The Mathematical Intelligencer

Letter to the Editors The Mathematical Intelligencer Letter to the Editors encourages comments about the material in ...

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Letter to the Editors

The Mathematical Intelligencer

Letter to the Editors

encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.

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n response to the delightfully intriguing ‘‘Locker Puzzle’’ by Eugene Curtin and Max Warshauer (Mathematical Intelligencer, vol. 28, #1, pp. 28–31 (2006). doi:10.1007/BF02986999), I thought your readers might enjoy the following variant of this puzzle:

The Return of Monty Hall In a new (‘‘Let’s-Make-A-Deal’’-type) game show for couples, there are 3 curtains behind which are hidden a car, a car key, and a goat. One member of the couple is designated ‘‘the carmaster’’ – the car-master’s goal is to find the car. The other member is designated ‘‘the key-master’’ – the keymaster’s goal is to find the key. If both partners succeed in their respective tasks, the couple drives away in their new car. If either one fails, the couple receives the booby prize, the goat. The game begins with the car-master (at this point, the key-master is led out of the room and cannot observe the proceedings). The car-master has

two tries to find the car (i.e., open any curtain; if the car isn’t there, then open another curtain). If the car-master succeeds in finding the car, all open curtains are reclosed, and the keymaster is brought back into the room. No communication whatsoever is permitted between the car-master and key-master at this point. The keymaster now has two tries to find the key (i.e., open any curtain; if the key isn’t there, open another curtain). Assuming the couple plays optimally, what are their odds of winning the car? Using the clever strategy outlined by Curtin and Warshauer, readers may verify that the couple will be able to drive away in a new car a remarkable 2/3rds of the time!

A. S. Landsberg Joint Science Department Claremont McKenna, Pitzer, and Scripps Colleges Claremont, CA 91711 USA e-mail: [email protected]

Ó 2008 Springer Science+Business Media, LLC, Volume 31, Number 2, 2009

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Note

Erroneous Euler and Venn Diagrams A. W. F. EDWARDS

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y guilt at having erroneously copied one of Venn’s own diagrams for my book Cogwheels of the Mind [2; p. 30] was only partially assuaged by including a corrected one of Peirce’s on the next page. Now I find that Venn himself made a mistake in the shading of a five-set diagram in Symbolic Logic [11; p.352], so I am in good company. Better still, Euler [4] also gave a mistaken diagram in Letters to a Princess. Here I examine these errors in chronological order.

Euler (1768) Euler published the original ‘Euler diagrams’ in 1768 in the second volume of Lettres a` une Princesse d’Allemagne [4] as a preliminary to his enumeration of syllogisms. He made a mistake in drawing one of his diagrams for three sets. When he added a third set C to a two-set diagram with overlapping sets A and B so as to overlap A partially, he drew the cases C wholly in B and C wholly outside B, but muddled the case C partially in B, which would have given him a traditional Venn diagram for three sets. Figure 1

shows the original of this group of three diagrams [4; p.112]. The second and third diagrams cover the cases C wholly in B and C wholly outside B, but in the first, Euler has overlooked the fact that C is also supposed to be partly in A as well as partly in B. An 1823 English reprint [6] of the erroneous diagram can be seen at www.math.dartmouth.edu/*euler/, E343 Plate III Figure 27, which is the same as Figure 26. Venn owned a copy of this book, but I have examined it, and he did not mark the error. The editors of Euler’s Opera omnia [8] noticed Euler’s mistake and replaced the erroneous diagram with the correct ‘Venn’ threecircle form, recording the change in a footnote. At least one modern edition [7] relying on the Opera omnia does not include the footnote however, thus giving the impression that Euler once presented Venn’s three circles. Hamburger [9] cited both [4] and [5] as containing the Venn form, an error I pointed out in [3].

Venn (1881) Venn’s principal account of what came to be known as ‘Venn diagrams’

Figure 1. Euler’s attempt (1768) to add a third set C to the diagram with overlapping sets A and B. The second and third diagrams show the cases C wholly in B and C wholly outside B correctly, but the first should show C partly in A as well as partly in B. 2

THE MATHEMATICAL INTELLIGENCER Ó 2009 Springer Science+Business Media, LLC

was in his book Symbolic Logic [11]. (For a detailed history see [2].) In it, Venn applied his diagrammatic method to a problem involving five ‘properties’ given by Boole in The Laws of Thought [1]. Venn changes Boole’s notation and condenses his wording into:

Figure 2. Venn’s attempt (1881) to solve a problem of Boole’s. He omitted shading of the area ‘y but not x + z + w + u’.

Figure 3. Venn’s corrected figure (1894) for Boole’s problem.

1. Wherever x and z are missing, u is found, with one (but not both) of y and w. 2. Wherever x and w are found whilst u is missing, y and z will both be present or both absent. 3. Wherever x is found with either or both of y and u, there will z or w (but not both) be found; and conversely. Boole had invited his readers to ascertain what the presence of the property x might indicate for the presence or absence of y, z and w, and whether any relations exist independently among the last three. ‘Secondly, what may be concluded in like manner respecting the property y, and the properties x, z and w’. Boole gave his results after two pages of symbolic algebra, adding, ‘I have not attempted to verify these conclusions’. Here was Venn’s chance to demonstrate the advantages of the graphical method for a case with five

properties, having in the preceding pages used his four-ellipse diagram to deal with four properties. For five sets, Venn added the fifth set to his four-set diagram in the form of an annulus [11; p. 281] and proceeded to shade out the sections corresponding to absent combinations of properties (Figure 2). Alas, in his attempt to demonstrate the superiority of his graphical method, the complexity of his diagram defeated him, and in the second edition of Symbolic Logic he gave a corrected one [11; p. 352] (Figure 3) and ruefully commented in a footnote ‘I may remark that, as Schro¨der and others have pointed out, the figure in the first edition was not quite accurate.’ Venn’s annulus for the fifth set meant that ‘absence of the fifth property’ was represented by two disjoint areas, an unwelcome breach of the convention that each set should be bounded by a single closed curve. It is certainly very confusing to work with a set with a hole in its middle. An improved five-set diagram, shaded to correspond to Boole’s example, is shown in Figure 4.

Peirce (1839–1914) Peirce, at a date unknown, persuaded himself of the possibility of always

Figure 4. The Edwards–Venn diagram for Boole’s problem. The label for each set is inside the set and adjacent to the set boundary. Thus, the lettered area corresponds to xyzwu. Note that the bottom left area is ‘not x + y + z + w + u’ which exists in Venn’s diagram only as the entire outside, ‘Which we have not troubled to shade’, he says in a footnote. Ó 2009 Springer Science+Business Media, LLC, Volume 31, Number 2, 2009

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three-circle form. Finding his figures too small for direct reproduction in Cogwheels of the Mind, I elected to redraw them: Disastrously, as it turned out. The five-set diagram [2; p. 30] (Figure 6) is wrong, as was pointed out by Richard Kemp, a graduate student, when I showed it in a lecture at Gonville and Caius College, Cambridge (Venn’s old college). A new printing of the book allowed the mistake to be corrected (Figure 7).

REFERENCES

1. G. Boole, An Investigation of The Laws of Thought, Walton and Maberly, London, 1854; reprinted Dover, New York, 1958. 2. A.W.F. Edwards, Cogwheels of the Mind: The Story of Venn Diagrams, Johns Hopkins

University

Press,

Baltimore,

2004. 3. A.W.F. Edwards, ‘‘Euler diagrams and Venn diagrams’’ (letter), The Mathematical Intelligencer (2006), 28(3). 4. L. Euler, Lettres a` une Princesse d’Allemagne, Imperial Academy of Sciences,

Figure 5. A corrected redrawing of Peirce’s demonstration that sets may be added indefinitely to a Venn diagram.

St. Petersburg, 1768. 5. L. Euler, Letters to a German Princess, transl. H. Hunter, Murray, London, 1795. 6. L. Euler, Letters to a German Princess, ed. D. Brewster, transl. H. Hunter, Tait, Edinburgh, 1823. 7. L. Euler, Lettres a` une Princesse d’Allemagne,

ed.

S.D. Chatterji,

Presses

Polytechniques et Universitaires Romandes, Lausanne, 2003. 8. L. Euler, Opera omnia Leonhardi Euleri, Series Tertia, Vols. 11 and 12, ed. A. Speiser, Teubner, Turici, 1960. 9. P. Hamburger, Cogwheels of the Mind. The

Story

of

Venn

Diagrams

by

A.W.F. Edwards (review), The Mathemat-

Figure 6. My failed attempt (2004) at redrawing Venn’s five-set figure.

Figure 7. Figure 6 corrected.

ical Intelligencer (2005), 27(4):36–38. 10. J. Venn,

On

the

diagrammatic

and

mechanical representation of propositions and reasonings. London, Edinburgh and Dublin Philosophical Magazine and Jour-

being able to add a further set to a Venn diagram by drawing a figure adding fourth, fifth, sixth and seventh sets to Venn’s three-circle form. Confronted with the difficulty of obtaining a photograph of the original to reproduce in Cogwheels of the Mind [2], I set about redrawing it from a poor copy I possessed, only to discover that Peirce had made a mistake. He had failed to 4

THE MATHEMATICAL INTELLIGENCER

make the sixth set divide all the regions of his five-set diagram, thus getting the seventh set wrong as well. Figure 5 shows the corrected figure [2; p. 31].

Edwards (2004) In the original paper on his diagrams, Venn [10] showed how fourth and fifth sets could be added directly to the

nal of Science [Fifth Series] (1880), 10:1– 18. 11. J. Venn,

Symbolic

Logic,

Macmillan,

London, 1881; second ed. 1894. Gonville and Caius College Cambridge University Cambridge CB2 1TA UK e-mail: [email protected]

Note

An Arm Trick for Unarmed Performers JU´LIUS KORBASˇ

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ne of the most popular facts in mathematics surely is that the fundamental group (invented by Poincare´ in the 1890s) of the threedimensional rotation group SO(3) has just two elements. So two times the generator—represented by a loop in the space of rotations—is zero, and you can entertain your nonmathematical (sometimes perhaps also mathematical) friends, or less experienced students, for instance, with the soup-plate trick (a simple description can be found in 8.10.3 of M. Berger’s Geóme´trie. 1. Cedic/Nathan, Paris 1977), if you are ‘‘armed’’ with a plate. But what if you are, so to say, an unarmed person, with no plate and no similar object handy? Is there any chance that even then you can entertain your audience, using the fact mentioned here? The answer is positive, as you will see in the following text. (Editors’ Note. The Editors present this ‘‘trick’’ somewhat hesitantly: we are not able to perform it convincingly, and the referee has not altogether persuaded us that it ‘‘works.’’)

An arm trick You are invited to perform the following experiment. To describe it unambiguously, I shall speak about your right arm. STEP 1. Relax your arms. STEP 2. Stand firmly, looking (for instance) at a wall in front of you, so that your back is parallel to the wall. During the experiment, you will move the right arm, but the rest of your body should remain fixed (if possible, support your back by some vertical object). Put your right arm straight out horizontally, with the palm in the vertical position (hence your thumb is at the top and the little finger is at the bottom), with all the fingers tight and straight, so that the end of your longest finger is softly touching the wall.

STEP 3. Keeping your back always parallel to and at a constant distance from the wall, and keeping your right arm always straight with all the fingers tight and straight, slowly rotate your arm in the direction downward-backward-upward-forward, alongside your body (roughly speaking, in a plane perpendicular to the wall) so that the end of your longest finger draws (in the air) one complete circle. After the rotation, your palm is again to be in the vertical position. Of course, on doing the rotation as described, there comes a moment, on going backwards with your hand, when you are no longer able to keep your palm in the plane perpendicular to the wall: you feel a torsion in the muscles of your rotating arm and you are pressed to rotate the palm, eventually by 360 degrees, the axis of the palm-rotation passing along your straight arm. What do you find after the rotation? You feel that your muscles remain in a certain torsion, tension, somehow contracted. Even more: after having done the rotation, you see that your arm became shorter, by some centimeters. STEP 4. After completing the rotation from Step 3, do such a rotation immediately for a second time. What a surprise! After the second rotation, your muscles are not any longer in tension; they are again normalized. Even more: after this second rotation, your arm has again the same length as prior to the experiment! The experiment is illustrated by Figure 1. It consists of nine partial photos, placed in three rows and three columns; the partial photo in the ith row and jth column is referred to as (i, j). For the first rotation, see (1, 1) ? (1, 2) ? (1, 3) ? (2, 1) ? (2, 2); for the second rotation, see (2, 2) ? (2, 3) ? (3, 1) ? (3, 2) ? (3, 3). If you succeeded in the experiment, congratulations, you can start looking


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Figure 1. Two consecutive rotations (photographed by Rafael Korbasˇ).

for a suitable audience to share your joy and teach others. If you tried it many times, strictly following our description, analysing your possible faults, and in spite of all you did not observe the desired phenomena, then you are free to blame the author. But please do not be excessively cruel: if nothing else, you can perhaps recognize that by doing the arm-rotations, you followed your doctor’s longstanding advice to take more exercise!

A mathematical explanation of the arm trick As indicated at the beginning, our experiment has the same background

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as the soup-plate trick (or similar tricks, such as the Philippine wineglass dance or the spinor spanner; I was told about the latter two by Jim Stasheff to whom I am grateful for encouragement). Indeed, the first rotation of your arm realizes a loop r in SO(3), that is, in the space of rotations of R3 . Denoting by I the 0-degree rotation, the loop r is not homotopic (i.e., not suitably deformable) to the constant loop at I (that is why you feel a torsion or contraction in the muscles after the first rotation), and therefore represents the generator [r] of the fundamental group pðSOð3Þ; I Þ ¼ Z=2Z. Because 2[r] = 0 in p(SO(3); I), the two

consecutive rotations as described above give the concatenation of the loop r with itself, r * r. Of course, since [r * r] = 2[r] = 0, the loop r * r must then be homotopic to the constant loop at I (that is why you feel your muscles again relaxed, after the second rotation). Department of Algebra, Geometry, and Mathematical Education, Faculty of Mathematics, Physics, and Informatics Comenius University Mlynska´ dolina, 842 48 Bratislava 4 Slovakia e-mail: [email protected]

Note

Curiosum ROBERT J. MACG. DAWSON

F

or unriddling of this gadget, if that be needed, and for its history, see ‘‘What Is It?’’ by Robert J. MacG. Dawson.

Department of Mathematics & Computing Science St. Mary’s University Halifax NS B3H 3C3 Canada e-mail: [email protected]


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Viewpoint

Desperately Seeking Mathematical Proof MELVYN B. NATHANSON

The Viewpoint column offers mathematicians the oppurtunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editors-inchief endorse or accept responsibility for them.

This paper elaborates one of the themes in Nathanson [4]. I thank Saul Kripke for many useful discussions of various topics in the philosophy of mathematics.

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ow do we decide if a proof is a proof? Why is a random sequence of true statements not a proof of the Riemann hypothesis? That is, how do we know if a purported proof of a true theorem is a proof? Let’s start with remarks on language. The phrase ‘‘true theorem’’ is redundant, and I won’t use it again, since the definition of ‘‘theorem’’ is ‘‘true mathematical statement,’’ and the expression ‘‘false theorem,’’ like ‘‘false truth,’’ is contradictory. I shall write ‘‘theorem’’ in quotation marks to denote a mathematical statement that is asserted to be true, that is, that someone claims is a theorem. It is important to remember that a theorem is not false until it has a proof; it is only unproven until it has a proof. Similarly, a false mathematical statement is not untrue until it has been shown that a counterexample exists or that it contradicts a theorem; it is just a ‘‘statement’’ until its untruth is demonstrated. The history of mathematics is full of philosophically and ethically troubling reports about bad proofs of theorems. For example, the fundamental theorem of algebra states that every polynomial of degree n with complex coefficients has exactly n complex roots. D’Alembert published a proof in 1746, and the theorem became known as ‘‘D’Alembert’s theorem,’’ but the proof was wrong. Gauss published his first proof of the fundamental theorem in 1799, but this, too, had gaps. Gauss’s subsequent proofs, in 1816 and 1849, were okay. It seems to have been difficult to determine if a proof of the fundamental theorem of algebra was correct. Why? Poincare´ was awarded a prize from King Oscar II of Sweden and Norway for a paper on the three-body problem, and his paper was published in Acta Mathematica in 1890. But the published paper was not the prizewinning paper. The paper that won the prize contained serious mistakes,

H

THE MATHEMATICAL INTELLIGENCER 2009 Springer Science+Business Media, LLC

and Poincare´ and other mathematicians, most importantly, Mittag-Leffler, engaged in a conspiracy to suppress the truth and to replace the erroneous paper with an extensively altered and corrected one. There are simple ways to show that a purported proof of a false mathematical statement is wrong. For example, one might find a mistake in the proof, that is, a line in the proof that is false. Or one might construct a counterexample to the ‘‘theorem.’’ One might also be able to prove that the purported theorem is inconsistent with a known theorem. I assume, of course, that mathematics is consistent. To find a flaw in the proof of a theorem is more complex, since no counterexample will exist, nor will the theorem contradict any other theorem. A proof typically consists of a series of assertions, each leading more or less to the next, and concluding in the statement of the theorem. How one gets from one assertion to the next can be complicated, since there are usually gaps. We have to interpolate the missing arguments, or at least believe that a good graduate student or an expert in the field can perform the interpolation. Often the gaps are explicit. A typical formulation (Massey [2, p. 88]) is: By the methods used in Chapter III, we can prove that the group p(X) is characterized up to isomorphism by this theorem. We leave the precise statement and proof of this fact to the reader. There is nothing improper about gaps in proofs, nor is there any reason to doubt that most gaps can be filled by a competent reader, exactly as the author intends. The point is simply to emphasize that proofs have gaps and are, therefore, inherently incomplete and sometimes wrong. We frequently find statements such as the following (Washington [5, p. 321]): The Kronecker-Weber theorem asserts that every abelian extension

of the rationals is contained in a cyclotomic field. It was first stated by Kronecker in 1853, but his proof was incomplete …. The first proof was given by Weber in 1886 (there was still a gap …). There is a lovely but probably apocryphal anecdote about Norbert Wiener. Teaching a class at MIT, he wrote something on the blackboard and said it was ‘‘obvious.’’ One student had the temerity to ask for a proof. Wiener started pacing back and forth, staring at what he had written on the board, saying nothing. Finally, he left the room, walked to his office, closed the door, and worked. After a long absence he returned to the classroom. ‘‘It is obvious,’’ he told the class, and continued his lecture. There is another reason why proofs are difficult to verify: Humans err. We make mistakes and others do not necessarily notice our mistakes. Hume [1, Part IV, Section I,] expressed this beautifully in 1739: There is no Algebraist nor Mathematician so expert in his science, as to place entire confidence in any truth immediately upon his discovery of it, or regard it as any thing, but a mere probability. Every time he runs over his proofs, his confidence increases; but still more by the approbation of his friends; and is raised to its utmost perfection by the universal assent and applauses of the learned world. This suggests an important reason why ‘‘more elementary’’ proofs are better

than ‘‘less elementary’’ proofs: The more elementary the proof, the easier it is to check and the more reliable is its verification. We are less likely to err. ‘‘Elementary’’ in this context does not mean elementary in the sense of elementary number theory, in which one tries to find proofs that do not use contour integrals and other tools of analytic function theory. On elementary versus analytic proofs in number theory, I once wrote [3, p. ix], In mathematics, when we want to prove a theorem, we may use any method. The rule is ‘‘no holds barred.’’ It is OK to use complex variables, algebraic geometry, cohomology theory, and the kitchen sink to obtain a proof. But once a theorem is proved, once we know that it is true, particularly if it is a simply stated and easily understood fact about the natural numbers, then we may want to find another proof, one that uses only ‘‘elementary arguments’’ from number theory. Elementary proofs are not better than other proofs,…. I’ve changed my mind. In this paper I argue that elementary (at least, in the sense of easy to check) proofs really are better. Many mathematicians have the opposite opinion; they do not or cannot distinguish the beauty or importance of a theorem from its proof. A theorem that is first published with a long and difficult proof is highly regarded. Someone who, preferably many years later, finds a short proof is ‘‘brilliant.’’ But if the short proof had been obtained in

the beginning, the theorem might have been disparaged as an ‘‘easy result.’’ Erd} os was a genius at finding brilliantly simple proofs of deep results, but, until recently, much of his work was ignored by the mathematical establishment. Erd} os often talked about ‘‘proofs from the Book.’’ The ‘‘Book’’ would contain a perfect proof for every theorem, where a perfect proof was short, beautiful, insightful, and made the theorem instantaneously and obviously true. We already know the ‘‘Book proofs’’ of many results. I would argue that we do not, in fact, fully understand a theorem until we have a proof that belongs in the Book. It is impossible, of course, to know that every theorem has a ‘‘Book proof,’’ but I certainly have the quasi-religious hope that all theorems do have such proofs. There are other reasons for the persistence of bad proofs of theorems. Social pressure often hides mistakes in proofs. In a seminar lecture, for example, when a mathematician is proving a theorem, it is technically possible to interrupt the speaker in order to ask for more explanation of the argument. Sometimes the details will be forthcoming. Other times the response will be that it’s ‘‘obvious’’ or ‘‘clear’’ or ‘‘follows easily from previous results.’’ Occasionally speakers respond to a question from the audience with a look that conveys the message that the questioner is an idiot. That’s why most mathematicians sit quietly through seminars, understanding very little after the introductory remarks, and

.................................................................................................... AUTHOR

MELVYN B. NATHANSON is widely known for his books and

articles on additive number theory and related topics. But did you know that before doing his doctoral work in mathematics he was an undergraduate in philosophy and a graduate student in biophysics? Or that among his visiting positions he has been a post-doc with I.M. Gel’fand in Moscow and later an assistant to Andre´ Weil at the Institute for Advanced Study? Department of Mathematics Lehman College (CUNY) Bronx, NY 10468 USA CUNY Graduate Center New York, NY 10016 USA e-mail: [email protected]

2009 Springer Science+Business Media, LLC, Volume 31, Number 2, 2009

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applauding politely at the end of a mostly wasted hour. Gel’fand’s famous weekly seminar, in Moscow and at Rutgers, operated in marked contrast with the usual mathematics seminar or colloquium. One of the joys of Gel’fand’s seminar was that he would constantly interrupt the speaker to ask simple questions and give elementary examples. Usually the speaker would not get to the end of his planned talk, but the audience would actually learn some mathematics. The philosophical underpinning to this discussion is the belief that ‘‘mathematical’’ objects exist in the real world, and that mathematics is the science of discovering and describing their properties, just as ‘‘physical’’ objects exist in the real world and physics is the science of discovering and describing their properties. This is in contrast to an occasionally fashionable notion that mathematics is the logical game of deducing conclusions from interesting but arbitrarily chosen finite sets of axioms and rules of inference. If the mathematical world is real, then it is unlikely that it can be encapsulated in any finite system. There are, of course, masterpieces of mathematical exposition that develop a deep subject from a small set of axioms. Two examples of books that are perfect in this sense are Weil’s Number Theory for Beginners,

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which is, unfortunately, out of print, and Artin’s Galois Theory. Mathematics can be done scrupulously. Different theorems can be proven from different assumptions. The compendium of mathematical knowledge, that is, the collection of theorems, becomes a social system with various substructures, analogous to clans and kinship systems, and a newly discovered theorem has to find its place in this social network. To the extent that a new discovery fits into an established community of mathematical truths, we believe it and tend to accept its proof. A theorem that is an ‘‘outsider’’ – a kind of social outlaw – requires more rigorous proof, and finds acceptance more difficult. Wittgenstein [6, p. 401] wrote, If a contradiction were now actually found in arithmetic – that would only prove that an arithmetic with such a contradiction in it could render very good service; and it would be better for us to modify our concept of the certainty required, than to say it would really not yet have been a proper arithmetic. This passage (still controversial in the philosophy of mathematics) evidences a pragmatic approach to mathematics that describes how mathematicians behave in the privacy of their offices,

in contrast to our more pietistic public pronouncements. Perhaps we should discard the myth that mathematics is a rigorously deductive enterprise. It may be more deductive than other sciences, but hand-waving is intrinsic. We try to minimize it and we can sometimes escape it, but not always, if we want to discover new theorems.

REFERENCES

[1] David Hume, A Treatise of Human Nature, Barnes and Noble, New York, 2005. [2] William S. Massey, A Basic Course in Algebraic Topology, Graduate Texts in Mathematics, vol. 127, Springer-Verlag, New York, 1991. [3] Melvyn B. Nathanson, Elementary Methods in Number Theory, Graduate Texts in Mathematics, vol. 195, Springer-Verlag, New York, 2000. [4] Melvyn B. Nathanson, Desperately seeking mathematical truth, Notices Amer. Math. Soc. 55:7 (2008), 773. [5] Lawrence C. Washington, Introduction to Cyclotomic Fields, 2 ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1996. [6] Ludwig Wittgenstein, Remarks on the Foundations of Mathematics, MIT Press, Cambridge, 1983.

