t o 1: i t i s supposed t h a t , f o r every " t e s t " f u n c t i o n b ( x ) , i . e . f o r every s u f f i c i e n t l y smooth d ( x ) which vanishes a t t h e e x t r e m i t i e s 0 and 1, w e f i n d t h a t

11 f ( x ) b ( x ) d x 0

=

0.

The c r u c i a l lemma a s s e r t s t h a t no such n o n - i d e n t i c a l l y vanishing f (x) can e x i s t . I f w e r e p l a c e t h e continuous v a r i a b l e by a v a r i a b l e w i t h only two p o s s i b l e values 1 and 2 , s o t h a t f ( x ) becomes t h e p a i r of v a l u e s f ( l ) , f ( 2 ) , i . e . t h e v e c t o r f = f i , f , and s i m i l a r l y (6 = b l , 62, t h e corresponding a s s e r t i o n i s t E a t i f t h e s c a l a r product f b v a n i s h e s f o r e v e r y d , t h e n t h e v e c t o r f i s t h e n u l l -

112

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v e c t o r . An e q u i v a l e n t formulation i s t h a t a p o i n t of t h e p l a n e , s i t u a t e d on every s t r a i g h t l i n e through t h e o r i g i n , can only be t h e o r i g i n i t s e l f . A c t u a l l y , we need h e r e only t h e weaker h y p o t h e s i s , t h a t t h e p o i n t l i e s on both t h e c o o r d i n a t e axes. There a r e o t h e r i n s t a n c e s i n which E u l e r f a i l e d t o s e a r c h f o r a deeper meaning t o h i s many b e a u t i f u l r e s u l t s , r e l a t i o n s and t r a n s f o r m a t i o n s . Only 30 y e a r s a f t e r E u l e r d i e d , a very minor mathematician was a b l e t o g i v e a p a r f e c t l y a c c e p t a b l e i n t e r p r e t a t i o n t o imaginaries : t h a t was t h e Swiss Argand, known f o r t h e Argand diagram. Indeed h e was one of s e v e r a l persons t o do s o independently. S u r e l y Euler should have found t h i s ? I t was v i r t u a l l y suggested by h i s formula eix = cosx

+

i sinx.

He was n o t l o o k i n g f o r such t h i n g s . Indeed, without adequate d e f i n i t i o n s , much t h a t he d i d remained a s meaningless a s t h e imaginaries had been. The g r e a t e r t h e t r e e , t h e g r e a t e r i t s need f o r deep r o o t s . Perhaps E u l e r , t h e most c r e a t i v e of mathematicians, was handicapped by having l e f t school s o soon: he could have used so w e l l t h e g r e a t e r thoughtfulhess t h a t one a c q u i r e s i n l a t e r y e a r s a t s c h o o l , He might have produced a l i t t l e l e s s , b u t i t might n o t have been d i s c a r d e d s o soon, j u s t a s a g r e a t t r e e , w i t h shallow r o o t s , w i l l c r a s h i n t h e f i r s t w i l d storm. He had e x c e l l e n t methods of c a l c u l a t i o n : he d i d n o t r e a l i s e tnat a calculation i s valueless u n t i l the quantity calculated i s known t o e x i s t . For i n s t a n c e , he understood very w e l l how t o transform a s e r i e s i n t o one which converges r a p i d l y , o r i s e a s i l y e v a l u a t e d . But i n many c a s e s , t h e o r i g i n a l s e r i e s had no meaning a s h e i n t r o d u c e d i t . Without a s a t i s f a c t o r y n o t i o n of cbnvergence, s e r i e s o r i n t e g r a l s summed t o i n f i n i t y were empty symbols. To p u t t h e s e t h i n g s r i g h t seems today a simple e x e r c i s e ; b u t i t took 100 y e a r s . I n a book e n t i t l e d "Calcul de G k n k r a l i s a t i o n , " by a p r o f e s s o r G . Oltramare, dean of t h e Faculty of Science of t h e U n i v e r s i t y of Geneva, published i n 1899 by t h e very r e p u t a b l e f i r m of Herniann--that l a t e r was t o p u b l i s h Schwartz's Theory of Distributions--we f i n d on p . 27 t h e q u a n t i t i e s I T ~ @ +( aX m ) and .rra-'$(x + a J 7 ) equated ( f o r an a r b i t r a r y @ ) t o t h e express ions

lom ( @ ( x+t ) - 0 ( x- t ) ) / ( a' +t ') t d t

jam< + (x+t) +$ (x-t ) ) / ( a 2 + t 2) t d t

, ,

which a r e c e r t a i n l y r e a l ! We f i n d t h i s h o r r o r long a f t e r Cauchy, long a f t e r Poisson, r i g h t i n t h e p e r i o d o f W e i e r s t r a s s i a n r i g o u r ! And t a l k i n g of Poisson, w e f i n d on p , 4 3 two formulae quoted from h i s paper i n P . 1 3 , pages 295 and 33--P being presumably t h e J o u r n a l de 1 ' E c o l e Polytechnique-- , which

IV.

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113

a g a i n g i v e complex v a l u e s t o two r e a l i n t e g r a l s . The a u t h o r informs us t h a t P o i s s o n ' s e v a l u a t i o n s have been c r i t i c i s e d , i t having been suggested t h a t t h e imaginary p a r t s of t h e v a l u e s given by Poisson be d e l e t e d . T h i s , w e a r e t o l d , would b e i n c o r r e c t , s i n c e P o i s s o n ' s formulae agree w i t h a formula given by Cauchy, and t h e r e f o r e unquestionably t r u e . This l a s t formula, quoted from Cauchy's paper p r e v i o u s l y r e f e r r e d t o by t h e a u t h o r (Sav. E t r . 1827, 1241, s t a t e s t h a t , a g a i n f o r an a r b i t r a r y $,

/"0

(y2+b2t2)-'($(x+t)+$(x-t))dt

=

n(yb)-'$(x-b-'yGr),

a r e s u l t f u l l y as n o n s e n s i c a l as t h e preceding ones. What i s amazing i s n o t j u s t t h a t t h e g r e a t Cauchy was c a p a b l e of p u b l i s h i n g nonsense, b u t t h a t t h i s nonsense could b e b e l i e v e d long a f t e r , w i t h a t r u l y A r i s t o t e l i a n f a i t h , i n a c i t y once proud of t h e r i g o u r and s t e r n n e s s of i t s c r e e d , ' a n d i n a F a c u l t y of Science t h a t has s i n c e i n c l u d e d Karamata and de Rham. Nor was t h i s a time when S w i t z e r l a n d lacked good mathematicians. I t i s t r u e t h a t Minkowski i n Zurich was a f o r e i g n e r , b u t i n Bern, f o r i n s t a n c e , t h e p r o f e s s o r from 1853 t o 1891 had been L . S c h a f l i (1814-1895), now mainly remembered f o r h i s study of t h e r e g u l a r s o l i d s i n h i g h e r s p a c e s . Geneva i t s e l f was n o t w i t h o u t i t s mathematical t r a d i t i o n . I t had had a s s t u d e n t (1818-1822) t h e Sturm ( C . F. 1803-1855) of SturmL i o u v i l l e d i f f e r e n t i a l e q u a t i o n s . H i s t e a c h e r was S . A . G. L h u i l l e r (1750-1840), p r o f e s s o r from 1794 t o 1823, who had won i n 1784 t h e B e r l i n P r i z e f o r h i s memoir on t h e i n f i n i t e : " l ' i n f i n i e s t un g o u f f r e o h s e p e r d e n t nos p e n s g e s . " E a r l i e r s t i l l , from 1724 t o 1752, t h e Geneva Chair was h e l d by G. Cramer (1704-1752), p i o n e e r of a l g e b r a i c c u r v e s : "il n ' y a r i e n de p l u s remarquable dans l e s courbes que les branches i n f i n i e s e t l e s p o i n t s s i n g u l i e r s . " The paradox o f Cramer i s t h a t through t h e n 2 p o i n t s common t o two curves of o r d e r n , t h e r e p a s s an i n f i n i t y of curves of t h e same o r d e r , Of t h e many t h i n g s t h a t E u l e r d i d , perhaps t h e most s t r i k i n g were h i s two l i t t l e c o n t r i b u t i o n s t o topology: one concerns the seven b r i d g e s of Koenigsberg, t h e o t h e r t h e E u l e r charact e r i s t i c of a polyhedron. Some people may a l s o be i n t e r e s t e d i n h i s P h i l o s o p h i c a l Reflexions on Space and T i m e . The E u l e r c h a r a c t e r i s t i c i s t h e number V - E + F , where V i s t h e number of v e r t i c e s , E t h a t of edges, F t h a t of f a c e s . For a cube, t h e c h a r a c t e r i s t i c i s 2 . With t h e seven b r i d g e s of t h e f i g u r e , t h e problem i s t o f i n d a p a t h c r o s s i n g each of them e x a a t l y once: E u l e r proved t h e r e i s none. Fig. 3

+ -

E u l e r d i d n o t f o l l o w t h e s e t h i n g s up. H i s f a i l u r e t o do s o was no doubt due t o t h e v a s t amount he p u b l i s h e d , and t o t h e v a r i e t y of t o p i c s he i n v e s t i g a t e d . Like Newton, he deserved a whole s c h o o l , such as P l a t o had i n Ancient Athens: young men

114

Introduction

of r e a l promise, capable n o t only of following up h i s i d e a s , b u t of s e e i n g deeper. Base1 should have r e c a l l e d him, a t l e a s t t o resume h i s a s s o c i a t i o n w i t h t h e B e r n o u l l i ' s , i n s t e a d of allowing him t o b e h u m i l i a t e d i n B e r l i n . U n f o r t u n a t e l y , n o t every i n s t i t u t i o n l i k e s t o appoint someone who i s unhappy where h e i s : t h e q u e s t i o n asked i s "What's wrong w i t h him?". Only t h e United S t a t e s , i n i t s i n s p i r i n g message of t h e S t a t u e of L i b e r t y , f r e e d i t s e l f from such a p r e j u d i c e . H i s t o r i c a l events a f f e c t e d E u l e r a g r e a t d e a l . Russia d u r i n g h i s f i r s t s t a y was i n a s t a t e of f l u x . P e t e r t h e Great had made g r e a t e f f o r t s t o modernise and e d u c a t e , h i s sudden d e a t h i n 1725 l e f t everything i n t h e b a l a n c e . Two y e a r s l a t e r , when Catherine I d i e d , t h e r e was q u i t e a r e a c t i o n i n t h e s h o r t r e i g n o f P e t e r I1 (1727-1730). The r e i g n of Empress Anna (1730-1740) was more s u c c e s s f u l , and E u l e r was reasonably happy. Anna's i n f a n t nephew then r u l e d one y e a r under t h e regency of h i s mother, b u t t h e y were overthrown by P e t e r t h e G r e a t ' s daughter E l i z a b e t h i n 1 7 4 1 , and t h i s i s when E u l e r l e f t f o r B e r l i n . E l i z a b e t h d i e d i n 1 7 6 2 , and h e r s u c c e s s o r P e t e r 111 was soon a f t e r overthrown by Catherine t h e G r e a t , who i n v i t e d E u l e r a g a i n and had t h i n g s w e l l i n hand. I n t h e i n t e r v e n i n g B e r l i n p e r i o d , t h i n g s were made d i f f i c u l t f o r Euler a f t e r 1756, when Maupertuis l e f t ; however, t h a t was t h e s t a r t of t h e seven y e a r war between P r u s s i a and s e v e r a l o t h e r c o u n t r i e s , s o t h a t F r e d e r i c of P r u s s i a h a r d l y had much t i m e i n which t o h u m i l i a t e E u l e r . 'In his second public lecture, Littlewood describes equally shocking things occurring also in Cambridge. (See, for instance, his comments on Forsyth and on Ward, Chapter 1 1 1 section 32.)

19.

The r i s e of mathematics i n P a r i s .

We come t o a second p e r i o d of h i s t o r y i n which t h e important events were due t o mathematicians. They played a v i t a l p a r t i n t h e s u r v i v a l of t h e French Revolution, and i n t h e m i l i t a r y v i c t o r i e s o f Napoleon. The g r e a t e s t of t h e s e mathematicians i s La r a n e: he came t o P a r i s s h o r t l y b e f o r e t h e Revolution, y e a r s as E u l e r ' s s u c c e s s o r i n B e r l i n . H e counted a s a f t +e2+ r I t a l i a n and had t o be n a t u r a l i s e d ; h e a l s o had t h e s p e l l i n g of h i s name changed t o sound l e s s a r i s t o c r a t i c . Joseph Louis de l a Grange (1736-1813), l a t e r known a s Lagrange, was born i n Turin. The family w a s French--an o l d family of Touraine l i k e t h a t of D e s c a r t e s , t o which i t was a l l i e d . However, French n a t i o n a l i t y laws a r e no b e t t e r than t h o s e of o t h e r c o u n t r i e s . The I t a l i a n s a r e most happy eo claim ' L u i g i Lagrange:' I was i n v i t e d t o Turin f o r t h e 150th a n n i v e r s a r y o f h i s d e a t h , A s a m a t t e r of f a c t , Lagrange i s s a i d t o have spoken French v o l u b l y , w i t h a s t r o n g I t a l i a n accent--not

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115

u n f a s h i o n a b l e under Napoleon. Lagrange a t t r i b u t e d h i s mathematical c a r e e r t o two t h i n g s : f i r s t , t h a t h i s f a t h e r , a s u p e r i n t e n d e n t of p u b l i c o f f i c i a l s , l o s t h e a v i l y i n s p e c u l a t i o n s ; and second, t h a t he had R e v e l l i a s High School t e a c h e r , who i n t e r e s t e d him i n geometry, and i n r e a d i n g works, j u s t a v a i l a b l e i n p r i n t , of John and James B e r n o u l l i , Newton, correspondence of John B e r n o u l l i and L e i b n i z , and papers o f Leibniz and E u l e r . The f i r s t o r i g i n a l paper h e r e a d i s s a i d t o have been by t h e astronomer H a l l e y . Sometime between t h e ages of 1 6 and 1 9 , he was appointed t o t e a c h mathematics a t t h e Turin A r t i l l e r y School. A t 18 he began t o correspond w i t h G . C . Fagnano and E u l e r . Why go i n t o d e t a i l s ? Because of l e s s o n s t o b e drawn, Lagrange was p r o f e s s o r a t t h e A r t i l l e r y School a t 1 9 o r e a r l i e r : t h a t makes him a g e n i u s . Newton, a f t e r a v e r y l a t e s t a r t , became p r o f e s s o r a t 2 6 : s u r e l y a genius t o o . E u l e r , although h e l d up by t h e confusion i n R u s s i a , was p r o f e s s o r a t 2 3 . Are we n o t g e t t i n g r a t h e r a l o t of g e n i u s e s ? There w i l l b e o t h e r s , a s we s h a l l see: who s a i d g e n i u s i s r a r e ? Mathematics i n those days seems t o us n o t s o v e r y h a r d : w e e x p l a i n i t t o undergraduates. Perhaps, e x a c t l y a s I s u g g e s t e d i n t h e c a s e of p r e c o c i t y , what we c a l l genius i s sometimes no more than t h e coura e of n o t allowing o n e s e l f t o b e i n t i m i d a t e d by supposed d i i c u t i e s ?

-7T-f-

Another l e s s o n t h a t we observe a g a i n , i s t h e v a l u e of l e a r n i n g t o READ, as Newton had done, and t h e v a l u e of a good t e a c h e r , such a s E u l e r had. F i n a l l y we come t o t h e m i s f o r t u n e t h a t b e f e l l Lagrange's f a t h e r . By a c o i n c i d e n c e , my g r a n d f a t h e r had a s i m i l a r e x p e r i e n c e : my f a t h e r would o t h e r w i s e , as h e t o l d m e h i m s e l f , have been chained t o t h e family banking. I n s t e a d he won a s c h o l a r s h i p t o Cambridge, and ended up as a mathematician. Such m i s f o r t u n e s , t h a t t u r n i n t o b l e s s i n g s i n d i s g u i s e , t y p i f y what happens a l s o i n mathematical r e s e a r c h . Who, i n d e e d , h a s n o t experienced t h e disappointment of f i n d i n g t h a t a b e a u t i f u l theorem, dreamt up o v e r n i g h t , remains unproved, o r even f a l s e ? But t h i s i s when h e f i n d s o u t t h a t mathematics i s deeper and more b e a u t i f u l than a clumsy f i r s t a t t e m p t . Lagrange and h i s s t u d e n t s founded t h e Academy o f T u r i n , and t h i s i s where he f i r s t p u b l i s h e d . One i n i t i a l memoir concerns Maxima and Minima: t h e t o p i c r e c u r s throughout many of h i s r e s e a r c h e s . I n t h e t h e o r y s u b j e c t t o c o n s t r a i n t s , he i s known f o r h i s m u l t i p l i e r s ; they r e a p p e a r i n h i s work on S t a t i c s , Dynamics and t h e Calculus of V a r i a t i o n s . A l l t h i s i s fundam e n t a l , b u t u n f o r t u n a t e l y n o t q u i t e sound. A s i n E u l e r , t h e r e i s a b a s i c confusion i n Lagrange and i n o t h e r mathematicians up t o t h a t p e r i o d : no proper d i s t i n c t i o n between a c o n d i t i o n which i s n e c e s s a r y , and one which i s s u f f i c i e n t . I n t h e Calculus of V a r i a t i o n s t h e n a t u r e of Lagrange m u l t i p l i e r s w a s n o t c l e a r e d up, and s a t i s f a c t o r y p r o o f s found, u n t i l t h e papers of McShane about 1940. One d i f f i c u l t y i s t h a t m u l t i p l i e r s a r e n o t always s i n g l e - v a l u e d : i n S t a t i c s , t h e m u l t i p l i e r s become

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r e a c t i o n s of t h e c o n s t r a i n t s - - i n d e t e r m i n a t e even i n t h e c a s e o f a t a b l e , s t a n d i n g on f o u r l e g s . From t h e s t a r t , t h i s work of Lagrange i s i n s e v e r a l v a r i a b l e s . This means t h a t geometry of a s o r t was beginning t o be taken i n t o account. I n t h i s , t h e development of mathematics p a r t l y resembles t h e much l a t e r development o f economics. The o l d e r economists understood a f u n c t i o n of a s i n g l e v a r i a b l e : s i n c e they were d e a l i n g w i t h f u n c t i o n s of a number of v a r i a b l e s , they kept saying "This i s what happens when such and such a v a r i a b l e i s a l t e r e d , o t h e r t h i n g s being e q u a l . The corresponding ciiange i n t h m n c t i o n , they c a l l e d a tendency. Since d i f f e r e n t v a r i a b l e s were c o n s t a n t l y changing t o g e t h e r , t h e r e would b e a number of such t e n d e n c i e s , and economists used t o v a i n l y a t t e m p t t o d e s c r i b e t h e r e s u l t by waving t h e i r arms. U n f o r t u n a t e l y , i t must be admitted t h a t mathematicians l i k e w i s e have n o t always understood t h e f i n e r p o i n t s regarding s e v e r a l v a r i a b l e s . Here, t o o , Lagrange i s n o t always e n t i r e l y sound. A t y p i c a l d i f f i c u l t y t h a t people l i k e Lagrange were a p t t o overlook was t h e f o l l o w i n g : i t i s p o s s i b l e f o r a f u n c t i o n f ( x , y ) of two v a r i a b l e s t o b e d i s c o n t i n u o u s , even though i t i s continuous on every s t r a i g h t l i n e of t h e x , y p l a n e . (More p r e c i s e l y : even though i t s r e s t r i c t i o n t o a s t r a i g h t l i n e L i s a continuous f u n c t i o n on L , f o r each L . ) L e t C be t h e upper h a l f of a c i r c l e , and suppose t h a t t h e diameter d which determines i t i s t h e segment (0,l) of t h e xa x i s . We denote by A , B t h e upper halves of two e l l i p s e s , f o r which d i s , r e s p e c t i v e l y , t h e minor and t h e major a x i s . I n t h e c r e s c e n t bounded by t h e p a i r of a r c s A , B , we d e f i n e f ( x , y ) t o be 0 on A and B , Fig. 4 t o b e 1 on C except a t i t s e n d s , and t o b e a l i n e a r f u n c t i o n on each v e r t i c a l segment j o i n i n g C t o A o r B . We complete t h e d e f i n i t i o n of f ( x , y ) by making i t 0 o u t s i d e t h e c r e s c e n t , The f u n c t i o n thus d e f i n e d i s c l e a r l y discontinuous a t t h e p o i n t s 0,O and 1,0, s i n c e i t vanishes a t t h e s e p o i n t s and takes t h e v a l u e 1 on t h e a r c C which has t h e s e p o i n t s as e x t r e m i t i e s . However, on any l i n e n o t p e n e t r a t i n g t h e i n t e r i o r of t h e c r e s c e n t , t h e f u n c t i o n i s continuous, s i n c e i t i s i d e n t i c a l l y 0 t h e r e ; while on any l i n e through t h e c r e s c e n t , t h e f u n c t i o n goes smoothly from 0 t o 1 and back a g a i n , a t most twice. Hence f ( x , y ) i s continuous on every s t r a i g h t l i n e , ( I n p a r t i c u l a r , t h i s f u n c t i o n i s continuous i n x, f o r each y , and continuous i n y f o r each x.)

%@I

I n o r d e r r e a l l y t o grasp t h e n a t u r e of f u n c t i o n s of s e v e r a l v a r i a b l e s , we need XX-th century methods. Even i n t h e e a r l y 1 9 0 0 ' ~t h ~ e g r e a t Lebesgue admitted t h a t he had n o t r e a l i s e d t h a t a f u n c t i o n of x , y can b e continuous i n each v a r i a b l e ,

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The New Beginning

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without being continuous i n t h e p a i r . One o f t h e f i r s t a p p l i c a t i o n s of C a n t o r ' s Theory of S e t s of P o i n t s was t h e proof by Ren6 B a i r e t h a t such d i s c o n t i n u i t i e s a r e , i n a s e n s e , " r a r e . 'I By modern s t a n d a r d s , Lagrange made many e r r o r s : t h e s e were, i n f a c t , more s e r i o u s and h a r d e r t o c o r r e c t than t h o s e of E u l e r . Y e t , i n h i s own t i m e , h i s work was c o n s i d e r e d t h e u l t i m a t e i n meticulousness and elegance. He took p r i d e i n t h e f a c t t h a t h i s book on A n a l y t i c a l Mechanics c o n t a i n s n o t a s i n g l e diagram, He was much more conscious than E u l e r , of t h e need f o r mathemat i c a l r i g o u r : i t made him proceed most o b s t i n a t e l y i n t h e wrong d i r e c t i o n , w i t h t h e wrong n o t i o n of f u n c t i o n . I t i s n o t t o b e l i t t l e t h e p a s t , t h a t I have chosen t o c a l l p r e h i s t o r y e v e r y t h i n g p r i o r t o t h e French Revolution and i t s h o r r o r s . The whole s p i r i t of t h e e v o l u t i o n of man h a s been towards g r e a t e r freedom; I speak of t r u e freedom, n o t t h e freedom t o s h o u t and k i l l . The key t o t r u e freedom I have s a i d i s mathematics; and mathematics i t s e l f needs t h e freedom o f wider concepts, t h a t we enjoy today. I n t h i s r e s p e c t , t h e mind of Lagrange was s t i l l r o o t e d i n p r e h i s t o r y . T h i s l e d him t o r e j e c t vehemently t h e i d e a s of F o u r i e r , and t o r e f u s e t o l e t t h e P a r i s Academy p u b l i s h them, a s long a s h e could s t o p i t . H e saw t h a t t h e s e i d e a s would l e a d t o d i f f i c u l t i e s - i n f a c t they d e s t r o y e d h i s whole p o s i t i o n ; b u t they needed t o t a k e r o o t f i r s t . The t r o u b l e was t h a t Lagrange, although h e f e l t uncomfortable about l i m i t i n g p r o c e s s e s , took over from E u l e r t h e b a s i c n o t i o n t h a t a f u n c t i o n i s i n e f f e c t a power s e r i e s , i t s s o - c a l l e d Taylor expansion; h e even went s o f a r a s t o imagine he had "proved" t h i s .

