No title

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

Wolf Prize Reading the lively account (Mathematical Intelligencer, vol. 11, no. 2, 1989) of the 1988 Wolf Prizes at a moment when the Birzeit mathematician Taysir al Aruri is appealing a deportation order from the West Bank, I am moved to recall the remarks I made at the Knesset, Jerusalem, in response to the award of the 1982 Wolf Prize in Physics. , "I am deeply grateful to the Wolf Foundation for awarding me this prize, and to you, Mr. President and Mr. Speaker, for the hospitality of this house. I hope we may all remember that Ricardo Wolf was a revolutionary as well as a capitalist, a fighter for justice as well as a benefactor of science. I hope we may also remember that this land which we love is a homeland for two peoples and two cultures, for Arabs as well as for Jews, and that we have a responsibility to foster education and learning in Bethlehem and Birzeit as well as in Rehovoth and Jerusalem." Freeman J. Dyson The Institute for Advanced Study Princeton, NJ 08540 USA

Complex Numbers In opening the Spring 1989 issue of the Mathematical Intelligencer, I was struck by that lovely quote on page 3 from Huygens to Leibnitz. It reminds me of an experience I had many years ago, perhaps worth sharing with the readers of the Intelligencer. As a boy of 14, barely beyond my first course in algebra, I was told by m y uncle, an engineer-businessman by profession, about imaginary and complex numbers. Changing sign is just rotation by a halfcircle, he explained, ti~o such rotations amounting to a complete turn, and this was supposed to be an explanation of the multiplication rule for signs. So it seems logical, he continued, that for the forbidden square root of - 1 one should take rotation by a

quarter-circle. The familiar picture emerged, including adding the new "imaginary" numbers to the old real ones. I went away satisfied, and there was enough conventional math homework to do, so that the conversation was soon forgotten. Recalling it some months later, it suddenly struck me (perhaps because in algebra class we were n o w told that not only square roots but also fourth roots of negative numbers were forbidden) that if it is legitimate to "create" new, formerly forbidden, numbers, I can go on to introduce fourth roots of negatives and so enlarge the domain of n u m b e r s still further. With smug self-satisfaction I anticipated impressing my uncle with m y smartness at our next meeting. But a few days later an anxious and disturbing thought surfaced, namely, that following my uncle's earlier explanation, an eighth-turn should create these new fourth roots of negatives, but then they would already be expressible by one single operation of taking the square root of a negative. All my instincts rebelled against the idea that this should be possible, and so I gave up, depressed by my naivete. Yet a few more days later, a blindingly beautiful late summer afternoon that will always remain in my memory, I had a sudden inspiration. Try the impossible! I r u s h e d for pencil and writing pad, and with mounting excitement proceeded to exercise my freshman algebra skill on the expression (1 + V"U'-I)4. When after a few moments the number - 4 emerged, I was struck and amazed such as I never was before, and rarely since. That moment of triumph and mystery, that feeling of plumbing incomprehensible depths, is hard to transmit now after many years of living with complex numbers as my second nature. But there it was; for a moment in my life I knew whereof speaks Huygens when in his letter to Leibnitz he declares that when one is dealing with imaginary quantities "there is

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4 9 1989 Springer-Verlag New York 3

something h i d d e n therein which is incomprehensible." History and mathematics have gone a long way since the time of Huygens and Leibnitz, and perhaps we are all too easily taught, and we all too readily accept the admirable and the marvelous, even if not incomprehensible, things. Perhaps others too had their similarly enlightening moments of wonder and mystery, and it would be interesting to hear some described. But for myself, I shall never forget that day many summers ago when suddenly, with utter surprise and certitude, complex numbers were mine. Andrew Lenard Department of Mathematics Indiana University Bloomington, IN 47405 USA - - - A n I n t e r n a t i o n a l L a n g u a g e for M a t h e m a t i c s m

In a few respects the opinions of Strichartz and the DLT designers differ. Linguists know that no artificially constructed symbol system can ever possess the expressive power needed for conveying the full content of human-language texts. Thus the intermediate language cannot be a constructed one. So w h y not take English or German for that function? Unfortunately, computational linguists know by painful experience that fully automatic translation from a human language is impossible. For many practical reasons, however, Strichartz is correct in suggesting that only the first half of the translation process n e e d be human-aided, while the second should be fully automatic. DLT's solution is an intermediate language which was an artificial symbol system when launched a century ago but has become a fully expressive language by being used in a human language community for over a hundred years. The Esperanto text in DLT is "designed in advance for easy translation." This is so thanks to the characteristics of Esperanto and because the text being translated into Esperanto is enriched with disambiguating knowledge from the system's artificial-intelligence components and from the user's dialogue answers. Our opinions differ in another important detail: The DLT designers do not believe in building a machine translation system for a single thematic domain with the hope of extending it to other domains later. The experience of the Canadian Meteo system, which translates w e a t h e r reports b e t w e e n English and French but could not be extended to another domain despite costly efforts, is a warning. Therefore, from the beginning DLT has pursued the aim of generalpurpose translation. DLT is a long-term research and d e v e l o p m e n t project. An operational prototype translates English Esperanto ~ French. A version for restricted language is scheduled to be marketed after 1992; the generalpurpose system will be available by the end of the next decade.

In vol. 11, no. 1, pp. 12-13, 1989, of the Mathematical Intelligencer Professor Strichartz calls for a computer system that would translate mathematical texts into many languages by means of an artificial intermediate language which he calls Intermath. The BSO software house in Utrecht, Netherlands, is building a computer system that translates nonliterary texts into many languages by means of a no longer artificial intermediate language, namely Esperanto. The coincidence is striking. And it is not only these superficial features that happen to coincide. The foll o w i n g a c c o u n t of the BSO m a c h i n e translation system, called Distributed Language Translation (DLT), should be c o m p a r e d with Strichartz's Opinion (I borrow phrases from Strichartz, to emphasise the parallelism): The DLT system is designed for translating nonliterary texts between a large number of languages. Since fully automatic high-quality translation is impossible and because of the danger of combinatorial explosion, DLT does not link its source and target languages directly but translates each text first into an intermediate language. The user is prompted to "answer a few queries about ambiguous sentences." This inter- Klaus Schubert active disambiguation dialogue is carried out in the BSO/Research language of the text so that the user need not know Postbus 8348 NL-3503 RH Utrecht any foreign language. "No one would read the text in" Netherlands DLT's intermediate language, but the intermediate text is ready for automatic translation into one or more 9A n o t h e r I n t e r n a t i o n a l L a n g u a g e target languages. Because the computer is "not allowed to cheat by peeking at the original," the inter- This letter is a response to Robert Strichartz's essay mediate language is the one and only link between (Mathematical Intelligencer, vol. 11, no. 1, 1989) on An source and target. Therefore the intermediate lan- International Language for Mathematics. In that essay, guage should possess the full and unrestricted expres- Dr. Strichartz proposes the development of an artifisive power of a h u m a n language. A poor system of cial language, which he calls 'Intermath,' as an interprefabricated sentence patterns would be completely lingua that could be used in computer-aided translainsufficient. The intermediate language should "em- tion of mathematical text. body all the different ways of expression used by Dr. Strichartz identifies some properties that such a mathematicians in their normal modes of writing." language might possess: 4

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

9 It should be modelled on the formal language of mathematical logic; 9 It should extend beyond that formal language "to embody all the different ways of expression used by mathematicians in their normal modes of writing"; 9 It should have a vocabulary including technical terms which can be readily expanded to include newly coined words; 9 It should embody the 'unambiguous meaning' underlying mathematical text, without necessarily preserving stylistic features; 9 It should be amenable to machine translation techniques. I am pleased to report that a language has already been developed that meets the criteria described by Dr. Strichartz. Lojban (pronounced LOZH-bahn) is the latest version of the language first described by Dr. James Cooke Brown in "'Loglan," Scientific American, June 1960. Dr. Brown modelled his grammar on predicate logic and specifically included grammatical techniques of clearly and unambiguously expressing logical connectives and other concepts that are not easily expressed in English and other languages. Dr. Brown also attempted to devise a mathematically-provable unambiguous grammar. Computational techniques of proving computer languages unambiguous were first applied to Loglan in the mid-70s. Lojban is the latest version of this language, develo p e d by The Logical Language G r o u p from Dr. Brown's seminal research. It is the first version of the language to have a completely specified unambiguous grammar. Lojban's grammar includes a grammar for mathematical expressions that goes well beyond simple mathematical operations. This grammar includes a metalanguage capability for defining new operators, redefining the precedence grouping of operators ( u n p a r e n t h e s i z e d Lojban defaults as leftgrouping), and intermingling of "normal language expression" with mathematical expressions. The Lojban design allows for standard notations as well as Polish, reverse Polish, and an indefinite variety of other s y s t e m s - - a feature vital for supporting the many varieties of mathematical expression used in specialized disciplines. Lojban also incorporates a scheme for borrowing words from the international scientific vocabulary, as well as an extremely rich combinatorial technique of building words from Lojban roots. What does this mean in terms of Dr. Strichartz's proposal? Lojban is certainly capable of serving as 'Intermath.' A Lojban parser is under development, the first step in a machine translation system. Artificial intelligence

experts have started looking at techniques that are especially suited for AI-based translation from natural languages into Lojban. Lojban's syntax resembles that of LISP and PROLOG, which are often used in storing knowledge-base information for AI processing. The Logical Language Group, Inc. is a non-profit organization dedicated to developing Lojban, teaching it, and encouraging its practical use. We have teaching materials available in addition to computer-aided tools. A quarterly journal, Ju'i Lobypli, discusses the project goals and technical issues in the language, and provides some amount of Lojban text in a variety of styles. Recent issues have dealt with both mathematical expression and machine translation, and reprint copies are available. A short newsletter, lojbo karni, is also published to inform interested but less active participants. Bob LeChevalier, President The Logical Language Group, Inc. 2904 Beau Lane Fairfax VA 22031 USA

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989 5

Allen Shields*

Lejeune Dirichlet and the Birth of Analytic Number Theory: 1837-1839 It is approximately 150 years since Dirichlet's fundamental paper [18371] appeared, in which he proved that a n y arithmetic p r o g r e s s i o n {a + nq}, n = 0,1,2 . . . . . for which a and q are relatively prime, contains infinitely m a n y prime numbers. He presented his result to the Royal Prussian Academy of S c i e n c e s in Berlin on 27 July 1837, and it was published in the Abhandlungen of the Academy for that year. H o w e v e r , the Abhandlung w a s not actually printed until 1839, at which point it became available to a wider public. Also, as Davenport [1980] points out (page 1), the proof in this first paper of Dirichlet was complete only in case q was a prime. For the general case, Dirichlet had to assume his class number formula (for the number of inequivalent quadratic forms in two variables over the integers, with fixed discriminant). He published the proof of this formula in [1839-40]. The theorem on the infinitude of primes in arithmetic progressions had been conjectured earlier by Legendre, and used by him as an unproved lemma in some of his researches (see Dirichlet [18371], Werke, p. 316). Davenport begins his book with the words: "Analytic number theory may be said to begin with the work of Dirichlet, and in particular with Dirichlet's memoir of 1837 on the existence of primes in a given arithmetic progression." Dirichlet based his proof in part on a proof by Euler of the infinitude of primes; Euler's method proved the stronger result that Y~p - ~ = ~, where the summation is over all prime numbers (see Euler [1748], Chap. 15). Dirichlet proved the corresponding result that the sum of the reciprocals of the prime numbers in an arithmetic progression is infinite. (See Dirichlet [18372], 309-310.) * Column editor's address: Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003USA

Euler argued as follows. First, if p is a prime number then (1 - p-1)-1 = y~ p - , (0 ~ n < o0). Let M be a given (large) positive number, and choose N so that Y~n-1 > M, where the summation is for 1 ~ n ~ N. Form the product 11 (1 - p-1)-1 over all primes p ~ N. If we multiply the corresponding infinite series, then we may rearrange their product in any order because of the absolute convergence. In particular, we see that the product of the series contains all reciprocal integers n-~, n ~ N. Thus the product is greater than M. T a k i n g the r e c i p r o c a l w e see that the p r o d u c t II (1 - p-l) over all primes is zero. This is equivalent (by a standard relation between products and series) to the divergence of ~ p-1. These ideas lead to Euler's formula: II (1 - p-S)-1 = E n - S , s > 1. Dirichlet had the idea of introducing what today are called group characters and using them to obtain analogous formulae. If q is a prime number, then the integers 1,2 . . . . . q - 1 form a group under multiplication mod q. Let X be a character of this group. We define • for all positive integers n by putting x(n) = 0 if n is a multiple of q, and x(n) = x(j) if n and j are congruent m o d q. Dirichlet formed the series Lx(S) = x(n) n-s, s > 1 (summed over all positive integers n) and obtained the analogue of Euler's formula: L•

= 1-I (1 - •

where the product is over all primes p (p # q). Dirichlet used the letter L for these functions, and it has been used ever since. The reader wishing more precise details should consult Chapter I of Davenport [1980]. After some preliminary simplification the crucial point in the proof consists in showing that, if • is not the identity character (that is, the character such that • = 1, n not a multiple of q), then Lx(s) remains bounded as s approaches 1.

