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Letters

to

the

Editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.

W h a t Piero Painted I was thrilled to see Piero della Francesca on the cover of The Mathematical InteUigencer (vol. 19, no. 3). Having worked with perspective for years, I found Mark Peterson's exposition of Piero's perspective construction of particular interest. He points out that, in his own treatise, Piero led readers from the construction in perspective of a square lying flat on the ground to "objects constructed in the square (tilings, for example, to represent a tiled floor)." A perfect illustration of such a tiling is found in the cover illustration, Piero's Flagellation of Christ. The two bays in front of Christ are depicted as being tiled in a uniform material (it appears to be either red marble or terracotta), while the three bays under the loggia where Christ is being whipped are given special emphasis by receiymg richer pavements. Extremely foreshortened eight-pointed stars (symbolizing immortality and resurrection), appear in the bays directly in front of and behind Christ, while He stands in the middle of a circle, symbol of divinity. It might seem at first that Piero's emphasis on perspective constructions for pavement designs are merely exercises illustrating the technique, but in fact they contribute to the representation of the pictorial space. Art historian Erwin Panofsky has noted that by depicting the floor plane as decorated even by a pattern as simple as the checkerboard, in essence the artist creates a coordinate system by which the location of objects and the distance between them may be found by merely counting the tiles.t So Christ is not just standing somewhere in the middle of the picture; we can actually defme his

1Erwin Panofsky, La Prespettiva come "Forma Symbolica' e Altri Scritti, Guido di Neff, editor (Milan: Feltrinelli Editore, 1960), see p. 62. 2Leone Battista Alberti, The Ten Books of Architecsture, 1755 (reprint New York: Dover, 1986), book VII, chapter X, see p. 150.

4

THE MATHEMATICALINTELLIGENCER9 1998 SPRINGER-VERLAGNEW YORK

location within the bounds of the picture space by referring to the pavement design. This articulation of the floor plane also allows us to perceive certain proportional relationships within the composition. For example, we can see that the architectural bays are perfect squares. In all, there are (at least) three mathematical ideas present in the Flagellation of Christ: a coordinate system, proportional relationships in the distances between objects, and the depiction of geometrical forms. Piero was by no means alone in favoring mathematical pavement design. Leon Battista Alberti, a contemporary of Piero and author of the most famous fifteenth-century treatise on architecture, wrote: "And I would have the Composition of the Lines of the Pavement full of Musical and geometrical Proportions; to the Intent that whichsoever Way we turn our Eyes, we may be sure to find Employment for our Minds. "2 Employment, indeed. Kim Williams, Architect Via Mazzini 7 50054 Fucecchio (Fi), ITALY e-mail: [email protected]

Lex Talienis Jet Wimp's excellent triple review "Mathematics Not" (InteUigencer, vol. 19, no. 3, 70-75), caused much entertainment in our coffee room--and some hunting through a dictionary (stenotopic, kef, gongorisms, gussy). In the spirit of lex talionis, might one ask if Berlinski's lexiphanic prose infected Wimp with logodmdaly? Thomas Ward School of Mathematics University of East Anglia Norwich NR4 7TJ, UK

JET WIMP REPLIES: My sesquipedalian proclivities antecede Berlinski's book. Most likely they are the result of a library accident I suffered as a youth.

MONTE J. ZERGER

A Quotc a Day Educatcs ow do w e instill i n students an appreciation f o r the " h u m a n i n the m a t h e m a t i cian," as well as the " m a t h e m a t i c i a n i n the h u m a n " ? For the p a s t 10 years I have used one gently persistent approach. Each day I post a short quotation on m y ofce door. A brief, yet penetrating, quote can surely be compared to a Japanese haiku, a s p a r s e 17-syllable p o e m which one w r i t e r has des c r i b e d as bursting o r b l o o m i n g in y o u r mind.

T y p e s of Q u o t e s The s t a t e m e n t I p o s t is m o s t frequently one m a d e by a m a t h e m a t i c i a n a b o u t s o m e a s p e c t o f mathematics. S o m e t i m e s it is a s t a t e m e n t by a m a t h e m a t i c i a n a b o u t s o m e t h i n g o t h e r than mathematics. A n d s o m e t i m e s I use statements by nonmathematicians about mathematics--or at least m a t h e m a t i c s as t h e y p e r c e i v e it. In this c a s e I try to d o c u m e n t the p e r s o n ' s profession. It s e e m s to a d d dim e n s i o n to the statement. Quotes can range from p e r e n n i a l favorites like Kronecker's "God c r e a t e d the natural numbers, and all the rest is the w o r k of man," to less well k n o w n s t a t e m e n t s such as a favorite of mine, "It will be a n o t h e r million years, at least, before w e u n d e r s t a n d the primes" from Paul Erd6s. I o c c a s i o n a l l y use light and h u m o r o u s quotes like GianCarlo R o t a ' s "God c r e a t e d infinity, and man, unable to und e r s t a n d infinity, had to invent finite sets," or Charles Darwin's "A m a t h e m a t i c i a n is a blind m a n in a d a r k r o o m looking for a b l a c k cat w h i c h isn't there." Some exhibit only the s u b t l e s t o f c o n n e c t i o n s to mathematics, like "Everything tries to b e round," from the n o t e d Oglala Sioux m e d i c i n e man, Black Elk. The M e c h a n i c s F o r visual a p p e a l and to a t t r a c t readers, I use multic o l o r e d 3 x 5 cards. An i n e x p e n s i v e h o l d e r can be o b t a i n e d

from m o s t s t a t i o n e r y stores; simply slide y e s t e r d a y ' s card out a n d t o d a y ' s in. I k e e p t h e m filed by the y e a r used, thus allowing for recycling after several years. I believe there is a defmite advantage to putting up one quote at a time, a n d replacing it daily. Several of m y colleagues p l a s t e r their d o o r s with clippings a n d cartoons, and then only change t h e m sporadically. It s e e m s if would-be r e a d e r s c h e c k a c o u p l e o f times and see nothing new, they m a y n o t b o t h e r to c h e c k anymore. If r e a d e r s k n o w to exp e c t a fresh quip every day, they are m u c h m o r e likely to m a k e stopping b y a d o o r a p a r t of their routine. Of course, the first time, the r e a d e r j u s t h a p p e n e d to b e n e a r the d o o r and r e a d the c a r d b e c a u s e it w a s there.

Why?. Acquainting Students with Mathematicians: I believe m o s t students of m a t h e m a t i c s have an i n c o m p l e t e picture o f the personalities w h o have crafted their subject. They m a y k n o w t h e "what's, when's, and where's" s u r r o u n d i n g the contributions of t h e s e individuals, a n d p e r h a p s even the "why's." But do t h e y k n o w h o w they felt a b o u t mathematics? Making it e a s y for s t u d e n t s to r e a d carefully chosen quotes can help t h e m gain a c c e s s to the h u m a n within the mathematician. Planting Seeds: Well s e l e c t e d quotes s o w tiny s e e d s that can germinate into fresh n e w w a y s of viewing various asp e c t s of mathematics. F o r example, h o w m a n y s t u d e n t s w o u l d think o f a p r o o f as a form o f p o e t r y ? Morris Kline o n c e said, "An elegantly e x e c u t e d p r o o f is a p o e m in all

9 1998 SPRINGER-VERLAG NEW YORK, VOLUME 20, NUMBER 2, t998

5

b u t the form in which it is written," and Karl W e i e r s t r a s s quipped, "It is true that a m a t h e m a t i c i a n w h o is n o t s o m e w h a t o f a p o e t will n e v e r b e a p e r f e c t mathematician." Challenging Students' View of You: I have had s t u d e n t s r e a d one and exclaim incredulously, "You p u t that up?" or, "Do y o u really believe that?" It w a s a p p a r e n t their p e r c e p tion o f m e had j u s t t a k e n a different twist. Educating Colleagues: It can also be a w a y to "educate" colleagues--mathematicians a n d n o n - m a t h e m a t i c i a n s a l i k e - - t o humanistic m a t h e m a t i c s . A n u m b e r o f faculty with offices in m y building m a k e it a dally routine to "read m y door." Conciseness Impacts: The w o r l d t o d a y is fast paced, a n d getting faster. We are b o m b a r d e d with information to s u c h an e x t e n t that it can all t e n d to s e e m equally import a n t - o r unimportant. Often a concise s t a t e m e n t can b e m o r e p e n e t r a t i n g than s c o r e s o f pages, if it can b e m a d e to s t a n d out. Spurring Discussions: A quote can spur a discussion a n d even controversy with a s t u d e n t o r a colleague that w o u l d not otherwise have happened. This can e x p a n d horizons. Triggering Further Explorations: Finally, a brief, b u t potent, s t a t e m e n t from an individual m a y l e a d a r e a d e r to e x p l o r e further the s o u r c e from which the quote came. Reading George Boole's s t a t e m e n t "It is not the e s s e n c e o f m a t h e m a t i c s to b e c o n v e r s a n t with the ideas of n u m b e r a n d quantity" could s p u r the curious r e a d e r to investigate Boole, a n d w h a t he b e l i e v e d that e s s e n c e was. To encourage this, I m a k e every effort to include a reference with the quote. Bonuses

T h e r e are several b o n u s e s to initiating such a program. One builds up a wonderful collection o f quotes w h i c h can be called u p o n to lighten up a lecture, a d d c o l o r to a syllabus, o r e n h a n c e p e r s o n a l writing. I also fred that I n o w r e a d m a t h e m a t i c a l literature with n e w enthusiasm. I a m always looking with eager anticipation for n e w items to a d d to m y collection. Sources

Books of Mathematical Quotes: Two b o o k s o f m a t h e m a t i cal quotes and e x c e r p t s have r e c e n t l y b e e n p u b l i s h e d b y MAA, a n d are listed in the bibliography. One of t h e s e is new, a n d the o t h e r is a r e p u b l i c a t i o n o f an old classic,

Memorabilia Mathematica. Other Mathematical Books: Many writers on m a t h e m a t i c s begin each c h a p t e r with one or m o r e quotes. This p r a c t i c e has even b e c o m e p o p u l a r with t e x t b o o k writers. Mathematical Journals: Various m a t h e m a t i c a l j o u r n a l s use quotes as fillers at the e n d s of articles, and in otherwise awkward-to-fill spaces. Internet: A fine collection of m a t h e m a t i c a l quotes, maint a i n e d b y Mark W o o d a r d at F u r m a n University, can be found on the w o r l d w i d e web. Point y o u r b r o w s e r to math.furman.edu/- mwoodard/mquot.html General Collections: Most general collections of q u o t e s c o n t a i n s o m e m a t h e m a t i c a l quotes. In addition to m a n y

6

THE MATHEMATICALtNTELLIGENCER

books, t h e r e are several s o u r c e s for general quotes on the internet. Gathering Locally: I have even u s e d p a s s i n g r e m a r k s m a d e by others. I c a n always d e t e c t their p r i d e w h e n they notice one o f their r e m a r k s has b e e n recognized. A couple of y e a r s ago a colleague, f r u s t r a t e d by a teaching experience, muttered, "Happiness is thinking everything is linear." I p o s t e d it the next day. Once it b e c a m e c o m m o n l y k n o w n that I c o l l e c t e d quotes, p e o p l e b e g a n bringing t h e m to me. I have r e c e i v e d several gems in this manner.

Conclusion I often p o n d e r on m y duties as a t e a c h e r of the subject I love. I feel I a m r e s p o n s i b l e for m o r e t h a n simply transmitting knowledge. I wish I could help m y students see m a t h e m a t i c s from various vantage points. One of t h e s e should be from a p o i n t high enough to afford a full, sweeping view of the m a t h e m a t i c a l valley b e l o w - - m a y b e missing the details w e strive to c o n v e y in c l a s s - - b u t seeing the landscape of mathematics. Claude B r a g d o n said, "Mathematics is the handwriting on the h u m a n c o n s c i o u s n e s s of the very Spirit of Life itself." I w a n t m y s t u d e n t s to consider that such a b o l d s t a t e m e n t might actually be true. BIBLIOGRAPHY Moritz, Robert (1993). Memorabilia mathematica. Washington, D.C.: Mathematical Association of America. Rose, Nicholas (Ed.). (1988). Mathematical maxims and minims. Raleigh: Rome Press. Schmalz, Rosemary (1993). Out of the mouths of mathematicians, Washington, D.C.: Mathematical Association of America.

STEVE SMALE

Mathematical Problems for the Next C e n t u r y 1

V

I. Arnold, on behalf of the International Mathematical Union, has written to a number of mathematicians w i t h a suggestion that they describe some great 0 problems f o r the next century. This report is m y response.

Arnold's invitation is inspired in part by Hilbert's list of 1900 (see, e.g., [Browder, 1976]) and I have used that list to help design this essay. I have listed 18 problems, chosen with these criteria:

P r o b l e m 1: T h e R i e m a n n Hypothesis Of the zeros of the R i e m a n n zeta function, defined by analytic continuation f r o m

~(s) 1. Simple statement. Also preferably mathematically precise. 2. Personal acquaintance with the problem. I have not found it easy. 3. A belief that the question, its solution, partial results, or even attempts at its solution are hkely to have great importance for mathematics and its development in the next century. Some of these problems are well known. In fact, included are what I believe to be the three greatest open problems of mathematics: the Riemann Hypothesis, the Poincar~ Conjecture, and "Does P = NP?" Besides the Riemann Hypothesis, one below is on Hilbert's list (Hilbert's 16th Problem). There is a certain overlap with my earlier paper "Dynamics retrospective, great problems, attempts that failed" [Smale, 1991]. Let us begin.

_s -

1, -

Re(s) > 1,

-

n=l us

are those which are in the critical strip 0 <- Re(s) -< 1 all on the line Re(s) = L? 2 This was Problem #8 on Hilbert's list. There are many free books on the zeta function and the Riemann hypothesis which are easy to locate. I leave the matter at this. P r o b l e m 2: T h e P o i n c a r 6 C o n j e c t u r e Suppose that a compact connected 3-dimensional manifold has the property that every circle in it can be deformed to a point. Then must it be homeomorphic to the 3-sphere? The n-sphere is the space n+l

Sn

: {X (~ ~ n + l

I Ilxll

-- 1},

Ilxll2 = T,

2

i=1

A compact n-dimensional manifold can be thought of as a closed bounded n-dimensional surface (differentiable and non-singular) in some Euclidean space.

1Lecture given on the occasion of Arnold's 60th birthday at the Fields Institute, Toronto, June 1997,

9 1998 SPRINGER-VERLAGNEWYORK, VOLUME20, NUMBER2, 1998

7

The n-dimensional Poincar~ c o n j e c t u r e a s s e r t s t h a t a c o m p a c t n-dimensional m a n i f o l d M having the p r o p e r t y that e v e r y m a p f : S k ~ M, k < n (or equivalently, k<-n/2) c a n b e d e f o r m e d to a point, m u s t be h o m e o m o r p h i c to S n. Henri Poincar~ s t u d i e d t h e s e p r o b l e m s in his pioneering p a p e r s in topology. Poincar~ in 1900 (see [Poincar~, 1953], pp. 338-370) a n n o u n c e d a p r o o f o f the general n-dim e n s i o n a l case. Subsequently (in 1904) he found a countere x a m p l e to his first v e r s i o n o f the s t a t e m e n t ([Poincar~, 1953], pp. 435-498). In the s e c o n d p a p e r he limits h i m s e l f to n = 3, and states the 3-dimensional c a s e as the p r o b l e m a b o v e (not actually as a "conjecture"). My o w n relationship with this p r o b l e m is d e s c r i b e d in t h e s t o r y [Smale, 1990]. There I wrote, I first h e a r d of the Pioncar~ conjecture in 1955 in Ann A r b o r at the time I w a s writing a thesis on a p r o b l e m o f topology. Just a s h o r t time later, I felt that I h a d found a p r o o f (3 dimensions). H a n s S a m e l s o n w a s in his ofrice, a n d very e x c i t e d l y I s k e t c h e d m y i d e a s to him . . . . After leaving the office, I realized that m y "proof.' h a d n ' t used any h y p o t h e s i s on the 3-manifold. In 1960, "on the b e a c h e s o f Rio," I gave an affirmative a n s w e r to the n - d i m e n s i o n a l Poincar~ c o n j e c t u r e for n > 4. In 1983, Mike F r e e d m a n gave an affirmative a n s w e r for n = 4. (Note: for n > 4, I p r o v e d the s t r o n g e r result that M w a s the s m o o t h union of t w o balls, M = D n U Dn; that result is u n p r o v e d for n = 4, today.) F o r b a c k g r o u n d on t h e s e matters, besides the a b o v e references, s e e [Smale, 1963]. Many o t h e r m a t h e m a t i c i a n s after Poincar~ have c l a i m e d p r o o f s o f the 3-dimensional case. See [Taubes, 1987] for an a c c o u n t o f s o m e of t h e s e attempts. A r e a s o n that Poincar~'s conjecture is f u n d a m e n t a l in the history of m a t h e m a t i c s is that it h e l p e d give focus to a m a n i f o l d as an o b j e c t o f study. In this way, Poincar~ inf l u e n c e d m u c h of 20th-century m a t h e m a t i c s with its att e n t i o n to geometric objects, including eventually algebraic varieties, Riemannian manifolds, etc. I h o l d the conviction that t h e r e is a c o m p a r a b l e phen o m e n o n t o d a y in the n o t i o n of a "polynomial-time algorithm." Algorithms a r e b e c o m i n g w o r t h y o f analysis in t h e i r o w n right, not m e r e l y as a m e a n s to solve o t h e r p r o b l e m s . Thus I a m suggesting t h a t as the study of the set o f solutions o f an equation (e.g., a manifold) p l a y e d s u c h an imp o r t a n t role in 20th-century mathematics, the s t u d y of finding the solutions (e.g., an algorithm) m a y p l a y an equally i m p o r t a n t role in the n e x t century. Problem

3: Does

P = NP?

I s o m e t i m e s c o n s i d e r this p r o b l e m as a gift to m a t h e m a t ics from c o m p u t e r science. It m a y be useful to p u t it into a f o r m which l o o k s m o r e like traditional m a t h e m a t i c s . T o w a r d s this end, first c o n s i d e r the Hilbert Nullstellensatz over the c o m p l e x numbers. Thus let f l , . . . ,fk b e c o m p l e x p o l y n o m i a l s in n variables; w e are a s k e d to d e c i d e if t h e y

8

THE MATHEMATICAL{NTELUGENCER

have a c o m m o n zero ( E C n. The Nullstellensatz asserts that this is n o t t h e case if and only if t h e r e are c o m p l e x p o l y n o m i a l s gl, 9 9 9 gk in n variables satisfying k

gifi = 1

(1)

1

as an identity of polynomials. The effective Nullstellensatz as established by Brownawell (1987) a n d others, states that in the a b o v e one m a y a s s u m e that the d e g r e e s o f the gi satisfy deg gi -< max(3, D) n,

D = m a x deg fi.

With this d e g r e e b o u n d the decidability p r o b l e m b e c o m e s one of linear algebra. Given the coefficients of the fi, one can c h e c k if (1) h a s a solution in the coefficients of the gi. Thus one has an algorithm to d e c i d e the Nullstellensatz. The n u m b e r of arithmetic s t e p s required g r o w s e x p o n e n tially in the n u m b e r of coefficients of thef~ (the input size). C o n j e c t u r e ( o v e r C). There is no p o l y n o m i a l - t i m e algorithm f o r deciding the Hilbert Nullstellensatz over C. A p o l y n o m i a l - t i m e algorithm is one in which the numb e r of a r t h m e t i c s t e p s is b o u n d e d b y a p o l y n o m i a l in the n u m b e r o f coefficients of the fi. To m a k e m a t h e m a t i c a l sense o f this conjecture, one has n e e d of a formal definition of algorithm. In this context, the traditional definition of Turing m a c h i n e m a k e s no sense. In [Blum-Shub-Smale, 1989] a s a t i s f a c t o r y defmition is p r o p o s e d , a n d t h e a s s o c i a t e d t h e o r y is e x p o s e d in [BlumCucker-Shub-Smale (or BCSS), 1997]. Very briefly, a machine over C has as inputs a finite string of c o m p l e x numbers, a n d the s a m e for states a n d outputs. C o m p u t a t i o n s on states include arithmetic operations a n d shifts on the string. Finally, a b r a n c h o p e r a t i o n on "xl = 0?" is provided. The size o f an input is the n u m b e r o f e l e m e n t s in the inp u t string. The time of a c o m p u t a t i o n is t h e n u m b e r of machine o p e r a t i o n s u s e d in the p a s s a g e f r o m input to output. Thus, a p o l y n o m i a l - t i m e algorithm over C is well-defined. Note that all t h a t has b e e n said a b o u t the m a c h i n e s and the c o n j e c t u r e use only the s t r u c t u r e o f C as a field, and h e n c e b o t h the m a c h i n e s a n d the c o n j e c t u r e are meaningful over any field. In particular, if the field is ~2 of tWO elements, w e have t h e Turing machines. Consider the decision problem: I n p u t k polynomials i n n variables w i t h coefficients i n ~-2. Is there a c o m m o n zero ~ (~_~)n? C o n j e c t u r e . There is no p o l y n o m i a l - t i m e algorithm over •2 deciding this problem. This is a simple reformulation o f the classic c o n j e c t u r e P C NP. In the a b o v e I have b y p a s s e d the b a s i c ideas and theor e m s r e l a t e d to NP-completeness. F o r the classic case of

Cook and Karp, see [Garey-Johnson, 1979], and for the theory over an arbitrary field, s e e BCSS. Problem 4: Integer Zeroes of a Polynomial Let m e start b y defining a diophantine invariant Tmotivated b y c o m p l e x i t y theory. A p r o g r a m for a p o l y n o m i a l f ~ ;7[t] of one variable with integer coefficients is the o b j e c t (1, t, ul, 9 9 9 Uk), w h e r e Uk = f, a n d for all ~, ue = ui o uj, i , j < e, and o is + o r - or • Here u0 = t, u - 1 = 1. Then T ( f ) is the m i n i m u m o f k over all such programs. Is the n u m b e r o f d i s t i n c t i n t e g e r z e r o e s o f f p o l y n o m i n a U y b o u n d e d by r I n other w o r d s , is Z a ( f ) <- 7 ( f ) c

all f E 2~[t],

w h e r e Z a ( f ) is the n u m b e r o f d i s t i n c t i n t e g e r z e r o s o f f and c is a universal constant?

Mike Shub and I d i s c o v e r e d this p r o b l e m in our complexity studies. We p r o v e d that an affirmative a n s w e r implied the intractibility of the nullstellensatz as a decision p r o b l e m over C and thus P r NP o v e r C. See [Shub-Smale, 1995] a n d also BCSS. Since the d e g r e e o f f is less t h a n o r equal to 2 ~+1, ~ = 9 ( f ) , t h e r e a r e no m o r e than 2 ~+1 zeros altogether. F o r C h e b y s h e v polynomials, the n u m b e r of distinct real zeros g r o w s e x p o n e n t i a l l y with 7. Many o f the classic diophantine p r o b l e m s are in t w o or m o r e variables. This p r o b l e m a s k s for an estimate in j u s t one variable, and n e v e r t h e l e s s s e e m s not so easy. Here is a r e l a t e d problem. A p r o g r a m for an integer m is the o b j e c t (1, m l , . 9 me), w h e r e m e = m , m o = 1, mq = m i ~ mj, i , j < q, and o = +, - or x . Then let T(m) be the m i n i m u m of ~, over all such programs. Thus ~-(m) repres e n t s the s h o r t e s t w a y to build up an integer m starting from 1 using plus, minus, a n d times. Problem: Is t h e r e a c o n s t a n t c such that 7(k!) --< (log k) c for all integers k? One might e x p e c t this to be false, so that k! is "hard to compute"; see [Shub-Smale, 1995].

Problem 6" Finiteness of the Number of Relative Equilibria in Celestial Mechanics G i v e n p o s i t i v e real n u m b e r s m l , 9 9 9 ,mn as the m a s s e s i n the n - b o d y p r o b l e m o f celestial m e c h a n i c s , is the number of relative equilibria finite?

The p r o b l e m is in Wintner's b o o k (1941) on celestial mechanics. A relative equilibrium is a solution to N e w t o n ' s equations w h i c h is i n d u c e d by a plane rotation. F o r the 3-body p r o b l e m there are five relative equilibria: three found by Lagrange, t w o by Euler. F o r 4 b o d i e s the finiteness is unknown. In [Smale, 1970], I i n t e r p r e t e d the relative equilibria as critical points of a function i n d u c e d by t h e p o t e n t i a l of the p l a n a r n - b o d y p r o b l e m . More precisely, t h e relative equilibria c o r r e s p o n d to t h e critical p o i n t s o f (z : (Sk - h ) / S O ( 2 ) - - ) R

w h e r e Sk = {x E (R2)n I Z m i x i = O, 1/2Zmillxill 2 = 1}, h = {x E Sk I Xi = x j s o m e i ~sj}. The r o t a t i o n group SO(2) acts on Sk -- A, and V is i n d u c e d on the quotient from the potential function V(x) = ~

mim/

IIxi - xjll Note that V : Sk --) R is invariant u n d e r the r o t a t i o n group SO(2) and that the quotient s p a c e Sk/SO(2) is h o m e o m o r phic to c o m p l e x p r o j e c t i v e s p a c e o f d i m e n s i o n n - 2. Mike Shub (1970) has s h o w n that t h e set o f critical p o i n t s is compact, a n d P a l m o r e (1976) that even for n = 4, V m a y have d e g e n e r a t e critical points. Relative equilibria p l a y an i m p o r t a n t role in celestial mechanics, for example, as in the b i f u r c a t i o n of the angular m o m e n t u m map. M o r e o v e r there are "the Trojans" in the solar system, w h i c h c o r r e s p o n d to the Lagrange relative equilibria. K u z ' m i n a (1977) h a s found explicit u p p e r b o u n d s in the generic case. F u r t h e r b a c k g r o u n d m a y b e f o u n d in [AbrahamMarsden, 1978].

Problem 5- Height Bounds for Diophantine Curves C a n the f e a s i b i l i t y o f a d i o p h a n t i n e e q u a t i o n f ( x , y ) = 0 (inputfE 7/[u, v]) be decided i n t i m e 2 sc w h e r e c is a universal constant?

Here s = s ( f ) is the size o f f , defined b y

s(f) =l~d~ (logla.I + 1),

f(x,y)

= ~

1,7-
a,x"ly"2,

a = ( a i , a 2 ) , a n d lal = oLl -4- o~2, o/i __~0. Moreover, f is feasible if t h e r e exist integers x, y with f ( x , y ) = 0. The Turing m o d e l of c o m p u t a t i o n is s u p p o s e d . This p r o b l e m is p o s e d essentially in [Cucker, Koiran, a n d Smale, 1997]. The size s ( f ) is a v e r s i o n of the "height" o f f i C o n j e c t u r e d height b o u n d s for e x a m p l e as in [Lang, 1991] could p r o v e helpful in settling this problem. See also [Manders-Adleman, 1978].

Problem 7: Distribution of Points on the 2-Sphere Let VN(X) = ~l<_i<j<_g log I~:~-1 x~t~'where x = (Xl, 9 9 9 XN), the xi are distinct points on the 2-sphere S 2 C ~3, and ]~i -- X3t] is the distance in R 3. Denote m i n x VN(X) b y VN. F i n d (Xl, . . . , XN) s u c h that VN(X) -- VN <-- c log N,

c a universal constant.

(2)

To 'Tmd" m e a n s to give an algorithm w h i c h on input N o u t p u t s distinct xt, 9 9 XN on the 2-sphere satisfying (2). To be p r e c i s e one c o u l d t a k e a real n u m b e r algorithm in the sense of BCSS (adjoining a square r o o t c o m p u t a t i o n ) with halting time p o l y n o m i a l in N. This p r o b l e m e m e r g e d from c o m p l e x i t y theory, jointly with Mike Shub [Shub a n d Smale, 1993]. It is m o t i v a t e d by finding a g o o d starting p o l y n o m i a l for a h o m o t o p y algorithm for realizing the F u n d a m e n t a l T h e o r e m of Algebra.

VOLUME20, NUMBER2, 1998 9

An ( x l , . . . , xN)= x such that VN(X)= VN is called an N-tuple of elliptic Fekete points (see [Tsuji, 1959]). The function VN as a function of N satisfies _

It is natural also to consider the functions VN(X,S) = i<j ~ ]lXi --1 Xj]]s'

VN,s = man VN(X,S),

x as before and 0 < s < 2. The original VN(X), VN correspond in a natural way to s = 0, and for s = 1, VN(X, 1) is the Coulomb potential, and VN(1) corresponds to an equilibrium position of N electrons constrained to lie on the two-sphere. I had asked Ed Saff for some help in dealing with the main problem above. Subsequently, he and his colleagues produced a number of f'me papers dealing with the subject and its ramifications. See [Kuijlaars and Saff, 1997] and [Saff and Kuijlaars, 1997] for background and fi~rther references. In [Rakhmanov, Saff, and Zhou, 1994], one can find numerical evidence (iV = 12,000) to support these problems. Another way of looking at our main problem here is to optimize the function WN(X) = (exp VN(X)) -1 : H I]Xi -- Xj] I. i<j However, as was written in [Shub and Smale, 1993], "... this may not be so easy since there are saddle points of index N (on a great circle in S 2, evenly spaced N points, xl . . . . , XN). Also the various symmetries that WN possesses will confuse the picture."

Problem 8: Introduction of Dynamics into Economic Theory The following problem is not one of pure mathematics, but lies on the interface of economics and mathematics. It has been solved only in quite limited situations. E x t e n d the m a t h e m a t i c a l model o f general e q u i l i b r i u m theory to include price adjustments. There is a (static) theory of equilibrium prices in economics starting with Walras and firmly grounded in the work of Arrow and Debreu (see [Debreu, 1959]). For the trivial case of one market this amounts to the equation "supply equals demand," and a natural dynamics is easily found [Samuelson, 1971]. For several markets, the situation is complex. There is a function called the excess demand, Z(p) = D(p) - SO)), from the space of prices to the space of commodities. Both demand D and supply S are defined by aggregation over the individual agents. Economics justifies conditions on individual behavior which lead to axioms on Z. These axioms for the excess demand map Z : Re+ ~ R e are:

1. Z()tp) = Z(p), all p = (Pl . . . . , Pe), P~ >- O, A E ~, ~ > O. 2. Z~= 1 piZi(P) = O, Walras's Law (the total value is zero). 3. Zi(p) > O ifpi = 0 (positive demand for a free good).

10

THE MATHEMATICALINTELLIGENCER

By (1), (2), (3), Z may be interpreted as a vector field on the intersection of the (2 - 1)-sphere with the positive orthant, pointing inward on the boundary. The existence of an equilibrium price vector pC follows from H o p f s theorem, so that Z(p*) = O, and "supply equals demand." Problem 8 asks for a dynamical model, whose states are price vectors (perhaps enlarged to include other economic variables). This theory should be compatible with the existing equilibrium theory. A most desirable feature is to have the time development of prices determined by the individual actions of economic agents. I worked on this problem for several years, feeling that it was the main problem of economic theory [Smale, 1976]. See also [Smale, 1981] for background.

Problem 9: The Linear Programming Problem Is there a p o l y n o m i a l - t i m e algorithm over the real n u m b e r s w h i c h decides the f e a s i b i l i t y o f the linear system of inequalities A x >_ b? The algorithm requested by this problem is that given by a real number machine in the sense of BCSS (see also Problem 3). The system Ax _> b has as input an m x n real matrixA and vector b E ~m, and the problem asks, is there some x ~ ~n with Z~= 1 aijxj >-- bi for all i = 1 , . . . , m? Time is measured by the number of arithmetic operations. This problem is in BCSS. This is a decision version of the optimization problem of linear programming: Given A, b as above and c E ~n, decide if m a x c.x x~ n

subject to Ax -> b

exists, and ff so, output such an x. The famous simplex method of Dantzig provides an algorithm for both problems (over ~), but Klee and Minty showed that it was exponentially slow in the worst case. On the other hand, Borgwardt and I, with subsequent important support from Haimovich, showed that it was polynomial time on the average. For all of these things, see [Schrijver, 1986]. In terms of the Turing model of computation, using rational numbers Q, and cost measured by "bits," there is a parallel development. Starting with ideas of Yudin and Nemirovsky, Khachian found a polynomial time algorithm (the ellipsoid method) for the linear programming problem. Subsequently Karmarkar with his "interior point method" found a practical algorithm for this problem, which he showed ran in polynomial time in the Turing model. For all these things one can see [Gr6tschel, Lovfisz, and Schrijver, 1993] as well as [Schrijver, 1986]. Closer to the main problem above over R is a similar problem asking for a "strongly polynomial algorithm" over Q for solving these linear programming problems. This notion of algorithm demands that the number of arithmetic operations as well as the number of bit operations be polynomial in the size of the input (m • (n + 1)). Partial

results are due to Megiddo and especially Tardos (see [GrStschel-Lovfisz-Schrijver, 1993]). For the problem over R there are also references [Barvinok-Vershik, 1993] and [Traub-Woz~liakowski, 1982].

Problem 10: The Closing Lemma Let p be a n o n - w a n d e r i n g p o i n t o f a d i f f e o m o r p h i s m S : M---)M o f a compact manifold. Can S be arbitrarily well a p p r o x i m a t e d w i t h derivatives o f order r (C ~a p p r o x i m a t i o n ) f o r each r, by T : M--) M so that p is a periodic p o i n t of T? A non-wandering point p ~ M is one with the property that for each neighborhood U of p there is a k E 7/such that S k U A U 4- 4). Here S k is the kth iterate of S; p is a periodic point of period m if Tm(p) = p. This is the discrete form of the famous "closing lemma," which in the CLcase has been solved affirmatively by Charles Pugh (1967). There is an easy C~ with the desired property. Peixoto observed that this argument failed for C tapproximations, correcting a mistake of Ren~ Thorn (which Ren~ has told me was his biggest mistake). Pugh and Robinson (1983) have proved the closing lemma with CLapproximations for the Hamiltonian version. Peixoto gave an affirmative answer with Cr-approximations (any r) for the circle, as well as the continuoustime version for orientable 2-dimensional manifolds. Recently the closing lemma has been given additional importance by the work of Hayashi (1997); see also [Wen-Xia, 1997].

Problem 11: Is One-Dimensional Dynamics Generally Hyperbolic? Can a complex p o l y n o m i a l T be a p p r o x i m a t e d by one o f the s a m e degree w i t h the property that every critical p o i n t tends to a periodic s i n k u n d e r iteration? This is unsolved even for polynomials of degree 2. Here a polynomial map T: C ---) C is considered a discrete dynamical system by iteration. So if z ~ C, its orbit in time, z = z0, Zl, z 2 , . . , is defined by zi = T(zi-1) and i may be interpreted as time (discrete). A fixed-point w of T (T(w) = w ) is a s i n k if the derivative T ' ( w ) of T at w has absolute value less than 1. A periodic sink of T of period p is a sink for Tp. A critical point of T is just a point where the derivative of T is zero. While the problem is now made precise, it is useful to see it in the framework of hyperbolic dynamics from the 1960's. A fkxed-point x of a diffeomorphism T : M--->M is hyperbolic if the derivative D T ( x ) of T at x (as a linear automorphism of the tangent space) has no eigenvalue of absolute value 1. If x is a periodic point of period p, then x is hyperbolic if it is a hyperbolic fixed-point of T p. The notion of hyperbolic extends naturally to ~, the closure of the set of non-wandering points (see Problem 10). A dynamical system T E Diff(M) is called hyperbolic (or

satisfies Axiom A) if the periodic points are dense in t2 and t2 is hyperbolic [Smale, 1967] or [Smale, 1980]. We assume also a no-cycle condition. The work of many people, especially Ricardo Mafi~, has identified hyperbolic dynamics with a strong notion of the stability of the dynamics called structural stability. There is even the beginning of a structure theory for this class of dynamics. While hyperbolic systems constitute a large set of dynamics, an even larger set, including applied chaotic dynamics, lies beyond. The concept of hyperbolicity extends from the invertible dynamics to the case of our problem above, polynomial maps from C to C. Classical complex variable theory permits recasting the problem to an equivalent one: Can a p o l y n o m i a l m a p T : C ---* C be a p p r o x i m a t e d by one w h i c h is hyperbolic? The theory of complex one-dimensional dynamics was begun by Fatou and Julia towards the beginning of this century. In the 1960's I asked my thesis student John Guckenheimer to look at this literature and try to solve the above problem (among other things). His thesis (see [Chern and Smale, 1970]) contains the affirmative answer, but with a gap in the proof. Now the problem stands open as the fundamental problem of one-dimensional dynamics. John's paper is one of many on the problems of this essay with a wrong proof, starting with Poincar& Complex 1-dimensional dynamics has become a flourishing subject and includes important contributions of Douady and Hubbard, Sullivan, Yoccoz, McMullen, among many others. See [McMullen, 1994]. There is a parallel field of real 1-dimensional dynamics of a smooth map T : I--->/, I = [0, 1]. Problem: Can a smooth m a p T : [0, 1] ---) [0, 1] be app r o x i m a t e d i n C r, a / / r > 1, by one w h i c h is hyperbolic? About the time of Guckenheimer's thesis, I asked Ziggy Nitecki to study this problem. My earlier negligence was compounded in not catching the mistake in Nitecki's thesis (see [Chern and Smale, 1970]), which purported to give an affirmative proof. Subsequently, Jakobson (1971) answered the problem for Cl-approximations, but the general case remains open. See [de Melo-van Strien, 1993] for background.

Problem 12: Centralizers of Diffeomorphisms Can a d i f f e o m o r p h i s m o f a compact m a n i f o l d M onto itself be Cr-approximated, all r >- 1, by one T: M - o M w h i c h c o m m u t e s w i t h only its iterates? Thus the centralizer of T in the group of diffeomorphisms, Diff(M) should be {T k I k E ~_}. I had started thinking about the centralizer in [Smale, 1963], but it was after Nancy Kopell's thesis with me (see [Chern and Smale, 1970]), answering the question affirmatively in case dim M = 1, that I proposed this problem

VOLUME 20, NUMBER 2, 1998

11

[Smale, 1967]. Today it remains unsolved even for 2-dimensional manifolds M. One may also ask if the set of diffeomorphisms of M with trivial centralizer is dense and open in Diff(M) with the C~-topology. The main work on these problems has been done by Palis and Yoccoz (1989), with almost complete answers in the case of hyperbolic dynamics (see Problem 11) for any manifold. I wrote in [Smale, 1991], "I find this problem interesting in that it gives some focus in the dark realm, beyond hyperbolicity, where even the problems are hard to pose clearly." Problem 13: Hilbert's 16th Problem Consider the differential equation i n [R2 dx dt - P(x,y),

dy = q(x,y), dt

= - 1 0 x + 10y

= y -f(x),

dt

dy

= -x,

(4)

dt

where f ( x ) is a real polynomial with leading term x 2k+ 1 and satisfying f(0) = 0. I f f ( x ) = x 2 - x, then (4) is van der Pors equation with one limit cycle. More generally, it can be easily shown that all the solutions of (4) circle around the unique equilibrium at (0,0) in a clockwise direction. By following these curves, one defines a "Poincar~ section," T : R + ---) R + where R + is the positive y-axis. The limit cycles of (4) are precisely the fixed-points of T. In various talks I raised the question of estimating the number of these fixed-points (via some new kind of fixed-point theorem?). In response, Linz, de Melo, and Pugh (1977) found examples with k different limit cycles and

12

THE MATHEMATICALINTELUGENCER

~] = 28x - y - x z = - 3 z + xy.

(3)

This is a modern version of the second half of Hilbert's 16th Problem. Except for the Riemann hypothesis, it seems to be the most elusive of Hilbert's problems. In fact, since a paper of Petrovskfi and Landis (1957) purporting to give a positive solution, the progress seems to be backwards. Earlier, Dulac (1923) claimed that the system (3) always has a fmite number of limit cycles. After a gap in Petrovskii and Landis was found (see [Petrovskfi and Landis, 1959]), Ilyashenko (1985) found an error in Dulac's paper. Moreover Shi Songling (1982) found a counterexample to the specific bounds of Petrovskii-Landis for the case d = 2. Subsequently, two long works have appeared, independently, giving proofs of Dulac's assertion [l~calle, 1992] and [Ilyashenko, 1991]. These two papers have yet to be thoroughly digested by the mathematical community. Thus one has the fmiteness, but no bounds. We will consider a special class where the finiteness is simple, but the bounds remain unproved. The following corresponds to Li~nard's equation (see, e.g., [Hirsch and Smale, 1974]) dx

Problem 14: Lorenz Attractor Is the d y n a m i c s o f the o r d i n a r y differential equations o f L o r e n z that o f the geometric L o r e n z attractor o f Williams, Guckenheimer, and Yorke?

The Lorenz equations are:

where P and Q are p o l y n o m i a l s . Is there a bound K on the n u m b e r o f l i m i t cycles o f the f o r m K <- d q, where d is the m a x i m u m o f the degrees o f P and Q, and q is a u n i v e r s a l constant?

--

conjectured this number k for the upper bound. Still no upper bound of the form (degf)q has been found. Because T is analytic, (4) has a fmite number of limit cycles for eachf. For more background see [Browder, 1976], [Ilyashenko and Yakovenko, 1995], [Lloyd and Lynch, 1988], and [Smale, 1991].

Lorenz (1963) analysed these equations by computer to fmd that most solutions tended to a certain attracting set, and in so doing, he produced an important early example of "chaos." However, mathematical proofs were lacking. This numerical work inspired the rigorous mathematical development of a geometrically defined ordinary differential equation which seems to have the same behavior (Yorke, [Williams, 1979], [Guckenheimer and Williams, 1979]). This geometric attractor has been analysed in detail and one can fairly say that it is understood. Problem 14 asks if the dynamics of the original equations is the same as that of the geometric model. The most complete positive answer would be to describe a homeomorphism of R 3 to R 3 which would take solutions of the Lorenz equations to solutions (a member of the family) of the geometric attractor---or rather, of a member of the twoparameter family of geometric attractors. An answer to this problem would be a step in establishing foundations for the field of applied dynamical systems. Up to the present time, in the equations of engineering and physics, chaos has only been established in a weaker sense, that of proving the existence of horseshoes (e.g., Melnikov, Marsden, and Holmes; see [Guckenheimer and Holmes, 1990]). Geometric, structurally stable, chaotic attractors in dynamics are in my paper [Smale, 1967]. But these did not arise from any physical system. Some partial results on problem 14 are due to Rychlik and Robinson, see [Robinson, 1989]. Problem 15: Navier-Stokes Equations Do the Navier-Stokes equations on a 3-dimensional dom a i n l'l i n R 3 have a u n i q u e s m o o t h solution f o r all time?