The Ancients thought mainly i n terms of l i n e a r f u n c t i o n s and q u a d r a t i c s ; cubics and h i g h e r polynomials g r a d u a l l y came i n a f t e r t h e Renaissance. By comparison, a power s e r i e s i s a wider concept. However i t s d e f e c t i s brought o u t by an important example (due t o Cauchy), which h a s become of fundamental importance i n Schwartz's theory of d i s t r i b u t i o n s , and which i n c i d e n t a l l y answers what w a s once a f a v o u r i t e q u e s t i o n f o r Ph.D. c a n d i d a t e s i n Physics o r Astronomy, w i t h a minor i n mathematics. ( I f a f u n c t i o n h a s a convergent Taylor expans i o n , does t h e Taylor s e r i e s r e p r e s e n t t h e f u n c t i o n ? The answer i s t h a t i t need n o t . ) I n f a c t t h e f u n c t i o n , v a n i s h i n g a t x = 0 , defined f o r x # 0 by the expression e- 1 / x 2 , h a s such c l o s e c o n t a c t w i t h t h e x - a x i s a t t h e o r i g i n , t h a t a l l i t s d e r i v a t i v e s t h e r e v a n i s h . I t s Taylor s e r i e s f o r t h e neighbourhood of 0 t h u s v a n i s h e s i d e n t i c a l l y . ( I t c e r t a i n l y converges, b u t n o t t o t h e f u n c t i o n . ) I want t o g i v e a l s o a f u r t h e r example, of h i s t o r i c a l importance: i t i l l u s t r a t e s t h e p i t f a l l s o f s e v e r a l v a r i a b l e s ,

b u t does n o t i n t r o d u c e any complicated f u n c t i o n s , only t h i n g s

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that you meet everyday. It is due to H. A. Schwarz (note the different spelling). The function concerned is now defined on a square of the x,y plane, and is constant in y. It is f(x,y)

=

J1-xz (-l<x 0 , and i d e n t i c a l l y 0 f o r x 5 0 , i s wh et h er , as Lagrange b e l i e v e d , e v e r y function possessing a l l i t s derivatives i s a n a l y t i c , i . e . expressible a s a power s e r i e s . *For u s i n t h e XX-th c e n t u r y , it i s a s a p i o n e e r o f f u n c t i o n a l a n a l y s i s and f o r e r u n n e r o f Hamilton, t h a t Legendre s t a n d s o u t w i t h h i s Legendre t r a n s f o r m a t i o n , j u s t a s Laplace s t a n d s o u t w i t h h i s Laplace t r a n s f o r m a t i o n . I n t h i s way, i n a l l f a i r n e s s , Laplace and Legendre can t o day no l o n g e r be ranked below Lagrange, o r below Gauss. You see why I do n o t s u b s c r i b e t o p u t t i n g p e o p l e i n o r d e r of e x c e l l e n c e , e x c e p t i n t h e c o n t e x t o f some v e r y narrow f i e l d .

3They d e s i r e d it, as one d e s i r e s an a d v e n t u r e , a c h a l l e n g e , o r t h e way o u t o f an impasse, when t h e outcome i s s t i l l unknown. I t was, o f c o u r s e , a v e r y d i f f e r e n t s t o r y when t h e y began t o s e e what it r e a l l y meant.

Chapter I THE ROMANTIC PERIOD

20. The Flouting of Arithmetic The times we are mainly interested in begin with the French Revolution. They would seem strange to us if we were not still living in them: we would find it difficult to understand the apparent contradiction berween the noblest of principles and the most barbaric of acts. We might go so far as to echo the words of a victim: "Liberty, what crimes are committed in your name!" Or we might find ourselves quoting the historian and Egyptologist Ernest Renan, whom I have mentioned, and whose portrait suggests benevolence and common sense: one of the few to refer to those heroic enough to commit crimes for the sake of noble principles, as "consciences troubles, incapables de distinguer leurs grossiers appetits de passions que leur fr6ncSsi.e leur reprhsentait saintes." However, from the mathematical point of view, we should l o o k at these things dispassionately, as if it were a matter of saying, on a picnic, that fingers were made before forks. The persistence of barbarism is basically of the same nature, and so is the law of conservation of ignorance that I spoke of earlier. These are instances of the fact that historical events depend, not merely like many natural phenomena on their immediate past, but on the whole past from -m. onwards. It is for this very reason, that ediucation is so necessary; this is also why we had to have a long introduction, dealing with earlier times. Dependence on the whole past involves a more complex mathematical apparatus than is used, say in mechanics; we need concepts developed in functional analysis, and especially in stochastic processes in statistics. Non-mathematical readers may relax: I shall not go into these theories. The little I shall say of the mathematics of history will be no more than common sense. Here I want to stress rather the connection with education, a vital part of the formation of mathematicians. The story of the XIX-th and XX-th centuries is largely that of vast, and only partially successful, experiments in mass-education. The success or failure of these experiments is to be measured by the extent to which they have eradicated barbarism and ignorance. One great difficulty, in this connection, is the phenomenon of the bends, that I mentioned earlier. Another, that I referred to briefly, consists in the removal of traditional safeguards, such as the moderating influence of the priesthood and of 124

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a s t r o l o g y , which c e a s e t o f u n c t i o n when people begin t o t h i n k f o r themselves. A f u r t h e r t r a d i t i o n a l s a f e g u a r d , t h e concept of honour, i s e q u a l l y endangered, a s I s h a l l e x p l a i n i n conn e c t i o n w i t h t h e War of 1 8 7 0 and i t s consequences. Humanity has l e a r n t t o l i v e w i t h c e r t a i n checks and b a l a n c e s , and when t h e y a r e d i s t u r b e d , t h e r e may be u n p l e a s a n t consequences. The bends a r e a s p e c i a l problem: e d u c a t i o n provides g r e a t e r freedom. Even a bad education provides t h e i l l u s i o n of freedom. I n e i t h e r c a s e , people imagine t h a t t h e y can now suddenly o b t a i n anything they want. This i s when, a s I exp l a i n e d , suppressed d e s i r e s , d a t i n g from t h e most p r i m i t i v e times, suddenly r e a p p e a r : barbarism, t h a t e d u c a t i o n was t o have e r a d i c a t e d , comes r i g h t back. The bends a r e one of t h e g r e a t f o r c e s of h i s t o r y , and w e must l e a r n t o c o n t r o l them, j u s t a s we now do i n t h e c a s e of d i v e r s . They were c l e a r l y n o t c o n t r o l l e d a t t h e t i m e of t h e French Revolution: t h e y a r e s t i l l f a r from c o n t r o l l e d t o d a y . W e may wonder what our f i v e mathematicians thought of i t a l l . I presume t h a t l i k e most educated persons they a t f i r s t hoped f o r a r e v o l u t i o n , o r something s i m i l a r . Normally i n such m a t t e r s I would n o t , a s a mathematician, wish t o g i v e t h e impression t h a t people i n my p r o f e s s i o n n e c e s s a r i l y see more c l e a r l y , and t h i n k more l o g i c a l l y , t h a n t h e i r contemporaries. However I am s u r e t h a t t h e y must have had some r e s e r v a t i o n s , when t h e y saw t h e a b s u r d i t i e s around them begin t o v i o l a t e elementary a r i t h m e t i c . Mathematicians do n o t n e c e s s a r i l y e x c e l i n a r i t h m e t i c : but they a r e not t o t a l l y b l i n d .

We saw t h a t i n t h e s o - c a l l e d age of r e a s o n i t was n o t only i n England a f t e r Newton, t h a t mathematics s u f f e r e d a slump. The n e g l e c t of a r i t h m e t i c was even more n o t i c e a b l e , and t h e South Sea Bubble was n o t t h e only i n s t a n c e of t h i s , by a n a t i o n i n t o x i c a t e d by i t s d e l u s i o n of r i c h e s f o r a l l . The two t h i n g s go t o g e t h e r : t h e atmosphere of s u p e r f i c i a l i t y and w i s h f u l t h i n k i n g t h a t a l l o w s a r i t h m e t i c t o be i g n o r e d , i s h a r d l y one i n which a n a t i o n can expect t o form g r e a t mathematicians. I n France, a c o u n t r y t h a t had been r u n w i t h an ever i n c r e a s i n g d i s r e g a r d of a r i t h m e t i c , c h a r a c t e r i s t i c of r e v o l u t i o n s i n t h e making o r i n b e i n g , i t i s h a r d l y s u r p r i s i n g t h a t p r i o r t o t h e r e t u r n of Lagrange t o t h e land of h i s f o r e f a t h e r s t h e r e had been no r e a l mathematician of n o t e f o r 1 0 0 y e a r s . Lagrange must have experienced some r a t h e r mixed f e e l i n g s on h i s r e t u r n . The mere f a c t t h a t he d i d r e t u r n , s u g g e s t s t h a t , l i k e many people who have mainly l i v e d abroad, he was r a t h e r p a t r i o t i c . Moreover he was s p e c i a l l y honoured: who would n o t b e a p a t r i o t , when on h i s long longed-for r e t u r n t o h i s beloved France, honours a r e showered upon him by a l l - - t h e King and Queen, t h e Revolution, Napoleon? Yes, b u t f i r s t he must be n a t u r a l i s e d . H e had been born i n t h e x c h y o t Savoy, an independent m u l t i l i n g u a l s t a t e , l i k e Luxembourg, Belgium, S w i t z e r l a n d , today. The French p a r t of t h e duchy i s now i n France, t h e r e s t i n I t a l y . This means t h a t today t h e s u b j e c t s of t h e former dukes can count e i t h e r a s French o r a s I t a l i a n : L o r i a counts a s I t a l i a n Lagrange's p u p i l d e Foncenet, who was born a t Thonon on Lake L h a n The s a i d p u p i l , on t h e s t r e n g t h of a memoir

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i n h i s name presumably w r i t t e n by h i s t e a c h e r , a t t a i n e d i m p o r t a n t p o s i t i o n s i n t h e army and navy, u n t i l d i s g r a c e d f o r s u r r e n d e r i n g V i l l a f r a n c a t o French r e v o l u t i o n a r y f o r c e s i n 1 7 9 2 . The d i s g r a c e was d o u b t l e s s a s unmerited as t h e whole c a r e e r . With such p u p i l s , we can w e l l imagine t h a t Lagrange was n o t happy -- h e was a c t u a l l y anxious t o l e a v e T u r i n a t any c o s t , according t o a l e t t e r i n h i s c o l l e c t e d works. However P a r i s was n o t ready f o r him u n t i l a f t e r h i s long appointment i n B e r l i n , and t h e n , a s I s a i d , P a r i s welcomed him - - provided he was f i r s t n a t u r a l i s e d . He was even given an apartment a t t h e Louvre. The n e x t t h i n g w e hear i s t h a t he i s a f t e r a l l t o be e x p e l l e d a s a f o r e i g n e r , i n v i r t u e of a law passed i n 1 7 9 3 , it: i s only thanks t o L a v o i s i e r t h a t an exception t o t h a t law i s made i n h i s f a v o u r . Imagine then how p a t r i o t i c a l l y French Lagrange must have f e l t , when t h a t same L a v o i s i e r was g u i l l o t i n e d ! Does France expect of i t s g r e a t ones t h e i n s e n s i b i l i t y and t h i c k s k i n of a r h i n o c e r o s ? N a t i o n a l i t y , p a t r i o t i s m , enthusiasm f o r noble p r i n c i p l e s , a r e not m a t t e r s of c o l d geography: they come from t h e h e a r t . But t h e r e a r e times when t h e h e a r t no longer c a r e s f o r honours . . . The f e e l i n g s of Lagrange were t h o s e of many Frenchmen. I t i s a d e l u s i o n of h i s t o r i a n s t h a t a r e v o l u t i o n i s a popular movement, o r a t l e a s t t h a t i t remains one f o r v e r y long. The American War of Independence was a war, n o t a r e v o l u t i o n : i t was popular -- n o t n e c e s s a r i l y w i t h a l l c o l o n i s t s , b u t w i t h q u i t e a few of t h e i r many s u p p o r t e r s i n B r i t a i n i t s e l f . However i n Cromwell's r e v o l u t i o n , Parliament was c o n s t a n t l y f l o u t e d . I n L e n i n ' s , t h e p e o p l e ' s r e p r e s e n t a t i v e s were simply s e n t home. I n t h e French r e v o l u t i o n , a f t e r 8 y e a r s of T e r r o r during which 85% of t h o s e g u i l l o t i n e d were n e i t h e r n o b l e s nor p r i e s t s , an e l e c t i o n gave a m a j o r i t y t o monarchists. THIS ELECTION WAS DECLARED I N V A L I D . Even American independence had i t s a b s u r d i t i e s , a s most wars have: on one s i d e , Hessians were made easy t a r g e t s by b r i g h t r e d fancy d r e s s , and f o r a King n o t t h e i r own; on t h e o t h e r s i d e t h e f i g h t was a g a i n s t some t i n y u n j u s t t a x , which has s i n c e been r e p l a c e d by t a x e s w i t h r e p r e s e n t a t i o n , t h a t a r e g e n e r a l l y agreed t o be t h e most i n e q u i t a b l e ever devised - - American P r e s i d e n t s , even when n o t a s s a s s i n a t e d , i n v a r i a b l y f a i l t o reform them. Of c o u r s e a r e v o l u t i o n i s n o t t h e only occasion i n which a r i t h m e t i c o r common s e n s e a r e f l o u t e d . It i s a l l a q u e s t i o n of degree: t h e bends can appear i n a m i l d e r form. For i n s t a n c e , t h e s t o r y of King Canute can r e p e a t i t s e l f : i t i s then i n v a r i a b l y proclaimed a g r e a t 'moral v i c t o r y ' f o r t h e powers t h a t be -- ONLY AN INFINITESIMAL PROPORTION OF THE WATERS OF THE GLOBE DARED DISOBEY THE ROYAL COMMAND. T h i s o f f i c i a l s t u p i d i t y i s a r e c u r r e n t B r i t i s h i l l n e s s , n o t t o t a l l y unknown elsewhere a s w e l l . I t happened t o be p a r t i c u l a r l y marked i n t h e XVIII-th c e n t u r y , i n a s o c i e t y engaged i n hunting and b e t t i n g . I c o n s i d e r i t lucky t h a t i t l e d t o American independence: without t h i s , Europe would n o t have survived our XX-th c e n t u r y wars. However i n France t h e r e was supposed t o be ' r e a s o n ' , t h e r e was supposed t o be l o g i c . I t i s t h e r e f o r e n e c e s s a r y t o go back a l i t t l e i n French h i s t o r y .

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The r e i g n of Louis X I V had begun w i t h two c i v i l w a r s , t h e ' F r o n d e s ' . They had somehow been patched u p , and i n t h e end t h e government was a l l t h e s t r o n g e r . Mazarin was c l e v e r enough f o r t h a t -- he could make r i n g s round anyone e l s e . I n p a r t i c u l a r , t h e people of P a r i s , who had more t h a n once shown t h e i r s t r e n g t h and t h e i r f e r o c i t y i n t h e p r e v i o u s c e n t u r i e s , were thoroughly subdued. What's more, Mazarin was t h r i f t y : when he d i e d , t h e S t a t e c o f f e r s w e r e f u l l . Louis X I V was a b l e t o u n i f y France under h i s a b s o l u t e r u l e , by t h e simple expedie n t of making everybody dependent on h i s money: t h e Huguenots and t h e J a n s e n i s t s were e x c e p t i o n s , and he g o t r i d of them. The n o b l e s , who a s h o r t time b e f o r e had stood up t o Mazarin, turned i n t o domestic p e t s -- g l o r i f i e d lackeys and l o r d s of t h e bedchamber - - , when he p a i d them pensions and r e q u i r e d them t o d r e s s up i n fancy c l o t h e s . The r e s t of t h e p o p u l a t i o n l i v e d on t h e p r o f i t s of providing t h e fancy c l o t h e s , t o g e t h e r w i t h t h e c a r r i a g e s and h o r s e s , t h e p a r k s and p a l a c e s , t h a t went w i t h such e x a l t e d p o s i t i o n s . Banks and money-lenders were e s p e c i a l l y prosperous, and c o n t r i b u t e d a l l t h e more t o t h e dependence of t h e p e t s on t h e i r p e n s i o n s . The system worked, everybody was happy, b u t i t c o s t money.

I t worked w h i l e C o l b e r t was i n charge of f i n a n c e : i t worked l e s s w e l l when a p p e t i t e s had grown, e s p e c i a l l y i n t h e subsequent r e i g n s , and when t h e people of P a r i s began t o show once more t h a t they had minds of t h e i r own. A f t e r England and France had been a t w a r 1 5 o u t of 23 y e a r s between 1740 and w i t h one a n o t h e r and w i t h one h a l f o r o t h e r of 1763 Europe - - , and t h e i r P a r l i a m e n t s had f a i l e d t o f o o t t h e b i l l , England passed on some of i t s d e b t t o t h e American c o l o n i a l s , whose wicked r e j e c t i o n of t h e generous g i f t we a l l know. F r a n c e ' s e q u a l l y smart s o l u t i o n was t o farm o u t t a x e s f o r y e a r s ahead: t h i s was bound t o end i n bankruptcy. The time came when everybody knew i t : nobody c a r e d -- "AprBs nous, l e d6lugeI" What f i n a l l y wrecked French f i n a n c e s was t h e American War of Independence -- any enemy of B r i t a i n deserved a l l t h e h e l p France could squeeze o u t of f u t u r e t a x - y e a r s . Things became s o bad t h a t t h e S w i s s banker Necker was brought i n t o f a l s i f y accounts t o . t h e t u n e of 50 m i l l i o n l i v r e s i n 1781, by d e c l a r i n g a pretended s u r p l u s . He was a b l e t o borrow: i n t h e long r u n , t h i s merely meant t h a t p u b l i c debt payments r o s e by 1789 t o 300 m i l l i o n l i v r e s -- t r i p l e t h e amount p r i o r t o t h e American r e v o l u t i o n .

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A s t o t h e people being happy, t h i s i s of c o u r s e a r e l a t i v e term! Under C o l b e r t t h e y were n o t p r e c i s e l y i d l e : C o l b e r t was f a r from i d l e himself - - h i s working hours compare w i t h t h o s e of Napoleon, who i s s a i d t o have worked 18 hours a day. Under Mazarin, I s a i d t h a t t h e people of P a r i s were subdued; b u t under C o l b e r t , t h e people of France were p u t t o work. I t was a system of r e g i m e n t a t i o n unequalled even by t o d a y ' s comunism: a t o t a l l y s u c c e s s f u l S t a t e s o c i a l i s m , h a t e d by a l l , which brought t o France an enforced p r o s p e r i t y and a dominant economic p o s i t i o n . However, o n l y two y e a r s a f t e r C o l b e r t d i e d , many of t h e b e s t French craftsmen were f o r c e d t o f l e e France, because of t h e r e v o c a t i o n of t h e E d i c t of Nantes:

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France l o s t 2 5 0 , 0 0 0 P r o t e s t a n t c i t i z e n s . I n a d d i t i o n , t h e system lacked t h e automatic i n c e n t i v e t o p r o g r e s s , which i s t o some e x t e n t a v a i l a b l e under f r e e e n t e r p r i s e : i t needed, f o r i t s continued s u c c e s s , S t a t e - s u p p o r t e d s c i e n t i f i c and t e c h n i c a l r e s e a r c h . Colbert himself understood t h i s v e r y w e l l , and a l s o t h e importance of supporting t h e a r t s f o r t h e same purpose. H e was t h e one who founded t h e Acadgmie d e s S c i e n c e s , and who brought Huygens t o P a r i s ; he a l s o founded t h e Acadgmie des I n s c r i p t i o n s e t B e l l e s L e t t r e s . But t o keep t h e system going, a succession of C o l b e r t s would have been needed: u n f o r t u n a t e l y -- f o r t u n a t e l y , some may s a y - - such men a r e r a r e . I n any c a s e , t h e system was n o t designed f o r t h e purpose of c o n s t a n t wars, such a s France indulged i n . A s t o what a c t u a l l y happened, I may mention t h a t i n t h e 50 y e a r s s t a r t i n g i n 1730 t h e c o s t of l i v i n g doubled; but wages d i d n o t i n c r e a s e . Even s o , people i n France were probably b e t t e r o f € t h a n i n most of t h e r e s t of Europe. They had seen b e t t e r d a y s , b u t how many r e main t o remember t h a t , a f t e r 50 y e a r s ? People do n o t remember a c o s t of l i v i n g index: t h e y remember h a t r e d ; t h e y remember r e g i m e n t a t i o n ; t h e y remember being subdued. P a r i s remembered a t i m e when i t had held i t s own, and been f e a r e d ; and s t i l l f u r t h e r back i n t h e c e n t u r i e s , a time when i t had massacred t h o s e i t chose t o h a t e -- t h e Huguenots i n S t Bartholomew's n i g h t , t h e Armagnacs i n t h e e a r l y XV-th c e n t u r y . This i s how t h e d i s t a n t p a s t most e a s i l y c o n t i n u e s t o i n f l u e n c e c u r r e n t e v e n t s ; and how t h e p r i m i t i v e u r g e s keep r e t u r n i n g . These t h i n g s a r e held i n check by f e a r , by a s t r o n g government, by e x t e r n a l p r e s s u r e . They come back a l l t h e s t r o n g e r and more overwhelming, when t h e e x t e r n a l p r e s s u r e i s removed. This i s t h e phenomenon of t h e bends. It i s what we have t o l e a r n t o c o n t r o l . It i s t h e c e n t r a l problem of t h e mathematics of h i s t o r y . There can be l i t t l e doubt t h a t t h e s o l u t i o n , t h e c u r e , l i e s i n e d u c a t i o n , i n enlightenment. However i n t h e XVIII-th c e n t u r y t h e p r i n c i p l e s of t h e ' E n l i g h t e n ed Age' were q u i t e t h e o t h e r way: t h e y were t o t a l l y a t v a r i a n c e w i t h t h e e x i s t i n g s t a t e of a f f a i r s . They proclaimed t h e e q u a l i t y of man. They accentuated t h e d i s c o n t e n t among t h e poor; and they undermined t h e a u t h o r i t y of t h e government, j u s t when t h e c y n i c a l despotism of Louis XV had ended and t h e t h r o n e had passed t o t h e much more sympathetic Louis X V I . I n any c a s e , i t can be w e l l understood t h a t a government known t o be heading f o r bankruptcy w i l l n o t command e x c e s s i v e r e s p e c t . The l a s t straw, what r e a l l y crumbled t h e r e s p e c t f o r r o y a l t y , was t h e a f f a i r of t h e diamond necklace: t h e p r i n c e - c a r d i n a l de Rohan was t r i c k e d i n t o g i v i n g a diamond necklace t o a woman he believed t o be t h e Queen. When t h e p u b l i c l e a r n t t h a t a p r i n c e - c a r d i n a l thought t h e Queen could b e bought and seduced, t h e d i s c r e d i t t o t h e monarchy became immense. Why t h i s d i s c r i m i n a t i o n a g a i n s t a woman? Our age f i n d s t h i s d i f f i c u l t t o understand: t h e amorous a f f a i r s of Kings had long been a byword, and were accepted a s a m a t t e r of c o u r s e . Those of Louis X I V had s e t t h e f a s h i o n f o r t h e whole n o b i l i t y , and had c o n t r i b u t e d , t h e r e f o r e , t o i t s expenses, and d o u b t l e s s t o t h e u l t i m a t e bankruptcy; but of course h i s l i a i s o n w i t h Mme d e

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Maintenon, t h e r e p e n t a n t grand-daughter of t h e most r u t h l e s s of m i l i t a n t r e f o r m e r s , had done incomparably more harm by b r i n g ing about t h e r e v o c a t i o n of t h e E d i c t of Nantes, and by causi n g t h e b e s t c r a f t s m e n , t h e Huguenots, t o f l e e t h e c o u n t r y . However t h e French Revolution was h a r d l y t h e only o c c a s i o n when t h e long arm of a d i s t a n t c a u s a l i t y has been given f a n a t i c a l s t r e n g t h by a sudden s e l f - r i g h t e o u s demand f o r m o r a l i t y . A t any r a t e , t h e r e v o l u t i o n came, and i n e v i t a b l y w i t h i t , t h e bends. I t was freedom of a s o r t : w i t h g r e a t enthusiasm, a few c r i m i n a l s w e r e f r e e d from t h e B a s t i l l e . Then f o r 8 y e a r s of T e r r o r , masses of innocent people w e r e executed i n t h e i r s t e a d . A s f o r t h e o t h e r s , t h e o r d i n a r y people lucky enough t o be i n s i g n i f i c a n t , t h e ones now enjoying t h e g l o r i o u s ' L i b e r t y , E q u a l i t y , F r a t e r n i t y and t o whom S a i n t J u s t e , a s mouthpiece of t h e Revolution, promised t h e r e would be n e i t h e r r i c h nor p o o r , they merely l o s t t h e i r l i v e l i h o o d . I n May 1 7 9 4 , S o c i a l Secu r i t y was voted i n t o law: i t was a s v a l u e l e s s a s t h e paper currency - - t h e a s s i g n a t was w o r t h l e s s by t h e end of 1 7 9 5 , t h e mandat, i t s s u c c e s s o r , dropped t o 1%i n 6 months. Without wicked a r i s t o c r a t s t o supply w i t h t a p e s t r i e s and l a c e , and w i t h c a r r i a g e s and h o r s e s , t h e people w e r e f r e e - - t o s t a r v e . . . I n t h i s nightmare of misery and d e a t h , t h e a r c h - c r i m i n a l s are r e p u t e d t o have been poor Louis X V I and h i s Marie-Antoinette. L e t me t e l l you about t h e s e a r c h - c r i m i n a l s -- babes i n t h e wood who loved France. H e would have been happy a s a carpent e r -- i t was h i s hobby. H e d i d n ' t have 14 i l l e g i t i m a t e c h i l d r e n , l i k e Charles I1 of England. On h i s wedding n i g h t , n e i t h e r he nor Marie-Antoinette knew what t o do: h i s h i g h l y experienced g r a n d f a t h e r had shown him pornographic p a i n t i n g s , but they j u s t made him s i c k . A s f o r t h e Queen, of whom t h e people were so ready t o b e l i e v e t h e w o r s t , and f o r whom any nobleman would have given h i s l i f e , s h e loved t h e c o u n t r y s i d e and simple t h i n g s : she dreamt of being a shepherdess . . , e l e g a n t l y d r e s s e d , of c o u r s e . I am no King's champion; n e i t h e r do I b e l i e v e i n t h e d i v i n e r i g h t of Kings. But I do n o t condemn a human being f o r what a whole n a t i o n d e s i r e d him, and t r a i n e d him, t o b e . The B r i t a n n i c a s a y s Louis X V I was s t u p i d : he should have p u t hims e l f a t t h e head of t h e Revolution. Napoleon s a i d Louis X V I was s t u p i d : w i t h a few cannon s h o t s , he could have broken t h e Revolution. Louis d i d n e i t h e r : he was no Mazarin, t a u g h t t o b e t r a y o r k i l l s u b j e c t s under h i s p r o t e c t i o n . But what of wearing f i n e c l o t h e s and dancing t h e minuet, when t h e poor were s h o r t of bread? What of t h e Queen d e c l a r i n g "Let them e a t cake"? I t was a crime: so w e a r e t o l d by t h e e n l i g h t e n e d r e v o l u t i o n , whose generous paper-money gave no work t o p a s t r y cooks, and whose S o c i a l S e c u r i t y made France a thousand times more bankrupt. I am s o r r y t o say t h a t i f t h e King and Queen, and a l l t h e n o b l e s , had suddenly t a k e n t o buying t w i c e a s many f i n e c l o t h e s and d e l e c t a b l e p a s t r i e s , t h e r e might have been work f o r a l l : b u t t h i s , of c o u r s e , would have been a crime. You and I may argue t h a t you do n o t save a d i v e r by r u s h i n g him o u t of deep w a t e r , and t h a t you do n o t save a c a r , heading f o r a p r e c i p i c e , by p r e s s i n g t h e a c c e l e r a t o r ; b u t t h e e n l i g h t e n e d r e v o l u t i o n s know b e t t e r .