Viewpoint

Kronecker’s Algorithmic Mathematics HAROLD M. EDWARDS

The Viewpoint column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. Viewpoint should be submitted to the editor-inchief, Chandler Davis.

This essay is a lecture presented at ‘‘Computability in Europe 2008,’’ Athens, June 19, 2008.

I

wonder if it is as widely believed by the younger generation of mathematicians, as it is believed by my generation, that Leopold Kronecker was the wicked persecutor of Georg Cantor in the late nineteenth century and that, to the benefit of mathematics, by the end of the century the views of Cantor had prevailed and the narrow prejudices of Kronecker had been soundly and permanently repudiated. I suspect this myth persists wherever the history of mathematics is studied, but even if it does not, an accurate understanding of Kronecker’s ideas about the foundations of mathematics is indispensable to understanding constructive mathematics, and the contrast between his conception of mathematics and Cantor’s is at the heart of the matter. It is true that he opposed the rise of set theory, which was occurring in the years of his maturity, roughly from 1870 until his death in 1891. Set theory grew out of the work of many of Kronecker’s contemporaries—not just Cantor, but also Dedekind, Weierstrass, Heine, Me´ray, and many others. However, as Kronecker told Cantor in a friendly letter written in 1884, when it came to the philosophy of mathematics he had always recognized the unreliability of philosophical speculations and had taken, as he said, ‘‘refuge in the safe haven of actual mathematics.’’ He went on to say that he had taken great care in his mathematical work ‘‘to express its phenomena and truths in a form that was as free as possible from philosophical concepts.’’ Further on in the same letter, he restates this goal of his work and its relation to philosophical speculations saying, ‘‘I recognize a true scientific value—in the field of mathematics—only in concrete mathematical

truths, or, to put it more pointedly, only in mathematical formulas.’’ Certainly, this conception of the nature and substance of mathematics restricts it to what is called ‘‘algorithmic mathematics’’ today, and it is what I had in mind when I chose my title ‘‘Kronecker’s Algorithmic Mathematics.’’ Indeed, these quotations from Kronecker show that my title is a redundancy—for Kronecker, that which was not algorithmic was not mathematics, or, at any rate, it was mathematics tinged with philosophical concepts that he wished to avoid. At the time, I don’t think that this attitude was in the least unorthodox. The great mathematicians of the first half of the nineteenth century had, I believe, similar views, but they had few occasions to express them, because such views were an understood part of the common culture. There is the famous quote from a letter of Gauss in which he firmly declares that infinity is a façon de parler and that completed infinites are excluded from mathematics. According to Dedekind, Dirichlet repeatedly said that even the most recondite theorems of algebra and analysis could be formulated as statements about natural numbers. One needs only to open the collected works of Abel to see that for him mathematics was expressed, as Kronecker said, in mathematical formulas. The fundamental idea of Galois theory, in my opinion, is the theorem of the primitive element, which allowed Galois to deal concretely with computations that involve the roots of a given polynomial. And Kronecker’s mentor Kummer—whom Kronecker credits in his letter to Cantor with shaping his view of the philosophy of mathematics—developed his famous theory of ideal complex numbers in an altogether algorithmic way. It is an oddity of history that Kronecker enunciated his algorithms at a time when there was no possibility of


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implementing them in any nontrivial cases. The explanation is that the algorithms were of theoretical, not practical, importance to him. He goes so far as to say in his major treatise Grundzu¨ge einer arithmetischen Theorie der algebraischen Gro¨ssen that, by his lights, the notion of irreducibility of polynomials lacks a firm foundation (entbehrt einer sicheren Grundlage) unless a method is given that either factors a given polynomial or proves that no factorization is possible. When I first encountered this opinion of Kronecker’s, I had to read it several times to be sure I was not misunderstanding him. The opinion was so different from my mid-twentieth century indoctrination in mathematics that I could scarcely believe he meant what he said. Imagine Bourbaki saying that the notion of an nonmeasurable set lacked a firm foundation until a method was given for measuring a given set or proving that it could not be measured! But he did mean what he said and, as I have since learned, there are other indications that the understanding of mathematical thought in that time was very different from ours. Another example of this is provided by Abel’s statement in his unfinished treatise on the algebraic solution of equations that ‘‘at bottom’’ (dans le fond) the problem of finding all solvable equations was the same as the problem of determining whether a given equation was solvable. It would be explicable if he had said

that the proof that an equation is solvable is ‘‘at bottom’’ the problem of solving it, but he goes much further: If you know how to decide whether any given equation is solvable, you know how to find all equations that are solvable. To be honest, I don’t feel I fully understand these extremely constructive views of mathematics—I am a product of my education—but knowing that a mathematician of Abel’s caliber and experience saw mathematics in this way is an important phenomenon that a viable philosophy of mathematics needs to take into account. So Kronecker did mean it when he said that a method of factoring polynomials with integer coefficients is essential if one is to make use of irreducible polynomials, and he took care to outline such a method. I won’t go into any explanation of his method—I doubt that it was original with him, but his treatise is the standard reference— except to say that it is pretty impractical even with modern computers and to say that in his day it was utterly out of the question even for quite small examples. This observation makes it indisputable that the objective of Kronecker’s algorithm had to do with the meaning of irreducibility, not with practical factorization. It is a distinction that at first seems paradoxical but that arises in many contexts. If you are trying to find a specific root of a specific polynomial, Newton’s method is almost certainly the best approach, but if you want to prove that every polynomial

AUTHOR

.................................................................................................... HAROLD M. EDWARDS was a founding co-editor of The Mathe-

matical Intelligencer in 1978. Now Emeritus Professor at New York University, he has lived in New York since graduating from the University of Wisconsin in 1956, except for five years at Harvard and one year at the Australian National University. He has received both a Steele Prize and a Whiteman Prize from the AMS. He lives in Manhattan with his wife, journalist and author Betty Rollin. Courant Institute of Mathematical Sciences New York University New York NY 10012 USA e-mail: [email protected]

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has a complex root, Newton’s method is useless. In practice, it converges very rapidly, but the error estimates are so unwieldy that you can’t prove that it will converge at all until you are able to prove that there is a root for it to converge to, and for this you need a more plodding and less effective method. More generally, we all know that in practical calculations clever guesswork and shortcuts can play important roles, and Monte Carlo methods are everywhere. These are important topics in algorithmic mathematics, but not in Kronecker’s algorithmic mathematics. I am not aware of any part of his work where he shows an interest in practical calculation. Again, his interest was in mathematical meaning, which for him was algorithmic meaning. I have always fantasized that Euler would be ecstatic to have access to modern computers and would have a wonderful time figuring out what he could do with them, factoring Fermat numbers and computing Bernoulli numbers. Kronecker, on the other hand, I think would be much cooler toward them. In my fantasy, he would feel that he had conceived of the calculations that interested him and had no need to carry them out in any specific case. His attitude might be the one Galois expressed in the ‘‘preliminary discourse’’ to his treatise on the algebraic solution of equations: ‘‘... I need only to indicate to you the method needed to answer your question, without wanting to make myself or anyone else carry it out. In a word, the

calculations are impractical.’’ (... je n’aurai rien a` y faire que de vous indiquer le moyen de re´pondre a` votre question, sans vouloir charger ni moi ni personne de le faire. En un mot les calculs sont impraticables.) This somewhat provocative statement was omitted from the early publications of Galois’s works. See page 39 of the critical edition (1962) of Galois’s mathematics, like Kronecker’s, was algorithmic but not practical. That’s why it is not so surprising that all of this algorithmic mathematics—we could call it impractical algorithmic mathematics—was developed at a time when computers didn’t exist. This, in my opinion, was Kronecker’s conception of mathematics—that which his predecessors had accomplished and that which he wanted to advance. What generated the oncoming tide of set theory that was about to engulf this conception? Kronecker wrote about the rising tendency in very few places, but when he did write about it, he identified the motive for its development: Set theory was developed in an attempt to encompass the notion of the most general real number. In 1904, after Kronecker had been dead for more than a dozen years, Ferdinand Lindemann published a reminiscence about Kronecker that has become a part of the Kronecker legend and that is surely wrong. According to Lindemann, Kronecker asked him, apparently in a jocular way, ‘‘What is the use of your beautiful researches about the number p? Why think about such problems when irrational numbers do not exist?’’ We can only guess what Kronecker said to Lindemann that Lindemann remembered in this way, but I am confident that he would not have said that irrational numbers did not exist. To be persuaded of this, one only needs to know that Kronecker refers in his lectures on number theory (the ones edited and published by Kurt Hensel) to ‘‘the transcendental number p from geometry,’’ which he describes by the formula p4 ¼ 1 13 þ 15 17 þ : Note that Kronecker introduces p in his first lecture on number theory. Note also that he accepts p not only as an

irrational number but as a transcendental number; the proof of the transcendence of p was of course the achievement for which Lindemann was, and remains, famous. (His later belief that he had proved Fermat’s Last Theorem is benignly neglected.) Kronecker, as one of the great masters of analytic number theory, made frequent use of transcendental methods and would have had no qualm about real numbers. His qualm—and he stated it explicitly— had to do with the conception of the most general real number. My colleague Norbert Schappacher of the University of Strasbourg has discovered a document that states Kronecker’s qualm about the most general real number in a different way and confirms Kronecker’s statement to Cantor that his notions about the philosophy of mathematics were taught him by Kummer. The document is a letter of Kummer in which he states that he and Kronecker are in agreement in their belief that the effort to create enough individual points to fill out a continuum—that is, enough real numbers to fill out a line—is as vain as the ancient efforts to prove Euclid’s parallel postulate. (The quotation occurs in a letter from Kummer to his son-in-law H. A. Schwarz, dated March 15, 1872, in the Nachlass Schwarz of the archives of the Berlin-Brandenburg Academy of Sciences, folder 977.) In our time, when young students are routinely told that ‘‘the real line’’ consists of uncountably many real numbers and that it is ‘‘complete’’ as a topological set, this opinion of Kummer and Kronecker is heresy in the most literal sense—it denies the truth of what young people are told has the agreement of all authorities. So Kronecker, along with Kummer, saw what was going on—saw the push to describe the most general real number, saw, as it were, the wish on the part of his colleagues to talk about ‘‘the set of all real numbers.’’ Moreover, he responded to it. His response was: It is unnecessary. I have said that Kronecker says very little about the foundations of mathematics in his writings. But in the few words he does say, this message is

clear: It is unnecessary. One of the main goals of his mathematical work was to demonstrate that it was unnecessary by, as he told Cantor, expressing the truths and phenomena of mathematics in ways that were as free as possible from philosophical concepts. That would most certainly exclude any general theory of real numbers. He wished to show such a theory was unnecessary by doing without it. In view of the Kummer passage found by Schappacher, we see that he also believed there was a special importance to his belief that the construction of the set of all real numbers was not necessary, because he believed it was doomed to fail. In all likelihood you are now hearing for the first time the opinion that ‘‘the real line’’ may not be a well-founded concept, so I probably have no realistic hope of convincing you that this view may be justified. I won’t make a serious effort to do so. I will let it pass with just a brief reference to complications like Russell’s paradox, Go¨del’s incompleteness theorem, the independence of the continuum hypothesis and the axiom of choice, nonstandard models of the real numbers, and, coming at it from a different direction, Brouwer’s free choice sequences. There is a long history of unsuccessful efforts to wrestle with infinity in a rigorous way, efforts which, so far as I have ever been able to see, have been consistently frustrated. As Kummer and Kronecker foresaw. But even if one accepts that one day it will succeed—or that it long ago did succeed, except for uninteresting nitpicking—it seems to me that Kronecker’s main message is still worth hearing and considering: It is unnecessary. Mathematics should proceed without it to the maximum extent possible. Kronecker was confident that in the end its exclusion would prove to be no impediment at all. Well, of course modern mathematics has painted itself into a corner in which dealing with infinity in a rigorous manner is necessary. If mathematics is defined to be that which mathematicians do, then dealing with the real line is essential to mathematics. If mathematics insists on talking about


13

‘‘properties of the real line’’ as though the real line were a given, there is no room for the belief that it is unnecessary. Inevitably, then, Kronecker’s assertion is an assertion about the nature and domain of mathematics itself. It asserts that that which lies outside the Kroneckerian conception of mathematics is unnecessary. (Instead of the Kroneckerian conception, I would prefer to call it the classical conception of mathematics in deference to Euler and Gauss and Dirichlet and Abel and Galois, but somehow ‘‘classical mathematics’’ has come to mean the Cantorian opposite of this; therefore I am forced to call it the Kroneckerian conception.) With this meaning of ‘‘Kronecker’s algorithmic mathematics’’ in mind, we can perhaps agree that it is unnecessary to attempt to embrace the most general real number—to embrace ‘‘the real line.’’ What is lost by adopting this view of mathematics? I often hear mention of what must be ‘‘thrown out’’ if one insists that mathematics needs to be algorithmic. What if one is throwing out error? Wouldn’t that be a good thing rather than the bad thing the verb ‘‘to throw out’’ insinuates? I personally am not prepared to argue that what is being thrown out is error, but I think one can make a very good case that a good deal of confusion and lack of clarity are being thrown out. The new ways of dealing with infinity that set theory brought into mathematics can be seen in the method used to construct an integral basis in algebraic number theory. Kronecker gave an algorithm for this construction. You could write a computer program following his plan, and the program would work, although it might be very slow. Hilbert in his Zahlbericht approaches the same problem in a different, and outrageously nonconstructive, way. He imagines all numbers in the field written as polynomials with rational coefficients in a particular generating element a. The polynomials are then of degree less than m, where m is the

14


degree of a. Moreover, there is a common denominator for all the integers in the field when they are written in this way. Hilbert has the chutzpah to say: For each s = 1, 2, …, m, choose an integer in the field which is represented as a polynomial of degree less than s, and in which the numerator of the leading coefficient is the greatest common divisor of all numerators that occur in such integers. Such a choice is to be carried out for each s; the m integers in the field ‘‘found’’ in this way are an integral basis. Let me try to state in as simple a way as possible the process he is indicating: The integers in the field are a countable set, so it is legitimate to regard them as listed in an infinite sequence. The entries in the sequence are polynomials in a of degree less than m whose coefficients are rational numbers with a fixed denominator D. For each s, Hilbert wants us to first strike from the list all polynomials of degree s or greater, and, from among those that remain, choose one in which the numerator of the coefficient of as-1 is nonzero, but otherwise is as small as possible in absolute value. (Hilbert looks at the greatest common divisor of the numerators rather than the absolute value, but the effect is the same.) So, not once but m times, we are to survey an infinite list of integers and pick out a nonzero one that has the smallest possible absolute value. To put this in perspective, let me describe an analogous situation. Imagine an infinite sequence of zeros and ones is given by some unknown rule. Would it be reasonable for me to ask you to record a 1 if the sequence contains infinitely many ones and otherwise to record a 0? In twentiethcentury mathematics, one was asked to do such things all the time. Therefore, it is perhaps difficult to deny, as I would like to do, that it is a reasonable thing to ask. But surely no one would contend that it is an algorithm. No doubt Hilbert regarded his as a simplification of Kronecker’s construction. But only someone indoctrinated in the nonconstructive

Hilbertian orthodoxy, as I was, and as many of you surely were, could hear it called a ‘‘construction’’ without leaping from his or her chair in protest. To ‘‘throw out’’ from mathematics arguments of this type should be regarded as ridding it of ideas that are at best sloppy thinking and at worst delusions. And in this particular case, the argument for throwing out Hilbert’s argument is all the stronger because Kronecker had already shown many years earlier that it was, in truth, unnecessary. This contrast, between Kronecker’s algorithm for constructing an integral basis and Hilbert’s nonconstructive proof (can it be called a proof?) of the existence of an integral basis, illustrates the fork in the road that mathematics encountered at the end of the nineteenth century: To follow Kronecker’s algorithmic path, or to choose instead the daring new set-theoretic path proposed by Dedekind, Cantor, Weierstrass, and Hilbert. You all understand very well which path was taken and you all understand as well how I feel about the choice that was made. But now, in the twenty-first century, I hope mathematicians will begin to reconsider that fateful choice. Now that there are conferences devoted to ‘‘Computability in Europe’’ and mathematicians in their daily practice are dealing more and more with algorithms, approaching problems more and more by asking themselves how they can use their powerful computers to gain insight and find solutions, the climate of opinion surely will change. How can anyone who is experienced in serious computation consider it important to conceive of the set of all real numbers as a mathematical ‘‘object’’ that can in some way be ‘‘constructed’’ using pure logic? For computers, there are no irrational numbers, so what reason is there to worry about the most general real number? Let us agree with Kronecker that it is best to express our mathematics in a way that is as free as possible from philosophical concepts. We might in the end find ourselves agreeing with him about set theory. It is unnecessary.

An Analysis of Nonpositional Numeral Systems* CHRISTOPHER HOLLINGS

O

ver its many centuries of civilisation, the human race has produced a fantastic variety of numerical notations, as even a quick glance through texts such as 1 Cajori [3], Ifrah [9] and Menninger [12] will reveal. These numeration systems are usually broadly classified as being either positional or nonpositional. However, this dichotomous classification is somewhat coarse. To see this, consider the Roman numerals. These are often cited as the standard example of a nonpositional numeral system. However, if the Roman numerals were truly nonpositional, then we would expect IV and VI to represent the same value, which they do not. On the other hand, Roman numerals certainly do not form a positional system; they appear to be a broadly nonpositional system with some positional features. Conversely, there exist positional systems which display some of the features of nonpositional systems—specifically, the Babylonian sexagesimal [9, Chapter 26] and Maya vigesimal [9, Chapter 28] positional systems. Unlike the Hindu-Arabic numerals, each of these systems can have more than one symbol per position. The Babylonian system has dedicated symbols for 1 and for 10 which are then combined additively to make up the value (B59) in any given position. Thus, if we confine our attention to a single position, we find a nonpositional numeral system at work. This is evidence that the Babylonian sexagesimal positional system evolved from an earlier Sumerian decimal nonpositional system (which we will see in the next section). We find a similar phenomenon with the Maya system: there are dedicated symbols for 1 and for

5, which are used to make up the value in each position, hinting at the earlier use of a quinary nonpositional system. Clearly, a finer classification than the traditional positional-versus-nonpositional scheme is called for. The existence of such ‘‘mixed’’ numeral systems is acknowledged in [1], and a simple system of classification appears in [9, p. 429], into positional, nonpositional, and so-called hybrid systems; details can be found in [10, pp. 34–63]. The classification in the present paper will be rather different from that of Ifrah. The goal of this paper is to obtain a finer classification for nonpositional systems. Note that my analysis and classification will be of the mathematical and symbolical structure of numeral systems; a linguistical analysis of numeral systems has already been given in [15], for example. I will use the abbreviation ‘‘NNS’’ throughout to mean ‘‘nonpositional numeral system’’. Throughout this article, the word ‘‘number’’ will refer to a quantity, whilst the word ‘‘numeral’’ will describe a symbol. Thus, for example, ‘‘2’’ is a numeral denoting the number 2. However, since the Hindu-Arabic numerals will not be subject to analysis, there is no danger of confusion in our using these to denote numbers. Nonpositional numerical notations are by far the most common throughout history. They are characterised by the feature that the position of a particular symbol within the representation of a number is irrelevant. We may regard this as a consequence of the ‘‘additive’’ nature of such systems: juxtaposition of symbols denotes addition. For example, if h and j stand for 1 and 10, respectively, in a

*This article was written whilst the author was an EPSRC-funded research student at the University of York, UK. I will adopt [9] as my ‘‘standard’’ reference. I have also found invaluable the bibliography (posted online at http://phrontistery.info/nnsbib.html) of Stephen Chrisomalis’s doctoral dissertation The Comparative History of Numerical Notation (McGill University, 2003). 1


15

hypothetical NNS, then hh and hjhjjj both represent the number 42. It is prudent, however, to arrange numerals in a ‘‘sensible’’ order. In this article, I will always adopt the ‘‘standard ordering’’ which has been chosen most often throughout history: (SO)

group like-symbols together and arrange the groups (from left to right) in descending order of the values of the symbols.

Thus jjjjhh would be our ordering of choice for the representation of 42 in the above system. In contrast with the situation in an NNS, the placement of a symbol within a positional numeral system (also known as a place-value system) is vitally important. We can take our own Hindu-Arabic numerals as an example: 42 and 24 do not represent the same value. Thus positional numeral systems may be characterised as those systems of numerical notation in which the value of a symbol depends not only on the nature of the symbol, but also on its position with respect to any surrounding symbols. Such systems are much less common than nonpositional systems but are usually more succinct. In the case of our own Hindu-Arabic numerals, this is due in large part to the fact that our system employs ciphers: numbers are written using abstract symbols, rather than direct representations of the numbers themselves. For example, we write four as ‘‘4’’, rather than ‘‘||||’’. The discovery of the principle of ‘‘cipherisation’’ was a significant step forward in the development of numeral systems, both positional and nonpositional—see [2]. I will revisit the concept of cipherisation in a later section. Let me say a little more about the Roman numerals as motivation for the analysis to come. The precise nature of the symbols used in this system varied over the many centuries of Roman civilisation [9, Chapter 9]. For various theories as to the origins of the Roman numerals, see [11]. I will usually consider Roman numerals in the form in which they are most often used today (Figure 1). In addition to the symbols in Figure 1, we have the principle that placing a bar over a given numeral multiplies its value by 1000, thus M ¼ 1; 000; 000, etc. In fact, the Romans employed other similar devices; for example,

AUTHOR

......................................................................... CHRISTOPHER HOLLINGS was educated

at the University of York, England, where he got his PhD in 2007 in semigroup theory. He is now a post-doc at the University of Lisbon. Aside from semigroup theory and history of mathematics—and, of course, writing about mathematics—his pursuits are reading and photography. Centro de Algebra, Universidade de Lisboa, 1649-003 Lisbon, Portugal e-mail: [email protected]

2

Figure 1. Contemporary Roman Numerals.

placing u around a symbol would multiply its value by 100,000—see [12, p. 44]. Roman numerals are, of course, written additively. Thus, for example, MCCLV ¼ M þ C þ C þ L þ V ¼ 1255: However, as already observed, Roman numerals fall short of being truly nonpositional because of the subtractive principle, whereby a smaller numeral written to the left of a larger one acts subtractively. (Though this principle is commonly employed with Roman numerals today, it was not widely used by the Romans themselves. For a discussion of the subtractive principle, see [3, p. 31].) This allows us to write 4 as IV, 9 as IX, 40 as XL, and so on. If we accept the Roman numerals as a nonpositional system, then our aim must be to derive a means of classifying NNSs which admits something like the subtractive principle. Before I begin my analysis, I present four further examples of NNSs from history. Then in the following section I take one of these systems as our working example, and analyse the structure of NNSs in simple language-theoretic terms, leading up to a formal definition for an NNS. In a separate section I identify further important features of such systems. Next I return to the examples, applying the findings to analyse them more formally. I conclude the paper with a section where the analysis is applied to the motivating example of the Roman numerals.

Examples I now present a number of examples of NNSs from history. Each of these systems is written additively and employs (SO). Note also that each of these numeral systems varied during its period of usage; in each case, I therefore adopt a ‘‘standardised’’ version. Some of these systems also had provisions for expressing fractional quantities, but I will confine attention to the representation of (positive) integers.

E XAMPLE 1.1 S UMERIAN

2

The ancient civilisation of Sumer flourished in the Middle East from the 4th to the 2nd millennia BC. The Sumerians may have devised the sexagesimal positional numeral system which we now most often associate with the Babylonians [13]. That positional system had evolved from an earlier nonpositional system, the first four numerals of which are given in Figure 2—see also [8, p. 28]. Since this system will be our ‘‘working example’’ throughout the next section, for ease of type-setting, let us write: NUMERALS

For a (literally) very colourful account of the development of our understanding of Sumerian and Babylonian numerals, see [6]. See also [9, Chapter 11].