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4 9 1989 Springer-Verlag N e w York

7

Gustav Peter Lejeune Dirichlet

We mention that for certain special arithmetic progressions (for example, {4n + 3}), a simple proof can be given, similar to Euclid's proof that there are infinitely many primes. See Chapter 2, Section 3, of Hardy and Wright [1938, 1960]. In van der W a e r d e n ' s History of Algebra [1985], Chapter 12, he says that the theory of group characters begins with Gauss's Disquisitiones Arithmeticae, Sections 228-233. Dirichlet (who studied Gauss's book very closely, see below) carried the theory further, especially in the first paper cited in Dirichlet [183940]. Dedekind (who was Dirichlet's student) gave an exposition of the results of this paper in the fourth s u p p l e m e n t to Dirichlet's Vorlesungen iiber Zahlentheorie, 3rd edition, 1879. In the famous tenth supplement Dedekind carries the theory further. In a letter to Frobenius he wrote: "After all this [i.e., after Dirichiet's investigations], it was not much to introduce the concept and name of characters for every Abelian group, as I did in the third edition of Dirichlet's Zah-

lentheorie." We turn n o w to some biographical information about Dirichlet. Much of this is taken from Biermann [1959], Kummer [1860], Rowe [1988], and Klein [1926]. The reader would do well to commence with Rowe. Dirichlet's grandfather came from Verviers (Belgium) and emigrated to D/iren, between Aachen and K61n, in Germany (about 50 kilometers from Verviers), becoming a B/irger of D/iren in 1753. He made cloth. The name Dirichlet comes from "de Richelet" (Biermann, p. 8). Johann Peter Gustav Lejeune Dirichlet was born 13 February 1805 in DLiren; his father was the postmaster (Postkommissar) for the town. 8

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In May 1822 he went to Paris to study mathematics; Paris was then the world center for mathematics. Leg e n d r e (1752-1833) was still active, and Fourier (1768-1830) and Poisson (1781-1840) were leading members of the Academy. One year later he was invited by General Foy, the leader of the Opposition in the Chamber of Deputies, to teach mathematics to his children. In addition to attending lectures, Dirichlet spent much time studying Gauss's Disquisitiones Arithmeticae. Kummer (p. 315-316) states that Dirichlet read it not once, or even several times, but throughout his life he reread it. It was never in his bookcase, but always on the table w h e r e he worked. More than twenty years after this extraordinary book had appeared (in 1801) no one had yet understood it. Even Legendre had to confess in the second edition of his Th~orie des nombres that he would have liked to have enriched his book with Gauss's results; however, the methods of this author were so peculiar that this was impossible without great digressions, or merely assuming the role of a translator. Dirichlet was the first to understand this work thoroughly, but beyond that he found more natural proofs, thereby making the results accessible to others and enabling him to find deep new results. Dirichlet met Alexander yon Humboldt (1769-1859) in Paris in 1825 and told him that he would like to return to Germany, if there were a position for him. Von H u m b o l d t had heard of Dirichlet's talent from Fourier. Fourier liked to gather talented young men around himself, and Dirichlet had been a part of this circle. Gauss (1777-1855) wrote a strong letter of recommendation for Dirichlet. Von Humboldt wrote the Prussian Minister of Education, who replied that Dirichlet could go to the University of Breslau for his Habilitation with an annual salary of four hundred talers as a Privatdozent. In addition he would receive seventy-five talers to cover the expense of moving to Breslau. Von Humboldt was not satisfied and wrote again suggesting that Dirichlet be given the title of professor, with a salary of six to seven h u n d r e d talers. Nothing came of this, however. Before appointing Dirichlet (who was n o w twenty-one years old) as a Privatdozent the Minister felt compelled to check his background. It was not a liberal period and it would have been dangerous to appoint an unknown y o u n g man merely on the recommendation of yon Humboldt, w h o was himself considered somewhat liberal. The Minister of Education wrote to the Minister of the Interior asking if the police had anything against the appointment. They were suspicious of Dirichlet's connection to General Foy and inquiries were made in Paris. But in the end they decided there was nothing against him, and his connection with Foy had been for scientific rather than political purposes. He took up his duties in Breslau in April 1827. In Breslau a curious problem awaited him. The fac-

We digress to say a few words about A. L. Crelle ulty rules for the Habilitation required a dissertation written in Latin, which he could do, together with an (1780-1855), who had wide interests in technology as oral presentation and defense, also in Latin, which he well as mathematics. Klein [1926; p. 94-96] states that could not. However, at Dirichlet's request the Minister Crelle never gave up mathematics despite his other excused him from the oral defense. Some of the fac- wide-ranging interests, but that his mathematical ulty were angered at this intervention from above and work was without real significance. It had an encyclopedic character, which was really a tradition of the wrote the Minister (see Biermann, pp, 21-30). On 1 April 1828 he was given the title of ausseror- previous century but which was still widespread in dentlicher Professor in Breslau with the same salary of Germany in that his work touched on many different four hundred talers. He wished to go to Berlin, which topics in mathematics without going deeply into anywas the center of mathematical life in Prussia. A posi- thing. He was a member of the Prussian Academy of tion opened up at the Royal Military School in Berlin, Sciences. However, his real talents and contributions and he was given leave from Breslau to teach there for lay elsewhere. First of all he had a good eye for mathea year. Faculty from the University of Berlin some- matical talent, and he labored to secure a position for times taught at the Military School to supplement their Abel in Berlin (when the offer was finally sent in 1829 income. Dirichlet's position had been held by Pro- Abel had just died in Christiana). He also wrote the fessor O h m (1787-1854) (of " O h m ' s Law" in elec- Prussian Minister in 1830 supporting Dirichlet for a tricity), who had been teaching six hours a week for permanent position in Berlin. Finally in 1839 he wrote supporting Dirichlet's promotion to Professor (Ordinine months for an annual salary of six h u n d r e d talers. N o w the teaching load was increased to eight narius). But Crelle is especially r e m e m b e r e d for hours a week with no increase in salary. Professor founding the Journal fiir die reine und angewandte MatheOhm announced that he would not teach eight hours matik, still sometimes called Crelle's Journal, and u n l e s s he w a s paid eight h u n d r e d talers. T h e y editing the first fifty volumes. At the beginning there couldn't do this so he left, thereby opening the posi- were serious financial difficulties, which he met partly -tion for Dirichlet. from his own resources. After much effort he secured In Easter 1829, he became a Privatdozent at the Uni- a state subsidy. The first volume contained five papers versity of Berlin. In July 1831, he was promoted to aus- by Abel, one by Jacobi, and several by Steiner. In serordentlicher Professor, with a salary of four hundred volume 3 (1828) there are papers by Dirichlet, M6bius, talers; in addition, he continued teaching at the Mili- and Pl~icker. tary School. In May 1832 he married Rebekka MendelssohnBartholdy, the sister of the composer Felix. They had three sons and a daughter. Through the Mendelssohns he was connected to a number of other mathematicians. For e x a m p l e , K u m m e r ' s wife Ottilie (maiden name Mendelssohn-Bartholdy) was a cousin of Rebekka, and later H. A. Schwarz (1843-1921) married Kummer's daughter. Finally, Kurt Hensel (18611941) was a great nephew of Rebekka. 'As far as I can tell from Biermann [1959] (see his list of names at the end of the article), Dedekind (18311916) and Kronecker (1823-1891) were students of Dirichlet, and Eisenstein (1823-1852), Kummer (18101893), and Riemann (1826-1866) were influenced by his lectures. Riemann came over from G6ttingen for a year to attend lectures in Berlin. Biermann (pp. 34-39) lists the courses Dirichlet taught at Berlin with the number of students in each course. For example, in the summer semester of 1829 he taught "The theory of series, as an introduction to higher analysis" to five students. He also offered a course called "Indeterminate analysis" b u t only two students showed up and the course did not run. He seems to have offered two courses per semester (with two semesters per Gustav Peter Lejeune Dirichlet year), but if only two or three students reported, the course would not run. Mostly there were fewer than 20 students. THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989 9

learned this from him. When Gauss says he has proved In 1832 Dirichlet was elected a m e m b e r of the something, it is very probable . . . . when Cauchy says it, Academy of Sciences and became the first top-rank you can bet equally well pro or contra, but when Dirichlet mathematician in the Academy since Lagrange left in says it, it is certain. I prefer to leave myself out of this Deli1787 to go to Paris. In 1834 his University salary was katessen. increased, in view of his "laudable effectiveness" (beiIn the end these efforts were successful: Dirichlet's fallswerthen Wirksamkeit), to six hundred talers. He still taught at the Military School for an additional six salary at the University was raised from 800 talers to hundred talers. In 1839 he was promoted to Professor 1500 talers and he stayed in Berlin. Biermann [1959; p. 69] notes that in March 1848 Diat the University, still with the same salary of six richlet stood guard before the Palace of Prince William of Prussia as a member of the citizen's militia. The The Minister replied that in v i e w of Di- palace had been declared to be "national property." richlet's scientific achievements and his ac- The EncyclopaediaBrittanica (11th edition, 1911, vol. 11, tivity as a teacher for many years the min- p. 866), in the article on Germany describes this period. In February 1847 Friedrich William IV of Prussia istry had the sincere wish to grant him a suit- s u m m o n e d a united diet of Prussia to meet. But as able increase in salary, but that unfortunately Metternich (the Austrian premier) predicted, this only this was not possible at the present time. served as a forum for demands for a constitution, and deadlock ensued. In February 1848 revolution broke out in Paris; in May 1848 Metternich fell from power. hundred talers. Finally in 1842 this was increased to A riot broke out in Berlin, which was suppressed on eight h u n d r e d talers. He still taught at the Military March 15 by troops with b u t little bloodshed. AcSchool, though the constant routine lectures there cording to the encyclopedia article the king had an were b u r d e n s o m e . Apparently the salary of eight "'emotional and kindly temperament" and "shrank hundred talers at the University was below the official with horror from the thought of fighting his 'beloved scale for a Professor, as Dirichlet wrote the Minister of Berliners'." When on the night of the eighteenth of Education in August 1846 requesting that he be paid March the fighting was renewed he entered into negothe official salary. He says he will not comment on his tiations with the insurgents, which led to the withresearch, but that he feels his teaching has been suc- drawal of the troops. cessful. When he came to Berlin there were no lectures The next day Friedrich Wilhelm, with characteristic hisin many of the most important areas of mathematics. trionic versatility, was heading a procession round the Dirichlet worked to change all this, and it cost a great streets of Berlin, wrapped in the German tricolour, and deal of effort but he feels he was successful. The Minextolling in a letter to the indignant Tsar the consummation of 'the glorious German revolution'. ister replied that in view of Dirichlet's scientific achievements and his activity as a teacher for many years the ministry had the sincere wish to grant him a suitable increase in salary, but that unfortunately this Jacobi said that it would be impossible to rewas not possible at the present time. place Dirichlet since, aside from Gauss in Just at this point a new element was introduced into G6ttingen and Cauchy in Paris, his equal was this situation. At the end of 1846 the government of n o t to be found. Baden wrote to Dirichlet asking if he would consider receiving a call to a position in Heidelberg University. The faculty at Berlin instructed the rector to write the Dirichlet found the teaching at the Military College Minister, urging him to raise Dirichlet's salary, for too burdensome and asked to be excused from it, but otherwise they might lose him to a foreign university this was not done. Then after Gauss's death in 1855 and his loss could not be replaced. Jacobi wrote to the the government of Hanover wrote him asking if he Minister and to the King urging that Dirichlet be kept. would consider coming to G6tfingen as Gauss's sucIn the letter to the King Jacobi said that it would be cessor. He replied that he would, unless Prussia eximpossible to replace Dirichlet since, aside from Gauss cused him from teaching at the Military School. The in G6ttingen and Cauchy in Paris, his equal was not to Prussian Minister of Education, however, decided to be found. Dirichlet's loss and Bessel's recent death wait for an official offer from H a n o v e r before rewould mean that Prussia would no longer be so com- sponding. When the offer came, he excused Dirichlet petitive in the exact sciences. from further duties in the Military School, and raised Jacobi also wrote von Humboldt giving him argu- his salary. But it was too late--Dirichlet felt bound by ments to use with the Minister. We quote one passage: what he had wrriten to Hanover, and he left. Kummer He alone, not I, not Cauchy, not Gauss knew what a com- was called as his successor in Berlin with a salary of pletely rigorous mathematical proof is, indeed we first fifteen hundred talers. 10

THE M A T H E M A T I C A L INTELLIGENCER VOL. II, NO. 4, 1989

Commenting that the total number of Dirichlet's publications was not so large, Gauss wrote that the works of Dirichlet are jewels, and that jewels are not weighed on a grocery scale. Dirichlet taught at GOttingen until 1858 w h e n he fell ill o n a trip to Switzerland. Returning to G6ttingen he died of severe heart sickness on 5 May 1859. C o m m e n t i n g that the total n u m b e r of Dirichlet's publications was not so large, Gauss w r o t e that the works of Dirichlet are jewels, and that jewels are not w e i g h e d on a grocery scale (Biermann, p. 5).

ed.), Graduate Texts in Mathematics 74, SpringerVerlag, New York. (1st ed. 1967, Markham Publishers, Chicago.) MR 82m:10001 3. P. G. L. Dirichlet [18371], Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enth/ilt. 4. 5.

6. 7.

Bibliography

8.

MR = Mathematical Reviews; JFM = Jahrbuch iiber die Fortschritte der Mathematik; ZBL = Zentralblatt fiir Mathematik.

9.

1. K.-R. Biermann [1959], Johann Peter Gustav Lejeune Dirichlet, Dokumente fiir sein Leben und Wirken (zum 100. Todestag), Abhandlungen der Deutsch. Akad. der

Wissen. zu Berlin, Klasse ffir Math., Physik und Technik, Jahrgang 1959, Nr. 2, Akademie Verlag, Berlin. 2. H. Davenport [1980], Multiplicative number theory (2nd

10. 11. 12.

Abhandlungen der KfJniglich Preussischen Akademie der Wissenschaflen zu Berlin, 45-81; Werke I, 313-342. - [18372], Beweis eines Satzes fiber die arithemetische Progression, Werke I, 312. - [1839-40], Recherches sur diverses applications de l'analyse infinit6simale a la th6orie des nombres, J. reine angew. Math. 19, 324-369; 21, 1-12 and 134-155; Werke I, 411-496. - [1889], Werke I, ed. L. Kronecker; II (1897), ed. L. Fuchs, Verlag Georg Reimer, Berlin. JFM 21, 16-17. L. Euler [1848], Introductio in analysin infinitorum, vol. I, Bousquet, Lausanne. G. H. Hardy and E. M. Wright, An Introduction to the theory of numbers, Oxford, University Press, London; 1st ed. 1938, 4th ed. 1960. F. Klein [1926], Vorlesungen fiber die Entwicklung der Mathematik im 19. Jahrhundert, Teil I, ed. R. Courant, O. Neugebauer, Julius Springer Verlag, Berlin. JFM 52, 22-24. E. E. Kummer [1860], Ged/ichtnisrede auf Gustav Peter Lejeune Dirichlet, Abh. K6nig. Akad. Wissen. zu Berlin. Reprinted in Dirichlet's Werke, 309-344. D. E. Rowe [1988], Gauss, Dirichlet, and the law of biquadratic reciprocity, Math. Intellig. 10, no. 2, 13-25. B. L. van der Waerden [1985], A history of algebra, Springer-Verlag, Berlin, Heidelberg. MR 87e:01001.

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THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

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The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreements and controversy are welcome. An Opinion should be submitted to the editor-in-chief, Sheldon Axler.