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TheKing was convicted of t h a t most heinous and outrageous of crimes: a t t e m p t i n g t o p u t on t h e b r a k e s . The Queen was convict e d o f s t a n d i n g by h e r husband. They were executed: s o were thousands of o t h e r s , i n c l u d i n g a l l t h e K i n g ' s former t a x - c o l l e c t o r s , L a v o i s i e r among them. The t a x - c o l l e c t o r s had t r i e d t o r e f o r m t a x e s , t h i s was a crime t o o . Meanwhile France dropped fromone bankruptcy t o a n o t h e r . The King and Queen had t r i e d t o e s c a p e i n 1 7 9 1 , and were brought back: troops l o y a l t o them, s e n t t o e s c o r t them out of France, had f a i l e d i n t h e i r m i s s i o n , delayed one crucial. hour, i t i s s a i d , by an impassable c a n a l n o t i n d i c a t e d on t h e i r map. When governments abroad l e a r n t t h a t t h e babes i n t h e wood were now p r i s o n e r s , , they became alarmed. A u s t r i a and P r u s s i a s e n t what they thought a t a c t f u l n o t e , t h e D e c l a r a t i o n of P i l n i t z , a p p e a l i n g t o o t h e r European r u l e r s t o seek w i t h them t h e b e s t way of enabling t h e King of F r a n c e t o c o n s o l i d a t e i n p e r f e c t l i b e r t y t h e b a s i s of monarchicalgovernment e q u a l l y s u i t a b l e t o t h e r i g h t s of s o v e r e i g n s a n d t o t h e w e l f a r e of t h e French n a t i o n . I n France t h i s was i n t e r p r e t e d a s a t h r e a t . Things looked very bad: on t h e t o p of a l l t h e i n t e r n a l t r o u b l e s , t h e r e was t h e danger of f o r e i g n invasion, I t w a s t h e mathematicians who saved t h e country.

21. The Founding of t h e Ecole Polytechnique Our f i v e mathematicians had f o r t u n a t e l y escaped e x e c u t i o n . I n t h e m i d s t of p a s s i o n s and madness, t h e r u l e of reason was s t i l l p a i d l i p - s e r v i c e t o : mathematicians were s l i g h t l y s a f e r than o t h e r s , even i f e l e c t e d t o o f f i c e . It was a c t u a l l y a n e a r t h i n g : t h e r e v o l u t i o n a r y Marat h a t e d academicians, because h e (Similarly H i t l e r hadbeen r e j e c t e d by t h e Academy of Sciences. h a t e d p r o f e s s o r s , because he had been denied e n t r a n c e a s a s t u dent i n Vienna. 1 F o r t u n a t e l y , Marat was a s s a s s i n a t e d , b u t even s o h i s vengeance f i n a l l y caught up w i t h L a v o i s i e r , although t h i s w a s then s i x months a f t e r M a r a t ' s own d e a t h . Another of M a r a t ' s v i c t i m s was t h e Secr6tai.re Perpgtuel of t h e Academy of Sciences , Condorcet ( 1 7 4 3 - 1 7 9 4 ) , who was spared execution only by t a k i n g poison. Condorcet's claim t o fame i s r a t h e r l i m i t e d : wewould c l a s s i f y him today a s a s o c i a l s c i e n t i s t , b u t he then c o u n t e d a s a mathematician-- he had made a p p l i c a t i o n s of t h e most crude p r o b a b i l i t y t o e l e c t i o n r e s u l t s . H e was a c t u a l l y a Marquis, and member of t h e Gironde, and he had voted a g a i n s t t h e d e a t h p e n a l t y f o r Louis X V I , b u t b a s i c a l l y he was s a c r i f i c e d , l i k e L a v o i s i e r , t o t h e ghost of Marat. S t i l l he counted a s a mathematician,and h i s d e a t h was a blow t o t h e R e v o l u t i o n ' s supposed enlightenment. Therefore t h e N a t i o n a l Convention r e v e r s e d i t s e l f , and made a h e r o of him: i t s u b s c r i b e d t o 3000 c o p i e s of Condorcet's Esquisse d ' u n Tableau h i s t o r i q u e des Progre'sde 1 ' E s p r i t humain, a work n o t e n t i r e l y of high s t a n d a r d o n popular s c i e n c e . It might have been b e t t e r t o have l e t theMarquis l i v e . Even s o , h e was what some people would now c a l l a n " a r i s t o c r a t of i n t e l l e c t , " and i t i s remarkable t h a t t h e F r e n c h Revolution should have honoured such persons a t a l l . T h e t r u t h i s t h a t t h e l e a d e r s were most anxious t o prove t h a t t h e i r s r e a l l y was t h e r u l e of r e a s o n , and n o t t h a t of t e r r o r .

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For i n s t a n c e , they involved people l i k e Lagrange, Laplace, Legendre, i n s p e c i a l p r o j e c t s , such a s t h e i n t r o d u c t i o n of t h e m e t r i c system, w h i l e Monge and Carnot became a c t i v e i n p u b l i c affairs. Monge was a c t u a l l y M i n i s t e r f o r t h e Navy and Colonies i n 1 7 9 2 1 7 9 3 , and he was t h e one who f i r s t had occasion t o p r a i s e t h e m i l i t a r y s k i l l of Napoleon Bonaparte i n r e t a k i n g Toulon from t h e B r i t i s h ; he became a f r i e n d of Napoleon, who never f o r g o t anyone who had helped him. However i t was through Carnot, h i s former p u p i l , t h a t Monge was of g r e a t e s t s e r v i c e t o t h e Revol u t i o n . Carnot was e l e c t e d t o t h e N a t i o n a l Assembly a s r e p r e s e n t a t i v e of t h e Pas-de-Calais i n 1 7 9 1 , b u t soon turned o u t t o b e t o o v a l u a b l e t o a t t e n d many meetings. H e was an i n s p i r i n g f i g u r e , h i s f a m i l y background was i n law, and he had himself f r e q u e n t e d l i t e r a r y c i r c l e s ; he was a l s o one of t h e few army o f f i c e r s who had n o t l e f t France. He was t h e r e f o r e assigned t o a number of m i s s i o n s , e i t h e r i n h i s c a p a c i t y a s an o f f i c e r , o r a s a member of t h e d i p l o m a t i c and p u b l i c e d u c a t i o n committees. One may wonder t h a t Carnot and Monge should have allowed thems e l v e s t o have any s i g n i f i c a n t p a r t i n t h e R e v o l u t i o n , a f t e r i t had shown t h a t a noble d i s r e g a r d f o r a r i t h m e t i c and common sense i s f a r from being a p u r e l y Royal P r e r o g a t i v e . I f we ask such q u e s t i o n s , we must s u r e l y t a k e f o r g r a n t e d t h a t t h e r e a r e no s i m i l a r ones about our own t i m e s , and t h a t t h e e l e c t e d o f f i c i a l s of today a r e so s u p e r i o r t h a t t h e y do n o t , f o r i n s t a n c e , solemnly d e c l a r e , t h e day b e f o r e d e v a l u a t i o n , t h a t t h e B r i t i s h pound w i l l never be devalued. But w e must n o t f o r g e t t h e tremendous enthusiasm w i t h which t h e Revolution began, and w i t h which France g r e e t e d t h e D e c l a r a t i o n of t h e Rights of Man. People from a l l walks of l i f e suddenly had a c a u s e , t o which t o d e d i c a t e t h e i r l i f e . Even t h o s e who d i e z i n t h e e r r o r b e l i e v e d t o t h e end i n t h e famous D e c l a r a t i o n . Their l a s t moments were l i k e t h o s e of t h e C h r i s t i a n m a r t y r s , n o b l e r t h a n what t h e r e s t of u s a c h i e v e i n a l i f e t i m e . They had n o t l o s t hope t h a t i n t h e end a t r u e r humanity would p r e v a i l . They remembered t h e enthusiasm w i t h which t h e p r i v i l e g ed c l a s s e s had g i v e n up t h e i r s p e c i a l r i g h t s , and t h e q u i x o t i c g e s t u r e of t h e C o n s t i t u e n t Assembly, i n making i t s own members i n e l i g i b l e f o r t h e N a t i o n a l Convention t h a t was t o succeed i t . There were no d u t i e s i n t h e g r e a t D e c l a r a t i o n , t o b a l a n c e t h e r i g h t s i t proclaimed: t h i s was i t s weakness, and y e t a t t i m e s , u n w r i t t e n laws can be more s c r u p u l o u s l y observed t h a n w r i t t e n ones -- a Frenchman's d u t y was t o devote h i s l i f e , i f necess a r y , t o support t h e g r e a t D e c l a r a t i o n . Carnot and Monge were among t h o s e who took t h i s view, and soon, as w e s h a l l s e e , t h e i r help was badly needed. On t h e d i p l o m a t i c f r o n t , rel a t i o n s w i t h f o r e i g n c o u n t r i e s d e t e r i o r a t e d r a p i d l y , and France d e c l a r e d war on A u s t r i a and i t s a l l i e s i n A p r i l 1792. A f t e r t h e a b o r t i v e escape-attempt by t h e King and Queen, t h e N a t i o n a l Assembly had s t a r t e d t o r e o r g a n i s e t h e army, by adding 1 0 0 , 0 0 0 v o l u n t e e r s from t h e N a t i o n a l Guard, and by t r a i n i n g s u i t a b l e o f f i c e r s , of whom t h e r e was a s h o r t a g e , s i n c e t h e y had formerly

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belonged t o t h e upper c l a s s e s . Carnot was kept v e r y busy, f i r s t w i t h t h e Army of t h e Rhine, t h e n i n t h e Pyrenees t o o r g a n i s e defense a g a i n s t a p o s s i b l e a t t a c k by S p a i n , then i n t h e North and t h e Pas-de-Calais.

A t Valmy, on t h e 3 0 - t h June 1 7 9 2 , t h e French were lucky t o reform t h e i r l i n e a s i t was about t o b r e a k , and t o hold on u n t i l r a i n made t h e enemy d e c i d e t o c e a s e t h e a t t a c k s . There was a t e n day s t a l e m a t e , t h e n t h e enemy withdrew, a s t h e weather was making l i n e s of communication d i f f i c u l t , and illn e s s was t a k i n g i t s t o l l . I t had been touch and g o , and now f o r some months t h e f i g h t i n g had i t s ups and downs. The French marched i n t o t h e N e t h e r l a n d s , but were badly d e f e a t e d t h e r e t h e following March, and t h e i r Commander, General Dumouriez, d e f e c t e d w i t h h i s whole army, a s he was being r e l i e v e d of h i s command. I t had been h i s i n t e n t i o n f i r s t t o b e a t t h e A u s t r i a n s , and then t o march on P a r i s and r u l e i n t h e K i n g ' s name. The K i n g ' s e x e c u t i o n , and h i s own d e f e a t , put a s t o p t o such p l a n s . By t h e n France had d e c l a r e d war a l s o on Great B r i t a i n and Holland, who were shocked by t h e King's e x e c u t i o n . War was f u r t h e r d e c l a r e d on S p a i n , so t h a t France was now faced by a formidable c o a l i t i o n , j u s t when i t had l o s t a whole army. I n t h e d a r k e s t days of t h e Revolution, Carnot was brought i n t o t h e Revolutionary Government, t h e Committee f o r General Def e n s e , which soon a f t e r became t h e Committee of P u b l i c S a f e t y . On t h e 9 - t h March 1793, he had t h e Committee send into t h e v a r i o u s p a r t s of t h e country 82 r e p r e s e n t a t i v e s , t o e x p e d i t e t h e u r g e n t c o n s c r i p t i o n of 300,000 men. The proclamation went I t a l s o appealed t o s c i e n t i s t s o u t "La p a t r i e e s t en danger". and mathematicians t o a s s i s t w i t h t e c h n i c a l m a t t e r s , w i t h t h e supply of arms and gunpowder, and w i t h anything e l s e w i t h i n t h e i r competency. One of t h o s e who responded most e f f e c t i v e l y was Monge. He supervised foundry o p e r a t i o n s , and wrote a number of f a c t o r y handbooks: "Advice t o I r o n Workers i n S t e e l " , "Description of t h e A r t of Manufacturing Cannon", . . . A t t h i s c r i t i c a l time, Carnot was v e r y l i t t l e i n P a r i s : i n m i l i t a r y m a t t e r s , he was l e f t v i r t u a l l y i n c h a r g e ; i n e v e r y t h i n g e l s e , h i s c o n s t a n t absences prevented him from having much s a y . He was i n t h e Pyrenees from September 1792 t o January 1793, t h e r e by missing a l l b u t t h e f i n a l v o t e i n t h e t r i a l of t h e King, when he c a s t h i s v o t e w i t h t h e m a j o r i t y - - he could h a r d l y do o t h e r w i s e , although he was g e n e r a l l y a moderating i n f l u e n c e . Then, a f t e r g e t t i n g approval of h i s army-reorganization p l a n s , he was absent a g a i n , w i t h t h e Army of t h e North, p u t t i n g h i s p l a n s i n t o e f f e c t , d i s c u s s i n g s t r a t e g y - - which he r e v o l u t i o n i z e d -- and g e n e r a l l y i n s p i r i n g t h e new t r o o p s , and t h e i r o f f i c e r s and g e n e r a l s , w i t h h i s own boundless energy and enthusiasm. He r e t u r n e d b r i e f l y t o P a r i s i n August 1793 t o become member of t h e new Committee of P u b l i c S a f e t y , and then back t o t h e Army of t h e North, while t h e enemy was now b e s i e g i n g Maubeuge. His m i s s i o n t h e r e ended i n t h e v i c t o r y of W a t t i g n i e s on t h e 1 6 - t h October, when t h e r e s u l t s of h i s t i r e l e s s e f f o r t s became a p p a r e n t . This was a good t h i n g f o r him:

i n May 1794 d i s s e n s i o n s began

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t o a r i s e i n t h e Committee of P u b l i c S a f e t y . H i s a s s o c i a t e s i n t h a t Committee were e l i m i n a t e d by a coup i n J u l y 1794, and t h e T e r r o r ended. I t ended f o r o t h e r s , b u t n o t e n t i r e l y f o r him . . . The following year i t was proposed t h a t a l l former members of h i s Committee be executed. The sanctimonious r u l e of thumb l o g i c on which t h i s was based p e r s i s t s t o t h i s d a y , a s a g l a r ing i n s t a n c e of t h e Law of Conservation of Ignorance: i n t h e B r i t a n n i c a , we a r e a s s u r e d t h a t Carnot, simply because he had been a member of t h e Committee of P u b l i c S a f e t y , must SHARE w i t h t h e r e s t of t h e Committee, b o t h t h e c r e d i t f o r r e o r g a n i s i n g t h e army, and t h e g u i l t f o r t h e T e r r o r . F o r t u n a t e l y f o r Carnot, t h e N a t i o n a l Convention used no such automatic e q u a l i t a r i a n r u l e of u i l t by a s s o c i a t i o n . Things might have gone badly f o r him, S t h o u g h he a s s u r e d t h e Convention t h a t h i s r e s p o n s i b i l i t y had been only f o r m i l i t a r y a f f a i r s : b u t another deputy shouted t h a t Carnot was t h e " o r g a n i s e r of t h e Thus, f o r once, t h e d o c t r i n e of e q u a l i t y was n o t victor 'I. misuse : t h o s e who do so by making i t an excuse f o r r u l e of thumb condemnations, have never p r o p e r l y r e a d t h e D e c l a r a t i o n of t h e Rights of Man, which i s an i n s p i r i n g and w e l l thought o u t document, and perhaps t h e b e s t t h i n g t h a t has come down t o u s from t h e s o - c a l l e d age of r e a s o n , and from t h e Revolution i t s e l f . Rule of thumb automatic condemnations a r e on t h e c o n t r a r y s i g n s of t h e worst forms of t y r a n n y , t h a t t h e document sets out t o fight against.

-4

A s i t w a s , Carnot was e l e c t e d t o t h e D i r e c t o i r e , which r u l e d France f o r t h e n e x t f o u r y e a r s . H e was s t i l l a moderate, even i n t h a t more moderate a d m i n i s t r a t i o n . He was a l s o n o t one t o f l o u t a r i t h m e t i c : i n 1 7 9 7 , when a n e l e c t i o n produced a moilarchist m a j o r i t y , he r e f u s e d t o a g r e e t o quashing t h e e l e c t i o n , and he had t o f l e e t h e c o u n t r y f o r h i s l i f e . H e r e t u r n e d t o P a r i s when Napoleon s e i z e d power i n 1 7 9 9 . However he opposed Napoleon i n v a r i o u s t h i n g s : he w a s a g a i n s t any e x t e n s i o n of Napoleon's power as f i r s t c o n s u l , a g a i n s t h i s being made consul f o r l i f e , a g a i n s t h i s becoming emperor. When Napoleon r e t u r n e d from E l b a , he served a s m i n i s t e r of t h e i n t e r i o r , and he urged Napoleon t o c o n t i n u e t h e f i g h t a f t e r Waterloo. A f t e r t h e R e s t o r a t i o n of 1 8 1 5 , Carnot was e x i l e d , and h i s son S a d i was d i s c r i m i n a t e d a g a i n s t a l l h i s l i f e . To make up f o r t h e s e t h i n g s , t h e t h i r d r e p u b l i c honoured t h e e l d e r Carnot by p l a c i n g h i s a s h e s i n t h e Panthbon and by e l e c t i n g h i s grandson S a d i I1 p r e s i d e n t of t h e Republic. The honours w e r e perhaps n o t u n j u s t i f i e d : i n t h e whole Revolutionary and Napoleonic p e r i o d , I can t h i n k of no one who deserved them more.

Remarkably, w i t h a l l t h a t w a s going o n , Carnot and Monge were a b l e t o do mathematics: t h e y founded t h e French School of Geometry. A s a c r e a t i v e mathematician, Monge i s t h e b e t t e r known, b u t i n a s e n s e l e s s i n t e r e s t i n g , because a t t h a t p e r i o d mathematics d i d n o t need r e s u l t s , nor even methods, so much a s new d i r e c t i v e s . F e l i x K l e i n a t t a c h e s C a r n o t ' s name to a n i c e l i t t l e theorem of elementary p l a n e g e o m e t r y ,

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Given a t r i a n g l e ABC, and a l i n e meeting i t s s i d e s (prolonged i f necessary) BC, CA, AB i n t h e p o i n t s X , Y , Z , then t h e products of t h r e e d i s t a n c e s and Fig. 6 :

Not C a r n o t ' s .

XC,ZB,m

a r e equal.

The r e s u l t , u n f o r t u n a t e l y , was known t o Ptolemy, and i s u s u a l l y r e f e r r e d t o a s Menelaus's theorem. However Carnot wrote i n 1 7 9 7 a c a r e f u l c r i t i c a l study of t h e C a l c u l u s , e n t i t l e d ~ 6 f l e x i o n s s u r l a Mgtaphysique du Calcul I n f i n i t g s i m a l : i t was widely c i r c u l a t e d and t r a n s l a t e d , and went t o a number of e d i t i o n s . I t c o n t a i n s one g l a r i n g minor e r r o r , which s u g g e s t s t h a t i t may have been based on a s t u d e n t ' s n o t e s of a course Carnot gave; moreover i t does n o t , by any means, a n t i c i p a t e t h e exact d e f i n i t i o n s t h a t we owe t o Cauchy. Nevertheless i t d i d a g r e a t d e a l t o c l e a r up t h e r o l e of such t h i n g s a s L e i b n i z ' s i n f i n i t e s i m a l s : t h e b a s i c i d e a , a s Carnot s a y s , comes from Lagrange, b u t Lagrange r e a l i s e d t h a t t o reform t h e Calculus i n t h i s way, would t a k e time and e f f o r t and c a r e f u l thought. I n t h i s way it became r e l a t i v e l y simple f o r S t o l t z l a t e r t o g i v e a s a t i s f a c t o r y i n t e r p r e t a t i o n of d i f f e r e n t i a l s , once t h e Cauchy d e f i n i t i o n s had taken r o o t . Another h i g h l y i n t e r e s t i n g book by Carnot i s ''G&om&r-e de p o s i t i o n " : F e l i x Klein says t h a t i t i s i n p a r t elementary t o t h e p o i n t of t r i v i a l i t y , and i n p a r t ultra-modern. I t appeared i n 1 8 0 3 . I t was remarkable, and a l l t h e more so c o n s i d e r i n g t h e e v i d e n t admiration Carnot shows i n t h e e a r l i e r book f o r Lagrange. One can imagine no g r e a t e r c o n t r a s t than t h a t between t h e p o l i s h e d meticulousness of Lagrange, w i t h h i s t o t a l absence of diagrams, and t h e c r u d e , but b a s i c a l l y c o r r e c t , c r i t i c i s m s of t h e r a d i c a l Carno-t. For Lagrange, i t was geometry t h a t was s u s p e c t . For Carnot, on t h e c o n t r a r y , i t was a n a l y s i s t h a t was t o t a l l y misleading and misused. C a r n o t ' s example of t h e misuse of a n a l y s i s i s n o t convincing: it i s -a=/=a

,

/=a

=

fi

=

a.