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Figure 2. Sumerian Numerals.

j ¼ 1;

h¼ 10;

Figure 3. Minoan Numerals.

j ¼ 100;

hj ¼ 1000:

Further powers of ten could be obtained in this system by repeatedly prefixing h|- with copies of h, thus hh|- = 10,000, hhh|- = 100,000, and so on; that is, h acted multiplicatively. For example,

Figure 4. Attic Numerals I.

hhj hj hj j j j hhhh jjjjj ¼ 12; 345:

E XAMPLE 1.2 M INOAN

3

These numerals (Figure 3) were used by the Minoan civilisation of Crete in around 1400 BC and are rather more compact than the Sumerian numerals of Example 1.1. Unlike the Babylonians, however, the Minoans had no way of writing higher powers of 10. NUMERALS

E XAMPLE 1.3 G REEK A TTIC N UMERALS 4 This system is one of the earliest of the Greek numeral systems, and was in use from the 5th to the 1st centuries BC. These numerals are often named ‘‘Herodian’’, or ‘‘Herodianic’’, after the Greek grammarian Herodian who described them in the 2nd century AD. (Menninger [12, p. 268] argues against calling these numerals ‘‘Herodian’’, as they had already been in use for over five centuries by Herodian’s time. He suggests the alternative name of ‘‘row numerals’’. The name ‘‘acrophonic’’ has also been applied to them [18], because they represent the first letters of the corresponding number-words.) There are at least two different versions of the Herodian numerals: the ‘‘Attic’’ version, so called because it was principally used in Attica, and the ‘‘Boeotian’’ version, used elsewhere. The differences between them were purely cosmetic; the two systems operated in exactly the same way. The ordinary symbols of the Attic system are given in Figure 4 (see [8, p. 30] for the Boeotian forms). There were also a further four composite symbols in common use (Figure 5). (Strictly speaking, the D in the symbol for 50 should be placed under the ‘‘hook’’ of the C. Similarly, the H, X and M. The forms used in Figure 5 are a concession to type-setting. See [8, p. 30] for the correct forms.

E XAMPLE 1.4 G REEK M ILESIAN N UMERALS 5 This is the well-known system of Greek ‘‘alphabetic’’ numerals which, by the 1st century BC, had replaced the earlier system of Example 1.3. The Milesian system of numerals (Figure 6) is related to the standard Greek (Ionic) alphabet and is named after Miletus, the most important ancient

Figure 5. Attic Numerals II.

Ionic city. Numerical values were assigned to each of the 24 letters of the Ionic alphabet, as well as to three additional signs: (digamma), (koppa) and (sampi). (It has generally been thought that the Ionic alphabet developed first and that the Milesian numerals came later, hence the need to include three extra symbols. However, there is some evidence that alphabet and numeral system developed side by side—see [14]. It has also been suggested that these numerals developed from the earlier (Egyptian) demotic numerals—see [4]. Incidentally the Phoenician-derived letters digamma and koppa, though little used in Classical Greek, survived as the Etruscan and Latin F and Q). There were two different ways in which multiples of 1000 could be written in this system. The first way was to prefix any of the symbols in Figure 6 with a 0 , e.g., 0 a = 1000, 0 b = 2000, etc. The other was to write the relevant symbol as an exponent of , thus a ¼ 1000, b ¼ 2000, etc. There were a number of ways in which tens of thousands could be written: 1 by writing the relevant symbol as an exponent of M (for myriad), e.g., Mb ¼ 20; 000; 2 by prefixing M! (also for myriad) by the same symbol, e.g., bM! ¼ 20; 000; 3 by using the ‘‘dot’’ notation, e.g., b ¼ 20; 000. Example: Ma 0 b sl ¼ 12; 345: Before moving on to the analysis of NNSs in general, let me make some observations about the examples in this section. First, they are all decimal systems; base 10 is by far the most common base for human numeral systems, for the obvious anatomical reasons (for a wider discussion of decimal systems, see [17]). Furthermore, these various numeral systems did not necessarily develop independently; the Minoan numerals of Example 1.2, for instance, may have been influenced by earlier Egyptian (decimal)

3

See [9, p. 217] and [16]. See [8, p. 30] and [9, p. 225]. 5 See [8, p. 31] and [9, Chapter 17]. 4


17

over A (the empty string e has no part to play here) will be denoted by A+. In the case of the Sumerian system, we can take A to be the following set: fj; h; g:

Figure 6. Milesian Numerals.

numerals [5, p. 33]. However, not all of history’s NNSs have been decimal; the Aztecs possessed a vigesimal NNS [9, p. 224], whilst, at one stage, the Sumerians employed a sexagesimal NNS [9, p. 171]. The second point relates to the way in which the numerals in each system are deployed. The demand for (SO) is not the only rule affecting the way the numerals of a given system are set out. In the Attic system, for example, we would never write 6 as IIIIII; it would always be written in its simplest form: CI. Similarly, we would never write CDCD for 100, and so on. We therefore demand that numerals always be written in their simplest form. Moreover, continuing with the example of the Attic numerals, although there is no dedicated symbol for 100,000, we still do not allow this to be written as CM CM , for the simple reason that this is not ‘‘in keeping’’ with the rest of the numeral system: if none of CD CD ; CH CH , and CX CX is permitted, then CM CM should not be either. In a given NNS, any string of numerals which satisfies the rules vaguely indicated in this paragraph will be said to be in proper form. I will formalise these ideas in the next section. I note finally that the Minoan, Attic, and Milesian numerals are all finite: in each case, there is an upper bound to the values which the numeral system is capable of representing in a proper form (99,999 in each case). The reason for these bounds is simply that the original users of these numeral systems had no use for numbers beyond that point. It is only the Sumerian numerals which feature an in-built method for representing arbitrarily high numbers: the principle of repeatedly prefixing |- by h to obtain successive powers of 10. It is clear, however, that this system would be impractical for large powers of 10, so the Sumerians presumably had a practical upper bound to their numerals.

Analysis In this section, I will identify the important features of an NNS and thereby arrive at a formal definition. As already commented, the Sumerian numerals of Example 1.1 will be our ‘‘working example’’ throughout this section. The most fundamental feature of a numeral system is surely the set of symbols from which all numerals are constructed. Using the terminology of formal language theory, I will refer to this set as the alphabet of the numeral system and denote it by A. The set of all nonempty strings 18


Observe that this alphabet is finite; a numeral system with an infinite alphabet would be far too impractical to use. Notice also that - is meaningless on its own; it must appear in conjunction with another symbol, specifically h, if it is to represent a number. Thus an element of the alphabet of a numeral system does not necessarily have a value assigned to it, and so the alphabet on its own is not enough to define the system; I will return to this point shortly. First, however, I make an important distinction between the different types of string which can appear in a numeral system. Let us continue with the example of Sumerian numerals and consider the following two strings: j j ¼ 101 and hj ¼ 1000: The first of these has been obtained as the ‘‘additive juxtaposition’’ of |- and |; the second, on the other hand, has been built up without recourse to this form of addition, as indeed have the |- and | which make up |- |. I will reserve the word numeral for any grouping of symbols from A which has been built up without using additive juxtaposition. Thus, in the Sumerian system, h|- is a numeral, but |- | is not. Denote the set of numerals of a numeral system by N. In the case of the Sumerian numerals, N is the following set: fj; h; j; hj; hhj; hhhj; . . .g: In order to construct the set N for a given NNS, we need a set R of rules which tell us which strings in A+ are admitted as numerals. For the Sumerian numerals, R contains the following rules: • | and h are numerals but - is not; • |- is a numeral; • |- preceded by finitely many of h is a numeral. The subset of A+ which consists of all strings which are admitted as numerals by the rules in R will be denoted by A+/R, so that A+/R = N. It is clear that any combination of symbols (such as, for example, |- | in the Sumerian system) which represents a numerical value is a string over N, i.e., is an element of N+. Any such string in N þ nN will be referred to as a numeral string. Note that unlike the usual concatenation of strings in formal language theory (and indeed the concatenation used to obtain A+), the concatenation of strings used to obtain N+ is commutative, by the additive nature of NNSs. It is conceivable that the alphabet A and the set of numerals N may coincide (in which case, we regard R as being empty); indeed, the Minoan numerals of Example 1.2 possess this feature. Any numeral system with this property will be called a simple system. I have perhaps been a little hasty in stating that any element of N+ represents a number in the given numeral system. Whilst this is certainly true, there are some combinations of numerals which we would simply never write;

for example, in the Sumerian system, we would never write ten of | to represent 10. There would be no ambiguity in us doing so but it would be rather unwieldly; it is much better to write this as the more compact h. Thus in our description of a numeral system, we must include a set S of relations which hold between the various numerals. In the case of the Sumerian numerals, S would of course contain the following relations: • • • •

ten ten ten ten

of | are equal to one of h; of h are equal to one of |-; of |- are equal to one of h|-; of h h j are equal to one of h h j , where |ffl{zffl} |ffl{zffl} n times

nþ1 times

n C 1. I will always assume that a string of numerals in N+ is as short as it can possibly be, in accordance with the relations in S. If this is the case, then I will say that the string is reduced, or that it is in reduced form. For example, a reduced string of numerals in the Sumerian system can contain no more than nine of |, no more than nine of h, etc. The maximum number of times which a numeral x can appear in a numeral string will be called the potential multiplicity of x and will be denoted by pm(x). In the Sumerian system, all numerals have potential multiplicity 9. Any numeral string which is in reduced form and which obeys (SO) will be said to be in proper form, as discussed at the end of the last section. In general, the subset of strings over N which satisfy the relations in S will be denoted by N+/S. I have so far made no mention of how a numeral system represents actual numbers. The fact that we have made it this far without explicitly mentioning such a fundamental feature is due to the utter clarity of the representation. However, if we are to obtain a formal definition of an NNS, then this idea too must be formalised. I do so by introducing a function f : N þ =S ! N which takes any numeral string in the given NNS to the positive integer which it represents; f will be called the interpretation of the system. I will borrow some terminology from [7]: if f is onto, then I will call the corresponding numeral system complete, because such a system is capable of representing any positive integer. Otherwise, I call the system incomplete. If f is one-one, then I will call the system nonredundant; a system with a many-one f is redundant, for it permits more than one representation of a given positive integer. Thus a nonredundant numeral system is more efficient than a redundant one; I will always choose nonredundant systems over redundant. Indeed, I have already taken steps to guard against redundancy by insisting that numeral strings be in proper form. I observed at the end of the last section that most of our examples of numeral systems have an upper bound. Let us now make this observation more precise by using the interpretation f: the (upper) bound of a numeral system is the least positive integer L such that f (x) \ L, for all x [ N+/ S. Let us also assume that f maps onto the set fn 2 N : n \ Lg, i.e., there are no ‘‘gaps’’—the system in

question is capable of representing all positive integers below its bound. Any numeral system for which the range of f in N is finite will be called a finite system. It is clear that a finite system has a bound and that this bound is M + 1, where M is the maximum element in the range of f. The range of f will also be termed the range of the system. With the assumption that there are no ‘‘gaps’’, it is also clear that finite NNSs are precisely incomplete NNSs. We can now use f to give a formal expression to the major features of an NNS which I noted initially. First, the notion of ‘‘additive juxtaposition’’ can be expressed as follows: (AJ) for all x,y [ N+/S, f (xy) = f (x) + f (y). Second, the nonpositional nature of an NNS can be expressed thus: (NP) for all x,y [ N+/S, f (xy) = f (yx). Further, we can now express (SO) in terms of f: (SO) let x1,...,xn [ N be such that x1 x2 xn 2 N þ =S; if i [ j, then f (xi) B f (xj). A nonredundant NNS is regarded as being nonredundant up to (SO) and the relations in S. Let us suppress f, except when making a general argument, or whenever the reinstatement of f will aid clarity; there is, I think, little danger of confusion in writing things like ‘‘| = 1’’, indeed, this is precisely what I did throughout the presentation of examples! Observe that if we agree that x n ðx 2 A; n 2 NÞ denotes the string consisting of n x 0 s; then (AJ) yields the following: f ðx n Þ ¼ n f ðxÞ: Beware the difference between f (xn) and f (x)n! Notice that (NP) does not appear to be satisfied by Roman numerals, for f ðIVÞ 6¼ f ðVIÞ. I will address this ‘‘problem’’ below. Thanks to (AJ) and to the relations in S, we need only specify the effect of f on the most basic of numerals, e.g., f ðjÞ = 1 for the Sumerian system; the values of f h , f jÞ, etc., follow easily. In the case of the Sumerian numerals, we can also use f to give a more elegant rendering of the relations in S: • • • •

f h = 10 9 f j ; f jÞ = 10 9 f h ; f hjÞ = 109 f jÞ; for n 2 N; f hn jÞ ¼ 10n f ðjÞ ¼ 10nþ2 :

By way of summary, let me gather together all of the foregoing considerations into a formal definition for an NNS:

D EFINITION 2.1 A nonpositional numeral system (NNS) is a quadruple S ¼ ðA; R; S; f Þ, where • A is the alphabet of S, that is, the set of symbols from which the system is constructed; • R is a set of rules which tell us how the elements of A may be concatenated to make numerals; the set of numerals is denoted by N = A+/R;


19

• S is a set of relations which tell us how the elements of N are interrelated and which numeral strings are in proper form; • f is a function N þ =S ! N, called the interpretation of the system, which satisfies conditions (AJ), (NP) and (SO). Further, a nonpositional ðA; R; S; f Þ is called: • • • •

numeral

system

the Attic numerals, of course, have base 10. Indeed, as observed earlier, all of our examples have base 10. Continuing with the example of the Attic numerals, observe that, whilst these numerals do not have base 5, there do exist dedicated symbols for various other multiples of 5. This observation leads to the following definition:

S¼

D EFINITION 3.2 Let S ¼ ðA; R; S; f Þ be an NNS with base b, and suppose that there exists an x 2 N ¼ Aþ =R with f ðxÞ ¼ a 2 N n f1g; for some a \ b. If there also exists a y [ N with f (y) = ab, then we will call a an auxiliary base for S:

simple if R = 6 0 and A = N; complete if f is onto; nonredundant if f is one-one; finite if the range of f is finite.

A simple numeral system will be regarded as a triple (N, S, f ), with N, S and f as above.

Thus the Attic numerals, for example, have auxiliary base 5.

Further features

D EFINITION 3.3 Let S ¼ ðA; R; S; f Þ be an NNS with base

In this section, I consider other noteworthy features of NNSs. One such feature is the base of the system. The base of an NNS is perhaps not so well-defined a concept as the base of a positional system, but, as we can see from the examples above, it is still a valid concept. Here is a concrete definition:

D EFINITION 3.1 Let S ¼ ðA; R; S; f Þ be an NNS and let x [ N = A+/R be such that f ðxÞ ¼ b 2 N n f1g. If there also exists a y [ N with f (y) = b2, then we will call b a base for S. (I do not insist that there exist numerals for higher powers of b, since these may be beyond the bound of the system.) Whilst each of our examples of NNSs has only one base, it is conceivable that a numeral system could have more than one. Let me demonstrate this by an artificial example:

E XAMPLE 3.1 Extend the example given in the introductory section by putting ( ¼ 1;

M ¼ 5;

¼ 10;

N ¼ 25:

So, by Definition 3.1, this numeral system has two bases: 5 and 10. However, this leads immediately to the problem of redundancy: 42 ¼ (( ¼ NM((: As already commented, nonredundant systems are to be preferred over redundant systems. If we were making practical use of the system given in this example, it is likely that we would quickly adopt one base or the other by, for example, discarding N. This is presumably also the reason why all of our examples of historical nonpositional numeral systems have only one base each. I tentatively suggest that these practical considerations would mean that a (nonpositional) numeral system with more than one base could never develop naturally. As an application of Definition 3.1, consider the Attic numerals. The first symbol after I is C = 5, so we might be tempted to think that the Attic numerals have base 5. However, there is no dedicated symbol for 25, so this cannot be;

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b. Any numeral x 2 N ¼ Aþ =R with f (x) = bk, for some k 2 N; will be called a basic numeral. The set of basic numerals of S will be denoted by B N : A basic NNS is an NNS in which all numerals are basic (i.e., B = N). (If we were allowing numeral systems with more than one base, we could, of course, refer to numerals as being basic with respect to a given base.) Notice that in a basic NNS with base b, every numeral has potential multiplicity b-1. A glance through the examples above reveals that the Attic numerals are basic but that the Milesian numerals are not. Indeed, in some sense, the Milesian numerals are as far away from being basic as it is possible to be. Let us make this statement more precise, via the following definition. (The terminology introduced here is inspired to some extent by that used by Archimedes in his Sand-reckoner— see [12, p. 140].)

D EFINITION 3.4 Let S ¼ ðA; R; S; f Þ be an NNS with base b and set of numerals N ¼ Aþ =R . The k th period of S is the set PðkÞ ¼ fx 2 N þ =S : bk1 f ðxÞ bk 1g: Note that the size of P(k) increases with k. In a decimal NNS, P(1) contains representations of all integers from 1 to 9, inclusive, P(2) from 10 to 99, inclusive, and so on. In [14], the periods of the Milesian numerals are referred to as enneads, from the Greek word for ‘‘nine’’; I adopt this as the term for the periods of any decimal NNS. Note that if a given finite numeral system has base b and bound L = bk, then the system has k periods. An NNS need not consist of full periods; in the last section, we will see that the Roman numerals, as defined in this paper, do not. For each system, we may ask how many numerals it has per period:

D EFINITION 3.5 Let S ¼ ðA; R; S; f Þ be an NNS with base b. The number of numerals in the k th period of S is called the k th rank of S: We denote the k th rank of S by r(k). If r(k) is the same for each k, then call S a regular NNS, and refer

to r(k) := r as the rank of S: Otherwise, we call S irregular. Notice that rðkÞ ¼ jPðkÞ \ N j: Thus, the Attic numerals are regular with rank 2, whilst the Milesian numerals are regular with rank 9. Notice that a basic NNS is necessarily regular and may be characterised as a system with rank 1. In the introductory section, I mentioned the concept of cipherisation. This is the process by which numerals come to be written by abstract signs which have little or no connection with the quantity being represented. Our own numeral 2, for example, is a cipher, since the only connection it has with the quantity ‘‘two’’ is the connection which we impose upon it; contrast this with the Roman II. If cipherisation is taken to extremes, then we obtain an NNS in which there is a separate symbol for each numeral within a given period; there is no longer any need to repeat numerals additively.

D EFINITION 3.6 Let S ¼ ðA; R; S; f Þ be an NNS with base b. If S is regular with rank b-1, then we call S completely ciphered. Thus we may regard the rank of an NNS as a measure of its ‘‘degree of cipherisation’’. The Milesian numerals, which are completely ciphered, are ‘‘more ciphered’’ than the Attic numerals, which are, in turn, ‘‘more ciphered’’ than a basic NNS. A basic NNS is the ‘‘least ciphered’’ numeral system we can have. Note that a completely ciphered NNS with base b will have auxiliary bases 2, 3, . . .; b 1: In a completely ciphered NNS, every numeral has potential multiplicity 1. Observe further that in such an NNS, if s is a numeral with f(s)2 \ b, then s is not just an auxiliary base, but a fully-fledged base of the system, in accordance with Definition 3.1. The fact that the system is completely ciphered means that such a redundancy problem as illustrated above does not arise.

The examples analysed Let us proceed with the application of these general ideas to the examples of numeral systems above. In some cases, there is more than one way of applying our earlier observations; wherever possible, I will take the simplest, or most intuitive, option.

E XAMPLE 4.1 As observed earlier, the Sumerian numerals form a basic numeral system with base 10 and rank 1. They are complete and, like all our other examples, nonredundant. E XAMPLE 4.2 The Minoan numerals (Example 1.2) form

A ¼ N ¼ fI; C; D; CD ; H; CH ; X; CX ; M; CM g: The relations are: I5 ¼ C; C2 ¼ D; D5 ¼ CD ; and so on, so that I, D, H, X and M all have potential multiplicity 4, whilst C, CD ; CH ; CX and CM all have potential multiplicity 1. The interpretation is f ðIÞ ¼ 1 . The Attic numerals have auxiliary base 5. Just like the Minoan system, this system is finite, with bound 100,000. It is regular with rank 2.

E XAMPLE 4.4 Let us turn next to the Milesian system (Example 1.4). Notice first that if we were to allow both of the possible ways of writing multiples of 1000, this system would be redundant. Let us therefore use only the ‘‘0 a’’ convention. Similarly, let us choose to write all multiples of 10,000 in the ‘‘Ma ’’ notation. Let G denote the set of letters in the standard Greek (Ionic) alphabet. If we then put H ¼ G [ f , , g we have the alphabet A ¼ H [ f0 ; Mg: The rules in R are then as follows: • the elements of H are numerals; • for any x 2 H; 0 x and Mx are numerals. In order to regard Mx as an element of A+, it is convenient to agree that it is simply a special way of writing the string Mx; this particular string has no other meaning attached to it, so no ambiguity is entailed. (We could not have made the analogous convention with CD in the Attic system, for example, for CD ¼ 15 6¼ CD : It worked better to regard CD as an element of A from the first.) To give the most succinct rendering of the relations in S for this system, I begin by giving those of the form an = x, remembering that an denotes the string of n a0 s: • a2 ¼ b; a3 ¼ c; . . .; a10 ¼ i: Now that we know how to express i in terms of a, I give the relations of the form in ¼ y: • i2 ¼ j; i3 ¼ k; . . .; i10 ¼ q: Next, the relations of the form qn = z: • q2 ¼ r; q3 ¼ s; . . .; q10 ¼ 0 a: The remaining relations can be expressed more succinctly by using f: • for each x 2 H; f ð0 xÞ ¼ 1000 f ðxÞ 10; 000 f ðxÞ:

and

f ðMx Þ ¼

a system which is both simple and basic, with A = N being the set of numerals given in Figure 3. The relations contained in S are clear: ten of are equal to one of , etc. With these relations established, define f by putting f ð Þ = 1. As there is no (proper) representation for any number greater than 99,999, this system is finite, with bound 100,000.

We of course also need f (a) = 1. Using these relations, we can now evaluate f (x) for any x 2 N þ =S; though in a rather more involved way than for our other examples. For instance:

E XAMPLE 4.3 Let us revisit the Attic numeral system of

Note that if one chose to write multiples of 1000 in the ‘‘ x ’’ notation, then one would not need to include the symbol 0 in A. However, one could not then regard

Example 1.3. We see straight away that this system is simple, with

f ðlbÞ ¼ f ðlÞ þ f ðbÞ ¼ f ði4 Þ þ f ða2 Þ ¼ f ða40 Þ þ f ða2 Þ ¼ 40f ðaÞ þ 2f ðaÞ ¼ 42:


21

a ¼ 1000; for example, as a special way of writing a; since the latter already stands for 901. The only way around this would be to include a, b, etc. in A. That is why I adopted the ‘‘0 x’’ notation. The Milesian system is finite, with bound 100,000, and, as already observed, it is completely ciphered. Consequently, this system has auxiliary bases 2, 3, 4, 5, 6, 7, 8 and 9. Strictly speaking, 2 and 3 are also bases of the system.

The Roman numerals revisited At last I return to the Roman numerals (Figure 1) and the ‘‘IV-vs.-VI’’ problem from which this analysis began. The easy way around this problem would of course be to insist that 4 always be written as IIII, 9 always be written as VIIII, etc., as many Romans did; but the ‘‘subtractive principle’’ can be handled using the analysis of this paper. Let us take the alphabet for the system of Roman numerals to be the following: A ¼ fI, V, X, L, C, D, M, g: Next come the rules in R which determine the numerals of the system. It is at this stage that we take care of the tricky combinations, such as IV. Recall that the nonpositional nature of an NNS was expressed in (AJ) as follows: f (xy) = f (x) + f (y), for all x; y 2 N þ =S: We see then that if we define IV to be a numeral in this system, then our problems disappear, since the string ‘‘IV’’ cannot then be broken down into ‘‘I’’ and ‘‘V’’, so (AJ) no longer applies. Furthermore, IV cannot be confused with VI, thanks to (SO). In full, the list of rules in R is as follows: • everything in A n f g is a numeral; • IV, IX, XL, XC, CD and CM are numerals; • if x 2 A n fI; g; then x is a numeral. (Of course x is simply a shorthand for the string x-.) Note that I have excluded I as a numeral in the interests of nonredundancy. Our next step is to specify the relations in S: 5

2

5

2

5

2

• I ¼ V; V ¼ X; X ¼ L; L ¼ C; C ¼ D; D ¼ M; • I4 ¼ IV; VI4 ¼ IX; X4 ¼ XL; LX4 ¼ XC; C4 ¼ CD; DC4 ¼ CM; 5 2 2 5 2 • M5 ¼ V; V ¼ X; X ¼ L; L ¼ C; C ¼ D; D ¼ M; thus I, X, C, M, X; C and M all have potential multiplicity 4, and all other numerals have potential multiplicity 1. We also have: ¼ 1000 f ðxÞ; for any x 2 A n fI; g: • f ðIÞ ¼ 1 and f ðxÞ There is one remaining difficulty which requires care: what value do strings such as VIV represent? This numeral string can be read in two different ways: either as VI + V, or as V + IV. Note however that VI + V = V + I + V does not satisfy (SO), whereas V + IV does. We should therefore read VIV as V + IV. It is then an easy task to write this string in reduced form: we write IV in terms of I and then apply the above relations to give VIV = VIIII = IX. We see then that the Roman numerals form a decimal system with an auxiliary base of 5, since there exist 22


dedicated numerals for 5, 50, 500, 5000, 50,000 and 500,000. This system is nonredundant and finite; given that we have only allowed a bar to be placed over the individual numerals V, X, L, C, D and M, the highest-valued numeral string which can be written down is MMMMDCCCCLXXXXV M M M M D C C C C X C I X ¼ 4; 999; 999; giving the Roman numerals a bound of 5,000,000. (Given that this numeral string is in proper form, there is no ambiguity in the final four symbols: they must be XC = 90 and IX = 9.) Roman numerals are neither simple not basic; we have B ¼ fI; X; C; M; X; C; Mg: Roman numerals are not regular either; the system, as defined here, consists of 6 full enneads (beginning with I, X, C, M, X and C, in turn) and one ‘‘half-ennead’’ (running from M to the above representation of 4,999,999). We have r(1) = r(2) = r(3) = 4, whilst r(4) = r(5) = r(6) = 2. If we chose to allow such strings as MV and X L as numerals, then we could force the Roman numerals to be regular.