This is perhaps the most celebrated problem in partial differential equations. Let us be a little more precise. The Navier-Stokes equations may be written oqU

--

at

+ ( u . V ) u - v A u + grad p = 0,

div u = 0,

where a C~-map u : ~+ • 1~/--~ ~ 3 a n d p : ~ ---) R are to be found satisfying these equations, u prescribed at t = 0 and on the boundary 0~. Here R+ = [0,~), u.V is the operator ~13 u z.~0x/' and v is a positive constant. See, e.g., [ChorinMarsden, 1993] for details. Many mathematicians have contributed toward the understanding of this problem. An affirmative answer has been given in dimension 2 and in dimension 3 for t in a small interval [0, T]. See [Temam, 1979] for background. The solution of this problem might well be a fundamental step toward the very big problem of understanding turbulence. For example, it could help realize the ideas of [Ruelle and Takens 1971], which put the notion of a chaotic attractor into a model of turbulence. See also [Chorin, Marsden, and Smale, 1977]. In [Smale, 1991] I asked if the solutions of the 2-dimensional Navier-Stokes equation with a forcing term on a torus must converge to an equilibrium as time tends to infinity. Babik and Vishik (1983) had given some evidence to the contrary. Subsequently Liu (1992) provided examples to show convergence to a more complicated attractor.

Problem 16: The Jacobian Conjecture Suppose f : C n --> C n is a p o l y n o m i a l m a p w i t h the p r o p e r t y that the d e r i v a t i v e at each p o i n t is n o n - s i n gular. Then m u s t f be one-to-one?

Here f ( z ) = ( f l ( z ) , . . . , f n ( z ) ) , and eachj~ is a polynomial in n variables. The derivative of f at z, D f ( z ) : C n --) C n, may be thought of as the matrix of partial derivatives, and the non-singularity condition as Det D f ( z ) r O. I f f is indeed injective, then it is surjective and has an inverse which is a polynomial map. For an elementary proof of this see [Rudin, 1995]. The problem goes back to the 1930's. See the excellent surveys [Bass, Connell, and Wright, 1982] and [van den Essen, 1997] for the importance, background, and related results.

Problem 17: Solving Polynomial Equations Can a zero o f n c o m p l e x p o l y n o m i a l equations i n n u n k n o w n s be f o u n d a p p r o x i m a t e l y , on the average, i n p o l y n o m i a l t i m e w i t h a u n i f o r m algorithm?

The final theorem in my five-paper series with Mike Shub [Shub and Smale, 1994] is exactly this result without the word "uniform." I review the definitions. Considerf : cn---->Cn,f(z) = ( f l ( z ) , ... ,f~(z)), z = (zl, 9 Zn) where eachfi is a polynomial in n variables of degree di. It is reasonable to make theJ~ into homogeneous polynomials by adding a new variable z0, work in the corresponding projective space, and then translate the algorithm and results back to the initial affine problem. "Approximately" can be defined intrinsically using Newton's method, and is necessary because of Abel, Galois, et al. Time is measured by the number of arithmetic operations and comparisons, -<, using real machines (as in Problem 3), if one wants to be formal.

A probability measure must be put on the space of all such f, for each d = (dl,. 9 9 dn), and the time of an algorithm is averaged over the space offl Is there such an algorithm where the average time is bounded by a polynomial in the number of coefficients o f f (the input size)? In [Shub and Smale, 1994] it is proved that this can be done, but the algorithm is different for each d and even for each desired probability of success. A uniform algorithm is one that is independent of d (d is part of the input). Certainly finding zeros of polynomials and polynomial systems is one of the oldest and most central problems of mathematics. This problem asks if under some conditions, specified in the problem, it can be solved systematically by computers. If there is no polynomial-time way of doing it, then no computer will ever succeed. There is a more recent development putting the problem of zeros of polynomials into a universal role. The Hilbert Nullstellensatz (as a decision problem) is NP-complete over any field (see Problem 3). A similar, more difficult, problem may be raised for the real numbers.

Problem 18: Limits of Intelligence What are the l i m i t s o f intelligence, both a r t i f i c i a l and human?

Penrose (1991) attempts to show some limitations of artificial intelligence. His argumentation brings in the interesting question whether the Mandelbrot set is decidable (dealt with in [Blum and Smale, 1993]) and implications of the GSdel incompleteness theorem. However, a broader study is called for, one which involves deeper models of the brain, and of the computer, in a search of what artificial and human intelligence have in common, and how they differ. I would look in a direction where learning, problem-solving, and game theory play a substantial role, together with the mathematics of real numbers, approximations, probability, and geometry. I hope to expand on these thoughts on another occasion.

REFERENCES Abraham, R. and Marsden, J. (1978). Foundations of Mechanics. Addison-Wesley Publishing Co., Reading, Mass. Babin, A.V. and Vishik, M.I. (1983). Attractors of partial differential evolution equations and their dimension. Russian Math. Surveys 38, 151213. Barvinok, A. and Vershik, A. (1993). Polynomial-time, computable approximation of families of semi-algebraic sets and combinatorial complexity. Amer. Math. Soc. Trans. 155, 1-17. Bass, H., Connell, E., and Wright, D. (1982)~The Jacobian conjecture: reduction on degree and formal expansion of the inverse. Bull. Amer. Math. Soc. (2) 7, 287-330. BCSS: Blum, L., Cucker, F., Shub, M., and Smale, S. (1997). Complexity and Real Computation, Springer-Verlag. Blum, L., Shub, M., and Smale, S. (1989). On a theory of computation and complexity over the real numbers: NP-completness, recursive

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functions and universal machines. Bulletin of the Amer. Math. Soc. (2) 21, 1-46. Blum, L. and Smale, S. (1993). The G6del incompleteness theorem and decidability over a ring. Pages 321-339 in M. Hirsch, J. Marsden, and M. Shub (editors), From Topology to Computation: Proceedings of the Smalefest, Springer-Verlag. Browder, F. (ed.), (1976). Mathematical Developments Arising from Hilbert Problems, American Mathematical Society, Providence, RI. Brownawell, W. (1987). Bounds for the degrees in the Nullstellensatz. Annals of Math. 126, 577-591. Chern, S. and Smale, S. (eds.) (1970). Proceedings of the Symposium on Pure Mathematics, vol. XIV, American Mathematical Society, Providence, RI. Chorin, A., and Marsden, J. (1993). A Mathematical Introduction to Fluid Mechanics, 3rd edition, Springer-Verlag, New York. Chorin, A., Marsden, J., and Smale, S. (1977). Turbulence Seminar, Berkeley 1976-77, Lecture Notes in Math. 61,5,Springer-Verlag, New York. Cucker, F., Koiran, P., and Smale, S. (1997). A polynomial time algorithm for Diophantine equations in one variable. To appear. Debreu, G. (1959). Theory of Value, John Wiley & Sons, New York. Dulac, H. (1923). Sur les cycles limites. Bull. Soc. Math. France 51, 45-188. Fcalle, J. (1992). Introduction aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac. Hermann, Paris. van den Essen, A. (1997). Polynomial automorphisms and the Jacobian conjecture, in Algebre non commutative, groupes quantiques et invariants, septieme contact Franco-Belge, Reims, Juin 1995, eds. J. Alev and G. Cauchon, Soci6t6 math~matique de France, Paris. Freedman, M. (1982). The topology of 4-manifolds. J. Diff. Geom. 17, 357-454. Garey, M. and Johnson, D. (1979). Computers and Intractability, Freeman, San Francisco. Gr6tschel, M., Lov&sz, L., and Schrijver, A. (1993). Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, New York. Guckenheimer, J. and Holmes, P. (1990). Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, third printing, Springer-Verlag, New York. Guckenheimer, J. and Williams, R.F. (1979). Structural stability of Lorenz attractors. Publ. Math. IHES 50, 59-72. Hayashi, S. (1997). Connecting invariant manifolds and the solution of the C 1-stabilityconjecture and Q,-stabilityconjecture for flows. Annals of Math. 145, 81-137. Hirsch, M. and Smale, S. (1974). Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York. Ilyashenko, J. (1985). Dulac's memoir "On limit cycles" and related problems of the local theory of differential equations. Russian Math. Surveys VHO, 1-49. Ilyashenko, Yu. (1991). Finiteness Theorems for Limit Cycles, American Mathematical Society, Providence, RI. Ilyashenko, Yu. and Yakovenko, S. (1995). Concerning the Hilbert 16th problem. AMS Translations, series 2, vol. 165, AMS, Providence, RI. Jakobson, M. (1971). On smooth mappings of the circle onto itself. Math. USSR Sb. 14, 161-185. Kuijlaars, A.B.J. and Saff, E.B. (1997). Asymptotics for minimal discrete energy on the sphere. Trans. Amer. Math. Soc., to appear.

14

THE MATHEMATICALINTELLIGENCER

Kuz'mina, R., (1977). An upper bound for the number of central configurations in the plane n-body problem. Sov. Math. Dok. 18, 818-821. Lang, S. (1991). Number Theory III, vol. 60 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, New York. Linz, A., de Melo, W., and Pugh, C. (1977), in Geometry and Topology, Lecture Notes in Math. 597, Springer-Verlag, New York. Liu, V. (1992). An example of instability for the Navier-Stokes equations on the 2-dimensional torus. Commun. PDE 17, 1995-2012. Lloyd, N.G. and Lynch, S. (1988). Small amplitude limit cycles of certain Lienard systems. Proceedings Roy. Soc. London 418, 199-208. Lorenz, E. (1963). Deterministic non-periodic flow. J. Atmosph. Sci. 20, 130-141. Manders, K.L. and Adleman, L. (1978). NP-complete decision problems for binary quadratics. J. Comput. System ScL 16, 168-184. McMullen, C. (1994). Frontiers in complex dynamics. Bull. Amer. Math. Soc. (2) 31, 155-172. de Melo, W. and van Strien, S. (1993). One-Dimensional Dynamics. Springer-Veriag, New York. Palls, J. and Yoccoz, J.C. (1989). (1) Rigidity of centralizers of diffeomorphisms. Ann. Scient. Ecole Normale Sup. 22., 81-98; (2) Centralizer of Anosov diffeomorphisms. Ann. Scient. Ecole Normale Sup. 22, 99-108. Palmore, J. (1976). Measure of degenerative relative equilibria. I. Annals of Math. 104, 421-429. Petrovskif, I.G. and Landis, E.M. (1957). On the number of limit cycles of the equation dy/dx = P(x,y)/Q(x,y), where P and Q are polynomials. Mat. Sb. N.S. 4,3 (85), 149-168 (in Russian), and (1960)Amer. Math. Soc. TransL (2) 14, 181-200. Petrovskii', I.G. and Landis, E.M. (1959). Corrections to the articles "On the number of limit cycles of the equation dy/dx = P(x,y)/Q(x,y), where P and Q are polynomials." Mat. Sb. N.S. 48 (90), 255-263 (in Russian) Penrose, R. (1991). The Emperor's New Mind, Penguin Books. Poincar6, H. (1953). Oeuvres, Vl. Gauthier-Villars, Paris. Deuxieme Compl~ment & L'Analysis Situs. Peixoto, M. (1962). Structural stability on two-dimensional manifolds. Topology 1, 101-120. Pugh, C. (1967). An improved closing lemma and a general density theorem. Amer. J. Math. 89, 1010-1022. Pugh, C. and Robinson, C. (1983). The C 1 closing lemma including Hamiltonians. Ergod. Theory Dynam. Systems 3, 261-313. Rakhmanov, E.A., Saff, E.B., and Zhou, Y.M. (1994). Minimal discrete energy on the sphere. Math. Res. Lett. 1,647-662. Robinson, C. (1989). Homoclinic bifurcation to a transitive attractor of Lorenz type. Non-linearity 2, 495-518. Rudin, W. (1995). Injective polynomial maps are automorphisms. Amer. Math. Monthly 102, 540-543. Ruelle, D. and Takens, F. (1971). On the nature of turbulence. Commun. Math. Phys. 20, 167-192. Samuelson, P. (1971). Foundations of Economic Analysis, Atheneum, New York. Saff, E. and Kuijlaars, A. (1997). Distributing many points on a sphere. Mathematical Intelligencer 10, 5-11. Schrijver, A. (1986). Theory of Linear and Integer Programming, John Wiley & Sons. Shi, S. (1982). On limit cycles of plane quadratic systems. Sci. Sin. 25, 41-50.

Shub, M. (1970). Appendix to Smale's paper: Diagrams and relative equilibria in manifolds, Amsterdam, 1970. Lecture Notes in Math. 197, Springer-Verlag, New York. Shub, M. and Smale, S. (1993). Complexity of Bezout's theorem, II1: condition number and packing. J. of Complexity 9, 4-14. Shub, M. and Smale, S. (1994). Complexity of Bezout's theorem, V: polynomial time. Theoret. Comp. Sci. 133, 141-164. Shub, M. and Smale, S. (1995). On the intractibility of Hilbert's Nullstellensatz and an algebraic version of "P = NP", Duke Math. J. 81, 47-54. Smale, S. (1963). Dynamical systems and the topological conjugacy problem for diffeomorphisms, pages 490-496 in: Proceedings of the International Congress of Mathematicians, Inst. Mittag-Leffler, Sweden, 1962. (V. Stenstr6m, ed.) Smale, S. (1963). A survey of some recent developments in differential topology. Bull. Amer. Math. Soc. 69, 131-146. Smale, S. (1967). Differentiable dynamical systems. Bull. Amer. Math. Soc. 73, 747-817. Smale, S. (1970). Topology and mechanics, I and II. Invent. Math. 10, 305-331 and Invent. Math. 11, 45-64. Smale, S. (1976). Dynamics in general equilibrium theory. Amer. Economic Review 66, 288-294. Smale, S. (1980). Mathematics of Time, Springer-Verlag, New York. Smale, S. (1981). Global analysis and economics, pages 331-370 in Handbook of Mathematical Economics 1, editors K.J. Arrow and M.D. Intrilligator. North-Holland, Amsterdam. Smale, S. (1990). The story of the higher-dimensional Poincar~ conjecture. Mathematical Intelligencer 12, no. 2, 40-51. Also in M. Hirsch, J. Marsden, and M. Shub, editors, From Topology to Computation: Proceedings of the Smalefest, 281-301 (1992). Smale, S. (1991). Dynamics retrospective: great problems, attempts that failed. Physica D 51,267-273. Taubes, G. (July 1987). What happens when Hubris meets Nemesis? Discover. Temam, R. (1979). Navier-Stokes Equations, revised edition, NorthHolland, Amsterdam. Traub, J. and Wo~iakowski, H. (1982). Complexity of linear programming. Oper. Res. Letts. 1, 59-62.

Tsuji, M. (1959). Potential Theory in Modem Function Theory, Maruzen Co., Ltd., Tokyo. Wen, L. and Xia, Z. (1997). A simpler proof of the C ~ connecting lemma. To appear. Williams, R. (1979). The structure of Lorenz attractors. Publ. IHES 50, 101-152. Wintner, A. (1941). The Analytical Foundations of Celestial Mechanics. Princeton University Press, Princeton, NJ.

VOLUME 20, NUMBER 2, 1998

15

BERND KAWOHL

Symmetry or Not?* ~

f

~

any problems in analysis appear symmetric, yet in fact their solutions are sometimes nonsymmetric. I shall present a number of exramples for this. One of them is the opaque square problem: Given a unit square,find the shortest curve C in the square such that every

line that intersects the square also intersects the curve C. Another one is Newton's body of minimal resistance, one of the oldest problems in the calculus of variations. I present also a collection of eight tricks on how to prove symmetry for solutions of PDE or of variational problems; and one of the tricks solves a problem left open in [2]. No S y m m e t r y Example 1. Cylinders of Constant Thickness Imagine that a producer of horizontal circular objects like pizzas, buttons, or CD's wants to monitor their circular

shape by an optical reading device, which measures the length of their horizontal shadows from all possible angles. For circular objects all these shadows are equally long, but are circles the only objects having this property? Intuition might suggest this, however the answer is definitely no, as Figure 1 shows. A regular (2N + 1)-gon is "rounded" by circular arcs of suitable radius. Mechanical engineers know the left part of Figure l a from the Wankel engine; and the shape corresponding to N = 3 is used for the 50p coin in Britain. Curves bounding such cross sections have long been known as curves of constant breadth, see [23].

Figure 1. (a) C u r v e s o f constant breadth

* Dedicated to Wolfgang Wendland on the occasion of his 60th birthday, September 1996.

16

THE MATHEMATICALINTELLIGENCER9 1998 SPRINGER-VERLAGNEWYORK

(b) P r o o f o f an isoperimetric result.

Incidentally, the British Mint is saving material b y shaping coins in a n o n c i r c u l a r way. The following result has a p r o o f w h i c h I c a n n o t resist presenting. A m o n g all plane d o m a i n s o f outer d i a m e t e r 1 the circular one has m a x i m a l area. F o r the p r o o f [21] one o b s e r v e s that d o m a i n s o f maxim a l a r e a will b e convex. Then one l o o k s at Figure l b and c o n c l u d e s swiftly as follows area = 2 .0

r2(0) + r 2 0

dO,

b u t for all 0 the integrand is Op2 + OQ2 = pQ2 <_ 1. Example 2. Newton's Body of Minimal Resistance:

S u p p o s e you w a n t to s e n d a b o d y t h r o u g h a rarefied liquid. The p a r t i c l e s o f the liquid have a large m e a n free-path length a n d do n o t interact with each other. They may, however, b o u n c e into the b o d y and undergo a perfectly elastic shock, in w h i c h case s o m e of their m o m e n t u m is u s e d up to slow the b o d y down. If the b o d y ' s front surface is par a m e t r i z e d as a function u ( x ) with x E B(0,1) C R 2, w h e r e B(0, 1) d e n o t e s the unit disk, the r e s i s t a n c e of the b o d y can be m e a s u r e d b y the functional R ( u ) :=

1 (o,1) 1 + IUul 2 dx.

This p r o b l e m was s t u d i e d m o r e t h a n 300 years ago b y Newton, see [6,8]. One can discuss various classes of admissible functions u, over w h i c h the n o n c o n v e x functional R should be minimized. A class of functions for w h i c h one can p r o v e the e x i s t e n c e o f a minimizer is 1,co

A := {v E Wloc (B(0,1)) I 0 -< v(x) <--M; v concave].

Isaac N e w t o n minimized R o v e r radially s y m m e t r i c functions from A, b u t is the minimizer o v e r A n e c e s s a r i l y radially s y m m e t r i c ? I tried to p r o v e this with m y c o a u t h o r s [8] and m a n y a t t e m p t s failed; at last w e could prove t h e opposite [6]. F o r a p r o o f w e p e r t u r b e d the circular level sets of N e w t o n ' s solutions to regular p o l y g o n s with a large numb e r of c o r n e r s a n d s h o w e d that this d e c r e a s e s R. We a d d e d an angular derivative, so to speak, to the gradient. This inc r e a s e d the gradient a n d d e c r e a s e d R. Example 3. The Opaque Square

The o w n e r of a square p i e c e of land, s a y of unit size, k n o w s that a straight t e l e p h o n e line is running a c r o s s his land at a d e p t h o f 30 cm. To fmd t h e line, he c a n dig a ditch (of total length 4) o n c e a r o u n d the square. But digging up three sides of the square will also u n c o v e r the t e l e p h o n e line. w h a t is the s h o r t e s t ditch C that will serve this p u r p o s e ? After a little thought m a n y p e o p l e n o t i c e that two diagonals (total length 2.82 . . . ) are shorter; and those r e a d e r s w h o a r e familiar with minimal s u r f a c e s will f'md the shortest p a t h c o n n e c t i n g all four c o r n e r s in Figure 2 (of total length 2.73). The angle 27r/3 at w h i c h the line s e g m e n t s m e e t can also b e found in the h e x a g o n a l s h a p e of honeycombs.

Figure 2. Shortest path connecting four corners.

But an even b e t t e r ditch is known. It h a s t w o c o n n e c t e d c o m p o n e n t s ; its total length 2.64, is b e l i e v e d to b e optimal. It has only one reflection symmetry, while the square has four.

i

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~

~

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',

: ! !

'I,

:/

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I

I

: I

s"

.-

I s"

."

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./ #

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Figure 3. Conjectured shortest path for a square.

What if the square is r e p l a c e d by a regular p o l y g o n with N corners? What i f N - ~ o~? Then the p o l y g o n converges to a circle, s a y of r a d i u s 1. N o w intuition tells us that to unc o v e r a straight t e l e p h o n e line crossing the circle w e have to dig a ditch o f length 2~r once a r o u n d the circle. But a ditch of length ~r + 2 will do the s a m e trick. It is o b t a i n e d after a limiting p r o c e s s in Figure 4. F o r m o r e details I refer to [19].

VOLUME 20, NUMBER 2, 1998

17

i

/

II

C0

/

I'

II

I

I I

/ I

F o r any fixed t > 0 there is a unique spatial m a x i m u m point of u, the hot spot, see [18]. Suppose this hot spot is stationary and sits at the origin. Then w e m a y e x p e c t the h e a t to flow with the s a m e rate in direction x as in the opposite direction - x . Consequently &2 should be equal to -s or, equivalently, ~ should be centrosymmetric. Thus it was asked [20] w h e t h e r only centrosymmetrie d o m a i n s could have a stationary h o t spot. A c o u n t e r e x a m p l e in ~3 is provided b y a regular tetrahedron, see also [16,18] for related results.

/

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Example 6. Nonsymmetric Ground States

~ l //

/

I II /I Ii ii / / / 11 I I,',' / .'" ff p t / 1/ // 9 !,',"..

~

PN Figure 4. Conjectured Shortest Path for an N-gon.

Example 4. Eigenfunctions

Let s b e a circular disk in ~2, a n d c o n s i d e r the eigenvalue problem

hu + h u = O in~, u = 0 on OD_,.

mize f , {]Vv] 2 + a[v] 2} d x on the set (1)

The p r o b l e m is symmetric, the Laplace o p e r a t o r is invariant u n d e r rotations, but the solutions need not be radial. Of c o u r s e the first eigenfunction o r "ground state" is radial, as is well known. This follows from various arguments which will b e mentioned in the s e c o n d p a r t of m y paper. A ground state is a function u that minimizes the Dirichlet integral fa I~u] 2 d x on the s e t A : = {u 9 H~(s ]fa u2 d x = 1}. Problem (1) is a crude approximation to a vibrating elastic membrane. What if the two-dimensional m e m b r a n e is replaced b y a three-dimensional linearly elastic ball? Then the deformation u is vector-valued, the Laplace operator is replaced b y the Lam6 operator, and (1) is replaced by the elliptic system h u 4- a grad div u 4- A u = 0 u = 0

THE MATHEMATICAL INTELLIGENCER

Example 7. Poincarb-type Inequalities

Consider t h e one-dimensional p r o b l e m of finding the b e s t c o n s t a n t Cp in t h e estimate

IlvllLp<-l,,)-- c~ IEv'IIL~(-1,,) for v E H i ( - 1 , 1 ) with

v(x)dzc = 0. 1

Example 5. Stationary Hot Spot

18

The a s s o c i a t e d Euler equation r e a d s - A u + au = h]ulP-tu in ~ . Is the minimizer u radial? If ~ is a ball, this is true, and one can prove it for instance through symmetrization. But ff ~ C R 2 is an annulus [10] or ~ C ~'~, n >-- 3 is a spherical shell [3], this c a n b e false. See also [t4, p. 95 and 99] and references therein, o r [15, 19] for the c a s e o f systems.

on 0~,

Imagine a c o n v e x b o d y ~ C R 3 at uniform initial t e m p e r a t u r e Uo(X) ~- 1 for x 9 ~ d i p p e d into ice water. A c c o r d i n g to the laws of heat conduction, its t e m p e r a t u r e will d e v e l o p a s p a t i a l n o n s t a t i o n a r y profile a n d will eventually go d o w n to zero. In o t h e r w o r d s u(t,x) solves in R+ x s one+ xa~, in l~.

a := Iv e H i ( a ) I fo Ivlp+' dx = 1}.

ins

with a = (h +/z)//z o b t a i n e d from the Lain6 c o n s t a n t s h a n d / z . Again one can define g r o u n d states, this time as minimizers of f a ([ Vu[ 2 + a (div u ) 2) d x on the c o r r e s p o n d i n g set A of vector-valued functions. Are g r o u n d s t a t e s still radial, i.e., do they d e p e n d on r only, or can t h e y b e repres e n t e d as u ( x ) = xg(r) with a s c a l a r function g? To b o t h questions the a n s w e r is negative, as was s h o w n in [15].

ut-hu=O u=0 u(O,x) = 1

We have s e e n in E x a m p l e 4 that one m a y e x p e c t scalar-valu e d g r o u n d s t a t e s to be s y m m e t r i c if the g e o m e t r y allows t h e m to b e so. E v e n this e x p e c t a t i o n m u s t b e t a k e n with a grain of salt, b e c a u s e M. E s t e b a n h a s e x h i b i t e d nonsymmetric g r o u n d s t a t e s on e x t e r i o r s y m m e t r i c d o m a i n s in [11]. In h e r c a s e g r o u n d states w e r e minimizers of the functional fa {]VU]2 4- U2} dx o n A : = {v E H i ( a ) ] J'a ]vlP+l dx = 1},withn--3andl 0; and mini-

(2)

The e x i s t e n c e of Cp, and of a c o r r e s p o n d i n g function Up for w h i c h equality occurs, is a simple e x e r c i s e in the calculus o f variations. If p = 2, the b e s t c o n s t a n t is a c h i e v e d for the (odd) function u2 = sin(~rx/2) and C2 = (2/~r) e --~ 0.4057. Notice that oddness, i.e., u2(x) = - u 2 ( - x ) , is s o m e sort o f symmetry. What i f p = ~? Is the function t h a t l e a d s to Ca (i.e., the function for w h i c h the a b o v e inequality is an equality) still an o d d function? By m o n o t o n e r e a r r a n g e m e n t m e t h o d s as in [14] one can s h o w that a function u w h i c h minimizes the quotient IIv'l1241vll~m u s t be m o n o t o n e . Without loss o f generality w e m a y a s s u m e that 9 it is m o n o t o n e increasing ( o t h e r w i s e r e p l a c e u ( z ) by

u(-x)), 9 it attains its m a x i m u m m o d u l u s at 1 (otherwise r e p l a c e u ( x ) by - u ( - x ) ) , 9 Ilull = u ( 1 ) = 1 (otherwise rescale).

So u has e x a c t l y one zero in ( - 1 , 1 ) , b u t this zero is not located, as one might expect, at the origin, and u is n o t odd! In fact, minimizing Ilu'll 2 on the m a n i f o l d of functions with m e a n value zero formally l e a d s to the Euler equation u " + A = 0, so t h a t a minimizer is e x p e c t e d to be a second-degree polynomial. Then a fmite-dimensional variational analysis identifies a minimizer u ~ a s u ~ ( x ) = (3x 2 + 6x 1)/8. This function is definitely n o t odd, see Figure 5. The b e s t c o n s t a n t C~ is t h e r e f o r e Ca = X/-2~ ~ 0.8165. Notice that m o n o t o n e r e a r r a n g e m e n t is n o t n e c e s s a r y for the proof. One can for i n s t a n c e use p e r i o d i c continuations of u a n d shifts of x to s h o w that w i t h o u t loss of generality w e may assume : u(1) = 1.

[lu[l

Method 2. Symmetry via Convexity

S u p p o s e u minimizes a strictly c o n v e x functional J ( v ) on a c o n v e x set o f a d m i s s i b l e functions v. M o r e o v e r v is defined on a s y m m e t r i c set ~ , i.e., 12 is invariant u n d e r s o m e group action. I f g is an e l e m e n t o f the group, g(12) = 12 and consequently u ( x ) = u(g(x)), i.e., u is invariant u n d e r the group action; o t h e r w i s e the c o n v e x c o m b i n a t i o n w ( x ) = [u(x) + u ( g ( x ) ) ] / 2 w o u l d have s m a l l e r "energy" J ( w ) < J ( u ) , a contradiction. It is easily s e e n that Method 2 is less general than Method 1. Method 3. Symmetry via R e a r r a n g e m e n t

S u p p o s e u minimizes a functional of t y p e J ( v ) = fa {IVvl2 + F(v)} d x

U~

1

Figure 5. The Extremal u~(x).

This n o n s y m m e t r y r e s u l t w a s b r o u g h t to m y a t t e n t i o n b y C. S c h w a b and is p r o v e d in [13] w i t h o u t r e f e r e n c e to m o n o t o n e r e a r r a n g e m e n t . A f t e r w a r d s F. Brock, H.G. Reschke, a n d G. Talenti e a c h f o u n d o t h e r ( u n p u b l i s h e d ) d e r i v a t i o n s i n d e p e n d e n t l y a n d c o m m u n i c a t e d t h e m to me. W h a t h a p p e n s f o r p E (2,oo)? Is t h e e x t r e m a i f u n c t i o n odd? I b e l i e v e it is not, b u t this a p p e a r s to be an o p e n p r o b lem. Symmetry

After seeing so m a n y c a s e s o f n o n s y m m e t r y I hope that the r e a d e r is c o n v i n c e d that s y m m e t r y p r o p e r t i e s of solutions n e e d a proof. There are n u m e r o u s w a y s to prove s y m m e try. Let m e list s o m e of t h e m and illustrate t h e m for v e r y simple e x a m p l e s . Of c o u r s e t h e s e m e t h o d s apply to m o r e general p r o b l e m s , b u t p r e s e n t i n g t h e m in the m o s t general c o n t e x t w o u l d only o b s c u r e the b a s i c ideas. M e t h o d 1. Symmetry via U n i q u e n e s s

S u p p o s e a p r o b l e m like (1) is invariant u n d e r s o m e nontrivial (reflection or rotation) group action, b u t its solution is unique. Then the solution is symmetric. Otherwise t h e r e w o u l d be m o r e than one solution.

(3)

on a certain set A of admissible functions v which are defmed on a Steiner-symmetric d o m a i n 12 C R n. These are domains which have a reflection s y m m e t r y and are c o n v e x in directions orthogonai to the plane P of symmetry. Is every global minimizer u Steiner-symmetric, i.e., does u symmetrically d e c r e a s e as w e move a w a y from P? The a n s w e r is in general positive, even for m o r e general functionals. F o r details I refer to m y m o n o g r a p h [14]. The i d e a is to deform o r rearrange a n o n s y m m e t r i c alleged minimizer into a symmetrically decreasing function u* with s m a l l e r energy. Of course this m e t h o d w o r k s only if u E A implies u* E A. The s a m e question c a n be r a i s e d for local minimizers of functionais. Then the p r o o f s are m u c h harder, b e c a u s e the r e a r r a n g e m e n t u* o f a n o n s y m m e t r i c function c a n b e far a w a y in function s p a c e from the original function u, a n d so the two m u s t b e c o n n e c t e d b y s o m e s o r t of homotopy. This c o n s t r u c t i o n w a s successfully p e r f o r m e d b y F. B r o c k in [4] by a variational v e r s i o n o f the so-called moving plane method, s e e also [7] for further extensions. Method 4. Symmetry via the M a x i m u m Principle I

F o r simplicity of exposition, c o n s i d e r a s e m i l i n e a r elliptic p r o b l e m on the unit ball B(0,1) in En: - Au +f(u)

= 0

u > 0 u = 0

in B ( 0 , 1 ) ,

in B(0,1), on OB(O,1).

(4)

I f f satisfies certain regularity a s s u m p t i o n s , if it is for ins t a n c e Lipschitz continuous, t h e n u is k n o w n to be radially symmetric. This is the f a m o u s result of [12], which is b a s e d on showing t h a t u < Ur in every s p h e r i c a l cap Ct = B(0,1) n {Xl > t} with t E (0,1) containing the right endp o i n t of B(0,1) b u t n o t its center. Here uT is the value of u in the reflected cap: Ur (xl, 9 9 .) = u ( 2 t - x l , 9 9 .). Let us see w h a t happens. By the mean-value t h e o r e m - h ( u - u r ) + f ' ( ( z ( x ) ) ( u - u~) = 0 for suitable z~ in Ct and u - u~ < 0 on OCt. Moreover, if Ct is very small, w = u - Ur <- 0 in Ct. We w a n t to s h o w w < 0 in Ct, b e c a u s e then we can c o n c l u d e that w -< 0 in Ct-~. To this end, set c ( x ) := f ' ( ~ t ( x ) ) a n d n o t e that -Aw

+ c+(x)w = -c-(x)w

<- 0

in Ct.

VOLUME 20, NUMBER 2, t998

19

Here c+(x) := m a x {c(x),0} a n d c - ( x ) := min {c(x),0}. N o w the m a x i m u m principle implies w < 0 in Ct, and by a continnity argument w e can p u s h t d o w n to zero, at w h i c h p o i n t w --= 0. This trick is k n o w n as the moving plane method. It is v e r y general and l e a d s to m a n y o t h e r interesting results also in n o n s y m m e t r i c settings. F o r a related variational m e t h o d see [4]. Without Lipschitz continuity of f the m e t h o d falls. C o u n t e r e x a m p l e s s e e m to be w e l l - k n o w n a m o n g s o m e e x p e r t s in this field. F o r n = 1 one can find a c o u n t e r e x a m p l e in [12, p.220]; for n -> 2 F. B r o c k h a s given a c o u n t e r e x a m p l e in [5]. Method 5. Symmetry via the Maximum Principle II

In [2] the authors s t u d i e d classical solutions of h u + IVul m = f ( r ) u = const,

in B(0,1), on 0B(0,1)

(5)

with m E (0,1) a n d s h o w e d t h a t radially s y m m e t r i c solutions h a d to be unique. The question w h e t h e r all classical s o l u t i o n s are radial w a s left as an open problem. The following trick s e e m s to b e not so well known. F o r simplicity a s s u m e that n = 2 a n d that 0 is the angular comp o n e n t o f the p o l a r c o o r d i n a t e s of x. Set v = u0 a n d differentiate (5). Then

Does u have the s a m e reflection s y m m e t r y as Ft? The ans w e r is positive, a n d the p r o o f quite elegant. Let us split 12 i n t o tWO p a r t s ~'~r : = ~'~ n {x I > 0} a n d ~ t which are reflections o f e a c h other. Then the energy J ( u ) is equipartitioned a m o n g t h o s e two subdomalns, in o t h e r w o r d s

1

ilvul 2 +

= 0 v = 0

in 12 on 012,

"~tr

ilvul

+

(7)

(u)l

In fact, o t h e r w i s e a s s u m e that the left-hand side is domih a t e d b y the right. Then the function

U(Xl'

X2' * " " ' Xn)

:

I U ( X l , X 2 , . . . , Xn)

if x E 12t,

[U(--Xl,X2,

if X ~

. . . , Xn)

12r

(8)

w o u l d have the p r o p e r t y J ( ~ ) < J ( u ) , a contradiction. This p r o v e s (7). If u minimizes J, then so d o e s the reflections y m m e t r i c function ~ defined by (8). But ~ coincides with u on haft of 12. This implies that ~ c o i n c i d e s with u on all of 12, b e c a u s e minimizers of (3) solve an Euler equation w h o s e solutions have the unique c o n t i n u a t i o n property. Thus u = ~ a n d u is reflection symmetric. How's that?

Method 7. Symmetry via Stability

Consider the following problem: -

hv + mlVulm-2VuVv

=

Au + f(u)

= 0

u = 0

in 12, on O~

(9)

(6)

w h e r e Ft := B(0,1) \ {x E B(0,1) I 1~Tu(x)l = 0}. Equation (6) c~ is a linear equation for v with a coefficient in L~c(12 ) in front o f Vv. Consequently, b y t h e m a x i m u m p r i n c i p l e v ~0, i.e., u is n e c e s s a r i l y radial. This trick solves the o p e n p r o b l e m in [2] as well as m a n y others. Method 6. Symmetry via the Unique Continuation Principle

I l e a r n e d this trick from [22]. Let u be a minimizer of (3) a n d let 12 have a reflection symmetry, without loss of generality with r e s p e c t to Xl. This is a little less than Steiner symmetry, b e c a u s e it i n c l u d e s d o m a i n s like the M e r c e d e s logo; s e e Figure 6.

for t2 = B(0,1). We call a solution u stable if the s e c o n d variation o f the a s s o c i a t e d functional (3) is strictly positive, i.e., if

f~(0,1)(Ivr

+f'(u)~b2)dx > 0

(10)

for every nontrivial test-function r E H~(B(0,1)). (Here I use the c o m m o n convention that F is a primitive of f.) Stable s o l u t i o n s m u s t be symmetric. I l e a r n e d this from Alikakos a n d Bates [1]; s e e also [17] for e x t e n s i o n s to quasilinear equations. F o r a p r o o f one sets 4) = uo, an angular derivative of u, a n d finds out that (10) is violated. Therefore uo -~ 0 and u is symmetric.

Method 8: Symmetry via the P-function

}

Consider again p r o b l e m (9), a n d s u p p o s e one can s h o w for s o m e positive c o n s t a n t a that P(x)

:=

rvu(x)l 2 -

=

eonst.

in 12. Then this implies s o m e s y m m e t r i c s h a p e of u, n a m e l y a) radial symmetry, as s h o w n in [25] for 12 = B(0,1) C R n, a = 2/n a n d f ~ - 1 , o r b) d e p e n d e n c e o f u on a single real variable, as s h o w n in [9] for 12 = R n, a = 2 and fairly general f, e.g., f ( u ) = U 3 --U.

Figure 6. A Domain with Reflection Symmetry.

20

THE MATHEMATICALINTELLIGENCER

Let us explain b o t h situations in m o r e detail and reveal w h y the first one d o e s n o t easily c a r r y o v e r to n o n c o n s t a n t f. In the first c a s e a) one can argue that P is harmonic, so that

AP =

~.

2

uij +

n-1

i,j= 1

f'(u)]~Tul 2 -

1

n

f 2 ( u ) = 0.

n

If w e a s s u m e t h a t f ' --> 0, then Uij i,j=l

(U) =

-

Uii

-

n

i=l

<~ i=l

U i i <~ Uij. i,j=l

Here w e have u s e d Schwarz's inequality. Since the righth a n d side equals the left, w e m a y c o n c l u d e that u i j (x) = 0 a n d u i i ( x ) = u j j ( x ) if i #=j, as well a s f ' ( u ) = 0. Hence 1 f m u s t be constant, u i i = ~ f ( u ) , a n d u is radial. Another w a y to address situation a) is the observation that the outer normal derivative Pv vanishes on aB(0,1), but Pv = UvUw - ~ f ( U ) U v = O. Assuming that Uv r 0 and recalling that A u = U w + ( n - 1)Uv = f ( u ) on 0B(0,1), w e obtain 1 1 Uw = - - f ( u ) = f ( u ) - ( n - 1)uv or Uv = - - f ( u ) on 0B(O,1). n n

But n o w w e have t w o m o r e derivatives uv a n d u ~ pres c r i b e d as c o n s t a n t on the boundary, and a l r e a d y one of t h e m f o r c e s t h e solution to b e radial, p r o v i d e d it is positive. This follows from a delicate analysis due to Serrin [24]. In this p a p e r Serrin laid the f o u n d a t i o n for Method 4 and the w o r k o f [12]. In c a s e b) let v again d e n o t e the direction of - V u , a n d s u p p o s e Uv r 0 in ~ n and n -< 8. The fact that P i s c o n s t a n t implies P v = 2UvUw - 2 f ( U ) U v = O. Recalling t h a t Au = u w + ( n - 1 ) H ( x ) u v = f ( u ) on level surfaces o f u, we obtain that H ( x ) , their m e a n curvature, vanishes. In o t h e r words, level surfaces are minimal surfaces. But n o w b y a result of Bernstein the level surfaces are planes. F o r general n one has to w o r k a little h a r d e r to get the s a m e result; s e e [9, T h e o r e m 5.1]. Acknowledgment It is m y p l e a s u r e to thank F. Brock for m a n y stimulating discussions on the subject o f s y m m e t r y and M. Mester for his expertise in getting the figures into electronic f o r m - - a s well as the Royal Mint for supplying a picture of their coin. Notes A d d e d Oct. 1 9 9 7 A r e c e n t a n d e x c e l l e n t survey on E x a m p l e 1 can b e found in G.D. C h a k e r i a n & H. Groemer: Convex b o d i e s of cons t a n t width, in C o n v e x i t y a n d i t s A p p l i c a t i o n s . Eds. P.M. Gruber, J.M. Wills, Birkhfiuser (1983) 49-96. A story relating E x a m p l e 1 to the Challenger d i s a s t e r is c o n t a i n e d in Richard F e y n m a n ' s b o o k S u r e l y y o u ' r e j o k i n g , M r . F e y n m a n , W.W. N o r t o n (1985), New York. E x a m p l e 7 has also b e e n a d d r e s s e d in B. Dacorogna, W. Gangbo and N. Subia in "Sur une generalisation de l'inegalit~ de Wirtinger," A n n . I n s t . H e n r i P o i n c a r d A n a l . N o n L i n e a i r e 9 (1992), 29-50.