I gave b e t t e r examples e a r l i e r , from t h e i n c r e d i b l e book of Oltramare. But Carnot was r i g h t t o p o i n t t o t h e dangers of many-valued f u n c t i o n s , and t o attempt t o curb t h e previous l i g h t h e a r t e d u s e of meaningless o r many-valued symbols: we s h a l l s e e i n connection w i t h Gauss t h a t i t had e s c a l a t e d beyond a l l bounds. The remedy, Carnot s a i d , was t o argue g e o m e t r i c a l l y , and t o f r e e geometry from t h e h i e r o g l y p h i c s of a n a l s i s . To u s , t h i s i n s i s t e n c e on geometry, o t a l l t h i n g s , &ne r e a l l y sound t h i n g i n mathematics, may seem extreme. Yet, a s we s h a l l s e e , it was t h e guiding p r i n c i p l e i n t h e theory of Riemann s u r f a c e s , h a l f a c e n t u r y l a t e r . Moreover i t bore f r u i t almost immediately, o r perhaps independently. A v e r y few y e a r s l a t e r t h e S w i s s Argand produced a g e o m e t r i c a l

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i n t e r p r e t a t i o n o f t h e imaginary and complex numbers, u s u a l l y r e iy, f e r r e d t o a s t h e Argand diagram: t h e complex number x where i denotes t h e s q u a r e r o o t of -1, i s r e p r e s e n t e d by t h e p o i n t x , y of t h e p l a n e , and t h e o p e r a t i o n s of a d d i t i o n , i w l t i p l i c a t i o n , e t c , on complex numbers a r e s i m p l y c e r t a i n c o r r e s ponding g e o m e t r i c a l o p e r a t i o n s i n t h e p l a n e .

+

Before t h e French R e v o l u t i o n , mathematics, l i k e e v e r y t h i n g e l s e , savoured of an atmosphere of u n r e a l i t y and faded e l e g a n c e . Carnot was one o f those who gave t h e impetus t o change t h i s : now t h e s q u a r e r o o t of -1 needed t h e meaning i t had lacked f o r 200 y e a r s . The i n t e r p r e t a m f o u n d by Argand w a s found a l s o by a number o f o t h e r s , Gauss among them. According t o Cauchy, a s t u d e n t Henri True1 came upon i t a s e a r l y a s 1786. I t was f i r s t p u b l i s h e d by t h e Dane Caspar Wessel i n t h e Royal Academy of Copenhagen i n 1 7 9 9 . Wessel's memoir t r e a t s analogous t h i n g s on a s p h e r e and i n t h r e e dimensions, r e l a t e d t o s p h e r i c a l t r i g o n o metry and t o a beginning of q u a t e r n i o n s , and a l l i n Danish. Mathematicians were j u s t n o t ready f o r i t , e s p e c i a l l y from a c a r t o g r a p h e r who wrote n o t h i n g f u r t h e r . The Genevese Argand, who p u b l i s h e d h i s r e p r e s e n t a t i o n anonymously i n 1806, was lucki e r : he d i d n ' t c l a i m a u t h o r s h i p u n t i l 1813, when i t had somehow been u n e a r t h e d by t h e e d i t o r Gergonne of t h e Annales de I.Iath6matiques a f t e r b e i n g quoted i n a l e t t e r o f Legendre, which had reached Gergonne t h i r d hand. Even t h e n , t h e r e s u l t a n t publ i c i t y had l i t t l e e f f e c t : people had managed w i t h o u t an i n t e r p r e t a t i o n f o r 300 y e a r s . L o r i a t e l l s us t h a t t h e same i n t e r p r e t a t i o n was p u b l i s h e d a l l over a g a i n i n 1828, and independentl y by two p e o p l e , n e i t h e r of whom was aware of i t s b e i n g known b e f o r e . The work of one of them, Warren, on t h e g e o m e t r i c a l representation e t c , m e t great opposition i n tradition-bound England. That of t h e o t h e r , Mourey, "Vraie Th6orie . . , , I ' had a t l e a s t one r e a d e r - - L i o u v i l l e . F i n a l l y a s i n g l e page by Gauss i n 1831, i n a memoir i n t h e G o e t t i n g e r g e l e h r t e n Anzeigen which c i t e s n e i t h e r Argand, n o r Wessel, n o r anyone e l s e , seems t o have achieved t h e g e n e r a l acceptance of t h e i n t e r p r e t a t i o n t h a t h i s p r e d e c e s s o r s had f a i l e d t o p u t a c r o s s t o more t h a n a s e l e c t few . What t h i s whole s t o r y shows, i s t h e remarkable t e n a c i t y of t h e

law of c o n s e r v a t i o n of i g n o r a n c e . We begin t o understand t h a t some h i s t o r i a n s f e e l v a s t human s u f f e r i n g t o b e t h e o n l y way in which humanity can escape from i t s p r i s o n of p r e j u d i c e . On t h e same b a s i s , a deep-sea d i v e r could n e v e r r e t u r n t o t h e s u r f a c e w i t h o u t s u f f e r i n g t h e t e r r i b l e bends. However he can avoid t h e bends by r e g u l a t i n g t h e r a t e a t which he comes u p , o r equival e n t l y by b e i n g rushed i n t o a decompression chamber, and by r e g u l a t i n g t h e r a t e a t which t h e p r e s s u r e t h e r e i s reduced. My own f e e l i n g i n such m a t t e r s , and t h a t of most r e a s o n a b l e men, c e r t a i n l y most s c i e n t i s t s , i s probably v e r y c l o s e t o what Carnot a l l the s u f f e r i n g ; l e t us a t l e a s t f e l t : w e have been-through shed t h e p r e j u d i c e s , and n o t s t a r t t h e p r o c e s s a l l over a g a i n . What Carnot showed, and what t h e French Revolution showed, i s t h a t , i n mathematics j u s t l i k e i n everyday l i f e , sooner o r l a t e r humanity can no l o n g e r o p e r a t e by r u l e s of thumb, o r r u l e s o f e t i q u e t t e , i n v o l v i n g symbols whose meaning i s n o t c l e a r . Carnot

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was no more concerned with patching up analysis, than with patching up the monarchy. Geometry was to be independent. A s for analysis, the task of forging it afresh was soon, as we shall see, to begin. Monge was more accommodating: he had no more objection to using analysis when it suited his purpose, than to remaining on good terms with Napoleon. He wrote two books on geometry: G6om6trie descriptive, and L'application de l'analyse 21 la gbombtrie. The second is partly what we now call Differential Geometry: just the opposite of the "synthetic'' geometry freed from analysis (or as it came to be called, Projective Geometry), that Carnot was asking for, and that his pupils developed. Still Monge was in a sense the father of it all, since Carnot was his pupil too. In the history of mathematics, outside of France, Carnot hardly even exists: only Felix Klein could see the great importance of his ideas. This is because, for Felix Klein, research and teaching went together. Carnot and Monge were the teachers of our age, the pioneers who provided for mathematics the new directives it s o badly needed, and that others, after them, formalised in the manner we are now accustomed to. "C'est le premier pas qui cotite." The first step, the first little unsure, faltering totter, makes possible the great march, the surge forward, that is to come. Monge and Carnot had, as we shall see, a number of pupils: they were the founders of a great French School, and here the word "school" can be taken literally. They founded the Ecole Polytechnique. The founding of this great school was part of Carnot's plan for the Army, authorized by the Assembly in 1794. It had an immense influence on the subsequent development of mathematics, as we shall see. Monge also participated in the founding of the Institut de France in 1795: it took over, until the Restoration in 1815, 'the functions of the Paris Academy of Sciences, that had been dissolved in 1792, owing to the attacks of Marat. After the fall of Napoleon, the old Academy of Sciences was revived, and the institute abolished; Monge was stripped of all honours, and died a few years later of a stroke. However the Ecole Polytechnique remained: it had proved far too valuable to be scrapped. The military successes of Revolutionary and Napoleonic France after Wattignies were the direct result of Carnot's reorganisation of the Army and of the foundation of the Ecole Polytechnique. Of course the credit must go in part to Monge and to the other mathematicians and scientists, who responded to Carnot's appeal, and who assisted with the Ecole Polytechnique. Incidentally, the military successes also solved for the time France's financial problems: France simply looted its defeated neighbours. This explains why Frenchmen remained willing to engage in more wars. At any rate, one early result of all this was that mathematicians were in high esteem, and this too has to a certain extent remained the case to this day in France. Even in the worst Terror, equalitarian Revolutionary France was far from anti-elitist and far from ungratef u l . Scientists and mathematicians were honoured and privileged: they were not rewarded with a blood-bath. The value

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of their collaboration had come to be suddenly and dramatically realised, probably far more so, than even in World War 11 in America. They were a privileged class in an otherwise equalitarian nation, almost like lawyers and journalists today, perhaps even like professional ball-players or like the purveyors of advertisements and unmusical noise. Carnot and Monge had demonstrated the need for mathematicians, and the Ecole Polytechnique was founded to produce them. Previously an attempt had been made to found the Ecole Normale, but it was abandoned in its second year and not resuscitated for some time. The Ecole Polytechnique was part of a gigantic educational plan, that involved all the schools of France. Each year, 150 new students were selected by an extremely stiff competitive examination, for which they were prepared by special courses in mathematics, called "Math6matiques sp6ciales," of UP to 16 hours a week. I am not defending competitive examinations; neither am I recommending Napoleon's use of the best young minds of France - the "polytechniciens" -- as cannon fodder. I merely describe the system as it was. Competitive examinations have been beautifully ridiculed by Ernest Bramah, who represents them as having come to us from Ancient China. He describes, in one of his stories, an examination in which the questions, by order of the Emperor of China, concerned exclusively a previously quite unknown and mediocre poem by the Emperor himself, entitled In Cambridge, I once took a Modern Langu"Concerning Spring. age Scholarship Examination: Z had spent most of my life up to then in the best French-Swiss schools. To test my knowledge of the spirit and character of the French language and literature, and of the clarity and logic of its thought, I was to translate three short passages: the first, entitled "A night in the country," proved to be an excuse for a long list of French birds; the second, similarly for one of architectural items interior to a church; the third, remarkably and no doubt as a surprise after the first two, made of a dead XVIII-th century soldier a catalog of French articles of that period's military clothing. I was asked about arranging for an examination in German: not being an ornithologist, a church architect, a period tailor, I excused myself. Examinations in mathematics may test more worthwhile qualities, but however biased I may be in favour of my own subject, I shall not deny that one can have too much of "a good thing," if you insist on calling it good. In the competition for entrance to the Ecole Polytechnique, a score of 2000 was the theoretical maximum: the greatest score ever attained so far is Hadamard's 1875 - - in spite of which Hadamard elected to enter the Ecole Normale. Things may have changed of late, but at the times we speak of, the High School preparation for the entrance competition was intense, and involved an enormous amount of problem-solving. This was still nothing compared to the concentrated regime of mathematical cramming that the successful candidates then had to undergo at the Ecole Polytechnique itself, where a truly military timetable was enforced, starting at five in the morning, and

Chapter I continuing until late at night. This involved some 20 hours a week of high-powered lecture-material, and endless hours of problem-solving and of repetition of the lecture-material, I mentioned earlier the Arab mathematician Avicenna's "Liberation,' I and only a short time ago I spoke of humanity's need to escape from its prison of prejudice. Was this drudgery a "liberation?" Can we expect a creative human being to survive the soul-killing force-feeding of clever little tricks? I fear that the great problem of education has no such simple solution, and yet, as I indicated, its solution is becoming urgent if we are to escape the suffering we cause ourselves: we, after all, are the ones who make history; our own brains determine future events. Surely we should control what we make? And surely this is the prime purpose of education? However, to those of you who will be teaching, I need hardly say as a teacher, that there will be moments of great discouragement: I have known them, though I have had the great good fortune of knowing also the opposite. Therefore let me say just this once something to encourage teachers, - - something perhaps out of line with the objectivity we expect of lectures. I like to imagine some Providence trying to help us learn: a fond hope, no doubt, but we need all the help we can get. I do not myself picture Providence as all-powerful - - this is too crude - - , but simply as our wouldbe teacher. We, the teachers, are then, in our discouragements, closest to what I can only call "the tears of Paradise." The Ecole Polytechnique was a heroic effort, and it reflected the character and the drive of its founder, Carnot. Superficially, what made the long ordeal of its students more attractive, was that at entrance they immediately and automatically became, as polytechniciens, paid officials within the French civil service, and that ultimately the highest positions would be assigned to them. But what was really vital to the scheme, was that the professors, and even the coaches who assisted with the p'roblems and repetitions, were the top mathematicians of France: only their encouragement and inspiration could breathe lifento that deadest of cramming systems, the prototype of today's motorcar assembly-lines. -What also helped, was- that the profession of mathematician had suddenly become, as I explained, highly honoured. Still we cannot expect miracles from an educational system imposed from above or below, on teachers who have to do as they are told. It did not produce 150 mathematicians a year - - on the average, it did not even produce one. It produced, over the years, I am told, many public officials; in the early years it provided Napoleon's finest cannon-fodder. Even so, with all its glaring defects, it gave to mathematics a tremendous push forwards. It reduced the mathematics prior to the Revolution largely to the level of exercises. It put an end to the period in which a person could fancy himself a mathematician after a few month's study, and become a professor at a very early age. The study of mathematics had become a serious business, involving very hard work. We must also bear in mind that a differently planned Ecole Polytechnique would not have survived after the Restoration: it had to have military value. This was also the capacity in which it had served Napoleon. Indeed some of its defects were due to him: he did what he

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could to accelerate the flow of army engineers to help in his campaigns. Fortunately he realized that 'lone must not kill the hen that lays the golden eggs." The Ecole Polytechnique consisted of several specialised schools: the Ecole des Ponts et Chaussges, the Ecole des Mines, and the strictly military schools - Ggnie, Artillerie. They were ranked in order of prestige, and candidates were admitted to them in order of merit: the best were free to choose their specialisation, the weaker ones were limited to one of lower rank. One of the remarkable things about the Ecole Polytechnique was that, in spite of everything I have said, it developed mathematical intuition. This was partly due to its great emphasis on geometry as a tool of engineering. Not only was there engineering drawing - - which in our totally different century is generally considered by the better students a complete waste of time - - there was also engineering model-making. Monge was very particular about this: it was not enough that students should see beautiful models of surfaces in three dimensions, such as, at one time, all mathematical departments in universities came to possess: they had to construct them with their own hands. It goes without saying that in our age model-making would be even less popular than drawing. Nevertheless, Felix Klein, one of the great teachers of all time, maintains that Monge was right: he states in his book on the Development of Mathematics in the XIX-th century that intuition should be encouraged in this manner. However it can be done at an earlier stage: with Felix Klein's blessing, my father and mother developed, more than 100 years after Monge, such a plan for young children in their "First book of geometry," which was successful in translation into German and other languages. A s to doing these things at the Ecole Polytechnique, I more than suspect that it was the enthusiasm of Monge and Carnot, that kept their students from being bored. Enthusiasm is catching, and they knew how to inspire, even at second hand, through their successors. The total number of mathematicians who issued from the Ecole Polytechnique, was, as already mentioned, not great; however it included nearly alL the French mathematicians o f the XIX-th century. The main exceptions were Galois and Hermite. Typical polytechniciens were Poisson and Cauchy, and the geometers Poncelet and Chasles. They were not only students there, they also became professors. Another polytechnicien was Fourier; however he was used by Napoleon as an administrator. A sideeffect of the founding of the Ecole Polytechnique was the close relation of mathematics to government circles, on account of the fact that these circles were also largely staffed by polytechniciens. The number of mathematicians who came from the Ecole Polytechnique is of course greatly increased, if we include names now associated with Physics, Engineering, or Chemistry. For 30 years, everything achieved in France in these fields came out of the Ecole Polytechnique. There was Sadi Carnot (1796-1832) , the son o f our Lazare, and much better known, outside of France, than his father. He wrote the famous "R6flexions sur la puissance motrice du feu," a little essay

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of great significance in the development of Steam Engines, but which remained unnoticed for 10 years after its publication in 1824. There was Fresnel (1788 - 1827), and there was Malus (1775 - 1812): we can count them as mathematicians, quite as much as experimentalists. They discovered polarisation and aberration of light, highly mathematical phenomena; and the theorem of Malus in Optics is as basic in the Calculus of Variations and in Control Theory, as in its original setting. Another great name is that of Navier (1785 - 1836), a civil engineer whose work is familiar to hydrodynamicists and aerodynamicists. The influence of the Ecole Polytechnique was by no means limited to its pupils: it extended to Europe as a whole. Lectures were attended by outsiders as well, engineers and others who wished to brush up their knowledge. But above all, it created a relatively wide mathematical public, in the French civil service and elsewhere, consisting of former polytechniciens, outside auditors, special mathematics teachers in the schools, not forgetting their gifted pupils who would later qualify for entrance. France became the land of mathematical textbooks read throughout Europe. The Restoration might strip Monge.of his honours, and exile Carnot, but their work continued to bear fruit. The first writers of the new textbooks were naturally our other three mathematicians from before the Revolution. Legendre wrote, in addition to the works already mentioned, Exercices de calcul integral (1811 - 1819) in three volumes, and Trait6 des fonctions elliptiques (1825 - 1827). Laplace became president of the Bureau des Longitudes, and Napoleon made him a marquis. He produced Me'canique c6leste (five volumes 1798 - 1827), Thborie analytique des probabilitgs (1812), together with two popular books on the same subjects -- beautifully written models of French prose, the first.of which include2 Laplace's famous "Nebular Hypothesis," - - Exposition du systeme du monde (1796) and Essai philosophique sur les probabilit6s (1814). Lagrange became professor at the Ecole Polytechnique, and remained a likeable figure, a little crusty and obstinate, as old people can be, but always genuine. Napoleon made him a count and a senator, but such things had ceased to matter to him. At a meeting at which he had announced a paper, he said simply: "I1 faut que jly songe encore." This means: I must reflect, dream of it, further. Many mathematical papers would be improved if their authors would do this. In Lagrange's case, it was a good thing that he withdrew the paper: he thought he had solved the quintic! He was not the last to imagine this, as we shall see. Lagrange's books, in addition to the two-volume second edition of his MBcanique analytique published after his death (1811 - l815), were his lectures at the Ecole Polytechnique: Thcorie des fonctions analytiques (1797), LeSons sur le calcul des fonctions (1804). Felix Klein comments that the books are a joy to read, but that there is one thing missing from them as compared with those of Carnot and Monge, who could stimulate and inspire. Lagrange, Laplace, Legendre, count only as great writers, great teachers, great mathematicians: their

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crystal-clear presentation does not give the reader that greatest of all pleasures that learning can bring, the excitement of actually witnessing the birth of a mathematical discovery. In this respect Euler was unsurpassed: he would follow the route he had taken himself, and warn of the pitfalls he had fallen in, and of the mistakes he had been careless enough to make; he would describe the vain attempts he had engaged in, before hitting on the right way. He would also write of unsolved difficulties, and suggest ways of trying to overcome them. Lagrange and Laplace represented, at the time of Napoleon, mathematics of the past, raised to its greatest perfection. It was very proper for them to write textbooks: this could raise the whole level of mathematics by making the material common knowledge among mathematicians. By contrast, Carnot and Monge were more concerned in pointing the way for the future. Legendre was somewhere in between. Not unnaturally, Lagrange and Laplace received the highest honours: there is nothing controversial in perfecting methods and problems of the past. Lagrange, throughout his life, was above politics, above nationality. He had been honoured in Italy and in Berlin. He had been honoured, as I said, by Louis XVI, by the Revolution, by Napoleon, and cared little for these honours. Even so, he had been honoured, perhaps, a shade too much. One could not help liking him, but no man is free from error, not even in the later years and after a breakdown in health. At the Institut de France, in the end, he set the tone of arbitrarily rejecting whatever he personally disagreed with, or was unable to digest. This high-handed tradition was followed after his death, notably by Cauchy, with the result that anything involving new ideas was likely to be held up in the files of the Paris Academy. This affected not only the work of Fourier and Cauchy, but also that of Abel, and tragically that of Galois. I wish I could say that today all publications, or even all American publications, are more receptive. Laplace is already slightly more modern than Lagrange. He has been accused of being a time-server and a politician: we could do with a few more such time-servers. The mathematician Kahane recently quoted from a book entitled "MBlanges scientifiques et litt&aires," published in Paris in 1858. The author, J.-B. Biot, tells us how he came to Laplace with a small manuscript, and was encouraged to submit it, but warned that any attempt to extend the work would lead to great difficulties. After the manuscript had been accepted, Laplace showed Biot a manuscript of his own, which he had suppressed to allow Biot to publish his: the difficulties of extending the work had also been there. To us Laplace is known for the Laplace transform, which first appeared in his probability and is fundamental in many branches of analysis, and even more for Laplace's equation in potential theory, AV = 0, where A is the operator az/ax' + a2/ay2 + a2/az2. This equation plays a crucial part in many fields, because its solutions, the harmonic functions, have properties that generalise those of linear functions of a single variable. On the other hand, in astronomy and probability

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L a p l a c e ' s work belongs r a t h e r t o t h e previous e r a . I t goes much beyond what had been done: h e went i n t o such m a t t e r s as t h e p r o b a b i l i t y of a c a u s e , and i n t o v i t a l and o t h e r s t a t i s t i c s ; i n astronomy he developed p e r t u r b a t i o n methods, and made l e n g t h y c a l c u l a t i o n s i n q u e s t i o n s of s t a b i l i t y , p a r t i c u l a r l y a s r e g a r d s t h e Moon. By modern s t a n d a r d s t h i s work i s unduly o p t i m i s t i c , even a p a r t from h i s many s e n t e n c e s beginning "I1 e s t f a c i l e de v o i r e . " Except f o r s h o r t - r a n g e p r e d i c t i o n t h e p r a c t i c e of astronomers i n n e g l e c t i n g "Higher o r d e r terms" i n t h e e c c e n t r i c i t y , i s unsound: i t cannot d e c i d e q u a l i t a t i v e q u e s t i o n s f o r a l l time - - t o t a l l y d i f f e r e n t approaches a r e needed. I n any c a s e t h e b a s i c g r a v i t a t i o n a l theory has been modified by modern views of Physics and Chemistry, by R e l a t i v i t y , by R a d i a t i o n .

22.

Gauss

In Germany t h e r e were perhaps even g r e a t e r changes i n t h e p e r i o d we have been d i s c u s s i n g . I t was of c o u r s e no Germany such a s we know i t , j u s t a s t h e r e was s t i l l no I t a l y . There w a s a v a s t anachronism, c a l l e d t h e Holy Roman E m p i r e , a warning of what t o d a y ' s world may be f a s t becoming: a d i s o r g a n i s a t i o n of d i s u n i t e d n a t i o n s , a chaos of p e t t y p r i n c i p a l i t i e s . A few were l a r g e enough t o count a s s t a t e s , b u t t h e r e were some 300.more, n o t counting 1500 t i n y e n c l a v e s . Local l o r d s h i p s r u l e d i n a t a n g l e d web of diplomacy and i n t r i g u e , p l o t t i n g a g a i n s t one a n o t h e r , o r a g a i n s t t h e Holy Roman Emperor, w h i l e t h e i r c o u r t i e r s and c o u n c i l l o r s c o n s p i r e d a g a i n s t one a n o t h e r o r t h e chanc e l l o r , o r a g a i n s t h i s l o r d s h i p h i m s e l f . A l l t h i s had endured f a r beyond t h e p o i n t of having become t o t a l l y r i d i c u l o u s , s o l o n g t h a t i t seemed hopeless t o t r y t o change i t . N e v e r t h e l e s s , i n t e l l e c t u a l l i f e was emerging, t o g e t h e r w i t h a l i t e r a r y language which c o n t r a s t e d w i t h t h e l o c a l d i a l e c t s s t i l l used today i n country d i s t r i c t s . German a r t s and l e t t e r s , and German t h o u g h t , began t o f l o u r i s h i n t i n y p r i n c i p a l i t i e s and s l e e p y townships, q u i e t backwaters away from t h e t u r b u l e n t storm. One speaks i n German l i t e r a t u r e o f a p e r i o d of storm and s t r e s s - Sturm und Drang - - but i t i s no more t h a n a storm i n a t e a c u p , a p u r e l y l i t e r a r y r e b e l l i o n a g a i n s t former l i t e r a r y s t a n d a r d s . Heads d i d n o t r o l l , blood was n o t s h e d , no one s u f f e r e d t h e t i n i e s t s c r a t c h . The only c u t s were t h o s e of t h e meters of v e r s i f i c a t i o n . I d e a s , t h o u g h t s , d i d n o t change: i t was t h e time o f German " i d e a l i s m . " This meant t h a t German thought was d i r e c t e d inwards, and ignored t h e v a s t and a p p a r e n t l y h o p e l e s s t a s k of reforming s o c i e t y and t h e body p o l i t i c . I t was t h e time o f t h e p h i l o s o p h i e s of Kant and F i c h t e , t h e g l o r i f i c a t i o n of o n e ' s i n n e r s e l f , t h e n o b i l i t y of t h e human s p i r i t , t o t h e e x c l u s i o n of t h e world o u t s i d e . Amid t h e abs u r d i t i e s of everyday l i f e , i t can be p l e a s a n t t o say t h a t what m a t t e r s i s o n e ' s own mind: " S i f r a c t u s i l l a b i t u r o r b i s , impavif durn f e r i e n t r u i n a e . " I t i s a n o b l e d o c t r i n e , l i k e t h a t o one i n a p o s i t i o n of t r u s t , who promises w i t h t e a r s i n h i s e y e s , t o do h i s whole d u t y . But t h e r e i s something m i s s i n g from i t , a s i n a r e l i g i o n where no one p r a y s . I t has no b r a k e s . There i s no room i n i t f o r s e l f - c r i t i c i s m . Each of u s becomes h i s own e g o c e n t r i c God: t h e E a r t h i s no l o n g e r t h e c e n t r e of t h e

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universe, the centre is in each of us. In this philosophy, we cannot wonder that Kant claimed, as stated earlier, to possess a proof of Euclid's axiom of parallels. After Kant and Fichte, rival philosophies had their sway: they sensed that something was missing before, and they called it "the absolute." This was defined by Schelling as the unity of subjective and objective, until Hegel objected that such a unity would be that of a night in which all cows are black. Hegel's followers eventually occupied virtually every Chair of Philosophy in Germany. His absolute was a God who fashions mankind's history: this provides a justification of many things! The escalation of absurdity all around was now quite all right, since it was history; the same applied to what might be called the planting of the dragon's teeth, a planting from which our century reaped a bitter harvest.