Further testing Note that whilst I have tried to ensure that this analysis is as general as possible, I may have overlooked some features which will come to light only as further examples are analysed. Also, all my examples have been decimal NNSs, yet I mean the analysis to be applicable to NNSs of any base. Examples of non-decimal NNSs are rare, but I have already mentioned two: a Sumerian sexagesimal NNS, from around the 26th century BC, and an Aztec vigesimal NNS, which was in use up until the Spanish conquest. I have applied the foregoing analysis to each of these numeral systems, and it is indeed still valid. However, I leave these analyses as exercises for the reader!

REFERENCES

[1] W. F. Anderson, Arithmetic in Maya numerals, American Antiquity 36 (1971), 54–63. [2] C. B. Boyer, Fundamental steps in the development of numeration, Isis 35 (1944), 153–168. [3] F. Cajori, A History of Mathematical Notations, Volume I: Notations in Elementary Mathematics, Open Court, 1974. [4] S. Chrisomalis, The Egyptian origin of the Greek alphabetic numerals, Antiquity 77 (2003), 485–496. [5] S. Dow, Mycenaean arithmetic and numeration, Classical Philology 53 (1958), 32–34. [6] J. Friberg, Numbers and measures in the earliest written records, Scientific American 250 (2) (1984), 78–86. [7] H. L. Garner, Number systems and arithmetic, Advances in Computers 6 (1965), 131–194. [8] T. L. Heath, A History of Greek Mathematics, Volume I: From Thales to Euclid, Clarendon Press, Oxford, 1921. [9] G. Ifrah, Histoire Universelle des Chiffres, 1981 (translated by Lowell Bair as From One to Zero: A Universal History of Numbers, Viking Penguin, New York, 1985).

[10] G. Ifrah, A Universal History of Numbers III: The Computer and the

[14] D. K. Psychoyos, The forgotten art of isopsephy and the magic

Information Revolution, The Harvill Press, London, 2000. [11] P. Keyser, The origin of the Latin numerals 1 to 1000, American

number KZ, Semiotica 154 (2005), 157–224. [15] Z. Salzmann, A method for analyzing numerical systems, Word 6

Journal of Archaeology 92 (1988), 529–546. [12] K. Menninger, Number Words and Number Symbols: A Cultural History of Numbers (translation by Paul Broneer), MIT Press, 1969. [13] M. A. Powell, The antecedents of Old Babylonian place notation and the early history of Babylonian mathematics, Historia Mathematica 3 (1976), 417–439.

(1950), 78–83. [16] G. Sarton, Minoan mathematics, Isis 24 (1936), 375–381. [17] G. Sarton, Decimal systems early and late, Osiris 9 (1950), 581– 601. [18] M. N. Tod, The Greek numeral notation, Annual of the British School at Athens 18 (1911–12), 98–132.


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Mathematical Entertainments

Really Good Answer MICHAEL KLEBER This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, surprising, or appealing that one has an urge to pass them on. Contributions are most welcome.

Michael Kleber and Ravi Vakil, Editors

D

ear Readers: I need your help. I have a really good answer, and I’m desperately seeking the question that goes with it. Let me explain. A few years ago, I ran into a venerable recreational math puzzle. Here is the version I first encountered: The Buried Cable Problem. A company has just buried a ten-mile-long cable containing 120 individual wires. Unfortunately, the project was overseen by a summer intern, and the sad result is that the wires are entirely unlabelled. Your job is to fix this mess: Label each wire with the same name on both ends. The tools you have available are: 1. Plenty of patch cord: You can connect any wires to any other wires at the same end of the cable. 2. A battery and light bulb, so that you can check whether or not two wire ends are connected to each other. 3. Your feet. When you are at one end of the cable, you can do all the patching and testing and labeling that you want—but then you need to walk the whole 10 miles to the other end before you can continue. Question: In how few trips down the length of the cable can you label the wires? With just a hint of shame and bitterness in my voice, let me proclaim that I came up with a great solution. And it’s not just mine: A number of friends and colleagues came up with this solution independently. I’m sorry, am I starting to sound defensive? All right, enough disclaimers, here it is.

Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University, Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA e-mail: [email protected]

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The Triangle Number Solution Suppose that the number of wires n is a triangle number (like n = 120, conveniently). Then you can label all the wires in only two trips.


By way of example, suppose n = 10, and think of the wires corresponding to the dots in the picture:

At one end of the cable, patch together all wires that are in the same row of the triangle, then walk to the other end. By testing to see how many wires each one is connected to, you can reconstruct the equivalence classes. Now patch together any two wires that are in the same column and walk back to the first end, remove your earlier patches, and again test to learn the size of each wire’s equivalence class. You’ve now labelled each end of each wire by the pair (s1, s2) of sizes of its equivalence classes, and all the labels are distinct. Success! And what if n is not triangular? Here’s the really great part of my solution. Notice the coordinates that we used in the 10-wire example:

They all sum to 5, 6, 7, or 8. If we had, say, three more wires, then we could apply this technique to them at the same time:

There’s no problem confusing the two triangles with each other, since the coordinate sums do not overlap. In general, then, we can use this method for n wires any time n can be written as a sum of triangle numbers such that if we use the r th triangle number T (r) in the sum, we can’t use

any T (s) for r \ s \ 2r. Let’s call this a Non-Overlapping Triangle Sum representation of n, or say that n is a NOTS for short. So which n have NOTS representations? I was delighted to prove that this condition holds for all but finitely many n: There are only 39 numbers which are not NOTS; the largest is 720.1

P ROOF . Observe that for any ‘‘sufficiently large’’ value of n, you can write it as a NOTS by using any triangle number that is at least 4n/5 and then recursing. The remainder is less than 1/4 of the triangle number you just used, therefore less than the smallest triangle number which is illegal to use in the sum. Now we just need a base case for our induction, and brute force calculation through n = 3960 will do the trick. Starting at 3961, the function ‘‘n minus the smallest triangle number larger than 4n/5’’ is always 721 or larger. Once we verify that every number from 721 to 3960 can be expressed, all larger values come for free. If you feel like you’ve gotten your money’s worth out of this problem, by all means stop reading here. There’s sad news ahead. The Easy Way The first person who gave me a different way to label all the wires with only two trips was my then-colleague Jade Vinson. And after a little research, I discovered that Martin Gardner had already written about this other, better, solution. Gardner had used the problem in his ‘‘Mathematical Games’’ column in Scientific American, reprinted in Hexaflexagons and Other Mathematical Diversions (1959), his first collection published in book form.2 That version had 11 wires running from the basement to the attic, and an electrician trying to avoid too many flights of stairs. Here’s the simple solution for n = 11, but all odd n work the same

way. Starting in the attic, patch wires together in pairs, leaving the odd wire not attached to anything—call this one 0. Now go to the basement and identify which wire is 0, and identify the other pairs, arbitrarily numbering the pairs 1 through 5 and labeling the individual wires 1a and 1b, 2a and 2b, 3a and 3b, etc. Still in the basement, you now patch 0 to 1a, 1b to 2a, 2b to 3a, and so on, leaving 5b unattached. Back in the attic, you remove the old patch cords (but remember the pairings) and look for the only wire connected to 0. That must be 1a, and the other in its pair must be 1b. The only wire connected to 1b must be 2a, and so on. A small variant works when the number of wires is even: Start by leaving two wires unpatched, so in the attic you have two candidates for 0. In the basement, you pick one of them arbitrarily to play the role of 0 above and leave the other one out of the loop entirely. When you get back to the attic, you’ll have no trouble telling the true 0, now attached to 1a, from the fake 00 ; still not attached to anything. (Well, unless n = 2, in which case you are out of luck: There is no way to tell just two identical wires apart. Maybe you can find a cold water pipe to use as ground.) Several of Gardner’s readers also offered a second solution, this time based on a triangular pattern—but with no triangle-number restriction on n. For 13 wires, for instance, he patches wires into equivalence classes based first on the rows, and later on the columns, of this diagram:

The incomplete triangle means that when you inspect the results of the patching-by-rows, you find two equivalence classes of size 3. When you patch by columns, you arbitrarily choose which size-3 group will play

which role, and back at the other end, it’s easy to tell the size-3 groups apart: Only one of them has two wires patched by columns to the unique group of size 2

Desperately Seeking At this point you may well be asking yourself: If Martin Gardner published two perfectly good solutions that work for all n 6¼ 2; why did I waste your time telling about my Non-Overlapping Triangle Sum solution? I’m holding out hope. While it isn’t the answer to the buried cable problem, I’d like to think that the all-but-39 NOTS solution is the answer to some other problem. Well, some other problem with appeal, of course, something better than ‘‘Find all numbers that can be written as a sum of triangle numbers T ðk1 Þ þ T ðk2 Þ þ þ T ðkr Þ with ki 2kiþ1 .’’ For a while, I thought I knew what that problem was. In the Buried Cable problem, you can probe the system by performing one simple operation: Asking ‘‘Is this wire patched to this other wire?’’ The NOTS solution, though, is built on asking a strictly weaker question: ‘‘To how many other wires is this wire connected?’’ But as Wellesley College prof, and my wife, Jessica Polito pointed out, you can identify the wires of an 11-strand cable using just the ‘‘how many wires’’ question, by patching them together according to the rows and columns of this nontriangular recipe:

Here we have removed one wire (at ) from the triangular solution and replaced with two wires, one in the same row but a new column, and one in the same column but a new row. Repeating this transformation (or adding one unpatched wire) gives a solution for n = 12, a number not attainable with the NOTS approach.

1

For the record, the complete list of exceptions is: 2, 5, 8, 9, 12, 17, 20, 23, 26, 30, 33, 38, 41, 44, 53, 54, 57, 60, 63, 64, 75, 83, 86, 87, 90, 114, 117, 122, 125, 128, 162, 243, 251, 317, 363, 408, 411, 618, 720. 2 Now conveniently available on ‘‘Martin Gardner’s Mathematical Games’’ (CD-ROM, Mathematical Association of America, 2005) containing scans of all 15 books covering 25 years of his column.


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So it is time to abandon the buried cable and look elsewhere. I leave it in your hands, dear reader. Is there a natural problem to which this Non-

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Overlapping Triangle Sum cofinite set is the solution? Or will it be relegated to the dustbins of too-clever puzzle solution history? Let me know.

Google 5 Cambridge Center, Cambridge, MA 02142 USA e-mail: [email protected]

Years Ago

David E. Rowe, Editor

A Look Back at Hermann Minkowski’s Cologne Lecture ‘‘Raum und Zeit’’ DAVID E. ROWE

Send submissions to David E. Rowe, Fachbereich 08, Institut fu¨r Mathematik, Johannes Gutenberg University, D-55099 Mainz, Germany. e-mail: [email protected]

A

century ago, David Hilbert stepped to the podium at a special meeting of the Go¨ttingen Academy of Sciences to recall the achievements of his close friend Hermann Minkowski. Just one month earlier, on 12 January 1909, the 43year-old Minkowski had died unexpectedly after suffering a ruptured appendix, leaving those close to him in a state of shock. No colleague was more deeply affected than Hilbert, whose memorial lecture [Hilbert 1910] reflects his deep sense of personal loss: Our science, which we loved above everything, had brought us together. It appeared to us as a flowering garden. In this garden there are beaten paths where one may look around at leisure and enjoy one self without effort, especially at the side of a congenial companion. But we also liked to seek out hidden trails and discovered many a novel view, beautiful to behold, so we thought, and when we pointed them out to one another our joy was perfect [Hilbert 1910, 363]. Hilbert loved to think about mathematics while tending the spacious garden behind his house, a sanctuary that only a few were allowed to enter. Minkowski, who lived just a stone’s throw away, was a regular visitor. So far as we know, Minkowski was not, like Hilbert, a passionate gardener. But both loved nothing more than to meet and talk about mathematics. Minkowski’s tragic death came at a time when he and his friends were still celebrating one of the high points of his career, the lecture ‘‘Raum und Zeit’’ [Minkowski 1909] he delivered in Cologne on 21 September 1908. Before a mixed audience of seventy-one

mathematicians, physicists, and philosophers [Wangerin 1909, 4], Minkowski made it clear from the outset that he had something important to say: The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. Their tendency is radical. From this moment onward, space by itself and time by itself will totally fade into shadows and only a kind of union of both will preserve independence [Minkowski 1909, 431]. The physics Minkowski had in mind was itself radically new. It sprang from Lorentz’s electron theory, Einstein’s ether-free electrodynamics based on the principle of relativity, and Planck’s further extension of Einstein’s findings regarding the inertia of energy. Minkowski not only found this line of physical research promising, he sought to convince his audience that it demanded a totally new conception of space and time, one that departed fundamentally from the premises of classical Newtonian physics. His lofty rhetoric in ‘‘Raum und Zeit’’ no doubt made an impression on his audience, even if few could appreciate the deeper points that propelled his argument forward. But, fortunately, Max Born and Arnold Sommerfeld were there.1 Both went on to pursue Minkowski’s ideas immediately after his death, thereby bringing space-time physics into the relativistic arena. Hilbert, too, was hardly a disinterested party; he was also determined to help ensure that his friend’s contributions to relativity theory would not be forgotten.2 In his memorial lecture, he recalled watching Minkowski put the last touches on the final page proofs as he lay

1 Born recalled attending Minkowski’s lecture in his autobiography [Born 1978, 131]; Sommerfeld was listed as one of the discussants after the lecture in Wangerin [1909, 9]. 2

Hilbert took charge of publishing Minkowski’s collected works [Minkowski 1911] soon after the latter’s death.


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Photo 1. This photo, taken around 1905, shows Minkowski on an excursion with the Hilberts: Ka¨the, David, and their son Franz. Flanking the group are two of Hilbert’s many gifted doctoral students: Alfred Haar, to the left, and Ernst Hellinger. The young woman between them was probably the Hilberts’ housekeeper.

on his deathbed. He further related how Minkowski took solace in the thought that his passing might induce others to study his contributions to the new physics more carefully. During their lifetimes, Hilbert and Minkowski were likened to Castor and Pollux, the inseparable twins [Born 1978, 80]. Their careers first became intertwined in the 1880s as fellow students at the university in Ko¨nigsberg, the Albertina, where Jacobi had once taught. From this remote outpost in East Prussia, they made their way into the German mathematical community that was then taking form [Rowe 2003]. They were among the small group of mathematicians who gathered in Bremen in 1890 at a meeting of the Society of German Natural Scientists and Physicians in order to launch a new national organization, the Deutsche Mathematiker-Vereinigung (DMV). The DMV remained, in 28


fact, for some time under the umbrella of this larger scientific society, which helps explain why several physicists were in Minkowski’s audience in Cologne. Go¨ttingen’s Felix Klein, who also attended that first meeting of the DMV, came away from it firmly convinced that Hilbert was the ‘‘rising man’’ among the younger generation of German mathematicians [Rowe 1989, 196]. Klein was not merely a talent scout for his university. Already well past his prime as a researcher, he saw himself as uniquely qualified to judge prospective appointments throughout the entire German system of higher education. Soon after the Bremen meeting he imparted his opinion to Friedrich Althoff, the key figure in the Prussian Ministry of Culture who would later play a decisive role in promoting the careers of both Hilbert and Minkowski. After 1902, as colleagues in Go¨ttingen, both would

put their unmistakable imprint on the mathematics of the new century. One of those most deeply affected by this Go¨ttingen atmosphere was Hermann Weyl. A month after the German army surrendered to the Soviets in Stalingrad in early February, 1943, Weyl wrote a letter to Minkowski’s widow recalling the mathematical world he had known in his youth: Last Sunday while attending a mathematical meeting in New York I learned that Hilbert has died. There was a short report about this, dated Bern, 19 February, which appeared in the New York Times on 20 February, but I missed this and perhaps you did, too. The news brought back powerfully again the whole Go¨ttingen past . . . . The two friends’ activities, their influence on the younger generation complemented one another in the most harmonious way, Hilbert perhaps

the more blazing and buoyant (Lichtere und Leichtere), your husband the warmer . . . and certainly the kinder . . . . Nowhere today is there anywhere anything even remotely comparable . . .3 Soon thereafter, Weyl paid tribute to Hilbert in two obituary articles, the longer of which appeared in the Bulletin of the American Mathematical Society [Weyl 1944]. It began on the same note, though without mentioning Minkowski’s role in Hilbert’s success story. ‘‘In retrospect,’’ he reflected, ‘‘it seems to us that the era of mathematics upon which [Hilbert] impressed the seal of his spirit and which is now sinking below the horizon achieved a more perfect balance than prevailed before and after . . . . No mathematician of equal stature has risen from our generation’’ [Weyl 1944, 130]. In recalling these events and circumstances, one cannot help but notice a certain imbalance in the literature documenting the lives of these two stellar figures. Whereas accounts of Hilbert’s singular career abound (for example [Blumenthal 1935] and [Reid 1970]), relatively little has been written about Minkowski’s life, personality, or mathematical style. Yet despite some striking affinities, one can hardly overlook the contrasts between them. As a researcher, Hilbert pursued clearly posted programs undertaken during periods of intense activity. Most of his work focused on building a theory from the ground up: starting with invariant theory and algebraic number fields, he moved on to Euclidean and non-Euclidean geometry, then variational methods, integral equations, and foundations of physics, and finally proof theory and the axiomatization of arithmetic. He was a master builder, a systematizer who only seldom broke entirely new ground. Minkowski no doubt appreciated that approach, yet his own style was anything but systematic. More original and highly inventive, he had the temperament of an artist, one who sought to commune with his material rather than impose a preconceived design on it. One can sense these differences in the letters he wrote to Hilbert, especially

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those that pertain to their nearly abortive joint venture from the 1890s, the Zahlbericht. Yet despite this contrast in their styles, Hilbert had a deep understanding of Minkowski’s work; indeed, his memorial lecture [Hilbert 1910] remains even today the best single overview of the latter’s mathematical achievements. As a human portrait, on the other hand, it appears to tell us more about Hilbert’s personality than about Minkowski’s. Insight into the latter’s persona, however, can be gained from his letters to Hilbert [Minkowski 1973], published by Minkowski’s daughter Lily Ru¨denberg, who also added a short family biography. These letters shed considerable

light on Minkowski’s relationship with Hilbert, although unfortunately only this half of their correspondence has survived. Hilbert’s letters, a few of which were cited in Blumenthal [1935], disappeared sometime after Hilbert’s death. Another useful resource is Max Born’s autobiography [Born 1978], which abounds with lively reminiscences of Go¨ttingen in the era of Klein, Hilbert, and Minkowski, though Born is not always reliable regarding historical details. Drawing on these and other sources, I begin this essay with some key episodes in the partnership that developed between Hilbert and Minkowski, a familiar enough theme, although one

Photo 2. Minkowski made these five sketches for the illustrations that appear in the published version of his ‘‘Raum und Zeit’’ (Courtesy of Niedersa¨chsische Staats- und Universita¨tsbibliothek Go¨ttingen). Compare the spacetime diagram shown in colour on our cover.

Hermann Weyl to Auguste Minkowski, March 1943; transcription courtesy of Gunther Ru¨denberg.

Ó 2009 Springer Science+Business Media, LLC 2009, Volume 31, Number 2, 2009

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that nevertheless deserves closer attention.4 In particular, I would like to focus on the symbiotic nature of their spiritual relationship as devotees of pure mathematics, a calling of special significance within the context of German culture. Both were intent on promoting their purist vision of mathematical knowledge, at first within the realm of algebra and number theory, but eventually permeating into the new physics. Minkowski’s ideas were to have an especially deep and lasting impact on relativity theory, although it would be a mistake to regard this final chapter of his life in isolation from the rest. Indeed, the episodes related here suggest that Minkowski’s Cologne lecture on ‘‘Space and Time,’’ the culminating triumph of his career, was shaped by earlier experiences he and Hilbert shared in public mathematical arenas.