[3]

[4]

[5] [6] [7]

[8]

[9]

REFERENCES

[1] Alikakos, N.D. & P.W. Bates: On the singular limit in a phase field model of phase transitions. Analyse Nonlin6aire, Ann. Inst. Henri Poincar~ 5 (1988), 141-178. [2] Barles, G., G. Diaz, & J.l. Diaz: Uniqueness and continuum of fo-

[10] [11]

liated solutions for a quasilinear elliptic equation with a nonlipschitz nonlinearity, Comm. PDE 17 (1992), 1037-1050. Brezis, H. & L. Nirenberg: Positive solutions of nonlinear equations involving critical Sobolev components. Commun. Pure AppL Math. 36 (1983), 437-477. Brock, F.: Continuous polarization and symmetry of solutions of variational problems with potentials, in: Calculus of Variations, Applications and Computations, Pont b Mousson 1994 Eds.: C. Bandle et al., Pitman Res. Notes in Math. 326 (1995), 25-35. Brock, F.: Continuous Rearrangement and Symmetry of Solutions of BVPs. Manuscript, 1996. Brock, F., V. Ferone, & B. Kawohl: A symmetry problem in the calculus of variations, Calculus of Variations and PDE 4 (1996) 593-599. Brock, F. & A. Solynin: An approach to symmetrization via polarization. Manuscript (1996), currently in WWW under http://www. mi.uni-koeln.de/cgi.bin/preprint.out. Buttazzo, G., V. Ferone, & B. Kawohl: Minimum problems over sets of concave functions and related questions, Mathematische Nachrichten 173 (1995), 71-89. Caffarelli, L., N. Garofalo, & F. Segala: A gradient bound for entire solutions of quasilinear equations and its consequences, Commun. Pure AppL Math. 47 (1994), 1457-1473. Coffmann, C.V.: A nonlinear boundary value problem with many positive solutions. J. Differ. Equations 54 (1984) 429-437. Esteban, M.: Nonsymmetric ground states of symmetric variational problems. Commun. Pure AppL Math. 44 (1991), 259--274.

VOLUME 20, NUMBER 2, 1998

21

[12[ Gidas, B., W.M. Ni, & L. Nirenberg: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), 209-243. [13] Gerdes, K. & C. Schwab: Hierarchic models of Helmholtz problems on thin domains. Math. Models & Methods Appl. Sci., to appear [14] Kawohl, B.: Rearrangements and Convexity of Level Sets in PDE. Springer Lecture Notes in Math. 1150 (1985). [15] Kawohl, B. & G. Sweers: Remarks on eigenvalues and eigenfunctions of a special elliptic system. J. Appl. Math. Phys. (Z_AMP) 88 (1987), 730-740. [16] Kawohl, B.: A conjectured heat flow problem, Solution. SIAM Review 37 (1995), 105-106. [17] Kawohl, B.: Instability criteria for solutions of second order elliptic quasilinear differential equations. In: Partialdifferentialequations and applications. Eds. P. Marcellini, G. Talenti, E. Vesentini, Marcel Dekker, Lecture Notes in Pure and Applied Math. 177 (1996), 201-207. [18] Kawohl, B.: A short note on hot spots, Zeitschr. Angew. Math. Mech. 76 (1996) Suppl. 2, 569-570.

22

THE MATHEMATICAL1NTELLIGENCER

[19] Kawohl, B.: The opaque square and the opaque circle, in: General Inequalities VII, Int. Ser. Numer. Math. 123 (1997), pp. 339-346. [20] Klamkin, M.S.: A conjectured heat flow problem, Problem. SlAM Review 36 (1994), 107. [21] Littlewood, J.E.: A Mathematician's Miscellany, Methuen & Co. Ltd., London (1953). [22] Lopez, O.: Radial and nonradial minimizers for some radially symmetric functionals. Electronic Journal of Differential equations 1996 (1996) no. 3, 1-14. On WWW under URL:http//ejde.math. swt.edu. [23] Rademacher, H. & O. Toeplitz: Von Zahlen und Figuren, Springer, Berlin (1930); transl. The enjoyment of mathematics, Princeton Univ. Press, Princeton (1994). [24] Serrin, J.: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43 (1971), 304-318. [25] Weinberger, H.: Remark on the preceding paper of Serrin. Arch. Ration. Mech. Anal 43 (1971), 319-320.

[12[ Gidas, B., W.M. Ni, & L. Nirenberg: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), 209-243. [13] Gerdes, K. & C. Schwab: Hierarchic models of Helmholtz problems on thin domains. Math. Models & Methods Appl. Sci., to appear [14] Kawohl, B.: Rearrangements and Convexity of Level Sets in PDE. Springer Lecture Notes in Math. 1150 (1985). [15] Kawohl, B. & G. Sweers: Remarks on eigenvalues and eigenfunctions of a special elliptic system. J. Appl. Math. Phys. (Z_AMP) 88 (1987), 730-740. [16] Kawohl, B.: A conjectured heat flow problem, Solution. SIAM Review 37 (1995), 105-106. [17] Kawohl, B.: Instability criteria for solutions of second order elliptic quasilinear differential equations. In: Partialdifferentialequations and applications. Eds. P. Marcellini, G. Talenti, E. Vesentini, Marcel Dekker, Lecture Notes in Pure and Applied Math. 177 (1996), 201-207. [18] Kawohl, B.: A short note on hot spots, Zeitschr. Angew. Math. Mech. 76 (1996) Suppl. 2, 569-570.

22

THE MATHEMATICAL1NTELLIGENCER

[19] Kawohl, B.: The opaque square and the opaque circle, in: General Inequalities VII, Int. Ser. Numer. Math. 123 (1997), pp. 339-346. [20] Klamkin, M.S.: A conjectured heat flow problem, Problem. SlAM Review 36 (1994), 107. [21] Littlewood, J.E.: A Mathematician's Miscellany, Methuen & Co. Ltd., London (1953). [22] Lopez, O.: Radial and nonradial minimizers for some radially symmetric functionals. Electronic Journal of Differential equations 1996 (1996) no. 3, 1-14. On WWW under URL:http//ejde.math. swt.edu. [23] Rademacher, H. & O. Toeplitz: Von Zahlen und Figuren, Springer, Berlin (1930); transl. The enjoyment of mathematics, Princeton Univ. Press, Princeton (1994). [24] Serrin, J.: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43 (1971), 304-318. [25] Weinberger, H.: Remark on the preceding paper of Serrin. Arch. Ration. Mech. Anal 43 (1971), 319-320.

IJ~l~ii|r:inr.li[eile'~,]=,t,i!!lilii[:-]-ll

Ferran Sunyer i Balaguer (1912-1967) and Spanish Mathematics After the Civil War Antoni Malet

This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We include "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

Please send all submissions to the Mathematical Communities Editor, Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01063, USA; e-mail: [email protected]

Marjorie Senechal,

Editor I

Ferran Sunyer i Balaguer was an autodidact, or self-taught, mathematician working in Spain from the mid-thirties to his death in 1967, probably the most unenlightened decades of Spain's modem history. Although he was one of the most productive and internationally known mathematicians working in Spain, he always occupied a marginal position in Spanish academic institutions. This was due to a combination of factors, chief among them his Catalan origins, his being self-taught, his physical disability, and the peculiar values and workings of the Spanish mathematical community. His main contributions belong to classical analysis, to the study of functions represented by lacunary Taylor and Dirichlet series. However, his best-known contribution is a minor one---his theorem (proved with Ernest Corominas) on pointwise conditions for polynomiaiity of infinitely differentiable functions.

process, the doctors said. When it had become obvious that Ferran's mental abilities were not seriously damaged, they still recommended that he be spared mental stimulation in order not to tire or overload his impaired nervous system. When he was four, his family discovered that he had learned to read by himself, and then his mother decided that he would get as good an education as she could give him. The family moved to Barcelona, where Sunyer was to live for the rest of his life. He inherited rural property in Vilajoan, near Figueres, a place he loved deeply and where he spent all the long, hot months of the Catalan summer. Throughout his life Ferran Sunyer was lucky enough to enjoy the help and caring support of close relatives. His rather unusual family was made up of his mother and three cousins more or less his age. His father died when he was two. Shortly thereafter misfortune T h e M a n and t h e C i r c u m s t a n c e struck the family of his mother's only Ferran Sunyer was born in 1912 in the sister. The head of the family, one Mr. Catalan town of Figueres, near the Carbona, abandoned his wife and three French border, famous for being also children, malting no provision for their the birthplace of Salvador Dali, the lu- material needs; a few months later, minary of surrealist art. He was from a Ferran's aunt, the mother of the three landed family, the son, grandson, and children, died of tuberculosis. Ferran great-grandson of doctors. It was soon and his cousins, two girls and one boy, discovered that he grew up together as siblings. The boy, suffered from a severe Ferran Carbona (1909congenital nervous atrophy. All through 1979), was to become a chemical engineer his life he had to be who openly sided with moved around in a leftist parties during wheelchair and was Spain's CivilWar (1936otherwise physically 1939). When Franco's disabled, could not eat by himself, and troops occupied Barspoke with difficulty. celona, in January 1939, At first, doctors behe fled to Paris, where he settled, married, lieved that the child would not survive, and lived the rest of his life. In the years then that he was desof Spain's internationtined to die early. He was surely not fit for ai isolation following Ferran Sunyer i Balaguer, c a . 1 9 3 5 . any kind of learning World War II, he

9 1998 SPRINGER-VERLAG NEW YORK, VOLUME 20, NUMBER 2, 1998 2 3

helped Ferran Sunyer to get in contact with prominent members of the French mathematical community. Sunyer's two female cousins, Angels and Maria Carbona, lived with him all his life, and took care of him after his mother's death in 1955. They preserved Sunyer's papers and correspondence, and did their best to keep his memory alive in a society which traditionally has not rated scientific achievement very highly. With great determination they overcame bureaucratic interference and institutional indifference to set up a Ferran Sunyer i Balaguer Foundation, and endowed the Ferran Sunyer i Balaguer Prize for in the best book-length monograph, in English, in any mathematical field that has known important contributions in recent years. 1 They were extremely helpful to me when I was writing a study of Ferran Sunyer and the Spanish mathematical community in the Franco years. 2 Introducing Himself to Mathematics Ferran Sunyer introduced himself to mathematics and physics through the books his cousin Ferran Carbona studied in high school and then in engineering school. His ever-growing appetite for mathematics was fed by books his cousin fetched from the university library. The study of pure mathematics had never been strong in Barcelona, so the library was short of good, up-to-date books. Nor was there anyone around with whom to talk about what was going on in mathematics at the time. Sunyer's first attempt to contribute something original to mathematics reveals the isolation in which he was working. In early 1934, apparently unaware of the revolution algebra had undergone, he sent to Emile Picard (18561941), then the 78-year-old secretaire perpdtuel of the French Acad~mie des Sciences, a note correcting an error he had found in J.oA. Serret's Cours d'Alg#bre Supdrieure (in its 1928, seventh edition)--hoping, the covering

Vilajoan, early 1940s. From the left, A n g e l s Carbona, Angela Balaguer, Ferran Sunyer, M o d e s t a Masdevall.

letter said, to see the note published somewhere or to receive Picard's opinion about it. He never heard from Picard, but he was undaunted and kept studying mathematics. In December 1938 he tried again, this time sending two notes to Jacques Hadamard (1865-1963), and this time succeeding. Hadamard wrote back encouragingly and arranged for one of the notes ("Sur une classe de transformations des formules de sommabilit~") to be published in the

Comptes Rendus de l'Acaddmie des Sciences in 1939. Then came WW II and Hadamard went to the U.S.A. In 1946 Sunyer reestablished contact with Hadamard, again in Paris, sending him an important memoir on lacunary Taylor series. The 81-year-old Hadamard put Sunyer in touch with Szolem Mandelbrojt (1899-1983), his successor both as professor of mathematics and mechanics in the Coll~ge de France and as one of the leading authorities in analytic functions. Mandelbrojt helped Sunyer by providing him with bibliography and

advice about matters of style--concerning both mathematics and French. Two notes with results from the memoir were published in 1947 in the Comptes Rendus, and the memoir itself eventually appeared in Acta Mathematica in 1952 with the title "Sur la substitution d'une valeur exceptionelle par une propri~t~ lacunaire." Mandelbrojt also introduced Sunyer to the French and American mathematical societies and to young French mathematicians. From then until his untimely death, of a heart attack, in 1967 Sunyer worked in close contact with members of the international mathematical community, including mathematical correspondence with Wac~aw Sierpifiski and fmancial support from the Office of Naval Research (ONR). In 1960 Sunyer applied for financial support to the ONR with reference letters from A.J. Macintyre (1908-1967), then the Charles P. Taft Professor of Mathematics in the University of Cincinnati, and R.P. Boas Jr. (1912-1992), former editor-in-chief of Mathematical Reviews (MR) and former editor of the Pro-

ceedings of the American Mathematical Society (PAMS). Both knew Sunyer's papers well, most of which they had reviewed in MR. Macintyre, when a conjecture of his was proved by Sunyer, had had it published in the PAMS. 3 In 1959 he suggested that Sunyer visit the U.S. in exchange for someone who would teach at the University of Barcelona. Sunyer was interested in the project but did not get support from the Spanish academic authorities. (In the strongly bureaucratized and centralized Spain of the Franco regime, universities had no autonomy to decide on matters of appointments or visiting positions.) Boas had been impressed by the simplicity of the result on polynomiality provided in 1954 by Corominas and Sunyer, and decided to write his popular introduction A primer of real functions so that it would contain everything that is

1The winners of the Sunyer i Balaguer Prize have been so far, A. Lubotzky, with Discrete groups, expanding graphs and invariant measures; K. Schmidt, with Dynamical systems of algebraic origin; and M.R. Murty and V.K. Murty, with Non-vanishing of L-functions and applications; A. B6ttcher and Y.K. Arlovich, with Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. All the monographs are published by Birkhfiuser Verlag. 2A. Malet, Ferran Sunyer i Balaguer (1912-1967). Barcelona: IEC, 1995 (in Catalan). All correspondence and exchanges between mathematicians mentioned in the present article are duly referenced in this book. 3"A theorem on overconvergence," Proc. Amer. Math. Soc. 12 (1961), 495-497.

24

THE MATHEMATICALINTELLIGENCER

n e e d e d to p r o v e the Sunyer-Corominas t h e o r e m (on which see the n e x t section), a The r e s e a r c h p r o j e c t s Sunyer realized u n d e r a Office of Naval R e s e a r c h c o n t r a c t w e r e all on p u r e m a t h e m a t i c s , a c o n t i n u a t i o n of the w o r k he had b e e n doing in the 1950s. They m o s t l y dealt with a p p r o x i m a t i o n s of functions b y linear c o m b i n a t i o n s of e x p o n e n t i a l s and entire functions r e p r e s e n t e d b y Dirichlet series. In B a r c e l o n a in 1964, and p e r h a p s on o t h e r o c c a s i o n s too, Sunyer w a s visited b y ONR liaison scientists. Sunyer's career is rather unusual for someone who had had no access to formal mathematical education and enj o y e d little institutional support throughout his life. It is all the m o r e impressive w h e n a c c o u n t is t a k e n of his p h y s i c a l limitations. Unable to write, he h a d to rely on his m e m o r y m o r e t h a n is usual even for m a t h e m a t i c i a n s w h o w o r k "in their heads". In the c o u r s e of time he gained limited control o f his arms, so he w a s able to r e a d and shuffle p a p e r s a r o u n d b y himself, b u t he always n e e d e d his relatives' help to p u t his letters and p a p e r s in writing. On a typical day he w a s left alone in his carrel with the b o o k s a n d pap e r s he w o u l d use handy. Then, w h e n he n e e d e d to p r e s e r v e a result, he a s k e d s o m e o n e to write it d o w n in a n o t e b o o k . W h e n the p a p e r w a s "ready" in his head, he w o u l d dictate it. At a time w h e n having a s e r i o u s p h y s i c a l disability m e a n t social marginalization, Sunyer m a n a g e d to arrange everything that w a s n e e d e d for him to participate in national and international meetings. He a t t e n d e d the annual m e e t i n g s o f the Spanish m a t h e m a t i c a l society and travelled to Nice, to Florence, a n d to Oberwolfach. In m o s t o f t h o s e p l a c e s he s u b m i t t e d papers, w h i c h he w o u l d p r e s e n t orally while s o m e b o d y w r o t e the f o r m u l a e on the blackboard. The w a y Sunyer first met Wac]~aw Sierpifiski (1882-1969) illustrates his de-

Vilajoan, early 1940s. From the left, Modesta Masdevall (grandmother of Ferran Sunyer), Ferran

Sunyer,

Maria

Carbona,

Angela

Ba|aguer.

Mathematical Contributions S u n y e r was especially p r o u d o f the results he o b t a i n e d or intuited at the beginning of his career, b e t w e e n 1939 and 1945, w h e n H a d a m a r d e n c o u r a g e d him to devote himself to mathematics. Even before that he h a d o b t a i n e d two m i n o r results. The first (in the Comptes Rendus in 1939, t h e n in an e x t e n d e d v e r s i o n in 1948 in the first n u m b e r o f CoUectanea Mathematica) constructed a class of m e t h o d s o f analytic continuation. M s o in 1939, he p u b l i s h e d in the Revista Matemdtica HispanoAmericana a p r o o f o f the little Picard t h e o r e m relying on a novel a p p l i c a t i o n of the t h e o r y o f n o r m a l curves. The m o s t i m p o r t a n t results of the r e s e a r c h p r o g r a m initiated in 19391945 are in Comptes Rendus notes of 1947 (proofs followed in 1952 in Acta Mathematica), and in p a p e r s published in 1948 by the A c a d e m y of Barcelona, in 1949 in CoUectanea Mathematica, in 1950 by the A c a d e m y of Sciences of Zaragoza, and in 1953 in the Proceedings

of the American Mathematical Society. termination and self-confidence. At Nice in 1957, at the first R~union des Math~maticiens d'Expression Latine, he let Mandelbrojt k n o w that he would like to point out to Sierpifisld a non-trivial mistake he had found in Hypoth~se du continu, Sierpi~ski's famous b o o k first published in 1934. Mandelbrojt doubted that Sunyer had found such a mistake in a 23-year-old b o o k and recoiled at the idea of letting an unknown autodidact bother Sierpifiski--who with more than 50 years of professional career and m o r e than 600 publications was by then one of the most r e s p e c t e d and honored European mathematicians. Eventually Sunyer talked to Sierphlski and the result was a note in Fundamenta Mathematicae in which Sierpifiski corrected the mistake and w a r m l y credited Sunyer for finding it. 5 The mathematical correspondence that ensued focused on ordinal numbers and was the origin of Sunyer's only contribution to this subject. ~

M1 these w o r k s start, Sunyer explained, with the f a m o u s t h e o r e m of H a d a m a r d asserting that a lacunary Taylor series can n o t b e c o n t i n u e d bey o n d its circle of convergence. Other t h e o r e m s h a d s h o w n that gaps force the d i s a p p e a r a n c e of w h a t one might call e x c e p t i o n a l c a s e s in the location o f singularities. The general intuition underlying t h e s e p a p e r s of Sunyer is the principle that in Taylor series, gaps m a k e e x c e p t i o n a l c a s e s disappear. In the 1952 Acta Mathematica article, Sunyer p r o v e d t h a t if F, an entire fimction of integral order, has a Taylor series in w h i c h the m a x i m u m density of non-zero t e r m s is less t h a n a certain c o n s t a n t ( d e p e n d i n g on the order), t h e n the distribution of the zeros of F - f can not b e exceptional, relative to the e x a c t o r d e r o f F, for a n y merom o r p h i c function f o f l o w e r order. Similarly w h e n F is o f infinite order. In the 1948 p a p e r Sunyer e x t e n d e d these results to m e r o m o r p h i c functions. In

4Boas's book was first published as a Carus monograph by the Mathematical Association of America (New York, 1960), and had a second edition in 1966, a third in 1981, and a fourth in 1996. The Sunyer-Corominas theorem was first published in the Comptes Rendus ("Sur des conditions pour qu'une fonction infiniment derivable soit un polyn6me", 238 (1954): 558-559), and appeared in full in the same year in the Revista Matematica Hispano-Americana. 5"Sur un theoreme de S. Saks concernant les suites infinies de fonctions continues," Fund. Math. 46 (1958), 117-21. 6"Sur les types d'ordre distincts dont les n-i~mes puissances sent equivalentes", Fund. Math. 46 (1958), 221-224.

VOLUME20, NUMBER2, 1998 25

the 1950 paper, he proved similar results for Dirichlet series. When these are specialized to Taylor series, they differ from those obtained directly earlier: the conclusions are stronger, but the hypotheses are more restrictive. In the memoir in CoUectanea in 1949, Sunyer extended results of Nevanlinna and P61ya. Following Nevanlinna's suggestion (used by Milloux in 1935 on entire functions of infinite order) that the rate of convergence of a function to an asymptotic value could be used to study the relation between asymptotic values and exceptional values, Sunyer was able to improve P61ya's sufficient conditions (on the density of the sequence of exponents) for non-existence of asymptotic values of functions represented by lacunary series. The same idea is used in his 1953 Comptes Rendus note on the Denjoy-CarlemanAhlfors theorem (proofs published in 1956): the estimate of the number of asymptotic values of an entire function given by their theorem is sharpened under a hypothesis on the speed of convergence. In the last years of his life, Sunyer returned, with powerful functional-analytic methods, to the problem of exceptional values of entire functions of infinite order. In "Sobre un espai de funcions enteres d'ordre infinit," published by the Institut d'Estudis Catalans in 1967, Sunyer defined a topological space Sw comprising a class of entire fimctions of infmite order in the topology of uniform convergence on compact subsets of C; he proved that Sw is of second category and that certain subsets are of first category, in particular the set of all functions having exactly one exceptional value (in the sense of Borel) and the set of all functions having at least one nonexceptional direction (in the sense of Borel-Valiron). Still in the same area, Sunyer improved a theorem of H. Milloux of 1951 saying that for an entire function of positive order there is at least one Borel direction in common between the function and all its derivatives and integrals. Sunyer, in a Comptes Rendus note of 1953 and in the memoir which won the prize of the Madrid Academy in 1954 (published in 1956), showed the same for the Borel direc-

~)6

THE MATHEMATICALINTELLIGENCER

tions of maximal type, and related the number of Borel directions of maximal type with the number of exceptional values. At the beginning of his career Sunyer set out on a second line of work which did not turn out as fruitful as the first: generalizing the concept of almost periodic function. Sunyer (in the memoir which won the "Prat de la Riba" prize in 1948) replaced Ilzt - z21[, in the definition of almost periodic (a.p.) function in a strip, by spherical distance, and applied it to meromorphic functions, not just to holomorphic functions. In case the strip is all of C and almost-periodicity holds in two directions, Sunyer gave a generalization of the notion of elliptic function and studied the form Liouville's theorems take for these functions. Although Sunyer got original results and carried this idea farther than those who had made the same attempt earlier, it was always a serious inconvenience that the sum and product of these a.p. functions in a strip need not belong to the class. The other important line of research in Sunyer's mathematical career originated in the late 1940s, in a series of works which vastly generalized (as Mandelbrojt himself said), Szolem Mandelbrojt's inequality on adherent

series and introduced new ideas on "logarithmic precision" (of the adherence) enabling him to apply it to Dirichlet polynomiaks---"adherent polynomials." In his notes of 1950-51 and his memoir of 1952, Sunyer defined the logarithmic precision (of a representation of a function) more generally than Mandelbrojt had done in 1947, extended the so-called "fundamental" inequality of Mandelbrojt, and applied the result to the representation of analytic functions as a limit of exponential polynomials. In Sunyer's reformulations, it is no longer required that the approximants to a function be partial sums of a Dirichlet series, the only restriction on them is a pre-assigned sequence of exponents. Then in a 1959 Comptes Rendus note (proofs came out in 1961), Sunyer gave another major reformulation of these same ideas, leading to further improved results. In place of logarithmic precision he introduced b-logarithmic precision: Let A = {An} be a sequence of nonnegative real numbers and let r be the set of functions of the form an e-xns, n=

Let A be the horizontal half-strip (or > ~r0, Itl < 7rg(cr)), where g is a continuous function of bounded variation for or> (r0 such that g(vr) > 0 and lim g((r) > 0. Let ~(x) be the intersection of h with the vertical strip x - b < cr < x. Let the function F b e holomorphic in A, and p(x) a nondecreasing function tending to infinity. Then F is said to be represented in h with logarithmic bprecision p(x) by 0 E (b ff inf sup oer

Ferran Sunyer i Balaguer, in the early 1940s.

s = ~r + it.

l

IF(s) - O(s)l-< e -p(x).

s ~ 8(x)

In the important 1965 memoir "Approximation of functions by linear combinations of exponentials," Sunyer was able to give necessary and sufficient conditions, subject to some restrictions on p(x) and on the sequence {~}, for a function F to be representable with b-logarithmic precision. An especially original work of Sunyer won the prize of the Academia

de Ciencias of Madrid in 1957 (published in 1959). Here he investigated the expansion of analytic functions in series of iterated integrals of a given entire function. When the latter function is a constant, the primitives are polynomials; this theory was studied by L. Gontcharoff in 1937. The general case had been treated briefly by Rey Pastor in 1935, but (as R.P. Boas said in Mathematical Reviews) Sunyer "carried the subject much farther." Beginning with a suitable convergent sequence {Zn} ~ Z, and an entire function 0 such that 0(Z)= 1 and O(Zn) ::/: O, Sunyer defines 0n by On(n)(z) = n! O(z) and on(k)(Zk) = O (k < n). He shows that the series ZCnOn(Z) have essentially the same convergence properties as power series in (z - Z) n, and extends to them various basic results (on zeros of partial sums, on overconvergence, etc.). In particular, much of the theory of lacunary power series goes over to the general case. Sunyer also made contributions to function theory in response to works of other mathematicians. One is the note "On moments of functions bounded and holomorphic in an angle" (1953) on a problem which interested San Juan. Another, already mentioned, is his "Theorem on overconvergence" proving a conjecture of his colleague A.J. Macintyre. Sometimes these contacts with colleagues led him outside of classical function theory. We have already mentioned one of these "excursions" outside his primary domain, his contribution to the theory of ordinals in Fundamenta Mathematicae. Doubtless the most famous and important of Sunyer's works outside his specialty is the characterization of polynomial functions published with Ernest Corominas in 1954. They prove that i f f is infinitely differentiable on R, and if V x E R 3 n(x) such that f(n(x))(x) = 0, then f is a polynomial. They also prove for L a subset of R that the hypothesis V x E L 3 n(x) such that f(n(x))(X)= 0 f o r c e s f t o be a polynomial if and only if R - L contains no perfect set. They prove, finally, that if V x E R 3 n(x) such thatf(n(x))(x) E H, where H is a denumerable subset of R not containing a perfect set, t h e n f i s a polynomial.

Nice, 1957, at the I R6union des Math6maticiens d'expression latine. Seated in the first row (r-I) are Ferran Sunyer, Ferran Carbona, Maria Carbona, Antbnia Miravitlles (wife of Jaume Sunyer), and Angels Carbona.

Sunyer and the Spanish Mathematical Community Apart from the respect that Sunyer's character inspires and the interest that his contributions may have now, his career much illuminates the functioning of the mathematical community in Franco Spain. Sunyer's productivity, the range and quality of his foreign connections, and the international impact of his work put him in the top class of Spanish mathematicians. Yet when it came to institutional recognition, academic authorities neglected him, and with few exceptions his colleagues ignored his mathematical stature. In 1948 Sunyer applied to the Consejo Superior de Investigaciones Cientificas (Higher Council for Scientific Research, known in Spain by the acronym CSIC) to get a paid position there. Founded after the Civil War, the CSIC was the huge, strongly centralized institution solely responsible for scientific policy--a role it kept until the late 1960s. It maintained several research institutes formally independent of university institutions, but also allocated research funds--in the form of doctoral fellowships, technical assistants, librarians, grants to travel abroad and to invite foreign researchers, grants for books and material, and so o n - - t o university laboratories and departments. Only a few full-time research positions were available in the CSIC--about 150 for all dis-

ciplines in the whole country in the late 1950s. The preferred policy was to offer salary complements to university lecturers and professors who did research. In any case a full-time research position in the CSIC was not as coveted as a university chair, which carried higher salary, social prestige, and academic influence. Prejudices being what they then were toward people with physical disabilities (by no means only in Spain), it was out of the question for Sunyer to aspire to a university chair. On the other hand, his attempts to get a flfll-time position in the CSIC were plagued with bureaucratic obstacles and official indifference. The ordinary procedure for obtaining a CSIC position was closed to Sunyer because he lacked university degrees. The CSIC statutes allowed its presidency to appoint prominent researchers to full-time positions directly, that is to say, overlooldng ordinary requirements. A few colleagues strongly supported Sunyer for such appointment before top authorities in the Ministry of Education--most notably Julio Rey Pastor (1888-1962) and Ricardo San Juan (1908--1969), both holding chairs in the University of Madrid, then the most prestigious one nationwide. They also suggested that Sunyer submit papers for the prizes annually offered by the Spanish Academy of Sciences and the CSIC, and successfully campaigned in his behalf to the prize committees.

VOLUME20, NUMBER2, 1998 27

Thus it was that Sunyer won the 1954 and 1957 prize of the Academy of Sciences and the 1956 "Premio Nacional de Ciencias Francisco Franco," at the time the highest scientific award nationally. But to no avail---Sunyer could not be appointed CSIC researcher because he did not have a bachelor degree and a doctorate, in 1957 he decided to get them and registered in all the courses necessary to major in mathematics at the University of Barcelonr where he was of course well known. Eventually he got his degrees, but in the process many mathematics professors gave him examinations--and two of them gave him a very low grade. It was not until 1962 that he was appointed "junior collaborator" (the lowest research position in the CSIC academic hierarchy), and it was only weeks before his death in 1967 that he could join the top category of "researcher." To the end of his life Sunyer resented the lack of proper recognition by his Spanish colleagues. It was expressed most dramatically through his low status within the academic hierarchy, but was also made obvious in other ways. In 1962 P.K. Kamthan, then teaching mathematics at Birla College, in Pilani (India), initiated a mathematical correspondence with Sunyer. In January of the following year he expressed his earnest interest in being a postdoctoral fellow in Barcelona, to work with Sunyer, if he could find financial support. Sunyer did his best to get it through the Consejo Superior de Investigaciones Cientificas as well as other agencies--to no avail. Eventually, in late 1966 Kamthan moved to Canada with a postdoctoral fellowship. There is a similar story involving A.R. Reddy, who first contacted Sunyer in April 1965, when he was a postgraduate student in the Ramanujan Institute of Mathematics of the University of Madras (India). In 1967, after Sunyer had evaluated his doctoral dissertation, he was interested in working with Sunyer, but once again Sunyer was unable to fmd any financial support. Reddy ended up in Canada too. As far as I know, from 1940 through the 60s these were the only qualified mathematicians who ever wanted to study in Spain. Remarkably enough, the

28

THE MATHEMATICALINTELLIGENCER

Consejo Superior de Investigaciones Cientificas boasted of its international links and of the foreign scientists wanting to study in Spain, and presented it as proof of the scientific achievements of the Franco r e g i m e - - a n d the CSIC had indeed the money to support foreign visitors when their Spanish connection was the right one. Sunyer directed no Spanish doctoral dissertation, nor was he ever asked to be the reader of any of t h e m - - b u t the University of Madras asked him to be the chief examiner of A.R. Reddy's Ph.D. thesis in 1967. Sunyer was often asked to collaborate with Collectanea

Mathematica and the Revista Matemdtica Hispano-Americana, then the two main mathematical journals published in Spain, and he did so in many w a y s - - b y publishing in them, refereeing papers, and soliciting papers for publication from his international acquaintances. However, he was never asked to be on any editorial board. In December 1966 he received a letter from the Spanish Royal Mathematical Society, which edited the Revista Matemdtica Hispano-Americana, acknowledging his substantial contribution to the journal's life and rewarding it . . . with a Christmas gift! In every annual meeting of the Spanish Mathematical Society some mathematicians were honored as chairs, or as plenary lecturers or by some other gesture. Sunyer never received that sort of recognition either. It is easy to point to some factors that may help "explain" Sunyer's marginalization within Spanish mathematics. He was an autodidact and an outsider, and remained independent of lobbies or "schools" all his life. He was physically disabled at a time when the notion of respect for disabled people had made little headway. Moreover, Sunyer had to pay a political price for being a Catalan and an open Catalan nationalist. Catalan nationalism was one of the three big political foes the Franco regime wanted to exterminate, the other two being freemasonry and communism. A language with a rich literary past, Catalan was the main language among Catalan people, and the only language in many rural areas. The Franco regime banished its use from all public life, in-

cluding schools, press, printed books, entertainment, and of course legal documents and the administration. The Catalan learned i n s t i t u t i o n s . . , created and maintained by the Catalan regional government before the Civil War were deprived of their buildings, libraries, and so on, and left without financial support. Sunyer participated in the subdued academic life of these institutions by submitting papers for their devaluated scientific prizes, by presiding over the Catalan Mathematical Society, and by having, as early as the 1940s, some of his mathematical papers published in Catalan. This may have negatively influenced his academic advancement, but it hardly tells us why his own colleagues did so little to acknowledge him as one of the country's leading mathematicians.

The Mathematical Community and the Franco Regime Historians agree that Spanish scientific and scholarly life under Franco's rule was poor and provincial, suffering badly from scarcity of material means and international isolation. This view seems to me to be largely correct, although I would make two caveats, both inspired by evidence from Sunyer's career. First, the main problem with financial support was not that it was scarce but that it was misdirected and wasted. Second, international isolation was not a consequence of Spanish mathematicians being unable to visit foreign centers, or to be in contact with them, or to receive visitors. On the contrary, there were grants available to do such things, and indeed most Spanish mathematicians getting professorships in the 1950s and 60s had spent months if not years in foreign institutions. It is true that the Spanish mathematical community as a wholc cxcluding the individual cases of Sunyer, Sixto Rios, Germfin Ancochea, Ricardo San Juan, and a few other mathematicians well-known and appreciated a b r o a d - - w a s not in tune with the main developments shaping mathematical life abroad, and the gap was recognized by leading Spanish mathematicians. But the isolation of Spanish mathematics was not so much enforced or imposed on the mathe-

N u m b e r of items

Authors

1-10

13

N u m b e r of items

Authors

11-20

9

0

16

21-30

7

1-5

13

31-40

3

6-10

3

41-50

2

11-15

1

51-60

2

I5-20

2

61-70

1

21-25

2

matical community from outside, by agents foreign to the community itself, as it was an expression of the character of a community shaped by the culture of the peculiar country that kept alive the last dictatorship of western Europe. Most Spanish mathematicians, including the ones holding institutional power, turned their backs on the values and working standards of other, more productive and creative mathematical communities. I shall illustrate the peculiar character of the Spanish mathematical community between 1940 and 1970 with data concerning publications. Table 1 contains data from M a t h e m a t i c a l R e v i e w s on the production of 37 leading Spanish mathematicians. It shows that 13 Spanish mathematicians each had, between 1940 and 1972, 10 or fewer articles or books reviewed in M R ; 9 of them had between 11 and 20 items reviewed; and so on. 7 Table 2 contains data concerning articles and books published in nonIberian languages by the same mathematicians. (Equivalent numbers, albeit not exactly the same, result from counting items published in nonIberian journals.) s What is most interesting in these numbers is what they hide. First, they hide the fact that a group of substantial mathematicians just worked for the internal market. That is, many mathe-

maticians who were highly productive or productive in Spanish published nothing or marginally in any foreign language. For instance, one mathematician appears with 47 items in Table 1 but has just 1 in Table 2 (Cuesta Dutari). Another has 57 articles in Table 1 but only 5 in Table 2 (Roddguez Salinas). Others have 33 v e r s u s 1 (Plans), 27 v s 0 (Botella), 26 v s 0 (Orts), 39 v s 5 (Abellanas). And so on. This is in contrast to what we see today in small and peripheral countries: high scientific productivity is always linked to international integration. 9 In Franco Spain however we have a negative correlation, with most of the people in the productive segment of Table 1 appearing in the non-productive segment of Table 2. Perhaps the most interesting thing implicit in the tables above is the mathematical careers of the 8 people most productive in Table 2, those publishing more than 5 items in non-Iberian languages between 1940 and 1972. They all were included among the deepest and most research-oriented Spanish mathematicians, and yet only two of them happily concluded their careers there, ending up heading big, well-endowed departments and as powerful and influential voices in the Education Ministry. 1~The careers of the other six top performers offer interesting evidence about the forces and values pre-

dominant in the Spanish mathematical life. Three of them left Spain in the fifties, one becoming a professor in Argentina (M. Balanzat), the other two to professorships in the United States (Gil Azpeitia and F. Gaeta). The remaining three are Ferran Sunyer, Ernest Corominas and Ricardo San Juan. Ernest Corominas (1913-1992) left Spain in 1939 as a political exile--in fact as an officer of the Republican Army. After a few years in Argentina and a doctorate in Paris with Arnaud Denjoy, he came back to Spain in 1952 and got a temporary appointment in the CSIC, helped by Rey Pastor. In 1955 he was allowed to take the salary with him to spend one year at the Institute for Advanced Study in Princeton. From 1956 to 1960 he was again in Spain, always a badly paid fellow in the CSIC. In 1960 he left, eventually getting a professorship at the University of Lyons in 1964, where he died a French citizen and an emeritus professor in 1992. After leaving Spain in 1960 it would have been possible for him to get a tenured position there, but he refused to apply for it. The main reason was, he explained to Sunyer, the kind of academic neglect in which he lived while in Spain, which Sunyer was still enduring. Our second biographical sketch is perhaps the most enlightening one.

7Table 1 features data gathered by the author from the 20-Volume Author Index of Mathematical Reviews 1940-1959 (2 vols., Providence, RI: American Mathematical Society, 1966), the Author Index of Mathematical Reviews 1960-1964 (2 vols., Providence, RI: American Mathematical Society, 1966), and the Author Index of Mathematical Reviews 1965-1972 (4 vols., Providence, RI: American Mathematical Society, 1974). The 37 mathematicians whose production is here surveyed are P. Abellanas, R. Aguil6, G. Ancochea, J. Auge, M. Balanzat, J. Botella, R. Cid, E. Corominas, N. Cuesta Dutari, A. Dou, J.-J. Etayo, B. Frontera, F. Gaeta, A. Gil Azpeitia, E. Lines, R. Mallol, F. Navarro Borr#.s, J. Ninot, R. Ortiz Fornaguera, J.-Ma Orts, P. Pi Calleja, A. Plans, P. Puig Adam, D. Ramfrez Dur6, J. Rey Pastor, S. Rfos, T. Rodriguez Bachiller, B. Rodr[guez Salinas, R. Rodrfguez VidaI, F. Sales, R. San Juan, Sancho Guimer&, Sancho San Roman, F. Sunyer, J. Teixidor, J. Vaquer, E. VidaI Abascal. 8Source: the same as Table 1. 9T. Schott, "Scientific productivity and international integration of small countries: Mathematics in Denmark and Israel, Minerva 25 (1987), 3-20. l~ Rios and G. Ancochea.

VOLUME20, NUMBER2, 1998 2 9

Ricardo San Juan is the most productive man in both Table 1 and Table 2. As a brilliant, hard-working young mathematician, he won a mathematical chair in Salamanca before the Civil War. He stayed aloof from the military conflict, and just at the end of it, in 1940, he got a chair in the University of Madrid. He twice won the "Francisco Franco" national scientific prize, in 1949 and 1954, as well as many other prizes and honors, and was elected to the Spanish Academy of Sciences in 1956. For many years his research was supported by the U.S. Air Force. Among the well-established Spanish mathematicians he was the one who most decidedly and tirelessly promoted Sunyer's career. San Juan was no doubt an idiosyncratic character. A misanthropic man, he lived in growing isolation, scientific as well as personal. But all this in no way explains why, from the late fifties to the end of his life in 1969, he was an ever more marginal figure. From occupying a prominent position in the forties and early fifties, he ended up a sick, secluded man who felt unjustly forgotten. The fact is that he was publicly attacked and humiliated by young---or not so young--figures who had managed to get powerful positions in the CSIC and the University of Madrid. His chief opponent was one of the leading mathematicians of the "internal market" school. Most interestingly, San Juan was a c c u s e d - - o f all things--of being an incompetent oldfashioned mathematician. Taking ammunition from the fad of the "new maths"--what the French used to call mathdmatiques modernes--university

30

THE MATHEMATICALINTELLIGENCER

committees publicly cast aspersions on San Juan, whose specialty was classical function theory. They managed to reduce his teaching to a few token classes, and eventually he completely withdrew from academic life. A concluding remark. It has been observed that underdeveloped countries may have scientists but lack scientific communities. We can recognize many features of the life of science in underdeveloped countries in the Spain of Franco--scanty means, ignorance of what is going on outside, internal isolation of the few active scientists, institutions oriented towards education rather than research, values and social preferences that lead the best intelligences towards the civil service and away from science, the predominance of powerful local "scientists" or "science personalities" whose source of power is only political and independent of scientific achievements. All these were present in Spain from 1940 through the 1960s. Yet there really was a full-fledged mathematical commun i t y in Spain. It had academic journals and an official society that convened annually to hear its members' papers. It awarded prizes and honors. There were many university positions and departments, and two research institutes within the CSIC. So everything was there--only it worked in an anom- foreign colleagues during Sunyer's alous way. Instead of elevating its life. The barriers were not material nor more prominent members (prominent physical nor imposed by politicians. according to explicitly endorsed stan- Rather they were intrinsic to the dards), most of them were "centrifu- Spanish mathematical community-gated" towards marginality or exile. built in, so to speak, as it grew rooted Here is the nature of the gap separat- in an idiosyncratic, authoritarian miing Spanish mathematicians from their lieu.

G. GELBRICH AND K. GIESCHE

I-ractal 1 sober Salamanders and Other Animals

9 n this article, we deal with a very special kind of tiling of the Euclidean plane. We will ~

give a method to construct all tilings of this particular type, as well as the procedure to visualize their tiles on the computer screen.

A tiling is a covering of the plane by a set of tiles which do not mutually overlap. We require that each tile be a b o u n d e d subset of R 2, which is equal to the closure of its interior, and that the family of tiles be locally finite. Let us start from mathematical art. Perhaps the reader already knows some of the tilings designed by Escher; for example, the one shown in Figure 1. Besides the remarkable fact that the tiles look like animals, we observe very special properties. It is fairly obvious that all tiles are congruent. Moreover, the tiling is invariant under the action of a lattice of translations. In addition, a rotation by 120 ~ around a point where the heads of three salamanders meet takes the tiling onto itself. So its symmetry group s is the crystallographic group p 3. This group acts transitively on the set of tiles. In other words, we can choose one of the animals as the "prototile." The copies of the prototile under the action of the elements of F yield the entire tiling. The tiles themselves have no symmetry, so if the prototile is fLxed, there is a one-toone correspondence of tiles and group elements. In this case, we will speak of a crystallographic tiling, or r-tiling, where F is the underlying group. In general, if the symmetry group F of a tiling acts transitively on the family of tiles, one can treat it as a F-tiling for an appropriately chosen subgroup F c F.