It is remarkable that mathematics was about to flourish in a land whose thinking was being formed by such philosophies. In contrast to Revolutionary France, it was philosophy, and not mathematics, that was important in Germany. However it must be admitted that the competence of the occupants of German University Chairs in mathematics, prior to Gauss, was not overwhelming. A whole school, to which many of them belonged, was headed by a Leipzig physicist and philosopher, a certain Hindeburg (1741 - 1 8 0 8 ) , editor and founder of a mathematical journal that ultimately became the Archiv der reinen und angewandten Mathematik. They promoted for decades the delusion of a formal inversion for an arbitrary power series. We may well ask, what could there have been, in Germany, of all places, to make possible at such a time the appearance of a great mathematician? I have gone out of my way to explain logically how the French Revolution and the founding of the Ecole Polytechnique helped mathematics. But we shall see that the next great mathematician was f a r f r o m being helped by the French Revolution. History is like that: you produce a nice logical explanation, and the first case you come to appears to contradict it. The truth is that, between cause and effect, there can be a time-lag: light moves with finite velocity, but the rapidity of enlightenment is rarely very great. It is not at all clear what should be done to produce mathematicians: this question I raised in the introduction, perhaps only the history of mathematics can help to answer it. Cesari once told me of his first meeting with Leonida Tonelli. Two students were with Tonelli: to the one, Tonelli was most kind and encouraging; to the other, he was severe to the point of bitter sarcasm. When they left, Tonelli said, 'You noticed how differently I treated those two? They are equally good." This shows the fallacy of any pre-conceived system of education: human beings are not bars of soap that have to come out alike. According to Pascal, it is the commonplace person who sees no difference between men. Each individual must develop in the way best suited to him. Some need encouragement, like delicate plants in a greenhouse; others must be exposed to wind and rain.

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Chapter I

Carl Friedrich Gauss (1777 - 1855) was born in the German city of Braunschweig, or Brunswick, from parents in extreme penury. By all accounts he was a very precocious child. In spite of continual discouragements at home, and the toughest of demands on his time, he managed to find the energy and a few stolen hours, in which to develop his mind: this helped prepare him for discouragements and incomprehension later in life. He learnt to keep to himself knowledge not appreciated around him. Eventually his mother and his primary school teachers were able to obtain for him financial assistance from the duke of Brunswick, the same Charles Ferdinand of Brunswick who later commanded the Prussian troops at Valmy and elsewhere. Gauss was thus able to attend secondary school or Gymnasium, consisting of five years at the Catherineum and two at the proper University pre-school, or Carolinum, and to go on to spend three years at the University of Goettingen. There he suddenly switched from a strong interest in philology to mathematics. I mentioned that the story about adding the numbers 1 to 1000 migrates from Euler to Gauss and others; it is not in character in the case of Gauss - - not because his interests were then in philology, but because he kept his knowledge to himself. We know a great deal about Gauss: a great deal more than he allowed his contemporaries, with a few exceptions, to know of Him. The material took a long time and much effort to be made available in an intelligible form. Ten volumes of his collected works appeared in 1870 - 1930, considerably after his death in 1855. Further material was brought out in 1911 - 1922, under the direction of Felix Klein and M. Brendel. Then there is the Gauss Gedenkband of 1955, to which must be added studies prepared for Gauss's second centenary in 1977. Much of this material is available in any good mathematical library: I even have in my private collection several works of Gauss. What is more, it is possible to consult Gauss's Tagebuch, and also his letters to Bessel, to Bolyai, to von Humboldt, to Nicolai, to Oblers, to Repsold, to Schumacher, and to Sophie Germain. This still does not include articles, such as Pizetti's "Hohere Geodesie" in the German Encyclopedia. Above all, there is the oral tradition in Goettingen, mainly through Felix Klein : his competence was unique, and he devotes to Gauss a long chapter in his Entwicklung. Klein was hardly able to go anywhere, without coming across traces of this tradition: he imbibed it through Clebsch and Gordan, through his friend Stoltz - - a relative of Gauss's student von Staudt - - , and he found it in Erlangen, where he succeeded Gauss's student Moebius, and where he was surrounded by students come to him from Clebsch. With such a treasure-house of material, I hope the reader will take t o heart my advice about going to the original sources, and about thinking and judging for himself. Much of the material was not easily come by. The story of the Tagebuch is characteristic: this diary dates virtually all Gauss's discoveries from 1796 to 1801, and many more up to 1814. Yet it was retained with other "private" papers by the Gauss family, when Gauss's Nachlass was bought by the State. By the merest good fortune, the Tagebuch was discovered in 1899; even then the owner was most reluctant to allow it to be studied. Klein, in

145

The Romantic Period

the Entwicklung, wonders what further treasures may have been lost to us? A s for the deciphering and interpreting, which, for Gauss's works, took 70 years or more, a typical setback was the appearance, in the printed version of the Tagebuch, of a supposed portrait of Gauss, which turned out, in spite of a persistent family tradition, to be one of Bessel! So much for family tradition! The true Gauss tradition, as Klein implies, undefaced by a denigration of intellectual values, was passed on, not to descendents now mostly in Missouri, but to the great names that succeeded Gauss, and so, in the famous years, to Goettingen, which became incidentally my parents' place of pilgrimage and my own birthplace. In my mind's eye, Gauss's name stands over that of George I1 of England, high on the University building.

In his first year at Goettingen, at the age of 19, Gauss succeeded in proving that a regular 17-sided polygon can be constructed with ruler and compass - - a tremendous step forward in the 2000-year old problem of the construction of regular polygons. Soon after, he solved this problem completely: an n-sided regular polygon can be constructed with ruler and compass, if and only if n is a prime number of the form n

=

2zk

+

1.

This was Gauss's first publication, a short note in the "Intelligenzblatt" - - intelligence-leaflet - - of the University of Jena. By then, as we see from the Tagebuch and early rough notes, Gauss had done many things. Like Newton, he saved every bit of paper and wrote very small, using every square inch. One thing to excite his curiosity was the "arithmetic-geometric mean:" from two positive real numbers a,b form the arithmetic mean a' and the geometric mean b'; then from a',b' form similarly the means a",b", and so on. Here a' b"

=

=

b' =%(ab)%; S(a+b) (%(a+bjS(ab) .

a"

=

%(a+b)

+

S(ab)',

Gauss noticed, what is not at all plausible from these formulae, that the two sequences a,a',a", . . . and b,b'b,b", . . . converge to a same limit. This limit, the arithmetic-geometric mean, turned out, as if it were some curious fact revealed at the beginning of a novel, to be connected with what came later. Another tool was his method of least squares, which he found, according to his own statementi in 1795. After this, he became interested in number theory, and rediscovered many known results; he also made tables of primes, and of their reciprocals lfp for p between 1 and 1000 - - with as many decimals as needed to complete the period, in some cases several hundred. He made tables of quadratic residues and non-residues, and conjectured, as Euler had done, the quadratic reciprocity law, the "theorema aureum:" this, as we know, was proved by Legendre, but the young Gauss proved it also, which gave rise to disputes. These discoveries were still prior to coming to Goettingen in the autumn of 1795, when he had for the first time access to the works of Euler and Lagrange. Now comes the diary: at first

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Chapter I

there is much that Gauss rediscovered. This was no waste of time: it is more important to be able to prove things for oneself, than to be familiar with the latest wrinkle. However there were soon new things as well, beginning, on the 30-th March 1796, with the construction I mentioned o f the regular 17-sided polygon, and continuing on the 8-th April, with his complicated first proof in 8 parts of the theorema aureum. A year later, the diary shows that Gauss was already working with elliptic functions, and that he had connected them to the arithmetic-geometric mean. The theorema aureum, about which there were bitter disputes with Legendre, may be described as a "Yes -- No" theorem, except that instead of Yes and No we write +1 and -1. In other words, it is in the jargon of today's computers. Suppose that m,n are primes greater than 2. If there is an integer x such that x 2 is congruent to n(mod m), i.e. such that x2-n is divisible by m , then WE say "Yes'', and write in shorthand notation

(g)

=

+l.

Here the symbol on the left denotes, not a fraction, but an abbreviation for the relevant property of the primes n , m , while the symbols = +1' mean in words "is true." Similarly the symbols ' = -1' would mean "is false;" so that if there is no such integer x , we say "NO," and we write in the same notation n (-) = -1. m However, if we take a second look at our two equations, we can say that they define a certain numerical function of m,n, denoted by the lefthand side, and that this function takes only the values +1,-1. The theorema aureum tells us how this function aIters when we interchange m and n, where m,n are primes greater than 2. It asserts that the value of the product " =m-1 - T .n-1 -T.

(i) ):( is

(-l)N, where

The statement does not seem too hard, once we know what it is all about: the hard thing is to produce a nice elegant proof. Gauss was in no hurry to publish his complicated one in 8 parts. What he did publish, as his dissertation, was both sensational and beautiful. Gauss's dissertation consisted of a proof at long last of the Fundamental Theorem of Algebra. He bbtained with it a Ph.D. in absentia from the University of Helmstedt (1799). It was already characteristic of Gauss that he carefully disguised his deep underlying mathematical ideas, in order to think about them more at leisure. Then in 1801, he published his first great work, the Disquisitiones Arithmeticae: I shall say more about it. However 1801 was also when Gauss produced a sensational piece of work in astronomy, on the minor planet Ceres

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147

which had been discovered on the 1-st January by Piazzi, and lost again soon after in the rays of the Sun. Gauss quickly computed the orbit so accurately that Ceres was found again without difficulty, far from where the approximate circular orbit rule, used until then by astronomers, would have placed it. Gauss developed the method he used for Ceres in his Theoria Motus (1809), which became a standard text for computational astronomy. He went on to attempt similar work for the minor planet Pallas, discovered on the 28-th March 1802: this turned out to be a task of immense difficulty, involving perturbations by Jupiter and Saturn, and it seems to have been abandoned after many years. Why was this work abandoned, or left unpublished? There has been much speculation. Another, perhaps not unconnected, question concerns a note in pencil: "Der Tod ist mir lieber als ein solches Leben." What are we to make of it? The note appears between purely mathematical matters, written between 1807 and 1810 on elliptic functions, at a time when he had moved to Goettingen. Until 1806, he had been content to depend on the goodwill of his protector, the duke, and he had refused several offers to go to Goettingen and elsewhere. However the duke was mortally wounded at Auerstadt, and Germany was overrun by Napoleon's troops. So Gauss agreed to go to Goettingen as director of the observatory and professor of astronomy. There he had to teach, which he found irksome, since it took time, but at least he had, over the years, excellent pupils, who became well-known astronomers for the most part. Unfortunately there were other drawbacks. The observatory did not exist - - its construction was begun in 1811 under Napoleon's brother Jerome, who became for a few years King of Westphalia, but even then the work was interrupted, and was not resumed until 1816 when the Hanoverian regime had been restored. Moreover Gauss was having to pay taxes to the French occupation authorities, without receiving the salary due to him. He lived in a miserable house in the Turmstrasse, next to the old fortifications, and his sole observatory was no more than he could erect from it himself. His nights were spent constantly observing. By day he was always at his calculations, except for irksome interruptions. His family viewed him as some sort of crank: he was wearing himself out at useless things,.instead of providing better for his dependents, like a good German family man. Gauss had progressed very far with the work on Pallas: some remarkable conclusions were presented in the form of a mysterious magram, a partial explanation of which is given in a letter to Bessel, written in 1812, where Gauss asserts that the ratio of the mean motions of Jupiter and Pallas oscillates around the rational value 7/18. What Gauss did publish was his second major astronomical work "On Secular Perturbations" (1818), a standard theory of first order approximations. However the astronomer Struve showed much later that for Pallas one needs at least third order approximations. This gives some idea of the magnitude of the task that Gauss faced with Pallas, and of the still larger task that preparing the work for publication would have entailed. He was a perfectionist: he wished to publish "nothing unless perfect." Long after his

.

148

Chapter I

d e a t h , h i s papers were s o r t e d o u t and s t u d i e d , many of them j u s t s c r a p s of paper covered w i t h c a l c u l a t i o n s . I n h i s a s t r o n o mical work, h i s a n a l y t i c a l a p p a r a t u s i s i m p r e s s i v e : a s he s a i d himself i n a l e t t e r t o B e s s e l , by comparison t h e volumes of Laplace seem t h e work of a c h i l d . Today, w i t h e l a b o r a t e comp u t e r technology, i t i s n o t c e r t a i n t h a t we have g o t as f a r a s Gauss had. Even s o , t h e r e i s a n o t h e r p o i n t t o be borne i n mind. As a mathematician, Gauss a c q u i r e d t h e r e p u t a t i o n of extreme r i g o u r : i n t h i s he was g r e a t l y ahead of h i s t i m e . Many y e a r s l a t e r , J a c o b i could d e c l a r e , on coming i n t o c l a s s , "Gentlemen, f o r Gaussian r i g o u r we have today no t i m e . " Such remarks would n o t endear a person t o Gauss, b u t i t was what many people f e l t : i n 1 8 1 0 , t h e textbook w r i t e r Lacroix b o a s t e d i n t h e p r e f a c e t o h i s T r a i t 6 du C a l c u l D i f f g r e n t i e l e t du Calcul I n t e g r a l of n o t r e q u i r i n g " t h e p u n c t i l i o u s n e s s t h a t had plagued t h e Ancient Greeks." J u s t t o l i v e among such P h i l i s t i n e s , and t o f i n d t h a t they were being honoured, can be a t r i a l . Napoleon, a s we know, honoured Lagrange and Laplace, b u t t o Gauss he brought only L i b e r t y , E q u a l i t y , F r a t e r n i t y : t h e L i b e r t y t h a t deprived him of h i s b e n e f a c t o r , t h e E q u a l i t y t h a t p r e v e n t e d h i s r e c e i v i n g h i s s a l a r y , and t h e F r a t e r n i t y t h a t f o r c e d him t o pay t a x e s t o h i s French b r o t h e r s , t o h e l p make t h e honours p o s s i b l e . What I f i n d remarkable, i s t h a t we know only of t h a t one p l a c e where Gauss goes so f a r a s t o w r i t e f a i n t l y - - i f h e was t h e one - - t h e " c r i du coeur" about d e a t h being p r e f e r a b l e . Before t h a t , t h i n g s had n o t been i d e a l e i t h e r : he must have been v e r y one bad r e p o r t t o t h e duke conscious o f being v u l n e r a b l e might p u t him r i g h t back where he had come from.

--

Gauss went h i s own way, i n s p i t e o f e v e r y t h i n g ; he w a s n o t w i l l i n g t o be s a c r i f i c e d on t h e s h r i n e s of a m i l i t a n t mediocrit y . Abel speaks of Gauss's u n a p p r o a c h a b i l i t y ; Legendre of h i s c h u r l i s h n e s s . People who s t a r t w i t h t h a t opinion do n o t always f i n d a g r e a t man c o o p e r a t i v e : a r e h i s own problems l e s s worthw h i l e t h a n t h e i r s ? Perhaps he f i n d s i t irksome t o descend t o t h e p e d e s t r i a n l e v e l of t h e i r t h i n k i n g , which i g n o r e s f i n e r p o i n t s as " p u n c t i l i o u s n e s s , " s o t h a t t h e i r b a s i c assumptions a r e a l r e a d y i n s u l t i n g . Without t h e " p u n c t i l i o u s n e s s , I 1 G a u s s ' s Fundamental Theorem of Algebra could be a t t r i b u t e d t o d'Alembert who gave a "proof of s o r t s " i n t h e H i s t o i r e de 1'Acaddmie de B e r l i n i n 1 7 4 6 , o r even t o t h e u n d i s t i n g u i s h e d A l b e r t G i r a r d , who guessed t h e theorem i n 1 6 2 9 l (cf end s e c t . 2 3 ) . I t took mathematicians a good p a r t of t h e XIX-th c e n t u r y t o c a t c h up w i t h Gauss. For him, such p i o n e e r s as Legendre, Abel, G a l o i s , Cauchy h i m s e l f , s t i l l h a r d l y come up t o h i s conception of a p r o o f : we t o o , p e r h a p s , would n o t do s o , we a r e t o o n e a r our own times t o judge of them i m p a r t i a l l y . We a r e a t l a s t emerging from t h e s t a g e of two k i n d s of mathematics - - one i n which t h i n g s a r e proved, and one i n which they a r e " v e r i f i e d . " Mathematicians proper r e c o g n i s e only t h e former, f o r them i t would be a s i f I had spoken o f two medicines - - t h e orthodox and t h e quack. However, a t t h e time of Gauss t h i n g s were n o t so s i m p l e . For him t h e r e was only one mathematics - - i t i s what I t h i n k t o o . He worked i n many of i t s f i e l d s : t h e r e a r e people who r e g r e t t h i s , and who f e e l t h a t i n some of them he

The Romantic Period

149

sacrificed rigour. But for him, perhaps, rigour was not so much our rigidly inflexible techniques of a now routine and mechanical formal logic, as a constant, and flexible, checking and reinforcing, a perpetual probing for weak spots and improvements, what we in Wisconsin like to call "sifting and winnowing.''

In astronomy Gaussian rigour took the form of great numerical accuracy: like all astronomers since Ptolemy, Gauss used, in his approximations, infinite series, such as trigonometric ones, of which he simply calculated as many terms as he thought necessary for the appropriate numerical approximation. Such series are deceptive: there is no certainty that they converge; if they do, the initial terms may still be substantially smaller than some group of much later terms taken together. For us this is not altogether satisfactory. Whether Gauss was aware of the difficulty, we do not know: such fundamental questions did not bother his colleagues in astronomy, but they might bother him; they might also act as a deterrent to publication, since the time and effort needed would be greatly increased, if he was to put his work on a completely sound mathematical basis, and since this would not be appreciated by astronomers. We do not know how much Gauss held back of his astronomical work. We know that he held back a great deal in other fields, things only found very late in his Nachlass, after they had been rediscovered independently. Indeed a good deal of the work of the XIX-th century consisted in rediscovering what Gauss knew. It appears, for instance, that Gauss had early given thought to non-Euclidean geometry: in 1818 he wrote to a friend about a new geometry that a certain Schweikart claimed to have discovered, and that he termed "Astralgeometry." This was non-Euclidean geometry, and Schweikart went so far as to propose an empirical test to decide which geometry occurs in our world. Gauss praised Schweikart's ideas, but warned against publication: there would be a great clamour of Beotians', the wasps would buzz and fly all around his ears (cf end sect. 23).

I mentioned that Gauss's first astronomical work began in 1801, when he had just published his Disquisitiones Arithmeticae. Today the two topics are not nearly so far apart as they must have seemed in 1801: astronomers have used.trigonometric series ever since Ptolemy, and trigonometric sums are now basic for estimates in the theory of numbers. Gauss's Disquisitiones Arithmeticae are in three parts. Parts I and I11 deal with matters I have already discussed, except that the proof of the "Theorema aureum" is much improved, and that its appearance caused the priority fight with Legendre. It is Part I1 that is exciting by modern standards. Two of its topics involve long-standing problems, solved only in my time. Let n be, for suitable integers x,y, the value of the quadratic form ax2

+

bxy

+

cy2,

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Chapter I

where the coefficients a,b,c are given integers. If we substitute x*

=

Lx

+ My,

y" = Nx

+ Py,

where L,M,N,P are integers subject to LP - MN = 1, then for these integers xJc,y*, the integer n will also be the value of a*x>k2 + b>kxJk + cfy7k2, obtained from the original quadratic form by the inverse substitution x

=

px>k - My>k, y = -Nx*

+ Ly".

For this reason, the two quadratic forms are termed equivalent3 (cf end sect. 23). They then have the same discriminant b2

- 4ac

=

b>'' a n d N . B o h r , Jess e n , L a n d a u , C o u r a n t , B o r n , C a r a t h e o d o r y , F r e c h e t , H . Hasse ( 1 8 9 8 - 1 9 7 9 ) , von Neumann, V e b l e n , t h e t w o B i r k h o f f s , W i e n e r ,

294

C h a p t e r I11

N . L e v i n s o n ( 1 9 1 2 - 1 9 7 5 ) , A . D r e s d e n (1882 - 1 9 5 4 ) , S p e n c e r , Kaczmarz, t r u l y a g a l a x y of m a t h e m a t i c i a n s ! And t h e r e were young p e o p l e , l a t e r well-known: S . B o c h n e r , P . E r d o s , R . Baer, H . H e i l b r o n n ( 1 9 1 1 - 1 9 7 5 ) , K . M a h l e r , B . Neumann, W . Rogosins k i , R . Rado, K . H i r s c h , K . H a r t l e y , F . W o l f , A . E r d e l y i ( 1 9 0 8 1 9 7 7 ) , S . U l a m , S . E i l e n b e r g , O l g a T a u s s k y ( l a t e r M r s . John T o d d ) . Some were i n Cambridge o n l y f o r f r e q u e n t s h o r t v i s i t s -E r d o s a n d Mahler were i n M a n c h e s t e r w i t h t h e v e r y a c t i v e s p o n s o r s h i p o f L . J . M o r d e l l (1888 - 1 9 7 2 ) , who h i m s e l f d i d a t r e m e n d o u s amount f o r B r i t i s h m a t h e m a t i c s . T h e r e were f i n e young Camb r i d g e s t u d e n t s , t o o : I o u g h t t o know, s i n c e I g a v e , f o r h a l f a d o z e n y e a r s , a d v a n c e d l e c t u r e s t o c l a s s e s o f some 2 5 o f t h e b e s t a n a l y s t s . I h a v e m e n t i o n e d t h e most f a i t h f u l , A . J . Ward, and t h e r e w a s my p u p i l E . R . Love. O t h e r f i n e a n a l y s t s , my s t u d e n t s o r c o n t e m p o r a r i e s , were S m i t h i e s , P i t t , Miss C a r t w r i g h t , J o h n Todd. From o t h e r f i e l d s , C o x e t e r , C . A . CGulson ( 1 9 1 0 1 9 7 4 ) , Temple, DuVal, J a e g e r , A . H . W i l s o n , S e m p l e , J . A . Todd, R . H . S t o y , J . C . P . M i l l e r , h e l p e d t o make a f i n e m i x t u r e of v e r y d i f f e r e n t i d e a s and p o i n t s o f view. A w e a l t h of t a l e n t s t a r t e d t o bloom: w e c a n i m a g i n e t h e f e e l i n g s of a man l i k e H a r d y , who had d o n e so much t o make i t p o s s i b l e . B u t t h e h u r r i c a n e s t r u c k : much p r o m i s e d b l o o m i n g r e m a i n e d what m i g h t h a v e b e e n . And what of t h e f e e l i n g s of L i t t l e w o o d ? For him w e had a s p e c i a l a f f e c t i o n , t h e more s o f o r h i s h a v i n g handed over t h e m a j o r r o l e t o H a r d y , j u s t a s my r e s p e c t f o r F o w l e r was a l l t h e g r e a t e r a f t e r h i s p u p i l Dirac had b e e n made p r o f e s s o r i n p r e f e r e n c e t o hims e l f . The c h o i c e of Hardy and Dirac w a s s u b s e q u e n t l y more t h a n j u s t i f i e d by t h e i n t e r n a t i o n a l r e p u t a t i o n t h a t Cambridge acq u i r e d . I t a l s o a d d e d a l l t h e more t o t h e s t a t u r e o f t h e o t h e r s who had worked f o r t h i s .