Fractal Geometry Steven G. Krantz

Editor's note: The following articles by Steven G. Krantz and Benoit B. Mandelbrot have an unusual history. In the fall of 1988, Krantz asked the Bulletin of the American Mathematical Society Book Reviews editor, Edgar Lee Stout, whether he could review the books The Science of Fractal Images (edited by Heinz-Otto Peitgen and Dietmar Saupe) and The Beauty of Fractals (by Heinz-Otto Peitgen and Peter Richter) for the Bulletin. Subject to editorial approval, Stout agreed. Krantz submitted the review in mid-November. The editor requested a few changes, they were made, and the piece was accepted. Krantz received the galley proofs in mid-January of 1989. Meanwhile, Krantz circulated copies of the review to a number of people, including Mandelbrot, who took strong exception to the review and wrote a rebuttal. Stout encouraged Krantz to withdraw his review from the Bulletin and to publish it in a forum that accepted rebuttals. Krantz refused to withdraw his review, but he suggested that the Bulletin publish Mandelbrot's rebuttal along with the review. However, the policy of the American Mathematical Society (AMS) prohibits responses in the Bulletin to reviews. Stout then asked Krantz to make a number of revisions to soften the review. Krantz made the requested changes. After further thought, Stout decided that even the revised review (printed here) was not appropriate for the Bulletin, and he retracted his acceptance of the review. Krantz appealed the matter to the Council for the American Mathematical Society, which decided to support Stout"s editorial prerogative. The AMS Council suggested that Krantz's review and Mandelbrot's response be published in the Notices of the AMS. Krantz felt that the Bulletin should not reject a previously accepted review. Because Krantz was dissatisfied with his treatment by the AMS, he did not agree to have his review printed in the Notices of the AMS. The Mathematical Intelligencer, which welcomes controversy, is happy to publish Krantz' s review and Mandelbrot' s response.

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A recent cocktail party conversation at my university was concerned with the question of w h e t h e r academics are more eccentric or more depressive than the average functioning adult. At one point a clinical psychologist joined in and asserted that the matter had been studied in detail and the answer is "no." In fact, no profession seems to have more eccentric and depressive people than any other. The only exceptions, he w e n t on to say, are mathematicians and oboe players. Apparently the property that mathematicians and oboe players have in common is that both do something that is quite difficult and which few others ap-

THE MATHEMATICAL INTELL1GENCER VOL. 11, NO. 4 9 1989 Springer-Verlag New York

preciate. Be that as it may, we have all struggled with (or chosen to ignore) the problem of explaining to nonmathematicians w h a t it is that we do all day. Other scientific professionals can throw up a smoke screen with phrases like "genetic engineering," "black holes," "plasma physics," and "string theory." Although listeners are no better informed after hearing these phrases than before hearing them, they are at least comforted by having heard them before. We mathematicians could not hope for a similar effect with phrases like "exotic cohomologies," "EinsteinK/ihler m e t r i c s , " " p s e u d o c o n v e x d o m a i n s , " or "weakly strongly singular integrals." Nonmathematicians have no forum for encountering the terminology of mathematics. That is the nature of the beast: most of the deep ideas in mathematics are couched in technical language. But a consequence of the rarification of our subject is that the public tends to be intensely unaware of us. The history of mathematics in the popular press has until recently been virtually void. John von Neumann graced the cover of Time because of his work on stored program computers. Hans Rademacher was written up in Time for "proving" the Riemann Hypothesis. But few if any nonspecialists got even a whiff of the Kodaira Vanishing Theorem or Weil's proof of the Riemann Hypothesis for function fields over finite fields. In his Mathematician's Apology, G. H. Hardy crowed that he never had done nor would he ever do anything "useful." By implication he also would never do anything that anyone but a mathematician would care about. Times have changed and for several reasons. The American Mathematical Society (AMS) has an advocate in Washington. A public relations firm has been hired by the AMS to promote the cause of mathematics nationwide. One result: The U.S. Congress has d e c r e e d one w e e k per year to be " M a t h e m a t i c s Awareness Week." In addition, there have been administrative and pecuniary pressures for pure mathematicians to interact with the applied world. Conveniently, the ready availability of high-speed digital computing equipment has served as a catalyst and a common language in this collaborative process. And the collaboration prospers. It is also significant that several mathematicians, notably Ron Graham, have made a point of cultivating contacts with the press. Thus they can serve both as a sieve and a buffer between us and the world at large. On the whole, the effect of this effort has been positive. In particular, we owe to this the copious attention given to Freedman's sglution of the four-dimensional Poincar6 conjecture, Thurston's work on three-manifolds, the ill-starred solutions of Fermat's last theorem and the three-dimensional Poincar6 conjecture, and Karmarkar's algorithm. Charles Fefferman was even written up in People magazine!

Heady stuff, that. But n o w there is a mathematical development that threatens to dwarf all others for its potential publicity value: the theory of fractals. While the sets called fractals have been studied for many years (in harmonic analysis, in geometric measure theory, and in the theory of singularities, for instance), the term "fractal" was coined and popularized by Benoit Mandelbrot (1975). By his own telling "the first steps of the development of a systematic fractal geometry, including its graphic aspects, were taken at the IBM T. J. Watson Research Center, or wherever I happened to be visiting from the IBM base." In The Beauty of Fractal Images Mandelbrot elaborates on this theme: No more than six years ago! Only ten and twenty-odd years ago! On many days, I find it hard to believe that only six years have passed since I first saw and described the structure of the beautiful set which is celebrated in the present book, and to which I am honored and delighted that my name should be attached. No more than twentyodd years have passed since I became convinced that my varied forays into unfashionable and lonely corners of the Unknown were not separate enterprises. Hailed as a lingua franca for all of science, the theory of fractals is said by some to be the greatest idea since calculus. The subject of calculus has played a special role in the history of modern science: Most of physics and engineering, and important parts of astronomy, chemistry and biology, would be impossible without it. Thus it is a compliment of the highest order to compare any new development with the calculus. Let us discuss that subject for a moment. In the early days of calculus, it was practiced by a handful of fanatics. And so it had to be, for the theories of fluxions and fluents were virtually devoid of rigor and were full of internal contradictions. Bishop Berkeley's broadside The Analyst: A Discourse Addressed to an Infidel Mathematician, which ridiculed infinitesimals as "the ghosts of departed quantities," was a much needed breath of fresh air. It forced mathematicians to re-examine the foundations of analysis. There followed two hundred years of intense effort by the best minds in Europe. The result was the rigorous calculus we know today. What makes calculus important and what fueled in part Berkeley's frustration and fury is that calculus solves so many wonderful problems: The brachistochrone, Kepler's Laws, and many other deep properties of nature follow with calculus from a few elegant physical principles. Like the fathers of calculus, the founders of fractal g e o m e t r y constitute a cadre of dedicated fanatics. They should not be hampered by lack of rigor, for they share in the hard-won wisdom of the last 300 years. Yet there is not even a universally accepted definition of the term "'fractal." It seems that if one does not prove theorems (as, evidently, fractal geometers do THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

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One notable difference between fractal geometry and calculus is that fractal geometry has not solved any problems. not), then one does not need definitions. One notable difference b e t w e e n fractal geometry and calculus is that fractal geometry has not solved any problems. It is not even clear that it has created any new ones. This is a rather strong contention and requires elaboration. One definition of "'fractal" is that it is a set whose Hausdorff dimension exceeds its topological dimension. Many examples are self-similar sets: Pick a neighborhood of a point in the fractal, dilate the neighborhood, restrict the dilated set to the original neighborhood, and voila! the picture is unchanged. Fractals abound both in mathematics and in nature. The yon Koch snowflake curve is a fractal, as is (the construction of) the Peano space-filling curve. Perhaps the most famous example of a fractal in nature is the coastline of England, which has the property that the closer you look, the more it wiggles. Thus the coastline is nonrectifiable and has infinite length. Attached to many fractals is a numerical quantity called its fractal (similarity) dimension. If a fractal S can be divided into N congruent (in the sense of Euclidean geometry) subsets, each of which is an r-fold dilation of the original set, then the fractal dimension D of S is defined to be D =

log N log (l/r) "

This formula is emblazoned in 24-point type on page 29 of The Scienceof FractalImages. Even though it is but a pale shadow of the truly deep concept of Hausdorff dimension, fractal dimension is one of the big ideas in the subject of fractal geometry. Thus many (but certainly not all) fractals have a fractal dimension, and naturally we want to compute this quantity. We learn that the perimeters of projections of certain clouds are fractals and that their fractal dimension is 4~. That turns out to be the same fractal dimension as that of a certain Cantor set. What have we learned? Better still, it has the same fractal dimension as the staircase in a certain engraving of M. C. Escher. Does this demonstrate some intrinsic structure in the universe? Are we, like Thomas Hobbes, on the verge of a calculus of ethics? Or are we, like Erik yon Daniken in Chariots of the Gods, celebrating form over substance? My mention of Escher is not a frivolous one. The books under review invoke the names of Escher and Ansel Adams as a means of lending both charm and credence to their subject. Other august n a m e s m J o h n Milnor, Dennis Sullivan, and William Thurston (to enumerate but a f e w ) m a r e mentioned as examples of 14

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mathematicians whose work has apparently been inspired by fractal geometry. And this is a point worth noting explicitly: Some of the pictures of fractals have provoked the thoughts of Mandelbrot (who is good at dreaming up pretty questions) and of the aforementioned mathematicians. The latter have, as a result, proved some deep and interesting theorems in iteration theory. I don't think that Mandelbrot has proved any theorems as a result of his investigations, but that is not what he claims to do. By his own telling, he is a philosopher of science. There is an important issue implicit in this discussion that I w o u l d n o w like to examine. A famous counterexample (due to Celso Costa) in the theory of minimal surfaces was inspired by the viewing of a Brazilian documentary about samba schools--it seems that one of the dancers wore a traditional hat of a bizarre character that was later reflected in the shape of the example. I once thought of an interesting counterexample by lying on my back and watching the flight of seagulls. Whatever the merits of samba dancers and seagulls may be, they are not scientists and they are not mathematicians. Why should fractal geometers be judged any differently? Writings on fractal geometry find fractals in the work of m a n y fine mathematicians, but that is as much insight as the theory of fractals lends to pre-existing theory. What we have is a language which is sufficiently diluted that it says something (of a descriptive nature) about almost anything that you can think of. I would be foolish to accuse fractal geometers of poaching from other fields. What fractal geometry has to say about other fields is not sufficient to make that a viable possibility. An important ambiguity needs to be clarified at this stage. Some fine mathematics, such as the theory of sets of fractional dimension, H a u s d o r f f measure, nonrectifiable sets, currents, etc., has been s w e p t under the umbreUa of fractal geometry (see [2], which has the misleading word "Fractal" in its title but which actually describes some beautiful, pre-fractal mathematics). When I criticize fractal geometry I am criticizing specifically the activities described in the two books u n d e r r e v i e w - - n o t the substantive areas of mathematics that have been caught up in the whirlwind of publicity surrounding Mandelbrot. When one opens the books under review, it appears that fractal geometry is a science--evidently a mathematical one. However, nowhere in either book do I see a theorem, and there are few definitions. As noted above, there is no precise definition of the term "fractal." As a mathematician I find that this bodes no good. Look what happened to set theory when Russell's paradox was discovered. The trouble with any subject that relies more on computer output than on theory is that one has to think of something to say about it. The result is that

much of the writing turns out to be anecdotal. Alt h o u g h the following passage from The Science of Fractal Images is not representative of the best that fractal theory has to offer, it serves to illustrate my point: The overall outline is now reminiscent of a dog's head while just the upper portion could be the Loch Ness monster. Shapes with a fractal dimension D about 0.2 to 0.3 greater than the Euclidean dimension E seem particularly favored in nature. Coastlines typically have a fractal dimension around 1.2, landscapes around 2.2, and clouds around 3.3. I once heard a talk by an eminent mathematician about automata theory. He confessed at the outset that he had a lot of questions and no answers. The rest of the talk consisted of looking at a variety of computer printouts and saying "this looks like a gopher's hole" and "this looks like a thundercloud." All quite boring and disappointing. It seems to me that if a subject is to be called a science, then one should be able to say more about it than this. Of course, the books under review are not research journals, nor are they monographs. One hardly expects to see Theorem-Proof-Theorem-Proof. What one does expect to see is a development of ideas leading to some crescendo, the artful synthesis of concepts to give new insight, the formulation of precise mathematical discoveries accompanied by convincing arguments or proofs. I cannot find any evidence of these in the books under review. No discussion of fractals would be complete without due homage to the pictures. They are wonderful and are apparently the raison d'etre for all the uproar over fractals. Pictures of Julia sets and Mandelbrot sets are astonishing in their complexity and diversity. I do not accept the assertion (page 177 of The Science of Fractal Images) that the Mandelbrot set "is considered to be the most complex object mathematics has ever seen." This type of hyperbole may appeal to readers of pop-

ular magazines but rings untrue to the trained mathematician. However, my main point is somewhat different: I wish to establish a distinction between fractal computer graphics and some other computer graphics of recent note. Dave Hoffman, Jim Hoffman, and Bill Meeks at the University of Massachusetts have received considerable attention for the graphics they have generated in connection with the study of minimal surfaces. But the work of Hoffman, Hoffman, and Meeks was moti-

The hypotheses and conjectures that the fractal people generate are (like the objects which they study) self-referential. One generates the pictures to learn more a b o u t the pictures, not to attain deeper understanding. vated by a deep and important scientific question: Do there exist non self-intersecting minimal surfaces of high genus? The startling answer is "yes," and it was determined by generating models numerically, staring at the graphic realizations of the models, figuring out what is going on mathematically, and proving a theorem that answered the original question. In my view this type of work is a prime example of the most important n e w use of c o m p u t e r s - - n o t just for number crunching but for doing "what if" calculations that we could never do by hand. In the preface to The Science of Fractal Images, Mandelbrot suggests that fractal geometers also use computer graphics to develop hypotheses and conjectures. But the difference is that the hypotheses and conjectures that the fractal people generate are (like the objects which they study) self-referential. One generates the pictures to learn more about the pictures, not to attain deeper understanding. That the pictures have occasionally inspired fine mathematicians to prove good theorems seems serendipitous at best. It is this admittedly rather fine distinction that troubles my mathematical sensibilities. Good mathematicians do not always answer the questions they originally set out to study. Fritz John once said that w h e n the a n s w e r to your question is " y e s " then you've asked the wrong question. One expects good questions to open doors, and there is nothing more stimulating than following one's nose into new terrain. The assertion that the relationship between fractal theory and mathematics is symbiotic is Mandelbrot's - - n o t mine. But the true nature of the symbiosis is m u d d i e d by the terminology of fractal geometry: "Fractal" appears to be a n e w name for sets of fractional Hausdorff dimension; the "Weierstrass-Mandelbrot function" is a small variant of the Weierstrass nowhere differentiable function; the "Mandelbrot set" THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

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was not invented by Mandelbrot but occurs explicitly in the literature a couple of years before the term "Mandelbrot set" was coined (see [1, p. 68]). In fact, Fatou and Julia initiated the study of the iterates of the function z ~ z 2 + c but Mandelbrot, at least by association, receives much of the credit for it these days.