In Figure 2a we present a new salamander produced by our mathematical gene laboratory. You may object that this pitiable creature looks quite degenerate; it seems to have extra legs, the tail is brutally shortened, and so forth. Certainly this is true, but our salamander has a new mathematical property, namely serf-similarity. Figure 2b shows that this tile is the union of seven palrwise congruent and nonoverlapping sets, each of which is similar to the entire tile. These fractally-shaped salamanders, like the originals by Escher, tile the plane, and the symmetry group of this tiling is also p 3 (see Fig. 3). Self-similarity is not only a property of single tries but of the entire tiling, in the following sense. Consider the expanding similitude which carries tile n u m b e r 1 to the union of tiles 1-7. This map takes every tile to the union of seven tiles. The prototile of a self-similar crystallographic tiling is called a crystallographic reptile. The term "reptile" is a short form of "replicating tile" (a reference to the process of repeated subdivision and rescaling which constructs a self-similar set). On the other hand, it can be seen as a reflection of the appearance of these sets: the fractal boundary which is typical for self-similar shapes makes them look like wild animals or plants. Now we will describe the systematic construction of

9 1998 SPRINGER-VERLAG NEW YORK, VOLUME 20, NUMBER2, t998

31

crystallographic reptiles. Detailed proofs of the statem e n t s given here, as well as computations, can be found in [3] a n d [4]. For more information o n tilings and reptiles, see [1]-[5], [7], a n d [8], and the references therein. [5] offers also a concise a n d well-illustrated introduction to crystallographic groups. Let us fix the notations: A - - a crystallographic reptile; F--the underlying group of the associated tiling; lowercase Greek letters elements of F; k - - t h e n u m b e r of pieces of A; g - the e x p a n d i n g map of the plane that carries A to the u n i o n of k copies of A. So we have g(A)

Figure 1. Escher salamanders. M,C, Escher's "Symmetry Drawing E25" 9 1997 Cordon Art--Baarn-Holland, All rights reserved.

Figure 2. Salamander produced in mathematical gene laboratory,

32

THE MATHEMATICALINTELLIGENCER

= TI(A) U "-" U Tk(A).

for certain elements T1, 99 9 Tk E F. Note that the maps f i = g-1 Ti are contractions, and we have A = f l ( A ) U 9.. U f k ( A ) , so by a t h e o r e m of H u t c h i n s o n [6], the maps g, T1, 99 9 Tk determine the set A uniquely (among the n o n e m p t y compact subsets

Figure 3. Tiling by fractal salamanders.

o f R2). B e c a u s e of this, (g, ~/1, 99 9 ~/k) will be called the d a t a of A. So o u r aim is the c o n s t r u c t i o n of a p p r o p r i a t e data. Before doing this, w e s h o u l d d i s c u s s h o w to d r a w an image of A from k n o w n data, for a picture is m o r e pleas-

ing than a collection of affme maps. Simply c h o o s e an arbit r a r y point z E R 2 a n d plot the points f i l o f/2 . . . . . fire(z) for s o m e n a t u r a l n u m b e r m and all s e q u e n c e s with il, i2, 9 9 im E {1, . . . ,k}. I f m i s large enough, then y o u obtain a g o o d i m a g e of A on the c o m p u t e r screen. The r e a s o n is that the a s s i g n m e n t C F ( C ) = f t ( C ) U ... U f k ( C ) for c o m p a c t sets C is a contraction in s o m e space, and the sets Fro(C) converge to A b y B a n a c h ' s fLxed-point theorem. (This is the p r o o f of H u t c h i n s o n ' s theorem.) Coloring t h e points depending on it, t h e subdivision into p i e c e s is visible. There are b e t t e r algorithms for drawing self-similar sets, w h i c h shall n o t b e d i s c u s s e d here. To c o n s t r u c t "good-looking" reptiles, it is r e a s o n a b l e that we confme ourselves to sets A w h i c h are h o m e o m o r phic to a c l o s e d ball (we call such A disklike). The search for d a t a o f disklike crystallographic r e p t i l e s c o n s i s t s of the following steps.

Step 1. Select F to b e one of the 17 p l a n a r crystallographic groups. Suppose w e are given a F-tiling with disklike tiles. We say that two tiles a r e n e i g h b o r s if t h e y have a topological arc in common. We can assign to the tiling its n e i g h b o r h o o d graph ~ in the following way. In the i n t e r i o r of every tile ~(A) (~/E F), s e l e c t a reference p o i n t p~. The reference Figure 4. Fractals in the ocean. On the left: sea horse; on the right: coral reef.

VOLUME 20, NUMBER 2, 1998

Figure 5. Fractal plants. Don't touch the right one, it seems to be a carnivorous species.

points are the vertices of u3. For any two neighbors 7(A) and 7'(A), there is exactly one map ~? E F with 7~? = 7'. Draw a directed edge from p~ to p~, with label ~?. Notice two facts. First, u3 embeds into R 2. Second, edge labels belong to a fmite set S C F, namely ~? E S iff ~?(A) is a neighbor of A. As we can reach every tile starting at A and stepping from neighbor to neighbor, S generates s So (~ is nothing but the Cayley graph of s with respect to the generating set S. In case of our fractai salamanders, S consists of six rotations. Selecting tile number 1 in Figure 3 to be the prototile A, the m e m b e r s of S are the maps taking A to tiles 2, 3, 7, 8, 9, and 10.

that F must admit an isomorphism onto a subgroup of index k. Let us explain this. Let A be the prototile, and A ' some other tile. The map taking A onto A ' is in F, and the map taking g(A) onto g(A') belongs to s too. Suppose 7 ( A ) = A ' . Then, g ( A ' ) = gT(A) = gT(g-l(A)). So we have a map ap : F --~ F given by aP(7) = gTg -1, which is an isomorphism onto a subgroup. Because the copies of g(A) under the action of the subgroup (P(F) yield the expanded tiling (the original one, expanded by g), and area(g(A)) = k area(A), ~P(F) has index k.

S t e p 2. Choose a neighborhood graph. All possible neighb o r h o o d relations are listed in [5], pp. 288-290. You have to select one and find an appropriate graph. In fact, what is needed is the set S.

S t e p 4. Select the elements 71, 99 9 7k. We required A to be disklike, so g(A) = 71(A) U --. U 7k(A) is disklike, too. Consequently, the subgraph ~ ' C ~ spanned by the vertices p ~ must be connected. Furthermore, ~ ' must not contain any circles enclosing vertices which do not belong to ~'.

S t e p 3. Choose a n u m b e r k -> 2. Be aware that an unfortunate choice leads to an empty list of data. The point is

Taking the expansion 71-1 g instead of g moves only the expanded prototile but does not change A. So we will not

Figure 6. Fractal horaes--or dromedaries? This is left to the reader.

34

THE MATHEMATICALINTELLIGENCER

forget any reptile if w e a s s u m e T1 = id. Note that with this r e s t r i c t i o n w e have only a finite n u m b e r of choices in this step. S t e p 5. Finally, d e t e r m i n e an e x p a n s i o n m a p g from the h o m o m o r p h i s m r : F --->F. In m a n y cases, including F = p 3, g can b e c o m p u t e d directly from 4p. (See [3], e x a m p l e 3.3(b)). If not, g is d e t e r m i n e d up to s o m e parameter, which c a n be c h o s e n freely (this will only change the p o s i t i o n of A, b u t not its shape). The h o m o m o r p h i s m r is d e t e r m i n e d b y its values on the generating set S. In fact, w e only n e e d to k n o w t h e values on a minimal generating set S' C S. In the e x a m p l e from Figure 3, w e n e e d k n o w only the values of q) on S' = {~h, ~72}, w h e r e ~h (tile 1) = tile 2, and /)2 (tile 1) = tile 7. The m a p s (1 ~ 3) and (1 ~-* 8) are their inverses, a n d (1 ~-~ 9) and (1 ~-> 10) are c o m p o s i t i o n s of them. The value of qb(~?) for ~? E S' can b e c o m p u t e d from

g( ~?(A) ) = g~?g- l(g(A ) ) = (I)(~?)(TI(A)) U . " U r Since A a n d ~?(A) are neighbors, g(A) and g(~?(A)) are neighbors, too, so t h e r e m u s t be indices i a n d j such that ~i(A) and r are neighbors. Thus, t h e r e is s o m e /~ E S such that ~/i/~ = r or, equivalently, 4P(~7) = ~/d~/j-1. Select a function S' --* F with this p r o p e r t y and c h e c k if it e x t e n d s to a h o m o m o r p h i s m on F. S t e p 6. C o m p u t e the m a p s f i , d r a w the set A as d e s c r i b e d above, a n d e n j o y the pictures. You are n o t g u a r a n t e e d that every a r r a y of d a t a y o u find in this w a y l e a d s to a disklike reptile (or to a reptile at all).

However, the d a t a of all types o f disklike crystallographic reptiles in the p l a n e (up to affine change of c o o r d i n a t e s ) can be found by o u r algorithm. Because in Steps 2, 4, and 5, t h e r e are only finitely m a n y essentially different choices, for every k there is a finite n u m b e r of t y p e s of tiles. F o r example, if F = p 3, for k = 3 and 4, t h e r e are t h r e e and four tiles, respectively, [4]. Surprisingly, the n u m b e r i n c r e a s e s r a p i d l y for t h e next k which is p o s s i b l e w h e n F = p3, n a m e l y k = 7. While examining only 21 (out of h u n d r e d s of) configurations of the ~,i (Step 4), w e have found 82 disklike tiles, a m o n g t h e m the s a l a m a n d e r in Figure 2. Six m o r e o f t h e m a r e s h o w n in Figures 4-6. REFERENCES

1. C. Bandt, Self-similar sets 5. Integer matrices and fractal tilings of Rn, Proc. Am. Math. Soc. 112 (1991), 549-562. 2. C. Bandt and G. Gelbrich, Classification of self-affine lattice tilings, J. London Math. Soc. 50(3) (1994), 581-593. 3. G. Gelbrich, Crystallographic reptiles, Geometriae Dedicata 51 (1994), 235-256. 4. G. Gelbrich, Gruppen, Parkette und Selbst~hnlichkeit, Hamburg: Verlag Dr. Kova~, (1995). 5. B. GrQnbaum and G.C. Shephard, Tilings and Patterns, New York: Freeman (1987). 6. J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. 7. R. Kenyon, Self-replicating tilings, in Symbolic Dynamics and its Applications (P. Waiters, ed.), Contemporary Math. No. 135, Providence, RI (1992). 8. A. Vince, Rep-tiling Euclidean space, Aequationes Math. 50(1-2) (1995), 191-213.

VOLUME 20, NUMBER 2, 1998 3 5

i

~vj i~-'lL ;i i [~..I,

~ ii ~l [,.z-: u i

~ial(;-l"h~"~-Iiilii(=-~l]L~"-ll

Geometry Problems Revisited Alexander Shen

This column is devoted to mathematics

Alexander

Shen,

Editor

After publishing a column a b o u t twoand t h r e e - d i m e n s i o n a l g e o m e t r i c p r o b lems I got several letters a b o u t omissions in the column. F o r example: To The Editor: The Mathematical E n t e r t a i n m e n t s s e c t i o n (in Math. InteUigencer vol. 19, no. 3) tried to p r o v e geometrically t h a t if w e take three intersecting circles, the chords joining pairwise intersections have a p o i n t in c o m m o n . This is not quite true: the t h r e e lines are always in a pencil, b u t t h e y m a y be parallel r a t h e r t h a n concurrent. The intersecting circles

for fun. What better purpose is there for mathematics? To appear here, a theorem or problem or remark does not need to be profound (but it is allowed to be); it may not be directed

x 2 + y2 + x + 5y = 0

x2 + y2 + 2x + 5y = O x 2 + y2 + 5y + 2 = O, for instance, have all three c h o r d s parallel to the y-axis.

only at specialists; it must attract and fascinate. We welcome, encourage, and frequently publish contributions

William C. W a t e r h o u s e Department of Mathematics P e n n State University

f r o m readers--either new notes, or replies to past columns.

I have to confess that that is n o t the only i n a c c u r a c y in the column; a lot of details are omitted. One o m i s s i o n is of special interest. In the c h o r d p r o b l e m it is p o s s i b l e that chords do n o t intersect at all, only the lines to w h i c h t h e y belong do intersect:

Please send all submissions to the Mathematical Entertainments Editor, Alexander $hen, Institute for Problems of information Transmission, Ermolovoi 19, K-51 Moscow GSP-4, 101447 Russia; e-mail:[email protected]

THE MATHEMATICALINTELLIGENCER9 1998 SPRINGER-VERLAGNEWYORK

I

I

The p r o o f I gave d o e s not w o r k for this case. Indeed, it c o n s i d e r e d t h r e e spheres that have the given circles as d i a m e t e r sections, a n d t o o k the p o i n t in t h r e e - d i m e n s i o n a l s p a c e that belongs to all t h r e e spheres. And n o w there is no such point. However, there is a simple and rather general argument that allows us to extend the result to this case almost free. Let us consider the coordinates of the center points and the radii as variables. Then the coefficients of the chords' equations are functions of these variables, and the claim (three lines intersect in one point) b e c o m e s an equality. What functions are involved in this equality? It is clear that they are analytic (in fact, algebraic) functions. So if the equality is true on a set of configurations that has a non-empty interior, the analytic continuation principle guarantees that it is valid everywhere. And, of course, the set o f configurations with three intersecting circles has a non-empty interior, so w e are done. The s a m e r e m a r k c a n be applied to m a n y o t h e r g e o m e t r i c problems. As Vahe Y. A v e d i s s i a n p o i n t s out in his letter, the situation arises in a n o t h e r p r o b l e m m e n t i o n e d in the s a m e column: a b o u t the triangle and p e r p e n diculars. The p r o o f u s e d a t e t r a h e d r o n which d o e s not exist in all cases. But again the set o f configurations w h e r e such a t e t r a h e d r o n exists has a none m p t y interior a n d the claim is an equality b e t w e e n analytic functions, so w e can e x t e n d the result to all cases. However, s o m e caution is n e e d e d w h e n we use this argument: w e should carefully c h e c k t h a t all the functions a r e i n d e e d analytic. F o r example, t h e d i s t a n c e b e t w e e n a p o i n t a n d a line is not analytic w h e n the p o i n t c r o s s e s t h e line; it is only an a b s o l u t e value o f an

analytic function. This is why we often need to consider oriented lengths and angles to m a k e a general s t a t e m e n t that is true for all configurations. From a Special Case to a General Case This way of reasoning (first prove something for a special case and t h e n use some m e t a - a r g u m e n t to extend the result to the general case) is not limited to geometry. Here are some other nice examples. Hamilton-Cayley Theorem Let A be a linear operator i n n - d i m e n s i o n a l linear space. Let P(A) = det(A - A) be its characteristic polynomial. Then P(A) = O. To see why it is true (for matrices over C), we can use the following argument. First consider the case w h e n all the eigenvalues of A are distinct c o m p l e x n u m b e r s A1, 9 9 9 An. Then the operator A has an eigenhasis where the matrix is diagonal with At, 9 9 9 An on the diagonal. In this basis the e q u a t i o n P ( A ) = (A - A1) "-'(A - An) = 0 is evident, since the factors annihilate the coordinates one by one. Now we have to e x t e n d this result to the g e n e r a l case. Let us c o n s i d e r e l e m e n t s of the m a t r i x A ( a s s u m i n g s o m e basis is fixed) as variables. T h e n the coefficients of P are p o l y n o m i a l s in these variables, as are the entries of P(A). Therefore, the claim is j u s t a n equality saying that s o m e p o l y n o m i a l is equal to zero (actually we have n 2 equalities, n o t one). Since p o l y n o m i a l s are c o n t i n u o u s funct i o n s a n d o p e r a t o r s with d i s t i n c t eigenvalues form a d e n s e set, this equality m u s t b e true for all operators. Q.e.d. We can even make one step more and prove this theorem for a n y field. Indeed, these n 2 polynomials that form the equalities have integer coefficients, and these coefficients do n o t depend on the ground field. So if the equalities are true for complex n u m b e r s , it j u s t m e a n s that coefficients are all zero, so the s t a t e m e n t is true for any field (or ring).

Quantifier Elimination The term "meta-argument" r e m i n d s us of m e t a m a t h e m a t ics and mathematical logic. It is natural to ask logicians w h e t h e r they can justify some general scheme for this kind of reasoning. And indeed, several s c h e m e s of this type are well known. Here is one of them, the so-called quantifier elimination for algebraically closed fields of characteristic 0 (Seidenberg-Tarski). Consider formulas that c o n t a i n variables (whose range is some field), addition, multiplication, equality sign, logical operations ( a n d , o r , n o t , i f . . . t h e n ) and quantifiers (V, 3). Examples of formulas are

3 y ( x y = 1) and 3 x ( ( x 2 + p x + q = 0) a n d (x 2 + r x + s = 0)).

The first formula says that element x has a multiplicative inverse; the s e c o n d one says that two quadratic equations with given coefficients have a c o m m o n root. The quantifier elimination t h e o r e m guarantees that any formula is equivalent to a quantifier-free one. It is easy to see what are the quantifier-free formulas in our two examples: the first f o r m u l a is equivalent to x r 0 (here inequalities come into play); the s e c o n d one is true if a n d only if the resultant polynomial

det

l p q 0 0 l p q 1

r

s

0

0

1 r

s

is equal to zero. By "equivalent" we m e a n that both formulas are simultaneously true or false for any elements of any algebraically closed field of zero characteristic. Both examples are formulas with free variables (x in the first formula; p, q, r, s in the second one). F o r formulas without free variables the equivalent quantifier-free formula should be a logical constant t r u e or false, a n d we get the following statement ("completeness of the theory of algebraically closed fields of zero characteristic"): A n y theorem that can be expressed as a f o r m u l a and proved f o r some algebraically dosed f i e l d o f zero characteristic is a u t o m a t i c a l l y true f o r all such fields. To s h o w that this t h e o r e m is n o t s o m e t h i n g trivial, let us see w h y H i l b e r t ' s N u l l s t e l l e n s a t z is a corollary. One of the f o r m s of the N u l l s t e l l e n s a t z says that if the system PI(Xl, 9 9 Pn(Xl,

Xk) = 0 , Xk)

0

has no solution xl, 9 9 9 Xk E C, then there exist polynomials Q1, . 9 Qn such that P1Q1 + "'" + PnQn = 1. Let us assume for simplicity that the coefficients of P's are integers. Then the s t a t e m e n t that the system has n o solution can be expressed as a formula. Being true over C, this formula should be true for any algebraically closed field of zero characteristic (and therefore for any field of characteristic zero, for a solution in some field r e m a i n s a solution in the algebraic closure). So it is enough to show that if Q's with the desired properties do n o t exist, t h e n there is a field where the system has a solution. But this is easy: the nonexistence of Q's m e a n s that the ideal generated by P1, 99 9 Pk is not trivial, so it is contained in some m a x i m a l ideal I C C[xl . . . . , Xk]. Then C [ x l , . . . , Xk]/I is a field where the system has solution x l = [Xl], 9 9 Xk = [Xk], where [xl] is the image of the polynomial Xl u n d e r the factor-mapping C[Xl . . . . . Xk] ---) C[xl, 9 Xk]/I.

VOLUME 20, NUMBER 2, 1998

37

Make Your Dollar Bigger Now!!! Ivan Yaschenko Do y o u want to increase your dollar? To enter y o u need only one dollar bill. Education, employment, etc. aren't important. You can work in your spare t i m e . . , to make your dollar bigger NOW! The problem of getting m o r e money is rather old (probably as old as money itself). We even don't know the origin of the version of that problem considered here. I heard this question w h e n I was a 10th-grade student at Moscow High School no. 91. Quite recently, this question was mentioned in a sci.math.research article as "Margulis' Napkin Problem." As Jim Propp (MIT Math Department) wrote, R u m o r has it that all R u s s i a n graduate s t u d e n t s o f m a t h e m a t i c s k n o w how to solve this problem, but that as soon as they come to the U.S. (as so m a n y o f t h e m s e e m to do.O, they forget h o w it's done.

Here is the problem. Take a dollar bill and try to fold it somehow. The goal is to get a planar object (with multiple layers at some points) which has a perimeter bigger than the perimeter of the bill. You are not allowed to tear the bill (so it remains valid after you iron it back). Of course, this is not an exact formulation of the problem, since it is not clear what "folding" is. If you are lucky and suggest a solution that evidently satisfies any reasonable definition of folding, you are done. But if your folding is complicated enough, people may suspect that you've stretched the bill somehow (to a small extent, not tearing the bill). And you need an exact definition of folding if (after several unsuccessful attempts) you want to prove that such a folding does not exist. The first idea that comes to mind is the following: Take a line that intersects the bill's rectangle, and bend our dollar along that line (so the segment of that line bec o m e s a part of the perimeter, but some parts of the original perimeter line are n o w gone).

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THE MATHEMATICALINTELLIGENCER

This is the only allowed operation and it can be iterated. Surely this operation is a legal folding, but unfortunately it does not increase the perimeter of the figure. For Figure 1 this is an evident consequence of the triangle inequality:

Now let X be an arbitrary polygon (not necessarily convex, it can even have holes), and assume that we fold it along some (straight) line and get some polygon Y. We need to prove that perimeter of Y does not exceed perimeter of X.

Here is the sketch of a proof. We assume that X i s folded along a horizontal line. All new perimeter points belong to that line, and we have to prove that the total length of the added segments (there may be several of them) does not exceed the length of the disappeared part. Consider one of the segments (e.g., the segment AB in the picture). For each "new" point on that segment we find an "old" point that is replaced. To do that, consider a line perpendicular to AB and choose a point of the boundary of X that is closest to AB. (It may lie on either side of the line AB). This point is not included in the perimeter of Y, because it is covered by the other part of X after folding. Therefore, for any new point we have found a corresponding disappeared point. Of course, for computing perimeter we have to take into account the slope of the disappeared segments, but the perimeter can only be increased. This has ignored the case when two boundary points of X are symmetric w.r.t, line AB, and therefore coincide after folding. The perimeter loss is the same, however, since instead of two segments in the perimeter we now have only one.

This argument can be applied to all the new segments. As the disappeared parts do not intersect, we see that perimeter of Y does not exceed the perimeter of X. This argument may be applied to any reasonable curve, but things are even easier in our case since a folded dollar bill is a polygon. However, our notion of folding is too restricted--there are some natural foldings that cannot be represented as iterated folding along a straight line. Here is a simple example of a folding which is not covered by the definition above, but nevertheless evidently should be allowed:

We don't have a planar object yet, we need to put the outstanding part back into the plane somehow. The simplest way is shown already (fig. 3). We do not yet achieve our goal (the perimeter is smaller than before), but look: we replaced AB by the two segments AE and EB, and A E + E B > AB, so this part of the perimeter did increase. If we only manage to deal with the remaining part somehow, not losing this (rather small) increase in the perimeter, we are done. Let us change temporarily the definition of folding and make it as broad as possible: define folding as any mapping of the rectangle (i.e., the bill) into the plane that does not increase distances. This requirement is weak enough to allow us to give an example of folding that increases the perimeter. Add two new folding lines to our map (see picture 4a), then fold the outstanding part along those lines (picture 4b) A

B

E /

This folding doesn't increase the perimeter, but definitely shows that our definition of folding was too restrictive. Now we are ready to show the folding (or something close to folding) that does increase the perimeter; the only problem is that it is not clear whether it is a legal folding. Here are the instructions. The five segments shown in Figure 3a are folding lines (draw them using a ball pen, applying it with enough pressure to make bending easier).

D

~

F

Figure 4a 0

E B

A~ A

Figure 4b

D

and consider the orthogonal projection of the resulting construction onto a plane. We get a mapping that does increase distances; its image has perimeter equal to AE + E B + B C + CF + FD + DA (OF is vertical). However, very few people will agree to consider such a projection as folding. Let us show something that looks more like a folding but nevertheless increases the perimeter. We can fold the outstanding part of a dollar further like this:

D Figure 3a

Now fold the bill as shown in Figure 3b:

B

A

Figure 3b

D

Figure 5a

VOLUME 20, NUMBER 2, 1998

39

until the zig-zags are small enough; after that w e can try to make them horizontal with perimeter loss that is smaller than a perimeter gain based on A E + E B > A B . (The smaller is the height of the outstanding part, the smaller is perimeter loss.) Measurements s h o w that the perimeter does increase. However, the author still cannot prove that he does not violate the rules of the game (i.e., does not stretch the bill).

Maybe you can analyze this construction? Or find another construction that is easier to analyze? Or prove an upper bound for the perimeter of the folded bill for the most liberal def'mition of f o l d i n g . . . You have almost nothing to l o s e - - t a k e you dollar and make it bigger, bigger, and b i g g e r . . . The author is very grateful to Alexander Shen for help in preparing the illustrations and for many useful suggestions. [Editor's note: One possible definition of folding is the following: a continuous mapping f : B ~ ~2 where B is the rectangle (the bill) is a folding if there exists a triangulation of B such that the restriction o f f to any triangle preserves distances. The folding s h o w n in Fig. 3 satisfies this definition; it is not clear whether the final example does.[ Moscow

Center for Continuing

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LENORE BLUM, FELIPE CUCKER, MICHAEL SHUB, and STEVE SMALE

COMPLEXITY AND REAL COMPUTATION The goal of this book is to develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing. The book opens with a foreword by Richard M. Karp. 1 9 9 7 / 4 5 3 PP./HARDCOVER $39.95/ISBN 0-387 98281-7

KARL E. GUSTAFSON and D.K.M. RAO

NUMERICAL RANGE The Field of Values of Linear Operators and Matrices "Ve O" readable, well-written introduction. Each chapter concludes with interesting notes and references. Highly recommended. Undergraduates attracted by linear algebra will enjoy this book. " AMERICAN MATHEMATICAL MONTHLY

1996/208 PP./S0FTCOVER/$34.95 ISBN 0-387-94835-X UNIVERSlTEXT

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A HISTORY OF CHINESE MATHEMATICS Translated from the French by S, WILSON

This book is uniquely accessible, both as a topical reference work, and also as an overview that can be read and reread at many levels of sophistication by both sinologists and mathematicians alike. Indeed anyone with an interest in Chinese culture or the history of ideas will derive great benefit from this book devoted to the history of Chinese mathematics from its origins to the beginning of the 20th century. 1997/450 PP./HARDCOVER/$59.00

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GEOMETRY: PLANE AND FANCY This text offers students a fascinating tour through parts of geometry they are unlikely to see in the rest of their studies while. at the same time, anchoring their excursions to the well known parallel postulate of Euclid. The author shows how alternatives to Euclid's fifth postulate leads to interesting and different patterns and symmetries. 1 9 9 8 / 1 7 6 PP., 117 ILLUS./HARDCOVER $34.95/ISBN 0-387-98306-6 UNDERGRADUATE TEXTS IN MATHEMATICS

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40

THE MATHEMATICAL INTELLIGENCER

Reference: H225

Innlit=~v~N~-1~,t~Ji~.k|t.~.mlnm~.1,nd~i

A Mathematics Treasure in California S.I.B. Gray

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafd 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 to this column 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]

Dirk Huylebrouck,

Editor

here should tourists go to see outstanding collections of historic mathematics books? Most of us would try Europe first, perhaps the Biblioth~que nationale de France in Paris, founded in 1367, before Gutenberg's time. At least since Mersenne, many outstanding mathematicians have enjoyed scholarly exchange in France. Thus, we might expect to find extensive collections of manuscripts and letters in Paris [http://www.bnf.fr/ bnfgb.htm]. We could also travel across the Alps to Italy where we would find another candidate for the world's finest mathematics library in the Vatican. Founded in 1447, for handwritten manuscripts, the Biblioteca Apostolica Vaticana is especially strong in pre-Gutenberg materials [http://sunsite.unc.edu/expo/vatican. exhibit/Main_Hall.html]. Although the Vatican has its antecedents in the Middle Ages and the Roman Empire, we note that the most famous center of learning in antiquity, the library in Alexandria, was completely destroyed by fire. Thus, the Vatican Library, which might be expected to have had access to extremely rare ancient materials, found few surviving manuscripts. We might also expect to find great collections in Florence, the home of the Renaissance with its rebirth of scholarship founded on antiquity. Modern tourists can view the Biblioteca Medicea Laurenziana, founded in 1444 by Cosimo de' Medici. He chose Michelangelo as the architect to house the great family collection. But its collections were limited by the size of the room. When a room was filled, a new collection was started and moved elsewhere. Also, the Laurentian Library was organized while Florence was a seat of power and wealth, but long before the great mathematics of the 17th century. What about the Teutonic world? The rise of Wissenschaften in the 19th century was coupled with the acquisition of major mathematics collections.

W

r

[

But these collections were dispersed. With GOttingen, Munich, Berlin, and the more modern Max Planck Institutes, for example, all sharing a strong interest in a limited number of manuscripts, no single library is dominant today. Also, pillaging after warfare, dating at least to the 17th century, has diminished the chances of finding any one outstanding collection intact in Germany. Another factor is that 19th-century historians and librarians in Germany valued antiquity. A Mommsen would write about the use of Roman coins, but probably never realized that his contemporaries-G. Cantor, Kronecker, and Hilbert-would have a major impact on future mathematical thought. When D.E. Smith arrived in Germany in the 1880s to study the history of calculus, his mentor, M. Cantor, said, "Well, Mr. Smith, if I were you, I should not go back much father than Antipho and Bryso,'--both of the fifth century B.C. (1, p. 310). M. Cantor, as a historian, was not focused on the contributions of Barrow, Newton, and Leibnitz. The English-spealdng world is similar to the German world in its complexity. Many excellent collections exist in Cambridge, at the Bodleian Library at Oxford, and at the British Museum, the Royal Society, and the Royal Institution in London. In Britain, one's choice of the premier library would depend on what century, what language, what translation, and whether one seeks books or manuscripts. (See Fig. 1.) The Bodleian, founded in 1602, is the oldest library in England and thus has a special position. Posterity in general and mathematicians in particular owe a special debt of gratitude to the Bodleian and the Vatican libraries. They house copies of the two oldest editions of Euclid's E/ements (888 and ca. 900 A.D., respectively). Most mathematicians would agree that if one had to select a single title as the most important contribution from our discipline to civilization, the E/ements would be named

9 1998 SPRINGER-VERLAG NEW YORK, VOLUME 20, NUMBER 2, 1998

41

Figure 1. From an illuminated medieval manuscript in the collection of the British Library and shared by the British Museum, This work was executed by a London scribe named Matthew in 1247, A woman is instructing Benedictine monks in geometry and mathematics.

first. Euclid is simply "the most successful textbook writer the world has ever known" (7, I, p. 103). Are there any great collections in North America? In fact, there are quite a few. The Widener at Harvard, the Beinecke at Yale, the Regenstein at Chicago, and the Bancroft at Berkeley all have wonderful collections. Columbia University has the awesome Plimpton Collection of clay tablets dating to the Old Babylonian period (ca. 1900 to 1600 B.C.)~ The Library of Congress has the Lessing J. Rosenwald Collection of 2,653 rare mathematics books [http://lcweb. loc.gov//cc/rarebook/guide/ra057001. jpg.]. The Artemas Martin Collection at American University is rich in 19th-century textbooks. Many others deserve a

42

THE MATHEMATICALINTELLIGENCER

visit. But a superb collection--arguably one of the finest in the world--is found in Southern California, of all places, near Los Angeles. The Collector

The collection was built with wealth-the wealth of Henry E. Huntington, the nephew of a 19th-century railroad "robber baron." As the heir of one of Oscar Lewis's Big Four (Leland Stanford, Mark Hopkins, Charley Crocker, and Collis P. Huntington), Henry shared in the forttme derived from building and joining t h e Central Pacific and Union Pacific railroads in Promontory, Utah Territory, 1869. Not one, but several spikes of gold, silver, and alloys were driven, representing the enormous

wealth of joining the West Coast to the cities of the eastern United States [see http://www.huntington.org]. Symbolically, the wealth of the American West combined with the more refined tastes of the eastern elite. When his uncle died, Henry was his favorite aide and confidant. He was managing the railroad-owned streetcar system that is so popular today with San Francisco tourists. Henry relocated to Los Angeles, and carried with him his knowledge of developing valley and foothill lands by controlling the rate and direction of public transportation. The business experience acquired in developing access to San Francisco's many hills worked with equal success in the vast Los Angeles basin. Like his uncle, Henry continued to make money--and to hold on to it. He even outdid his uncle by consolidating the family fortune. Henry married his Uncle Collis's widow, a woman slightly his junior. Both of them admired Britain, and the British passion for paintings, silver, porcelain, and furniture. Henry was especially fond of rare and elegant book collections. From the time (1910) he moved into his newly built mansion in the Los Angeles suburb of San Marino until his death (1927), he bought en bloc every important library that came on the market in Britain. The power of his dollar so dominated the bibliographical markets of the world that American, British, and Continental estate owners simply emptied their shelves of hereditary collections. In this brief interval of time he acquired a Gutenberg Bible on vellum, the Ellesmere manuscript of Chaucer's Canterbury Tales, an unsurpassed number of original early Shakespearean editions, and an Audubon double-elephant folio of The Birds of America. He also inadvertently acquired mathematics titles. The British and Continental landed gentry had been purchasing mathematics publications since the 17th century. The libraries consolidated by Henry Huntington reflected the educated tastes of connoisseurs who were aware of the achievements of their contemporaries. Later, Henry's collection was augmented by the efforts of astronomers who worked at Caltech's observatories.

Both George Ellery Hale and Edwin Hubble were devoted book collectors. Both were friends and colleagues of E. T. Bell. The Huntington's mathematics collection was Bell's primary source of scholarly references. In addition, Bell used the Huntington as his benchmark in guiding other collectors. For example, Bell strongly advised the eminently successful Kentucky actuarial lawyer, William Marshall Bnllitt, to try to acquire a copy of Niels Abel's Mdmoire, the now-famous eightpage pamphlet published at Abel's own expense in 1824. Bullitt managed to purchase a copy from the widow of an actuarial professor in Oslo (see [4]). The only other known surviving copies are in G6ttingen and in the Mittag-Leffier Library in Stockholm. Student Appreciation and Four Titles What might a late-20th-century student of mathematics, knowing no Latin, be able to appreciate upon viewing the collection at the Huntington? The illustrations and many of the calculations show clearly that mathematics is a universal language that overcomes the limitations of words. The illustrations and diagrams, drawn by hand, are superb examples of craftsmanship. In most cases, the implications and detail are left to the reader, a procedure prevalent today as well. Recently, my students, on viewing the Huntington collection in chronological order, remarked that Euler's Introductio was the first to look like a math book. By the 18th century, the equation and notation had evolved, as well as the printing techniques, to have math equations, not verbal explanations, embedded in the text. Calculus students will immediately recognize that Euler was worldng on infinite series. Let's now examine four titles from the Huntington's Collection to illustrate the value of showing our students works that in many instances predate Columbus's discovery of North America. The first is Ptolemy's Almagest, or "the greatest." Almagest Ptolemy wrote in Greek in the second century A.D. He produced the definitive

Greek work on determining the location of the planets. His mathematics and its geocentric theory stood unchallenged for 1400 years until Copernicus proposed his heliocentric theory in 1543. Viewing his page upon page of chord charts, representing essentially the first trigonometric tables, is an incredible experience. The charts are a compilation of small, meticulously hand-written numbers. Each number records a value painstakingly determined. This surviving copy uses the sexagesimal number system with Arabic [not Hindu-Arabic] numerals, the precursor of our modern counterparts of degrees, minutes, and seconds. It is thought to have been produced in the south of France, but the monastery, or atelier, is unknown. This translation is taken from Gerard of Cremona's work using an Arabic edition that entered Europe via the Moors in Spain. The charts raise many questions. Mathematicians wonder what techniques were used to extract square roots. How were the calculations made? What theorems were applied? Did Ptolemy really have the modern sum and difference formulas for chords? Claudius Ptolemy was a Roman citizen from a Greek family who is thought to have spent his entire adult life in Mexandria, Egypt; thus, Ptolemy most likely had a knowledge of Latin, Greek, and Aramaic. For careful study of his work, the sexagesimal, Greek, Roman, and modern number systems must be understood. Calculations for a circle are based not on 360~ but units which divide a circle into 120 parts. Values in modem trigonometric tables are ratios. Ptolemy, in the prevailing mode of his era, gave lengths of chords in a circle of radius 60, the base of his sexagesimal number system. The Huntington's copy of the Almagest is illuminated with gold and color adornment. The vellum is not thick but has an unforgettable texture of strength tempered with the appeal of the freest translucent paper. Elements The Huntington has more than 30 editions of the Elements, mostly in Latin, Greek, and English. But not all books

have survived in all editions. Erhard Ratdolt's translation in Venice in 1482 was the first to be printed. Greek and Latin versions started appearing in many European countries. From the 17th to the 19th century, mathematicians in Britain often produced their own versions. Marginalia, illustrations, and comments often become incorporated with the text in subsequent editions. Todhunter's edition is an example which is still often found in public libraries. For those familiar with a modern presentation of Euclid's axiomatic system, the 1482 edition contains some surprises. The manuscript opens immediately with 23 definitions, with the first three being those of the point, line, and plane! "A point is that which has no parts." "A line is without breadth." (What is a part? What is breadth?) Since David Hilbert's Grundlagen der Geometrie (1899), modern mathematicians have accepted these terms without definition in constructing an axiomatic system. Euclid never thought to do so. M a n y - - m o s t - - o f the illustrations are immediately recognizable to any student of geometry, e.g., "diameter, circulus, major, minor, semicirculus, eqlaterus, ppendicularia [sic]." Impressively, the Latin and illustrations of "punctus, linea, plana," are clear to English readers 500 years later. On the second page, Euclid immediately sets forth ten principles of reasoning, his five mathematical postulates, and five "common notions." In the lower left-hand margin is the illustration for the famous fifth, or parallel postulate. For a mathematician, the sight evokes the later drama of Saccheri, Gauss, Lobachevsky, Bolyal, and Riemann. In later pages, two illustrations clearly communicate Euclid's "windmill" or "bride's chair" proof of the Pythagorean Theorem (Figure 2); also to be found is his "elefuga" or "pons asinorum" proof of the equality of the base angles of an isosceles triangle. Some scholars admire a particular book of the Elements. E. T. Bell, for example, labeled Book V, or the rigorous introduction to the notion of continuity, a "masterpiece" (1, p. 311). Others (2, p, 131) have expressed

VOLUME 20, NUMBER 2, 1998

43

"fear" as well as admiration for Book X, which includes his geometric treatment of incommensurables, i.e., irrational numbers. The Huntington has two copies of the first published English translation of Euclid's E/ements (London, 1570). The copies, which are slightly different, purport to be the work of Sir Henry Billingsley, who later became Lord Mayor. Billingsley's E/ements contains pop-up, three-dimensional models embedded in the text. Among others, a reader may assemble a pyramid, a tetrahedron, and perpendicular planes. Also, we find delightful English expressions, e.g., "A cube number is that which is equally equal equally or which is contained under three equal numbers" (VII.19).

lished in the entire Western Hemisphere. Twenty-four of the 206 pages (103 folios) are devoted exclusively to arithinetic and algebra, while the rest

cover the purchase price of various grades of silver, the purchase price of gold, percents, exchange rates, taxes, and other monetary affairs.

Analyse des Infiniment Petits L'H6pital's Analyse is the first differential calculus book to appear in print. In the text appears his eponymous rule, which is almost certainly the work of John Bernoulli (3, p. 430). L'HSpital was an exceptionally clear and concise writer. There is no elaboration. A book that dominated mathematics as a principal text for most of one century needed only a few well chosen w o r d s - - n o t equations--to make its points. This stands in sharp contrast to today's calculus texts. It is noteworthy that the only re-issued second edition among our selections is L'H6pitars Analyse. The first edition was published in 1696, with the reprinted version appearing 12 years after his untimely death in 1704. All other books on our list are original first editions. Sumario Compendioso

The first mathematical work printed in the New World predates all North American settlement, i.e., Jamestown, Plymouth Colony, and Quebec City. (Figure 3.) The Sumario Compendioso was written in Spanish, not Latin, by Brother Juan Diez, "freyle," and published by Juan Pablos Bressano, in Mexico City in 1556. We read, "El qua] fue impresso en la muy grande y muy leal ciudad de Mexico." The Sumario was the fn~st textbook of any kind, other than religious instruction, pub-

44

THE MATHEMATICALINTELUGENCER

Figure 2. A marble, Derby porcelain, and ormolu "mathematics" mantle clock created in London, ca. 1787, by the Benjamin Vulliamy atelier. Vulliamy is known to have been a special favorite of George III. The clock was purchased by Henry Huntington in 1913 for his new home then under construction in the Los Angeles suburb of San Marino. The scroll (below) illustrates what is known variously as the Bride's Chair, Windmill, or Peacock proof (I, 47) of the

Pythagorean Theorem. (Courtesy of the Huntington Library, Art Galleries, and Botanical Gardens, San Marino, California.)

Proof 1

The "cosa" is 4~ and the square of the cosa is the quadrado.

This will bear comparison with Brother Juan's far bettter known French contemporary, Francois Vi~te (1540-1603). Vi~te solved a quadratic x 2+ax=b by using x = y ~ to eliminate the linear term, and then also taking a square root. Neither had the advantage of modern notation. Of the other two known copies of the Sumario, one is in the British Museum and the other at the Biblioteca Nacional in Madrid. Henry Huntington himself acquired the California copy in 1920, presumably not influenced by any philosophy of multiculturalism. David Eugene Smith of Columbia, a former MAA president, published a translation which you may borrow from UC Santa Barbara or UC Berkeley. [You may note Cajori borrowed the Berkeley copy several times.] Also in Henry Huntington's day, and presumably heedless of gender equity issues, the Library acquired two titles by the Marquise du Ch&telet. In fact, the Library has three copies of her famous translation of Newton's Pr~nc/p/a. It also has Maria Agnesi's Instituzioni Analitiche, with her famous illustration of the versed sine curve, the "Witch of Agnesi." --

Figure 3. The Sumario Compendioso (1556), the first book other than religious instruction published in the Western Hemisphere. The cover is of primitive leather. You are looking just inside the front cover. A piece of recycled printed material has been used to cover the rough interior of the skin. It is pasted with the printing turned vertically. Paper must have been in very short supply in Mexico City at that time. Beneath the magnificent frontispiece on the

right is written, " . . . questions of silver and gold in the kingdoms of Peru (sic) that are necessary in the markets and for general business. Also, some rules of arithmetic."