For my p a r t , t o h a v e s e e n i t a l l grow i n t h e g o l d e n y e a r s was a n i n t e l l e c t u a l e x p e r i e n c e , t h e l i k e o f w h i c h w i l l p e r h a p s neve r r e c u r . I t was a l s o much more t h a n t h a t . Along w i t h t h e t h e o r e t i c a l honesty t h a t , as I q u o t e d , an atmosphere o f i n t e l l e c t u a l d e d i c a t i o n b r i n g s , I had o c c a s i o n t o a d m i r e t h e n o b i l i t y o f a d v e r s i t y . I made many f r i e n d s among t h e r e f u g e e s : I n e v e r h e a r d them c o m p l a i n o f t h e i r l o t . I was s o r r y t o l e a v e f o r t h e c o m p a r a t i v e s e c u r i t y o f a d i s t a n t S o u t h A f r i c a and t h e H e a d s h i p o f a d e p a r t m e n t . The w o r d s o f Mercury s o u n d h a r s h a f t e r t h e s o n g s of A p o l l o : "You, t h a t way -- we, t h i s way." 'In Germany, the greater tensions were actually postwar. Behnke mentions two instances at the normally very comfortable University of Heidelberg. There, in 1924, E.J. Grumbel (1891 - 1966), professor of statistics, was dismissed for referring as pacifist to the "field of dishonour" on which World War I soldiers had fallen. And, in 1922, the physicist Lenard was fined for not dismissing classes on the day of mourning for the assassinated foreign minister Rathenau.

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3 4 . The b a t t l e g r o u n d o f i d e a s ,

Great t h o u g h i t w a s , t h e m a t h e m a t i c a l d e v e l o p m e n t o f Cambridge b e t w e e n t h e t w o wars f a i l e d t o do j u s t i c e t o t w o t h e m e s t h a t seem most i m p o r t a n t t o d a y . When a n i d e a h a s l o n g b e e n d e n i e d i t s p l a c e , i t i s n o t enough f i n a l l y t o a c c e p t i t a n d t o u s e i t , g r u d g i n g l y as i f d i r t y i n g o n e ' s h a n d s ; t h a t n e g l e c t e d i d e a i s where p r o g r e s s l i e s , where f u t u r e development h a s its best poss i b i l i t i e s . A few p e o p l e , my f a t h e r among t h e m , r e a l i s e d t h i s : i t w a s t h e d r i v i n g f o r c e of h i s many p a p e r s . H e s t r e s s e d a f u r t h e r p o i n t i n h i s 1924 r e t i r i n g P r e s i d e n t i a l a d d r e s s t o t h e London M a t h e m a t i c a l S o c i e t y : "The r e l u c t a n c e w i t h w h i c h t h e T h e o r y of S e t s w a s a l l o w e d i t s i n d i v i d u a l p l a c e i n t h e scheme o f m a t h e m a t i c s w a s b u t a symbol o f t h e same s p i r i t w h i c h , i n s c i e n t i f i c s t u d i e s g e n e r a l l y , was d i c t a t i n g t h e demand t o r e l e g a t e m a t h e m a t i c s t o t h e r a n k o f handmaiden o f t h e o t h e r s c i e n c e s ( H i l f s w i s s e n s c h a f t ) " . Once t h i s a t t i t u d e i s a c c e p t e d , t h e r e f o l l o w demands t h a t s c i e n c e become t h e handmaiden o f p o l i t i c s , a n d p o l i t i c s t h a t o f some c l i q u e , a n d s o o n t h e w h o l e w o r l d i s t h r e a t e n e d , T h u s e v e n t s i n m a t h e m a t i c s c a n n o t be s e p a r a t e d f r o m what g o e s on a r o u n d u s . We wonder a t t h e XIXth C e n t u r y d e v o t i o n t o f o r m u l a e i n E l l i p t i c F u n c t i o n s , which L i t t l e w o o d r e f e r s t o as a d i c t i o n a r y s u b j e c t . P e o p l e keep g r i n d i n g t h e s e o u t , p a g e s and p a g e s o f them, i n s t e a d o f r e g a r d i n g them a n d t h e i r a n a l o g u e s f o r A b e l i a n Funct i o n s , a s mere i l l u s t r a t i o n s o f R i e m a n n ' s m u l t i v a l u e d a n a l y t i c f u n c t i o n s and o f t h e i r Riemann s u r f a c e s . F o r m a l work w a s a h o l d o v e r , l i k e t h e f a d e d e l e g a n c e o f a Royal C o u r t amid s t t h e v i g o r o u s R o m a n t i c c r e a t i v i t y . Y e t E l l i p t i c F u n c t i o n s were resp o n s i b l e f o r a sudden acute i n t e r e s t i n s i n g l e - v a l u e d a n a l y t i c f u n c t i o n s , by l e a d i n g t o t h e q u i t e s t a r t l i n g theorem o f P i c a r d . A p p a r e n t l y , E l l i p t i c F u n c t i o n s were u s e d i n a n e s s e n t i a l way i n t h e p r o o f . E x p e r t s b e g a n , more and m o r e , t o d e v o t e t h e i r accumu l a t e d knowledge o f E l l i p t i c F u n c t i o n s t o s t u d y i n g a n a l y t i c f u n c t i o n s . Newcomers, w i t h t h e f u l l e n c o u r a g e m e n t o f P i c a r d h i m s e l f , t r i e d t o d o t h e same w i t h o u t E l l i p t i c F u n c t i o n s . The E l l i p t i c Function i n P i c a r d ' s proof ( t h e E l l i p t i c modular funct i o n ) , occurs s o l e l y . b e c a u s e it s o l v e s a l i t t l e problem, t h a t o f mapping c o n f o r m a l l y o n a d i s c a z e r o - a n g l e d c i r c u l a r t r i a n g l e , d e t e r m i n e d by 3 c i r c u l a r a r c s , t o u c h i n g i n p a i r s a t t h e v e r t i ces. I t w a s some t i m e b e f o r e p e o p l e r e a l i s e d t h a t t h e p r o b l e m was i m p o r t a n t , n o t t h e s o l u t i o n . P i c a r d found h i s theorem i n 1879. Bore1 gave t h e f i r s t " e l e m e n t a r y " p r o o f i n 1896. From 1904 onwards, r e f i n e m e n t s began t o a p p e a r , b y L a n d a u , S c h o t t k y , C a r a t h g o d o r y . Landau p r o v e d t h a t e v e r y power seriesCanz" whose f i r s t two C o e f f i c i e n t s a u , a l < a r eg i v e n is -- i n a f i x e d o p e n d i s c I zI R ( a o , a l ) -F i g . 9: Zero-angled somewhere, 0 or 1 or nonconvergent, circular triangle q u i t e i r r e s p e c t i v e o f how t h e r e m a i n i n g c o e f f i c i e n t s are c h o s e n .

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The a n a l y s t s who worked on s u c h q u e s t i o n s a b o u t 30 y e a r s o r more a g o a f t e r P i c a r d p r o d u c e d h i s t h e o r e m s t r i e d t o f r e e them

206

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from t h e now u n f a s h i o n a b l e t o p i c of E l l i p t i c F u n c t i o n s . A f e w t r i e d u s i n g i n s t e a d t h e l a t e s t S e t T h e o r y , Real V a r i a b l e a n d F u n c t i o n a l A n a l y s i s . Complex a n a l y t i c f u n c t i o n s became i n t h i s way a t e s t i n g g r o u n d f o r t h e new m e t h o d s . O l d - f a s h i o n e d E l l i p t i c F u n c t i o n s must h a v e some v a l u e i f t h e y were u s e f u l i n p r o v i n g P i c a r d ' s t h e o r e m : were t h e new m e t h o d s e q u a l l y v a l u a b l e ? I t was l i k e s a y i n g , "Our r a c e h o r s e s managed v e r y w e l l when t h e y were s h o d by t h e v i l l a g e s m i t h , c a n y o u r modern t e c h n o l o g y do a n y b e t t e r i n h e l p i n g t o win races?" What a t e s t f o r t e c h n o l o g y ! Any t e c h n i c i a n n o t h e l p i n g i n h o r s e races i s r e g a r d e d l i k e t h e s m i t h ' s h a l f - w i t t e d h e l p e r . N e v e r t h e l e s s , some i m p o r t a n t d e v e l opments o f S e t Theory and F u n c t i o n a l A n a l y s i s a r o s e from t h e s e a r c h f o r a p p l i c a t i o n s t o a n a l y t i c f u n c t i o n s . F o r t h e new f i e l d s a C i n d e r e l l a r o l e w a s b e t t e r t h a n n o n e , c o n s i d e r i n g how h a r d i t s o m e t i m e s i s t o g e t new i d e a s a c c e p t e d a t a l l . Some s e e d s p l a n t e d i n t h e m i n d s of men t a k e r o o t a n d s p r e a d a l most a t o n c e , a n d t h e r e i s n o c o n t r o l l i n g them. T h e r e a r e b e t t e r s e e d s , a n d t h e y m o s t l y c o n c e r n u s h e r e , b u t t h e y seem b a r e l y t o s u r v i v e , a n d p e r h a p s l a y d o r m a n t f o r l o n g empty p e r i o d s o f t i m e , o r i n t h e s t o r m s a n d stresses o f wars a n d r e v o l u t i o n s . Y e t i n t h e e n d t h e y may come t o d o m i n a t e o u r t h i n k i n g : w e may wonder t h a t t h e y d i d n o t l o n g b e f o r e . T h u s i t i s t h a t w h a t comes a f t e r g i v e s s i g n i f i c a n c e and v a l u e a t l a s t t o what came l o n g a g o . I n t h i s way t h e S e t T h e o r y a n d F u n c t i o n a l A n a l y s i s o f t o d a y a t long last b r i n g t o f r u i t i o n t h e philosophy of P l a t o and t h e p o l e s a n d p o l a r s of A p o l l o n i u s .

Even s o , t h e r e was s o m e t h i n g m i s s i n g f r o m b o t h P l a t o and Apoll o n i u s t h a t s l o w l y h a d t o e v o l v e . The d u a l i t y o f p o l e s a n d pol a r s w a s f o r g o t t e n w i t h many o t h e r t h i n g s u n t i l D e s a r g u e s , f o r g o t t e n a g a i n and r e i n t r o d u c e d s e v e r a l t i m e s , l a r g e l y b e c a u s e , i n e l e m e n t a r y g e o m e t r y , d u a l i t y becomes i n t e r e s t i n g o n l y a f t e r i t a t t a i n s a s y m m e t r y , t a n t a m o u n t t o t h e i n t r o d u c t i o n o f Ep l e x numbers. S i m i l a r l y , t h e n o t i o n o f s e t , t h a t P l a t o s t u d i e d i n a n a l y s i n g l i k e n e s s , becomes more i n t e r e s t i n g m a t h e m a t i c a l l y when c o u p l e d w i t h t h e c o n c e p t o f o r d e r . We t h e n h a v e n o t j u s t s e t s b u t o r d e r e d sets: w e c o u l d g o on t o p a r t i a l l y o r d e r e d s e t s , but l e t m e n o t c o m p l i c a t e t h i n g s . A s e t b y i t s e l f c o r r e s p o n d s t o P l a t o ' s n o t i o n o f l i k e n e s s , o r s i m i l a r i t y , or equivalence: w e can t h i n k o f i t a s r e p r e s e n t e d by t h e e q u a l i t y s i g n = , a n d s p e a k of i t a s a n e q u a l i t y r e l a t i o n b e t w e e n e l e m e n t s . Then a n o r d e r e d s e t becomes a n o t i o n o f i n e q u a l i t y . I t i s , of c o u r s e , as o l d as mathematics i t s e l f : i t o c c u r r e d i n t h e classical i s o p e r m e t r i c i n e q u a l i t y o f t h e l e g e n d o f t h e f o u n d a t i o n o f Cart h a g e , when Dido w a s a l l o w e d a s much l a n d as c o u l d b e e n c l o s e d by t h e h i d e o f a n o x . T h e r e i s a l s o , I a m t o l d , a n i m p l i e d i n e q u a l i t y f o r a t r i g o n o m e t r i c polynomial i n Ptolemy's p o s s i b l e c o n f i g u r a t i o n s o f t h e p l a n e t s . However, m a t h e m a t i c i a n s h a d t o h a v e a symbol f o r i n e q u a l i t y , a n d t h i s w e owe t o Thomas H a r r i o t (1560? - 1621), our o l d f r i e n d f r o m e a r l y i n t h e s e l e c t u r e s , I h a v e p u r p o s e l y p o s t p o n e d s a y i n g much a b o u t H a r r i o t : t h e i n t e r est i n him is r e c e n t , a n d t h e s t a n d a r d w o r k s o f m a t h e m a t i c s s a y almost n o t h i n g a b o u t him -- t h e y a t t a c h l i t t l e v a l u e t o h i s c o n t r i b u t i o n s . One book now e x i s t s w h o l l y d e v o t e d t o H a r r i o t ,

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b u t many d e t a i l s are c o n j e c t u r a l . H i s own w o r k s a r e m a i n l y unpubi s h e d b u t a v a i l a b l e a t t h e B r i t i s h Museum o r e l s e w h e r e . H a r r i o t i s much s t u d i e d by a " H a r r i o t S e m i n a r " , a g r o u p o f B r i t i s h a n d American h i s t o r i a n s o f m a t h e m a t i c s w h i c h h a s m e t f o r 1 0 y e a r s o r more. H a r r i o t s t u d i e d a t O x f o r d a n d j o i n e d , a f t e r g r a d u a t i o n , t h e h o u s e h o l d o f young Walter R a l e i g h , whom h e a s s i s t e d a n d adv i s e d . H e w a s s e n t t o t h e N e w World i n 1585 a n d w r o t e ' A B r i e f e

and T r u e R e p o r t o f t h e New Found Land o f V i r g i n i a ' ( L o n d o n , 1588), one of t h e main s o u r c e s used i n subsequent a c c o u n t s of American l i f e a n d p r o d u c t s f o r 2 c e n t u r i e s , About 1 5 9 3 , when S i r Walter R a l e i g h ' s f o r t u n e s were a t a low e b b , H a r r i o t came u n d e r t h e p a t r o n a g e o f Henry P e r c y , E a r l o f N o r t h u m b e r l a n d , a n d seems t o h a v e b e e n p a i d a handsome a l l o w a n c e f o r t h o s e d a y s , a m o u n t i n g t o some 8 0 p o u n d s p e r annum, f o r t h e rest o f h i s l i f e . H e d i v i ded h i s t i m e b e t w e e n t h e h o u s e h o l d s o f h i s two f r i e n d s , R a l e i g h a n d N o r t h u m b e r l a n d , b u t h e w a s f r e e t o p u r s u e h i s own mathernatical and s c i e n t i f i c i n t e r e s t s . H e had s e r v a n t s , s p a c e f o r e x p e r i m e n t a l l a b o r a t o r i e s , a n d an a b l e a s s i s t a n t , named T o o k e . H e bequeathed h i s p a p e r s t o t h e E a r l of Northumberland, w i t h i n s t r u c t i o n s t h a t t h e y b e g o n e o v e r by a c l e r g y m a n named T o r p o r l a y , who w a s s o m e t h i n g of a m a t h e m a t i c i a n h i m s e l f . A c o m p l e t e e d i t i o n o f H a r r i o t ' s c o l l e c t e d w o r k s s h o u l d c e r t a i n l y b e p u b l i s h e d . Here I s h a l l c o n t e n t myself w i t h b r i e f i n d i c a t i o n s . H a r r i o t w a s t h e f i r s t d i s c o v e r e r o f t h e l a w s of r e f l e c t i o n a n d r e f r a c t i o n o f l i g h t , g e n e r a l l y known a s S n e l l ' s l a w s . T h e s e S n e l l d i s c o v e r e d , no d o u b t i n d e p e n d e n t l y , i n 1 6 2 1 , t h e y e a r Harr i o t d i e d ; w h i l e Descartes' D i s c o u r s d e l a MBthode, w h e r e t h e l a w s are l i k e w i s e g i v e n , appeared o n l y i n 1637. Harriot d i s c o v e r e d t h e l a w s i n 1601. I n astronomy, Harriot observed s u n s p o t s and v e r y a c c u r a t e l y c a l c u l a t e d t h e p e r i o d of s o l a r r o t a t i o n ; h e a l s o drew o n e o f t h e f i r s t maps o f t h e Moon, a n d h e t r a c e d t h e moons o f J u p i t e r . One work b y H a r r i o t w a s f i n a l l y p u b l i s h e d i n 1 6 3 1 , by T o r p o r l e y a n d o t h e r s , i n h i s name: t h e A r t i s a n a l y t i cae p r a x i s , a d a e q u a t i o n e s a l g e b r a i c a s n o v a e x p e d i t a , e t g e n e r a l i m e t h o d o , r e s o l v e n d a s . T h e r e i s f u r t h e r m a t e r i a l , on m i l i t a r y m a t t e r s , on p r o j e c t i l e s and i m p a c t , on s p e c i f i c g r a v i t i e s a n d weighing i n f l u i d s . Moreover, a l e t t e r from B r i g g s t o a f r i e n d i n d i c a t e s t h a t H a r r i o t possessed i n 1603 t h e formula f o r t h e area of a s p h e r i c a l t r i a n g l e , a r e s u l t p u b l i s h e d 26 y e a r s l a t e r by Albert G i r a r d . I mentioned t h a t Harriot d i d a g r e a t d e a l t o improve t h e a c c u r a c y of n a v i g a t i o n a l a s t r o n o m y . I n t h i s h e s t i l l u s e d t h e i n c o r rect P t o l e m a i c a s t r o n o m y , a l t h o u g h h e h i m s e l f w a s a C o p e r n i c a n . The error i n v o l v e d w a s s m a l l enough n o t t o l e a d n a v i g a t o r s a s t r a y . I t i s j u s t f o r s u c h estimates o f s m a l l n e s s t h a t i n e q u a l i t y s i g n s h a v e b e e n most u s e f u l f o r a l o n g p e r i o d : t h e y o c c u r , some 200 y e a r s a f t e r H a r r i o t , i n t h e Cauchy d e f i n i t i o n s o f conv e r g e n c e a n d c o n t i n u i t y ; t o s t a t e i n p r e c i s e terms what p e o p l e had p r e v i o u s l y b e e n u n a b l e t o refer t o e x c e p t i n v a g u e p i c t o r i a l terms, was q u i t e a n a c h i e v e m e n t , a n d o n e t h a t c o u l d h a r d l y h a v e b e e n a r r i v e d a t w i t h o u t H a r r i o t ' s s p e c i a l symbol o f i n e q u a l i t y . A s i m i l a r remark a p p l i e s t o t h e s u b s e q u e n t d e f i n i t i o n s by S t o k e s a n d H e i n e of u n i f o r m c o n v e r g e n c e and u n i f o r m c o n t i n u i t y . An e q u a l l y f u n d a m e n t a l u s e o f i n e q u a l i t y o c c u r r e d e a r l y i n

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t h e n o t i o n s o f maximum and minimum; t h e s e l e d i n t h e i r t u r n t o t h e C a l c u l u s o f V a r i a t i o n s a n d Modern O p t i m a l C o n t r o l , and t o a number o f b a s i c t o o l s o f a n a l y s i s t r e a t e d i n t h e famous book 'Inequalities' by H a r d y - L i t t l e w o o d - P o l y a , m e n t i o n e d e a r l i e r . Another fundamental use o c c u r r e d i n t h e l a t t e r h a l f o f t h e XIXth C e n t u r y , as a n o r d e r i n g o f r e a l n u m b e r s , i n t h e d e f i n i t i o n o f i r r a t i o n a l numbers by Dedekind " c u t s " . I m p o r t a n t t h o u g h t h e y were, t h e s e u s e s o f i n e q u a l i t y s i g n s were r a t h e r f e w : m a t h e m a t i c s had b e e n much l e s s c o n c e r n e d w i t h i n e q u a l i t i e s than with e q u a l i t i e s . This is logically r a t h e r s t r a n g e . What would w e t h i n k o f b e i n g s f r o m a n o t h e r w o r l d who happened t o v i s i t E n g l a n d a n d F r a n c e f o r a few c e n t u r i e s and who, i n t h a t t i m e , had c o l l e c t e d i n f o r m a t i o n o n l y a b o u t t h o s e o f u s who happened t o b e b o t h E n g l i s h a n d F r e n c h ? A r a t h e r e l u s i v e c l a s s o f ambiguous p e r s o n s , e q u a l l y E n g l i s h a n d F r e n c h ! A c l a s s f i t t i n g l y r e p r e s e n t e d b y t h e e l u s i v e i d e a l t h a t w e term e q u a l i t y ! But why s t u d y o n l y what may n o t e x i s t ? Why n o t t h e more n a t u r a l c l a s s of a l l E n g l i s h m e n , o r a l l Frenchmen? To a l low f o r t h e ambiguous c a s e , w e c a n a g r e e t o c a l l E n g l i s h someo n e a t l e a s t a s E n g l i s h a s F r e n c h , and s i m i l a r l y F r e n c h someone a t l e a s t a s F r e n c h as E n g l i s h , s o t h a t t h e two c l a s s e s would be a p p r o p r i a t e l y a s s o c i a t e d w i t h t h e s i g n s 5 , ? , T h e s e are sugg e s t i o n s e a s i l y made t o v i s i t o r s , b u t i t t a k e s a few h u n d r e d y e a r s t o a p p l y similar o n e s t o o u r s e l v e s . I n t h e case o f i n e q u a l i t y , t h i s h a s m e a n t , i n m a t h e m a t i c s , a whole p r o g r a m t h a t w e can r e g a r d a s t h e r e a l l e g a c y o f Thomas H a r r i o t : f o r e v e r y problem, theorem, a l g o r i t h m or d e f i n i t i o n i n v o l v i n g e q u a l i t y t o l o o k f o r a more o r l e s s a u t o m a t i c a n a l o g u e i n v o l v i n g o n e o f t h e a b o v e i n e q u a l i t y s i g n s . I c a n n o t q u i t e s a y when t h i s p r o g r a m s t a r t e d c o n s c i o u s l y , b u t a r o u n d 1900 i t was p r o c e e d i n g w i t h a r u s h , and t h i s w a s c e r t a i n l y l o n g a f t e r H a r r i o t ! I t i s sometimes a l o n g t i m e b e f o r e m a t h e m a t i c s b e n e f i t s i n f u l l from a r a t h e r simple invention. The most p o p u l a r items i n t h e program h e l p e d w i t h p a s t p r o b l e m s . I have a l r e a d y mentioned Minkowski's d e f i n i t i o n o f c o n v e x i t y : t h i s g e n e r a l i s e s l i n e a r i t y , j u s t a s .: g e n e r a l i s e s t h e = s i g n -it is s u f f i c i e n t t o s u b s t i t u t e , a t t h e a p p r o p r i a t e p l a c e i n t h e d e f i n i t i o n of l i n e a r i t y , t h i s i n e q u a l i t y s i g n f o r e q u a l i t y . I n f a c t , f o r c o n t i n u o u s r e a l f u n c t i o n s , l i n e a r i t y is e x p r e s s e d by t h e r e l a t i o n * f ( x ) + f r f ( y ) = f ( + x + By) , c o n v e x i t y b y h e r e s u b Alternatively, for a twice-differentiable s t i t u t i n g 1. f o r = f u n c t i o n f of a real x , a n d without t w i c e - d i f f e r e n t i a l i t y i f we use t h e language of Schwartz d i s t r i b u t i o n s , l i n e a r i t y is exp r e s s e d b y f"=O , c o n v e x i t y by f t t > 0 . For r e a l f u n c t i o n s o f s e v e r a l v a r i a b l e s , t h e r e i s a l s o a n a n a l o g u e d u e t o F . Riesz: f i s h a r m o n i c i f i t s L a p l a c i a n Af v a n i s h e s , s u b h a r m o n i c if A f > 0 . L i k e c o n v e x i t y , s u b h a r m o n i t y i s a c o n c e p t w i t h most v a l u z b l e a p p l i c a t i o n s , recent ones t o o , f o r i n s t a n c e , t h o s e o f B a e r n s t e i n , a f o r m e r W i s c o n s i n s t u d e n t . You see how s i m p l e i t a l l is: H a r r i o t f o r e v e r !