Minkowski’s Partnership with Hilbert Minkowski and Hilbert grew up in Ko¨nigsberg where they developed a deep aesthetic appreciation of mathematical ideas. Both became numbertheorists, which meant that they tended to identify good mathematics with beautiful mathematical ideas. In his memorial lecture, Hilbert described his friend’s tastes in these words: ‘‘Minkowski was deeply captivated by the significance of number theory, as reflected in the works and enthusiastic expressions of its heroes: Fermat, Euler, Lagrange, Legendre, Gauss, Hermite, Dirichlet, Kummer, and Jacobi. At all times he felt its charms in the liveliest way, for the virtues of number theory—the simplicity of its foundations, the precision of its concepts, and the purity of its truths— accorded completely with his innermost sensibilities’’ [Hilbert 1910, 351]. Hilbert shared Minkowski’s sense of belonging to an exalted elite: they were not just pure mathematicians but enthusiasts who cultivated that lovely corner of their garden in which the delicate flowers of number theory grew. Only a small circle of mathematicians belonged to this group because, 4

in the opinion of Hilbert and Minkowski, most of their colleagues lacked the requisite qualifications. Yet even if the noble creations of Gauss and other greats were ‘‘too sublime’’ to be appreciated by most mathematicians, this did not prevent them from reaching down to ‘‘the masses.’’ Indeed, Minkowski’s lecture course from 1903-1904, later published in book form under the title Diophantische Approximationen, had this very objective: to open the ears of its Zuho¨rer to this ‘‘gewaltige Musik,’’ as he called it [Hilbert 1910, 351-352]. If number theory stood at the center of Minkowski’s research interests, this hardly kept him from exploring other intellectual terrain. He took up a serious interest in physics during the early 1890s while teaching as a Privatdozent in Bonn,5 even asking his brilliant colleague, Heinrich Hertz, for permission to take part in the laboratory exercises offered for beginning students. After donning a lab frock for a while, he quickly sensed the vast gulf that separated the world of the pure mathematician from that of the natural scientist. Shortly before Christmas, he sent Hilbert a humorous account of his activities, in which he described himself as a constructor of instruments, toiling away in his blue lab uniform, a ‘‘Praktikus of the worst sort imaginable’’ [Minkowski to Hilbert, 22 December 1890, Minkowski 1973, 39]. He let Hilbert know that he would not be returning to Ko¨nigsberg for the holidays, but consoled his friend with the thought that he and Adolf Hurwitz would have found him unfit for proper mathematical discourse. For he had become ‘‘thoroughly physically contaminated’’ to such a degree that he would have ‘‘perhaps even had to undergo a 10-day quarantine’’ before they would have ‘‘found him sufficiently pure and unapplied mathematically to be allowed to take part in their walking excursions’’ (ibid.). By 1891 Minkowski’s letters began to take on a more intimate tone as he and Hilbert became ‘‘Dutzfreunde’’ and close allies. From now on they

would share all kinds of news, both mathematical as well as purely personal, including innocent gossip. Many of the allusions in these letters are obscure or even totally incomprehensible today, but their witty lightheartedness nevertheless makes for delightful reading. Minkowski undertook many strolls through favorite paths in their impressive mathematical garden, but most of the time he reported about people and events of mutual interest. Thus he writes often about their former teachers in Ko¨nigsberg, Ferdinand Lindemann and Adolf Hurwitz, especially the latter after Minkowski became his colleague in Zu¨rich from 1896 to 1902. Hurwitz’s mathematical tastes had much in common with Minkowski’s; they also occasionally shared painful experiences as German Jews who had managed to rise to prominence in a highly competitive academic environment. Minkowski’s brother, Oscar, was a distinguished medical researcher who discovered the role of pancreatic dysfunction in diabetics [Minkowski 1973, 12]. His work, however, came under attack and he had already reached age 50 by the time he became a professor. Hurwitz was Klein’s most gifted pupil, yet even this was not enough for him to become an Ordinarius in Prussia [Rowe 2007]. Still, such milder forms of anti-Semitism were so pervasive at this time that both Hurwitz and Minkowski seem to have taken such slights for granted; in fact, this theme rarely surfaces in their correspondence. What they wrote to Hilbert and to each other reflects instead their deep love for the mathematical quest and a strong personal ambition to crack a problem or create a beautiful theory. Hurwitz and Minkowski were artists who lacked Hilbert’s ferocious career ambition or his striking ability to reform a subject and then conquer it. Hurwitz had a genuinely serene personality that won him many friends, but he also suffered from serious health problems that sometimes left him incapacitated. Minkowski enjoyed his

For a thoughtful assessment of how Hilbert’s views on axiomatics related to Minkowski’s approach to the foundations of physics, see Corry [1997] and Corry [2004]. Here, the example of Peter Gustav Lejeune Dirichlet readily comes to mind. As the successor of C. F. Gauss in Go¨ttingen, Dirichlet managed to combine his research interests in number theory with impressive contributions to mathematical physics. Minkowski documented his close affinity with Dirichlet in Minkowski [1905], a glowing tribute delivered before the Go¨ttingen Academy to celebrate the centenary of his birth. 5

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company thoroughly, but he was nevertheless eager to leave Zu¨rich, as he took little pleasure teaching there. After only one semester he complained to Hilbert that the students at the ETH expected their professors to spoon-feed them [Minkowski to Hilbert, 31 January 1897, Minkowski 1973, 94], an attitude he had seldom encountered at the two Prussian universities, Bonn and

Ko¨nigsberg, where he had taught earlier. Minkowski tried to accommodate his lecture style to this new scene, but he soon gave up. Unlike Hurwitz, he never showed much talent as a lecturer, and even in Go¨ttingen his courses were not a great success. His position at the ETH, by the way, remained vacant for many years after his departure in 1902, but it was eventually filled by a former

Photo 3. This portrait of Minkowski, reproduced by Teubner for the publication of his Gesammelte Abhandlungen, was made at the time he was teaching in Zu¨rich, 1896-1902 (Courtesy of Niedersa¨chsische Staats- und Universita¨tsbibliothek Go¨ttingen).

student whom Minkowski had found particularly lazy, a certain Albert Einstein [Fo¨lsing 1993, 281]. By this time, Minkowski and Hilbert were together again in Go¨ttingen, where they found themselves deeply immersed in the new physics of the electron. But back in the early 1890s the friendship between them had only begun to intensify as they came to think of themselves as collaborators. Minkowski’s early letters to Hilbert reveal that he followed the latter’s work on invariant theory closely and with enthusiasm. He was, of course, well aware of the controversy that erupted when Paul Gordan took issue with Hilbert’s nonconstructive methods [Rowe 2003]. So he was delighted when he saw how his friend had managed to find a new approach that was less objectionable on these grounds. ‘‘For a long while,’’ Minkowski wrote, ‘‘it has been clear to me that it would only be a question of time before you settled the old invariant questions so that hardly more than the dotting of the ‘‘i’’ remained. But that it all went so easily and simply makes me truly happy and I congratulate you on this’’ [Minkowski to Hilbert, 9 February 1892, Minkowski 1973, 45]. And so he went on: the smoke from Hilbert’s earlier paper might still have been bothering Gordan’s old eyes, but now that Hilbert had invented smokeless gunpowder it was high time to decimate the castles of the robber knights like Gordan, who went around capturing the individual invariants and stowing them in their dungeons, before the danger mounted that no new life would ever emerge from these quarters again. Minkowski went on teasing his friend that if he were not ‘‘so sehr radikal’’ he would consider doing his fellow mathematicians the favor of setting forth his new results in the form of a monograph, a book that would enable them to build on the newly reformed theory. He had in mind something similar to the edition of Paul Gordan’s lectures on invariant theory prepared by Georg Kerschensteiner, but jested that if Hilbert waited for his ‘‘Kerschensteiner’’ to come along he might find that his book contained a lot of cherry stones that could spoil the reader’s appetite.


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Needless to say, Hilbert never entertained the idea of writing such a book, nor did a young Kerschensteiner emerge to write it for him. Six months later he completed his research program in invariant theory by submitting a definitive 70-page paper to the Mathematische Annalen [Hilbert 1893]. With it, he wrote to Minkowski, he would leave the field and turn instead to the theory of algebraic number fields. Indeed, the methods he used in this paper were largely taken from algebraic number theory together with powerful new results in algebraic geometry. The following year Hilbert offered some general reflections on invariant theory in the form of a short survey article [Hilbert 1896]. This had been solicited by Felix Klein, who packed it in his luggage along with the several other papers he later presented at the Mathematical Congress held in conjunction with the Chicago World’s Fair. In this article, Hilbert describes three phases in research on classical invariants: a naı¨ve period (Cayley and Sylvester), followed by the development of sophisticated formal methods (Clebsch and Gordan), and finally a critical phase, dominated of course by his own work. No one who read this article in 1893 could have doubted that Hilbert’s immense talent was matched by his formidable chutzpah. While Klein was busy in Chicago showcasing his own research and that of his German colleagues, Hilbert and Minkowski were attending the DMV meeting in Munich. From its founding, the society had signaled that a major goal would be the preparation by its membership of official reports on recent and not-so-recent research in various fields. Thus long-time collaborators Alexander Brill and Max Noether, who were charged with the task of writing a compendium on algebraic geometry, would soon unveil a monumental work on this vast field of research [Brill and Noether 1893]. Clearly, they had splendid credentials for taking on such a task. Hilbert and Minkowski were still young and had never collaborated at all. Nevertheless, they agreed to prepare a report on recent developments in number theory, although Hilbert had yet to publish a single paper in this field. Minkowski, on the other hand, had as a teenager made a name for himself as a number-theorist 32


when the Paris Academy awarded him first prize for his essay on the representations of a number as the sum of five squares. Hilbert evidently had more to gain from this difficult venture than did his friend. Minkowski’s letters over the next several years refer to his efforts to make headway on this project, although the sometimes exasperated tone of the letters also made it clear that he viewed his efforts as a largely thankless task. His main interest was to finish his monograph on the geometry of numbers, a topic he pursued with the encouragement of the leading French number-theorist, Charles Hermite. Still, he did not neglect his homework for the DMV report. In fact, he managed to produce 75 abstracts for the Jahrbuch u¨ber die Fortschritte der Mathematik on papers from the mid 1890s. His report, so he wrote Hilbert, would highlight results in his forthcoming book as well as his earlier papers. ‘‘Whether beyond that I bring people closer to comprehending the results of several till now hardly appreciated works, which after all is the purpose of such reports, this is for me a pleasant enough task, even though not the type that I value the most’’ [Minkowski to Hilbert, 10 February 1896, Minkowski 1973, 78]. By early 1896 Minkowski’s book was still not finished, whereas Hilbert’s portion of the report, dealing with the modern theory of algebraic number fields, was nearing completion. Not surprisingly, he grew impatient and suggested to Minkowski that he might want to consider publishing his part of the report later, a plan his friend greeted with open arms. As it turned out, Hilbert’s report did not appear until the following year whereas Minkowski managed to finish a truncated version of his Geometrie der Zahlen in 1896 [Minkowski 1896]. His report for the DMV, on the other hand, never saw the light of day. The following autumn Minkowski joined Hurwitz in Zu¨rich, and both read the galley proofs of Hilbert’s Zahlbericht with great care and considerable enthusiasm. Minkowski was shocked to see that Hilbert had neglected to thank his wife for all her work in copying the entire manuscript and preparing the index [Minkowski to Hilbert, 17 March 1897, Minkowski 1973, 98]. After receiving this

thorough scolding, Hilbert added a note of thanks to her in the preface along with these words: ‘‘My friend Hermann Minkowski has read the proofs of this report with great care; he also read most of the manuscript. His suggestions have led to many significant improvements, both in content and presentation. For all this help I offer him my most hearty thanks’’ [Hilbert 1897, 64]. When he received the published text a month later, Minkowski wrote to congratulate Hilbert, offering the prognosis that it would not be long before he too would be counted among the great classical authors in the field of number theory, the discipline both men revered so much [Minkowski to Hilbert, 14 May 1897, Minkowski 1973, 100]. But in this same letter, Minkowski also cautioned that when such recognition came it might be confined to a select group of connoisseurs. In this connection, he related his impressions of Go¨sta MittagLeffler, the politically astute pupil of Weierstrass who had recently visited Zu¨rich in connection with the forthcoming International Congress. Minkowski found him dull and unimpressive, the more so because he evinced no real appreciation for Minkowski’s new book; nor did he imagine that Hilbert’s report would fare any better in the eyes of this worldly-wise Swede. But as a consolation he offered Hilbert the thought that ‘‘the facility for mental abstraction indeed causes many people headaches’’ (ibid.).

Plans for Hilbert’s Paris Lecture Minkowski’s role as Hilbert’s silent collaborator during the production of the Zahlbericht ought not to be exaggerated, but he certainly did offer his friend a great deal of intellectual and moral support. The same pattern can be observed a few years later when Hilbert called on his advice again during a time when both were making plans to attend the Second International Congress of Mathematicians in Paris. It was on this occasion, of course, that Hilbert delivered his famous lecture on ‘‘Mathematical Problems’’ [Hilbert 1901], a performance that soon brought him lasting fame. Had it not been for Minkowski’s wise counsel, however, he might well have chosen a more controversial, but less ambitious topic for this lecture. Once again, Minkowski’s letters

shed considerable light on the motivations and behaviors of both friends. As the newly-elected president of the DMV, Hilbert received an invitation in December 1899 to deliver a plenary address in Paris on a theme of his choosing. This was a splendid opportunity, or at least so Hilbert thought, to throw down the gauntlet to the dominant mathematician of the era, Henri Poincare´. At the previous ICM, held in Zu¨rich in 1897, Poincare´ had been designated as one of the plenary speakers, and for this occasion he took as his theme the role of physical conceptions in guiding fertile mathematical research. With that speech still in mind, Hilbert thought he should counter by singing a hymn of praise to pure mathematics. No doubt he also recalled how Jacobi had once rebuked his French contemporary, Joseph Fourier, for failing to recognize that the highest and only true purpose of mathematics resided in nothing other than the quest for truth; this alone reflected honor on the human spirit [Klein 1926, 114]. But Minkowski strongly counseled his friend to abandon this plan [Minkowski to Hilbert, 5 January 1900, Minkowski 1973, 119]. After rereading Poincare´’s text, he found that its assertions in no way compromised the integrity of pure mathematics. Furthermore, he felt that the Frenchman’s views were formulated so cautiously that Hilbert and he could easily subscribe to them. Moreover, he recalled how Poincare´ had not even been present in Zu¨rich, so his text was actually read by another party; few would remember what he had said on that occasion. Minkowski then contrasted this rather dull speech with the stirring lecture delivered by Ludwig Boltzmann at the annual meeting of German scientists held in Munich just four months earlier [Boltzmann 1900]. Hilbert clearly took this all to heart, even if he never said so. Indeed, he had returned from that Munich meeting in high spirits, writing to Hurwitz that this event was the best attended and most stimulating of all the DMV meetings held thus far.6 Furthermore, he was clearly pleased by the response to his two talks (one on the axioms of

arithmetic, the other on Dirichlet’s principle). Still, these could hardly be compared with Boltzmann’s dazzling performance [Boltzmann 1900], which left the spellbound audience literally buzzing with excitement afterward. Clearly it left a strong impression on both Hilbert and Minkowski, not only theatrically but also thematically; in fact the parallels between Boltzmann’s speech and Hilbert’s far more famous Paris lecture are simply impossible to overlook (although it seems nearly everyone since has done just that). Both texts offered sweeping accounts of past developments, undertaken with an eye toward unsolved problems that would test the steel of the younger generation. In Munich, Boltzmann surveyed major developments in theoretical physics over the course of the nineteenth century, ending his talk with a string of open questions, most of them connected in one way or another with the fate of the mechanical world picture. In closing, he beckoned the younger generation on—‘‘a Spartan war chorus calls out to its youth: be even braver than we!’’— expressing hope that the new century would bring even greater surprises than the outgoing one had seen. But Boltzmann was unwilling to stick his neck out terribly far either; he made it clear that he would not attempt to ‘‘lift the veil’’ [Boltzmann 1900, 73] covering the face of future events by making prognostications about the course of physical research. Hilbert would do just that one year later in Paris (see [Hilbert 1901]), which brings us back to Minkowski’s letter. After dowsing Hilbert’s original idea with cold water, Minkowski threw out some thoughts of his own about how his friend might best take advantage of this unusual opportunity. ‘‘Most alluring,’’ he wrote, ‘‘would be the attempt to look into the future, in other words, a characterization of the problems to which the mathematicians should turn in the future. With this, you might conceivably have people talking about your speech even decades from now. Of course, prophecy is indeed a difficult thing’’ [Minkowski to Hilbert, 5 January 1900, Minkowski 1973, 120].

Minkowski went on to reflect about the audience, assuring Hilbert that a lecture with real substance, such as the one delivered by Hurwitz at the ICM in Zu¨rich, was far more effective than a merely general presentation, such as the one given by Poincare´. He even mentioned two earlier lectures that he thought might prove useful for Hilbert to read, one a speech delivered by Hermite in 1890, another by H.J.S. Smith entitled, ‘‘On the Present State and Prospects of some Branches of Pure Mathematics.’’ Grand visions swept before Minkowski’s eyes, yet he assured Hilbert that success depended less on the theme of his address than the quality of his presentation. Still, he realized well enough that the ‘‘frame of the theme could have the effect that twice as many listeners turn out’’ as might otherwise. Hilbert vacillated, and then came a long pause when Minkowski heard nothing more from him about any plans for Paris. The months went by, and soon it was June. A letter arrived from Paris containing a preliminary program for the congress, but when Minkowski opened it, he saw that Hilbert’s name was not included among the speakers. Deeply disappointed, he shot off a letter to his friend, lamenting that he had apparently given up on speaking at the Paris ICM [Minkowski to Hilbert, 22 June 1900, Minkowski 1973, 126-127]. Minkowski had clearly been looking forward to nothing so much as Hilbert’s lecture, and now he suddenly lost all interest in attending: ‘‘The participation from here [the German contingent] will be almost zero. Mostly one will get to see French school teachers and exotic mathematicians, Spaniards, Greeks, etc., for whom Paris in August still feels cool in comparison with their home countries’’ (ibid.). This was all just a misunderstanding, of course. It seems Hilbert vacillated a bit too long, leaving the congress organizers in the dark about his intentions. At any rate, Minkowski’s infectious enthusiasm quickly returned when Hilbert wrote back, saying that he had, in fact, been working all along on his lecture for the Paris congress. He then turned to

6

Hilbert also lamented that Lindemann failed to attend the Munich conference, despite the fact that it was held at his university. Hilbert to Hurwitz, 5–12 November 1899, Mathematisches Archiv, Niedersa¨chsische Staats- und Universita¨tsbibliothek, Go¨ttingen.


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Minkowski to inquire whether he knew of appropriate physical theories with open foundational problems. Pointing to Boltzmann’s work on statistical mechanics and thermodynamics, Minkowski noted that this terrain offered ‘‘many interesting mathematical questions also very useful for physics,’’ [Minkowski to Hilbert, 10 July 1900, Minkowski 1973, 128]. This tip clearly influenced Hilbert’s formulation of his sixth Paris problem, which addresses the axiomatization of physics. Only one week later, Minkowski found himself staring at page proofs of various portions of Hilbert’s lecture, including drafts of what ultimately became the second and twenty-third problems. He and Hurwitz continued to make various suggestions, including cuts that would be necessary to stay within the time limits allowed. When he finished reading the final draft, Minkowski was nothing less than ecstatic: I can only wish you luck on your speech; it will certainly be the event of the congress and its success will be very lasting. For I believe that this speech, which probably every mathematician without exception will read, will cause your powers of attraction on young mathematicians to grow still more, if that is even possible. . . . Now you have really wrapped up the mathematics for the twentieth century and in most quarters you will gladly be acknowledged as its general director [Minkowski to Hilbert, 28 July 1900, Minkowski 1973, 129-130]. These words could hardly have been more prophetic.

Boltzmann and the Energetics Debates Soon after Minkowski’s appointment in 1902 to a new chair in Go¨ttingen, he and Hilbert began pursuing various topics in the foundations of physics. In 1905—the very year Einstein published his famous paper, ‘‘On the electrodynamics of moving bodies’’ [Einstein 1905] marking the birth of special relativity—they conducted a joint seminar on electron theory, a field that was already buzzing

7

with activity in Go¨ttingen. Two years later, Minkowski unveiled his own 4-dimensional approach to electrodynamics in Minkowski [1908], a paper widely regarded by contemporaries as unreadable [Walter 2008, 214, 229]. Einstein and his young collaborator, Jakob Laub, rewrote some of its key results in the language of 3-vectors, while disputing some of Minkowski’s physical claims. In all likelihood, Minkowski took little or no notice of this critique. Einstein, after all, was still just an obscure patent clerk in Bern. A far more significant rival was Poincare´, who had already used a 4-dimensional formalism in connection with the Lorentz transformations. Poincare´ was, in fact, the first to note that these form a group, thereby calling attention to the importance of group invariants for electrodynamics. On the other hand, Poincare´’s conventionalist views led him to attach far less importance to space-time physics, a viewpoint that Minkowski meant to challenge as forcefully as possible.7 It is not difficult to imagine that Hilbert’s Paris address was in the back of Minkowski’s mind when he began thinking about his Cologne lecture, ‘‘Space and Time.’’ He knew that such opportunities were rare and that the time was ripe for promoting his recent work on electrodynamics. Undoubtedly disappointed by the cool reception his work had received, he clearly saw this as a chance to make a real splash. Hilbert’s speech in Paris had done just that and its lofty rhetoric was surely inspirational, but it could hardly serve as a model for Cologne. Minkowski’s topic was the foundations of physics, and on that score he surely remembered the brilliant public performances of Ludwig Boltzmann. The Austrian physicist had recently committed suicide while vacationing with his family. Boltzmann was in many ways a singular figure. During his lifetime, no physicist in Germany cultivated contacts with mathematicians with the same intensity as did he. A decade prior to his death, Minkowski and Hilbert were both present at a longremembered conference in Kiel at

which Boltzmann caused a great stir over the energetics program promoted by the chemist Wilhelm Ostwald and others. The two friends had met in Go¨ttingen shortly before the conference convened to prepare a preliminary report on their number theory project. This was presented by Hilbert during the first days of the meeting, after which he apparently left. Minkowski thus described the final Friday [Minkowski to Hilbert, 24 September 1895, Minkowski 1973, 70-71], when he bumped into Klein and had lunch with him at their hotel, surrounded by three physicists from the anti-energeticist camp: Boltzmann, Walther Nernst, and Arthur Oettingen. All five continued their passionate discussion. Minkowski even suggested to Hilbert that they should consider ending their Zahlbericht with some general theses regarding the essence and significance of number theory, just as the physicists had done in closing their debates on energetics. Another eye witness at the Kiel conference was young Arnold Sommerfeld, who was then Klein’s assistant in Go¨ttingen. Sommerfeld left this brief, but memorable account of the climactic confrontation in Kiel: [Georg] Helm from Dresden gave the report on energetics, behind him stood Wilhelm Ostwald and behind both stood the Naturphilosophie of Ernst Mach, who was not present. The opponent was Boltzmann, seconded by Felix Klein. The duel between Boltzmann and Ostwald resembled, both outwardly and inwardly, a fight between a bull and a subtle fencer. But on this occasion, despite all his swordsmanship, the toreador was defeated by the bull. Boltzmann’s arguments won out. At that time all of us younger mathematicians stood on the side of Boltzmann [Deltete 1999, 56]. This version is about all that remains of this debate in the collective memory of physicists, though one can easily reconstruct many details that reveal a far more complicated and interesting story (see Deltete [1999]). Indeed, the debates that took place during that particular meeting echoed for many years afterward, particularly

Scott Walter has pointed out that Minkowski paid very close attention to Poincare´’s publications [Walter 2008, 223].

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in Go¨ttingen circles. One of the critical issues then under discussion—the link between conservation of energy and the laws of classical Newtonian mechanics—was of direct and immediate relevance for Minkowski when he composed his 1908 lecture. Boltzmann had contested Georg Helm’s claim that the laws of motion in classical mechanics could be deduced from the energy conservation law. Yet what Minkowski later wrote in ‘‘Raum und Zeit’’ flies in the face of Boltzmann’s earlier claim: ‘‘At the limiting transition . . . to c = ?, this fact [that the laws of motion follow from energy conservation] retains its importance for the axiomatic structure of Newtonian mechanics as well, and has already been appreciated in this sense by I. R. Schu¨tz.’’ In a footnote, Minkowski cited Schu¨tz’s paper on ‘‘The Principle of the Absolute Conservation of Energy,’’ published in the Go¨ttinger Nachrichten [Schu¨tz 1897]. Here he highlighted how the principle of relativity held the key to understanding the role of energy conservation not only in classical mechanics but in the new physics inaugurated by Lorentz, Poincare´, and Einstein. Oddly enough, this highly significant passage from Minkowski’s lecture has been largely overlooked in recent studies and commentaries. Historians have likewise ignored this paper by Ignatz Schu¨tz, himself a forgotten figure. Schu¨tz was a young Dozent in Go¨ttingen at this time, a prote´geé of the physicist Woldemar Voigt. He spoke at a number of meetings of the Go¨ttingen Mathematical Society and was also a regular speaker at the annual Naturforscher conferences. So Minkowski was not the only one who knew and appreciated Schu¨tz’s work; his paper on conservation of energy was later highlighted by Max von Laue in a survey article on the energy concept [Laue 1955, 374]. Moreover, Felix Klein had a longstanding interest in these matters. Indeed, shortly after the Kiel

conference Klein reported on the energetics debates at a meeting of the Go¨ttingen Mathematical Society. His protocol notes indicate that on this occasion he explained the alleged mistake Boltzmann had found in Georg Helm’s argument deducing the laws of motion in Newtonian mechanics from the energy principle. Since Ignatz Schu¨tz also attended the Kiel conference, it seems safe to assume that his work was part of a larger Go¨ttingen effort aimed at resolving this particular point of contention.8

Minkowskian Physics Minkowski was guided by a unified view of physics based on the universal validity of Einstein’s principle of relativity. In ‘‘Raum und Zeit’’ he made no pronouncements about the ultimate inertial properties of matter. Moreover, his whole approach led to a deeper understanding of how in special relativity the laws of motion are wedded to energy conservation and vice-versa. In his memorial lecture, Hilbert describes how Minkowski was guided by his world postulate, according to which the laws of physics are based on Lorentz covariant concepts. Since the current density components together with the density transform just like the spacetime coordinates, these four entities form a 4-vector. Likewise, the electric and magnetic 3-vectors, taken together form an anti-symmetric tensor; they therefore transform just like the six Plu¨cker coordinates familiar from line geometry. Thus these two pairs of physical concepts—current density and density, electricity and magnetism—cannot be separated, but rather each pair depends on the choice of a spacetime coordinate system. These insights naturally emerge as consequences of Minkowski’s world postulate, in fact so naturally that it is perhaps difficult to realize how fundamentally new this way of thinking was in 1908.