What type of algebra was being published in 1556? I quote from the first problem.

its own square root." The problem is followed by the "regla," or rule, and the proof, similar to the following:

Primera quistion [cuesti6n]

Regla (Rule)

"Da me un numero quadrado que restando del, 15, y3, quede fu propria rays [raiz]." or, "Find a square number from which if 15~auis subtracted, the difference is

Let the number be cosa. One-half of cosa squared is 4 of the 3 zenso (quadrado). Adding 15 and ~to -1. makes 16, of which the root is zt 1 4, and this plus ~ is the root of the required number.

a

Visiting the Huntington

Figure 4. Mathematics students visiting the Huntington Library. Notice the darkened room and gloves worn by Dr. Ronald Brashear, Curator, History of Science.

These editions are priceless and fragile. (Figure 4.) The Huntington Library seldom displays more than one or two mathematics and science books in one exhibit. Degradation associated with light and humidity is a particular concern. Security is another. At the present time should you, as a member of the general public, become one of the half million tourists who visit the Huntington Library each year, you would fmd only two works on display: Hubble's copy of Coperuicus's De Revolutionibus Orbium Coelestium and Galileo's first illustrations of moon craters from Sidereus nuncius: Venice, 1610. To search more freely, you will

VOLUME 20, NUMBER 2, 1998

45

need to make an appointment with the Library staff.

Acknowledgment I thank Dr. Ronald Brashear, Curator, History of Science, Huntington Library, for his assistance and encouragement. REFERENCES

For supplementary materials, see 1. E.T. Bell, Mathematics, Queen and Servant of Science, McGraw-Hill, New York, 1951. 2. C. B. Boyer and U. Merzbach, A History of Mathematics, 2nd ed., John Wiley & Sons, New York, 1989. 3. R. Calinger, ed., Vita Mathematica, Mathematical Association of America, Washington, DC, 1996. 4. R. M. Davitt, William Marshall Bullitt and his Amazing Mathematical Collection, Mathematical Intelligencer 11:4 (1989), 26-33.

46

THE MATHEMATICAL INTELLIGENCER

5. W. Dunham, Journey Through Genius: The Greatest Theorems of Mathematics, John Wiley & Sons, New York, 1990. 6. G. F. Simmons, Calculus Gems: Brief Lives and Memorable Mathematics, McGrawHill, New York, 1992. 7. D. E. Smith, History of Mathematics, Ginn, Boston, 1923. 8. C. Tanford and J. Reynolds, The Scientific Traveler, John Wiley & Sons, New York, 1992.

For further reading on Henry Huntington, see 9. O. Lewis, The Big Four, Alfred A. Knopf, New York, 1945. 10. J. Thorpe, Henry Edwards Huntington: A Biography, University of California Press, Berkeley, 1994.

For teaching a course on the History of Mathematics, see

11. F.J. Swetz,Some Not So Random Thoughts about the History of Mathematics--Its Teaching, Learningand Textbooks, PRIMUS, 5 (1995) 97-107.

For translations, samples of original sources, and extensive bibliographical essays, see 12. D.E. Smith, The First Work on Mathematics Printed in the New World, American Mathematical Monthly, 28 (1921), 10-15. 13. N.G. Wilson, From Byzantiumto Italy.: Greek Studies in the ItalianRenaissance,The Johns Hopkins UniversityPress, Baltimore, 1992. Departmentof Mathematicsand Computer Science CaliforniaState University, Los Angeles Los Angeles,CA 90032-8204 e-mail: [email protected] http://web.calstatela.edu/faculty/sgray/sgray.htm

KARL GUSTAFSON AND TAKEHISA ABE

(Victor) Gustave Robin: 1 855-1 897 ustave Robin, a professor of m a t h e m a t i c a l p h y s i c s at the Sorbonne i n Paris, died 100 years ago. In c o m m e m o r a t i o n o f his work i n m a t h e m a t i c s and physics, we look at Robin's contribution to science. In particular, we describe all o f his published works as w e know them. To our knowledge, these indeed are all o f his written works, except those which he burned shortly before his premature death. We also attempt a portrait of his life, although we have precious little factual data about it. Part of the difficulty is that Robin was only 'Chargd de cours h la facult~ des sciences de Paris,' never Professeur Titulaire. We were astonished that our searches through the archives of Paris turned up absolutely nothing about him, beyond his birthdate, May 17. We are still hopeful that more information might turn up, possibly as a result of the publication of these essays. In the first part [1] of this two-part essay, we discussed primarily the third boundary condition c~U

dn of potential theory. This is what led us to Robin. The first and second boundary conditions of potential theory carry the well-known names Dirichlet and Neumann, but Robin's name on the third boundary condition occurs only more recently, sporadically, and even problematically: to our knowledge, Robin never used this boundary condition. Yet as we explained in [1], there are good reasons for Robin's name to be so remembered from his work in potential theory. Now we turn to his work in general. Beyond potential theory, Robin's most notable contributions were in thermodynamics. Robin's Contributions to Science We have listed all of Robin's published works in the bibliography. It may be observed that these works represent

three principal thrusts. First, there was Robin's method of single- and double-layer potentials for boundary value problems of electrostatics. This is Robin's thesis [3R] and is reproduced as the main part of [7R]. We may describe this thrust somewhat loosely as the work of Robin's graduate student days, although the results of course show a maturity beyond that level. Second, there are Robin's Sorbonne lectures on thermodynamics, compiled by Louis Raffy in [8R]. According to Raffy, Robin "never ceased to think about (thermodynamics), from the age of 20 until the last months of his life." We may nonetheless loosely describe this thrust as that of his professor days. Third, there is the somewhat mystifying development of real analysis in [9R]. According to Raffy, he and Robin already convinced themselves in the Lyc~e that analysis must be based solely upon the rational numbers, excluding the irrationals. We could, therefore, somewhat irreverently describe this thrust as that of Robin's high-school days. That would be unfair, however, inasmuch as Robin as instructor presented elements of this approach to analysis to students in physics and chemistry at the Sorbonne in 1892-1893. Robin then lost interest in this development, but Raffy persevered, and [9R] is written from Raffy's notes, not Robin's. One can safely say that Raffy felt more strongly about publishing this than Robin did. Overall, it appears that mathematics was more a method than an end for Robin. When he speaks of the need for rigor in science, he does not mean existence proofs. His goal seems to have been to clarify certain topics in physics and chemistry by placing them in a clear mathematical frame-

9 1998 SPRINGER-VERLAG NEW YORK, VOLUME 20, NUMBER 2, 1998

47

work; at the same time, he insisted that the only theories of value were those that could be directly validated by physical experiment. Mathematical rigor in the sense of clarity, then, and physical rigor in the sense of reality. Robin was what we now call a mathematical physicist.

Robin and Potential Theory [3R, 4R, 6R, 7R] We have already described in [1] the essentials of Robin's method for solving electrostatic boundary-value problems by use of single- and double-layer potentials. An important physical motivation for this is the notion that static electricity equidistributes itself on the outside of a perfect conductor. Then there is no electromagnetic field, and the electric potential becomes stable. This coexistence of physical reasoning with mathematical results is characteristic. Although Robin's work in potential theory is remembered (see below) in a rather pure mathematical form, Robin's thinking was firmly grounded in physical experience. The f'ffst-named author (K.G.) would like to offer an anecdote here. Shortly after being nearly hit by lightning on top of E1 Diente, a 14,000 ft. peak in the Colorado Rockies, he recounted the incident to Dr. James Tuck, at that time head of the Los Alamos fusion project. Dr. Tuck's hobby was lightning in all its forms, and he suggested the practice that climbers should carry along something like very heavy battery cable, and when in a lightning predicament, they should just squat inside a perfect conductor formed from the cable. The conclusion of the conversation was that it would be best if Dr. Tuck would try this idea out himselfi. Robin would have sympathized. He always tried hard to match theory to experiment. For example, his four posthumously published notes (Fragments divers of [7R]) all try to place important physical experiments in electromagnetism (e.g., those of Thomson, Hertz) on a fnan mathematical basis. In Figure 1 below we show Robin's diagram from his Thdorie du "replenisher" de Thomson in [7R]. This illustrates Robin's standard for physical, as well as mathematical, clarity in his thinking and work. Without doubt Robin's dissertation work in potential

A

§ (-'~§162

B'

Figure 1. Diagram of Thomson's electrocharge supplier. (From [7R], Fig. 2, p. 129.)

48

THEMATHEMATICALINTELLIGENCER

theory was his most important contribution to mathematics. The Dirichlet and Neumann boundary-value problems of partial differential equations were reduced to integral equations for electric densities and to power series expansions for potentials. The single-layer equilibrium potentials were later called the Robin-Steklov potentials. The perturbation power series expansions of unknowns were later called the Robin-Steklov method. The criteria for uniform convergence was later called the Robin principle. What is now called the Robin Constant in the theory of transfinite diameter in measure-theoretic potential theory is a logarithmic capacity determined by use of the Robin potential. The Russian school of V. Steklov (1864-1926), A. Lyapunov (1857-1918, Steklov's advisor), and N. Gtinter (1871-1941), and the French school of Picard and later Goursat, regarded the mathematical research of Robin as fundamental and of great merit. It is almost certain that the names of Robin and Steklov were eventually attached to the third boundary condition--Robin for the case a > 0, Steklov for the case a < 0--in recognition of their work in potential theory.

Robin and Hydrodynamics [5R, 7R] Oversimplified, this work may be described as the application of the methods of (electrostatic) potential theory to find the solution of propagating shock waves in fluids. This leads to some integral equation formulations for the vorticity equation for incompressible fluids, and some methods for construction of solutions to Euler's equations for compressible fluids. Hydrodynamics was apparently not a major interest of Robin. However, again we note that Robin tries to put the physics into a clear mathematical setting. In dealing with explosions in fluids, he considers the speed, force, energy modelled by use of Poisson's equation, following along the line of G. Green and G. Stokes. In particular he calculates a fluid potential for an ellipse-shaped boundary. When dealing with Helmholtz's vorticity theorem, which says roughly that if one ignores viscosity, a vortex remains always a vortex of the same strength, Robin recognized this as a condition of integrability for the velocity, and was therefore able to solve the problem by means of Poisson's equation. Robin and the Foundations of Analysis [9R] We are not comfortable in evaluating the importance of this work which, as we expressed above, seems to have been more to the liking of Raffy than to Robin. We also leave it to others to determine its originality. See Raffy's Preface to [9R] for his motivation. Perhaps the goal was simply to see how far one could go basing real analysis only on the rationals. In addition to the calculus, Fourier series and firstorder partial differential equations are treated in this way. The notion in [9R] that two fractious are equal if they each denote the same ratio of objects chosen from two different sets of objects is perhaps some attempt to reduce notions just to those of elementary set theory. It must be remembered that work on the foundations of the real number sys-

tern, e.g., the work of J.W.R. Dedekind and G. Cantor, was still in its infancy at the turn of the century. In any case, we recuse ourselves from further explanation of [9R].

Robin and Thermodynamics [1R, 2R, 8R] Beyond his early work on single- and double-layer potentials, clearly Robin's second major contribution to science was his effort in thermodynamics. However, because thermodynamics is such a contentious field, it is difficult to judge the quality and importance of his work. In the pungent words of C. Truesdell [2], thermodynamics was in its early history (the Caruot period) a "tragicomedy" and remains "a dismal swamp of obscurity." The problem with thermodynamics treated mathematically is that it needs to include all the effects such as friction, irreversibility, disorder, that we prefer to neglect in Hamiltonian systems. Robin as mathematician and in his lectures at the Sorbonne tried to put more rational sense into the foundations of thermodynamics. In so doing, as we shall describe below, he found himself in the middle of pervasive scientific and philosophical conflicts within the French intellectual community. About the possible effects of these disputes upon Robin's mood and work in the last years of his life, we can only wonder. However, there can be no doubt about his systematic approach toward science, and toward the foundations of thermodynamics in particular. Robin's first paper [1R], although published in 1879 when Robin was 24 years old, probably dates back to the age of 20 when (Raffy says) Robin become interested in thermodynamics. The work [1R] is included and elaborated by Raffy in [8R, Chapter XIV]. There [8R, P. 251] Raffy begins by presenting material from an (unpublished) letter of Robin, written in August 1879. However, on p. 252 of [8R] we find a footnote which refers to a paper Les gas parfaits suivent la loi de Dulong et Petit published in the Bulletin of the Soci~t~ Philomathique on 11 November 1876.* In his first paper [1R], Robin begins with reference to work of Clausius, who, to introduce temperature, assumes that the average of the molecular forces within a body depends only upon the temperature, and not upon the positions of the individual molecules nor upon the total volume. According to Robin [1R, p. 9], Malgrd la grande autoritd de M. Clausius, ce f a i t ne p a r a i t pas dvident. Si, m a i n t e n a n t u n corps d tempdrature constante, on f a i t varier la pression qu'il supporte, son volume change: les trajectoires des moldcules sont modifides; et rien, me semble-t-il, n'autorise d admettre que la force vive moldculaire reste la m#me dans ce nouveau m o u v e m e n t du systOme que dans l'ancien.

Robin then goes on to show that under Clausius's assumption, the heat q must be proportional to the absolute temperature, q = kT. From this, Robin derives Clausius's expression for the entropy. As elaborated in [8R, Chapter XIV], Robin's unpublished letter of August 1879 and the paper [1R] then lead to two main conclusions. The first is that the law of P. L. Dulong (1785-1838) and A. T. Petit (1791-1820), which was a law of caloric heat formulation for solids, holds as well for ideal gases. Secondly, in a perfect gas, the specific heat per volume is invariant. In his closely following second paper [2R], Robin deals with state changes, absorption or emission of heat, and the influence of pressure thereon. He shows that given a state change at predetermined temperature, there will be only one pressure for which the transformation will be reversible. This work is elaborated in [8R], where it is broken into two parts, that commencing on p. 160 in Chapter IX and that commencing on p. 186 in Chapter X. This work by Robin would eventually culminate in what is now commonly referred to in the literature as Robin's L a w [3, 4, 5]: When a s y s t e m is i n a condition o f either chemical or physical equilibrium, an increase of the pressure favors the s y s t e m f o r m e d w i t h a decrease i n volume: a reduction in pressure favors the s y s t e m f o r m e d w i t h an increase i n volume; and a change of pressure has no effect upon a s y s t e m f o r m e d without a change i n volume.

Let us now look at the totality of the tome [8R]. However, first we should recall how it came about. In a course in mathematical chemistry taught at the Sorbonne in 1892, Robin gave a rather long introduction about his own personal position vis-a-vis thermodynamics treated as a rigorous subject. Notes taken by M. Chassang were later used by Raffy to reconstruct these lectures. In 1896 Robin was named to inaugurate the teaching of physical chemistry at the Faculty of Sciences of the Sorbonne. In his two years of lectures, he tried to lay a new and more rigorous foundation for general thermodynamics. Key issues were, of course, work and reversibility. He became ill during his second year, which was his last. Raffy later used his own course notes and those of M. Couturat to reconstruct these contributions to the book [8R]. Raffy also found some very old papers and notes of Robin that he could incorporate into this book. It seems that with this book, and through his own promotional and writing efforts of more than two years, Raffy succeeded in placing Robin's name into the literature of thermodynamics. The first eight chapters of [SR] are Robin's version of a

*We take the opportunity here to clarify some errors in citation, including our own. In our book on Partial Differential Equations (see the account in [1]), from which our interest in Robin began, we credit Robin with this (1876) paper, stating only that it was published in "about 1877". Such vagueness requires a second look, and indeed somehow we missed the name of Moutier [8R, p. 252, last line], who is actually the author of this paper. Raffy's presentation in [8R] contributes to (but does not excuse) our error. We do find a paper by Moutier, Bull. Soc. Philomathique, 7th Series, Vol 1, (1876-1877), p. 3, entitled Sur une d~monstration de la Ioi du Dulong et Petit, which is presumably the paper to which Raffy refers (although the title is slightly different). Moutier published 72 short notes in the Bull. Soc. Philomathique between 1876 and 1880[ Robin was clearly following Moutier, to whom he refers elsewhere (e.g., [8R, p. 160}, [2R, p. 24]) as well. Raffy [SR, P. 160] states that Robin's second paper [2R] begins on p. 14; we find it beginning on p. 24.

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mathematical foundation for thermodynamics. These chapters introduce and treat, respectively, temperature and heat (I), work (II), reversibility (III), internal potential (which Robin accepts and uses extensively) (IV), Carnot's theories (V), Carnot cycles (VI), latent heats (VII), open cycles and adiabatic process (VIII). These are our abbreviations of Robin's actual chapter headings. It should be emphasized that Robin's abhorrence for abstractness not possessing experimental verification limited his treatment of thermodynamics. We shall return to this issue later. Chapter IX is an important chapter applying the thermodynamic foundations established above to problems of chemistry: we shall discuss these below. Chapter X returns to mathematics and develops what we would now call total differentials, path independence, and the like, for application to the theories of perfect gases and shock waves in Chapter XI. Chapters XII and XIII treat permanent deformations (e.g., as in metal fatigue) and other irreversible transformations. As previously noted, Chapter XIV elaborates the early work on the law of Dulong and Petit extended to perfect gases and clarification of results of Clausius on the heat capacity of solids. Let us highlight a few points here. In developing his version of the fundamental quantities and concepts in thermodynamics, Robin is following the development of the first law of thermodynamics as established by J. R. von Mayer, H. L. F. von Helmholtz, J. P. Joule, and others. Robin divides Mayer's principle (the first law) on the equivalence of heat and work into two parts: the principle of closed cycle and the principle of open cycle. In the case of a reversible process, the latter is regarded as a generalization of the former. From these principles he derives a series of propositions and Mayer's principle as a corollary. For the second law and questions of reversibility and irreversibility, Robin is following N. L. S. Carnot, R. J. E. Clausius, W. Thomson (Lord Kelvin), and others. He characterizes Carnot's first principle (the principle of impossibility of the second kind of perpetual motion) and Carnot's second principle (the principle of conservation of heat) as nonnegativity and positivity of work, respectively, by applying the ideas of reversibility. Thus the two Carnot principles were essentially distinguished by a mathematical characterization. According to Marcolongo [6], Robin was the first to do this. Further, he succeeded in deriving Helmholtz's free energy (which Robin called an invariant) by a new approach, namely, by combining the concepts of internal potential and work in an isothermal reversible process. Also, concerning internal thermodynamic potential issues, Robin provided much clarification, based on rigorous and clear mathematical reasoning. Thus one could argue that Mayer's principle and Carnot's principle could as well be called Mayer-Robin and Carnot-Robin principles, respectively. L. Raffy [7] insists that several such concepts as clarified by Robin deserve to have his name attached. Marcolongo [6] would appear to support this. Let us next turn to the applications of these fundamentals to chemistry given in Chapter IX. First Robin considers the principle of maximum work. Van't H o f f s Theorem

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P

Figure 2. Diagram of Faraday's disk experiment (from [8R], Fig. 17, p. f6s).

states essentially that the size of a chemical affinity is equal to maximum work done on a system. The notion of chemical affinity goes back to De Donder and is related to Gibb's free energy; it states that a certain force is present at a chemical reaction. Robin derives Van't H0ffs Theorem more rigorously than had been done before, assuming that thermal equilibrium is equivalent to maximum work being done by the system. Robin points out carefully that Van't Hoff reasoned rigorously only in a neighborhood of absolute zero where latent heat vanishes along with temperature, and only for perfect gases or very dilute materials. Next, p. 158, Robin shows the uniqueness of pressure if a change of state is to remain reversible. The second part (in Chapter X, p. 186) applies as well to the study of the variation of heat in irreversible as well as reversible state changes, provided that the pressure is allowed to vary. We have discussed this above in connection with Robin's paper [2R]. Then, p. 162, Robin verifies a theorem by Kirchhoff which is at the foundations of the theory of absorption and release of heat as a system changes state from thermal equilibrium. The heat gain or loss between two different systems is given by the Clapeyron (1799-1864)-Clausius (1822-1888) differential equation and is a rather general description of phase change, e.g., vaporization or solidification. Robin's formulation and derivation is more precise than the previous versions. Again he stresses the role of pressure. In the last two sections of Chapter IX, Robin presents arguments against, and then improves, the theory of the battery as developed by Helmholtz. Robin begins with a clear explanation of the Peltier (1785-1845) effect, the phenomenon of absorption or generation of heat beyond Joule heat which occurs when electricity flows to the junction of two different conductors. This is carefully described in terms of the Faraday apparatus shown in Figure 2 above. Then Robin explains conditions for the reversibility of the voltage under very small current changes. Finally, he derives the amount of chemical heat generated by the battery. His analytically calculated estimates are shown to agree quite well with physical measurements for 8 different batteries made of zinc, copper, etc. The same insistence on confirmation by experiment may

be seen in Chapter XI where Robin treats the state equation of Van der Waals and the balloon experiment of Clement and Desormes. There he also obtains the theorem of Hugoniot for the constancy of the speed of propagation of a shock wave caused by an explosion in a fluid, using thermodynamic arguments. Robin's Scientific Philosophy and Personal Life Robin's pedagogy and research, as illustrated above, were influenced by his general philosophy. Specifically, Robin's scientific philosophy was positivism. His position was near that of E. Mach. Accordingly, in treating thermodynamics, he rejected (to an extreme) intuitive concepts and deduction, and insisted on induction. His fLrst lectures at the physical chemistry course, see the Introduction to [8R], emphasized the importance of induction in science and his absolute confidence in it. Also, R. Marcolongo's review [6] testifies that Robin was an antimechanist, as opposed to the more deductive Gibbs school. We can easily see Robin's philosophy exemplified when fundamental concepts in thermodynamics are defmed by him. For example, he removes the concept of adiabatic change from the premises for the theory, regarding it as a nonverifiable abstraction, and replaces it with a realizable one. Furthermore, energy, entropy, and force are stated in terms of work caused by mass displacement. This means Robin sought a unique development of thermodynamic theory quite different from and more rigorous than the traditional way at that time. We know very little of Robin's personal life. Robin's father, Charles Pierre Robin, was a professor of chemistry and biology in Paris and a m e m b e r of the Academy of Sciences there. It may be surmised that the family shared the Third Republic's powerful anticlerical bias. The young Robin was somehow, in spite of a very limited set of publications, made instructor at the Sorbonne in 1892, and in 1896 Robin was made professor. A full chair was in the offing (see below) but Robin died before this full chair (the holder of which becomes Professeur Titulaire, a position for life) could be formally established. Robin was awarded the following prizes: The Francoeur Prize (1893) and the Poncelet Prize (1895): for work on mathematical physics. The Francoeur Prize (1897): for work on mathematics. The members of the nomination committees were: C. Hermite, J. L. F. Bertrand, H. Poincar6, C. E. Picard (Sarran, instead of Picard, in 1895), and J. G. Darboux. In spite of the fact that Robin was evidently well placed within the Paris establishment of those times, we have not been able to obtain his obituary, nor a photograph or portrait. The following archives were checked: Biblioth~que Nationale, Biblioth~que Mazarine, Biblioth~que Historique de la Ville de Paris, Biblioth~que de la Sorbonne, Archives Nationales, Biblioth~que de l'Institut H. Poincar6, Archives de l'Acad6mie des Sciences. After Robin's death in 1897, his post as lecturer in physical chemistry was given to Jean Perrin in 1898. This lectureship eventually became a full chair. As is well known,

Perrin received the Nobel Prize later for his research on Brownian motion. Robin and Duhem Seeking more details about Robin's fmal year, the nature of his death, and more about his life in general, we followed the trail of his friend and colleague Louis Raffy (18551909). This quickly led us to the important conflict between Robin and the celebrated French scientist and philosopher P. Duhem. This conflict must have created problems for the acceptance of Robin's scholarly work on thermodynamics. Pierre Maurice Marie Duhem (1861-1916) published more than 450 papers on many aspects of science and later in the philosophy of science. There are several treatises on the life and work of Duhem, and we recommend especially the book by Jaki [8]. Duhem wanted the chair in which Robin was tentatively installed, in the last year of his life, 1897. We do not presume to know the likelihood that Robin would have actually attained that chair, but the awarding of the Francoeur Prize (twice) and the Poncelet Prize would certainly bode well for his chances, in spite of Robin's strikingly meager publication record (6 papers, as compared to Duhem's 114 papers by 1896). On the other hand, Duhem was fervently Catholic. The Paris intellectual establishment managed to prevent Professor Duhem from ever attaining any kind of chair in Paris during his lifetime. Duhem and Robin had known each other for years, had attended each other's lectures, and both claimed some of the same results in thermodynamics. Thus the scientific and priority criticisms that Duhem [9] directs against the Robin/Raffy book [8R] almost certainly were already known by Robin in the 1890's as he tried to persevere in his more rigorous but perhaps less phenomenologically interesting treatment of general thermodynamics. We conjecture that those criticisms had greatly depressed Robin, and possibly contributed to his illness. Quite likely they were an important factor in leading Robin to burn most of his work on thermodynamics. Duhem [9] scoffs at Raffy's great admiration for Robin's Thermodynamique gdndrale, which Raffy had placed in the Avertissement at the beginning of the book [8R]. He challenges Raffy's insistence that Robin and only Robin had established all the claimed new foundational principles. Duhem also objects to Robin's presentation of Carnot's principle; he disputes the rigor claimed, and points out contradictions. And Duhem opposed Robin's scientific philosophy on certain key points. For example, the first chapter of the collection [10] is an excerpt from Duhem's famous book [11]. Although the references to Robin are not paramount there, we do find [10, p. 22] Duhem quoting Robin's scientific philosophy, from Robin's General Thermodynamics [8R], as follows:

The science we shall make will be only a combination of simple inductions suggested by experience. As to these inductions, we shall formulate them always in propositions easy to retain and susceptible of direct verification,

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never losing sight of the fact that a hypothesis cannot be verified by its consequences. Duhem, although politely, rejects this scientific philosophy as at most valid for pure mathematics but certainly inadequate for physics. He goes on (p. 23) to ridicule Robin's presumption that all chemistry can be axiomatized in accordance with verifiable experiment. In [10, p. 29] Duhem also criticizes Robin's Mathdmatiques [9R], the last of the Robin/Raffy Collected Works, and the idea that only ratiohal numbers be permitted in calculus: I f Professor Robin is intent on constantly and scrupulously satisfying this requirement, he would practically be unable to develop any calculation; theoretical deduction would be stopped in its tracks f r o m the start. After Duhem's 1901 attack [9] on Robin's Thermodyn a m i q u e gdndrale [8R], which had just been published by Raffy in 1901, Raffy [7] attempts to defend the work and thought of Robin against Duhem. In particular, Raffy claims that Duhem took ideas from the lectures of Robin. Here is Duhem's response [8, p. 291]: There is not a single topic in this (Robin's) book on general thermodynamics, to which I have not devoted some publication, and several of those topics had previously been a virgin field. Clearly Jaki [8] sides with Duhem. The recent thesis of Brenner [12] helps us to place the life and work of Robin within the politics of his time. We can do no better than quote [12, p. 82]: E x a m i n o n s quelques-unes des diffdrentes f o r m e s qu'a prises l'dnergdtisme. Gustave Robin est d'accord avec D u h e m pour critiquer l'emploi d'hypoth~ses mdcaniques en physique . . . . Mais la perspective mdthodologique est diffdrent. Robin met l'accent non sur la thdorie, mais sur l'observation et mOme sur la s e n s a t i o n , . . . To us, their scientific splitting of hairs both explains and obscures the real reasons for the power struggles, those of the politics, prizes, and positions in Paris in that era. If one admits and looks beyond personal motives, one may end on a somewhat positive note, that in science our belief system similarities are so much closer than are our personal differences. Quoting from [12, p. 83, footnotes]: Picard: Robin fut, comme Duhem, u n des fondateurs de la thermodynamique gdndrale, en se pla~ant d'ailleurs d u n point de vue philosophique tr~s diffdrent.

Acknowledgments We are grateful for the interest, patience, even the occasional impatience, of all whom we queried about Robin.

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We especially appreciate the sharp interest shown and crucial information provided by Professor Jean Dieudoim~ in France and Professor Clifford Truesdell in the United States. The first-named author (K.G.) would like to acknowledge library legwork assistance of two graduate students, Eric Larson in the late 1970's during the initial phase of the investigation, and Alejandro Spina in the late 1980's when we were looking further into the potential theory and thermodynamics aspects of Robin's work, and also that of one undergraduate student Julie Wiejaczka in the 1990's for finding further biographical information on Vladimir Steklov. The second-named author (T.A.) would like to express his hearty thanks to his colleague Isao Onda for their earlier collaborations on this research, and to K. Horie for valuable library research assistance. K.G. would like to express his gratitude to Professor I. Prigogine in Brussels, for all our discussions over the years about reversibility and irreversibility and the philosophies of science and thought. T.A. expresses his gratitude to Professor Y. Michiwaki for his interest and invitations to lecture about Robin at international conferences in Japan and China. We thank Professors Richard Kerner, Jean Eisenstaedt, and Jean Vignes at the Universit6 Pierre et Marie Curie, and Professor Franqoise Chatelin at CERFACS, France, for going beyond the call of duty to aid us in seeking a photograph of G. Robin from some archives in Paris. Special thanks for examining archive records is due Dr. Oscar Bandtlow and Professor Seiji Fujino. We appreciate our e-mall correspondences in 1994 with Mr. John Crow of CRAY Research, whose interest in Robin nearly matched ours. We consulted M. Andr6 Tuilier, author of a recent history of the University of Paris (see Chronicle of Higher Education, January 12, 1996, p. A 39). M. Tuilier makes the point that the history of the Sorbonne reflects the age-old cultural and religious conflicts of France. Certainly our glimpse of the life and work of Gustave Robin exemplifies this statement. We also consuited Professor P. Glansdorff in Brussels. Professor Glansdorff referred us to Professor Emeritus J. Chanu of the Sorbonne, who in turn queried descendents of the Perrin family currently residing in Paris. But none of them had any knowledge of Robin. The following eminent professors also entertained our query about Robin and tried to help: In France, P. L. Lions, H. Br6zis, J. Ginibre, R. Temam, A. Bensoussan, P. LeTallec; in Japan, K. Kobayashi, S. Nakanishi; in Russia, N. Bachvalov, V. Irin and G. Kobelkov. Countless others in the United States, Japan, and Europe were asked if they knew Robin's work. They owe no apology for not knowing Robin's work: even Professors Prigogine and Truesdell did not know of it. We would like to thank Elizabeth Stimmel for her typing of this manuscript, and Risa Masuda for translation assistance. Miss Masuda was supported by a grant from the University of Colorado Graduate School for JapaneseEnglish translation assistance. Finally, there are other re-

lated details about the Robin story and the third boundary condition which could not be included here, although we have tried to present all the key facts. Possibly these will eventually culminate in a monograph. Meanwhile we offer this recognition and celebration of Robin's contributions to mathematics and science on the 100th anniversary of his untimely death in 1897. REFERENCES

1. Karl Gustafson and T. Abe, The third boundary condition--was it Robin's?, The Mathematical Intelligencer, 20 (1998), no. 1,63-70. 2. C. Truesdell, The Tragicomical History of Thermodynamics 1822-1854, Springer, New York (1980). 3. The International Dictionary of Physics and Electronics, 2nd Ed., Van Nostrand, Princeton (1961). 4. Dictionary of Physics and Mathematics, McGraw-Hill, New York (1978). 5. T. Preston, Theory of Heat, McMillan, London (1919). 6. R. Marcolongo, Review of Thermodynamique generale by G. Robin, I'Enseig#ment Math. 6 (1904), 243-247. 7. L. Raffy, A Propos de la Thermodynamique generale de Gustave Robin, Bull. Sci. Math. 26 (1902), 87-92. 8. S. Jaki, Uneasy Genius: The Life and Work of Pierre Duhem, M. Nijhoff Publ., The Hague (1984). 9. P. Duhem, Analyse de I'ouvrage de G. Robin: Thermodynamique g~nerale, Bull. Sci. Math. 25 (1901 ), 174-203; Les 61ementaires b, I'usage des chimistes, in Thermodynamique et Chimie, Hermann, Paris (1902), 178-180. 10. S. Harding, (Editor), Can Theories be Refuted?, D. Reidel Publ., Dordrecht (1976). 11. P. Duhem, The Aim and Structure of Physical Theory (in French), Chavalier d. Riviere, Paris (1906). 12. A. Brenner, Duhem, Science, Realit# et Apparence, Mathesis, Paris, Vrin (1990). PUBLISHED WORK OF (VICTOR) GUSTAVE ROBIN

[1R] 1879. Sur la chaleur reellement continue darts les corps et sur la vraie capacite calorifique, Bull. Soc. Philomath., 7th Series, Vol 4, 25 October, 8-11. [2R] 1879. Sur les transformations isothermiques non reversibles, Bull. Soc. Philomath., 7th Seres, Vol 4, 23 November, 24-27. [3R] 1886. Sur la distribution de I'electricit6 & la surface des conducteurs fermes et des conducteurs ouverts, Th#se pour le Doctorat es science mathematique, Sorbonne, le 13 juillet. Published in the Annales scientifiques de I'E-coleNormale Superieure, 3 (1886), 31-58. [4R] 1887. Distribution de I'electricit6 sur une surface fermee convexe, C. R. Acad. Sci., Paris, t. CIV, 104, 1834-1836. [5R] 1887. Sur les explosions au sein des liquides, C. R. Acad. Sci., Paris, t. CV, 105, 61-64. [6R] 1887. Distribution de I'electricite induite par des charges fixes sur une surface fermee convexe, C. R. Acad. Sci., Paris, t. CVI, 413-416. [7R] 1899. Oeuvres Scientifiques de G. Robin, I-(1): Physique mathematique. E-lectricit~ Statique. This is a reproduction of [3R], presented in two parts.

Troisieme Partie: Additions et Complements. This has three short chapters dealing with topics in potential theory. The first part of Chapter I reproduces [4R]. The three remaining parts of Chapter I (pp. 63-71) deal with the distribution of electricity on a convex surface, a conductor of minimal electrical capacity, and the distribution of electricity on the faces of a parallelepiped induced by a point charge in the interior. Chapter II discusses Coulomb's taw from which all charge on a body goes to its exterior surface. Chapter III gives a solution to the Laplace equation &V = 0 in terms of a double charge layer on the surrounding surface. Hydrodynamique. This consists of two parts. The first is entitled Le Probleme General de I'Hydrodynamique (pp. 89-96) and follows work by Helmholtz dealing with limiting vorticity states for turbulent fluids. The second part is entitled Percussions et Explosions dans les Liquides and is a substantial extension (pp. 97-122) of [5R]. Robin had not published either of these two parts. Fragments divers. This contains four short articles about electomagnetic theory, written prior to 1889 and not published. The first is Principe des images dans les milieux dielectriques, the second is Theorie du 'replenisher' de Thomson, the third is Effet d'un obus aimant~ sur un galvanometre, the fourth is Theorie d'une experience de Hertz. All together comprise 23 pages (pp. 125-147). [8R] 1901. Oeuvres Scientifiques de G. Robin, I-(2): Thermodynamique generale. This is a reproduction of Robin's course notes from his lectures at the Sorbonne between 1892 and 1897, when Robin became ill. There is considerable editing here by Louis Raffy based upon his own notes and those of Fellow Students Maurice Chassang and Louis Couturat. This is a valuable tome (introductory comments plus pp. 1-262) on Robin's work and thoughts on thermodynamics. The final chapter (Chapter XlV, pp. 251-262) elaborates the paper [1R]. Chapters IX and X include (broken into two small parts) the paper [2R]. Then at the end there is a short Note sur un Theor~me d'electrostatique (pp. 263-265) explaining why a conductor in equilibrium has a constant potential, and how this may relate to a principle of Carnot in thermodynamics. [94 1903. Oeuvres Scientifiques de G. Robin, I1: Nouvelle theorie des fonctions, exclusivement fondee sur I'idee de nombre. [1 OR] 9... Oeuvres Scientifiques de G. Robin, II1: Legons de Chimie physique. This volume was never published, though advertised in [7R, 8R, 9R]. KARL GUSTAFSON Department of Mathematics University of Colorado Boulder, CO 80309-0395 USA e-mail: [email protected] TAKEHISA ABE Department of Mechanical Control Systems Shibaura Institute of Technology 307 Fukasaku, Ohmiya Saitama 330 Japan e-mail: [email protected]

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I'[:~-wa-tw-~[.nm

Jeremy

Gray

The Foundations of Geometry and the History of

Geometry*

Column Editor's address: FacuIty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England

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I

When historians of mathematics seek to explore the work of another mathematical culture, they naturally draw on their own experience and that of the mathematicians around them. This app r o a c h brings insights no careful reproduction of the texts can manage, but it brings the risk of misreading too. The celebrated example of nonEuclidean geometry, one of the m o s t actively discussed topics in 19th-century mathematics, is a case in point. Indeed, the early years of the axiomatisation of geometry produced a philosophy of geometry that had a marked effect on the writing of the history of mathematics at the time. It is usual to suggest that Hilbert's Grundlagen der Geometrie, first edition 1899, marks the start of the move towards axiomatising mathematics, or to be more precise, towards givhlg formal, "meaningless" axiom systems as the basis of each mathematical discipline and eventually of all mathematics (see Kline [1972], Ch. 43, w or Toepell [1986] Ch. 7, w who also quotes Hurwitz to this effect in 1903 (see p. 257)). This enterprise, which is also taken to expire with G6del's work in the 1930s, is often referred to as Hilbert's formalist programme. The bulk of Hilbert's work on it, a few early forays aside, is concentrated in the 1920s and deals with difficult questions about the relation of mathematics to logic. On this view, the importance of the work on the foundations of geometry is that it prefigures the later work, and, it is sometimes suggested, inspired similar treatments of other topics---group theory is often mentioned, along with Zermelo's axiomatization of set theory (see Moore [1982, p. 150]) and Steinitz's theory of fields. It is often observed that the Grundlagen tier Geometrie ran to 12 editions, changing its nature considerably over the years as several appendices were added, and that it was trans*To David Fowler on the occasion of his 60th birthday,

THE MATHEMATICAL INTELLIGENCER 9 1998 SPRINGER-VERLAG NEW YORK

lated into several l a n g u a g e s - - p r o o f of the esteem in which it was held. There are a number of problems with this view. The first edition does not fit the bill very comfortably: it is not about Euclidean geometry but about various non-Archimedean geometries, presented in the form of five families of axioms with particular attention to the question of what underlying continuum is presupposed. Only in the second edition are non-Euchdean geometry and the independence of the parallel postulate discussed. There are problems with the quality of the argument all the way up to the 8th edition, the first to be published after Hilbert's death in 1943. The edition that was translated into English was the second; the better-known second English translation corresponds to the 10th German edition. The source of the most rigorous critiques of Hilbert's work was Freudenthal, who also wrote one of the few good historical accounts of it, and it is with the history that I shall be most concerned. I shall take history here in two senses: I want to look, first, at the early years of the axiomatisations of geometry; second, at the effect this philosophy of geometry had on the writing of the history of mathematics at the time. These are connected, because the historians of mathematics (and it was a golden age for the history of mathematics) were professional mathematicians working in good mathematics departments. The awkward fact, which Freudenthal pointed out (see his [1957] and [1962]), is that around 1900 the axiomatisation of geometry is more an Italian than a Germany story; Kline's account concurs. The story does start in Germany: Moritz Pasch's Vorlesungen i~berneuere Geometrie [1882] is the first b o o k in which a thorough reworking of geometry is proposed. He sought to formulate rigorously every fact about plane

projective geometry, starting with the undefined or primitive concept of the straight line segment between two points. Results or properties about segments he felt necessary to assume without proof he called Grundsdtze. All Grandsdtze were, he said (Pasch [1882], 17), immediately grounded in observation, and he cited Helmholtz's paper "On the origin and significance of the geometrical axioms" at this point. Results he could deduce from the Grundsdtze he called Lehrsdtze. There were 8 Grundsdtze needed to base the theory of line segments, of which the first is "there is always a unique segment joining any two points." In general, Grundsdtze should be laid down until the mathematician could henceforth reason logically and without further appeal to sense perceptions. The rest of the book is devoted to showing that that can be done. Thereafter, the study of geometry from an axiomatic point of view was taken up most eagerly in Italy. One of the tricky topics, and one that most interested Hilbert, was the interdependence of the Theorems. One source for this interest was in the delicate business of sorting out projective geometry. A point D is called the harmonic conjugate of B with respect to A and C if the 4 points are collinear and a complete quadrilateral can be found such that 2 pairs of opposite sides pass through A and C and two diagonals through B and D. Desargues's Theorem was then used to prove the uniqueness of the 4th harmonic point. The fundamental Theorems of projective geometry are that a projective correspondence between two lines is completely determined when three distinct points on one are mapped onto the other, and that any correspondence between two lines that preserves harmonic conjugates is a projective correspondence. Similar statements about four points in the plane also hold. Christian Wiener had been the first to note (JDMV 1, 1890) that the proof of the first fundamental Theorem reties on the theorems of Desargues and Pappus, but he did not prove it. The

first to do so was Schur [1899]. After this, Hilbert had shown in his Grundlagen, Chapter 6 that indeed the first 15 axioms do not imply Pappus's Theorem. As Desargues himself had observed, the proof of Desargues's Theorem is automatic in projective spaces of dimensions 3 (or more), but in two dimensions it requires a special proof 1. Peano in his [1894] gave a proof in the three-dimensional case, but refrained from comment on the situation in two dimensions, which left the matter unresolved (as van der Waerden's [1986] confirms). Then Hilbert in his Grundlagen [1899], with a further simplification by Vahlen [1905] and most simply Moulton [1902], noted that one can have a projective space satisfying the first 12 axioms and the contradiction of Desargues's Theorem. (Moulton's argtanent replaced Hilbert's in the second English edition, pp. 74-5.) So Desargues's Theorem may be false in the plane, a rather shocldng result! Even the notion of harmonic conjugate is not as simple as it may seem. In his [1891] Fano noted, using the example of the finite projective plane with 7 points and 7 lines, that there need not be a harmonic conjugate at all, and using the finite space with 15 points he noted that the existence of a harmonic conjugate does not imply that A and C separate B and D. Building on this series of discoveries of novel, counter-intuitive theorems in projective geometry, Italian mathematicians did not constrain their projective geometry to the facts of everyday experience. The significant novelty in Mario Pieri's work, which marks it out from Pasch's, is the complete abandonment of any intention to formalise what is given in experience. Instead, as he wrote in [1895], he treated projective geometry "in a purely deductive and abstract m a n n e r , . . . , independent of any physical interpretation of the premises." Primitive terms, such as line segments, "can be given any significance whatever, provided that they are in harmony with the postulates which will be successively introduced. "2 In Pieri's presentation of plane projective geometry (Pieri

[1899]) nineteen axioms were put forward (typically: any two lines meet). It was the Italian work rather than that of Hilbert which travelled best, to the English-speaking world at least, as the citations in A.N. Whitehead's Cambridge Tract The Axioms of Projective Geometry (CUP Tract nr 4, 1906) show. One recalls that it was Peano's example that inspired Russell to take up mathematical logic. Whitehead's axiomatisation, citing the literature just described, used 12 axioms to describe the projective plane. Axiom 13 allowed for a point outside the plane. His treatment of order properties followed Pieri and his friend Bertrand Russell's Principles of Mathematics, Ch 24, 25. Then Whitehead introduced Fano's axiom, and soon had a system of 15, of which the last confined attention to 3-dimensional spaces. A further four axioms allowed the introduction of coordinates. Pieri's ideas spread to France, where they were summarised by Couturat in

his Principes

des

mathdmatiques

[1905]--a work openly inspired by Russell's work of the same name. In America Pieri's system of axioms for geometry was adopted by the mathematician J.W. Young, who learned them from Couturat. In his book [1911] he compared the systems of Hilbert, based on congruence, and Pieri, based on motion (in the sense of 1-1 transformation); finding the concept of congruence more abstract, he advocated that motion be taken as the fundamental concept and congruence derived from it. Meanwhile, Veblen had come independently to some of these ideas. Whitehead particularly acknowledged the mimeographed notes of Veblen's Princeton lectures "On the foundations of geometry" published by the University of Chicago in 1905. The later two-volume work by Veblen and J.W. Young is even-handed in its attributions to Hilbert and to the Italians. So it seems that in the early years of the 20th century Italian ideas, such as Pien's, met with a greater degree of acceptance than is commonly recognlsed today. The question then arises why the Italians have been so forgotten. The obvious answer is part of the correct

1Field and Gray [1986] 2Quoted in Bottazzini Storia, III.], 276

VOLUME20, NUMBER2, 1998

47

I PRINCIPII DELLA O E O M E T R I A DI POSIZIONE, ECC.