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Not a l l t h e items i n t h e p r o g r a m were a s p o p u l a r . I n t h e same s p i r i t about 100 y e a r s a g o , p e o p l e l i k e D i n i , du B o i s Reymond, D a r b o u x l , b r o u g h t i n , i n p l a c e o f l i m i t -- w h i c h n e e d n o t e x i s t -- t h e n o t i o n s o f u p p e r and lower l i m i t s . T h e n , i n p l a c e o f t h e

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Riemann i n t e g r a l , t h e r e were Darboux u p p e r and lower i n t e g r a l s ; and i n p l a c e o f d e r i v a t i v e s , D i n i u p p e r a n d lower d e r i v a t i v e s on t h e r i g h t , a n d s i m i l a r l y on t h e l e f t . By t h o s e who t o o k a s u p e r f i c i a l v i e w , t h i s w a s no more a p p r e c i a t e d t h a n J o r d a n ' s t h e o r e m -- i t w a s t o o l i k e d o t t i n g t h e i ' s a n d c r o s s i n g t h e t ' s . Y e t i t l e d t o o u r Real Variable t h e o r y : a f t e r a l l t h i s , i t d o e s not r e q u i r e a g r e a t e f f o r t of imagination t o define, i n place o f c o n t i n u i t y , n o t i o n s o f u p p e r and o f l o w e r s e m i c o n t i n u i t y : s i n c e w e a r e now w e l l i n t o t h e c o n c e p t u a l a g e , b e g i n n e r s a n d n o n - m a t h e m a t i c i a n s may o b t a i n much s a t i s f a c t i o n b y t a k i n g a l o o k a t t h e Cauchy c o n t i n u i t y d e f i n i t i o n s , and t r y i n g t o m o d i f y them i n t h i s way. I t c o s t s n o t h i n g t o f i n d p e r h a p s t h a t you might have been g r e a t mathematicians a c e n t u r y ago! But t h i s is j u s t w h e r e t h e r e was s o much o p p o s i t i o n : i t w a s c o n s i d e r e d f a r f e t c h e d r i g h t u p t o t h e e n d o f World War I . For a d m i r e r s o f P o i n c a r g , i t w a s u n w o r t h y o f a v a t h e m a t i c i a n t o waste t i m e o n such impractical things: d i s c o n t i n u i t y w a s contrary t o nature. The g r e a t Ren6 Baire ( 1 8 7 4 - 1 9 3 2 ) , who d e v i s e d u p p e r a n d lower s e m i c o n t i n u i t y , a?d who l e c t u r e d i n P a r i s o n d i s c o n t i n u o u s f u n c t i o n s at t h e C o l l e g e d e France i n t h e academic y e a r 1903 - 1904, was r e l e g a t e d t o a p r o v i n c i a l b a c k w a t e r o f a u n i v e r s i t y a t D i j o n , f r o m w h i c h h e s o o n had t o r e t i r e i n a n e x h a u s t e d s t a t e , a s a p o s s i b l e h o s t a g e , l i k e Georg C a n t o r , t o t h e C o n s e r v a t i o n of I g n o r a n c e . Y e t now s c i e n t i s t s h a v e c e a s e d t o b e l i e v e n a t u r e c o n t i n u o u s : t h e l a w s o f n a t u r e , Dirac s a y s , c o n t r o l a s u b s t r a tum o f w h i c h w e c a n n o t form a m e n t a l p i c t u r e w i t h o u t i r r e l e v a n cies.

D i r a c ' s statement w i l l r e m i n d u s o f t h e d i s t o r t i o n s t h a t t h e human b r a i n i m p o s e s on i n f o r m a t i o n from o u r s e n s e s , a s I e x p l a i n e d i n t h e i n t r o d u c t i o n . I t d o e s n o t mean t h a t t h e w o r l d o f what e x i s t s i s w i d e r t h a n t h e w o r l d o f t h e t h i n k a b l e . M e n t a l p i c t u r e s , t h e l i t t l e models w e c o n s t r u c t i n our m i n d s , are o v e r s i m p l i f i e d and padded w i t h i r r e l e v a n c i e s , B u t d i f f e r e n t m e n t a l p i c t u r e s need not contain t h e imperfections: otherwise o u r m i n d s would h a v e b u i l t i n c e r t a i n p e r m a n e n t p r e j u d i c e s , a n d t h e v e r y s u p p o s i t i o n would be f r e e o f s u c h p r e j u d i c e s , a n d c o n tradict itself.

Reng B a i r e w a s t h e most o r i g i n a l a n d c r e a t i v e o f a s m a l l g r o u p of S e t T h e o r i s t s , i n t h e f i r s t u n c e r t a i n b l o o m i n g o f t h e s e e d s sown b y Georg C a n t o r . T h e i r s w a s a c o m r a d e s h i p o f p i o n e e r s , n o t u n l i k e t h a t of t h e B o u r b a k i g r o u p some 3 0 y e a r s l a t e r i n F r a n c e . I t w a s not without a struggle t h a t t h e i r ideas, especially t h o s e o f B a i r e , were f i n a l l y a c c e p t e d . We c a n be happy when p i o n e e r s a r e r e c o g n i s e d , however b e l a t e d l y , d u r i n g t h e i r l i f e t i m e : t h e y p l a n t t h e s e e d s , t h e i r successors r e a p t h e honours. I s p e a k o f a c c e p t a n c e b y t h e i r own c o l l e a g u e s , t h e i r f e l l o w mathematicians o r f e l l o w s c i e n t i s t s . I h a v e n o t c o u n t e d h a r d s h i p s from o u t s i d e , i n p e r i o d s s u c h a s w e h a v e b e e n t h r o u g h i n t h i s c e n t u r y : I must a t l e a s t g i v e some i d e a o f t h e s e h a r d s h i p s , a l t h o u g h t h e y were s h a r e d by p e r s o n s i n many w a l k s o f l i f e . I h a v e n o w i s h t o overwhelm my r e a d e r s w i t h t r a g e d i e s a l l a t o n c e , b u t w e must know some r e l e v a n t f a c t s , u n p l e a s a n t t h o u g h t h e y a r e , and y e t r e t a i n t h e c a p a c i t y t o l o o k a t them d i s p a s s i o n a t e l y a n d t o f i n d l o g i c i n t h e i r a b s u r d i t y . O t h e r w i s e w e s h a l l never l e a r n t o c o n t r o l them.

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Those who worked i n S e t T h e o r y o r i t s a p p l i c a t i o n s b e f o r e t h e f i r s t war i n c l u d e d , b e s i d e s B o r e l , L e b e s g u e , d e l a Vallee Pouss i n , my p a r e n t s , L . E . J . B r o u w e r , C a r a t h 6 o d o r y and o t h e r s I h a v e m e n t i o n e d , t h e German f o l l o w i n g o f Georg C a n t o r , p e o p l e l i k e Arthur S c h o e n f l i e s s (1853 - 1 9 2 8 ) , F e l i x Hausdorff (1868 - 1 9 4 2 ) , E. Zermelo ( 1 8 7 1 - 1 9 5 3 ) , a n d E r h a r d S c h m i d t ( 1 8 7 6 - 1 9 5 9 ) . Most o f t h e s e p i o n e e r s I m e t , and I a l s o knew some p i o n e e r s o f t h e 1 9 2 0 ' s , among them my f r i e n d F . H a r t o g s ( 1 8 7 4 - 1 9 4 3 ) , o n e o f t h e f o u n d e r s of t h e modern t h e o r y o f complex a n a l y t i c f u n c t i o n s o f s e v e r a l complex v a r i a b l e s , a s u b j e c t t h a t l a t e r d e v e l o p e d g r e a t l y t h a n k s t o Behnke, B o c h n e r , and t h e i r p u p i l s a n d f o l l o w e r s . I w e l l remember m e e t i n g H a r t o g s . A f t e r t h e T r i p o s P a r t I i n 1 9 2 6 , I a r r i v e d i n Munich l a t e f o r t h e Summer S e m e s t e r , b u t e a r l y f o r H a r t o g s ' s s e c o n d l e c t u r e on h i s f a v o u r i t e s u b j e c t . The room w a s empty, I became i t s s o l e o c c u p a n t t i l l Tietze j o i n e d me; t h e n a n o s e a p p e a r e d , a s H a r t o g s p e e p e d i n . H e s a i d "Quick, t h e r e a r e no s t u d e n t s , f o l l o w me.'' H e g a v e h i s w h o l e c o u r s e t h a t a f t e r n o o n on a p a r k b e n c h , a n d w e wound u p i n a l i t t l e c a f 6 . T h i s same u n p r e t e n t i o u s p i o n e e r o f a g r e a t s u b j e c t was l a t e r o n e of t h e many t o commit s u i c i d e w i t h o u t w a i t i n g f o r arrest b y g o o s e - s t e p p e r s . So w a s o u r f r i e n d G . P i c k , t h e p u p i l of F e l i x K l e i n i n L e i p z i g , a f t e r b e i n g p r o f e s s o r a t Prague; so were a l s o t h e m o t h e r and s i s t e r o f P i c k ' s f r i e n d P . Kuhn, who had h i m s e l f e s c a p e d t o Norway and Sweden. I f you s t u d y modern m a t h e m a t i c s , you w i l l h e a r o f H a u s d o r f f s p a c ' e s , and p e r h a p s o f t h e Hausdorff-Young i n e q u a l i t i e s i n F o u r i e r s e r i e s . The s p a c e s were a c t u a l l y i n t r o d u c e d i n a famous H i l b e r t f o o t n o t e , and H a u s d o r f f ' s p a r t o f t h e i n e q u a l i t i e s was t h a t of a p r o f e s s i o n a l , r a t h e r t h a n a p i o n e e r . But h e was a g r e a t man, and h i s 1 9 1 4 book on S e t T h e o r y was a m a s t e r p i e c e compared w i t h t h e p r e v i o u s s u r v e y by S c h o e n f l i e s s . H a u s d o r f f was b o r n i n Bresl a u ; h i s s c h o o l a n d e a r l y s t u d e n t y e a r s were a t L e i p z i g ; h e w e n t on t o F r e i b u r g a n d t o h i s d o c t o r a t e i n B e r l i n i n 1 8 9 1 ; h e m a r r i e d i n 1899. H e w a s m u s i c a l and almost c h o s e m u s i c r a t h e r t h a n m a t h e m a t i c s ; h e a l s o had a g i f t f o r w r i t i n g , a n d p u b l i s h e d v e r s e s a n d a s a t i r i c p l a y u n d e r a p s e u d o n y m -- h i s sarcasm w a s l a t e r d r e a d e d whenever h e was an e x a m i n e r . Such d e t a i l s I h a v e a t second hand: I n e v e r m e t him. H e r e t u r n e d t o L e i p z i g i n 1902 as a s s o c i a t e , d e c l i n e d t h a t p o s i t i o n i n Goettingen, accepted it i n Bonn i n 1 9 1 0 , and e n d e d u p i n 1 9 1 3 w i t h a p r o f e s s o r s h i p a t G r e i s s w a l d , from w h i c h h e was f o r c i b l y r e t i r e d i n 1 9 3 5 . Greissw a l d was n o t t h e g r e a t e s t u n i v e r s i t y on t h e w o r l d , b u t H a u s d o r f f was w e l l r e g a r d e d , a n d when i n World War I1 t h e S w i s s Government o f f i c i a l l y a s k e d f o r p e r m i s s i o n f o r him t o come t o S w i t z e r l a n d , t h e y were a s s u r e d t h a t h e w a s i n no d a n g e r . A f o r t n i g h t l a t e r , an i n f o r m e r d e n o u n c e d him. Now n o t h i n g could s a v e h i m . L i t t l e p l a c e s l i k e Greisswald cannot f i g h t t h e machinery o f a S t a t e , c l a m o u r i n g f o r o n e J e w t h e l e s s , o n e fewer a r i s t o c r a t o f i n t e l l e c t , o n e l e s s e l i t i s t . A l l t h a t c o u l d be d o n e was t o h a v e him s e n t t o T h e r e s i e n s t a d t , a p l a c e f o r " b e t t e r " J e w s , a Czech town t h a t had b e e n c l e a r e d o f i t s i n h a b i t a n t s . When o n e J e w was t a k e n t h e r e , a n o t h e r w a s removed. T h e r e H a u s d o r f f and h i s w i f e c o m m i t t e d s u i c i d e . I t was d o u b t l e s s r e g r e t t e d , b u t what i s one Jew t h e l e s s , e v e n i f h e i s a b e t t e r J e w ? What c a n a m a t h e m a t i c i a n s a y t o such people? A s i m i l a r problem occured r e c e n t l y , when a young p h y s i c i s t was k i l l e d a t W i s c o n s i n i n a bombing,

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a n d it w a s o n e e l i t i s t t h e less. I n A m e r i c a , a t l e a s t , t h e r e p l y c a n be t h a t J e w o r n o t J e w , e l i t i s t o r n o t , h e i s one f e w e r p i o n e e r t o l e a d u s t o t h e b e t t e r w o r l d o f a s t i l l d i s t a n t tomorrow i n which w e s h a l l n o t need r e m i n d i n g o f t h e words a t t h e f e e t o f a g r e a t s t a t u e , "Give m e y o u r t i r e d , y o u r p o o r , y o u r h u d d l e d masses y e a r n i n g t o b e f r e e " . F o r a m a t h e m a t i c i a n o r a s c i e n t i s t , f o r a n y o n e who f e e l s a g r e a t d r i v e t o p r o g r e s s a n d t o e v o l v e , t h e freedom t h a t matters l i e s i n t h e e v o l u t i o n from a n a p e t o s o m e t h i n g b e t t e r t h a n a man: t h i s f r e e d o m is s e t b a c k b y t h e l o s s o f a p i o n e e r , whose crime i s t o h a v e a r i s e n m o m e n t a r i l y above a g e n e r a l a b s u r d i t y . Zermelo a n d E r h a r d S c h m i d t were c l o s e t o H a u s d o r f f i n S e t T h e o r y , and a l s o g r e a t f r i e n d s o f C a r a t h g o d o r y from t h e i r s t u d e n t d a y s i n G o e t t i n g e n . E r h a r d S c h m i d t h a d been w i t h C a r a t h 5 o d o r y a n d L. F e j e r i n B e r l i n , i n t h e S e m i n a r of H . A . S c h w a r z , a n d h a d i n d u c e d C a r a t h g o d o r y t o j o i n him i n G o e t t i n g e n i n 1902. C a r a t h e o d o r y made f r i e n d s e a s i l y , t h e r e i s a w i d e l y known s t o r y o f a g r o u p h o l i d a y i n t h e H a r z , a n d o f a b a t h b e i n g somewhere c h a r g e d f o r , t h a t e a c h member o f t h e g r o u p s u b s e q u e n t l y c l a i m e d t o h a v e t a k e n -- I m y s e l f o n c e h e a r d C a r a t h g o d o r y h e a t e d l y i n s i s t i n g t h a t h e w a s t h e one. A s i n Lord H a l d a n e ' s s t u d e n t d a y s , a b a t h w a s i n d e e d a n u n h e a r d - o f l u x u r y . . . E r h a r d S c h m i d t e v e n t u a l l y became a g r e a t a n d c r e a t i v e t e a c h e r : h e c o u l d see a n a l o g i e s a n d c o n n e c t i o n s , a n d b r i n g t o l i f e what f o r him w a s t h u s " v i s i b l e T 1 . To E r h a r d S c h m i d t , Zermelo s a y s , w e owe t h e main i d e a o f Zermelo's famous axiom o f c h o i c e , a n axiom f a m i l i a r t o S e t T h e o r i s t s a n d a b s o l u t e l y n e c e s s a r y b e f o r e an abstract S e t Theory c o u l d emerge from t h e n a i v e t h e o r y of C a n t o r . The Z e r m e l o axiom i s r e l a t e d t o i n e q u a l i t y : it w a s proved e q u i v a l e n t t o t h e e x i s t e n c e i n any set of a n o r d e r r e l a t i o n s i m i l a r t o t h e o r d e r o f t h e t r a n s f i n i t e numbers. T h i s is Zermelo's t h e o r e m a s s e r t i n g t h e " w e l l - o r d e r i n g " of a s e t . I t s importance l i e s i n making s o - c a l l e d t r a n s f i n i t e i n d u c t i o n a p p l y t o a w e l l - o r d e r e d s e t . Thus i n d u c t i o n , o n e o f t h e f i r s t t h i n g s w e l e a r n , becomes a v a i l a b l e i n a n a b s t r a c t s e t ting. A b s t r a c t n e s s i s n o t a l w a y s p o p u l a r . P e o p l e a s k , "Why have a n abs t r a c t t h e o r y ? " I f you make t h e m i s t a k e o f r e p l y i n g , "Why n o t ? " , t h e n you r e a l l y h e a r about i t -- how w e must be c o l d , c a l c u l a t i n g m o n s t e r s , a s I m e n t i o n e d b e f o r e ( a n d t h a t H a r t o g s a n d Hausd o r f f were d o u b t l e s s a c c u s e d o f b e i n g ) , who i g n o r e t h e i n d i v i d u a l and d e s t r o y h i s f r e e d o m . I make a p o i n t . o f r e p l y i n g w i t h a d i f f e r e n t q u e s t i o n . Have you h e a r d o f t h e g e o m e t e r , who s t a t e s a theorem, t h i n k s hard f o r a w h i l e , and t h e n suddenly announces ' ' I t l s o b v i o u s T 1 ?U n l i k e t h e s i m i l a r l e g e n d , w h i c h I s a i d was v a r i o u s l y a t t r i b u t e d t o W i e n e r , t o H a r d y , e t c . , w h e r e i t is o u t o f c h a r a c t e r , t h i s , i n g e o m e t r y , i s a n e v e r y d a y o c c u r e n c e . What i t i l l u s t r a t e s is t h a t g e o m e t r i c a l n o t a t i o n h a s n o t r e a c h e d t h e s t a g e o f making o b v i o u s r e s u l t s i m m e d i a t e . Geometry h a s s o f a r p r o g r e s s e d o n l y t o t h e b e g i n n e r ' s s t a g e i n C h i n e s e , when t h e simplest things take a considerable t i m e t o express. Similarly, i n m u s i c , a f t e r y e a r s of s t u d y , t h e e l e m e n t a r y a d d i n g o f a few semitones t o t r a n s p o s e t o a d i f f e r e n t k e y may be a n y t h i n g b u t i n s t a n t a n e o u s . We c a n n o t s p e a k o f s u c h t r i v i a l i t i e s a s " d i f f i c u l t " : t h e y are m e r e l y a n n o y i n g l y t i m e - c o n s u m i n g , l i k e t h e

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y e a r l y t a x f o r m s . They a r i s e almost s o l e l y from t h e u s e o f a cumbrous n o t a t i o n , c o m p a r a b l e t o a g r e a t c l a n k i n g m a c h i n e w i t h a l l manner o f u n n e c e s s a r y p a r t s . S i m i l a r l y , i n t h e e a r l y d a y s o f a r i t h m e t i c , c a l c u l a t i o n s had t o b e made on o n e ' s f i n g e r s , i n t h e " c o n c r e t e " . S u r e l y t h a t i s n o t "freedom"? I t i s p r e c i s e l y t o f r e e u s from s u c h i r r i t a t i n g d e l a y s t h a t m a t h e m a t i c i a n s dev e l o p e d a b s t r a c t t h e o r i e s , i n much t h e same way t h a t B e e t h o v e n c o n t i n u e d t o compose m u s i c a f t e r h e had become d e a f . I t was Zermelo who s t a r t e d S e t T h e o r y on t h e r o a d t o a b s t r a c t n e s s . H e t o o k h i s d o c t o r a t e i n B e r l i n i n 1 8 9 4 , m i g r a t e d t o Goettingen f o r h i s h a b i l i t a t i o n i n 1899, l e f t f o r Zurich as professor i n 1 9 1 0 . U n f o r t u n a t e l y , h e c o n t r a c t e d l u n g t r o u b l e d u r i n g t h e War, a n d had t o r e s i g n i n 1 9 1 6 . H e l i v e d p r i v a t e l y i n t h e B l a c k F o r e s t f o r 1 0 y e a r s , a n d w a s made h o n o r a r y p r o f e s s o r a t n e a r b y F r e i b u r g i n 1 9 2 6 . H e was d e p r i v e d of h i s p r o f e s s o r s h i p i n 1 9 3 5 , f o r n o t g i v i n g t h e H i t l e r s a l u t e , b u t h e was r e i n s t a t e d i n 1 9 4 6 , I may s a y t h a t Zermelo w a s f a r from b e i n g t h e o n l y o n e t o h a v e p r o f i t e d from i d e a s t h r o w n o u t by E r h a r d S c h m i d t , a m a t h e m a t i c i a n m a i n l y remembered t o d a y as a H i l b e r t p u p i l who

p r o d u c e d a n o r t h o g o n a l i s i n g p r o c e s s i n H i l b e r t s p a c e , and who s o l v e d some i n t e g r a l e q u a t i o n s i n terms o f e x p a n s i o n s i n v o l v i n g e i g e n v a l u e s and e i g e n f u n c t i o n s . I n h i s t i m e , E r h a r d Schmidt w a s w e l l known t o p u b l i s h v e r y s l o w l y h i m s e l f , and t o l e a v e h i s pup i l s , f r i e n d s , c o l l a b o r a t o r s , t o make h i s c o n t r i b u t i o n s known a t l a r g e ; u n f o r t u n a t e l y , t h e o r a l t r a d i t i o n , w h i c h alone c o u l d g i v e an a d e q u a t e i d e a o f h i s i m p a c t o n h i s t i m e , is l a r g e l y l o s t . The same a p p l i e s e v e n more t o t h e m a t h e m a t i c i a n G u s t a v H e r g l o t z ( 1 8 8 1 - 1 9 5 3 ) , who became p r o f e s s o r o f P u r e and A p p l i e d M a t h e m a t i c s a t G o e t t i n g e n i n 1925, a n d who w a s l i k e w i s e a g r e a t f r i e n d of C a r a t h g o d o r y . E r h a r d Schmidt was b o r n a t T a r t u ( D o r p a t i n German) i n E s t h o n i a ; a f t e r s t u d y i n g i n B e r l i n , h e t o o k h i s d o c t o r a t e w i t h H i l b e r t i n G o e t t i n g e n i n 1 9 0 1 , and h i s h a b i l i t a t i o n i n Bonn f i v e y e a r s l a t e r . H e w a s s u c c e s s i v e l y p r o f e s sor i n Zurich i n 1908, i n Erlangen i n 1910, i n Breslau i n 1911, and i n B e r l i n i n 1 9 1 7 . A s a young m a t h e m a t i c i a n , h e would be a s k e d how h e managed t o f i n d t i m e . f o r w o r k , c o n s i d e r i n g t h a t h e seemed t o b e g o i n g f o r w a l k s a t a l l t i m e s o f t h e d a y ? H e would r e p l y t h a t m a t h e m a t i c s i s so h a r d t h a t o n e c a n n e v e r work a t i t f o r more t h a n a v e r y s h o r t t i m e ( T h i s is a l e s s o n t h a t some s t u d e n t s , p a r t i c u l a r l y women, n e e d t o bear i n m i n d ) . Even s o , t h e work o f Zermelo a n d E r h a r d S c h m i d t , i m p o r t a n t t h o u g h it w a s , a n d t h e f i n e work o f H a u s d o r f f , d o n o t compare w i t h t h e e x t r a o r d i n a r i l y f r u i t f u l i d e a s i n t r o d u c e d by B a i r e . To p e o p l e l i k e L e b e s g u e , F r f k h e t , T o n e l l i , h i s n o t i o n s o f semicont i n u i t y were a g o d s e n d , because i n s p a c e s o f f u n c t i o n s , s p a c e s o f c u r v e s , o f s u r f a c e s , t h a t were b e i n g s t u d i e d i n c o n n e c t i o n w i t h l e n g t h , a r e a , a n d t h e C a l c u l u s o f V a r i a t i o n s , t h e r e are almost no i n t e r e s t i n g q u a n t i t i e s t h a t depend c o n t i n u o u s l y on t h e v a r i a b l e f u n c t i o n , o r t h e v a r i a b l e curve or surface, i n t h e t r a d i t i o n a l sense. The e x a m p l e m e n t i o n e d e a r l i e r , due t o H . A . S c h w a r z , o f t h e i n s c r i b e d p o l y h e d r a of a c i r c u l a r c y l i n d e r , is n o t a c o n t i n u shows t h a t , e v e n i n t h e s i m p l e s t c a s e s , o u s l y v a r y i n g q u a n t i t y . I n t h e case o f c u r v e s , J o r d a n showed t h a t t h e l e n g t h of i n s c r i b e d p o l y g o n s d o e s t e n d t o t h a t o f t h e