To appreciate the physical implications Minkowski drew for spacetime physics, one should turn to the fourth section of ‘‘Raum und Zeit.’’ There he proposed to reform the concepts of mechanics to ensure their compatibility with the principle of relativity pertaining to the Lorentz group. For this purpose, he took the invariant motive force vector from electrodynamics and set this equal to the mechanical force vector, thereby obtaining four relativistic laws of motion. The first three equations yield a generalization of the Newtonian laws associated with the conservation of center of mass, whereas the fourth equation expresses the relativistic version of conservation of energy. Since these two force vectors are both perpendicular to the velocity vector, Minkowski immediately deduced that the fourth equation, expressing energy conservation, is a consequence of the other three. But he also noted—and this was the decisive point that emerged from our story about the energetics debates in Kiel—that since the time axis can be taken freely in the direction of any time-like vector, one can also deduce the laws of motion from the energy principle alone. Hilbert underscored the importance of this result in his memorial tribute when he wrote that ‘‘Minkowski’s investigation led moreover to the principally interesting fact that by virtue of the world postulate the complete laws of motion can be deduced from the conservation of energy theorem alone’’ [Hilbert 1910, 359].9 Hilbert also attached great importance to the manner in which Minkowski derived his phenomenological equations for the motion of bodies in electromagnetic fields. This derivation required only three axioms: (1) that moving bodies can never attain the speed of light; (2) that Maxwell’s equations are valid for bodies in a state of rest; and (3) that when a single point of a body is motionless, then this point

8 Klein’s notes are in the Mathematisches Archiv, Niedersa¨chsische Staats- und Universita¨tsbibliothek, Go¨ttingen. In the second volume of his lectures on 19th-century mathematics, Klein noted how Schu¨tz had managed to clarify the relationships between the 10 first integrals of classical mechanics [Klein 1927, 58]. Klein went further, enlisting the support of Friedrich Engel, who published two papers on applications of Lie’s theory of infinitesimally generated groups to classical mechanics. 9 One could also cite Wolfgang Pauli, who in his report on relativity theory for the German encyclopedia [Pauli 1921] noted that similar consequences can be drawn for Minkowskian electrodynamics. Felix Klein also alluded to this in his commentary on Hilbert’s first note on the foundations of physics. Klein thereby launched the investigations into the status of energy conservation theorems in special and general relativity, the topic that led to Emmy Noether’s fundamental theorems on invariant variational problems (see Rowe [1999]).


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may be treated as if the entire body were at rest. Since the validity of Maxwell’s equations for matter at rest was considered beyond question, Minkowski’s derivation was based on a truly minimal set of assumptions that gave an impressive demonstration of the power of the principle of relativity. Hilbert was just one of several contemporaries who appreciated this. What he seems not to have noticed, however, is that Go¨ttingen’s Max Abraham had come up with a different set of equations that was nevertheless compatible with Minkowski’s axioms (see Walter [2007]; Fig. 2). Minkowski’s systematic reform of electrodynamics and mechanics involved important revisions to both fields. His fundamental equations for the electrodynamics of moving bodies were derived directly from the relativity principle and differed from the earlier results of Lorentz and Emil Cohn, neither of which was Lorentz-invariant. Later investigations showed that Lorentz’s derivation based on his electron theory was erroneous; afterward Minkowski’s equations gained general acceptance. In mechanics, the same was true of the Minkowski force, which was taken up by Lorentz in his Leiden lectures of 1910–1912 on special relativity, published in German translation in Lorentz [1929]. Max von Laue’s influential textbook on special relativity was even more decisively influenced by Minkowski’s new approach, which led Laue to formulate a new world tensor for expressing the totality of energy and matter in a physical system [Rowe 2008]. It did not take Einstein long to absorb these new developments, as can be seen from his unfinished draft of a review article on special relativity, written around 1912 [Einstein 1996]. Minkowski’s research program for mechanics was based on a strong analogy between mechanical concepts and those he had developed in his 4-dimensional approach to electrodynamics. Some historians, however, have regarded him as an exponent of the electromagnetic worldview, according to which the inertial properties of matter stem entirely from electromagnetic forces. This view, as Leo Corry has

10

emphasized, is a mistaken one [Corry 2004, 191]. Possibly Minkowski’s own rhetoric contributed to this faulty impression since in the closing paragraph of ‘‘Raum und Zeit’’ he wrote: ‘‘The unexceptional validity of the world postulate is, in my opinion, the true nucleus of an electromagnetic image of the world, which was discovered by Lorentz and further revealed by Einstein, and which now lies open in the full light of day’’ [Minkowski 1909, 444]. In later years, Einstein distinguished between two main types of physical theories: those that are constructive, such as Lorentz’s electron theory, and those of principle, thermodynamics being a prime example.10 The relativity principle, which later evolved into the special and general theories of relativity, was of the second type, which helps explain why Minkowski was so attracted to it. Still, one should not imagine that he and Hilbert were altogether out of sync with mainstream developments in theoretical physics. In his magisterial study, Electrodynamics from Ampere to Einstein [Darrigol 2000], Olivier Darrigol highlights the importance of a new trend in late19thcentury theorizing, one in which general physical principles began to displace the more concrete styles that had been prevalent before. Among the strongest representatives of this new approach were such luminaries as Helmholtz, Hertz, Poincare´, and Planck. Hilbert, on the other hand, swung to the far extreme by attaching great faith in axiomatic analysis and general mathematical methods, in particular variational principles. Minkowski’s success with his world postulate no doubt reinforced these convictions. Both he and Minkowski were extreme optimists in this regard, and their views with regard to the efficacy of mathematics in the pursuit of knowledge of the natural world differed sharply from the stance taken by Poincare´. Thus, while advancing the new physics, Minkowski parted company with Poincare´ regarding the importance of a 4-dimensional formalism. For Poincare´, who was always sceptical of arguments that purported to establish

the solute truth of a physical theory, the relativity principle was nothing more than a convenient hypothesis that helped the natural philosopher organize important knowledge. By contrast, Minkowski saw it as nothing less than a revealed truth; hence, he argued that the mathematical physicist should strive to develop a firmly grounded theoretical structure as a framework for guiding experimental work. His final rhetorical flourish in ‘‘Raum und Zeit’’ captures this vividly: ‘‘In the development of its mathematical consequences there will be ample suggestions for experimental verifications of the world postulate, which will suffice to conciliate even those for whom the abandonment of older established views is unsympathetic or painful by the idea of a preestablished harmony between pure mathematics and physics’’ [Minkowski 1909, 444].

On Canonizing a Classic Relativity theory was not the creation of one man, but rather of several, most notably Lorentz, Poincare´, Einstein, and Minkowski [Born 1959, 501]. Max Born was in an unusually advantageous position to judge the merits of Einstein’s and Minkowski’s respective contributions. He began his academic career in Go¨ttingen and was serving as Minkowski’s assistant at the time of his death. Later Born befriended Einstein when both were teaching in Berlin, and in Frankfurt he went on to write one of the most successful of the many popular books on relativity [Born 1920]. Though he made ample use of Minkowski’s spacetime diagrams throughout his text, Born entitled it Die Relativita¨tstheorie Einsteins. Still, Minkowski’s Cologne lecture continued to be read and cited, in large part because of the promotional efforts of Go¨ttingen allies and friends, and ‘‘Raum und Zeit’’ was on its way to becoming a classic by 1922. To see how this happened, let us turn back to that work and its reception. In 1910 Hilbert’s ever-faithful pupil, Otto Blumenthal, began a new series with the Leipzig publishing house of B. G. Teubner entitled Fortschritte der mathematischen Wissenschaften in Monographien. This venture turned out

He emphasized this distinction, for example, in his widely read article for the Times (London), [Einstein 1919b].

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to be a total flop, but initially it served its main purpose, namely to promote Minkowski’s contributions to physics. The first volume [Blumenthal 1910] contained a reprint of his 1907 paper, along with a related study on the properties of the electron prepared by his former assistant, Max Born, on the basis of Minkowski’s extant notes. Earlier, Teubner had also published ‘‘Raum und Zeit’’ as a separate brochure; this contained a brief introduction by the president of the DMV, August Gutzmer. By 1913, however, this brochure was already out of print, and when Arnold Sommerfeld learned of this he suddenly came up with a brilliant idea. Rather than requesting a new printing, he approached Blumenthal with a proposal for a second volume in his nearly still-born series. Sommerfeld’s idea was to publish a compendium of papers on relativity that featured Minkowski’s ‘‘Raum und Zeit’’ as its centerpiece. It did not take long for Blumenthal to see the light. After gaining the approval of Lorentz and Einstein, he was able to convince Teubner that this anthology [Blumenthal 1913], bearing the simple title Das Relativita¨tsprinzip, represented a promising commercial venture. Just how promising, no one could have imagined. After going through numerous printings and some major changes, it emerged in the early 1920s in its present-day form, a staple item in bookstores worldwide. To a considerable degree Das Relativita¨tsprinzip has since served to canonize the classics of relativity theory, especially Einstein’s works, although this was clearly not its original intent. The 1913 edition was, in fact, primarily conceived as another tribute to Minkowski, as evidenced by the inclusion of a photographic portrait of the deceased mathematician. Inside, the reader was exposed to the relevant prehistory: the two classic papers by Lorentz from 1894 and 1904 followed by Einstein’s famous 1905 article ‘‘On the Electrodynamics of Moving Bodies’’ as well as his brief note on the relationship between energy and inertial mass. These texts set the scene for Minkowski’s ‘‘Raum und Zeit,’’ which Blumenthal described in the foreword as the speech that set the great wave of popular interest in relativity theory in motion.

A reader in 1913 would presumably notice the difference between Minkowski’s soaring rhetoric and the lean and pithy style that marked Einstein’s presentations. Not that Minkowski took great pains to spell out the details, but for this the reader could turn to Sommerfeld’s notes for help, with the reassurance that these were merely added to remove ‘‘small, formal mathematical difficulties that could stand in the way of penetrating into the great thoughts of Minkowski.’’ Sommerfeld rounded out this slim volume with Lorentz’s Wolfskehl lecture, ‘‘Das Relativita¨tsprinzip und seine Anwendungen auf einige besondere physikalische Erscheinungen,’’ delivered in Go¨ttingen one year after Minkowski’s death. It was a fitting choice, as on that occasion the gracious Dutch physicist paid tribute to the distinguished mathematician’s contributions to his own field. Given the agenda Blumenfeld and Sommerfeld had in mind, this choice of texts hardly seems surprising, although in retrospect its omissions are striking. The most glaring absence, of course, was Poincare´’s work. Less surprising, but nevertheless noteworthy was the lack of any mention of Einstein’s generalized theory of relativity. By 1913, Einstein had been struggling for some six years to go beyond special relativity in order to link gravitation with the effects that arise in non-inertial frames. Yet the only relativistic treatment of gravitation to be found in the first edition of Das Relativita¨tsprinzip was the one given by Minkowski in the closing section of ‘‘Raum und Zeit’’ [Walter 2007]. There he briefly described how one can formulate a simple 4-dimensional force law between point masses that corresponds to Newton’s law for low velocities. Minkowski then concluded that this law of attraction ‘‘when combined with the new mechanics is no less well adapted to explain astronomical observations than the Newtonian law of attraction combined with Newtonian mechanics’’ [Minkowski 1909, 440]. Einstein had long been promoting a different approach to gravitation. In 1911 he published his calculations for the bending of light in the vicinity of the sun as well as for gravitational red shift. By 1915 he wrote an introductory article for the series Die Kultur der

Gegenwart in which he advances the opinion that the fundamental principle expressing the constancy of the speed of light had to be dropped since it appeared to be valid only in reference systems in which the gravitational potential was constant. A year earlier he published a 55-page article [Einstein 1914] on ‘‘The Formal Foundations of General Relativity,’’ a highly mathematical presentation that took great pains to show that the Einstein-Grossmann field equations were the only feasible candidates for a sound relativistic theory of gravitation. Interestingly, even in 1915 Einstein was still not satisfied with what he had written about general relativity. Only about a week after he delivered his Wolfskehl lectures in Go¨ttingen, he was contacted by Arnold Sommerfeld, who was then laying plans for an expanded second edition of Das Relativita¨tsprinzip and wanted to include texts by Einstein on general theory of relativity [Sommerfeld to Einstein, 15 July 1915, CPAE 8A 1998, 147]. But Einstein responded without enthusiasm for this plan, although he did note that for this purpose his paper [Einstein 1911] and the longer one [Einstein 1914] might have been the most suitable among his recent articles. Still, he preferred that the collection be reprinted without change since he was planning to write an introductory booklet that led to the general theory anyway. This letter appears to be the earliest evidence we have of Einstein’s own plan to produce a semi-popular textbook, an idea that evidently antedated the breakthrough he made in November 1915. Two years later Vieweg published [Einstein 1917b], an instant bestseller that soon went through numerous editions and translations. Sommerfeld waited until 1920 before introducing a new edition of Das Relativita¨tsprinzip, one that finally included writings by Einstein on general relativity. Since Lorentz’s 1910 lecture was by now thoroughly antiquated, he replaced this with the paper in which Einstein drew optical important consequences from his equivalence principle [Einstein 1911]. The new volume also contained Einstein’s canonical paper on the mature theory of general relativity [Einstein 1916a], his brief sequel on Hamilton’s


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principle from the same year [Einstein 1916b], the essay on relativistic cosmology [Einstein 1917a], and a speculative paper [Einstein 1919a], in which he modified the field equations to link gravity with an electromagnetic theory of matter. Clearly, this new anthology greatly enhanced Einstein’s reputation as the central, if not sole creator of the theory of relativity. This picture remained unaltered with the definitive 1922 edition, which added Hermann Weyl’s paper [Weyl 1918] on ‘‘Gravity and Electricity’’ to this compendium of classics. One year later, this text appeared in English translation, published by Methuen; this edition later served as the basis for the popular Dover paperback edition [Einstein et al., 1952], which can still be found in bookstores today. Ironically, the introductory passage in Einstein’s fundamental paper, [Einstein 1916a], was omitted from the text, although it was faithfully reproduced in the German original. Therein he acknowledged his debt not only to Minkowski but also to a whole series of other mathematicians, including his friend Marcel Grossmann. ‘‘The generalization of the theory of relativity,’’ he wrote, ‘‘was facilitated considerably by Minkowski, the mathematician who was the first to recognize clearly the formal equivalence of the space and time coordinates, and who utilized this in the construction of the theory’’ [Einstein 1916a, 283] Einstein noted that the ‘‘mathematical tools necessary for general relativity were readily available in the absolute differential calculus,’’ a theory developed by Ricci and Levi-Civita and which was based on earlier research on non-Euclidean manifolds undertaken by Gauss, Riemann, and Christoffel. He also pointed out that the Italian inventors of the absolute differential calculus had already applied it to a number of problems in theoretical physics. As for Grossmann, Einstein wrote that his ‘‘help not only saved me the effort of studying the pertinent mathematical literature’’ but ‘‘also aided me in my search for the field equations of gravitation’’ (ibid.). The omission of all this from the English language edition [Einstein et al., 1952], surely left a faulty impression, as the remainder of the text, that is, the portion that was 38


printed, contains only slight hints of the contributions of others. Yet Einstein clearly did make a sincere effort to discharge his intellectual debts. Much else slipped away with the passage of time, as key witnesses and their memories passed from the scene. Hilbert continued to pursue his dream of axiomatizing parts of physics until well into the 1920s, but his contributions to Einstein’s general theory of relativity were viewed by his contemporaries as being only of marginal importance. With the advent of the Nazi regime in 1933, Go¨ttingen became just another provincial German university, though the allure of its glorious past was by no means forgotten. In his obituary article for his former mentor, Hermann Weyl briefly alluded to Hilbert’s friendship with Minkowski during their Ko¨nigsberg days, and then went on to recall wistfully the atmosphere he remembered in Go¨ttingen after Minkowski’s arrival: The two friends became the heroes of the great and brilliant period which our science experienced during the following decade in Go¨ttingen, unforgettable to those who lived through it. Klein, for whom mathematical research had ceased to be the central interest, ruled over it as a distant but benevolent god in the clouds. Too soon was this happy constellation dissolved by Minkowski’s sudden death in 1909 [Weyl 1944, 131-132]. ACKNOWLEDGMENTS

This paper is based on a lecture delivered at the conference ‘‘Space and Time 100 Years after Minkowski’’ held from 7–12 September 2008 at the Physics Center in Bad Honnef, Germany. My thanks go to the conference organizers, Claus Kiefer and Klaus Volkert, as well as the other participants, especially Engelbert Schu¨cking and Scott Walter, who offered several insightful remarks. REFERENCES

Blumenthal, Otto, ed. 1910. Zwei Abhandlungen u¨ber die Grundgleichungen der Elektrodynamik. Mit einem Einfu¨hrungswort von Otto Blumenthal, Fortschritte der mathematischen Wissenschaften in Monographien, Heft 1, Leipzig: Teubner.

Blumenthal, Otto, ed. 1913. Das Relativita¨tsprinzip. Eine Sammlung von Abhandlungen. Mit Anmerkungen von A. Sommerfeld und Vorwort von O. Blumenthal, Fortschritte der mathematischen Wissenschaften in Monographien, Heft 2, Leipzig: Teubner. Blumenthal, Otto. 1935. ‘‘Lebensgeschichte,’’ in David Hilbert, Gesammelte Abhandlungen, Bd. 3, Berlin: Springer-Verlag, pp. 388–429. Boltzmann, Ludwig. 1900. ‘‘U¨ber die Entwicklung der Methoden der theoretischen Physik in neuerer Zeit,’’ Jahresbericht der

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Teaching the Kepler Laws for Freshmen MARIS

VAN

HAANDEL

AND

GERT HECKMAN

O

ne of the highlights of classical mechanics is the mathematical derivation of the three experimentally observed Kepler laws of planetary motion from Newton’s laws of motion and of gravitation. Newton published his theory of gravitation in 1687 in the Principia Mathematica [13]. After two short introductions, one with definitions and the other with axioms (the laws of motion), Newton discussed the Kepler laws in the first three sections of Book 1 (in just 40 pages, without ever mentioning the name of Kepler!). Kepler’s second law (motion is planar and equal areas are swept out in equal times) is an easy consequence of the conservation of angular momentum L = r 9 p, and holds in greater generality for any central force field. All this is explained well by Newton in Propositions 1 and 2. On the other hand, Kepler’s first law (planetary orbits are ellipses with the center of the force field at a focus) is specific for the attractive 1/r 2 force field. Using Euclidean geometry, Newton derives in Proposition 11 that the Kepler laws can hold only for an attractive 1/r 2 force field. The converse statement, that an attractive 1/r 2 force field leads to elliptical orbits, Newton concludes in Corollary 1 of Proposition 13. Tacitly he assumes for this argument that the equation of motion F = ma has a unique solution for given initial position and initial velocity. Theorems about existence and uniqueness of solutions of such a differential equation were formulated and mathematically proven only in the 19th century. However, there can be little doubt that Newton did grasp these properties of his equation F = ma [1]. Later, in 1710, Jakob Hermann and Johan Bernoulli gave a direct proof of Kepler’s first law, which is still the standard proof for modern textbooks on classical mechanics [14]. One writes the position vector r in the plane of motion in polar coordinates r and h. The trick is to transform the equation of motion ma = -kr/r3 with variable the time t into a second-order differential equation of the scalar function u = 1/r with variable the angle h. This differential 40


equation can be solved exactly, and yields the equation of an ellipse in polar coordinates [4]. Another popular proof goes by writing down the socalled Runge-Lenz vector K ¼ p L kmr=r with p = mv the momentum and F(r) = -kr/r3 the force field of the Kepler problem. The Runge-Lenz vector K turns _ ¼ 0 . This result can be derived out to be conserved, i.e., K by a direct computation as we indicate in the next section. An alternative geometric argument is sketched in the following section. Working out the equation r K ¼ rK cos h yields the equation of an ellipse in polar coordinates [4]. The geometric meaning of the Runge-Lenz vector becomes clear a posteriori: it is a vector pointing in the direction of the major axis of the ellipse. But at the start of the proof, writing down the Runge-Lenz vector seems an unmotivated trick. For an historical account of the Runge-Lenz vector, we refer to Goldstein [5, 6]. Goldstein traces the vector back to Laplace’s Traite´de Mećanique Ce´leste from 1798. Actually, the Runge-Lenz vector already appeared in a paper by Lagrange from 1781 [9], which as far as we know was the vector’s first use. Lagrange writes the vector down after algebraic manipulations and without any geometric motivation. It is more than clear by now that the name Runge-Lenz vector is inappropriate, but with its widespread use in modern literature it seems too late to change that. The purpose of this note is to present in the first section a proof of the Kepler laws for which a priori the reasoning is well motivated in both physical and geometric terms. Then, in the following section, we review the hodographic proof as given by Feynman in his ‘‘Lost Lecture’’ [7], and finally we discuss Newton’s proof from the Principia [13]. All three proofs are based on Euclidean geometry, although we do use the language of vector calculus in order to make the text more readable for people of the 21st century. We feel that our proof is really the simplest of the three, and at the same time it gives more refined information (namely

the length of the major axis 2a = -k/H of the ellipse E). In fact we think that our proof in the next section can compete both in transparency and in level of computation with the standard proof of Jakob Hermann and Johann Bernoulli, making it an appropriate alternative to present in a freshman course on classical mechanics. We thank Alain Albouy, Hans Duistermaat, Ronald Kortram, Arnoud van Rooij, and the referee for useful comments on this article. Note: Maris van Haandel’s work was supported by NWO.

motion for fixed energy H \ 0 is bounded inside a sphere with center 0 and radius -k/H. Indeed, V ðrÞ H , and so k=r H or equivalently r k=H . Consider the following picture of the plane perpendicular to L.

A Euclidean Proof of Kepler’s First Law We shall use inner (or scalar, or dot) products u v and outer (or vector, or cross) products u 9 v of vectors u and v in R3 , the compatibility conditions u ðv wÞ ¼ ðu vÞ w u ðv wÞ ¼ ðu wÞv ðu vÞw; and the Leibniz product rules ðu vÞ_ ¼ u_ v þ u v_ ðu vÞ_ ¼ u_ v þ u v_ without further explanation. For a central force field F(r) = f (r)r/r the angular momentum vector L = r 9 p is conserved by Newton’s _ thereby leading to Kepler’s second law of motion F ¼ p, law. For a spherically symmetric central force field F(r) = f(r)r/r, the energy Z H ¼ p2 =2m þ V ðrÞ; V ðrÞ ¼ f ðrÞdr is conserved as well. These are the general initial remarks. From now on, consider the Kepler problem f (r) = -k/r2 and V(r) = -k/r, with k [ 0 a coupling constant. If m is replaced by the reduced mass l = mM/(m + M ) then the coupling constant becomes k = GmM, with m and M the masses of the two bodies and G the universal gravitational constant. Using conservation of energy, we show that the

The circle C with center 0 and radius -k/H is the boundary of a disc where motion with energy H \ 0 takes place. Points that fall from the circle C have the same energy as the original moving point, and for this reason C is called the fall circle. Let s = -kr/rH be the projection of r from the center 0 onto the fall circle C. The line L through r with direction vector p is the tangent line of the orbit E at position r with velocity v. Let t be the orthogonal reflection of the point s in the line L. As time varies, the position vector r moves along the orbit E, and likewise s moves along the fall circle C. It is good to investigate how the point t moves.

AUTHORS

......................................................................................................................................................... graduated from Nijmegen University in 1993 with a thesis on Riesz spaces. He is a high-school teacher in RSG Pantarin at Wageningen. He has been working together with Gert Heckman for two years on a project on Newton and the Kepler laws, with the objective of writing a treatment suitable for high schools. He lives in a small village near Nijmegen, with his wife Yvette and their one-year-old son Ruben. RSG Pantarijn, Wageningen, Netherlands e-mail: [email protected]

graduated from Leiden University in 1980 with a thesis on Lie groups. He has been professor of geometry at Radboud University since 1999. His wife teaches Greek and Latin in high school; both their children are medical students. Heckman is an enthusiast for skating on the frozen canals in winter. He hopes that global warming will relent so that this hobby can continue.

GERT HECKMAN

MARIS VAN HAANDEL

2

IMAPP, Radboud University, Nijmegen, Netherlands e-mail: [email protected]

Ó 2009 The Author(s). This article is published with open access at Springerlink.com

41

T HEOREM . The point t equals K/mH and therefore is conserved.

P ROOF . The line N spanned by n = p 9 L is perpendicular to L. The point t is obtained from s by subtracting twice the orthogonal projection of s - r on the line N , and therefore t ¼ s 2ððs rÞ nÞn=n2 : Now s ¼ kr=rH ðs rÞ n ¼ ðH þ k=rÞr ðp LÞ=H ¼ ðH þ k=rÞL2 =H

T2 =a3 ¼ 4p2 m=k: The mass m we have used so far is actually equal to the reduced mass l = mM/(m + M), with m the mass of the planet and M the mass of the sun, and this almost equals m if m M. The coupling constant k is, according to Newton, equal to GmM with G the universal gravitational constant. We therefore see that Kepler’s (harmonic) third law, stating that T2/a3 is the same for all planets, holds only approximately for m M. It might be a stimulating question for the students to adapt the arguments of this section to the case of fixed energy H [ 0. Under this assumption, the motion becomes unbounded and traverses one branch of a hyperbola.

n2 ¼ p2 L2 ¼ 2mðH þ k=rÞL2 ;

Feynman’s Proof of Kepler’s First Law

and therefore t ¼ kr=rH þ n=mH ¼ K=mH ; where K = p 9 L - kmr/r is the Runge-Lenz vector. The _ ¼ 0 is derived by a straightforward computafinal step K tion, using the compatibility relations and the Leibniz product rules for inner and outer products of vectors in R3 .