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Figure 1, A page from Pieri's paper [1899] displays his special symbols, inspired by Peano, indicating how forbidding they could look,

one. H i l b e r t w a s Hilbert: a p o w e r f u l m a t h e m a t i c i a n at t h e c e n t r e o f t h e l e a d i n g m a t h e m a t i c a l d e p a r t m e n t in t h e w o r l d . H o w e v e r t r a n s i e n t his int e r e s t In t h e f o u n d a t i o n s o f g e o m e t r y ( a n d h e h a d at m o s t 4 P h D s t u d e n t s in t h i s area, o u t o f a total o f 6O in his life), h e l e n t t h e s u b j e c t his p o t e n t n a m e . Thence the numerous editions and the

5(}

THE MATHEMATICALINTELLIGENCER

translations. Add that he survived the War, a n d in t h e 1920s t o o k up t h e f o u n d a t i o n s o f m a t h e m a t i c s , w h e r e a s Pieri d i e d in 1913. A d d t h a t he w a s a l u c i d writer, w h e r e a s Pieri w r o t e in t h e unn a t u r a l l y c o n s t r i c t e d style o f his m e n tor, P e a n o (see Figure 1). In this argot, as m a n y w o r d s as possible of a natural language w e r e eliminated in f a v o u r of

symbols. Points of emphasis, o v e r v i e w s o f the argument, m o t i v a t i o n all c o u l d b e suppressed. P e a n o w o u l d s o m e t i m e s write a p a p e r in t w o halves, t h e first in this logical m a n n e r , t h e s e c o n d in italian prose. My s u b j e c t i v e e x p e r i e n c e w i t h s u c h p a p e r s is that t h e y are no easier o r h a r d e r to r e a d t h a n others, but a lot less exciting. After the w a r P e a n o l e c t u r e d m o r e and m o r e in this style, to the distress o f a D e a n of the F a c u l t y w h o c a m e to investigate (see Segre [1994, p. 321]). It is surely o b v i o u s w h i c h style will m a k e friends. So m a r k e d w a s t h e c o l l a p s e o f t h e Italian a x i o m a t i c g e o m e t e r s t h a t F r e u d e n t h a l calls t h e i r t r i u m p h a P y r r h i c victory. F o r a d i s c u s s i o n o f t h e strengths and w e a k n e s s e s in logical t e r m s of Peanian, as his m a t h e m a t i c a l language w a s colloquially called, t h e r e a d e r is r e f e r r e d to Segre's essay and Zaltsev [1994]. A x i o m a t i s i n g e l e m e n t a r y g e o m e t r y is a p r o c e s s that c o m e s to an end; it is, in m o r e s e n s e s t h a n one, a finite task, unless one is to p l u n g e into d e e p questions o f m a t h e m a t i c a l logic. The Italians purs u e d it as p a r t of P e a n o ' s p r o g r a m m e to write m a t h e m a t i c s in a c o m p l e t e l y unambiguous, logical fashion. Hilbert's taste w a s m u c h b r o a d e r , and that is also important. The A p p e n d i c e s to t h e Grundlagen der Geometrie include his f a m o u s t h e o r e m t h a t t h e r e is no s m o o t h isometric embedding of non-Euclidean t w o - d i m e n s i o n a l s p a c e into R 3. It d o e s n o t b e l o n g to a x i o m a t i c s , b u t it d o e s b e l o n g to m a i n s t r e a m m a t h e m a t i c s , and I think that mathematicians found t h a t r e a s s u r i n g o n c e t h e chill ~ n d o f a x i o m s h a d h a d its s a l u t a r y effect. T h e rise o f G6ttingen, t h e d i a s p o r a a f t e r t h e Nazis, a n d t h e c o n t e m p o r a r y d e c l i n e o f Italian g e o m e t r y i n t o a p r e s u m e d n e v e r - n e v e r l a n d o f h a z y i n t u i t i o n s all p r e s e r v e d o n e tradition, a n d c o v e r e d up a n o t h e r . T h e f o l k m e m o r y w a s est a b l i s h e d - a s history. On o n e i n t e r e s t i n g m a t t e r t h e folk m e m o r y is s t r a n g e l y silent. J u s t h o w m u c h did H i l b e r t k n o w o f c o n t e m p o rary Italian w o r k ? In his t h o r o u g h s t u d y o f Hilbert's r o u t e to his G r a n d / a g e n (Toepell [1986]) T o e p e l l s h o w s t h a t alt h o u g h Hilbert did n o t r e a d Italian easily, he listed P e a n o ' s b o o k on t h e G r a s s m a i m calculus in its G e r m a n trans-

latlon (Peano [1891]) as one of the This is a hint to the second difference: his, Roberto Bonola. His La geometria books on the axioms of geometry. for Hilbert, the study of geometry in non-Euclidea, which was published in Toepell therefore disagrees with Morris this new fashion led to different kinds 1906, grew out of an earlier essay writKline's remark that Hilbert "did not of arithmetic based on the addition or ten for a collection of monographs on know the work of the Italians" (quoted multiplication of segments, not all of geometry that Enriques had edited from Kline [1972], p. 1010). One might them equivalent to ordinary arithmetic, (Questioni riguardanti la geometria add that Italian geometers visited the segment arithmetic for Cartesian elementare, Bologna, Zanichelli, 1900). G6ttingen, where Hilbert became a pro- geometry. This novelty held out the It was translated into German, and fessor in 1895. Significantly, Hilbert did prospect of illuminating geometry over then into English by H.S. Carslaw, pubnot refer to Peano's much more ax- any algebraic number field, which was lished by Open Court in 1912, where it iomatic work, Peano [1889]. But it one of his prime purposes in pursuing appeared with a short Introduction by all this research. Only by compart- Enriques. The melancholy occasion of would be hard to argue that Hilbert could have remained ignorant of Italian mentalizing Hilbert's work for the the preface was Bonola's death in 1911, work after 1900. The International modern reader can he be represented at the age of 37. To produce the book, Congress of Mathematicians in Paris as so thorough-going a formalist; the which is a classic still worth reading that year not only carried two papers by palm for that belongs to Italian math- and has even been used as the basis of Alessandro Padoa on foundational ques- ematicians. 4 a course in geometry at Warwick tions, one was indeed devoted to ex(which is indirectly how I came to hispounding a new system of definitions The History of Geometry tory of mathematics, and so to be a stufor Euclidean geometry (Padoa [1902]). In an important paper of 1939, Ernest dent of David Fowler's), Bonola relied Padoa there made reference to much of Nagel argued that the reformulation of very sensibly on the work of the indethe earlier Italian work, including geometry was an important source of fatigable Friedrich Engel and Paul Pieri's. Moreover, the Congress of Math- modern logic. The kernel of his insight Stfickel, who did so much to publish ematicians came straight after the was that while duality in projective and publicise the work of Bolyai and International Congress of Philosophers, geometry puts points and lines in the Lobachevskii. Naturally Bonola, folat which many papers were presented plane on an exactly equal footing, in- lowing the lead of Beltrami and Segre, tuition must prefer points. So mathe- also played up the significance of on geometry, and several of the speakers attended both Congresses. It is maticians were forced away from in- Gerolamo Saccheri, the Italian mathehard to imagine that Hilbert would not tuition as the basis of geometry, and matician who had died in 1733 after have been drawn into conversadiscovering many results that tion on the topic. He and the now form the cornerstone of elNon-Euclidean geometry raises the ementary non-Euclidean geomeItalians may have begun sepaembarrassing question why mathe- try. His work had lapsed into alrately, but their interests now flowed together. A measure of maticians have been so wrong about most complete obscurity, and the degree to which Italian deBeltrami had recently trumpeted geometry for so long. velopments were not read may its merits upon rediscovering it be Poincar6's ignorance of them, in 1889. It was Engel, a former collaborator to which Freudenthal drew attention towards formalism and thence logic. I (Freudenthal [1962], pp. 620-1). would add that non-Euclidean geome- of Sophus Lie, and St~ckel, who also Two reason for the disparity between try promoted this tendency. It is the worked on editing the extensive Gauss Hilbert and the Italians may be worth geometry that raises the question of Nachlass under the direction of Felix noting. When Hilbert spoke at the the nature of space, and with it the em- Klein, who established the canonical International Congress of Mathematibarrassing problem of explaining why version of the story of non-Euclidean cians in Paris, he referred to signs as mathematicians had been so wrong geometry, through their editions of memory aids. "Geometrical figures," he about geometry for so long. Nagel's ex- work by Lambert, Schweikart, and said, "are signs or mnemonic symbols of ample of the formalist geometer was, Taurinus as well as Bolyai and space intuition and are used as such by of course, Hilbert--but in this he was Lobachevskii. Their book, Theor/e der all mathematicians." 3 Knowingly or not, unfair, in ways that affect our under- ParaUeUinien von Euklid bis auf this remark excluded Peano and those standing of Enriques. Indeed, as we Gauss, 1895, is a generous collection of shall see, the insight of Nagel owes a work by Wallis, Saccheri, and the later like him, for whom signs were, prelot to the original work of Enriques. writers through to Gauss, to which they cisely, formal symbols. Then Hilbert went on to speak of the necessity of If Enriques was the spokesman in- supplied a pertinent commentary. Engel giving a rigorous axiomatic investiga- ternationally for Italian geometers, the himself translated Lobacheskii's two tion of the conceptual content of geo- one who most securely grasped the early and extensive Russian papers into historical task was a former pupil of German (Lobachetschefskij [1899]), and metrical signs and their combinations. 3Hilbert [1976, p. 5]. 41 am indebted to David Rowe for bringing this point to my attention.

VOLUME20, NUMBER2, 1998 57

they tried to find as much as they could about the elusive figure of JSnos Bolyal. Subsequent writers have discovered a great many minor figures omitted by them, but with the arguable exception of Legendre, no Western mathematician has entered the Pantheon. One may conjecture that the reason is the progressive interpretation that can be placed on all this work, each author marking a significant advance until, while Schweikart hesitates and Taurinus looks backwards, Gauss takes the bold step into the non-Euclidean world. From such a perspective, Legendre's attempts are reactionary, and sometimes embarrassingly flawed. What did Engel and St~tckel have to say? The range of material they presented is impressive; 19th-century German mathematicians were well-educated scholars. The names fly by, however, as so many precursors of the chosen few. Commentary amounts to 15-20% of the book, the rest being biographical and bibliographical. The thesis, if there is one, is concealed in the choice of authors. Bonola's work, by contrast, makes more of an argument. With the Theor/e der ParalleUinien in print, he could content himself with summaries of the original sources, and tell a historical story of his own. He added a number of protagonists, expanded on the cryptic references to Arabic writers, included Legendre, and went on past Bolyal and Lobachevskii to Riemann. The first of five appendices considered the connection between the parallel postulate and the law of the lever; other appendices considered such topics as the independence of projective geometry from the parallel postulate, and the impossibility of proving the parallel postulate. The generic account in Bonola's book of a mathematician's contributions goes like this. The mathematician's definition of a straight line and of a parallel line is given, or, if none was supplied, one is uncovered from the use of the concepts. The original argument is then presented in something close to its own terms, and the fallacy, if any, is explained. So, when an argument that would seem to show that spherical geometry cannot exist is under discus-

THE MATHEMATICAL tNTELLIGENCER

sion, Bonola shows how the postulate of Archimedes or its consequence, the indefinite extendibility of the straight line, has been tacitly invoked. And when Legendre produces a fallacious argument using the postulate of Archimedes, Bonola also shows how it could have been avoided, the better to explain that this was not where Legendre erred. From first to last, Bonola's account of the origin and development of nonEuclidean geometry is rooted in an analysis of axioms: their equivalence and their independence. Indeed, an Italian geometer, and a pupil of Enriques, writing between 1900 and 1911, would naturally see geometry as organised in this way. It may be significant that Engel and St~ckel were, if anything, more in the orbit of Klein than Hilbert, and, being 40 in 1901-2, had several years of mature work behind them as differential geometers and analysts. At all events, Bonola's work is analytic where theirs is descriptive. For Bonola, geometry is a matter of axioms, so the history is a history of axioms. It need not have been so. The history of non-Euclidean geometry is open to other interpretations. Had it just been a question of exhibiting an axiom system for something fairly geometrical, then spherical geometry would have done. One needs, of course, to strike out two of Euclid's axioms: the parallel postulate and the indefinite extendibility of the straight line. That this was not done suggests that the ancients were not simply investigating axiom systems. It suggests, what a considerable amount of other evidence also suggests, that they were investigating something else: the geometry of physical space. The ongoing question was not "is the parallel postulate independent of the other axioms of geometry?", but "is the parallel posm/ate independent of the other axioms of geometry when giving an account of space?". This is a different enterprise from the much more overtly logical one in fashion around 1900. There is another problem with Bonola's analytic approach: it is insensitive to the methods originally used. More precisely, the old arguments are presented carefully and accurately, but their significance is ignored. The book

takes a dramatic turn on page 76 (in its English edition), with the frost appearance of an analytic formula. Thereafter, the whole flavour switches from Euclidean-style arguments about angles and lines to hyperbolic trigonometry. We have entered a Chapter called "The Founders of Non-Euclidean Geometry," the early work of Gauss is behind us, and that of Schweikart and Taurinus is upon us. It rapidly becomes clear that this new trigonometry is the vital ingredient that made the discovery possible. Indeed, it is well-known that the work of Bolyal and Lobachevskii falls short of carrying logical conviction: it is a coherent description, but based upon an assumption about lines that was not rigorously defended. Their accounts are full of a vivid analogy between hyperbolic and spherical trigonometry. Elsewhere, in the early papers translated by Engel, Lobachevskii gave his own way of deducing the trigonometry from an analysis of geometry, but that is missing from Bonola's account. It was first clearly supplied by Beltrami, in his "Saggio" of 1868. But if hyperbolic trigonometry is the vital ingredient, one might ask who discovered it. The answer, as Bonola said in a footnote on page 82, is Lambert. This should have caught Bonola's attention, but he ducked the issue, caught as he was by his axiomatic paradigm. If mathematicians from before Euclid to---shall we say--Poincar6 were trying to describe space, then it would be natural for them to use trigonometry or conventional ("Euclidean") geometry. Axioms would appear, as they usually do in geometry before the advent of modern logic, as the undeniable truths they were taken to be, mixed up with dermitions, stated explicitly or implicitly as is the way with statements of the obvious. Seen from this angle, which was obscured around 1900, the question at issue was not the logical status of the parallel postulate but something far more urgent: its truth. Paradoxically, the new-found clarity 100 years ago about the nature of mathematical reasolfing perhaps led Bonola to emphasise the logic of the original arguments at the expense of their purpose.

End Note

There is much more to be said about Hilbert and axiomatisation. In a forthcoming 'Years Ago' column, Leo Corry examines Hilbert's axiomatisation of physics, specifically, radiation theory. BIBLIOGRAPHY Beltrami, E., 1868: "Saggio di interpretazione della geometria non-Euclidea," Giomale di Matematiche Vl, 284-312. Bonola, R., 1906: La geometria non-Euclidea, Zanichelli, Bologna. Bonola, R., 1912: History of non-Euclidean geometry, tr, H.S. Carslaw, preface by F. Enriques, Open Court, Chicago. Bottazzini, U., 1988: Fondamenti dell'arithmetica e della geometria, pp. 253-288 in vol. 3 of Storia della scienza modema e contemporanea, Paolo Rossi editor. Unione Tipografico Editrice Torinese. Courturat, L., 1905: Principes des math6matiques, Alcan, Paris. Engel, F. and St&ckel, P., 1895: Theorie der Parallellinien von Euklid bis auf Gauss. Teubner, Leipzig. Enriques, F., 1898: Lezioni di geometria proiettiva, Zanichelli, Bologna. Enriques, F. (ed.), 1900: Questioni riguardanti ta geometria etementare, Zanichelli, Bologna. Fano, G., 1892: "Sui postulati fondamentali della geometria proiettiva," Giemale cfi Matematiche 30, 106-131. Field, J.V., and Gray, J.J., 1986: The Mathematical Work of Girard Desargues, SpringerVerlag, New York. Freudenthal, H., 1957: "Zur Geschichte der Grundlagen der Geometrie," Nieuw Archief veer Wiskunde (4) 5, 105-142. Freudenthal, H., 1962: "The Main Trends in the Foundations of Geometry in the 19th Century," in Logic, Methodology and Philosophy of Science,Proceedings of the 1960 International Congress (E. Nagel, P. Suppes and A. Tarski, editors), Stanford University Press, 613-621.

Gray, J.J., 1986: Ideas of Space, Euclidean, non-Euclidean and Relativistic, Oxford University Press. Helmholtz, H. von, 1870: "On the origin and significance of the axioms of geometry," tr. M.F. Lowe, in Hermann von Helmholtz, Epistemological Writings, P. Hertz and M. Schlick, eds., Boston Studies in the physics of science 37, Reidel, Dordrecht and Boston, 1977. Hilbert, D., 1899: Grundlagen der Geometrie, many subsequent editions. Hilbert, D., 1971 : Foundations of geometry, 10th English edition, translation of the second German edition by L. Unger. Hilbert, D., 1976: "Mathematical Problems," in Mathematical Developments arising from Hilbert Problems, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 2 vols. part 1. Kline, M., 1972: Mathematical Thought from Ancient to Modern Times, Oxford University Press, Oxford. Lobachetschefskij, N.I., 1899: Zwei geometrische Abhandlungen, tr. F. Engel. Teubner, Leipzig. Moore, G.E., 1982: Zermelo's Axiom of Choice, Springer-Verlag, New York. Moulton, 1902: "A simple non-Desarguesian Plane Geometry," Trans.American Mathematical Society, 3, 192-195. Nagel, E., 1939: "The formation of modern concepts of formal logic in the development of geometry," Osiris 7, 142-224. Padoa, A., 1902: "Un nouveau syst~me de definitions pour la geometrie euclidienne" Compte Rendu du Deuxieme Congres international des math#maticiens, Gauthier-Villars,Pads. Pasch, M., 1882: Vorlesungen Ober neuere Geometrie, Teubner, Leipzig. Peano, G., 1889: I principii di geometria Iogicamente espositi, Turin, rep. in Opere scelte, 2, Rome, 1958, 56-91. Peano, G., 1891; Die Grundz#ge des geometrischen KalkOls, tr. A. Schepp, Teubner, Leipzig.

Peano, G., 1894: "Sui fondamenti della Geometria," Rivista di matematiche 4, 73. Pied, M., 1895: 'Principii che reggiono la geometria di posizione', Atti Accademia Torino, 30, 54-108. Pieri, M., 1899:"1 Principii della Geometria di Posizione, compositi in sistema Iogico deduttivo," Memorie della Reale Accademia detle Scienze di Torino (2) 48, 1-62. Russell, B., 1903: Principles of Mathematics, Cambridge University Press, Cambridge. Schur, F., 1899: "Uber den Fundamentalsatz tier projectiven Geometrie," Mathematische Annalen 51,401-409. Segre, M., 1994: Peano's Axioms in their Historical Context, Archive for History of Exact Sciences, 48, 201-342. Steinitz, E. 1910 'Algebraische Theorie der K6rper', Journal f#r Mathematik 137, reprinted as Steinitz, [1930]. Steinitz, E., 1930: Algebraische Theorie der KSrper, R. Baer, H. Hasse (eds.), Walter de Gruyter, Leipzig. Toepell, M.-M., 1986: Ober die Entstehung von David Hilberts "Grundlagen der Geometrie", Vandenhoeck & Ruprecht, G6ttingen. Vahlen, T., 1905: Abstrakte Geometrie, Leipzig. Veblen, O., 1905: Princeton lectures "On the foundations of geometry," University of Chicago Press. Veblen, O. and J.W. Young, 1917: Projective Geometry, 2 vols, Ginn and Co, New York. Whitehead, A.N., 1906: The Axioms of Projective Geometry, Cambridge University Press Tract nr. 4, Cambridge. Wiener, H.L.G., 1890: "Ueber Grundlagen und Aufbau der Geometrie," Jahresbericht der Deutschen Mathematiker-Vereinigung 1, 45-48. Young, J.W., 1911: Lectures on Fundamental Concepts of Algebra and Geometry, Macmillan, New York. Zaitsev, E.A., 1994: "An Interpretation of Peano's Logic," Archive for History of Exact Sciences 46, 367-383.

VOLUME 20, NUMBER2, 1998

59

EDWARD

BURGER

Mathematics Beyond the Ivy-Covered Hall: Can We Read tho Writing on the Wall?

(Photograph by E.B. Burger@) Graffiti provide a w i n d o w into the soul of their creator. Given this observation, I w a s unable to contain m y smile u p o n viewing the a b o v e graffiti on the outside wall o f a

THE MATHEMATICAL INTELUGENCER 9 1998 SPRINGER-VERLAG NEW YORK

building in Austin, Texas, b a c k in t h e s u m m e r of 1987. On this t e n t h anniversary, I wish to c e l e b r a t e the w o r k of this wall artist a n d its wonderful h i d d e n life lessons.

9 Getting the person-on-the-street to take a look at mathematics. Certainly the artist's desire to deface p r o p e r t y is only e x c e e d e d by the artist's p a s s i o n for mathematics. In the eyes of our wall painter, m a t h e m a t i c s is w o r t h y o f the attention of the masses. This e n t h u s i a s m m u s t have b e e n the impulse for the artist to move b e y o n d scribbling m a t h on n a p k i n s to s p r a y painting m a t h on walls. W o u l d n ' t it be great if m o r e m a t h e m a t i c i a n s w o u l d actively s h a r e their p a s s i o n for m a t h e m a t i c s with the w o r l d at large? 9 S h a k i n g well before the f i n a l coat. Our wall artist emb a r k s u p o n a p r o b l e m without a d v a n c e k n o w l e d g e of h o w the issue w o u l d be resolved. Of course n e w discoveries are only m a d e after n u m e r o u s failed attempts. Our p a i n t e r has a firm grasp on b o t h the spray can and the p o w e r o f trying w i t h o u t fear of failing. Wouldn't it be g r e a t if m o r e t e a c h e r s w o u l d inspire their s t u d e n t s to be brave enough to e x p e r i m e n t (and even fail)? 9 Reveling i n the shock factor. WHOOPS! The artist s h a r e s with the o n l o o k e r the surprise of the realization that the p r o b l e m at h a n d was m o r e challenging than first thought. Our wall p a i n t e r is r a t h e r m a t h e m a t i c a l l y mature: i n s t e a d o f defacing the wall with a solution k n o w n to b e wrong, the p a i n t e r eagerly a d m i t s to all that the line of a t t a c k did not p a n out. Wouldn't it be great if m o r e s t u d e n t s w o u l d b e strong enough to curb the p o w e r f u l t e m p t a t i o n to r e c o r d an a n s w e r t h e y k n o w to b e incorrect for the sole p u r p o s e of writing s o m e t h i n g d o w n ? 9 H i t t i n g the brick wall. The wall artist has one o f the key ingredients to s u c c e e d in m a t h e m a t i c s : tenacity. Our p a i n t e r d o e s n o t give up or b e c o m e frustrated w h e n f a c e d with a m a t h e m a t i c a l impasse. Rather than a typical m o r e colorful epithet, w e s e e the thoughtful proclam a t i o n that further insights are required. Wouldn't it b e g r e a t if m o r e p e o p l e h a d a s e n s e that the j o u r n e y t h r o u g h m a t h e m a t i c s is an ongoing one t h r o u g h the u n c h a r t e d r e a c h e s o f thought? 9 P a i n t i n g n e w pictures. As all g r e a t m a t h e m a t i c s should, the artist's w o r k leads us to p o n d e r n e w and interesting questions. S u p p o s e w e define the graffiti curve to b e

y4 + 18

-

16x

16 - -

p a i n t e r is n o w an algebraic g e o m e t e r at s o m e university or serving time at s o m e o t h e r institution. In either case, the p a i n t e r a c c o m p l i s h e d something truly spectacular: all who w a l k e d b y that building, for one brief m o m e n t , tried to fact o r a polynomial. In the b e s t of all p o s s i b l e worlds, this writing on t h e wall w o u l d have r e m a i n e d to inspire interesting thoughts and conversations a m o n g generations of b o t h m a t h fans and m a t h foes. Sadly, this is n o t the b e s t of all p o s s i b l e worlds. I t o o k this p h o t o g r a p h early one Sunday morning. To m y surprise, t h r e e days later the b r i c k wall w a s c o m p l e t e l y p a i n t e d over with light blue paint to c o v e r the graffiti. That garish blue c o l o r c o v e r s the entire side o f the building to this very day. As p e o p l e stroll b y the wall today, t h e y are u n a w a r e of the b u r i e d t r e a s u r e w h i c h t h e y pass.

0.

Can y o u s h o w that there are no integer points on this curve? Are t h e r e any rational p o i n t s on the graffiti curve? Wouldn't it b e great if e a c h p i e c e of m a t h e m a t i c s d i s p l a y e d w o u l d inspire one p e r s o n to a s k one n e w question? Where is o u r graffiti artist today? P e r h a p s the wall

VOLUME 20, NUMBER 2, 1998

61

|:(:a,J[:a,,,~--~ Jet Wimp,

Editor

I

To Catch the Spirit: The Memoir of A. C. Aitken by A. C. Aitken With a Biographical Introduction by P. C. Fenton NEW ZEALAND: THE UNIVERSITYOF OTAGO PRESS, 1995, US$ $32.95 122 pp. ISBN 0 908569 99 8

Determinants and Matrices by A. C. Aitken EDINBURGH: OLIVER AND BOYD, 1939 (out of print)

F i g u r e 1. A. C. Aitken, 1912-1913. (From To

Catch the

Spirit Used by permission.)

The Case against Decimalisation by A. C. Aitken EDINBURGH: OLIVER & BOYD, 1962 (out of print)

Gallipoli to the Somme: Recollections of a New Zealand Infantryman by A. C. Aitken OXFORD: OXFORD UNIVERSITY PRESS, 1963 (out of print)

Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us,

REVIEWED BY JET WIMP

A haggard sky a bitter air We might have borne, when loss was new; When grief was fresh we could not bear The irony of heaven's blue.

telling us your expertise and your predilections.

Column Editor's address: Department of Mathematics, Drexel University, Philadelphia, PA 19104 USA.

62

Introduction "Each h u m a n being," Georg Btichner observes in his play Woyzech, "is an abyss into which one gets dizzy from looking." In some of us, those dizzying depths are closer to the surface than in

THE MATHEMATICALINTELUGENCER9 t998 SPRINGER-VERLAGNEW'YORK

others. The photograph (Fig. 1) in the first book above, taken in 1912-1913, of the deeply troubled yet spectacularly multi-talented New Zealand mathematician Alexander Aitken conveys a hint of the abyss: a youthflfl face that seems to gaze at something beyond the viewer, a brow slightly furrowed above handsome, aquiline features. Aitken is one of the most complex and provocative personalities in recent history, a man so baffling that he may, ultimately, remain for us "an enigma," as one of his students, Hans Schneider [sch2], has stated. When I arrived in Edinburgh in 1966 to begin studies for m y PhD, I knew that Aitken was, or had been, a lecturer at the University. Puzzled by the fact that I had not been introduced to him, I asked m y PhD supervisor, Arthur Erd61yi, about him. "He is ill," Erd61yi replied stonily. He did not elaborate, and Erd61yi's notoriously reserved manner did not invite further questioning. Aitken and I had worked in related fields. I had been much impressed by his work on summability methods and by his book, Determinants and Matrices [ait5]. On the one hand, I wondered how Aitken's many talents could have remained fallow for so many years even though he had been "ill," and on the other, my growing acquaintance with the British penchant for euphemizing what everyone agreed must not be directly addressed suggested to me that "ill" applied to faculties other than those of the body. As far as giving any insight into Aitken as a functioning human being, his obituaries have been distinctly unhelpful, for instance, the obituary printed in the Proceedings of the Edinburgh Mathematical Society, [mil]. Obituary writers, undoubtedly submitting to the dicta of professional discretion, often treat mathematicians as an emotionally neutered variety of humankind. They seem to have at hand a set of rubber stamps embossed with passages such as, "He was a m o s t inspiring teacher; his lecturing was superb, and his per-

sonality made an indelible impression on his students" [mill. Having learned something about the subject of Aitken, I wonder whether this description was intended to apply to those instances when the classroom and students appeared to him as a total void and he couldn't remember afterwards the content of his lecture. I have no doubt that those lectures made "an indelible impression" on his students. 1 Only with the publication of the astonishing book, To Catch the Spirit, have many of the secrets of Altken's life, and particularly, of his years in Edinburgh, been revealed. The book, assembled from papers in possession of his daughter Margaret Mott, consists in the main of Aitken's autobiographical memoir for the years 1923-1943 and a brief (two-page) but highly significant entry for the year 1958. I have never before been so enthusiastic about a mathematical biographical or autobiographical document; this book, with Peter Fenton's superb, insightful, lengthy introduction, is a completely successful attempt to restore to a deceased mathematician his humanity, and, by implication--since we all will one day be invoked only posthum o u s l y - t o vouchsafe us our own. The book's evocative title is taken from the legend inscribed below the proof of a theorem in one of Aitken's letters: "Q.E.D." and "A.C.A.," the latter acronymic for the Latin expression ad captandam a n i m a m - - " t o catch the spirit of the thing." What kinds of talents did Altken possess? Many. He was an intellectual golden boy: a polyglot (he rapidly acquired the Greek dialect, Lemnian, used on the island of Lemnos when stationed there for a few days during WWI), a poet, a musician of rare gifts (Delius's secretary, Eric Fenby, described Aitken as the most accomplished non-professional violinist he had ever known 2), a prodigious mental calculator (sometimes regarded as the greatest ever), a writer blessed with

Figure 2. Otago Harbour from the Peninsula. Photograph by Peter Fenton.

penetrating insights and an uncommon, poetical sense of language, a mathematician whose contributions to algebra, numerical analysis, and statistics secured for him at the age of 30 a fellowship in the Royal Society of Edinburgh, a fellowship in the Royal Society of London in 1936, and in 1946 the chair of Pure Mathematics at the University of Edinburgh. As a mathematician he possessed a highly individual style, and one of his discoveries, the so-called t~2-process, initiated the discipline of non-linear summability methods. I intend this article to be a review-with particular attention to the current b o o k - - o f the items in the above list, and more: I also want to give the reader a full account of Altken's life. To do so, I have drawn heavily on the books above, on Smith's book, The Great Mental Calculators [smi], on various obituaries, and on my own professional experience. I have been assisted by generous correspondence from those who have known Aitken, and by the librarians at the library of the University of Otago, whose Hocken collection contains a number of letters written by

Aitken and by people who knew him. I am also grateful to my colleague and former classmate at the University of Edinburgh, David Colton, and to my colleague Ivar Stakgold for a critical reading of a version of this article. Dunedin Alexander Craig Aitken was born on Apdl 1, 1895, in Dunedin, New Zealand, located on South Island. The environs of Dunedin include the lightly populated Otago peninsula--Otago being an anglicization of the Maori name Otakou (Fig. 2). The peninsula is very hilly, partly farmed, partly wasteland, and host to an abundance of wildlife: seals, cormorants, albatrosses. Further south is the exuberantly beautiful Catlins coast and an interior consisting of wild high country where occasional gorges slice through the landscape. Looking at a map of Dunedin, [aitl, p. 28], I was surprised at the resemblance of its layout to that of presentday Edinburgh. Many of the streets have the same names, "Leith," "Queen," etc., and even some of their parallelism has been retained. It reflects, of course, the influence of the first Scottish set-

1The tacenda in this account are illustrated by the reference to Aitken's last years of "indifferent" health. His health was not indifferent, it was homfic. As Hans Schneider observes, "[these obituaries] contain 1/2 to 2/3 the t r u t h . . . " [schl]. The note of Fenton [fen], though shorter, presents us with the picture of a real person. I had my own run-in with this tradition when I attempted to see into print an invited obituary of my mentor, Yudell Luke. The editors preferred to think of Yudell only in the abstract; they expressed dismay at my candid description of his difficulties with an unsympathetic employer. The piece, when it appeared, had been completely rewritten. 2Aitken could perform all of Bach's redoubtable sonatas and partitas for solo violin by heart. Music lovers will recognize this as an incredible feat.

VOLUME20, NUMBER2, 1998

tiers. Dunedin is the old Gaelic version of Edinburgh, and the city was founded by the Church of Scotland. Alec Aitken was the eldest of the seven children born of Scottish parents, William and Elizabeth Aitken. He had a nearly idyllic childhood, with indulgent and loving parents. His father, an assistant in a grocery shop, had an abiding interest in tending flowers. "Aitken's surprise," a chrysanthemum, is n a m e d after him. He had a life-long association with the Methodist Mission of Otago, but there was no fanaticism in his contemplative, completely decent character. He disciplined his children "by a look." Though no one called Aitken a prodigy, his exceptional gifts were obvious from his earliest years. He was a skillful reader and speller w h e n he entered primary school in March, 1901. His father introduced him to arithmetic via the shop accounts and found Alec could effortlessly add the figures, going either up or d o w n the column. When Alec helped his father make deliveries, some of the townspeople privately objected, saying the boy was too talented for such work. Altken spent m u c h of his b o y h o o d exploring the peninsula, at first with his siblings, as they paddled in the bay and fished and listened to stories of Scotland told by their Grandmother, but later alone. Solitude grew to bec o m e a deeply entrenched habit, one he would cling to all his life. Fenton's description of the peninsula in this b o o k evokes the scenery near Edinburgh, which Aitken later also came to love. He covered the entire peninsula around Dunedin, finally getting to know it better than the residents themselves. In his explorations, Alec was indefatigable. If it rained, he stuffed hay under his clothes and went on.

Each year Aitken was frost in his class. At the age of 13 he was awarded a scholarship to Otago Boys' High School. He became interested in sports while in high school, and he was very g o o d at them. He achieved the best performances in the hurdles and the high-

jump. Love of sports was to be a constant factor in his life. The javelinthrowing parties he would hold for his students in Edinburgh were famous [sch]. A signal event in his young life was his introduction to the violin, [aitl], p. 40: Alec [studied] w i t h an almost blind fiddler, who guided Alec's f i n g e r s by touch, but couldn't see what he w a s doing w i t h the i n s t r u m e n t - - a n d that was all the tuition he ever had w i t h the v i o l i n - - h e learned by listening to and watching other players, and by reading books such as Leopold Auer's How I Teach the Violin.

A childhood incident illustrates Aitken's lifelong inclination towards mysticism, though as early as his twenties he rejected all formal religion: My mother took me to Garrison Hall one evening to see Holman Hunt's "The Light of the W o r l d " . . . As we gazed the f i g u r e of Christ ceased to be a picture and seemed to stand out in the f r a m e . 3

This propensity for mysticism would proceed, for better or worse, to inform much of Aitken's mental life, even his renowned ability for mental computing. Later, he came to consider himself telepathic and to accept uncritically the validity of extra-sensory experiences. He even proposed his comparison with the English poet-mystic Blake and with the poet Wordsworth. Those w h o believe the war ineluctably changed Aitken for the worse will be surprised to learn that the personality he displayed in his b o y h o o d is pretty m u c h the same personality others found in him in his Edinburgh days, after the war. One Dunedin acquaintance characterized him as quiet and reserved, without many close friends. Although Aitken was not boisterous, he had a considerable measure of inteliectual pride. A fellow student termed him a pleasant, companionable fellow, a good talker especially w h e n the conservation reverted to himself. This same pride was manifest m u c h

later, and, when coupled with a cause cgl~bre like his obsession with duodecimalization, it could produce diastrous results. Aitken throughout much of his life possessed a beguiling charm, s o m e would call it elegance: the ability to deflate pomposity, and occasional selfdeprecating humor. He was, in Hazlitt's phrase, "in the world, not of it." Eva Erd61yi, the w i d o w of Arthur Erd61yi, says that the Aitken she knew in Edinburgh was the "most remarkable and distinguished as well as the m o s t lovable man I have been privileged to meet in m y life" [erd]. Nevertheless, his diaries s h o w clearly that, as Fenton points out, Aitken was to remain forever an essentially solitary person, "imbued with an overriding irony," one whose shyness might be mistaken for aloofness. Some of his classmates thought him a snob. It was his style, one of his cousins said, to make you feel down there while he was up here. The single traumatic incident during Aitken's bucolic Dunedin childhood was the death of his mother. She died a day before Alec's fifteenth birthday as a result of pressure on her trachea due to a goiter, a condition today both easily prevented and easily treated. The effect on the children was devastating; Alec and his sister Pearl hid in the space beneath the floorboards of the house and for the entire day refused to c o m e out. Subsequently Alec wrote a p o e m influenced by this experience, one stanza of which begins this article. Early in life Aitken showed little of his future gift for mathematics, and in his first two years in school he did not win the mathematics prize. "Arithmetic in primary school," he said, "since I recall hardly anything about it, must simply have bored me," [smi], p. 267. However, when he discovered algebra, his first year in high school, the whole picture changed. "I regard mental algebra as on a m u c h higher plane than mental arithmetic, and incomparably more rewarding." What got him started was the equation a 2 - - b 2 = ( a b)(a + b), see the final section of this article. Then he came under the men-

3Many would find this Pre-Raphaelite portrait rather sentimental, It depicts a crowned and aureoled and thoroughly Caucasian Christ knocking on a door and carrying a lantern.

THE MATHEMATICALINTELLIGENCER

torship o f William J o h n Martyn. Martyn, a gaunt, s t o o p e d y e t h a n d s o m e m a n with a h e a r t y d e m e a n o r a n d a d e e p b a s s voice, who was kindly and encouraging, though he h a d his o w n m e n t a l p r o b l e m s . Later, w h e n on the staff of Otago University, he suffered a n e r v o u s b r e a k d o w n and w a s unable to continue in his duties. Nevertheless he p r e s e n t e d a charismatic figure to his students, a n d p r o b a b l y m o r e t h a n any o t h e r p e r s o n w a s r e s p o n s i b l e for imparting to Aitken a love for m a t h e m a t ics that w o u l d eventually t r a n s c e n d the b o y ' s a t t r a c t i o n to q u o n d a m m o r e engaging interests, such as languages, literature, music. Later, in 1913, w h e n Aitken t o o k first p l a c e in the scholarship examinations, he w a s a w a r d e d a full scholarship to Otago university. This University, f o u n d e d on June 1, 1869, was, b y ordinance, "open to all classes o f Her Majesty's subjects." This unusually p r o g r e s s i v e university was the first in the British E m p i r e to a d m i t w o m e n (1872), but Aitken's m a t h e m a t i c a l exp e r i e n c e s there were u n h a p p y ones. He was s u b j e c t e d to the c l a s s r o o m peccancies of a p r o f e s s o r of m a t h e m a t i c s n a m e d D. J. Richards, a t e m p e r a m e n t a l a n d eccentric Welshman w h o w a s inc a p a b l e of imparting a taste for knowledge to his students. Richards grandly i n f o r m e d Aitken that he w o u l d never u n d e r s t a n d integral calculus. (In 1917 this n o n - t e a c h e r was p r e s e n t e d by the University with the option o f either resigning or being fired.) One consolation of Aitken's stay at the University w a s that he m e t his future wife, Winifred Betts, there. However, the world h a d its o w n p l a n s for Aitken, and at this time a university e d u c a t i o n w a s n ' t one of them. In August 1914 the G r e a t W a r b r o k e out.