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c o r r e s p o n d i n g c u r v e . However, when t h e c u r v e i s a p p r o x i m a t e d by polygons n o t i n s c r i b e d t o i t , t h e i r l e n g t h s need n o t t e n d t o t h a t of t h e c u r v e . T h i s s t a t e of a f f a i r s is e x p l a i n e d r a t h e r c h a r m i n g l y b y L e b e s g u e , i n h i s l i t t l e book " I n t h e m a r g i n of C a l c u l u s o f V a r i a t i o n s " , a n d t h e s t o r y may h e l p a n y r e a d e r who h a s h a d t r o u b l e f o l l o w i n g t h e r e m a r k s I h a v e made f r o m t i m e t o time a b o u t i n f i n i t e s i m a l z i g z a g s a n d g e n e r a l i s e d c u r v e s , a n d o r i g i n a l l y i n c o n n e c t i o n w i t h Descartes a n d F e r m a t , a n d w i t h S n e l l ' s l a w s ( t h a t w e now know t o h a v e b e e n f o u n d b y Thomas Harriot). I m e n t i o n e d t h a t , when L e b e s g u e w a s a t s c h o o l , a t t h e C o l l > g e d e B e a u v a i s , some classmates went a r o u n d c l a i m i n g t o p r o v e t h a t i n a n y t r i a n g l e ABC t h e s i d e BC i s t h e sum o f t h e o t h e r two s i d e s ! T h e i r " p r o o f " o f t h i s conundrum went a s f o l l o w s :

Let L be t h e broken l i n e BA + AC : l e t P , Q , R b e t h e m i d p o i n t s of t h e s i d e s BC,AC,AB , and l e t L' be t h e broken l i n e formed o f t h e 4 s eg men ts BR,RP,PQ,QC . By s i m i l a r t r i a n g l e s , RP a n d PQ e q u a l t h e h a l v e s AQ a n d AR o f AC a n d AB; h e n c e L ' h a s t h e same l e n g t h a s L . P r o c e e d i n g s i m i l a r l y i n each of t h e t r i a n g l e s RBP a n d QPC we o b t a i n 2 broken l i n e s o f 4 s i d e s e a c h , whose B P C sum L" h a s t h e same l e n g t h as L ' , a n d so a s L . I f w e c o n t i n u e i n t h i s manF i g . 9 : t h e t r i a n g l e conundrum n e r , we o b t a i n a sequence of broken l i n e s , a l l t h e same l e n g t h a s BA + CA , and i n t h e l i m i t t h e s e b r o k e n l i n e s become BC. A

L e b e s g u e g o e s o n t o s a y t h a t f o r h i s classmates t h i s w a s a l l a j o k e ; f o r him i t w a s m o s t d i s t u r b i n g -- h e c o u l d see no r e a l d i f f e r e n c e b e t w e e n i t a n d some l ' p r o o f s ' T t h a t h e w a s g e t t i n g e v e r y d a y i n c l a s s ( I n t h o s e d a y s t e a c h e r s were n o t p e r f e c t b u t a t l e a s t t h e y t u r n e d out p u p i l s able t o t h i n k c r i t i c a l l y . Today, t e a c h e r s are a i d e d by a l l a n d s u n d r y , w i l l y - n i l l y , w i t h somewhat less s a t i s f a c t o r y r e s u l t s ) . A t any r a t e , Lebesgue t h o u g h t l o n g a n d h a r d about t h e conundrum: t h e f i n a l e x p l i c a t i o n w a s i n t e r m s o f s e m i c o n t i n u i t y . The l e n g t h o f t h e a p p r o x i m a t i n g b r o k e n l i n e s t h e l e n g t h o f BC . L e n g t h i s n o t c o n t i n u o u s , b u t has a l i m i t semicontinuous. B e s i d e s l e n g t h and a r e a , o t h e r q u a n t i t i e s i n t h e C a l c u l u s of V a r i a t i o n s t u r n o u t t o be d i s c o n t i n u o u s . One p e r s o n who w o r r i e d a b o u t t h i s was T o n e l l i , of whom I s p o k e e a r l i e r : h e m e n t i o n e d h i s w o r r y t o my f a t h e r , who s u g g e s t e d "Can you u s e s e m i c o n t i n u i t y ? " T h i s became t h e theme of T o n e l l i ' s famous b o o k , F o n d a m e n t i a s i t became known, o r i n f u l l i n E n g l i s h , F o u n d a t i o n s of t h e

3 04

C h a p t e r I11

C a l c u l u s o f V a r i a t i o n s : h e r e , modern r e a l a n a l y s i s and t h e Leb e s g u e i n t e g r a l were f i r s t u s e d i n what h a d b e e n a s u b j e c t f u l l o f i n t u i t i v e a r g u m e n t s and c a l c u l a t i o n s , t h a t Weierstrass, H i l b e r t , C a r a t h e o d o r y , Hadamard, and o t h e r s had m e r e l y begun t o b r i n g u p t o d a t e . I n a whole c l a s s o f i m p o r t a n t p r o b l e m s , t h e q u a n t i t i e s t o be minimized t u r n e d o u t t o b e lower s e m i c o n t i n u o u s . In those days, people used t h e r a t h e r crude notions of l i m i t , d e v e l o p e d i n s p a c e s of f u n c t i o n s o r o f s u r f a c e s , c u r v e s , and s o o n , by F r 6 c h e t : I s p e a k o f t h e s e l a t e r . I n my " C a l c u l u s o f V a r i a t i o n s and Optimal C o n t r o l Theory", I i n t r o d u c e d e f i n i t i o n s , i n terms o f w h i c h c o n t i n u i t y c a n b e r e s t o r e d by t h e i n t r o d u c t i o n o f g e n e r a l i z e d c u r v e s . T h i s makes i t p o s s i b l e t o s t u d y , b e s i d e s t h e p r o b l e m s c o n s i d e r e d by T o n e l l i , o t h e r s i n w h i c h h i s s e m i c o n t i n u i t y d o e s n o t h o l d . N e v e r t h e l e s s , t h e work o f T o n e l l i r e p r e s e n t s an enormous s t e p f o r w a r d s , and i s t y p i c a l o f t h e way i n w h i c h t h e i d e a s o f Ren6 B a i r e , a t f i r s t c o n s i d e r e d f a r f e t c h e d , t u r n e d out t o be of importance. Baire c o n s i d e r e d h i m s e l f w i t h a d i f f e r e n t a s p e c t of s emico n tin u i t y , more c l o s e l y r e l a t e d t o S e t s of P o i n t s . I f w e r e p r e s e n t a s e t by t h e " i n d i c a t o r f u n c t i o n " ( o r c h a r a c t e r i s t i c f u n c t i o n , as i t was f o r m e r l y c a l l e d ) , whose v a l u e is 1 i n t h e s e t and 0 e l s e where, c o n t i n u i t y o f t h i s f u n c t i o n i n E u c l i d e a n s p a c e i s eq u iv al e n t t o its constancy, i . e . t o t h e set being e i t h e r empty.or t h e whole s p a c e . S e m i c o n t i n u i t y a p p l i e s t o less t r i v i a l cases: t h e i n d i c a t o r f u n c t i o n o f a c l o s e d s e t is u p p er s e m i c o n t i n u o u s , and t h a t o f a n open s e t is l o w e r s e m i c o n t i n u o u s . F o r t h i s r e a s o n , s e m i c o n t i n u i t y is e s s e n t i a l i n a Modern A n a l y s i s w h i c h d e p e n d s on S e t s o f P o i n t s . B a i r e w e n t on t o s t u d y more g e n e r a l f u n c t i o n s , a n d t o c l a s s i f y them by t h e i r g e n e r a l i t y . H i s f i r s t c l a s s , which i n c l u d e s , i n p a r t i c u l a r , t h e s emico n tin u o u s funct i o n s , c o n s i s t s o f t h e l i m i t s o f c o n v e r g e n t s e q u e n c e s o f continuous functions; generally, each successive class c o n s i s t s of l i m i t s of convergent sequences of f u n c t i o n s of p r e v i o u s classes. Baire f o u n d t h a t less and l e s s o f t h e o r i g i n a l c o n t i n u i t y carr i e s o v e r t o l a t e r classes. A f u n c t i o n o f t h e f i r s t c l a s s h a s a c t u a l p o i n t s o f c o n t i n u i t y : its' d i s c o n t i n u i t i e s f o r m a r a t h e r s p a r s e s e t , "of t h e f i r s t c a t e g o r y " . T h i s c a t e g o r y i s a s t r i k i n g c o n t r i b u t i o n o f B a i r e ' s . A n a l y s i s was b a r e l y u s e d t o Cant o r ' s d i s c o v e r y t h a t an i n t e r v a l h a s more t h a n c o u n t a b l y many p o i n t s : B a i r e showed t h a t i s n o t a c o u n t a b l e sum o f "nowhere d e n s e s e t s " , o r , as h e p u t i t , n o t o f t h e f i r s t c a t e g o r y ( A s e t A i s t e r m e d nowhere d e n s e -- o r , m i s l e a d i n g l y , non-dense -- i f t h e set B o f t h e p o i n t s n o t i n t h e c l o s u r e o f A i s " d e n s e " . T h i s c l o s u r e c o n s i s t s of A a n d o f t h e l i m i t p o i n t s o f A; a s e t i s s a i d t o b e " d e n s e " i f i t s c l o s u r e i s t h e whole s p a c e ) . B a i r e ' s t h e s i s a p p e a r e d i n 1899, when t h e L e b e s g u e i n t e g r a l h a d y e t t o b e i n v e n t e d . The o p p o s i t i o n t o sets was s u c h t h a t s e v e n y e a r s l a t e r L e b e s g u e h i m s e l f w r o t e a b o u t my p a r e n t s ' " S e t s of P o i n t s " , more t h a n s u g g e s t i n g t h a t a book on t h e s u b j e c t w a s n o t c a l l e d f o r , a t most an i n t r o d u c t o r y c h a p t e r ,

For s e t s o f p o i n t s , Borel had meanwhile d e v i s e d a t h e o r y s i m i l a r t o t h a t o f B a i r e , by i n t r o d u c i n g more and more g e n e r a l s e t s , b e l o n g i n g t o s u c c e s s i v e s o - c a l l e d B o r e l classes. H e had d e v e l o p e d f o r s u c h sets a t h e o r y o f m e a s u r e , a n d h e had p r o d u c e d h i s

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v e r s i o n o f t h e " B o r e 1 c o v e r i n g t h e o r e m " . A l l t h i s was i n c o n n e c t i o n w i t h a p r o b l e m of a n a l y t i c c o n t i n u a t i o n , on t h e f r i n g e o f classical f u n c t i o n t h e o r y . I t w a s d i f f i c u l t t o object t o i t ! E m i l e B o r e l ( 1 8 7 1 - 1 9 5 6 ) , s o n of a P r o t e s t a n t c l e r g y m a n i n t h e P r o v e n c e , w a s a t 18 f i r s t i n a n a t i o n a l c o m p e t i t i o n a n d i n t h e a d m i s s i o n s t o b o t h "Ecoles". H e c h o s e t h e Ecole Normale, g r a d u a t e d i n 1 8 9 2 , w a s f i r s t i n t h e a g g r 6 g a t i o n and w r o t e a s u b s t a n t i a l m e m o i r . H e t o o k h i s d o c t o r a t e i n 1 8 9 4 , w a s Army e n g i n e e r , ma?tre d e c o n f e r e n c e s a t L i l l e , r e t u r n e d a f t e r a f e w y e a r s t o P a r i s . I n 1 9 0 9 h e became p r o f e s s o r o f T h e o r y o f F u n c t i o n s , a n d d e p u t y d i r e c t o r of t h e E c o l e Normale. H e t h e n became immersed i n a d m i n i s t r a t i o n and government, w i t h h i s f r i e n d P a i n l e v 6 . H e wrote s e m i - p o p u l a r b o o k s , on A v i a t i o n ( w i t h P a i n l e v e " ) , on P r o b a b i l i t y , on a g e o m e t r i c a l i n t r o d u c t i o n t o P h y s i c s ( M a i n l y G a s e s ) . H i s m a i n i n f l u e n c e o n m a t h e m a t i c s came f r o m e d i t i n g m o n o g r a p h s , t h e C o l l e c t i o n B o r e l , i n c l u d i n g some b y h i m s e l f . B e s i d e s S e t T h e o r y , h e b r o u g h t o t h e r modern t o o l s t o F u n c t i o n T h e o r y , s u c h a s D i v e r g e n t S e r i e s . H i s p h i l o s o p h i c a l v i e w s on s e t s had a p r a c t i c a l o u t l o o k , i n w h i c h t h e most r a b i d c r i t i c s of s e t s were comp e l l e d t o see some r e a s o n a b l e n e s s . O t h e r s i n S e t T h e o r y a n d Real A n a l y s i s were f o r a t i m e l e s s f o r t u n a t e , n o t a b l y t h e f r i e n d of my p a r e n t s whom I h a v e m e n t i o n e d o c c a s i o n a l l y , H e n r i L e b e s g u e ( 1 8 7 5 - 1 9 4 1 ) , a man r a r e l y o u t s i d e P a r i s , o n c e e s t a b l i s h e d t h e r e , a l t h o u g h f o r t h e l o n g Summer, l i k e e v e r y o n e who m a t t e r e d , h e d o u b t l e s s d i s a p p e a r e d somewhere. I t i s q u i t e e v i d e n t t h a t B o r e l a n d L e b e s g u e were i n t e n d e d by n a t u r e t o f o r m a famous c o l l a b o r a t i o n l i k e t h a t o f Hardy and L i t t l e w o o d , u n f o r t u n a t e l y t h e y c o u l d n o t a b i d e one a n o t h e r . B o r e l ' s c o v e r i n g t h e o r e m , B o r e l ' s m e a s u r e , a n d t h e L e b e s g u e i n t e g r a l s h o u l d a l l be c o n s i d e r e d j o i n t c o n t r i b u t i o n s , w i t h o t h e r names a s s o c i a t e d a s w e l l . A number o f p e o p l e , e i t h e r "knew i t a l l b e f o r e " , o r e l s e " d i d i t much b e t t e r " , o r s o t h e y i m a g i n e d . B e t w e e n my f a t h e r a n d L e b e s g u e , a p r i o r i t y d i s p u t e w a s h a r d l y t h i n k a b l e : t h e y became c l o s e f r i e n d s , g e n e r o u s t o o n e a n o t h e r . O t h e r s , l i k e Osgood, a s w e s h a l l s e e , f e l t d i f f e r e n t l y , b u t t h a t is a n o t h e r s t o r y . L e b e s g u e w a s b o r n , a n d went t o s c h o o l , a t B e a u v a i s ; h e t h e n s t u d i e d a t t h e E c o l e Normale. H i s famous t h e s i s " I n t e g r a l e , l o n g e u r , a i r e " , a p p e a r e d i n 1 9 0 2 . H e t a u g h t from 1 9 0 6 t o 1 9 1 2 a t P o i t i e r s , a f t e r a s p e l l a t Rennes; t h e n he r e t u r n e d t o P a r i s , w h e r e h i s p r o f e s s o r s h i p w a s u n r e a s o n a b l y h e l d up -- a s l a t e a s 1 9 1 7 h e w a s p a s s e d o v e r i n f a v o u r o f Vessiot ( a m a t h e m a t i c i a n o f whom I s h a l l s a y n o t h i n g f u r t h e r ) . T h i s a b s u r d i t y w a s r e c t i f i e d t w o y e a r s l a t e r i n 1 9 1 9 , when L e b e s g u e w a s i n t h e m i d s t o f a b i g p r i o r i t y f i g h t w i t h B o r e l , i n t h e A n n a l e s d e 1 ' E c o l e Normale S u p g r i e u r e 1918 - 1 9 2 0 . L e b e s g u e became, n o t o n l y p r o f e s s o r a t t h e U n i v e r s i t y of P a r i s , t h e Sorbonne, b u t a l s o P r e s i d e n t of t h e F r e n c h m a t h e m a t i c a l s o c i e t y . My p a r e n t s , t o o , s h a r e d t h e r e s p e c t s u d d e n l y a c c o r d e d t o t h e p i o n e e r s of S e t T h e o r y m e t h o d s i n p o s t - W o r l d War I F r a n c e . They were h o n o u r e d b y t h e i r P a r i s f r i e n d s i n a v e r y t a n g i b l e way i n 1 9 2 4 , a t a t i m e when t h e y w e r e s u b o r d i n a t i n g a l l o t h e r e x p e n s e s t o t h a t of t h e e d u c a t i o n o f t h e i r c h i l d r e n : t h e y were t h e g u e s t s o f t h e F r e n c h d e l e g a t i o n t o t h e I n t e r n a t i o n a l Congress i n Toronto, and w e r e t r a n s p o r t e d i n a m a g n i f i c e n t c a b i n on t h e F r e n c h l i n e r , t h e " S u f f r e n " . I n P a r i s , i n R e a l A n a l y s i s , Lebesgue w a s King, and e v e r y o n e t h e r e was most a n x i o u s t o h o n o u r l i k e w i s e t h e p i o n e e r s who h a d f o r m e d

306

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w i t h him a c l o s e l y k n i t g r o u p i n l e s s g e n e r o u s t i m e s , p e o p l e l i k e my p a r e n t s a n d d e l a V a l l g e P o u s s i n . C h a r l e s J o s e p h d e l a Valle6 P o u s s i n ( 1 8 6 6 - 1 9 6 2 ) w a s b o r n i n L o u v a i n , and came f r o m a n o l d F r e n c h f a m i l y s e t t l e d i n B e l g i u m , a f a m i l y r e l a t e d t o t h e g r e a t p a i n t e r , P o u s s i n . The r e 1 a ; i o n s h i p w a s e m p h a s i z e d by m o d i f y i n g t h e o r i g i n a l name o f L a v a l l e e . D e l a Valle6 P o u s s i n s t u d i e d a t t h e J e s u i t C o l l e g e a t Mons, w h e r e h e o b t a i n e d t h e d i p l o m a o f e n g i n e e r , and h e r e t u r n e d t o L o u v a i n as t e a c h e r ( 1 8 9 1 ) , and soon p r o f e s s o r , o f m a t h e m a t i c s . I n s u c h p l a c e s , e n g i n e e r i n g a n d m a t h e m a t i c s s t i l l went v e r y much t o g e t h er -- you w i l l remember t h a t C a r a t h g o d o r y s i m i l a r l y s t u d i e d e n g i n e e r i n g i n B r u s s e l s . T h e n , f o r a t i m e , t h e two f i e l d s d i v e r g e d f r o m o n e a n o t h e r , b u t nowadays t h e g a p b e t w e e n them i s f a s t d i s a p p e a r i n g , a s i s t h a t b e t w e e n m a t h e m a t i c s a n d many o t h e r s u b j e c t s . D e l a Valleg P o u s s i n a c h i e v e d fame, j o i n t l y w i t h Hadamard, as I m e n t i o n e d e a r l i e r , by p r o v i n g t h e P r i m e Number t h e o rem, b u t w a s l a t e r c o n s i d e r e d t o h a v e " d e f e c t e d " t o t h e f r i n g e o f s e r i o u s m a t h e m a t i c s , by t a k i n g u p S e t T h e o r y a n d Real Analysis. However, h e o u t l i v e d a n y s u c h p r e j u d i c e by a g r e a t many y e a r s , a n d h e was s p e c i a l l y h o n o u r e d , and made B a r o n , by t h e King o f t h e B e l g i a n s i n 1 9 2 8 . I n L o u v a i n , d e l a Valle6 P o u s s i n had few p r o p e r p u p i l s : h e made u p f o r t h i s by p r o d u c i n g b e a u t i f u l l y w r i t t e n b o o k s . H i s Cours d ' A n a l y s e w e n t t o many e d i t i o n s , and was p o p u l a r e v e n i n s u c h p l a c e s a s C a m b r i d g e , w h e r e s t u d e n t s are n o r m a l l y f a r t o o p r e o c c u p i e d w i t h e x a m i n a t i o n s t o h a v e t i m e f o r s e r i o u s b o o k s i n a f o r e i g n t o n g u e . A n o t h e r b o o k , much i n d e mand, was t h e monograph ' I n t g g r a l e d e L e b e s g u e , f o n c t i o n s d ' e n sembles, classes d e B a i r e ' , p u b l i s h e d i n t h e Bore1 c o l l e c t i o n i n 1916 and e n l a r g e d i n 1 9 3 4 b y a n a p p e n d i x on A n a l y t i c S e t s and S t i e l t j e s i n t e g r a l . An a c c o u n t o f t h e L e b e s g u e i n t e g r a l had o r i g i n a l l y b e e n i n c l u d e d i n t h e 1 9 1 4 e d i t i o n o f t h e C o u r s d'Anal y s e , t h e e d i t i o n b u r n t by German t r o o p s . I n 1914 v a n d a l i s m s t i l l o u t r a g e d and s h o c k e d t h e c i v i l i s e d w o r l d i n Germany a s w e l l as elsewhere.

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I h a v e m e n t i o n e d i n t h e c o u r s e of t h e s e l e c t u r e s s e v e r a l k i n d s o f i n t e g r a l , and it i s time I s a i d s o m e t h i n g a b o u t t h e m , l e s t i t seem l i k e o n e o f t h o s e g h o s t s o f w o r d s t h a t f l i t i n a n d o u t of s e n t e n c e s i n t h e f i n e p r i n t o f q u e s t i o n a b l e d o c u m e n t s . I n t h e simpleminded a n a l y s i s t h a t Lagrange s t i l l b e l i e v e d i n , t h e i n t e g r a l o f a f u n c t i o n f of o n e r e a l v a r i a b l e x was a f u n c t i o n F w i t h f as i t s d e r i v a t i v e . Here F w a s d e t e r m i n e d e x c e p t f o r a n a r b i t r a r y a d d i t i v e c o n s t a n t . The d e f i n i t e i n t e g r a l from a t o b was s i m p l y t h e d i f f e r e n c e F ( b ) - F ( a ) , w h i c h i s i n d e p e n d e n t of t h e a d d i t i v e c o n s t a n t . I f t h e p o i n t b w a s a l l o w e d t o v a r y , and r e w r i t t e n a s x , t h e e x p r e s s i o n would a g r e e w i t h t h e o r i g i n a l i n t e g r a l , e x c e p t f o r t h e a d d i t i v e c o n s t a n t F ( a ) . T h i s was a l l t h e r e w a s t o i t , except for t h e geometrical i n t e r p r e t a t i o n of t h e d i f f e r e n c e F ( b ) - F ( a ) as t h e area between t h e l i n e s x = a and x = b , which l i e s below t h e c u r v e y = f ( x ) and above t h e a x i s of x . If t h i s c u r v e i s p a r t l y o r w h o l l y b e l o w t h e x - a x i s , a p p r o p r i a t e c o n v e n t i o n s of s i g n must be made. However, f o r Lag r a n g e , g e o m e t r i c a l i n t e r p r e t a t i o n s were u n n e c e s s a r y .

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The a r e a i n t e r p r e t a t i o n m e r e l y p r o v i d e d t h e h i s t o r i c a l j u s t i f i cation f o r denoting t h e difference F(b) - F(a) by t h e symbols b

la f ( x ) d x , w h i c h were t h o u g h t o f a s i n d i c a t i n g , i n a r o u g h s o r t o f way, a sum o f a g r e a t many a p p r o x i m a t e l y r e c t a n g u l a r a r e a s , s i t u a t e d between t h e l i n e s x = a and x = b and d e n o t e d g e n e r i c a l l y b y t h e p r o d u c t f ( x ) d x , b u t a c t u a l l y o b t a i n e d by d i v i d i n g t h e g i v e n area i n t o t h i n v e r t i c a l s t r i p s . The d e f i n i t i o n o f i n t e g r a l g i v e n b y Cauchy and Riemann c o n s i s t s i n r e p l a c i n g t h i s r o u g h i n t u i t i v e r e p r e s e n t a t i o n a s a sum, b y a p r e c i s e n o t i o n o f l i m i t . I s h a l l d e f i n e h e r e , i n t h i s way, t h e f o r m a l l y more g e n e r a l s t i e l t j e s i n t e g r a l f r o m a t o b , w h i c h i s written

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b

f(x)dg(x) ;

h e r e f , g are r e a l - v a l u e d f u n c t i o n s (The i n i t i a l symbol, h e r e and h i g h e r u p , w a s o r i g i n a l l y a c a p i t a l S , a n d s t o o d f o r t h e word ''sum"). For t h e p u r p o s e o f t h i s d e f i n i t i o n , l e t T d e n o t e t h e s u b d i v i s i o n of t h e i n t e r v a l ( a , b ) i n t o p a r t s ( x r - , , x r ) t o e a c h o f w h i c h w e a s s i g n a r e p r e s e n t a t i v e p o i n t x*,. , situated i n t h e p a r t ; w e s u p p o s e t h a t t h e r e a r e a f i n i t e number N o f s u c h p a r t s , and t h a t a = x

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