The enormous publicity of fractal theory has led to harmful government policies toward mathematics. As previously noted, fractal geometry has a special aura in part because it has roots in the work of Pierre Fatou and Gaston Julia and because it apparently serves as an inspiration to mathematicians like Adrien Douady and John Hubbard. Let me state quite plainly that the work of these researchers and others on dynamical systems, iterative processes, and related topics is truly excellent--it is some of the best mathematics being done today. But these mathematicians don't study fractals--they prove beautiful theorems. To the extent that fractal geometry has received praise from the mathematical community, it has been indirectly through praise of the work of Douady, Hubbard, Thurston, and others. In fractal geometry one uses some mathematics to generate a picture, then asks questions about the p i c t u r e - - w h i c h generates more pictures. Then one asks more questions about the new pictures. And so on. One rarely, if ever, sees a return to the original mathematics. What I find most bothersome, and this is no fault of the books under review, is that the public's perception of what mathematicians do these days is derived in large part from reading books about fractals, reading James Gleick's book Chaos, and reading about various incorrect proofs of long-standing conjectures. The latter item is just too bad, but the first two are terribly misleading. Both the theory of fractals and that of chaos are in their infancy. It is too soon to tell whether either will blossom into mature subjects. A point of pride for fractal theorists is that their language is being picked up by laboratory scientists such as physicists. One third of the submissions to Physics Review Letters concern themselves with, or at least mention, fractals. It is easy to come away with the impression that fractal theory must have something important to say about the world around us. Leo Kadanoff (Nobel Laureate, and Professor of Physics, University of Chicago) addresses this point in [3]: Unfortunately, although this single rather primitive measurement [fractal dimension] enables us to distinguish among objects, it never enables us to give a convincing case for their essential identity. Some progress has been made in identifying other qualities, beyond the fractal dimension, that might be universal. However, further progress in this field depends upon establishing a more sub16

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stantial theoretical base in which geometrical form is deduced from the mechanisms that produce it. Lacking such a base, one cannot define very sharply what types of questions might have interesting answers. One might hope, and even suspect, that eventually a theoretical underpinn i n g . . , will be developed to anchor this subject. Without that underpinning much of the work on fractals seems somewhat superficial and even slightly pointless. It is easy, too easy, to perform computer simulations upon all kinds of models and to compare the results with each other and with real-world outcomes. But without organizing principles, the field tends to decay into a zoology of interesting questions and facile classifications. Despite the beauty and elegance of the phenomenological observations upon which the field is based, the physics of fractals is, in many ways, a subject waiting to be born. Indirectly, the enormous publicity of fractal theory has led to harmful government policies toward mathematics. In some circles, it is easier to obtain funding to b u y hardware to generate pictures of fractals than to obtain funding to study algebraic geometry. Since algebraic geometry has withstood the test of time and fractal geometry has not, one must wonder what considerations led to such funding decisions. My o w n theory is that bureaucracies can cope with hardware more easily than they can cope with ideas. In any event, it is depressing to predict the long-term effects of such policy. The subject of fractal geometry is young, w e should watch its development closely. Who knows? In 300 years it may prove to be as important as calculus. Meanwhile, the books under review provide a delightful invitation to the subject. The prose is clean and dear, the illustrations profuse and attractive, and the concepts are enjoyable. One of the principal emphases in these books is the description of algorithmic techniques for generating fractal graphics images on a computer system. In this respect the books are a great success. However, as to the assertion that they provide a glimpse of a n e w science or the language for developing a new analysis of nature, I would say that any contribution that fractal theory has made in this direction has been accidental. In short, the emperor has no clothes.

References 1. Robert Brooks and J. Peter Matelski, The dynamics of 2-generator subgroups of PSL(2,C), Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (Irwin Kra and Bernard Maskit, eds.), Annals of Math, Studies 97, Princeton" Princeton University Press (1981). 2. K. J. Falconer, The Geometry of Fractal Sets, Cambridge: Cambridge University Press (1985). 3. Leo P. Kadanoff, Fractals: Where's the Physics?, Physics Today (February 1986), 6-7.

Department of Mathematics Washington University St. Louis, MO 63130 USA

Some "Facts" That Evaporate Upon Examination Benoit B. Mandelbrot As the mathematical community regains the flexibility and the pluralism that it showed over most of its history, every mood is entitled to be expressed. Thus, Steven G. Krantz is entitled to be heard, as long as he can be answered 9 My answer will not defend H.-O. Peitgen, P. H. Richter, and D9 Saupe for having failed to write books they did not intend to write9 Instead, I will counter the "facts" Krantz brings up to buttress his opinions concerning fractals. As I will show, these "facts," at first impressive, evaporate upon examination. The Brooks-Matelski reference. Krantz tells us "the 'Mandelbrot set' was not invented by Mandelbrot but occurs explicitly in the literature a couple of years before the term 'Mandelbrot set' was coined." As is well known, I " i n v e n t e d " that set in 1979-80 and fully published it in 1980 (in a widely quoted paper in J Annals of the N Y Academy of Sciences). Indeed, this happened "a couple of years" before 1982, when the term was coined by A. Douady and J. H. Hubbard. But let us examine the paper by R. Brooks and J. P. Matelski, which appeared in 1981, and to which Krantz refers. Here are all the relevant fragments in extenso. A) "Fatou-Julia allow one to draw by ~computer the 9. . region of C defined by {C:z2 + C has a stable periodic orbit}." [Note: This region is now denoted by M~ B) "We would like to thank Henry Laufer for suggesting and assisting us with the use of the computers."

c)

of M, bringing them close to something that was to prove special, but they gave no thought to the picture. In contrast, my thoughts "were provoked" over months and years, helped by long practice with analogous situations elsewhere. During 1979 and 1980, I lectured on M at Harvard, M.I.T., an A.M.S. summer meeting on W. Thurston's work, and (most important, perhaps) at Orsay (Paris-Sud) and Bures (I.H.E.S.). There I spent many hours and many meals describing M to Douady in as great detail as requested. Our discussion made him drop what he had been doing before, and he and Hubbard have made major contributions to iteration theory. Subsequently I spoke at D. Sullivan's CUNY Seminar and at Princeton to J. Milnor and W. Thurston. The set in question was not credited to anyone else at the time. A great recent paper by John Milnor lists several questions I had raised as having remained outstanding open problems. Eventually, all this has led to the est a b l i s h m e n t of the " G e o m e t r y S u p e r c o m p u t e r Project," of which I am a charter member, together w i t h . . . Douady, Hubbard, Milnor, and Thurston (to name but a few). Krantz now concedes that I am "good at dreaming up pretty questions," and allows that the work of the above mentioned men "is some of the best mathematics being done today." The Kadanoff quote. When read out of context, the text that L. P. Kadanoff published in the February 1986 Physics Today may sound like grounds for dismissing the contributions of the fractal point of view to

D ) " T h e set of c's such thatf(z) = z 2 + c has a stable periodic orbit." [Note: This is the caption for the figure shown in C).] This is all there is. A glance at C) shows that the region it illustrates (the original is larger) is not M ~ Indeed, the empirical "M~ (which is very hard to compute) is made of many large noncontacting patches. A second thought by a referee or the authors would have led to correction. But what is most interesting lies elsewhere: the authors' friend suggested a crude version THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4 9 1989 Springer-Verlag New York

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physics. But it is not an Olympian judgment by a disinterested party, and it can only be understood in context. Kadanoff was then, and continues to be, a prominent lecturer on fractals and a successful participant in the study of problems of physics that involve fractals in an essential way. Take, for example, the issue of the same Physics Today dated April 1986. A "news" piece signed by a staff writer is titled " N e w global fractal formalism describes paths to turbulence," and reports on some excellent experimental work by Kadanoff and his colleagues in Chicago ("some of the best physics being done t h e n " - - t o paraphrase Krantz). Here is an excerpt that describes the background of the mathematical techniques that motivated this work and helped its interpretation.

like the statement that "the Mandelbrot set is the most complicated object mathematics has ever seen." Actually, this statement may even be true if "to see" is taken literally to involve the eye, and surely it is preferable to its more emphatic original. The original ends by the words "the most complicated object in mathe-

Much of the geometric messiness of nature is too complicated even for fractals.

matics," and it is due to Hubbard (as quoted by A. K. D e w d n e y in the August 1985 Scientific American). Finally, I grab the chance of giving Krantz a good grade for something. He proclaims that "Fractals a b o u n d . . , in nature . . . . Perhaps the most famous example . . . is the coastline of England." This is one Starting in part with the pioneering studies of Benoit basic theme of my book The Fractal Geometry of Nature. Mandelbrot (IBM) a decade ago, work on fractal systems Therefore, the long uphill flight that led this "fanatic" has flourished in recent years. The basic idea of representing the global structure of fractals in terms of the for- to his book did seed at least one flower in this relucmalism described here originated [in 1985] with the work tant ground. A second basic theme Of my work is that of Uriel Frisch (Nice Observatory) and Giorgio Parisi (Uni- much of the geometric messiness of nature is too comversity of Rome) on well-developed turbulence. plicated even for fractals. Altogether, Krantz is overly concerned with deIt may be added that Frisch and Parisi have coined fining old labels to give them a narrow scope and then for their new formalism the term "multifractal," which attempting to fit n e w realities under these narrow has been generally accepted. Also, their paper de- labels. If I have proven only a few theorems, they scribes in limpid fashion how this formalism evolved were hard to guess, and they invariably concerned from the mathematics of a paper I wrote in 1974, and fields that were not yet opened at the time or were ends by stating that my mathematics remains more dormant. (See the text of my address to the Warsaw general than their formalism. International Congress of Mathematicians). One of my This is not to claim that fractals have conquered "easy" theorems answered a wish that has stood unphysics (they had never tried), but that physics also fulfilled since Poincard d e f i n e d the limit sets of has room for a diversity of opinion. Kleinian groups. Often, I only handled the simple The "'Weierstrass-Mandelbrot'" function. This function cases and working out the hard ones brought joy to a is indeed a small variant of the Weierstrass function, number of people. W(t), and it has little mathematical point; someone must But the issues of how to define mathematics today have written it d o w n previously. So w h y did the and of w h e t h e r or not I am to be categorized as a noted physicist Michael Berry add my name? Because mathematician are unimportant. On the other hand, this very natural variant has self-affinity properties i m p o r t a n t c o n s e q u e n c e s h a v e c o m e o u t of m y that W(t) fails to possess, while in many applications "dreaming up pretty questions" in different fields. they are essential. The trivial issue of h o w to name The technical impact of each question in its well-dethis variant of W(t) is one of many that Krantz ad- fined mainstreams is only part of the story. A different vances as if each were a devastating criticism, which part, which is attractive to very many but clearly not to they cease to be on examination. everybody, is that these questions are not discrete and "By his own telling, [BBM] is a philosopher of science." haphazard, but all seem to come from one spring. This, again, is one of the many assertions that Krantz To conclude, my comments in the Summer 1989 credits to m e - - d e s p i t e dear, published evidence to Mathematical Intelligencer are also relevant to the issues the c o n t r a r y - - a n d that the reader is well advised to raised by Krantz. In addition, the May 8, 1989 issue of disregard. No d o u b t Krantz damns philosophers of the Proceedings of the Royal Society (London), Series A science as commentators and critics of what others do, includes a long (but already out-of-date) list of books but I am nothing if not a doer. My work and the work that expound on diverse aspects of what Krantz asof those w h o m this work has inspired is not found in serts does not exist. Finally, those who read French philosophy journals but in those of working mathe- may care to know that my 1975 book Les objets fractals matics, science, and art. is having a third edition, in which it is followed by a I sympathize with Krantz on a few points. I also dis- survey (Survol). That text develops my introductory 18

THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 4, 1989

remarks in op. cir. in Proc. R. S. (London). Thus it presents (with evidence) my current views of the obvious and acknowledged limitations of fractal geometry and also, of course, its achievements. Physics Department IBM T.J. Watson Research Center Yorktown Heights, NY 10598 USA Mathematics Department Yale University New Haven, CT 06520 USA

The article by staff writer Barbara Levi in the April 1986 issue of Physics Today describes work of Leo Kadanoff that uses methods of statistical mechanics-not fractal geometry. Levi's decision to describe this deep scientific research using the language of fractals, and Mandelbrot's endorsement of it, is a perfect example of the fractal geometers' tendency to derive their sense of excellence vicariously from the excellence of others.

Steven G. Krantz Replies, The ideas of Brooks and Matelski were presented at a conference in 1978, and well pre-date any contribution that Mandelbrot may have made in 1979-80.

Editor's comment: The Letters-to-the-Editor column of the Mathematical Intelligencer welcomes comments about the issues raised by Steven Krantz and Benoit Mandelbrot.

THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 4, 1989 1 9

remarks in op. cir. in Proc. R. S. (London). Thus it presents (with evidence) my current views of the obvious and acknowledged limitations of fractal geometry and also, of course, its achievements. Physics Department IBM T.J. Watson Research Center Yorktown Heights, NY 10598 USA Mathematics Department Yale University New Haven, CT 06520 USA

The article by staff writer Barbara Levi in the April 1986 issue of Physics Today describes work of Leo Kadanoff that uses methods of statistical mechanics-not fractal geometry. Levi's decision to describe this deep scientific research using the language of fractals, and Mandelbrot's endorsement of it, is a perfect example of the fractal geometers' tendency to derive their sense of excellence vicariously from the excellence of others.

Steven G. Krantz Replies, The ideas of Brooks and Matelski were presented at a conference in 1978, and well pre-date any contribution that Mandelbrot may have made in 1979-80.

Editor's comment: The Letters-to-the-Editor column of the Mathematical Intelligencer welcomes comments about the issues raised by Steven Krantz and Benoit Mandelbrot.

THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 4, 1989 1 9

A Birthday Present Reinhard BOlling

"'I'd like to show you something if you have a minute to spare." These were the words spoken to me in the summer of 1986 by Mr. H. Hadan, Head of the Library of Humboldt University's Mathematics Department. What he showed me was a beautifully bound photo album (Figure 1); I could hardly believe my eyes when I read the inscription: Ehrenalbum dem Professor der Mathematik Karl Weierstrass zum 70 j/ihrigen Geburtstag am 31. Oktober 1885 ~iberreicht. There was no doubt about its genuineness. This was indeed the album Weierstrass mentions in a letter to

Sonya Kovalevskaya written just over 100 years ago, on 14 December 1885, in which he writes about the celebration marking his seventieth birthday: "The album for the photographs (more than 500 of them) is a magnificent work, which met with general acclaim" ([1], p. 129). On 44 pages there are a total of 294 photographs of about the same format--seven of them to a page, with a few gaps in b e t w e e n - - s h o w i n g pupils, friends, and colleagues of Weierstrass from m a n y countries in Europe, including renowned university professors, masters, headmasters, high school teachers, post graduate students, and undergraduates (including such names as L. Heffter, A. Gutzmer, and G. Wallenberg). Particularly conspicuous among the photographs is that of the only woman, his pupil Sonya Kovalevskaya (Figure 2), with w h o m he maintained a very close friendship until her untimely death in 1891. The photographs are arranged essentially according to country (in the German section, the photographs are grouped mostly according to university and region). One finds photographs of such people as (in the order of their appearance in the album) G. Zeuthen, A. Cayley (Figure 3), E. Picard, F. Casorati, Hj. Mellin, E. Phragm6n, S. Lie (Figure 4), L. Sylow, L. Schl/ifli, F. Schottky, G. Frobenius (Figure 5), C. Runge, R. Lipschitz, P. Gordan, M. Noether, J. Li~roth, L. Stickelberger, M. Pasch, H. A. Schwarz (Figure 6), O. H61der, A. Schoenflies, G. Cantor (Figure 7), M. Cantor, M. Planck (Figure 8), A. Hurwitz (Figure 9), F. Klein (Figure 10), F. Schur, H. Weber, A. Pringsheim, A. Brill, O. Schl6milch, and P. du Bois-Reymond, to name but a few of the prominent mathematicians and physicists.

Figure 1 (opposite). Front cover of the Weierstrass album. 20 THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4 9 1989 Springer-VeflagNew York

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21

Figure 2. Sonya Kovalevskaya

Figure 3. Arthur Cayley

B e c a u s e t h e r e is n o W e i e r s t r a s s e s t a t e , it was t h o u g h t likely that this p h o t o album had long since fallen p r e y to the c o n f u s i o n of the times. O n e can easily imagine the joy of discovery I felt w h e n leafing t h r o u g h the pages of the album, which was missing for decades only to re-emerge here in Berlin, on our v e r y doorstep, as it were, not far from the A c a d e m y a n d the University w h e r e Weierstrass once worked. Preparations for Weierstrass's 70th birthday on 31 October 1885 w e r e started by the beginning of 1884. A special committee was set u p for this purpose. It was h e a d e d by L. Fuchs, w h o succeeded E. E. K u m m e r as full professor at the Berlin University in April 1884. Its m e m b e r s i n c l u d e d S o n y a K o v a l e v s k a y a a n d the S w e d i s h m a t h e m a t i c i a n G. Mittag-Leffler. Kovalevskaya had just started working at the University (H6gskola) of Stockholm, which was still quite y o u n g then. Fuchs wrote her a letter on 3 February 1884 asking her to send 25 signed copies of an address that she was to deliver in h o n o u r of Weierstrass to various, mainly Russian, mathematicians ([2], p. 133). A collection was t a k e n u p in w h i c h m a t h e m a t i c i a n s at h o m e a n d a b r o a d took part. A c c o r d i n g to Mittag-Leffler, two thirds of the total s u m collected came from a b r o a d (letter to Kovalevskaya dated 20-22 July 1885 ([3], p. 110)). The m o n e y was n e e d e d to cover the costs of a bust, a medal, and the p h o t o album. The idea of presenting Weierstrass with an album seems to have origi n a t e d later on, as a letter Carl Itzigsohn w r o t e to Sonya Kovalevskaya dated 1 July 1885 suggests. An approximately literal English translation of the original is i n c l u d e d here* (a Russian t r a n s l a t i o n , parts of which are not entirely accurate, is r e p r o d u c e d in [3], pp. 252-253). 22

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Figure 4. Sophus Lie Berlin, July 1st 1885

Dear Madam, Forgive me for troubling you in your capacity as a member of the Weierstrass Committee with an inquiry. Contributions for the celebrations of the 70th birthday of our great master, teacher and dedicated researcher have been pouring in in ample quantity: at present we have five to six thousand marks at our disposal. We now intend to present Weierstrass not only with a bust and a medal, but also with an album containing plates of his pupils and admirers (visiting card or cabinet format). This idea is acceptable to both Prof. Fuchs and the Mathematical Society (Mathematischer Verein) of Berlin (the latter having contributed approx. 1000 marks). Because Prof. Fuchs is very busy at present, he has requested me to ask you if you would agree to having the costs of the album covered by money from the collected donations. In my o p i n i o n - - a n d this is only my personal opinion--the intention of the donors is that the fund should be used entirely for Prof. Weierstrass and that the cast medals should not be given away free to those desirous of obtaining them, but against a charge to cover the costs of their production. Dear Madam, feeling certain that it is your innermost desire to show, in every way possible, your deep admiration for this esteemed man on his forthcoming birthday--a desire which I share with you in full measure--I hope that you will not object to the idea of defraying the costs of the album from the donation. * The author is most grateful to the Mittag-LeffierInstitute in Djursholm, Sweden, for supplying him with a copy of this and two other letters (mentioned in the text) from C. Itzigsohn to G. Mittag-Leffler.

Figure 5. Georg Frobenius

Figure 6. Hermann Amandus Schwarz

If the funds should not be sufficient, I am prepared to .pay the costs of this present, because--and I hope my " assumption is correct--this will add to the birthday celebrant's joy. I only want the Committee to consider the whole matter as its own concern. I kindly ask you to inform Prof. Fuchs, whose address is Klein Beeren Str. No. 1

here [in Berlin]

or the undersigned, who will be glad to assume the task of conducting the collection, of your kind decision. Trusting you will accept my sincere greetings and assurances of my high esteem for you, I remain Sincerely yours Carl Itzigsohn Lothringer Str. No. 72 Itzigsohn sent a letter of the same content and bearing the same date to Mittag-Leffler, which the latter answered to the effect that the funds were to be used in the following order: 1. Marble bust; 2. A gold medal, which was to be presented to Weierstrass; 3. An album to be presented to Weierstrass; 4. Making a mold for making plaster casts, which would be available against a charge to those interested ([3], p. 110). Moreover, Mittag-Leffler offered his assistance in procuring photographs for the album. In his reply dated 16 August 1885, Itzigsohn referred to this question. With regard to France, Itzigsohn inquired whether it would be possible to ask Hermite to collect photographs and if Mittag-Leffler would be prepared to contact him regarding this matter. He also asked for advice in this matter concerning Italy and, if occasion arises, for the assistance of Casorati and Beltrami. Mittag-Leffier was sent 50 "circulars" for distribution to Scandinavian mathematicians. Photos of those living in Germany were compiled by the Mathematical

Figure 7. Georg Cantor

Society (Mathematischer Verein) of the Berlin University. Later on Weierstrass was to write about his birthday celebration: "Apart from my colleagues at home, the following (people) were personally present to present the gifts to me in the name of the committee: Cantor, Schwarz, Bruns, L i n d e m a n n , Killing, Thom6, P. Dubois" ([1], p. 129; corrected version, the name of Bruns, which occurs in the original, is missing). It is strange that in their personal recollections of the event, both W. Killing (in a speech delivered when he assumed the office of rector of the Royal Academy of M~inster on 15 October 1897 [4]) and E. Lampe (in a speech made at a session of Berlin's Mathematische Gesellschaft marking the 100th birthday of Weierstrass on 27 October 1915 [5]) only mention the presentation of the marble bust a n d - - a s they s a y - - o f the "gold memorial coin," not a w o r d being said about the album (which contains photographs of both of them). In contrast to these omissions, D. H. Kennedy, in his biography of Kovalevskaya, refers to photographs of Weierstrass's places of activities allegedly contained in the album ([6], p. 240). In any case, no such pictures are in the album. What happened to the album? After Weierstrass's death in 1897, it probably passed to his brother Peter (1820-1904) (of his two sisters, who lived with Karl-none of the four siblings ever married--Klara died in 1896 and Elisabeth in 1898, i.e., one shortly before, the other after their brother's death). In 1904 at the latest, the year of Peter's death, the album must have passed to Johannes Knoblauch (1855-1915), a mathematician working in Berlin (putting aside his own scientific ambitions, he distinguished himself by his dedicated THE MATHEMATICAL INTELLIGENCER VOL. 11, NO, 4, I989

23

Figure 8. Max Planck

Figure 9. Adolf Hurwitz

labor on the publication of the works of Weierstrass). That same year, Knoblauch, in any case, presented the album to Berlin's Kupferstichkabinett ( m u s e u m of copper engravings). But it is also possible that the album was given to Knoblauch right after Weierstrass's death. Mittag-Leffier reports that Knoblauch asked for and received a copy of a portrait in oil, which was among the papers left behind by the deceased ([7], p. 17). Be that as it may, for more than seventy years the album remained at Berlin's Kupferstichkabinett before it was offered to, and accepted by, the Mathematics Department of the Humboldt University of Berlin. But this is not the end of the story. It seems curious that Mittag-Leffler (himself a pupil of Weierstrass), w h o cherished the m e m o r y of his great teacher throughout his life, w h o compiled a historically valuable collection of items connected with Weierstrass's life and work at the institute (which today bears his name) founded by him in Djursholm, Sweden, and who, moreover, belonged to the festival committee, is not represented by a photograph in the album. This question led to an interesting answer. Namely, it was discovered that Berlin's Kupferstichkabinett had not one, but two similar albums in its possession. The circumstances leading to the separation of the albums and to their being kept at two different places have not been clarified. The second album is of a smaller format and its cover is less elaborate (the front cover bears the inscription XXXI OCTBR ANNO DOM MDCCLXXXV in three lines). The wording of the dedication inside is exactly the same as that of the first album described above. It contains a total of 40 photographs (the format of the photos is larger, approx. 10 x 14 cm, than in the 24

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO, 4, 1989

Figure 10. Felix Klein

first album, in which the average picture format is 6 x 9.5 cm). The photos include the one missing from the first album, namely that of G. Mittag-Leffler, as well as pictures of L. Fuchs, F. Prym, L. Cremona, G. K6nig, and P. L. Chebyshev. Thus it seems that the reason for producing two albums is simply that they were meant for two different formats of photographs. It should be noted, however, that the total number of 334 photos is still less than the 500 mentioned by Weierstrass. There are no tables of contents in the albums, but the pages and, in the first album, the photos (but not the gaps between them) are numbered, which means that any losses in the album could have h a p p e n e d only before the photos were numbered. Because there are only a total of 24 unnumbered gaps (14 in the first, 10 in the second album), loss can be ruled out as a possible explanation. In closing, the author would like to mention that he intends to publish a detailed edition of the albums. The preparations for the celebration were not altogether without snags. G. Cantor, for instance, refused to sign the address mentioned earlier on because in his opinion "its wording was cold, colorless, watery and shallow and promised to be a failure" (quotation from his letter to Kovalevskaya dated 30 December 1884 ([8], p. 687)). Because of Kronecker's well-known opposition to Weierstrass's analysis (and by extension also to Cantor's own set theory), Cantor wanted to see that appropriate homage was paid to Weierstrass for his a c h i e v e m e n t s . Cantor w r o t e that Weierstrass "holds by far the most eminent and distinguished position among all mathematicians who are currently acfive" (Cantor's letter to Fuchs dated 30 December 1884 ([81, p. 686)).

Kovalevskaya, who shared Cantor's opinion, on receiving the letter on 12 December 1884 immediately wrote to Mittag-Leffler, expressing her concern that the approaching birthday festivities might fan hostilities among various German mathematicians and might cause more grief than pleasure to Weierstrass. The presentation of the bust was not uncontested, because some regarded it as a slight to Kummer, who was not similarly honoured on his 70th birthday in 1880. In his letter to Mittag-Leffier of 16 August 1885,

The reader can easily imagine the joy of discovery I felt when leafing through the pages of the album, which was missing for decades only to re-emerge here in Berlin, on our very doorstep, as it were, not far from the Academy and the University where Weierstrass once worked. which was mentioned earlier, Itzigsohn writes that the desire for a "quite general" celebration "would probably not be completely realized, not all personalities .being guided by the same feelings as ourselves. I believe there will soon be a violent clash with opponents of Weierstrass's school; the air is already charged with it." (Also see Mittag-Leffier's letter to Kovalevskaya dated 30 August 1885 ([3], p. 118).) One can perhaps feel the stresses Weierstrass was exposed to when in a letter to Kovalevskaya dated 22 September 1885, just a few weeks before his 70th birthday, he wrote "I have decided to leave Berlin and settle in Switzerland.* I would probably have been gone already if extraordinary circumstances had not still kept me here; but I could not have anticipated that this would be such a drawn out affair [ . . . ] It might happen without much ado that I must still be here in October. You can guess what made me decide, finally, to leave this place for good" ([1], pp. 126-127). The only thing that remains to be told is how the birthday celebration went off. Weierstrass, at the beginning of his report to Sonya Kovalevskaya (who did not take part in the celebration), writes: "First of all I should like to admit to you without any reservations that the celebration of my 70th birthday organized by m y older a n d y o u n g e r listeners gave me m u c h pleasure indeed. Without any official trappings--only the minister of culture sent me a semi-official congratulatory m e s s a g e - - t h e festivities, even if not entirely devoid of exaggeration, assumed the form of a manifestation without anydissonances, proving that those taking part did so with their hearts" (Letter dated 14 December 1885 ([1], p. 129)).