The War: "Let These Things be Called by their Proper Names." Aitken enlisted for The First World War on his 20th birthday, April 1, 1915, along with m a n y o f his classmates. He u n d e r w e n t four m o n t h s of training

with the 6th Infantry Reinforcements of the N.Z.E.F., w h i c h sailed on 14th August on the t r o o p s h i p s Willochra and Tofua for Egypt. Aitken r e c o r d e d his w a r experiences in the v e r y moving and very graphic book, Gallipoli to the Somme: recollections of a New Zealand infantryman. Some have suggested that Aitken w r o t e this b o o k as a w a y o f exorcising his w a r t i m e experiences. This explanation s e e m s superficial. I susp e c t s u c h e x p e r i e n c e s cannot be exorcised, and I think that those w h o have e n d u r e d t h e m do not e x p e c t t h e m to be. The i m p o r t a n t fact is that this b o o k is one of the s t r o n g e s t p i e c e s of writing a b o u t the Great War. In its b r e a d t h a n d d e p t h it is very m u c h equal to the m o r e c e l e b r a t e d writings of Remarque, OrweU, Graves, Sassoon. The sonority of its prose, the perfectly wrought sentences, and the poetic sensibility proclaim the true reason for its authorship: Aitken, as Fenton so accurately concludes, intended to create a finished w o r k of art. No other mathematician, perhaps no other scientist, has written so well. Those who have read Paul Fussell's superb recent book, The Great War and Modern Memory [fus], already k n o w o f Aitken's wartime writing. FusseU quotes extensively from it. What m a k e s Aitken's b o o k such a compelling personal t e s t a m e n t is that the author so profluently addresses his feelings about his experiences, something rare in memoirs of the Great War. The British writer Graves's account [gra], certainly m o r e famous, maintains an icy--some would say characteristically B r i t i s h - - e m o t i o n a l d e t a c h m e n t from the ghastly panorama of the war. If G r a v e s w a s e v e r lonely, terrified, o r e v e n s a w a n y t h i n g of d a r k b e a u t y in his e x p e r i e n c e s , o n e w o u l d n't k n o w it f r o m his a c c o u n t . A i t k e n ' s b o o k is the greater, a n d it is lamentable t h a t it is o u t o f print. (One of m y a m b i t i o n s in w r i t i n g this p i e c e is to e n c o u r a g e O x f o r d U n i v e r s i t y P r e s s to r e p r i n t it.) Aitken finished off a first draft o f the b o o k in 1917, a n d a s h o r t e n e d ver-

sion a p p e a r e d in the Otago University Review in 1918. Much later, in 1962, he r e t u r n e d to it, s o t h e b o o k w e n o w have contains the exploits of youth as o b s e r v e d through the lens of midlife. I have not s e e n the first draft of the book, but the final version e m b o d i e s a m a t u r e d bitterness, a well-earned and fulminating anger t h a t I s u s p e c t was lacking in the original. Aitken railed against a military that p u t the personal aggrandizement of high-ranking officers above reason, above the welfare of the c o m m o n soldier. Battles were undertaken to secure m e d a l s and stripes, or even simply out o f high-ranking b o r e d o m , r a t h e r t h a n a s efforts to furt h e r clear-cut and r e a s o n a b l e strategic goals. Military o p e r a t i o n s b e c a m e a m e a n s of justifying t h e w a r a n d its horrific costs, not as m e a n s of forcing it to a conclusion. 4 He s c o r n e d the heroic literature of war:

Active service permanently removes any taste for the conventional poetry of war. Tennyson's "Charge of the Light Brigade" is so much painted cardboard, and Chesterton's "Battle of Lepanto" merely a cause of wonder that a grown m a n could write it. The New Zealand recruits b e g a n to p e r c e i v e the s h a p e of things to c o m e w h e n they w e r e a b o u t to d i s e m b a r k at Suez. The WiUochra t o o k on s o m e b a t t l e - s c a r r e d v e t e r a n s o f the campaigns of August, m a n y of t h e m w o u n d e d , w h o r e c l i n e d m u t e l y in their d e c k chairs, fully non-communicative, their eyes suggesting e x p e r i e n c e s they couldn't e x p e c t the recruits to understand. The a t m o s p h e r e on b o a r d grew even m o r e o m i n o u s w h e n s o m e of the recruits b e g a n to t a l k of G a l l i p o l i - - a l r e a d y the n a m e had acquired an air of menace. The next stop on the j o u r n e y was the G r e e k island of Lemnos. Never was a vestibule of hell so serene; Aitken fell u n d e r its enchantment:

I was content to walk the hill-side by Agriones, to watch the matrons sitting at house doors in the sun, gossiping and treadling their spinning

4Graves [gra] recounts how beguilingly complexthe war was, with dispatches and weapons drills and the demands of military protocol and the frenzied attacks and counterattacks and elaborate strategies. And the war possessed its own language. (Fussell [fus] provides a glossary of terms.) These factors served to distract all but the most cynical soldiers from any contemplation of the origins, strategies, or goals of the war. Aitken was one of the cynical ones.

VOLUME20, NUMBER2, 1998 65

w h e e l s . . . [I beguiled] the t i m e by exchanging vocabularies w i t h a darkeyed and highly intelligent small boy o f eight. It was nearing sunset; the unfenced white-washed houses, bluish i n the m o r n i n g light, were now cream i n the level rays, hens p i c k i n g peacef u l l y about, the other children were in two rows, approaching and receding i n a singing game exactly like 'Nuts i n May'. On an old O.A.S. I jotted down the tune . . . .

Unbeknownst to himself, Aitken was one of an impressive force of 78,000 men dispatched to help at Gallipoli. As the flotilla assembled near the peninsula, the commanding general Ian Hamilton discovered a ghastly error: guns and ammunition had been loaded on separate ships! Vessels had to return to Egypt to be supplied, and in the month-long delay the Turks were able to fortify their defenses. When the fighting began, the Allied forces found themselves pinned down on several unconnected beachheads. The Turks ringed the beachheads with reinforcements, and soon the fighting degenerated into trench warfare. After three brutal months of losses, Hamilton attempted a surprise attack on the western side of the peninsula. His soldiers lacked adequate naval support and the assault turned into a bloodbath. The Allied troops were ordered to take unassailable positions and were cut to pieces by the well-concealed Turkish machine-grinners. On January 8-9, 1916, the Allies withdrew from Gallipoli, accomplishing none of their objectives. They had suffered nearly 252,000 casualties. During the western assault, Altken slept on sliding earth, dug for cover into gullies among the roots of bushes. In the fatiguing climb up from the beaches, his comrades took turns carrying the precious violin up the slopes. He caught his first sight of death:

On November 11 they left Sarpi for Gallipoli. Aitken brought with him a violin which he had won in a shipboard raffle. He was afraid the violin would be confiscated as was all excess luggage, so he hid it during kit inspections, or, for security, arranged to have it covertly transferred from hand to hand among members of one of the rear platoons. The violin came to assume a talismanic significance both to Altken and to his fellow soldiers. It became their one link to beauty and to calm. Soon, Turkish bullets were skittering into the water around them. Abruptly, a man at the rail cried out, wounded. The innocence of the New Zealand recruits was soon to end. The Gallipoli campaign, the brainchild of then British First Lord of the Admiralty Winston Churchill, was one of the most misbegotten military ventures in history. Books and movies have been based on it; it has become synonymous with bungling, temerity, From the trench mouth, j u s t at hand, and short-sightedness. The Allied plan a stretcher p a r t y was emerging, carwas to take the pressure off the w i n g a m a n of the Wellington Western Front by forcing the Battalion, shot through the head and Dardenelles Strait and then go on to dying, a f r i e n d of m y own good Constantinople to dictate peace terms friend, Frank Tucker . . . . I date this to the Turks--one of the Central i n i t i a t i o n o f m i n e as 12th November, Powers. The Turks, though well- 1915. entrenched in the hillsides, were As Aitken stepped from the trench nearly out of ammunition and heavily mouth, a rogue enemy shell ricocheted damaged by the prelusive firepower into the dugout, killing the N.C.O. refrom 16 Allied battleships. Rear maining inside. Admiral John de Robeck, however, was unaware of their precarious A sequence o f blind accident . . . . B u t straits. Instead he called off the attack i f so, then m y own escape was equally and waited for reinforcements. accidental, since i f death falls i n this

fortuitous way, so does escape f r o m death. Thus it was useless to philosophize, as equally it was useless to be fatalistic.

Aitken was no military strategist but he easily perceived the dreadful military blunders to which he and his comrades were subjected: the time wasted in digging that should have been spent in advancing, the ludicrous choice of terrain for the attack--a labyrinth of deres and gullies, the antique unserviceable weapons or ineptly designed ones 5. One day, through his fieldglasses he spied a grisly vignette, seventy bodies, Indians, New Zealanders, Turks, lying in various attitudes of death, one Turk, quite dead, still seated lifelike in his aiming posture. It is conventional, even instinctive, to gloss over such scenes w i t h abstractions, lest they should grate too harshly on the soft susceptibilities o f civilian life. We take refuge in vaguen e s s . . , or i n noble phrases . . . . or i n something f o u n d i n such a non-combatant poet as Tennyson.

However, each evening provided a moment of respite from the horror. In the largest dugout Aitken gave a muted concert on the violin. Once the E-string broke. A resourceful compatriot successfully replaced it with the unravelled strands of a six-ply field telephone wire. Eventually, Aitken seemed to adjust stoically, as soldiers have always done, to the unbearable conditions. "I need not describe vermin," he says, laconically, of his trench accommodations. "Gallipoli had its several kinds, and even colours. "6 Though he was once very nearly killed, the war for Aitken did not end with the Allied evacuation. He was shipped immediately to the Somme in northern France. Like Gallipoli the Somme was a charnel house---a noman's-land along the Belgian border cob-webbed with trenches and nearly impregnable fortifications. In this nightmarish land victory was mea-

5"Cricket ball bombs," for instance, detonated by striking the head on a match striker sewn to the tunic, hence useless in wet weather. 6Rat stories constitute their own genre among legends of the war. Graves [gra] tells several. After a ferocious battle, a comrade heard a rustling as he entered his dugout and found on his bed two rats fighting over a severed hand.

66

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sured in terms of inches. For such intimate and personalized warfare new techniques were needed, and Aitken received bayonet training: We were taught the recent and business-like B r i t i s h s y s t e m . . , as simple as it was brutally efficient. It consisted in benumbing an e n e m y w i t h the foulest blows of foot or rifle butt, and then killing h i m economically w i t h the bayonet, the whole to be carried out i n a brisk f o r w a r d motion. To describe it accurately i n detail is to indict civilization. 7

He also received training in the use of gas and gas protection. (The Germans were the first to use gas, chlorine, in January 1915. The allies soon followed suit. s) There were numerous lulls in the fighting; and during these the environs of the struggle became another world. Some distance from the front, at a railw a y crossing in a brief m o m e n t of respite, Aitken saw a dogcart carrying two French girls who were waiting for the train to pass: They had distinction, and one was beautiful; and I had the sudden v i s i o n of a lost f o r m e r world, containing poetry and the m u s i c of Chopin, to which m u d and dust and khaki were strangers and unknown; the incident of a moment, a spark in the night at once extinguished.

Again, absent from the front for awhile, bicycling near Point-R~my, he experienced an alarming intimation, the first in his life, of the growing fragility of his sanity. He suddenly found himself trapped in a nightmare from which he could not wake: Walking in a trance, almost a levitation, the handle bars seeming to draw me on . . . . I had come to doubt the reality of externals and of m y own existence . . . . There is a type of dream that oppresses w i t h an unearthly sense of the i n f i n i t y of the past and of the elsewhere, a twilight world i n

which the s p i r i t knows that it is dreaming, even notes f o r rememb r a n c e - t h o u g h these dissolve at the m o m e n t of w a k i n g - - s t r a n g e features of the landscape, yet feels that i f there were only will-power enough it could break the shell and escape into the light of day. In such a dream, f a m i l iar yet unfamiliar, I was entangled . . . . At this moment, as i f i n answer to such an effort of will, a single white flash w i n k e d on the horizon, supernatural and unnerving; after an interval it opened and shut again like an eye . . . .

S o m e h o w Aitken was able to free himself, he described it as "waiting up," and The distant eye, somewhat east of Amiens, was still opening and shutting as I pedalled down f r o m Hallencourt to the billet, where all were asleep and I climbed quietly aloft.

Only later would he discover h o w truly ominous this experience was. The 27th of September, 1916, was Aitken's last day of active service. It was a day whose reverberations would plague him for the rest of his life. Despite a n u m b e r of factors that weighed heavily against success, his c o m p a n y was chosen to take the Gird Trench, the German fourth line. The c o m p a n y was allowed 27 minutes to cross 1,700 yards under the heaviest of fire with minimal cover. The action was a sort of military paroxysm, undertaken because nothing had happened on the front for awhile; the war had to be fed. The campaign was as monstrous as it was unreasoned; for the majority of the troops there was no chance of escaping the withering fire. Aitken's description of the offensive is the high point of the b o o k and I quote it in its entirety: At 2:15 the barrage, which seemed a perfunctory affair, moved f o r w a r d and the f i r s t two companies, in eight platoon waves, scrambled out and over to the right. M y Platoon followed

quickly up Goose Alley and climbed out on the right at the point where the s u n k e n road f r o m TTers to Eaucourt l'Abbaye crossed it. When the last m a n w a s clear we inclined left and hurried f o r w a r d after the other waves. This deployment was carried out without loss, so f a r as I could judge f r o m a quick survey, which gave me m y last glimpse in this world of Captain Herbert at the gap, directing the next wave. Like all the other officers I was wearing a private's tunic and equipm e n t and carrying a rifle and bayonet, the Spandau rifle I have mentioned. We had not gone a furlong when it became clear that there was something wrong ahead; the leading waves were not to be seen, except f o r isolated m e n straggling here and there. The reason became clear as we neared the road running due east and west f r o m Factory Corner to Eaucourt. M y f r i e n d Tucker, advancing a year later w i t h the stretcherbearers over to Gravenstafel in f r o n t of Passchendaele at d a w n on 4th October 1917, was haunted by Shakespeare's sonnet, "Shall I compare thee to a s u m m e r ' s day?", and heard all round h i m the slow movement, Largo e mesto in D minor, o f Beethoven's 7th Pianoforte Sonata. Such elevation w a s not mine; nothing extraneous or allusive distracted m y deadly attention. We were about half-way across w h e n German high explosive m i x e d w i t h shrapnel, of the greenish-black kind, began to fall thickly. Not ten yards ahead a group of the 8th Company vanished i n the smoke of a shell-burst, some falling where they stood, the others walking on dreamlike. I passed through the smoke. In a d i m w a y I wondered w h y I had not been hit by the f l y i n g pieces, but the m i n d would not trouble itself with problems at that moment, the overmastering impulse being to move on. On/On. t In an attack such as this, under deadly fire, one is as powerless as a m a n gripping strongly charged elec-

7Graves [gra] describes a sergeant bayonet instructor urging his trainees on with risible ferocity: "Hurt him now! In at the belly! Tear his guts outr' he screamed as the men charged the dummies. "Now let up a swing at his privates with the butt! Ruin his chances for life. No more little Fritzes!" 8Graves [gra] reveals that the instruments of gas warfare were euphemized; the word "gas" was not allowed to be used. The approved word was "accessory"; accessory cannisters, accessory masks, etc.

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trodes, powerless to do anything but Even then no thought of death came, go mechanically on, the f i n a l shield only some phrase like "sledgehammer f r o m death removed, the will is f i x e d blow," f r o m a serial read years before like the last thought taken into an in a boys' magazine. Pain came the anaesthetic, which is the f i r s t thought next moment; the spurious selfhyptaken out of it. Only safety, or the notism vanished and gave w a y to an shock of a wound, will destroy such overwhelming desire to run, anyautohypnosis. At the same time all where. Of three m e n - - a s I heard normal emotion is numbed utterly. later---crouching wounded in a shellClose upon this road now, I heard a hole that afternoon, one tried to keep voice abusing the Germans; crossing the other two in safety, but they broke the road, I realized that it had been away and made a wild dash in no particular direction, both being killed m y own. As I took the bank I looked left and by machine-gun fire w i t h i n a f e w saw Private Nelson, one of m y men, yards. I had the same wild wish, but fall forward on his knees and elbows, it was crossed and quelled by the his head between his hands. I men- resurgent rhythm of the f i r s t impulse, tally registered 23598, Nelson, W. P., so that I f o u n d myself walking on meand the terrific electromagnetic force chanically, yet wondering what I pulled m e on. He was dead. should do, disarmed now, i f I reached Now f r o m two directions, half- Gird Trench. A second wound disright, half-left, came the hissing of missed the question. As the right foot m a n y bullets, the herring-bone weave was coming to the ground a bullet of machinegun cross-fire. I saw some passed through in front of the ankle cut long straight scores in the ground, and fractured the several tarsal bones. sending up dust; some, as I f o u n d I crumpled and fell sideways to the later, cut m y tunic, f r a y e d the equip- right into a providential shell-hole, ment, and made rents in the cloth curling up like a hedgehog. Sleeve sopping, boot oozing blood cover of m y shrapnel helmet; m a n y seemed to w h i z z past m y ear, some to through the holes, I settled in as combury themselves under m y feet as I pactly as I could, lying back with head walked. Again these things are re- towards the enemy, tilting the shrapmembered as sometimes a deeply sub- nel helmet i n the direction o f the bulmerged dream m a y be recaptured in lets and waiting for the hours to pass. waking moments; at the time I took It was 2:23 p.m. s u m m e r time; the ataccount of them only dimly, and cer- tack had begun only eight m i n u t e s betainly did not think of death f o r a sin- fore; f i r i n g could not be expected to gle m o m e n t - - n o merit in this, we are cease before nightfall at least. The not responsible f o r what we do in wounds, n u m b at first, soon throbbed dream or hypnosis. Suddenly at m y left side m y platoon sergeant, Livingston, dropped on one knee and looked up at me in a curious doubtf u l gaze. "Come on, sergeant!" I said, stepping forward myself. He was killed, I think, the next instant; I never saw h i m again. All this occurred w i t h i n a f e w yards of crossing the road. I glanced right and left and saw the Platoon, thirty of them, crumple and fall, only two going on, widely apart, and no N. C.O. A f e w yards farther on I was nearly knocked down by a tremendous blow in the upper right arm and spun sideways; simultaneously the right hand unclenched and the Spandau riFigure 3. The violin. Held at Otago Boys' High fie and bayonet fell to the ground.

THE MATHEMATICAL INTELLIGENCER

recurrently. I drank very little water, reserving it f o r emergency. At one stage I wondered f o r a f e w seconds whether the arm-wound would affect the bowing of the violin, supposing I should ever get out of this; it seemed a foolish and unimportant thought. Using his left knee and elbow, Aitken c r a w l e d for four hours until he r e a c h e d safety. He h a d studied astronomy as a boy, a n d n o w that information s t o o d him in g o o d stead; he u s e d the stars to m a i n t a i n a southerly direction. Only one officer and twelve m e n r e a c h e d the Gird Trench, their objective. The a p p r a i s a l in the official historical r e c o r d states, "3 c o m p a n i e s o f I/Otago a l m o s t annihilated by shell fire and s t r e a m s of m a c h i n e gun bullets . . . . " O n l y t w o of 32 m e n in his platoon w e r e unscathed; five were killed. As to the violin, Aitken lost it at the Somme, b u t it w a s miraculously returned to him 18 m o n t h s later after having b e e n r e c o v e r e d by s o m e anonym o u s s o l d i e r a n d carefully p r o t e c t e d by the ranks. A n o t h e r soldier, on leave in London, h a d the violin p a c k e d a n d shipped to Aitken in Dunedin. Later, Aitken p r e s e n t e d it to the Otago Boys' High School, w h e r e it still r e p o s e s in a glass c a s e in the front hail (Fig. 3). The rolls o f Otago University h a d b e e n s a v a g e d b y the war. One councillor, 21 staff m e m b e r s , 554 s t u d e n t s or ex-students w e r e in the war; 94 o f t h e m w e r e killed, and many, m a n y m o r e p e r m a n e n t l y maimed.

School.

Some have attributed Aitken's later difficulties to what was once called battle fatigue or shell-shock and now is called post-traumatic stress syndrome, see [ame]. It is romantic to imagine so--nurture always appeals to us more than nature, it creates more interesting human dramas--but the explanation is unlikely; post-traumatic stress syndrome generally entails the reoccurrence in acute episodes of ideation drawn from the original traumatic event, g and Aitken seemed to have none of this. A story circulating among his colleagues was that his hallucinations were due to his having been gassed in the war. This, of course, is nonsense. I suspect the problem was biochemical, perhaps genetic in origin. In 1926 his brother Harry suffered a breakdown and was hospitalized in a nursing home near Dunedin. Aitken's illness seems to meet all the requirements of what is called today a chronic depression with psychotic episodes ([ame], p 292, and also [ben])--a condition that today is amenable to treatment and has a favorable prognosis. No doubt anyone witnessing the horrors Aitken did would be deeply damaged, but the damage is not always, or perhaps is seldom, mental. Rather, the inevitable disillusionment, alienation, and bitterness destroy the war veteran's ability to be spontaneous, to function as a free agent in society. The reference [orn] makes this point, and claims the failure to meet the needs of people so philosophically damaged constitutes one of the great indictments of contemporary society. Another chronicler of the war, Robert Graves [gra], apparently never met Aitken, although Graves knew many New Zealanders. Their narratives have much in common, and it is interesting to compare them. In his memoir, Goody-bye to all that, Graves lavishes contempt upon the civilian population by quoting from jingoistic newspaper articles and letters, including one grotesque but widely praised and reprinted letter from a British mother proclaiming her willingness to sacrifice the lives of her son and all other young men of Great Britain in the

war and then, if necessary, to take up arms herself, leading a mother's brigade against the infernal Germans. (Any reader wishing to experience the trauma of the WWI as transmuted into art could do no better than consult the war poetry of Siegfried Sassoon and others via the anthology [for], the trilogy of Pat Barker [barl], [bar2], [bar3], or the harrowing 1987 British film, A

Month in the Country.) Fenton claims that Aitken experienced periods of insomnia and depression that recurred at nearly fiveyear intervals and coincided with anniversaries of the battle of the Somme and which were exacerbated by involuntary memories of the war. This may be true, but Aitken experienced breaks with reality before the battle of the Somme, and the question of whether the war was at all causative is, I think, moot. After~a~l$ Aitken returned to New Zealand, and to his study of languages and mathematics at Otago University. He completed a first draft of his war book in mid-1917 and an extract from it appeared anonymously in the Otago University Review in 1918. For three months he had convalesced in London, and the aftermath of his wounds would always be with him. His right arm was partially paralyzed and the bones of his right heel shattered. He could walk, with the aid of a cane, but writing would be forever difficult. The mathematics department was in disarray. W. J. Martyn, who was in charge of the graduate students, had a breakdown and withdrew, leaving the students to fend for themselves. By corresponding with D. M. Y. Sommerville at Victoria College in Wellington, Aitken procured a continuation of his ~tition, but the texts, which were to be shipped from Great Britain, failed to arrive. The university wanted to nominate him for a Rhodes scholarship. Aitken declined ostensibly to give others from the University the chance to visit Europe. It is more likely that he was just exhausted, and needed the comfort of having a stable homebase.

At the end of that year, Aitken was awarded in his exams first class honors in French and Latin and second class honors in mathematics. The way the examinations were given was arrantly unfair, h o w e v e r - - a kind of academic colonialism prevailed. The exams were written and graded in Great Britain, and while there was a standardized syllabus the tastes of the examiners was, to someone living in New Zealand, an inscrutable quantity. No one in New Zealand got first in mathematics that year. Aitken, disillusioned, was convinced that his real aptitude was for languages. His reaction to his secondary honors was aberrant: he went into seclusion for two weeks [sch3]. He joined the staff of Otago Boys' High School in 1920, teaching Latin and French. However, he maintained some contact with the discipline of mathematics by enrolling at the University in a single course of study, Advanced Mathematics. R. J. T. Bell had been appointed to the Chair of Mathematics. He chose Aitken as his assistant to teach applied mathematics to honors students. Bell, a charismatic Scotsman recently from the University of Glasgow and an editor of the Proceedings of the

Edinburgh

Mathematical

Society,

reldndled Aitken's interest in mathematics. Bell and Aitken became close friends. When Aitken's brother Harry died in London in 1934, Aitken telegrammed Bell, not his father. In 1922 Aitken received a University of New Zealand scholarship to study mathematics under E. T. Whittaker at the University of Edinburgh. He left by train from Dtmedin in July. Aitken had married Winifred Betts on December 21, 1920, but Winifred finished out the school year before joining Aitken in Edinburgh in November. In Aitken's turbulent personal drama, his wife stays in the background. She and the children are infrequently mentioned in these diaries. But her role in his life was a result of her conscious decision. Though she briefly held a position teaching botany in Edinburgh, Aitken's daughter, Margaret, stated that

9[orn] states that "post-accident distress [is] closely linked to difficulties of cognitive assimilation of the traumatic event."

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All h e r energies . . . w e n t i n t o children, husband, h o m e . . . She had a t r e m e n d o u s belief i n m y f a t h e r ' s gen i u s , a n d I n e v e r heard h e r regret her o w n career and botanical work.

Aitken's diaries, the first book under consideration here, begin on July 30, 19239 He describes rapturously his journey from Dunedin to Auckland, New Zealand's major city, through moonlit forests and past immense mountains with magical Maori names, Ruapehu, Tongariro. At the hotel in Auckland before embarking he participated in some music making. A fellow traveller, an old gentleman, took Aitken aside and cautioned him solemnly: "The violin is a dangerous instrument. It puts thought into minds of many young women." Aitken embarked on August 7. On board ship, he was heartened by the unearthly beauty of the sunsets. His writing in the diaries is romantic and self-assured. He was no longer constrained by the deadly ambience of the war, and writing about unadulterated and tranquil beauty was a refreshing experience for him: I n the west, great cloud-castles o f f i r e , s h a d i n g through p a l e r h u e s to opal i n the north; t u r n i n g i n v o l u n t a r i l y w e r a n to the eastern rail, w h e r e the s k y s h o w e d f u l l o f even-sized s m a l l i s h c l o u d s . . . Then the p i l e d castles i n the w e s t a n d the f e a t h e r s i n the east both t u r n e d to grey, a n d then w e r e dissolved i n the dark blue o f f a l l i n g night.

There was a seven-day stayover in the United States, and then on to Great Britain. On October 1, Aitken met E. T. Whittaker, who immediately took him in hand and helped him to locate lodgings. Later he was introduced to E. T. Copson, then only 22. Aitken's brief, elliptical jottings during this period were occupied as much with musical events as with his mathematical studies: a lecture by the renowned musicologist Donald F. Tovey, a performance of a reduction of the Bach concerto in D minor for two violins, where he took one of the vio-

lin parts. He overcame his timidity and showed Tovey some sketches of one of his own compositions, two movements of a sonata9 Tovey remarked, noncommittally, that the score "showed promise," and suggested that Aitken take his class in composition9 Aitken began to explore Edinburgh and its environs. Perhaps they were kindred souls, city and man: mysterious, elegant, complex. He was delighted to discover that next door to a house he was visiting was the very residence where Chopin stayed in 1848. He visited Cramond, on the Firth of Forth, "very old-world," St. Andrews with its view of the North Sea, "pale blue," Portobello, with its beaches and dense crowds. Toward the end of 1924 Aitken found himself beset by moodiness and worry. The casual, disjointed jottings in his diaries abruptly cease; what follows is a retrospective account of those days that was written mostly in the 1940's and completed in 1958. The Aitkens spent the winter of 1925 in an old stone house. The house w a s i n i m p e r f e c t r e p a i r let i n r a i n at the back; the stone flagged floors, w o r n a n d u n e v e n i n the p a s sages, w e r e cold. A t n i g h t rats c a m e f r o m the m i l l s by the A l m o n d . . . and, e n t e r i n g the house by s o m e unplugged hole, left f o o t m a r k s i n the bathroom and s o m e t i m e s removed p i e c e s o f soap.

He was worried by the fact that his research was not prospering. Far into the night he performed Herculean computations with a crude calculator called the Archimedes multiplying machine and struggled to make sense of some divergent series which had resulted from his analysis of his thesis problem, one in statistics. He worried that he might not finish his thesis in time, he worried about the future, he worried about money. His daughter, Margaret, was born on April 10, 1925, and for a while was very ill. During the next four weeks, things began to look up. Margaret recovered nicely and, as Aitken states in a puz-

zling comment, success in his research project came with the "fortunate discovery of the clue--so simple when perceived, a s y m b o l i c L a u r e n t series instead of a Taylor s e r i e s . . . The thesis, therefore, took shape." The next several months he worked as many as 14 hours a day. Finally, he submitted the thesis, which was "lodged beyond recall or reconsideration, and I felt reaction, and then relief, after the calculations..." July 22 was the day of graduation. Greatly to his surprise, he was awarded the DSc degree, rather than the PhD, a great honor. 1~ On the platform was assembled an unusually august crowd of luminaries: Arthur Eddington, Ramsay MacDonald, G. K. Chesterton. Later, Aitken met Eddington at a garden party at Whittaker's. Aitken began teaching at the University on October 1. He was elected to the Royal Society of Edinburgh, and submitted several papers to the Proceedings 9 The school year of 1926 passed uneventfully, but in October he began to feel a languor and melancholy that alarmed him. In addition, he had just received bad news: his father had written informing him of the nervous breakdown of his brother, Harry, and it seemed that the prognosis was not good. Aitken's father believed the brother's health and career were permanently undone. Aitken found himself worrying about money again, and a dark and malign series of premonitions began, the first while he was visiting the Edinburgh Zoo on Costorphine Hill9 F r o m the h i l l . . 9 I w a s looking at the [Pentland m o u n t a i n s ] i n the f r e s h ness o f the s p r i n g sunlight. S u d d e n l y 9 I s a w t h e m take on a s i n i s t e r a n d u n e a r t h l y colour, the p l a i n between being dark and featureless, almost a moor, but s i m i l a r l y f o r b i d d i n g . . . . I descended a n d p a s s e d through the Reptile House. The s e n s a t i o n returned, i n t e n s i f i e d a n d bizarre. The obscure life o f these creatures, linked to eras antecedent to ours, and seemi n g to belong, even now, to a t i m e and space not ours, w a s disquieting; I f e l t

1~ the applicant is first awarded the PhD. The applicant then may apply, usually several years later, for the DSc, an advanced degree which requires the submission of another thesis.

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that the levels of existence were multiform and unstable.

This experience is very reminiscent of his vision at the Somme. When home, he dismissed the incident as a matter of nerves due to a transient physical upset. The next entry is in April, 1927. On the 18th of that month, a son, George, was born. Another dramatically unsettling incident occurred on Monday, May 30. In a deep depression, he walked into his classroom. Soon feelings like those that had occurred at the zoo descended on him. He felt as though he were drifting into another dimension. As he began his lecture, the students and the classroom vanished utterly. Yet, through a great act of courage, he continued to lecture. Just before I uttered the f i r s t word a student entered a fraction of a m i n u t e late, M i s s - - , placing a tennis racket by her on the desk as she sat down. The slight noise diminished, like a remote train receding at night, the lecture-room . . . disintegrated and became a brown mist, yet concentrated at the point of noise, in a f a i n t shimm e r of vibrating heliotrope. I pulled m y s e l f together, but was unable to see through the mist, unable to see m y lecture notes; so I spoke on as into a vacant a u d i t o r i u m or the radio, of the subject of Statistics as a young and developing Science . . . I remember that this was the topic, but do not rem e m b e r a single separate word. The students, I heard later, took no notes. Perhaps m y watch was, as usual, on the desk; at 4:55 1 ended, the room felt into focus in the sunlit afternoon, I gathered up m y papers and went into the Research Room opposite . . . The noise o f descending steps died quickly away; I was conscious of unprecedented silence and solitude.

For the first time, he nearly missed a performance that evening by the quartet of which he was a member. He arrived 10 minutes late and reacted to the music being played with a morbid hypersensitivity:

. . . and then, after coffee, the 2nd Rasoumovsky in E m i n o r . . . Now this, as I s a w afterwards, was f o r me a bad choice; f o r the work is closely a k i n to the Sonata Appassionata, Op. 57, and both of them express some psychological tragedy w i t h incomplete catharsis. This E m i n o r induced a dangerous excitement; it would have been better to keep to H a y d n . . .

Aitken and most of his biographers call 1927 the year of his "nervous breakdown," a commonly used but meaningless expression, the term implying a condition that either can or cannot be amended, but, once repaired, leaves the subject in a state of uninterrupted relief. Aitken experienced hallucinations, depression, hyperaesthesia, symptoms that were to continue for years. His diaries suggest he was never free from them. He suffered chronic insomnia; at night in his bed could observe scenes drawn from stories he had heard of family history, "enacted in the darkest colors." On June 4, 1927, he was to deliver a paper at the St. Andrews meeting of the Edinburgh Mathematical Society 11. I travelled to St. A n d r e w s by an early afternoon train. The country all the w a y was flooded w i t h sunshine, in all the beauty of early summer. Just beyond Markinch a strange thing occurred. A mother and daughter shared m y apartment, ordinary, unassuming persons, of w h o m I had taken no special notice--for I was thinking of how to present the paper later. These two now became transfigured before me, almost into angelic visitants; then, in a m i n u t e perhaps, they resumed their f o r m e r shapes.

He experienced distortions of time and of shape. It would take eons for the tramcar to go from the University to Haymarket, a short distance, and then in what seemed to be only seconds later but which in reality corresponded to a far greater distance, he would walk through the doorway of his house. Ordinary scenery would become transfigured and assume sinister, violent coloring and an other-dimensional,

apocalyptic suggestiveness. Eerie, disembodied melodies played on unrecognizable instruments sounded in his head. He transcribed many of these haunting melodies into the diaries contained in the present book. 12 During one four-week period of sleeplessness, he spent the nights with chords clashing and nonsense words exploding in his mind. His fitful nights began to impact his general health. Walter Ledermann recalls [led] how, after obtaining a PhD from St. Andrews, he applied in 1936 for a position at the University of Edinburgh. He was interviewed by Whittaker, who offered him a postdoctoral fellowship. "I have on my staff a Dr. A i t k e n . . . I advise you to study all the topics he can teach you." Aitken was nearby, and Whittaker pointed him out. Ledermann was shocked by his marmoreal appearance, and Whittaker added in an undertone, "Don't you see how pale he looks?" Of course, when one is battling one's personal devils, it's difficult to present a consistent face to the world. Whether any personal encounter with Aitken evoked his impatience or his tolerance, his distraction or his attentiveness, his charm or his moroseness, depended on the day and the time. We have a series of baffmgly inconsistent reports. Hans Schneider, his PhD student, says that he had virtually no contact with Aitken in the 20-month period in which he was his student. After a lecture, Aitken responded to a question with the somewhat dismissive command, "Read Frobenius!" by which he meant, Frobenius's papers on nonnegative matrices. Paradoxically this suggestion, the sole mathematical counsel that Hans received from his supervisor, proved to be the foundation of Hans's 45-year research career. Anyone who got to know Aitken in his first two decades or so at Edinburgh got past his superficial brittleness and found a warm, giving person underneath. Ledermann was a close friend and a fellow member of the Edinburgh Mathematical Piano Quartet. They took walks together, cel-

11To be distinguished from the Royal Society of Edinburgh. Both societies issue proceedings publications. 121 played these tunes. They are unlike anything I have ever heard.

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F i g u r e 4. The U n i v e r s i t y o f O t a g o t o d a y .

e b r a t e d holidays together. Ledermann, w h o h a d e s c a p e d from Nazi Germany, w a s p r o f o u n d l y grateful for Aitken's e n c o u r a g e m e n t and a s s i s t a n c e in fmding employment. However, L e d e r m a n n k n e w Aitken only in his b e s t years; he left Edinburgh in 1946, well b e f o r e the grim, final years of A i t k e n ' s life. The Human Calculator Aitken w a s one of the g r e a t e s t of the m e n t a l calculators. An entire c h a p t e r in the b o o k Great m e n t a l calculators

[smi] 13 is d e v o t e d to his life a n d to his talent for doing difficult calculations w i t h o u t the aid of writing materials. Like his m e n t a l affliction, this gift m a y also have h a d a genetic c o m p o n e n t : his father's e l d e r brother, though a farmer, p o s s e s s e d formidable skills at m e n t a l arithmetic. Aitken a t t r i b u t e d his facility at mental c o m p u t a t i o n partly to his i s o l a t e d New Zealand c h i l d h o o d and its beneficial effect on his concentration: "I g r e w up in a r e m o t e p a r t of the Empire, be-

fore the days of radio, when even the telephone, that m o d e r n n e c e s s i t y b u t chief a m o n g the interrupters o f thought, w a s a rarity." It w o u l d be interesting to k n o w w h e t h e r the relentless incursion o f television into o u r mental lives has r e s u l t e d in a s c a r c i t y of such talent. Calculating p r o d i g i e s fascinate us. Recently, the d e p i c t i o n of one in an American film g a r n e r e d for the a c t o r an A c a d e m y Award. A c o m m o n belief is that the talent is always allied with mental d e f e c t i v e n e s s - - t h e so-called idiot savant p h e n o m e n o n . As Smith points out, this is far from the truth. Most great m e n t a l calculators excel in other mental u n d e r t a k i n g s also: Euler, Gauss, Ampere, A i t k e n himself. Mental c a l c u l a t o r s t r e a t the integers as p e r s o n a l friends. Wim Klein 14, a Dutchman a n d a m o n g the g r e a t e s t 20th-century mental calculators, stated, "Numbers a r e friends for me, m o r e o r less. It d o e s n ' t m e a n the s a m e for you, d o e s it, 3,844? F o r you it's j u s t a three and an eight and a four and a four. But I say, 'Hi, 62 s q u a r e d . ' " This is r e m i n i s c e n t o f R a m a n u j a n ' s f a m o u s r e m a r k to H a r d y w h e n the latter visited him in the hospital. Of course, Ramanujan h i m s e l f w a s no m e a n calculator. F o r the m e n t a l calculator, n u m b e r s are as f e c u n d with associations as the n a m e s of friends. Calculating p r o d i g i e s have described the often labyrinthine, nonintuitive s c h e m e s a n d m n e m o n i c s t h e y use. 15 Their d e s c r i p t i o n s suggest to us that the mental c a l c u l a t o r has r e p l a c e d a straightforward t h o u g h tedious p r o b lem by a s e q u e n c e of m u c h m o r e arcane ones. Aitken w a s a s k e d once b y his children to multiply 987,654,321 b y 123,456,789. "I s a w in a flash," he said, "that 987,654,321 b y 81 equals 80,000,000,001; a n d so I multiplied 123,456,789 b y this, a simple matter, and divided the a n s w e r b y 81. Answer: 121,932,631,112,635,269. The w h o l e thing could h a r d l y have t a k e n m o r e

13Except when so noted, all quotations in this section are taken from this reference. 14At one time, Aitken participated in a radio contest with Klein. ~SThe most dramatic account of a mnemonist I know is given in [lur]. It seems there was nothing S. V. Shereshevskii, a professional mnemonist and the subject of this report by the distinguished Russian neuropsychologist A. R. Luria, could not memorize. The associations he used to do so were tortuous and bizarre. How, I wondered, was he able to remember his mnemonics? Shereshevskii could remember every word of every conversation he had ever had. When he became unable to distinguish the current conversation from one he might have had 30 years earlier he was committed to an asylum, where he spent the rest of his life. [hun2] has an intriguing comparison of the ways the memories of Shereshevskii and Aitken functioned.

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than half a minute." Why, though, d i d Aitken settle on 81? Those having the talent are impatient with scientists w h o a t t e m p t to p r o b e their abilities. One said with exasperation, "I j u s t do it. J u s t as it s e e m s natural to y o u to formulate a s e n t e n c e w i t h o u t consulting the rules of gramm a r o r tallying the meaning of e a c h word, so I calculate." When Aitken said of a number, "It feels prime," it invariably was. Their a t t e m p t s to explain their gifts are unhelpful, even baffling, and their m e t h o d s are mysterious. "The secret, to m y mind, is relaxation," Aitken o n c e wrote, "the c o m p l e t e antithesis of c o n c e n t r a t i o n as usually understood." Calculating p r o d i g i e s do not conceive o f n u m b e r s as s e q u e n c e s o f digits: t h e y t e n d to see t h e m as single entities, j u s t as w e see w o r d s as a distinctive whole, rather than a s e q u e n c e of letters n o r a chaotic m e n a g e o f s o u n d s ([smi], p.50). The w a y t h e y depict n u m b e r s internally is a p e r s o n a l affair. There are, generally, t w o different sorts o f mental calculators: t h e r e are t h o s e w h o s e e the n u m b e r s in their minds; m o r e surprisingly, m a n y mental c a l c u l a t o r s hear t h e m ([smi], Ch. 1). J a c q u e s Inaudi stated, "I h e a r n u m b e r s . . , a n d it is the e a r t h a t retains them," and to m o r e insistent questioning he said impatiently, " . . . h o w could I s e e them, since it w a s b a r e l y four y e a r s ago t h a t I l e a r n e d t h e m and I c a l c u l a t e d mentally long b e f o r e that time?" Aitken was p r i m a r i l y n e i t h e r a visual n o r a auditory c a l c u l a t o r - - r a t h e r , a combination of b o t h a n d s o m e t h i n g m o r e enigmatic. "I m y s e l f can visualize if I wish, and at intervals in a calculation, a n d also at the end, w h e n all is done, the n u m b e r s c o m e into focus; b u t m o s t l y it is as if t h e y w e r e hidden u n d e r s o m e medium," he said. "I think that it is neither seeing n o r hearing; it is a comp o u n d faculty of w h i c h I have n o w h e r e s e e n an a d e q u a t e description." Aitken h a d r e d o u b t a b l e memorization abilities, t h o u g h t h e s e are not alw a y s a s s o c i a t e d with calculating skills.