C OROLLARY . The orbit E is an ellipse with foci 0 and t, and major axis equal to 2a = -k/H.

P ROOF . Indeed we have jt rj þ jr 0j ¼ js rj þ jr 0j ¼ js 0j ¼ k=H : Hence E is an ellipse with foci 0 and t, and major axis 2a = -k/H. The above proof has two advantages over the earlier mentioned proofs of Kepler’s first law. The conserved vector t = K/mH is a priori well motivated in geometric terms. Moreover we use the gardener’s definition of an ellipse. The gardener’s definition, so called because gardeners sometimes use this construction for making an oval flowerbed, is well known to (Dutch) freshmen. In contrast, the equation of an ellipse in polar coordinates is unknown to most freshmen, and so additional explanation would be needed for that. Yet another advantage of our proof is that the solution of the equation of motion is achieved by just finding enough constants of motion (of geometric origin), whose integration is performed trivially by the fundamental theorem of calculus. The proofs by Feynman and Newton in the next sections on the contrary rely at a crucial point on the existence and uniqueness theorem for differential equations. We proceed to derive Kepler’s third law along standard lines [4]. The ellipse E has numerical parameters (the major axis equals 2a, the minor axis 2b and a2 = b2 + c2) a, b, c [ 0 given by 2a = -k/H, 4c2 = K2/m2H2 = (2mHL2 + m2k2)/m2H2. The area of the region bounded by E equals pab ¼ LT =2m; with T the period of the orbit. Indeed, L/2m is the area of the sector swept out by the position vector r per unit time. A straightforward calculation yields 42


In this section we discuss a different geometric proof of Kepler’s first law based on the hodograph H. By definition H is the curve traced out by the velocity vector v in the Kepler problem. This proof goes back to Mo¨bius in 1843 and Hamilton in 1845 [3] and has been forgotten and rediscovered several times, by Maxwell in 1877 [2] and by Feynman in 1964 in his ‘‘Lost Lecture’’ [7], among others. Let us assume (as in the picture in the previous section) that ivn/n = v, with i the counterclockwise rotation around 0 over p/2. So the orbit E is assumed to be traversed counterclockwise around the origin 0.

T HEOREM . The hodograph H is a circle with center c = iK/mL and radius k/L.

P ROOF . We shall indicate two proofs of this theorem. The first proof is analytic in nature, and uses conservation of the Runge-Lenz vector K by rewriting K ¼ p L kmr=r ¼ mvLn=n kmr=r as vn=n ¼ K=mL þ kr=rL; or equivalently v ¼ iK=mL þ ikr=rL: _ ¼ 0. Hence the theorem follows from K There is a different geometric proof of the theorem, discussed by Feynman, which, instead of using the conservation of the Runge-Lenz vector K, yields it as a corollary. The key point is to reparametrize the velocity vector v from time t to angle h of the position vector r. It turns out that the vector vðhÞ is traversing the hodograph H with constant speed k/L. Indeed we have from Newton’s laws m

dv dt kr dt ¼ ma ¼ 3 ; dh dh r dh

and Kepler’s second law yields r 2 dh=2 ¼ Ldt=2m: Combining these identities yields dv ¼ kr=rL; dh

so indeed v(h) travels along H with constant speed k/L. Since r = reih, a direct integration yields vðhÞ ¼ c þ ikr=rL;

c_ ¼ 0;

and the hodograph becomes a circle with center c and radius k/L. Comparison with the last formula in the first proof gives c ¼ iK=mL; _ ¼ 0 comes out as a corollary. and K All in all, the circular nature of the hodograph H is more or less equivalent to the conservation of the Runge-Lenz vector K.

T HEOREM . Let E be a smooth closed curve bounding a

Now turn the hodograph H clockwise around 0 by p/2 and translate by ic = -K/mL. This gives a circle D with center 0 and radius k/L. Since kr=rL þ K=mL ¼ vn=n ¼ iv; the orbit E intersects the line through 0 and kr/rL in a point with tangent line L perpendicular to the line through k r/rL and -K/mL. For example, the ellipse F with foci 0 and -K/mL and major axis equal to k/L has this property, but any scalar multiple kF with k [ 0 has the property as well. Because curves with the above property are uniquely charcterized after an initial point on the curve is chosen, we conclude that E ¼ kF for some k [ 0. This proves Kepler’s first law. A comparison with the picture in the previous section shows that E ¼ kF with k = -L/H. Indeed, E has foci 0 and -kK/mL = K/mH = t, and its major axis is equal to kk/L = -k/H = 2a. It is not clear to us whether Feynman was aware that he was relying on the existence and uniqueness theorem for differential equations. On page 164 of [7] the authors quote Feynman: ‘‘Therefore, the solution to the problem is an ellipse - or the other way around, really, is what I proved: that the ellipse is a possible solution to the problem. And it is this solution. So the orbits are ellipses.’’ Apparently Feynman had trouble following Newton’s proof of Kepler’s first law. On page 111 of [7] the authors write, ‘‘In Feynman’s lecture, this is the point at which he finds himself unable to follow Newton’s line of argument any further, and so sets out to invent one of his own’’.

Newton’s Proof of Kepler’s First Law In this section we discuss a modern version of the original proof by Newton of Kepler’s first law as given in [13]. The proof starts with a nice general result.

convex region containing two points c and d. Let r(t) traverse the curve E counterclockwise in time t, such that the areal speed with respect to the point c is constant. Likewise let r(s) traverse the curve E counterclockwise in time s, such that the areal speed with respect to the point d is equal to the same constant. Let L be the tangent line to E at the point r, and let e be the intersection point of the line M, which is parallel to L through the point c, and the line through the points r and d. Then the ratio of the two accelerations is given by j

d2r d2r j : j 2 j ¼ jr ej3 : ðjr cj jr dj2 Þ: ds2 dt

P ROOF . Using the chain rule, we get dr dr dt ¼ ds dt ds d 2 r d 2 r dt 2 dr d 2 t ¼ þ 2: ds2 dt 2 ds dt ds Because d2r/dt2 is proportional to c - r and likewise d2r/ds2 is proportional to d - r, we see that 2 2 d2r d2r dt d 2 r dr d 2 t dt d2r 2 j 2j : j 2j ¼ j 2 þ j : j 2j ds dt ds dt dt ds ds dt 2 dt jr ej : jr cj: ¼ ds Since the curve E is traversed with equal areal speed relative to the two points c and d, we get dr dr j j jr ej ¼ j j jr dj dt ds and therefore also dt ¼ jr ej : jr dj: ds In turn this implies that 2 d2r d2r dt j 2j : j 2j ¼ jr ej : jr cj ds dt ds ¼ jr ej3 : ðjr cj jr dj2 Þ; Ó 2009 The Author(s). This article is published with open access at Springerlink.com

43

which proves the theorem. We shall apply this theorem in case E is an ellipse with center c and focus d. Assume that r(t) traverses the ellipse E in harmonic motion, say d2r ¼ c r; dt 2 so the period for time t is assumed to be 2p.

The year 1687 marks the birth of both modern mathematical analysis and modern theoretical physics. As such, the derivation of the Kepler laws from Newton’s law of motion and law of universal gravitation is a rewarding subject to teach to freshmen students. In fact, this was the motivation for our work: we plan to teach this material to high-school students in their final year. Of course, the high-school students first need to become acquainted with the basics of vector geometry and vector calculus. But after this familiarity is achieved, nothing else hinders the understanding of our proof of Kepler’s law of ellipses. For freshmen physics or mathematics students in the university, who are already familiar with vector calculus, our proof given here is fairly short and geometrically well motivated. In our opinion, of all proofs, this proof qualifies best to be discussed in an introductory course. OPEN ACCESS

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited. Let b be the other focus of E, and let f be the intersection point of the line N , passing through b and parallel to L, with the line through the points d and r. Then we find jd ej ¼ je fj ; jf rj ¼ jb rj; which in turn implies that je rj is equal to the half major axis a of the ellipse E. We conclude from the formula in the previous theorem that the motion in time s along an ellipse with constant areal speed with respect to a focus is only possible in an attractive inverse-square force field. The converse statement, that an inverse-square force field (for negative energy H) indeed yields ellipses as orbits, follows from existence and uniqueness theorems for solutions of Newton’s equation F = ma and the previously mentioned reasoning. This is Newton’s line of argument for proving Kepler’s first law.

Conclusion There exist other proofs of Kepler’s law of ellipses from a higher viewpoint. One such proof by Arnold uses complex analysis, and is somewhat reminiscent of Newton’s previously described proof by comparing harmonic motion with motion under an 1/r2 force field [1]. Apparently Kasner had discovered the same method already back in 1909 [12]. Another proof, by Moser, is also elegant, and uses the language of symplectic geometry and canonical transformations [8, 10, 11]. However our goal here has been to present a proof that is as basic as possible, and at the same time is well motivated in terms of Euclidean geometry. It is difficult to exaggerate the importance of the role of the Principia Mathematica in the history of science.

44


REFERENCES

[1] V.I. Arnold, Huygens and Barrow, Newton and Hooke, Birkhaüser, Boston, 1990. [2] D. Chakerian, Central Force Laws, Hodographs, and Polar Reciprocals, Mathematics Magazine 74(1) (February 2001), 3–18. [3] D. Derbes, Reinventing the wheel: Hodographic solutions to the Kepler problems, Am. J. Phys. 69(4) (April 2001), 481–489. [4] H. Goldstein, Classical Mechanics, Addison-Wesley, 1980 (2nd edition). [5] H. Goldstein, Prehistory of the ‘‘Runge-Lenz’’ vector, Am. J. Phys. 43(8) (August 1975), 737–738. [6] H. Goldstein, More on the prehistory of the Laplace or RungeLenz vector, Am. J. Phys. 44(11) (November 1976), 1123–1124. [7] D.L. Goodstein and J.R. Goodstein, Feynman’s Lost Lecture: The motion of planets around the sun, Norton, 1996. [8] V. Guillemin and S. Sternberg, Variations on a Theme of Kepler, Colloquium Publications AMS, volume 42, 1990. [9] J.L. Lagrange, Theórie des variations sećulaires des e´le´ments des plane`tes, Oeuvres, Gauthier-Villars, Paris, Tome 5, 1781, pp 125–207, in particular pp 131–132. [10] J. Milnor, On the Geometry of the Kepler Problem, Amer. Math. Monthly 90(6) (1983), 353–365. [11] J. Moser, Regularization of the Kepler problem and the averaging method on a manifold, Comm. Pure Appl. Math. 23 (1970), 609–636. [12] T. Needham, Visual Complex Analysis, Oxford University Press, 1997. [13] I. Newton, Principia Mathematica, New translation by I.B. Cohen and A. Whitman, University of California Press, Berkeley, 1999. [14] D. Speiser, The Kepler problem from Newton to Johann Bernoulli, Archive for History of Exact Sciences 50(2) (August 1996), 103–116.

Orbital Anomalies FLORIN DIACU

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y the end of the 19th century, the famous CanadianAmerican mathematical astronomer and polymath Simon Newcomb, together with his collaborators at the United States Naval Observatory in Washington D.C., had achieved a precision of one arc second in their predictions of planetary motion [10]. This feat would not have been possible without the progress of celestial mechanics, which had come a long way since its birth in Newton’s Principia through the work of mathematical giants such as Euler, Lagrange, Laplace, Poisson, Jacobi, and Poincare´. But an issue still bothered Newcomb. Le Verrier—the co-discoverer of Neptune—had already pointed out the quandary several decades earlier [7]. The problem was the motion of Mercury. Its perihelion advance of about 476 arc seconds per century was by almost 43 arc seconds in excess of what celestial mechanics could account for. Two more decades passed before this disagreement between theory and observation was understood within the framework of general relativity. It then appeared that, at least from the descriptive point of view, gravitation had revealed its last secret, and predictions would pose no problems from then on. But a total victory over celestial motions was still far away.

Toward Better Approximations In the following years, general relativity achieved great success in cosmology and related fields, but it didn’t neglect its applications to celestial mechanics either. To explain the perihelion advance of Mercury, relativity had used only a 2body problem, so the need for a generalization to any number of bodies became a priority. The contributions of Chazy, [5] Levi-Civita [12, 13], Einstein [9], Eddington [8], and many others achieved this goal. Still, their discrete approximations of Einstein’s equations are complicated and difficult to handle other than numerically. In spite of this weakness, the current refinements of the post-Newtonian approximations have reached a high level, finding applications in fields such as geodesy, geophysics, and the Global Positioning System (GPS) [6]. Indeed, without these relativistic equations, the errors would render the GPS useless in urban traffic.

Newtonian celestial mechanics also continued to develop because they offered excellent approximations when the bodies moved at low speeds far away from each other. But the classical n-body problem was also needed to cope with the technological development of the 20th century. New challenges, such as the birth of space science and space voyages, forced mathematical astronomers to take into account many other parameters beyond gravitation, such as magnetic effects and solar wind. To know the exact positions of space shuttles, the experts had to introduce relativistic corrections, so the ‘‘classical’’ equations end up looking different from the Newtonian ones. Soon, however, even those highly sophisticated models could not explain some observations. New phenomena now make the experts wonder whether they understand gravity at all.

The Pioneer Anomaly On March 2, 1972, Pioneer 10 was launched from Cape Canaveral on an Atlas-Centaur rocket. NASA had invested great hopes in this mission, whose objectives were to study cosmic rays, magnetic fields, solar wind, neutral hydrogen, dust particles; the Jovian aurorae, radio waves and atmosphere, and Jupiter’s satellites, especially Io. Heading in the direction of Aldebaran, the brightest star in the constellation Taurus, Pioneer 10 was the first spacecraft ever to reach an outer planet. After passing the asteroid belt and surviving intense radiation, in December 1973 it came close to Jupiter on a hyperbolic orbit near the plane of the ecliptic. Ten years later, it passed beyond Pluto. Thus, Pioneer 10 was the first artificial object to reach the third cosmic speed, and therefore the first space probe to leave the solar system. Pioneer 11 followed its sister on April 5, 1973. Both missions fulfilled their objectives brilliantly, sending useful data back to Earth long after the expected lifespan of their functionality had ended. But there was a small problem. The two spacecraft were closer to the Sun than the computations predicted. 2009 Springer Science+Business Media, LLC, Volume 31, Number 2, 2009

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The data indicated the presence of a small Dopplerfrequency drift, interpreted as being caused by a tiny and (almost) constant acceleration of (8.74 ± 1.33) 9 10-10 m/s2 directed toward the Sun [1, 2]. After 30 years of travel, this acceleration delayed the space probes by about 8 hours, or 380,000 km, which is roughly the distance between Earth and Moon. Though this disagreement between theory and data may seem insignificant, our computations are so precise today that this discrepancy should not occur. Also strange is that this orbital anomaly took effect only after the spacecraft passed Saturn. Everybody suspected a single cause for this phenomenon, but nobody knew what it was.

The Model The determination of the distance to the space probe follows a two-way Doppler-tracking method [11]. According to special relativity, a signal of frequency m0 sent from Earth is seen by the probe as having frequency ð1 v=cÞm0 m1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 v2 =c2 where v is the probe’s velocity relative to Earth and c the speed of light. After reaching the spacecraft, the signal is sent back to Earth, where its frequency is perceived as ð1 v=cÞm1 m2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 v2 =c2 Consequently, m2 m0 2v=c 2v ; ¼ 1 þ v=c c m0 from which the observed velocity Vobs: = v of the probe can be computed. Then Vobs is compared to the computed velocity, Vcom, of the spacecraft, leading to the acceleration of (8.74 ± 1.33) 9 10-10 m/s2 mentioned earlier.

The computation of Vcom takes quite an effort within the framework of a sophisticated model, which combines the Newtonian and relativistic n-body problems with other forces and effects. These ingredients can be briefly described as follows: • Gravitational forces. The model uses a relativistic 11-body problem that contains the 9 planets (Pluto included) as well as the Sun and the Moon. The Newtonian gravity coming from large asteroids, the Earth tides, and the lunar librations are also taken into account. • Nongravitational forces. These forces are either external relative to the space probe, such as solar wind, solar radiation, and drag from interplanetary dust, or internal. The latter include control maneuvers and thermal radiation, and the torque that these two forces produce. • Model of ground stations. Since the observation stations are on Earth, several motions must be taken into account: precession, nutation, sidereal rotation, polar motion, tidal effects, and tectonic plates drift. • Model of signal propagation. The computations consider a relativistic model for light propagation with order up to v2/c2 and the dispersion of the signal because of interplanetary dust and solar wind. • Computational methods. Four independent algorithms are used to ensure the correctness of the computations. One is an orbit determination programme developed by Jet Propulsion Laboratory, the second a computer code from the Aerospace Corporation, the third an algorithm obtained at Goddard Space Flight Center, and the fourth a programme from the University of Oslo. In other words, the model used to compute the orbit and velocity of the spacecraft takes into account all the known forces and effects that could have even the slightest influence on the probe’s motion. But in spite of this care, the difference between the observed and the computed velocity is not negligible. No wonder that several researchers tried to explain this discrepancy.

......................................................................... AUTHOR

Attempted Explanations FLORIN DIACU is a long-time Professor of

Mathematics at Victoria, and also a former director of the Pacific Institute for the Mathematical Sciences (PIMS). During his tenure at PIMS, he helped set up the Banff International Research Station, which hosts more than a thousand researchers every year. His research interests are mainly celestial mechanics, dynamical systems, and history of mathematics. His non-research interests include writing for the general reader, reading, tennis, and gym. His popular books have had wide success. Department of Mathematics and Statistics University of Victoria, Victoria BC V8W 3P4, Canada e-mail: [email protected]

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A candidate for the slow-down of the probes’ motion is the interplanetary and interstellar dust. The former is known to have a density of less than 10-24 g/cm3 and the latter of less than 3 10-26 g/cm3. But the calculations show that only a density 3 105 larger than that of the interplanetary dust could account for the anomalous acceleration, therefore this attempt to explain the phenomenon failed. Another attempt used a result from general relativity. The post-Newtonian formalism is based on the assumption that the Sun’s centre of mass follows a geodesic. But this is not true in a general relativistic framework. So some authors assumed that the Sun has an acceleration orthogonal to the ecliptic, a result that follows from an exact solution of Einstein’s equations, first discovered by Levi-Civita in 1918 [4]. The computations show, however, that this acceleration can explain the Pioneer anomaly only if all radiation of the Sun is emitted in a single direction—a hypothesis that is obviously not satisfied.

The possibility of additional mass in the solar system also raised hopes for an explanation. Apart from dust, additional mass occurs due to larger particles, such as those that form rings around Saturn and Uranus. But again the computations could not account for the anomalous acceleration unless they assumed the additional mass exceeded 100 Earth masses, and that would contradict the observed and calculated orbits of several comets. A fancier attempted explanation was to link the Pioneer anomaly with cosmic expansion. Since the anomalous acceleration is approximately equal to cH, where c is the speed of light and H the Hubble constant, some researchers thought that the cosmic expansion influences the trajectory of the probes, the magnitude of the gravitational field, signal propagation, or the definition of the Astronomical Unit. But this attempt failed too. The computations showed that the acceleration resulting from cosmic expansion is tiny, namely VH = (V/c)cH, which is by a factor of V/c smaller than cH. Finally, researchers checked to see whether the problem is related to the drift of clocks on Earth. They went as far as to ask if a nonconventional physics might be necessary to explain the Pioneer anomaly. But this approach led to no better understanding either.

The Flyby Anomaly More recently, another anomalous behaviour was observed—this time close to home. Between 1990 and 2005, several missions were launched in the solar system, each with different objectives. All of them, however, started with flybys around the Earth for the purpose of attaining the right direction and velocity to engage on the desired orbits. For the first of them, Galileo, launched in December 1990, the NASA engineers detected a frequency increase, which corresponded to an accelerated motion that found neither a classical nor a relativistic explanation. When, two years later, Galileo passed again, it came within 300 km of Earth, so close that atmospheric drag impeded the detection of any anomaly. But two other missions—NEAR, launched to study an asteroid, and Rosetta, aimed at a comet—experienced the same accelerated motion during their flybys. Initially, thrusting maneuvers for the Cassini probe, whose goal was to reach Saturn, prevented the detection of any anomaly. The same thing happened with the MESSENGER mission, aimed at Mercury in 2005. But a team led by John D. Anderson from the Jet Propulsion Laboratory at the California Institute of Technology in Pasedena recently analyzed the data from all six flybys in great detail [3]. These researches found that each spacecraft experienced the same anomalous acceleration. The behaviour was so similar that they could capture its pattern in a single empirical formula, which can now predict future anomalies and help with placing probes on the desired orbits. Unfortunately, nobody knows yet why spacecraft experience this acceleration during flybys. As in the case of the Pioneer anomaly, several attempts have been made to explain this phenomenon.

Failed Attempts The flyby anomaly corresponds to a velocity increase of a few mm/s, ranging from 1.82 ± 0.05 for Rosetta to 13.46 ± 0.13 for NEAR [11]. These differences depend, of course, on positions and velocities relative to Earth. Again, these tiny values show how precise our measurements and calculations have become. A first attempt to explain this mysterious phenomenon was the drag of the atmosphere. The drag acceleration is given by the formula a ¼ K qv 2 A=m; where K is the probe’s drag coefficient, which can be safely approximated with 2, q is the density of the atmosphere, v the velocity of the spacecraft, A its effective area, and m its mass. For a density q& 10-14 kg/m3 at a height of 1000 km, a velocity v = 30 km/s, an effective area A = 2 m2, and a mass of 1 t, the drag acceleration is of the order 10-8 m/s2, much too small to explain the anomaly and of the wrong sign as well. Another idea was to check whether ocean or solid Earth tides have any impact on the change in velocity of the spacecraft. The acceleration caused by tides turns out to be at most 10-5 m/s2, again too small to provide an explanation. The solid Earth tides are much smaller than the ocean tides, so they cannot account for the flyby anomaly either. The Earth’s albedo accounts for an effect of 10-4 m/s2, the charging of the probe with electricity an effect of at most 10-8 m/s2, and the magnetic moment an effect of only 4 10-15 m/s2—all three of them much too small as compared with the unexplained acceleration. The effect from the solar wind is also negligible, exercising an influence of less than 3 10-9 m/s2. The Earth’s oblateness, the Moon and its oblateness, the Sun, and the gravitational attraction of the other planets were also taken into account, but all turned out to be of at least one order of magnitude smaller than the anomalous acceleration. Other researchers looked at alternative models. One considered a potential energy that contained the time explicitly. Another one used a non-Newtonian classical model for gravity. A third modified slightly the relativistic model, and a fourth succeeded in matching the data from the probes but failed to explain the current relative stability of the planetary orbits, so the model proved unsuitable for the solar system. The experts hope that future missions will provide more data and perhaps hint at a research direction that could explain both the Pioneer and the flyby anomalies. Whether gravity alone is responsible for these phenomena is not clear. The possibility remains at this time that only some new physics will provide an explanation.

REFERENCES

[1] J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, Indication from Pioneer 10/11, Galileo, and Ulysses data of an apparent anomalous, weak, long-range acceleration, Phys. Rev. Lett. 81 (1998), 2858.


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[2] J. D. Anderson, J. K. Campbell, J. E. Ekelund, J. Ellis, and J. F.

[8] A. Eddington and G.L. Clark, The problem of n bodies in general

Jordan, Indication from Pioneer 10/11, Anomalous orbital-energy changes observed during spacecraft flybys of Earth, Phys. Rev.

relativity theory, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 166 (1938), 465-475.

Lett. 100 (2008), 091102. [3] J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, Study of the anomalous acceleration of Pioneer 10 and 11, Phys. Rev. D 65 (2002), 082004.

[9] A. Einstein, L. Infeld, and B. Hoffmann, The gravitational equations and the problem of motion, Ann. of Math. 39, 1 (1938), 65-100. [10] M. Hoskin, R. Taton, C. Wilson, and O. Gingerich, The general History of Astronomy. Cambridge Univ. Press, Cambridge, 1995.

[4] D. Bini, C. Cherubini, and B. Mashhoon, Vacuum C metric and

[11] C. La¨mmerzahl, O. Preuss, and H. Dittus, Is the physics within the

the gravitational Stark effect, Phys. Rev. D 70 (2004), 044020. [5] J. Chazy, La theórie de la relativite´ et la mećanique ce´leste, Gauthier-Villars, Paris, 1930.

solar system really understood? arXiv:gr-qc/0604052v1, 11 Apr.

[6] T. Damour, M. Soffel, and C. Xu, General-relativistic celestial mechanics. IV. Theory of satellite motion, Phys. Rev. D 49, 2 (1994), 618-635. [7] F. Diacu and P. Holmes, Celestial Encounters—The Origins of Chaos and Stability, Princeton Univ. Press, Princeton, NJ, 1996.