Bibliography 1. Pisma Karla Veiergtrassak Sofye Kovatevskoy. Published and edited by P. Ya. Ko~ina-Polubarinova, Moscow (1973). 2. P. Ya. Ko~ina, Sofya Vasilevna Kovalevskaya, Moscow (1981). 3. Perepiska S. V. Kovalevskoy i G. Mittag-Leffiera. Published by A. P. Yu~kevi~. Edited and commented on by P. Ya. Ko~ina, E. P. O~igova. Nau~noye nastedstvo, vol. 7, Moscow (1984). 4. W. Killing, Karl Weierstrass. Natur und Offenbarung 43 (1897), 705-725. 5. E. Lampe, Zur hundertsten Wiederkehr des Geburtstages von Karl Weierstrass. Jahresber. Deutsch. Math.-Vereinigung 24 (1915), 416-438. 6. D. H. Kennedy, Little Sparrow: A Portrait of Sophia Kovalevsky. Athens and London (1983). 7. G. Mittag-Leffler, Die ersten 40 Jahre des Lebens von Weierstrass. Acta math 39 (1923), 1-57. 8. A. P. Yu~kevi~: Georg Cantor und Sof'ja Kovalevskaja. Ost und West in der Geschichte des Denkens und der kulturellen Beziehungen. Quellen und Studien zur Geschichte Osteuropas vol. 15, Berlin (1966), 683-688. Karl-Weierstrass-Institut fiir Mathematik Akademie der Wissenschaflen der DDR DDR-1086 Berlin, German Democratic Republic

* AS it is w e l l k n o w n , W e i e r s t r a s s d i d n o t d o so. THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

25

William Marshall Bullitt and His Amazing Mathematical Collection Richard M. Davitt

One of the world's most extraordinary collections of first editions of important mathematical works is contained in the William Marshall Bullitt Mathematical Collection of the University of Louisville. This article tells the story of how William Marshall BuUitt assembled this collection of rare mathematical manuscripts.

William Marshall Bullitt was a native of Louisville, Kentucky, where he was born on 4 March 1873. His familial connections to American and Kentucky history rival those of the principal characters of John Jake's The Kent Chronicles. Bullitt's ancestors Col. Thomas Bullitt, Dr. Thomas Walker, and Col. William Christian were prominent figures in eighteenth-century American history. Additionally, through his great-great grandmother, Anne Henry, William Christian's wife, William Marshall was a great-great nephew of both Patrick Henry, first governor of the Commonwealth of Virginia, and John Marshall, first Chief Justice of the United States Supreme Court. His great-grandfather, Alexander Scott Bullitt, was the first speaker of the Kentucky Senate and later first lieutenant-governor of Kentucky, serving from 1792 to 1804. All told, five Kentucky counties--Bullitt, Henry, Marshall, Logan, and Christian--are named after direct ancestors of William Marshall Bullitt. Bullitt was educated in private schools in Louisville and early in his scholastic career showed his prowess in mathematics by winning a gold medal in that subject at Trinity Hall. In his seventeenth year he was enrolled in LawrenceviUe School in New Jersey where he completed his preparation for college. In Sep26

tember 1890 he entered Princeton as a "special" student (one having had no Greek) and majored in mathematics, in which he continued to be very proficient; he graduated with a B.S. degree in 1894. Just one year later BuUitt obtained his law degree from the University of Louisville. Giving evidence of the remarkable energy which characterized his entire professional career, that very same year he passed his "bar examinations" and began his practice of law in his father's firm. Jillson [13] states:

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4 9 1989 Springer-Verlag New York

In 1956, the concealed wall safe to which W. M. Bullitt is pointing was burgled of $274,000, one of the largest cash thefts from an American home. The safe is located behind a paneled wall in the library at Oxmoor. Reprinted with permission from The Louisville Times.

The mental and personal traits which made his college paths through mathematics, astronomy, and physics highly creditable and attractive, when applied to problems, procedures, and practices of the law, within a few short years took him to the most select practice of corporate law in which his personal success was so marked as to bring him some of the most desirable clients in this county Uefferson County, Kentucky] and to provide the foundation stones of a very considerable personal fortune and widely recognized professional reputation. The "professional reputation" to which Professor Jillson refers i n c l u d e d such a c c o m p l i s h m e n t s as serving as the Solicitor General of the United States under President William Howard Taft. He was also a trustee of the American Surety C o m p a n y of New York, the Mutual Life Insurance C o m p a n y of New York, the Kentucky Home Life Insurance Company, the Carnegie Endowment for International Peace, and numerous other corporations at the state and national

levels. At the time of his death in 1957, only four or five lawyers had ever argued more cases before the United State Supreme Court than Bullitt did. On the mathematical front, Bullitt became an authority on insurance law and penned a number of precedent-setting pamphlets in actuarial mathematics. Over the years, he maintained an active membership in the American Mathematical Society, the Mathematical Association of America, and the American Association for the Advancement of Science. He was a close personal friend of m a n y famous twentieth-century mathematicians and scientists, such as Harlow Shapley, longtime director of the Harvard Observatory, R. D. Carmichael of the University of Illinois, G. D. Birkhoff, G. H. Hardy, and Albert Einstein. Although Bullitt often proclaimed himself to be just an amateur mathematician, he was more than that. He was familiar with the fundamental ideas of m o d e m matheTHE MATHEMATICAL INTELLIGENCER VOL, 11, NO. 4, 1989 2 7

matics and knew the lives of those who had developed the subject. Much of his correspondence with mathematicians deals with nontrivial mathematical topics about which Bullitt was seeking further enlightenment. Bullitt liked to tell how he once found an error in Bertrand Russell's Introduction to Mathematical Philosophy. It seems that Russell had quoted another mathematician's works incompletely. On a similar note, he had enough confidence in his mathematical ability to have offered Einstein an alternate, more concise mathematical explanation than the one Einstein himself had given concerning a particular point in one of Einstein's commentaries on his theory of relativity. Einstein subsequently acknowledged the reasonableness of Bullitt's point and thanked him for his suggestion. Bullitt held to the classical view of the teaching of mathematics in the schools--that it was an inducement to clear thinking. He believed no one was educated without knowing algebra, geometry, and some trigonometry. Over the years he participated in a number of projects designed to promote mathematics in the schools and in the public eye. Bullitt boasted that one of the first questions he asked any young lawyer applying for a job with him was "How much mathematics have you had?" Although Bullitt's home base of operations was Louisville, he spent a lot of time in New York and on the East Coast. During his stays at the Union Club in New York City in the 1930s and 1940s, he took numerous sidetrips to the Institute for A d v a n c e d Study at Princeton and to Harvard, where he was a member of the Visiting Committee to the Department of Mathematics for a period of time. As a result of his visits to the Institute and his contacts with Oswald Veblen, Einstein, John von Neumann, and others there, he was instrumental in obtaining a faculty position at the University of Louisville for the Czechoslovakian mathematical analyst Karl Loewner during World War II. As a token of his appreciation for this aid to Loewner, Einstein presented Bullitt with autographed presentation copies of five issues of the 1905 edition of Annalen der Physik in which Einstein had published his original findings in quantum theory and special relativity. (The Bullitt-Einstein c o r r e s p o n d e n c e and these autographed copies of Einstein's work today comprise an important and unique portion of the William Marshall Bullitt Mathematical Collection in the Rare Book Room of the Ekstrom Library at the University of Louisville.)

The 25 Greatest Mathematicians The story of the collection itself also begins with one of Bullitt's forays into East Coast scientific/academic circles. During the celebration of the Harvard Tercentenary in 1936, Bullitt was visiting with Shapley and Hardy. Hardy, as Bullitt told Landau [15], liked parlor 28

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

games. One of his favorites was to challenge a gathering to make up a cricket team of famous poets. Then by comparing the relative merits of the men who made up the two teams, the group could determine which team would be declared the victor. Bullitt said that he challenged Hardy to fight it out similarly with mathematicians. Hardy did not take up Bullitt's challenge at that time because, he explained, although there was no question that the three greatest mathematicians of all time were Archimedes, Newton, and Gauss, there would be quite an argument as to how to rate the others. Bullitt's interest, however, was piqued. He and his family had long been collectors of rare books in fields such as history, horticulture, horseracing, and literature. To house his book collections, Bullitt had added a magnificent two-story library wing, designed by architect Burrell Hoffman, to the mansion on the ancestral family estate, Oxmoor, outside of Louisville. As possible n e w acquisitions for this library, his friend A. S. W. Rosenbach, the world-famous book dealer, had already convinced Bullitt that science and mathematics books were likely to prove more valuable as investments than books on Napoleon or Henry Clay. Thus, the scene was set for Bullitt to begin purchasing rare mathematical editions for his collection, but first he set out to determine who were the 25 top mathematicians of all time and what was the most important work of each. In the fall of 1936, Bullitt was quite aware that E. T. Bell, another of his mathematical acquaintances, was in the process of readying his famous Men of Mathematics for publication. Consequently Bullitt wrote to Bell enlisting his aid in determining a canonical list of famous mathematicians and their most important work(s). BuUitt provided Bell with a "Tentative List of 25 Greatest Mathematicians (Excluding all Living Mathematicians)" which, not surprisingly, closely paralleled the table of contents of Men of Mathematics. The list consisted of 25 names in essentially chronological order: 1. Eudoxus, 2. Apollonius, 3. Archimedes, 4. Diophantus, 5. Descartes, 6. Fermat, 7. Pascal, 8. Newton, 9. Leibnitz, 10. Daniel Bernoulli, 11. Euler, 12. Lagrange, 13. Laplace, 14. Gauss, 15. Cauchy, 16. Abel, 17. Galois, 18. Lobatchewsky, 19. Jacobi, 20. Weierstrass, 21. Riemann, 22. Cayley, 23. Hermite, 24. Dedekind, and 25. Poincar4, and a supplemental list of another 12 mathematician/scientists. Both m e n agreed that it was wise to consider no living mathematicians in compiling the list of "greats.'" In his reply, Bell stated that his criterion for mathematical greatness was that "a man should have started something new that had to be of great and permanent interest to later generations of mathematicians." On this basis Bell asserted that Galois, for example, did indeed belong on the list, as did Zeno, Bolyai, and Cantor from the supplemental list, but that PoincarG

Dirichlet, and Kummer, among others, did not. Bell also excluded Copernicus, Galileo, and Kepler from the supplemental list as being (from the modern point of view) primarily astronomers and physicists. He also struck Euclid from that same list because he "certainly was not in a class with the 25 listed." Consciously or subconsciously appealing to the book collector in Bullitt, Bell added the remark that first (or first printed) editions of the works of Euclid and these astronomers were indeed rare and extremely valuable. A typical entry from Bell's eight-page annotated memorandum to Bullitt concerning the list is that regarding Lagrange. It reads: 12. LAGRANGE (1) Recherches sur la methode de maximis et minimis, Turin 1759 (in Miscellania Taurinensia).--Landmark in calculus of variations. (2) Sur les probl~mes indet&mines du s~conde degr~ (ibid.)Epoch in theory of numbers. (3) Trait~de la r~solution des ~quations num&iques de tous les degres (Paris. 1798).--Began the theory of equations. (4) Mdcanique analytique, Paris, 1788.--By most people judged his masterpiece, but the other items were equally important in the development of pure mathematics. The detail found in Bell's response to Bullitt clearly indicates how seriously he took Bullitt's request for assistance with this project. In 1937 Bullitt also sent his "Tentative List" and copies of Bell's memorandum to mathematicians, historians of science, and booksellers all over the world, requesting their reaction and input. He sought help from Harold Hancock, a longtime friend at the University of Cincinnati, R. Courant, J. von Neumann, G. H. Hardy, E. V. Huntington, G. D. Birkhoff, L. P. Eisenhart, O. Veblen, A. Dresden, H. W. Turnbull, D. E. Smith, R. C. Archibald, G. Sarton, T. L. Heath, L. C. Karpinski, the Library of Congress, the National Academy of Sciences, and such famous booksellers as H. Sotheran, Ltd., London; Maggs Bros., London; Gauthier-Villars, Paris; Ilse Brauer, Berlin; Friedlanders, Berlin; Jake Zeitlin, Los Angeles; and the previously mentioned A. S. W. Rosenbach of Philadelphia, amongst others. Almost everyone was enthusiastic about helping Bullitt with his project. One of the most interesting replies was that of G. H. Hardy and Harold Bohr (see box at right). G. Sarton, then editor of Isis, was perhaps the only individual noted above who pleaded that he could not help Bullitt with his project. Sarton replied wearily that he had neither the time, the patience, nor the good health to try tO determine the 25 greatest deceased mathematicians. He had received too many similar queries asking him to rank the 10 greatest physicists and the like. Additionally, he believed that the question really could not be answered because he THE MATHEMATICAL INTELLIGF_d'qCERVOL. 11, NO. 4, 1989

29

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Sarton's response and the expanded lists supplied by others combined with Bullitt's passion for collecting rare books so that soon Bullitt no longer restricted himself to a list of exactly 25 greatest mathematicians nor to just their first and greatest works. When the collection was finally catalogued for his estate after his death, it consisted of the principal works of no less than 60 "greats" in 370 separate title entries.