Given the page a n d a n y line n u m b e r of Virgil's Aeneid, h e could recite that passage. Psychologists s t u d i e d Aitken [hunl], [hun2], a n d r e p o r t e d his perf o r m a n c e on m e m o r y t e s t s [hun2]. He was tested, a m o n g o t h e r things, on his recall of m a t e r i a l l e a r n e d u n d e r experimental c o n d i t i o n s 27 y e a r s earlier. The a b s t r a c t of the r e p o r t says that his m e m o r y w a s "stronger t h a n average in all r e s p e c t s b u t n o r m a l in m o d e o f functioning." (I admit, I don't understand this sentence.) His m e m o r y w a s eidetic (the t e r m iconic is s o m e t i m e s used). He w o u l d s c a n a n e w issue o f a m a t h e m a t i c a l j o u r n a l as rapidly as w e w o u l d r e a d half-a-dozen lines a n d the material w a s fixed in his mind. He could r e p l a y it at his leisure ([altl], p. 26). Aitken did use t h e p r o c e s s of mental audition for m e m o r i z i n g numbers: "Memory in m y o w n c a s e is visual if I desire, t h o u g h in the main auditory, but resting on a r h y t h m i c foundation." In 1873, Dagbert c o m p u t e d qr to 707 decimal places. A i t k e n relied on this metrical a p p r o a c h to m e m o r i z e t h e figures. He f o u n d it e a s y to group the digits in r o w s of fifty each, and to group each fifty into ten g r o u p s of five, and then, r e a d the w h o l e off in s o m e privately c h o s e n cadence. An acquaintance of m i n e w h o h a d him for a class said Aitken one d a y effortlessly int o n e d for his s t u d e n t s the first thous a n d digits o f zr. "It w o u l d have b e e n a r e p r e h e n s i b l y u s e l e s s feat," Aitken once w r o t e o f s u c h a m e m o r i z a t i o n task, "had it not b e e n so easy." J. C. P. Miller r e p o r t e d t h a t w h e n 1,000 additional figures of 7r w e r e p u b l i s h e d in 1949, Aitken effortlessly i n c o r p o r a t e d t h e m into his p e r f o r m a n c e . 16 His m e m o r y w a s all-inclusive, even in wartime. On July 13, 1918, Aitken's 4th c o m p a n y w a s n e a r l y annihilated in the raid at Armentibres, and the roll b o o k of p l a t o o n 19, Aitken's old platoon, w a s missing. Battalion headquarters n e e d e d the n a m e s of the casualties a n d the conditions of the survivors. Aitken, lying e x h a u s t e d on

s o m e c o c o n u t m a t t i n g at regimental headquarters, offered t o help:

Apparently, [only] surnames were available . . . . I had no difficulty, having a well-trained m e m o r y now brought by stress into a condition almost of hypermnesia, in bringing the lost roll book before me, almost, as it were, floating; I imagined it taken away either by Mr. Johnson or perhaps in the pocket of Sergeant Bree in no man's land. Speaking f r o m the matting I offered to dictate the details; full name, regimental number, and the rest; they were taken down, by whom, I do not know. At that time a platoon numbered f r o m 50 to 100 troops. It is difficult to imagine a curse w o r s e than t h a t o f p e r f e c t recall for the v e t e r a n t r a u m a t i z e d b y the h o r r o r s Aitken h a d witnessed. Aitken often p e r f o r m e d publicly. In a 1951 a d d r e s s on m e n t a l calculation b e f o r e the Society o f Engineers he s q u a r e d four-digit n u m b e r s promptly, e x t r a c t e d five-place square r o o t s o f s e v e r a l three-digit n u m b e r s - - e a c h c o m p u t a t i o n t o o k t w o o r t h r e e seco n d s - c o m p u t e d the 96 digits of the recurring decimal for 1/97, identified as p r i m e or f a c t o r e d r a n d o m l y c h o s e n four-digit numbers. He c o n s i d e r e d his ability m o r e than j u s t a p a r l o r stunt. "Familiarity with n u m b e r s acquired by innate faculty s h a r p e n e d by a s s i d u o u s p r a c t i c e does give insight into the prof o u n d e r t h e o r e m s of a l g e b r a a n d analysis." He s t r e s s e d the a r t l e s s n e s s of this talent ([aitl], p. 26) w h e r e i n one didn't force the lock of discovery, one simply a l l o w e d it to open:

I believe we are surrounded the whole time by marvellous powers, are immersed in them, closer than breathing, and I think that all great music, poetry, mathematics, and real religion come f r o m a world not distant but right in the midst of everything, permeating it. When I wish to do a

16As of 1989, the record for memorizing digits of ~" was held by a Japanese, Hideaki Tomoyori (40,000 digits) [rei]. In this same article, Rajan Mahadevan, a recent doctoral student at Kansas State University, explains the mnemonics he uses to remember a string of digits. "1 scan the whole thing and start to make associations. For example there is '111 '; that's called a 'Nelson' because Adm. Nelson has one arm and one leg. I see a '312' in there: area code of C h i c a g o . . . ' And so on. See the previous footnote.

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f e a t of m e m o r y or calculation, or, sometimes, new mathematical discovery, I let slip some sort of cog and lie back in this world I speak of, not concentrating, but w a i t i n g in complete confidence f o r the thing desired to f l o w i n . . . F o r an o c c a s i o n a l c h a n g e of pace, Aitken enjoyed doing c o m p u t a t i o n s in t h e b a s e 12. He h a d a love affair with duodecimals, to his g r e a t detriment, as w e shall see.

The Mathematics It is a tribute to A i t k e n ' s laudable t o u g h n e s s of spirit t h a t he m a n a g e d to function p r o f e s s i o n a l l y in the y e a r s aft e r his 1927 b r e a k d o w n , dealing daily with vicissitudes as trying as any he h a d e x p e r i e n c e d in the war. Aitken and o t h e r such e x e m p l a r s - - v a n Gogh, Dostoevsky, S c h u m a n n - - s e r v e to rem i n d us that, paradoxically, an emotionaily e m b a t t l e d p e r s o n can also b e resilient and tough. Aitken w o r k e d in s e v e r a l areas: num e r i c a l analysis, statistics, geometry, classical algebra. He c o n s i d e r e d mathe m a t i c s only a n o t h e r interest, not the interest, in his life. Despite his manifest talent and creativity, qualities he b r o u g h t to all his m e n t a l activities, t h e r e w a s something of the dilettante a b o u t him. Many m a t h e m a t i c i a n s will n o t b e able to forgive him for his apost a s y from the true way, t h a t o f mathem a t i c a l abstraction, a n d s o m e of his obituaries t a k e on a d e c i d e d l y apologetic tone w h e n discussing his work. Aitken s a w no r e a s o n to c h o o s e a single a r e a of m a t h e m a t i c a l e x p e r t i s e and p u r s u e it with m o n o m a n i a c a l zeal. As a m a t t e r of fact, his m e n t a l complexion w a s much m o r e similar to that o f the great thinkers o f the r e n a i s s a n c e - the J a c k s of all t r a d e s - - - t h a n to that of t h e m o d e m specialized thinker. Aitken w a s n e v e r a friend to t r e n d s in m o d e m mathematics, and p r o b a b l y the comp o s i t i o n of the m a t h e m a t i c s departm e n t at Edinburgh suffered as a result o f it. On Aitken's final c o l l a p s e the de-

Figure 5. A. C. Aitken, September 1944. (From To Catch the Spirit. Used by permission.)

p a r t m e n t w a s in such d i s a r r a y a n d so lacking in direction that A r t h u r Erd~lyi, t h e n at the California Institute of Technology, w a s a s k e d to a s s u m e the l e a d e r s h i p and bring the departm e n t into the twentieth century. 17 A i t k e n ' s w o r k in statistics b e g a n with t h e p u b l i c a t i o n of his quirky little volume, Statistical Mathematics (1939). The book, as are all of A i t k e n ' s writings, w a s beautifully written, a n d in its 150 p a g e s contained an astonishing a m o u n t o f information, but t h e r e w a s

evidence that statistics w a s n o t Aitken's mdtier. The b o o k was a bit disorganized, a n d the crucial idea o f statistical i n d e p e n d e n c e was r e l e g a t e d to an appendix. However, it satisfied a clearly-felt need: it w a s almost the only reference w o r k on current statistical theories. The m o s t influential o f Aitken's statistical p a p e r s was "The estimation of statistical p a r a m e t e r s , " c o a u t h o r e d with H. Silverstone (1942). The p a p e r dealt with o n e - p a r a m e t e r estimation

17Erd61yi was a Jewish mathematician, a Hungarian, who lost three sisters in the holocaust. He fled to Great Britain late in the war and obtained a post at Edinburgh. E. T. Whittaker spearheaded a movement to keep Erdelyi in Great Britain, and primarily because of Whittaker's prestige, the effort was successful. I believe Erd61yi felt he owned Whittaker, and indirectly the University of Edinburgh, his life. Erdelyi worked at Edinburgh until the late forties, when he left to take a position at California Institute of Technology managing the Bateman Manuscript Project.

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p r o b l e m s w h i c h admitted, a m o n g the u n b i a s e d e s t i m a t e s of the p a r a m e t e r , one c h a r a c t e r i z e d b y m i n i m u m variance. The d e v e l o p m e n t in the p a p e r is quite difficult to follow, a n d t h o s e w h o struggled with it w e r e n o t especially h e l p e d b y t h e authors' enigmatic ass u r a n c e that the crux of the issue w a s "a minimal p r o b l e m in the calculus o f variations, of positive definite t y p e a n d formally simple." In s u b s e q u e n t publications, Aitken p u r s u e d the matter, generalizing the analysis to m a n y parameters. A b o u t a third of the over 200 p a p e r s p u b l i s h e d since the early forties that e m a n a t e from Aitken's w o r k mention his w o r k in statistics, in particular, this paper. In 1950, Aitken s t a r t e d to w o r k on M a r k o v chains. None of the w o r k w a s published. J. C. P. Miller [mil] has p o i n t e d out that in the statistical Library at Cambridge t h e r e is a p a c k e t of letters from Aitken to J o h n Wishart describing s o m e of the f o r m e r ' s discoveries. Aitken b e g a n with a twos t a t e M a r k o v chain with c o n s t a n t transition probabilities. He a s s u m e d t h a t a s c o r e s e q u e n c e So, $1, $2,. 9 9 w a s cons t r u c t e d b y starting with So = 0 and increasing S b y an a m o u n t x o r y dep e n d i n g on the n e x t state in the sequence. He was able to write d o w n an elegant m a t r i x for the F o u r i e r characteristic function of the s c o r e sequence a n d thus to d e t e r m i n e the asy m p t o t i c b e h a v i o r o f the s c o r e as n --* 0% including the derivation o f a central limit theorem. He generalized this analysis in several directions. In his o b i t u a r y contribution, Miller suggests the m a n u s c r i p t of this m a t e r i a l still exists s o m e w h e r e in Edinburgh. A i t k e n ' s w o r k in statistics n e v e r ext e n d e d into the realm o f the m o d e r n , w h e r e p r o b a b i l i t y fuses with m e a s u r e theory. P e r h a p s his p o w e r s o f m e n t a l c o m p u t i n g c a u s e d his interests to focus exclusively on m a t h e m a t i c s with a high algorithmic content, w h e r e explicit c o m p u t a t i o n s could b e m a d e a n d a b s t r a c t i o n w a s k e p t to a minimum. As his c a r e e r continued, m o r e a n d m o r e of A i t k e n ' s m a t h e m a t i c a l results w e r e c o m m u n i c a t e d b y m e a n s o f personal letters, e.g., [min]. Walter

L e d e r m a n n a n d P e t e r F e n t o n are in the p r o c e s s of a s s e m b l i n g these letters, and though the j o b is n o t nearly fmished, t h e y have s o far acquired over 300 p a g e s [led]. Publication p h o b i a is not a rare d i s o r d e r in the senior mathematician, b u t in A i t k e n ' s c a s e his app r e h e n s i o n a b o u t being j u d g e d b y his colleagues m a y have w o r s e n e d as his battles with his p e r s o n a l d e m o n s intensified. After m e e t i n g h e r at a conference in St. Andrews, he w r o t e to Olga T a u s s k y - T o d d asking for r e p r i n t s of her work. The l e t t e r has a plaintive, t i m o r o u s tone: Dear Miss Taussky: I felt I m u s t w r i t e to y o u to s a y how glad I a m that St. A n d r e w s has caused you to become, not a n a m e as y o u were before, but a personality. I began by being afraid o f you; I ended by f i n d i n g that there w a s no reason to be afraid o f a p e r s o n f o r w h o m one had acquired a liking and a respect . . . .

Aitken's g r e a t e s t mathematical strengths lay in n u m e r i c a l analysis. Here his w o r k w a s inspired. One wonders w h a t A i t k e n ' s p r o f e s s i o n a l develo p m e n t w o u l d have b e e n h a d he lived into the e r a of large-scale computers. It is p e r h a p s i m p o s s i b l e to p r e d i c t from the p a s t e x p e r i e n c e s of a m a t h e matician w h a t his o r h e r r e a c t i o n to c o m p u t e r s and their b o u n d l e s s potential will be. I k n o w o f a m a t h e m a t i c i a n w h o s e p r i m a r y i n t e r e s t is in surgery on manifolds, b u t w h o has written an imp o r t a n t b o o k on parallel processing. And I k n o w o f n u m e r i c a l analysts w h o are unfamiliar with any p r o g r a m m i n g language and t e a c h n u m e r i c a l analysis classes using only a p o c k e t calculator. Aitken h a d the m i n d of a classicist, a traditionalist o f a p e c u l i a r bent. Aitken in the w o r l d o f c o m p u t e r s ? P e r h a p s he would n o t have w e l c o m e d them. To m y mind, A i t k e n ' s m o s t impressive a c h i e v e m e n t in n u m e r i c a l analysis, one w h i c h h a d i m m e n s e ramifications a n d w h i c h virtually c r e a t e d the entire m o d e r n field of m o d e r n nonline a r s u m m a b i l i t y theory, grew out of his r e s e a r c h e s on the use of Bernoulli's

m e t h o d for c o m p u t i n g the largest root o f a p o l y n o m i a l equation. As m o s t of us know, the e r r o r in the Bernoulli method, i.e., the difference b e t w e e n the real r o o t of t h e p o l y n o m i a l equation and the a p p r o x i m a t e root, beh a v e s roughly exponentially, something like p", w h e r e p is the ratio of the next-to-largest r o o t to the largest root. When roots are close together, the conv e r g e n c e of the m e t h o d is poor. Aitken w o n d e r e d w h e t h e r one c o u l d modify the p r o c e s s and so i m p r o v e convergence. His reasoning w a s as follows. Let Sn d e n o t e the s e q u e n c e of iterates is in the method. S u p p o s e sn converges to its true value a like Sn ~ a + c 9 p'~. Assume, for the s a k e o f argument, that this equation holds exactly. Replacing n b y n 4- 1 and then b y n + 2 yields the t h r e e equations, S n -- Ol = C p n , S n + 1 -- OL -~ C p n + l , S n + 2 -- O~ = C p n + 2 .

Taking the ratio o f t h e left-hand sides o f the first two equations gives p on the right, b u t this is also w h a t the ratio of the left-hand sides o f the s e c o n d two equations gives. Setting t h e s e ratios equal to each o t h e r w e get Sn+ 1

--

S n -- Ol

OZ

Sn+ 2

--

OL

Sn + 1

--

OL "

We can easily solve this equation for the u n k n o w n r o o t a. Of course, in the general c a s e w h e r e the e r r o r isn't exactly an exponential, it is only an app r o x i m a t i o n to the r o o t a. F o r each value of n, then, w e have a sequence of a p p r o x i m a t i o n s to the r o o t a, call the sequence Sn~, w h i c h one can calculate readily from the first sequence s.r~: S~

-~

- - s~+I 2Sn+l 4- Sn '

SnSn +2 Sn+2

--

n = 0, 1 , 2 , . . . . It might be that the n e w sequence converges to the s a m e limit, only more rapidly. A clever idea, indeed. Blessed is the m a t h e m a t i c i a n w h o during his or her c a r e e r is privy to even a few insights such as this. W h e n a p p l i e d to

18Actually, one takes the ratic of successive iterates as approximations to the root.

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Bernoulli's m e t h o d the technique prov i d e s a n e w sequence w h i c h converges to t h e largest r o o t again exponentially, b u t p will be the ratio of next-to-nextto-largest root to the largest root. If the r o o t s are well-separated, the converg e n c e is substantially increased. The formalism of the m e t h o d is tantalizing: w h a t if one applies the m e t h o d r e p e a t e d l y ? or, w h a t if one a s s u m e s t h e e r r o r is a linear c o m b i n a t i o n of different exponential t e r m s ? The latter gives a m o r e general p r o c e d u r e going b y t h e name of the Shanks algorithm (actually due to E. Schmidt). P e t e r Wynn i n t r o d u c e d a c e l e b r a t e d m e t h o d called the e-algorithm w h i c h allows one to c o m p u t e the i t e r a t e s in the S h a n k s algorithm b y m e a n s of a twod i m e n s i o n a l lozenge p r o c e d u r e . Using t h e Shanks algorithm, one can comp u t e rational a p p r o x i m a t i o n s , called Pad~ approximants, to functions defined b y a p o w e r series. The Aitken p r o c e d u r e m a y be ext e n d e d to the solution of o p e r a t o r equations, and to sequences in a topological v e c t o r case [wim]. And so on and on and on. If one judges the merit of an idea by the n u m b e r of PhD theses it has directly or indirectly generated, the Aitken ~2_ process, as it is n o w called, m u s t occupy one of the most revered pedestals in the mathematical pantheon. Aitken didn't invent s u m m a b i l i t y m e t h o d s . 19 Methods for improving the c o n v e r g e n c e o f s e q u e n c e s can be t r a c e d to Euler, b u t a l m o s t all previous techniques were linear m e t h o d s . The ~2 p r o c e s s d e p e n d s nonlinearly on the t e r m s of the original sequence, Sn. Nonlinear m e t h o d s are vastly m o r e p o w e r f u l than the linear ones; for example, the Wynn algorithm can assign a "sum" to the vigorously divergent seq u e n c e of partial s u m s of the series 1! - 2! + 3! - 4! + .... The r e a d e r m a y s u s p e c t that the justification of such a

c o m p u t a t i o n is m o r e a m a t t e r of metap h y s i c s t h a n of mathematics. However, the value obtained is . 6 8 3 . . . , t h e value o f the integral

fo e-t +

braicists, it h a p p e n s time and again that the v e r y c o r e of an i m p o r t a n t advance rests u p o n an intricate formal relationship that w a s discovered b y o u r m a t h e m a t i c a l ancestors."

dt.

P e r h a p s the r e a d e r can see the connection. A n o t h e r productive line of r e s e a r c h was on algebraic problems. I d o n ' t m e a n m o d e r n a l g e b r a - - r i n g s , Galois groups, Lie a l g e b r a s - - I m e a n old-fashioned Chrystal-type algebra. Occasionally group-theoretic or c o m b i n a t o r i a l ideas w o u l d cross Aitken's path, b u t they n e v e r received his full attention. However, he w a s a p i o n e e r in the s t u d y of s y m m e t r i c functions, and his w o r k on d e t e r m i n a n t s had very useful consequences. The little b o o k Determinants and Matrices is, I feel, still the t r e a t m e n t o f the subject of m o s t value to the practicing analyst. An aggravating quality of this book, however, is the author's disinclination to state results in full. F o r instance, he gives the so-call Schweinsian determinantal identities, crucial for deriving the Wynn method, and the f o r m u l a for the inverse of a v a n d e r Monde matrix only for special cases, enjoining the r e a d e r to furnish the e x t e n s i o n to the general case. W. L e d e r m a n n in [mil] o b s e r v e d that "he invited his r e a d e r s to a c c e p t a description o f a typical case in p l a c e of a rigorous and c o m p l e t e argument. Whilst his m a t h e m a t i c a l style is thus occasionally s o m e w h a t sketchy, his English p r o s e is always lucid and pleasing." Nothing a b o u t v e c t o r s p a c e s o r linear t r a n s f o r m a t i o n s appears here, a n d little a b o u t eigenvalues or eigenvectors. Ledermann, quite rightly, p o i n t s out that A i t k e n s h o u l d n ' t be faulted for his taste. " . . . d e s p i t e the v a s t p o w e r t h a t a b s t r a c t i o n a n d generality o f a p p r o a c h has p l a c e d in the h a n d s o f alge-

The Dozenalists and the Crime of Digitism The technology that defines our society proceeds at a p h e n o m e n a l pace, yet most of us are unaware that among us is a cadre deeply dissatisfied with the numerical system that underpins it. They are the dozenalists, who maintain a pitched battle against metrication and the "digitism" of those who want to continue with the "primitive, finger-based" decimal counting system. They favor the base 12, and consider Great Britain's decision to a d o p t a ten-based m o n e t a r y system a giant step backwards. Aitken was a founding m e m b e r of the Dozenal Society of Great Britain [har], w h o s e battle cry is " M e a s u r e m e n t of the people, by the people, for the people." The society still flourishes, under the leadership of Arthur Whillock. The society points with pride to the movement's illustrious protogenists: Napoleon was a duodecimalist ("Twelve has always b e e n preferred to t e n a s a divisor. I can und e r s t a n d the twelfth p a r t of an inch, but not the t h o u s a n d t h part of a meter," he said), as w a s John Quincy Adams, w h o wrote, "Nature has no partialities for the n u m b e r ten, and the att e m p t to s h a c k l e h e r f r e e d o m with it will forever p r o v e abortive. "2~ One of Aitken's s t u d e n t s has m a d e the revealing c o m m e n t t h a t Aitken was a mathematical genius, b u t n o t a great mathematician. In an article in the Transactions of the Society of Engineers, d a t e d Nov. 1, 1954 [ait6], Aitken, speaking o f the d u o d e c i m a l system, states that "he s a w no future in it." However, he c h a n g e d his mind. In m y c o p y of this article, fur-

19Not even nonlinear ones. Claude Brezinski has discovered a surprising fact: Aitken's method can be found in the work of Japan's greatest classical mathematician, Seki Kowa. Let ci denote the length of the perimeter of the polygon with 2i sides inscribed in a circle of diameter 1. To derive a better approximation ~ to ~-, Seki in 1674 ([) used the formula "rr* = c16 + (c16 - c15)(c17 - c16) + ((c16 - c15) - (c17 - c16}), which is precisely the ~2-process. See Edel, p. x. 2~ a duodecimal (base 12) system, two additional digits are needed. The British prefer an inverted 2 for the 10 digit, an inverted 3 for the eleven, while some American dozenalists have chosen an * and a # respectively, and call * "dec" and #, "el" [vir]. Though the dedication of the members should not be questioned, they are not without a sense of humor; Whillock, when asked about the size of his membership, says it "is about a gross." Fred Newhall, an American dozenalist, has built an edifice called the Dozenalist House; all the measurements are multiples of 12: 12-foot-square rooms, a swimming pool with a 36-foot circumference. As might be predicted, the American society charges a $12 membership fee. The American movement has published a book [duo]. 21Tom Bussman [bus] reports there is an even lesser-known Baker's Dozenal Society, wedded to the idea of 13 to the dozen, so that three Baker's Dozenal feet are almost exactly one meter. My mind boggles at the implications of this.

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nished to me by the Dozenal Society, Aitken has asterisked this passage and in pen added, "I have revised this belief. A.C.A. 1958." Aitken mentions in the article his reading of the classic 1941 book of George S. Terry of Massachusetts, The Dozen System: A n Easier Method of Arithmetic [ter]. Perhaps Terry's arguments grew on him. Though most of us will dismiss the movement as a collection of ebullient but harmless eccentrics, the effect Terry's book had on Aitken was unfortunate; it fed into his pathology at a time of accelerating emotional impairment, and the duodecimal cause became a vehicle for his irascibility and paranoia. He wrote the 22-page 1962 pamphlet, now rather rare, The case against decimalisation, published by the sedate Edinburgh firm of Oliver & Boyd. Many of Aitken's arguments in this pamphlet are anchored to the British monetary system of 1962, so they have already lost their appositeness. The booklet has a discomfortingly fussy, driven tone. "Nothing stands still, not even arithmetic," Aitken wrote. "That arbitrary division of time, the second millennium, is app r o a c h i n g . . , and no doubt a few thousands of superstitious decimalists will sit up on that eve to await the new dawning of heaven and earth. In the interim there is bound to be incredible technological p r o g r e s s . . . Among these novelties, the transition from a defective system of numeration and metric to a new one, attainable by easy and gradual phase ..." He warns that this change was about to take place in both the United States and in Russia, leaving Great Britain far behind. "Political expediency is the ruin of science," he warned, in stern italics. A very long letter from Aitken, which must have embarrassed the faculty of Edinburgh University, appeared in The Listener on January 25, 1962 [ait7]. In it, his panegyrics to the base 12 reach dizzying heights: "... we must, unless we are deliberately blind, see what a prodigious release of time and energy, of human potential, this

would continuously ensure for the bet- with others, neighbors and professional colleagues, began to deteriorate ter and wider ends of mankind. Aitken managed to attract others to as he increasingly succumbed to parahis cause, such as the eminent noia and anger. John Todd recalls that Cambridge mathematician, J. C. P. when he visited Aitken during the early Miller, but, perhaps growing wary of years of the Second World War, Aitken the intensity of Aitken's obsession, complained bitterly about a neighbor's they soon lost interest. "I am fighting child who was learning the violin. He the government here, not quite alone, had concluded that her mistakes were but something very like Athanasius the cause of the great personal pain he contra m u n d u m 22 and I will shake was suffering. However, a former stuthem," he wrote in a sadly tremulous dent at Edinburgh states that Aitken's hand to H. P. Kidson on July 17, 1962. house was located much too far from "It is about decimalizing currency, and that of his neighbors for such a comI was sorry to see that N. Z. was watch- plaint to have a rational basis. There is ing the government here instead of a diary entry, dated 1958, describing a watching me. Since 1909 I have done situation that occurred about this time lightning calculation in decimal, since and involving a dispute with a neigh1936 in duodecimal as well, since 1952 bor, a Mr. Swanston. It is difficult to in octonary, and as a mere tour de take Aitken's account of it at face force in 1956-57 in Babylonian sexi- value. gesimal. The most suited to daily use For this defense of rights I incurred, is duodecimal, efficiency more than 3 as I have said, the e n m i t y of to 2 against decimal." Swanston. It would be tedious to deAitken had begun to pester governtail the various devices he used to ment functionaries about his cause. sicken me of the place; m a n u r e heaps "Further," he continues in the Kidson letter, "I have so frightened certain im- placed as close as possible to m y fence; dead carcasses of sheep planted in portant M.P.'s that my paper has been them... taken to the Chancellor of the Exchequer. Totum porcum ire paranHis growing suspicion of everyone tas 23 is my motto, and, in confidence, around him occasionally focused on I would slate the Prime Minister him- the press. On 23 May, 1956 he wrote to self if I thought he was wrong." Peter Kania [kan], who had requested Obviously, his illness was beginning to a biographical statement: overwhelm his characteristic restraint. Despite his efforts, he could not pre- These pages are f o r your own inforvent the encroachment of decimals mation, as they were drafted at your into British currency. The government request only. You will please not let decimaiized currency on February 15, the reporters, the press generally, get at them or make use of them. I have 1971. suffered a little now and then in that Aitken, and most other dozenaiists, line through exaggerated and ignosaw their struggle as a perpetual one. rant public statements, and I have "[We are] at the start of the metric f o u n d it annoying . . . slope," Whillock has stated [whi]. "We have warned there are no inalienable Aitken's request was honored. The rights . . . so Measurement of the peoletters were published in 1978, after his ple, by the people, for the people, death. In a later letter, February 22, could vanish from the Earth." 1961, he describes his insomniac nights, seized with a hallucinatory nosFinally talgia: Aitken's chronic condition involved more than just depression and occa- At times, lying in bed, late at night, sional hallucinations. His relationships or very early in the morning, every

22Literally, "Athanasius against the world." St. Athanasius, ca. 300 AD, a doctor of the church, was continually embroiled in political and doctrinal skirmishes. He applied this expression to himself. 23Literally, "the whole hog, ready to go." This must be Aitken's Latinization of the old slang phrase, "go the whole hog" (Amer. -1828, Br. -1850).

VOLUME20, NUMBER2, 1998 77

detail o f [New Zealand] is so close I could touch it w i t h m y hand; as indeed I did the other night. I had brought back to m i n d the w a s h i n g a n d scalding o f m i l k cans brought back f r o m the c r e a m e r y to m y grandf a t h e r ' s f a r m on Otago P e n i n s u l a . M e n t a l l y I p u t m y h a n d i n the copper boiler: at once every detail o f f a r m , byre, stable, cattle a n d the rest rose by magic... Aitken w a s f o r c e d to retire in 1964, n e i t h e r able to t e a c h n o r to do any sort of mathematics. There b e g a n a series o f hospitalizations, after e a c h o f which he s e e m e d w o r s e than before. In his last years, he was n e a r l y inert and n e e d e d constant attention. His care devolved upon his w i f e - - a d e m a n d i n g a n d appalling b u s i n e s s w h i c h she cond u c t e d without complaint. "I shave and b a t h him and have d o n e since he ret u r n e d from the hospital. He hasn't r e a d anything in all this time a n d w o n ' t even listen to the radio." Winifred herself w a s not well: she w a s crippled with arthritis and w a s 73 y e a r s old. Often, she had to rise as m a n y as 16 t i m e s a night to r e s p o n d to his demands. Yet she w a s d e t e r m i n e d to k e e p Aitken out of a nursing home. The last, m o s t selfless a c t i o n o f this courageous caregiver, p e r h a p s of m o s t caregivers, was to b e c o m e in h e r p e r s o n a l a p p e a r a n c e m o r e a n d m o r e like that of h e r w a r d [aitl]. She h a d c h a n g e d into a large, v e r y plain, elderly w o m a n i n long shapeless dresses a n d w i t h a careless bun o f g r e y h a i r . . . i n a household o f m o s t beautiful f u r n i s h ings, all o f w h i c h she h e r s e l f had created . . . . She w a s completely unpret e n t i o u s i n m a n n e r , w i t h o u t conceit or self-promotion. Arthur Erd~lyi returned to Edinburgh from the California Institute of Technology in 1964 to assume the reins of the mathematics department in the University. His wife Eva recalls that they m a d e only one visit to Aitken's bedside. Aitken

still s e e m e d to recognize the Erd~lyis, and gave Arthur his blessing as his successor. On May 27, 1967, Aitken s e e m e d to s h o w signs of s o m e i m p r o v e m e n t . Winifred, with s o m e assistance, got A i t k e n o u t o f bed, dressed, a n d installed in a chair by the fire. N o w he was listening once again to m u s i c on the radio, and soon he w a s reading. I p r o d u c e d h i s spectacles a n d said, quite casually, "Have a read at this book," a n d he began to read w i t h o u t a n y d e m u r a n d w e n t on till he h a d f i n i s h e d the book. He c o n t i n u e d to read, to listen to music, b u t his old vivacity w a s n e v e r to return. He d i e d on N o v e m b e r 3, 1967. The Faustian Dilemma H o w m a n y o f us w o u l d like to p l a y masterfully a musical instrument, to learn languages at a glance, to p e r f o r m in total r e l a x a t i o n the m o s t intimidating c o m p u t a t i o n s , to write the purest, m o s t beautiful prose, to m o v e in s u c h a swirl of m a t h e m a t i c a l i d e a s that w e n e e d only to r e a c h out to grab one? Having t h e s e manifold talents is the stuff o f fantasy. If we w e r e given the choice: to b e e n d o w e d with s u c h gifts a n d to suffer, on the one hand, o r to lead a life of h a p p y mediocrity, on the other, w h i c h w o u l d we c h o o s e ? 24 This q u a n d a r y - - o n e might call it the F a u s t i a n d i l e m m a - - h a s b e e n the subj e c t o f legend and literature. We s h o u l d all b r e a t h e a sigh of relief, p e r h a p s , that in real life the option is n e v e r offered, the choice is never ours to make. B e c a u s e few of us have A i t k e n ' s gifts, w e c o m f o r t ourselves with the belief that a n y o n e p o s s e s s i n g s u c h a b o u n t y m u s t find it as much a c u r s e a s a blessing. ~5 Concerning t h e gift for mental computation, Hans E b e r s t a r k , a great c a l c u l a t o r himself, writes, "Let m e w a r n y o u . . . [m]any o f the p e o p l e . . . b e c a m e o b s e s s e d by f i g u r e s . . , the n u m b e r s will k e e p such a tight grip on you t h e y will not allow you to sleep."

However, Aitken recalled with g r e a t fondness h o w as a b o y he first b e c a m e a w a r e of his potential: The [school]master chanced to s a y that y o u can u s e this f a c t o r i z a t i o n to square a n u m b e r : a 2 - b 2 - - ( a + b ) ( a - b). S u p p o s e y o u had 4 7 - - t h a t w a s h i s e x a m p l e - - h e s a i d y o u could take the b as a 3. So ( a + b ) is 50 a n d ( a - b ) is 44, w h i c h y o u can m u l t i p l y together to g i v e 2200. Then the s q u a r e o f b is 9 a n d so, boys, he said, 4 7 squared is 2209. Well, f r o m that m o ment, that w a s the light, and I n e v e r w e n t back. I w e n t straight h o m e a n d practiced a n d f o u n d that this reacted on every other branch o f m a t h e m a t ics. I f o u n d s u c h f r e e d o m . It w a s Erdds, I believe, w h o observed that m a t h e m a t i c s can be a great consoler. We m a y s u p p o s e that Aitken in his h a p p i e s t m o m e n t s was in t o u c h with the f r e e d o m m a t h e m a t i c s h a d p r o m i s e d him as a boy, and that it w a s this c o n s o l a t i o n t h a t h e l p e d him to secure w h a t e v e r intermittent p e a c e he enjoyed. Like all o f h i s t o r y ' s visionaries a n d mystics, all t h o s e w h o have navigated the p r o t e a n b o r d e r b e t w e e n sanity a n d madness, Aitken h i m s e l f w a s u n c e r t a i n w h e t h e r his m e n t a l state, as o n e r o u s as it was, w a s ultimately a curse o r a blessing, b u t he w a s convinced it offered him a privileged view of the world [altl]: It is as i m p o s s i b l e to recapture a n d describe [those years] as f o r the griefs t r i c k e n to describe h a p p i n e s s . . . or f o r the s a n e to k n o w the world o f ins a n i t y , f o r a M a c a u l a y to k n o w the m i n d o f a Blake or a Dostoievsky. This c o u n t r y o f m e l a n c h o l i a as the poets, s o m e o f them, have described it . . . . I could not but regard m y o w n breakdown as o f i m p o r t a n c e . . , a n d indeed as m a r k i n g a m i l e s t o n e i n the m i d w a y o f this m y m o r a l life. A n d a m i l e s t o n e I m u s t believe it to be; f o r s i n c e that t i m e there is not a tree, not

24Many have proclaimed the facilitative effect of mental instability on creativity. "Men have called me mad," Poe wrote, "But the question is not settled yet whether madness is or is not the loftiest intelligence, whether much that is glorious, whether all that is profound, does not spring from disease of thought." A less romantic point of view is expressed by Oxford psychologist Gordon Claridge." It could be that being pushy and domineering with high psychoticism allows you to persuade the unsuspecting public that your indifferent ideas are highly original," he suggests [bur]. Nevertheless, Aitken clearly possessed all eight of those traits that University of Kentucky psychiatrist Arnold M. Ludwig has discerned in what he calls "great creative achievers" [bow]. 25"Why is it," Dr. Johnson asked, "that we find no one endeavouring to convince us that the rich can also be happy?"

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THE MATHEMATICALINTELLIGENCER

a turn in a road, not a hill-top, not even a swaying reed, but speaks of the beauty, the at first terrible beauty and mystery of the world. BIBLIOG~L&FHY

[aitl] Aitken, A. C., To catch the spirit, University of Otago Press, Dunedin, N.Z. (1995). [ait2] Aitken A. C., The case against decimali.sation, Oliver & Boyd, Edinburgh (1962). [ait3] Aitken, A. C., Gallipoli to the Somme: recollections of a New Zealand infantryman, Oxford University Press, Oxford, England (1963). [ait4] Aitken, A. C., and Silverstone, H., "On the estimation of statistical parameters," Proc. Roy. Soc. Edinburgh (a) 61, 186-t 94 (1941 1943). [ait5] Aitken, A. C., Determinants and Matrices, Oliver and Boyd, Edinburgh (1939). [ait6] Aitken, A. C., "The art of mental calculation, with demonstrations," Trans. Soc. Eng. London 44, 295-309 (1954). [ait7] Aitken, A. C., "Twelves and Tens: A. C. Aitken on the case against the system of decimalization," in The Listener, Jan. 25 (1962). [ame] American Psychiatric Association, Diagnostic and Statistical Manual of Mental Disorders, 4th Edition, Washington, DC (1994). [bar1] Barker, Pat, Regeneration, Viking Press, NY (1991). [bar2] Barker, Pat, The Eye in the Door, Dutton, NY (1994). [bar3] Barker, Pat, The Ghost Road, Dutton, NY (1995). [ben] Conversation with Thomas Benfield, MD,

Jefferson University, Philadelphia, Dec. 7 (1996). [bow] Bower, Bruce, "Moods and the muse," in Science News, 147, 3478 (1995). [bur] Burne, Jerome, in The Sunday Telegraph, December 3 (1995). [bus] Bussman, Tom, in The Guardian, Sept. 16 (1995). [dell Delahaye, J.-P., Sequence Transformations, Academic Press, New York (1988). [duo] Duodecimal Society of America, Inc, Manual of the Dozen System, New York, NY. [erd] Erdelyi, Eva, letter to Jet Wimp, Dec. 31 (1995). [fen] Fenton, P. C., "A. C. Aitken (1895-1967)," Gazette Aust. Math. Soc. 22, 21-23 (1995). [for] ForchG Carolyn, ed., Against Forgetting, W. W. Norton, NY (1993). [gra] Graves, Robert, Good-bye to All That, new edition, revised with a prologue and an epilogue, Doubleday, Garden City, NY (1957). [fus] Fussell, Paul, The Great War and Modern Memory, Oxford University Press (1975). [bar] Hartson, William, Pastimes article, in The Independent, March 31 (1995), [hun1] Hunter, lan M., "An exceptional talent for calculative thinking," British Jour. Psy. 53, 243-258 (1968). [hun2] Hunter, lan M., "An exceptional memory," British. Jour. Psy. 68 (2), 155-164 (1977). [kan] Kania, Peter, "Dr. A. C. Aitken: Two letters to Mr. Peter Kania," New Zealand Mathematics Magazine 15, 188-194 (1978). lied] Ledermann, Walter, letter to Jet Wimp, Jan 18 (1996). [lur] Luria, A. R., The Mind of a Mnemonist: A Little Book about a Vast Memory, Avon Books, New York (1968).

[mil] Miller, J. C. P., et aL, "Obituary, A. C. Aitken," Proc. Edinburgh Math. Soc. (2) 16, 151-176 (1968/1969). [mini Minc, Henryk, "Six letters from Alexander C. Aitken," Linear and Multilinear Algebra 2, 1-12 (1974). [rei] Reid, T. R., article in The Los Angeles Times, July 12 (1989). [schl] Schneider, Hans, e-mail to Jet Wimp, Dec. 27 (1995). [sch2] Schneider, Hans, contribution to Proceedings, A. C. Aitken Centenary Conference, held in Dunedin N.Z. August 28-Sept. 1, 1995, to appear. [sch3] Schneider, Hans, e-mail to Jet Wimp, Dec. 19 (1995). [smi] Smith, Steven B., The Great Mental Calculators, Columbia University Press, New York (1983). [orn] Orner, Frederick J., "Post-traumatic stress disorders and European war veterans," Brit. Jour. Clinical Psych. 31,387-403 (1992). [ter] Terry, George S., The Dozen System: An Easier Method of Arithmetic, Longmans Green & Co., London (1941). [vir] Virag, irene, "Where dozen is not a dirty word," in Newsday, Oct. 26 (1990). [whi] Whillock, A. F., letter to Jet Wimp, Feb. 14 (1996). [wim] Wimp, Jet, Sequence Transformations, Academic Press, NY (1983). Department of Mathematics and Computer Science Drexel University Philadelphia, PA 19104 USA [email protected]

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VOLUME 20, NUMBER 2, 1998

79

B--'li~.n,,u,nn.o~.]-,,[=-]-,-

Robin

Chinese Mathematics II Raymond Flood and Robin Wilson

Wilson

I

he thirteenth century was one of the high-water marks in Chinese mathematics, with contributions to algebra and the development of numerical methods for solving equations. Among the most notable figures of that time was the mathematician and astronomer Guo Shoujing [Kuo ShouChing] ( 1 2 3 1 - 1 3 1 6 ) . While there is no extant treatise on mathematics that can be traced to him, there are records from the Ming period (1368-1648) that show his influence on astronomical calculation and calendar construction. The first Chinese work on spherical trigonometry is attributed to Guo.

T

1610), who disseminated knowledge of Western sciencc cspecially in mathematics, astronomy, and geography. He is shown here on a 1983 Taiwanese

Matteo Ricci

stamp with an armillary sphere. Perhaps his most important work was an oral translation into Chinese of the first six books of Euclid's E/ements; this translation was recorded by Xu Guangqi (1562-1633), Grand Secretary of the Wen Yuan Institute and '~the first man in China after the monarch himself'. By the time of Ricci's arrival, the calendar based

Guo Shoujing

The armillary sphere is an early astronomical device for representing the great circles of the heavens. The Chinese one pictured below dates from the fifteenth century, and appeared on a People's Republic of China stamp of 1953.

Xu Guangqi

on the work of Guo Shoujing was inaccurate, and Xu Guangqui was the principal figure in its reform at the end of the Ming dynasty. Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England

80

armillary sphere

The first missionary in China was the Italian Jesuit M a t t e o Ricci ( 1 5 5 2 -

THEMATHEMATICALINTELLIGENCER9 1998SPRINGER-VERLAGNEWYORK

Raymond Flood Department of Continuing Education Kellogg College Oxford, OX1 2JA UK

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