Why Fly By? We have all heard that a spaceship can use a close encounter with a planet to pick up energy enabling it to visit distant parts of the Solar System. Most accounts don’t explain the physics of this ‘‘boost’’. It makes one pause for thought, doesn’t it? We know that the spaceship passing Earth in empty space follows an orbit that is a conic section with Earth at one focus. That orbit is symmetrical: at any given distance as it is leaving the vicinity of Earth, the ship’s speed is just the same as when it was at that distance on its approach. No boost there! Let me try to sort this out. The effect is not deep, and no relativity is involved, still less the mysterious anomalies now perplexing experts. Let the mass of the Earth be m and that of the Sun be M. We can choose units, so let the mass of our spaceship be 1. We must assume 1 \\ m \\ M. Assume all three bodies are moving in a plane. Briefly during the flyby, we will be so close to Earth that the Sun’s effect on our course will be negligible: our path will be a hyperbola with Earth at one focus. During most of our flight, the Earth’s effect on our course will be small: our path will be approximately an ellipse with the Sun at one focus (not a hyperbola, for we are short of energy and surely are not bound for Aldebaran). These simpler situations are best parametrized in terms of the classical elements of the conic sections involved. The elliptical orbit around the Sun is characterized conveniently by its semi-major axis a and its semi-minor axis b. Better yet, let’s use a and the eccentricity pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e ¼ a2 b2 =a 2 ½0; 1Þ: The reason these are convenient parameters is that we can easily express the range of values for the speed. The ship’s potential energy is, as usual, V = -GM/r, and its kinetic energy K = v2/2; here G is the universal

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2006. [12] T. Levi-Civita, The relativistic problem of several bodies, Amer. J. Math. 59, 1 (1937), 9–22. [13] T. Levi-Civita, Le proble`me des n corps en relativite´ geńe´rale, Gauthier-Villars, Paris, 1950; or the English translation: The n-body Problem in General Relativity, D. Reidel, Dordrecht, 1964.

constant of gravitation, r the distance from the Sun, and v the speed. Though r and v vary, the total energy E = K + V remains constant throughout the orbit. As we are reminded elsewhere in this issue in ‘‘Teaching the Kepler Laws for Freshman’’ by Gert Heckman and Maris Van Haandel, this constant value is E = -GM/2a. Now at perihelion, where r takes its minimum value a(1 - e), K takes its maximum E + GM/(a(1 - e)), from which we compute 2 ¼ vmax

GM 1 þ e : a 1e

Similarly, the minimum of K and hence of v occurs at aphelion, and we compute 2 ¼ vmin

GM 1 e : a 1þe

(Of course, to describe the orbit completely needs more parameters: an angle to tell in which direction the ship would make its closest approach to the Sun (the ‘‘apsidal line’’), and something to tell when it would do what. But a and e suffice for specifying the shape of the orbit, and tell neatly what speeds are possible.) Thus before it passes by Earth our ship is on one ellipse, with parameters a, e, and after the encounter it is on another, with parameters a0 , e0 . What happens in between? When it is near the Earth, our path is a Kepler orbit around the Earth, but as the ship has enough energy to ‘‘escape’’ Earth’s field, the orbit is a hyperbola with the Earth at one focus. The potential energy relative to Earth is V = -Gm/r with the same G as previously described but a different mass m, and now r denotes the distance to the Earth. We approach, essentially, along one asymptote of the hyperbola and leave along the other, with energy unchanged. The flyby has switched us from one elliptical

orbit around the Sun to another one, with the same speed (relative to Earth) but a different direction. By the way, in putting it so simply I have ignored the Earth’s revolution about the Sun. This actually loses nothing: if we converted to a coordinate frame rotating with a period of a year, only minor changes in the parameters would result, the picture would stay the same. The only fraud in the preceding paragraph is that it said nothing about those transitional interludes when the ship is too close to Earth to ignore its gravitational field but not close enough to ignore the Sun’s. There, for a while, the problem is intractably the 3-body problem, and I am silent. This short-cut of skipping from one Newtonian orbit to another is known by the natural name of patched conics to specialists in designing space missions—so I am told by one of them, Jeremy Kasdin of Princeton. He also recommends two textbooks: Fundamentals of Astrodynamics by R.R. Bate, D.D. Mueller, and J.E. White; and Modern Spacecraft Dynamics and Control by M.H. Kaplan. Consider our flyby, then, as giving the spaceship only a change in heading, with speed (of ship relative

to Earth) staying fixed. That speed is given us, but we can choose the amount of change in heading almost freely, by really small changes in the line along which our orbit approaches Earth. If v and v 0 are the velocities (relative to the Sun) before and after, and u the velocity of Earth relative to the Sun, then the restriction is that |v 0 - u| = |v - u|, which leaves room for v 0 to exceed v by at most 2u. Still—the potential energy relative to the Sun has remained the same, whereas we can arrange for the speed to have increased by something on the order of |u|, increasing the kinetic energy; so the total energy will have increased. Also, we can arrange for the new eccentricity e 0 to be very close to 1. Now the farthest our ship will be able to get from the Sun on the new orbit is a 0 (1 + e 0 ). We have seen that a 0 may be boosted and that 1 + e 0 can be brought up close to 2. There is indeed something to be gained by this mysterious maneuver the flyby. Chandler Davis


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Right Triangle Li C. Tien

4412 Huron Drive Midland, MI 48642-3503 USA e-mail: [email protected]

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Mathematically Bent

The proof is in the pudding.

Colin Adams, Editor

The Lord of the Rings Part I: The NSF Fellowship of the Rings COLIN ADAMS

Opening a copy of The Mathematical Intelligencer you may ask yourself uneasily, ‘‘What is this anyway—a mathematical journal, or what?’’ Or you may ask, ‘‘Where am I?’’ Or even ‘‘Who am I?’’ This sense of disorientation is at its most acute when you open to Colin Adams’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be harmless.

Column editor’s address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267, USA e-mail: [email protected]

rodo was a somewhat short and relatively friendly algebraist with a postdoctoral position at the University of Nebraska, which had followed upon the completion of his graduate studies there. He had plentiful soft curly brown hair that seemed to sprout from every opening in his clothing. His demeanor was fundamentally kind, and students trusted him in a manner that they did not trust all faculty. Due to his general disorganization, his office was filled with towering stacks of paper and various and sundry items, so much so that it resembled less an office and more a hole dug out of a paper hillside. Frodo’s advisor, Bilbo Baggins, had also received his Ph.D. at Nebraska, but, as a young algebraist, he had left Nebraska for a postdoctoral position, traveling all the way to Berkeley. But he quickly discovered that adventure did not fit his demeanor, and soon he had returned to the safety of his office at Nebraska. Settling in for a quiet career, he published a paper here and there, but nothing too ambitious. Eventually, Bilbo came to the realization that he was easily the oldest member of his department, and it was his turn to retire. The math department

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put on a conference in his honor, which included 144 attendees, a big banquet, and drinks all around. At one point during the festivities, when the rest of the participants were dancing a cohomology dance, Bilbo motioned to Frodo to follow him, and the two of them disappeared from the banquet hall. Once outside the room, Bilbo pulled Frodo into an alcove. ‘‘Frodo,’’ he said. ‘‘There is something I want you to have. Many years ago, when I traveled to Berkeley, they gave me an office that was shared by several postdocs. On the day that I was leaving, I stumbled across this in one of the desks.’’ Bilbo pulled a sheaf of folded and wrinkled papers out of his pocket. ‘‘What is it?’’ asked Frodo. ‘‘It is the directions for the construction of a ring,’’ replied Bilbo, looking over Frodo’s shoulder to confirm they were alone. ‘‘But not just any ring.’’ Frodo glanced down at the papers. It appeared the ring was noncommutative with identity, but beyond that, he could not tell. It seemed awfully complicated. ‘‘I want you to take this ring. Guard it carefully,’’ said Bilbo. ‘‘It could destroy you and everyone you love. I must go live with the retired people in Florida. They will take care of me. But mark my words. The ring is very powerful. Use it carefully.’’ As he went to hand the pages to Frodo, a look of immense pain etched its way across Bilbo’s face. ‘‘Wait!’’ he wailed. ‘‘Give it back. Give it back!’’ Frodo immediately did so. Bilbo clutched the papers to his chest, mumbling incoherently. But presently he calmed down, and his grip on the papers loosened. Accessing some reserve of inner strength, he again thrust the papers into Frodo’s hands. ‘‘Please take it,’’ he pleaded. ‘‘Take it now, and do not let me see it again.’’ Bilbo then turned away and with an anguished cry, rushed back into the great hall, where the algebraists were now playing pin the tail on the geometer.


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For the next few weeks, Frodo ignored the papers and went about his business. But one day, as he was looking for a folder from amongst the stacks in his office, he noticed the sheaf of papers where he had laid them down on a pile of yellowed preprints. He smoothed them out on his desk and tried to see if he could follow the construction. Within minutes, he was deeply enmeshed in the work. He did not notice as the time for the algebra seminar passed, and, even more surprising, he worked straight through afternoon tea, missing the frosted pink cookies to which he so looked forward. As the shadows slowly lengthened, he came to the realization that this was no ordinary ring. It had properties he had never seen before. It could do things that most algebraists would have said no ring could do. By midnight he was still sitting in his office, unable to leave lest he miss some other marvel. It was as if he were handling an intricate gem that reflected light in myriad fascinating patterns as he turned it this way and that. Suddenly, the door to his office swung open. A tall thin figure with a greying beard and a walking stick stood at the door. ‘‘Who are you?’’ asked Frodo, as he instinctively covered the papers with his hands. ‘‘I am the algebraic geomancer Geisenbud,’’ replied the figure. ‘‘I am here to warn you. You must destroy this ring before it destroys you.’’ ‘‘I couldn’t possibly destroy this ring,’’ said Frodo. ‘‘Why on earth would I want to do that?’’ ‘‘Otherwise, it will corrupt you and take you over. Your entire life will be spent studying its properties.’’ ‘‘What’s wrong with that? It is so beautiful. I could happily spend my life studying this ring. I am the only one harmed. ’’ ‘‘You don’t understand,’’ said Geisenbud. ‘‘It is not just you who may suffer the consequences. For if this ring fell into the wrong hands, it could destroy the entirety of mathematics. The world as we know it would end.’’ ‘‘How could that be? It is just a ring.’’ ‘‘This is not just a ring,’’ thundered Geisenbud. ‘‘This is THE ring. The great ring, fashioned by Sauron. This 52


ring which controls the other four rings. Whoever possesses this ring, has immense power.’’ ‘‘You mean it is a power series ring?’’ ‘‘No, I mean it is one of a series of rings of great power. As a matter of fact, it is a quotient of a power series ring. But that is beside the point. That is not what makes it so powerful. ‘‘Within this ring are the seeds of destruction. For this ring contains a logical contradiction that could bring mathematics to its knees. If it is not destroyed, an immense conflagration will consume all of mathematics as we know it.’’ Geisenbud slammed his staff into the floor for emphasis. A shiver ran down Frodo’s spine. ‘‘But I am just a minor postdoc,’’ he said, trying to control the waver in his voice. ‘‘What can I do?’’ ‘‘You must take it to Purdue, and throw it into the math building there. Then it will be destroyed. They have no interest in noncommutative rings there.’’ ‘‘But how can I get to Purdue? It is very far away.’’ ‘‘I have arranged for you to give a series of talks at various institutions between here and West Lafayette, so as to allay suspicion. There are others who will help you along the way. But be very careful. For the minions of the Dark Lord seek the ring even as we speak.’’ ‘‘Who is the Dark Lord?’’ asked Frodo. But the great geomancer had already turned and disappeared behind the frosted glass of the door. The next morning, Frodo packed up a few preprints, stuffed the papers into his backpack and met a taxi down at the curb. As he opened the door, his good friend Sam came running over the hillside. Sam and Frodo had been undergraduate math majors together at a small liberal arts college. Sam had depended on Frodo to get him through some of the more difficult classes. As graduation loomed, they had both applied for NSF graduate fellowships. Much to their surprise, Sam received a fellowship while Frodo only received honorable mention. Frodo ultimately decided to attend Nebraska, and Sam eagerly followed suit, unable to conceive of doing mathematics without

the help of his friend. But Sam had yet to complete his PhD. ‘‘Hold it right there, Frodo Baggins. I saw that you had cancelled your classes today. Where are you off to?’’ ‘‘I cannot tell you, Sam, for fear I could get you into trouble.’’ ‘‘I doubt I could get into much more trouble than I already am. My teaching evaluations are in the dumpster, and my research is about as deep as a castrato’s voice,’’ he chuckled. ‘‘If you are leaving Nebraska on a trip, I reckon I will come with you.’’ Suddenly two other postdocs appeared running over the hill, carrying backpacks. They had also held NSF fellowships while graduate students at Northwestern. ‘‘If you are leaving on a trip, then we are coming, too,’’ said Merry. With that, Sam, Merry, and Pippin all hopped into the cab before Frodo could stop them. When they arrived at their first destination, Northern Illinois University, they discovered that the location for the talk was a large auditorium. ‘‘What kind of algebra seminar is this?’’ asked Frodo. As he spoke, Frodo’s host walked up. ‘‘Seminar? Who told you this would be a seminar? This is the Pi Mu Epsilon Honorary Mathematics Society Induction Ceremony Colloquium. You do have a talk that will be appropriate for a general audience, don’t you?’’ He gestured to the seats filled with cleancut undergraduates and their parents. Frodo gulped, his eyes widening with terror. But Pippin pulled him close and whispered in his ear, ‘‘Don’t worry, Frodo. Just talk extra slow and skip every other slide. You’ll be a hit for sure. Just wait and see.’’ The host stepped up to the podium. ‘‘Welcome, everyone. We are very pleased to have with us Frodo Baggins, an algebraist from the University of Nebraska. Rumors abound that he has amazing mathematical powers, and we expect great things of him in the future. But today, he has promised me that he will be giving an expository talk of the most understandable variety. I give you Frodo Baggins.’’ Frodo, shaking with fear, stepped to the lectern. He attempted a smile, and then said, ‘‘ Let R be a weak Armendariz ring.’’

The audience rustled uncomfortably in their seats. Frodo waved his hand reassuringly. ‘‘For those of you unfamiliar with weak Armendirez rings, just think of it as a reduced semiprime right Goldie ring.’’ Audience members turned to one another in consternation. Frodo cleared his throat nervously. He noticed a dark figure sitting immobile near the rear of the auditorium, staring back at him with sinister yellow eyes. As Frodo looked upon him, he suddenly found himself unable to move. The host, who was sitting in the front row, waved to get Frodo’s attention. ‘‘Baggins, what’s wrong with you?’’ he demanded. But Frodo was frozen. Suddenly a hand grabbed his arm. ‘‘It is a ringwraith. It seeks after the ring. It will destroy your career with one question at the end of your talk. We must leave at once.’’ The spell broken, Frodo turned to a tall figure in a low hat. ‘‘Follow me,’’ urged the figure as he steered Frodo away from the podium. ‘‘What are you doing?’’ cried the host in desperation, as the four

postdocs and their savior raced up the stairs of the auditorium, past the relieved-looking crowd, and disappeared through the back door. The ringwraith leapt up in pursuit. ‘‘Quick, duck in here,’’ said the figure. He pushed the four of them through a door into a classroom where a teaching assistant was running a review session. ‘‘Act like undergraduates,’’ he whispered. ‘‘No, the square root of a number is not negative,’’ the TA was saying to a student. ‘‘You have to put a negative sign in front of it to make it negative.’’ ‘‘That’s not what the professor said,’’ responded the student. Frodo and the others slipped into the last row. Frodo opened a book and held it in front of his face. They heard a noise outside the door. Glittering yellow eyes peered through the door window, surveying the room. Merry and Pippin began shoving each other. ‘‘You left your wet clothes in the washing machine, again,’’ yelled Pippin. ‘‘It’s not me. I haven’t washed my clothes all semester,’’ retorted Merry.

They slapped at each other. Sam raised his hand. ‘‘Excuse me,’’ he said to the TA, ‘‘But what’s on the exam?’’ ‘‘I can’t answer that,’’ replied the TA. ‘‘How about telling us what’s not on the exam?’’ persisted Sam. The ringwraith disappeared from the window. Frodo turned to the stranger. ‘‘Who are you?’’ he asked. ‘‘They call me Strider,’’ he replied. ‘‘I also had an NSF Fellowship while in graduate school.’’ ‘‘Actually, I only got honorable mention,’’ said Frodo. ‘‘Oh… That’s not what Geisenbud said. Well, no matter,’’ said Strider. ‘‘We must get to Bloomington, Indiana. There, we will be met by the rest of the NSF fellowship. They will help us to bring the ring to Purdue where it can be destroyed.’’ So the band of five fellows — well, four fellows and an honorable mention — set off to Bloomington to meet up with the rest of the members of the fellowship. Their adventures were many and varied, but that is another tale to tell.

2009 Springer Science+Business Media, LLC 2009, Volume 31, Number 2, 2009

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The Mathematical Tourist

Mathematics in The Marches, Italy KIM WILLIAMS

Does your hometown have any mathematical tourists attractions such as statues, plaques, graves, the cafe´ where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit an essay to this column. Be sure to included a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: [email protected]

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Dirk Huylebrouck, Editor

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he Italian region of Le Marche, or The Marches as it is sometimes anglicized, is well known as a holiday destination, but it is also a destination for the mathematical tourist. The region is located in Central Italy and is a kind of bent rectangle in shape. Its major northern border is with Emilia-Romagna, with the independent Republic of San Marino tucked between the two; it touches Tuscany on its northwestern corner; shares a long border with Umbria on the west; touches Lazio and borders on Abruzzo to the south; the Adriatic coastline runs the entire length to the east. The northernmost province of The Marches is formed of the two cities Pesaro and Urbino. These hold a special place for the history of mathematics: Pesaro was the city of Guidobaldo Del Monte (1545–1607), mathematician, astronomer and philosopher, and friend of Galileo. Urbino was the hometown of mathematician Bernadino Baldi (1533–1617) and Federico Commandino (1509–1575). It was where the Duke of Montefeltro kept court, attended by Piero della Francesca and Luca Pacioli from nearby San Sepolcro. For political history buffs, a ‘‘march’’ is a frontier region, usually militarized, ruled by a marquis, or, in England, a marquess. The Marches takes its name from the medieval March of Ancona. Pesaro and Urbino merit Mathematical Tourist columns on their own, but the present one is about a hidden mathematical treasure trove in the heart of the Pesaro-Urbino province, off the beaten track. Close to the border with Emilia-Romagna and the Republic of San Marino, in the small town of Pennabilli (pen-na-BEE-lee) is a fine museum dedicated entirely to the history of counting and calculation, and more generally to the history of mathematics. Named Museo Mateureka, it is situated in the former municipal palace, handsomely restored purposely


to house the museum, located just off the main square, by the Cathedral. The museum occupies all four floors of the palace. The itinerary begins on the top floor. The goal of the museum is to introduce the visitor to the fascinating world of mathematics and calculation, beginning with its most ancient origins, from counting first with fingers and toes and then with pebbles and sheep, and working its way through the historical development of operations, and theorems and applications of mathematics to related fields such as astronomy and commerce. Included are mathematical developments in ancient Egypt, Greece, and Rome, as well as China, India, Arabia, and Central America. On display are reproductions of calculating devices such as a Roman abaci, the Chinese suanpan, an Inca quipu, an Aztec calendar, the Russian schoty or stchote (cx/ns), and the Japanese soroban. After these ancient beginnings, the displays then pass chronologically through various stages in Western mathematics: Medieval, Renaissance, 1600s (with a beautiful reproduction of Galileo’s compass, the original of which is in Florence’s Museum for the History of Science, and another beautiful

Figure 1. The Museo Mateureka (Pennabilli, Italy) shows replicas from Galileo’s compass to the Pascalina (photo by the author).

reproduction of a Pascalina, invented by Blaise Pascal in 1642 for addition and subtraction), 1700s (with attention given to the application of mathematical concepts to mechanics, especially celestial mechanics, and some fine models of instruments such as armillary spheres), 1800s (new geometries, among other things) and 1900s (with Einstein’s relativity theory and the birth of modern computers) (Fig. 1). Not all of the objects on display are reproductions, however, and some of the pieces are stunning. The prize object is a 4,500 year old Sumerian terracotta foundation cone (Fig. 2). Foundation cones, sometimes called nails because of their flat heads, were impressed with cuneiform writing to record information relative to the founding of a building. After the widescale destruction of cultural artifacts in Afghanistan and Iraq, foundation cones have become even rarer. There is also a beautiful arithmometer, the device designed by Thomas de Colmar in 1822 as an improvement on Leibniz’s calculator, and widely manufactured and sold in Europe during the nineteenth century. This particular example was manufactured in Vienna towards the end of the century by H. Bunzell (Fig. 3). The third floor features collections of mechanical calculators, electric

calculators, and small electric calculators. The smallest of these is the Curta calculator, designed by Curt Herzstark of Austria while imprisoned in the Buchenwald concentration camp. The model in the collection is a Curta Type II, introduced in 1954, with 11 digits of slides, an eight-digit revolution counter, and a 15-digit result counter. The two lower floors feature rooms that are dedicated to mathematical themes such as the Platonic solids, pi, a working model of the Sieve of Eratosthenes, and a mechanized proof of the Pythagorean theorem. The lowest floor also has an exhibit of computers,

Figure 2. An original Sumerian foundation cone with cuneiform writing (photo by the author).

hardware, and software, from the earliest video games to virtual reality. The museum includes rooms for the projection of films, study rooms, and a library. The library is still being organized and includes a noteworthy collection of rare first editions of mathematics books. The exhibit displays are designed primarily to appeal to school children,

Figure 3. Late nineteenth-century reproduction of Thomas De Colmar’s arithmometer, invented in 1820 (photo by the author). Ó 2008 Springer Science+Business Media, LLC, Volume 31, Number 2, 2009

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and the itineraries follow the serpentine lines of the walls, which are colorful, graphically sophisticated and, well, fun. The museum design was awarded first place for exhibit design in the 2006 Index of the Italian Associazione per il Disegno Industriale. Perhaps the most surprising thing of all is that Mateureka is the work of one man, founder and director Renzo Boldoni. A high-school math teacher for many years (he retired at the end of the 2007 school year), Boldoni began his

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collection of mathematical objects on his own, came up with the idea for the museum, applied for and received funding from the European Union to finance the restoration of the municipal palace and the design and installation of the museum, and now oversees its operations. To reach Pennabilli, there is bus service from the train station in Rimini. One way to drive is to exit the A14 motorway at Rimini Nord, then pass via Santarcangelo and Novafeltria

(49 km), and another way goes from Urbino via Mercatale, Macerata Feltria, and Carpegna (57 km). A suggested route would be to arrive from Sansepolcro (after having visited Piero and Luca) via Badia Tedalda and Molina di Bascio (49 km). You can also be an armchair tourist and visit the website: http://www.mateureka.it. Via Cavour, 8, Turin 10123 Italy e-mail: [email protected]

Note

What Is It? ROBERT J. MACG. DAWSON

T

he illustration in ‘‘Curiosum’’ by Robert J. MacG. Dawson is not a photograph, but an image generated using the POV-Ray raytracer. It illustrates (in a slightly fanciful style) a slide rule for multiplying complex numbers. A simple ‘‘complex slide rule’’ could be built using a logarithmically labelled plane and a transparent overlay (say a sheet of paper and an acetate photocopy.) According to Whythe [6], the first known planar slide rule for complex numbers was devised by J. W. M. DuMond in the 1920s, although there is no evidence that it was marketed. FaberCastell patented a calculating device based on roughly the same concept, and with additional features, in 1952 (see [4]). It was manufactured between 1955 and 1960; somewhere between 100 and 200 appear to have been sold. However, this design is somewhat redundant. As most Intelligencer readers will be aware, the complex logarithm function is multivalued: if w is a logarithm of z in the sense that exp(w) = z, then so is w + 2npi for any integer n. At least when doing basic multiplication and division, we get no new results from these ‘‘extra’’ logarithms; but if we omit them, we run the risk of running ‘‘off the edge’’. This is a familiar problem with ordinary slide rules; one solution was the circular slide rule, which prevented overflow problems by wrapping the scales into circles, identifying x (on the main scales, at least) with 10x. Identifying all complex numbers that differ by a multiple of 2pi, we obtain an infinite cylinder as the quotient space, and the logarithm can be considered as a single-valued function from the nonzero complex numbers onto this cylinder. As z ? 0,

The Mathematical Intelligencer

The Mathematical Intelligencer

The Mathematical Intelligencer