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dentin addat parumper ip, fi D. Ex hoc maniftfffi ef~ qu6d cureterra fi1r in ,, maximum r Solis apogeum, in o minimum: in mcd/~aautcmcircumfcrmtijs ipfius, a epi, r162 facictipfum apogeum przccdcre udl~qui,aud'tum dimi nummfic, maius ant minus,& tk momm apparcrr diucrfum, mantca dc epicyclo& r162162 dcm6f~ratumr au, tern a ~c./rmmfcrcmia,&in ~cmtro rcfumaturcpicydus, & c6, mxa c zcxtendaturin rec'tam lincam c ~~, crit~ ~rz n angulus ~qualis ipfia c ~, proptcr rcuolutionum paritatcm,lg(tur ut fu peclus dcmonPcrau/mus,Dfignum dr r r homoccntro a s coa~qualemin z.cmtro,ar diffantia c z.,quaeip tip ~fuerit ~qualis, s quo~ fuum r fecundum diffan t/am c i. M~qualcm ip~ ~ o,,& o fimilitcr fr ~ ~, & c a d/lhmia~ xquaks,lmcrca fl cmtrfi tcrr~ Jamcmcnfum fucrit u tr Diagram showing Copernicus's revolutionary concept of epicycles in a sun-centered system (De Revolutionibus, page 91).

felt that there was no valid measurement for mathematical genius. He suggested that Bullitt try to answer questions such as "'Who are the 25 greatest American lawyers . . . . the 10 most beautiful w o m e n in the world . . . . the 15 toughest criminals?" to see h o w he viewed Bullitt's request to him for the mathematicians' list. Bullitt thanked Sarton for his "'nice note" and agreed with him that questions such as he and Sarton had posed could not be answered with any finality. However, he did suggest, in lawyer-like fashion, that the opinion of a majority of a jury of 10 to 15 thoroughly competent experts could at least approach some final determination and that this was the procedure he was following in determining w h o constituted the 25 greatest deceased mathematicians. Bullitt closed his note with an open invitation for Sarton to please visit him that summer at his vacation house in Stockbridge, Massachusetts. 30 THE MATHEMATICALINTELL1GENCERVOL. 11, NO. 4, 1989

The Collection In 1938 Bullitt and his wife traveled extensively in Europe visiting London, Berlin, Moscow, and Paris, where Bullitt's cousin, William Christian Bullitt, was American Ambassador to France. Through his political and bibliographic contacts, Bullitt was able to purchase m a n y notable and priceless editions of rare mathematical/scientific works at a time when an exceptional number of such items were being placed on the market. At this time, Bullitt also w e n t to great lengths to complete a subcollection of his mathematical collection dedicated to Galois, whose brief and extraordinary mathematical career had long held strong emotional appeal for him. He sought to obtain all that could be collected by or about Galois, even visiting his birthplace in Bourg-la-Reine and securing photostats from the National Archives in Paris when the original documents were unavailable to him. The Galois material in Bullitt's collection eventually came to be regarded as

"'Strangely enough, anyone wishing to write about Galois in Paris would do well to journey to Louisville, Kentucky.'" the most complete such source of information in the world. The last thing Leopold Infeld did w h e n researching his fictionalized biography of Galois, Whom the Gods Love, was to spend several days working with the collection in Louisville as Bullitt's guest [12:20-23]. Infeld acknowledges his debt to Bullitt and his Galois collection in the preface to that biography where he notes "Strangely enough, anyone wishing to write about Galois in Paris would do well to journey to Louisville, Kentucky." [11:xiv]. By the end of World War II, Bullitt regarded his collection as being essentially complete except for the acquisition of a copy of Abel's eight-page Mdmoire sur les

~quations alg~briques ou on d~montre l'impossibilitd de la r~solution de l'dquation gdn~rale du cinqui~me degr& Finally, in 1951, after a fourteen-year pursuit, through the unstinting efforts of Veblen, Paul Heegaard (a distinguished Norwegian mathematician), and Maggs Brothers, Bullitt was able to purchase a copy of the privately-printed 1824 pamphlet from the widow of a

Professor of Actuarial Mathematics at the University of Oslo. It is thought to be only one of three copies in existence, the other two to be found in G6ttingen and in the Mittag-Leffier Library at Stockholm. In a letter to Bell dated 10 December 1951, Bullitt writes I received today, in perfect order, the original Abel Md-

moire. It is true that I had to pay an outrageous price [$500] for it, but at least I have added almost the last item that I needed to complete my collection. I have Bolyai on Non-Euclidean Geometry, but Lobatchewsky's articles in the Kasan Messenger are still out of my reach. There were four articles, but the old magazine issues of that day are all gone, and I have no idea what Library even has the set. You may not be very proud of it, but still you are not the Godfather but the actual Father of my collection. I would like to get out a little printed catalog of them; and if I can manage to get it out, it will be dedicated on the Title Page to you. Bullitt never did have the collection catalogued. Instead, he kept it in his law offices in downtown Louisville and delighted in showing it to anyone interested in mathematics. In 1953, he sent the core of the collection to Harvard for an exhibition. Everyone, including his lawyer son, Thomas W., expected that Bullitt would bequeath the collection either to Harvard or to Princeton, his alma mater. However, in a handwritten will executed on 6 March 1951, he gave his law library to his son and directed that his other library and all his personal effects go to his wife. In 1958, a year after William Bullitt's death, Mrs. Bullltt gave the entire collection and related correspondence to the University of Louisville. The first items formally presented to the university by her were a copy of Newton's Principia Mathematica, Abel's M~moire, and the five pamphlets (author reprints) by Einstein. The Newton work was published in 1687 and is an original copy of the Principia known as the Halifax copy. It contains Newton's own manuscript corrections and, as expected, is one of the more valuable items in the collection. During the following two years, the university received about one hundred of the more valuable books in the collection. The remainder of the collection arrived in 1978. Two file boxes of correspondence relating to the compilation of the collection were rescued from the attic at Oxmoor by the author and Mr. Bullitt's personal secretary, Lurline H. Jochum, in 1980 and placed in the collection. The correspondence was arranged in chronological order beginning with the year 1936 and continuing to 1952. These files are meticulously complete containing the original letters, often handwritten, of Bullitt's correspondents, copies of Bullitt's letters to them, catalogues, telegrams, book orders, bills, and the like. Multiple copies of Bell's checklist and the Hardy-Bohr

letter (often with Bullitt's and others' handwritten notes scrawled on them) are also in the files as is a detailed inventory of the Mathematical Collection prepared by Mrs. Jochum after Bullitt's death. The inventory runs 111 typed legal pages. Many of the details found in this paper are based upon facts to be found only in this extraordinary set of reference materials. From a monetary viewpoint, the three items originally presented to the University of Louisville by Mrs. Bullitt are among the most valuable works in the collection. However, as Bell foresaw in making up his original checklist in 1936, the most valuable books acquired by Bullitt are indeed those of the early astronomers and Euclid. These include the Narratio Prima of Rheticus (Danzig, 1540), which contains the first announcement of the Copernican system of heliocentricity by his pupil and is one of five copies known to be extant; the first and second editions of Copernicus's De Revolutionibus Orbium Coelestium (1543 and 1566); a near-mint copy of the first printed edition of Euclid's Elements, which was the first mathematical book of

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31

ei ht-page p l' Memo tr. e Title page o~ am p h le t , an A b privately p rt te ; b~ Abel in 1824 g . n

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p .S ,) ;; ' T C" IlC Con I' (p + p. S{ ) etinn :I III -; i; = = v.,lt-urs diff Illarq1lont q" crenlc<. ii , e III CSI Ill ('I ll do ne ' no m hr q" e m = .' e pr el ll ic r. i en re . :'='1+<J.J O n au ro po -+<J'''''+ r ro ns eq ne 'l,J''+'1. nl 'I 9. II , N Y·=~P+P1SI C. cl on t ):lI ,k s fo nn io + (p+J'.SI co ns eq ue nt n. spnetri'Jl )-~. de s fo nn io lC:s de Y . ll s ra li on ne Y . y , elC • H te eq ua ti ll es de rt b '. el pa r on av ec r" 'ju c rl el .. . al io n pr op os ee I'r im ':e pa r E n eo m hi na , on en tir un e (oncli01l lll er a Ia vnlcnr ra li on n" "e tio ll es t to de :. rt {J c d ey cx · uj ou rs rc du tI el "; O et ih le n la r IlIIe tell I' fo rm e fonc. y = f' + R : + 1' , 1/ : + I" on P R 1' , 1' , el p . so R: +p .l I~ nt ,I,·s fo nc de s fonclio tin ns ,Ie I.• ns ration1lc f" rm e 1" 1' . ·Il, s ,Ie· " (, S" I' 1' . el C' tI el p. .\: "lalH In = l' .' " . +( D e C'eue va le1lr clc ," on (& ... -: +( (3 oit ~I·J+f(':Y lirt· .. +(fY~) = J) + 1' itA +1 (1 +( .( : + , S}·~ O r I", pr cm « + I = U ie r mem1,r" ;, 12 0 v.,l'·IIr ce ul em el lt S ,li ff ':r rn te 10 ; po r co s CI Ie seco ns e' ln el ll Y no nt de tr lld m em hr lie pe nt :lv ou ve r; mai e oir I.. "form s no ns ov on e q" e 1I01lS :lv oi r cc ue s ,le m on tr '; fo rm e si I'l \'~' qu e y -'l do ',ation pr op il neC'cssnire do ne : os ce ('Sl re m.cnt sol1l1,Ic, na ils co nd ll on "n ('Sl imposs ' ih le de reso Ie dll C'ill'l uclre pa r cle "i"lne dr S ''' s rndic,lII" :' r c' I" ' tion gC II su it imm 1lr,,,. edialell1cnt ,Ie ce II'l-or hi e de re so clne '1"'il ,.< ll dr e pa r \ ,II 'S ,Ie Inelne radicaux Ir superieurs il ll po "i . s "'I",llio1ls 3U dnqu ictnC'. r~"crale . ,,~. ,: ?gr~s

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THEMATHE "'T IC A I.. IN TE U GENCER VOl... 11 '0 .4 . 1989 I.. mr> .

any consequence to be printed, in original Venetian binding (Ratdolt. 1482); and the first English translation of the Elements (1570), the monumental Billingsley translation. In all, the collection contains 8 copies of the Elements published between 1482 and 1575, together with 1661 and 1703 editions of that work. In this article it is impossible to list all the first editions of the preeminent works of the "25" greatest deceased (as of 1936) mathematicians of all time that are to be found in the collection. A sampling of the items that I have judged to be of interest to mathematicians follows. The collection contains the aforementioned pamphlet of Abel and the Halifax copy of Newton's Principia; the first Latin translation of the Conic Sections of Apollonius (Venice, 1537); Archimedes's Opera (Basel, 1544); the 1832-33 treatise of Farkas Bolyai and its 26-page appendix on non-Euclidean geometry by his son Janos Bolyai; the first (1637) e d i t i o n of Descartes's Discours de la M ~ t h o d e . . . with its third appendix La Geomdtrie; multiple first editons of Euler's monumental eighteenth-century classics in analysis; Volume 11 (1846) of the Journal de Mathdmatiques Pures et Appliqudes ("Liouville's Journal") in which Liouville collected and published all the memoirs and manuscripts of Galois he knew to be extant 14 years after Galois' death; a presentation copy (1799) of Gauss's dissertation in which he proved the Fundamental Theorem of Algebra, the first (1801) edition of Gauss's single greatest publication, Disquisitiones Arithmeticae, and more than 25 other first editions of Gauss; the four works of Lagrange cited by Bell in his original memo; the Acta Eruditorum Anno 1684 in which Leibnitz first published his inventions of calculus in the journal he founded; an 1866 French translation of Lobatchewsky's 1826 and 1840 publications in the Kazan Messenger of this discovery of non-Euclidean geometry; a wrapper editon (1851) in slipcase of Riemann's brilliant dissertation in the field of complex function theory in which he presented such fundamental concepts as the "Cauchy-piemann equations" and a "Piemann surface"; and rare first editions of Weierstrass's seminal works in his program to "arithmeticize analysis." In closing, I simply wish to reiterate that Bullitt's own words in his letter to Bell show that he himself considered his collection to be as complete as was realistically possible when he obtained Abel's Memoir in 1951. At that point he had purchased virtually every publication that Bell and others had suggested he obtain for his collection, except perhaps for Fermat's own copy of Bachet's 1621 translation of Diophantus's Arithmetica in which Fermat made his famous marginal notes about his proof 'of his last "theorem." A glance at the 370 separate title entries in the checklist for the collection certainly reinforces Bullitt's judgement. The mathematical community and the University of Louisville owe an inestimable debt to William Marshall

Bullitt for the legacy he bequeathed to t h e m - - t h e Mathematical Collection which bears his name.

References 1. Eric Temple Bell, Men of Mathematics, New York: Simon and Schuster (1937). 2. "Bullitt Estate Set at 1.2 Million," Courier Journal [Louisville, KY] (CJ), 30 October 1957. 3. "Bullitt Funeral to be at Oxmoor Tomorrow Morning," CJ, 4 October 1957. 4. Thomas W. Bullitt, Personal interview, 27 January 1980. 5. - Oxmoor and the Bullitts, 1685-1980, Private publication based upon a 7 April 1980 address of Mr. Bullitt to the Filson Club, Louisville, KY. 6. William M. Bullitt, Personal correspondence. WMB Mathematical Collection, University of Louisville, Louisville, KY. 7. Ken Chumbley, "U of L's 'extraordinary' Bullitt collection," Potential [University of Louisville], Vol. 5, No. 24, 3 November 1979, 6. 8. Howard Eves, An Introduction to the History of Mathematics, Fifth edition, New York: Saunders College Publishing. (1982). 9. Joel Gwinn, "Epicycles of Ellipses? A Visit to the Bullitt Collection," Library Review [University of Louisville], Issue 22, June 1975, 6-9. 10. Sheila Strunk Harlan, "Rare and Valuable Books Donated to Library Are Bringing International Fame to the Collection," CJ, 1 February 1959. 11. Leopold Infeld, Whom the Gods Love: The Story of Evariste Galois, Reprint, with new introductory material, of the ed. published by Whittesey House, New York, Reston, VA 1978. 12. , Why I Left Canada: Reflections on Science and Politics, Pyenson and Lewis ed., Toronto: McGill--Queens U. Press. (1978), 20-23. 13. Willard Rouse Jillson, "William Marshall Bullitt, 1873-1957," Register of the Kentucky Historical Society, Vol. 56, No. 2, April 1958, 208-209. 14. Lurline H. Jochum, Personal interviews, 14 February, 28 February, 7 March, 27 September, 12 October 1980. 15. Joseph Landau, "The William Marshall Bullitt Mathematics Collection" and "A Personal Reminiscence of Mr. Bullitt," written privately for the University of Louisville Library at the personal request of Dr. Wayne Yenawine, Librarian, 30 May 1967. WMB Mathematical Collection, University of Louisville, Louisville, KY. 16. Ron M Linton, "Modest Baron of Oxmoor, Bullitt's Life Anything but Dull," CJ. 13 August 1957. 17. "Marshall Bullitt Dies at 84, Noted Lawyer Owned Oxmoor," Louisville Times, 3 October 1957. 18. George McWhorter, Checklist of the Bullitt Collection of Mathematics, (with covering memorandum), WMB Mathematical Collection, University of Louisville, Louisville, KY. 19. James S. Pope, Jr., "Bullitt's Library on Mathematics, Valued at $37,000 [sic], is Given U.L.," Louisville Times, 26 June 1958. 20. "Rare Book Chase: At the world's biggest book-and-art auction house, Parke-Bernet Galleries in New York," Vogue, 15 February 1952, 94-97, 117-118 (Photo of WMB on p.97). Department of Mathematics University of Louisville Louisville, KY 40292 USA THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

33

When Is a C Function Analytic? R. P. Boas

Students often wonder w h y their teachers insist on proving that the remainder in a Taylor expansion approaches zero before they will accept that the series represents the function from which it was obtained. The students tend not to be enlightened by the usual practice of invoking the Lagrange form of the remainder, with its mysterious unspecified intermediate point. It is not only students who are confused: a quite distinguished mathematician (A. Pringsheim) once made a mistake about remainders. This article is the story of the legacy of that mistake, which remained unnoticed for forty years. The Taylor series of a C" function f can have two kinds of singular behavior: the series may diverge except at its center, or it may converge, in a neighborhood of its center, to a function that differs from f in arbitrarily small neighborhoods of the center. There are many examples of the first kind of singularity, al-

34

though they are not easy to show to an elementary class. For the second kind, the standard example is the function F defined by F(x) = exp( - 1/x2) for x < 0; F(x) = 0 for x I-- 0. (As late as 1928 Osgood [5], p. 125, felt constrained to emphasize the triviality of the objection that F "is not really a function" because it is not defined by a single formula.) If we write the Taylor series for F a b o u t various points x, the radius of convergence p(x) of the series approaches 0 as x ~ 0 from the left. Suppose, however, that a function f has the property that p(x) is b o u n d e d away from 0: 0(x)/> ~ > 0 for all x in an interval. Is such a function real-analytic (equal to its Taylor series), or might it have a singularity of the second kind? This question was apparently first asked by Pringsheim [6] in 1893. He presented a proof that such a function f is necessarily analytic; I quote it in somewhat modernized notation. Suppose that

v a E MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4 9 1989 Springer-Verlag New York

oo

fC")(t)r"/n! n=O

converges for a ~ t ~ b and 0 ~ r ~ rl; then lim f(n)(t)r"/n! = 0

(A)

for a ~ t ~ b and 0 ~ r ~ r v Hence if 0 < s < r~, then

lira f(")(t + Os)s"/n! = 0 for a ~ t + 0s ~ b.

n---~oo

(B)

From (B), Pringsheim concludes that the remainder in the Taylor series of f about t approaches 0, so f is analytic on (a, b). Did you notice the fallacy? It is rather surprising that Pringsheim, who was an enthusiast for uniform convergence, failed to notice that nothing tells us that the limit in (A) or (B) is uniform in r (or 0), and hence that his proof was incomplete. To put the point more precisely, to say that p(x) > 0 is to say that lim sup {~(n)(x)]/n!}l/"< oo. yl---~oo

(1)

P r i n g s h e i m ' s hypothesis is lim sup {~(n)(x)[/n!}l/n< M,

(2)

for some finite M independent of x. It is well known

In 1932 J.J. Gergen had to give a course on Fourier series w i t h o u t using uniform convergence, because the H a r v a r d m a t h e m a t i c s dep a r t m e n t considered the concept to be too diff i c u l t f o r undergraduates. (and follows from the Lagrange form of the remainder in Taylor series) that the condition that ~(")(x)/n![1/" is uniformly b o u n d e d in a neighborhood of each point of an interval is sufficient for f to be analytic. Even today, uniform convergence is sometimes felt to be a difficult concept; fifty years ago it was considered even more difficult. In 1932 J. J. Gergen had to give a course on Fourier series without using uniform convergence, because the Harvard mathematics department considered the concept to be too difficult for undergraduates. I was one of those undergraduates. I had happened to read Pringsheim's paper, and was intrigued by the theorem. At first reading, I accepted Pringsheim's proof. However, duffng the summer vacation I tried to reconstruct the proof, but couldn't make it work. Eventually I gave up, drove 90 miles to Harvard, and looked up the proof. Then I read it more carefully, and realized that it was fallacious.

Alfred Pringsheim What do you do when you find a flaw in the proof of an interesting theorem? You might well write to the author, but that is not something that an undergraduate would be likely to do. In any case, I would probably have assumed that Pringsheim was dead after 50 years (actually he was only 82 in 1932, and he lived until 1941). But certainly you would try to find a correct proof. In 1932-33 I knew only the amount of set theory that occurs in introductory courses in real and complex analysis. In particular, I had never seen the Baire category theorem, but I managed to discover it for myself (only for the real line; it was at least two years before I was to be introduced to complete metric spaces). By using Baire's theorem, I was able at least to prove that if p(x) is strictly positive for every x in an interval J, then f is analytic in J except for a nowhere dense d o s e d subset. This result was new, and is interesting in itself, because it shows that a C= function cannot have singular points of the second kind at every point of an interval, or even at the points of a dense subset. [ realized that such a powerful result as Baire's theorem could hardly be new, but it was a long time before I could find it in a book. When I described the theorem to members of the faculty, they didn't recognize it either; however, it is quite likely that I didn't explain it very well. It was not until early in 1934 that I was able to find a convincing proof of Pringsheim's theorem. I worked it out in a garden on the island of Madeira, which I was visiting for nonmathematical reasons. Then I learned THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989 3 5

how difficult it is to convince people of the incorrectness of an alleged proof of a correct theorem. When I published my proof in 1935 [1], I ought to have included an analysis of Pringsheim's proof, but I was too naive to do so. I seem to have thought that of course all those experienced mathematicians out there would see that it was incorrect as soon as they looked at it. This was a fallacy: the reviewer for the Jahrbuch didn't believe that Pringsheim's proof was incorrect, and said so. I was eventually able to convince him, and he published a correction to his review. The reviewer for the Zentralblatt was more cautious, but I think he didn't believe me either. My original proof was unnecessarily long-winded; there is a more condensed proof in [7], and another proof [9] by Z. Zahorski, who independently discovered and corrected Pringsheim's error. Also there is a m o d e r n p r o o f of P r i n g s h e i m ' s t h e o r e m b y M. J. Hoffman and R. Katz in [3]. Recently A. Boghossian and P. D. J o h n s o n [2] rediscovered Pringsheim's theorem. They found numerous interesting generalizations a n d a n a l o g o u s results, a n d p l a c e d the theorem in a broader setting. Any future work in this area must start from [2]. Here I shall only outline a proof in order to indicate the principles on which all existing proofs depend. The argument actually establishes a more general result. THEOREM. Let f be C | on an open interval J and let p(x) be the radius of convergence of the Taylor series off about the point x. Suppose that (1) p(x) > 0 at each point x of J, and that (2) for every point p of J we have lim infx__,p p(x)/Ix - Pl > 1. Then f is analytic in J. This contains Pringsheim's theorem because if p(x) 1> 8 > 0 then l i m infx._~p p(x)/ix - p] = oo. In informal language, the hypotheses of the theorem say that if there is a point p at which f is not analytic then the interval of convergence of the Taylor series of f about points close to p must not extend beyond p. The proof falls naturally into several steps. I. Under hypothesis (1) alone, f is analytic on a dense subset of J. This is a straightforward application of Baire's theorem. The hypothesis (1) means that 1/0(x) = lim sup~_,|

TM <

~.

This amounts to saying that there is a finite function ix such that

[f(")(x)l ~< n![,(x)] ~, (n = 1, 2 . . . .

).

Let Em be the subset of J on which m ~< la(x) < m + 1, so that J = Um~oEm. Baire's theorem says that some Em fail to be nowhere dense in ]; that is, there are an integer m and an interval K C J such that Em is dense in K. Then (*) ~(~)(x)] ~< n! (m + 1)", n = 1, 2 . . . . .

for x E Era.

For x E K N E , the inequality (*) holds by continuity. 36

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

Hence the inequalities in (*) hold for all x E K. This implies that f is analytic in K. Arguing similarly for intervals in J \ K , we have f analytic on a dense subset of J. Let H be the relative complement of this open set. II. H contains no isolated points. This step is not essential, but it brings out the role of the second hypothesis in the theorem. Suppose that y is an isolated point of H. Then y is a common endpoint of two complementary intervals of H, say J1 on the left and J2 on the right. If z 1 is a point of ]1, sufficiently close to y, the Taylor series of f about zl converges in an interval that extends into J2; similarly for z2 E J2. We can calculate the Taylor coefficients

I learned how difficult it is to convince people of the incorrectness of an alleged proof of a correct theorem. of f at y from the Taylor series of f about zl, and this series represents f in an interval to the left of y. The coefficients of the same series (about y) can also be calculated from the Taylor series of f about z2, and this series represents f in an interval to the right of y. Thus f is represented, near y, by its Taylor series centered at y. In other words, y does not belong to H, contrary to assumption. Consequently H is a perfect set. III. Since H is closed, we can regard it as a complete metric space and apply Baire's theorem to it. We then find that there is a closed subset P of H and a finite h such that ~(n)(x)l ~< n! h ", n = 1, 2 . . . . .

(3)

for all x in the intersection of P with some interval Jv IV. We n o w have (3) satisfied at the points of a nowhere dense closed set P1. It is easy to see that f is not only analytic in each complementary interval of P1, but is in fact represented in these intervals by its Taylor expansion about an endpoint r of the interval. We n o w need to estimate ~(n)(x)/n!l TM in the complementary intervals to P1 in J1. This can be done by direct computation with series (see [2]). Alternatively, we can use an idea of Salzmann and Zeller [7]: extend f to the complex plane and apply Cauchy's estimates for derivatives. By either method, w e can obtain a uniform bound in J1 for ~(n)(x)/n!l TM, and this shows that f is analytic in an interval that contains points of H. This contradicts the definition of H, so H must be empty. That is, f must be analytic on J. This is the conclusion of Pringsheim's theorem. P r i n g s h e i m ' s t h e o r e m s e e m s not to be w i d e l y known; I have never seen it mentioned in a textbook. For half a century, I believed that it was a really "pure" theorem, with no applications, either in mathematics or elsewhere. However, a special case has recently found applications in physics ([3], [8]), although

the physicists had to discover it for themselves. The result t h e y n e e d e d is that if lim infn__~(2n+l)(t)[-11" / C > 0 for all t in an interval J, then f is the restriction of an entire function. This condition is equivalent to ~(2n+l)(t)[1/(2n+l) ~ L for all t ( J. A t h e o r e m of Hadam a r d ' s t h e n s h o w s t h a t f(2")(t) satisfies the corres p o n d i n g inequality (with a larger L). Rather than deducing this from H a d a m a r d ' s theorem, I shall simply p r o v e it in the r e q u i r e d form. By Taylor's t h e o r e m with r e m a i n d e r of order 2, if t and t + h belong to J, w e have

f(2n)(t) = f(2n-1)(t + K) - f(2n-1)(t) K - -f(2"+l)(t + 0h)K, [0[ < 1, so that ~(2")(t)[ ~ 2[K[-1L2"-I + 1/2KL2"+1. Let h be a positive real n u m b e r less than half the length of J. We m a y suppose that L > 2/h. Then for t E J one of t -+ 2/L J, so we m a y take k = --+2/L to obtain ~(2n)(t)[ ~ 2L 2n, ~(2n)(t)I1/(2n) ~ 21/(2n)L < 2L. I am indebted to Professor Johnson for letting me see the m a n u s c r i p t of [2], and also for helpful comments on an earlier draft of the present paper.

References 1. R. P. Boas, A theorem on analytic functions of a real variable, Bull. Amer. Math. Soc. 41 (1935), 233-236. 2. A. Boghossian and P. D. Johnson, Jr., Pointwise conditions for analyticity and polynomiality of infinitely differentiable functions, J. Math. Analysis and Appl. 140 (1989), 301-309. 3. M. J. Hoffman and R. Katz, The sequence of derivatives Amer. Math. Monthly 90 (1983), of a C| 557-560. 4. P. Kolar and J. Fischer, On the validity and practical applicability of derivative analyticity relations, J. Math. Phys. 25 (1984), 2538-2544. 5. W. F. Osgood, Lehrbuch der Funktionentheorie, vol. I, 5th. ed., Leipzig and Berlin: Teubner (1928). 6. A. Pringsheim, Zur Theorie der Taylor'schen Reihe und der analytischen Funktionen mit beschr/inktem Existenzbereich, Math. Ann. 42 (1893), 153-184. 7. H. Salzmann and K. Zeller, Singularit/iten unendlich oft differenzierbarer Funktionen, Math. Z. 62 (1955), 354-367. o 8. I. Vrkoc, Holomorphic extension of a function whose odd derivatives are summable, Czechoslovak Math. J. 35(110) (1985), 59-65. 9. Z. Zahorski, Sur l'ensemble des points singuliers d'une fonction d'une variable r6elle admettant les d6riv6es de tousles ordres, Fund. Math. 34 (1947), 183-245; supplement, ibid. 36 (1949), 319-320.

Department of Mathematics Northwestern University Evanston, IL 60208 USA T H E M A T H E M A T I C A L I N T E L L I G E N C E R VOL. 11, N O . 4, 1989

37

the physicists had to discover it for themselves. The result t h e y n e e d e d is that if lim infn__~(2n+l)(t)[-11" / C > 0 for all t in an interval J, then f is the restriction of an entire function. This condition is equivalent to ~(2n+l)(t)[1/(2n+l) ~ L for all t ( J. A t h e o r e m of Hadam a r d ' s t h e n s h o w s t h a t f(2")(t) satisfies the corres p o n d i n g inequality (with a larger L). Rather than deducing this from H a d a m a r d ' s theorem, I shall simply p r o v e it in the r e q u i r e d form. By Taylor's t h e o r e m with r e m a i n d e r of order 2, if t and t + h belong to J, w e have

f(2n)(t) = f(2n-1)(t + K) - f(2n-1)(t) K - -f(2"+l)(t + 0h)K, [0[ < 1, so that ~(2")(t)[ ~ 2[K[-1L2"-I + 1/2KL2"+1. Let h be a positive real n u m b e r less than half the length of J. We m a y suppose that L > 2/h. Then for t E J one of t -+ 2/L J, so we m a y take k = --+2/L to obtain ~(2n)(t)[ ~ 2L 2n, ~(2n)(t)I1/(2n) ~ 21/(2n)L < 2L. I am indebted to Professor Johnson for letting me see the m a n u s c r i p t of [2], and also for helpful comments on an earlier draft of the present paper.

References 1. R. P. Boas, A theorem on analytic functions of a real variable, Bull. Amer. Math. Soc. 41 (1935), 233-236. 2. A. Boghossian and P. D. Johnson, Jr., Pointwise conditions for analyticity and polynomiality of infinitely differentiable functions, J. Math. Analysis and Appl. 140 (1989), 301-309. 3. M. J. Hoffman and R. Katz, The sequence of derivatives Amer. Math. Monthly 90 (1983), of a C| 557-560. 4. P. Kolar and J. Fischer, On the validity and practical applicability of derivative analyticity relations, J. Math. Phys. 25 (1984), 2538-2544. 5. W. F. Osgood, Lehrbuch der Funktionentheorie, vol. I, 5th. ed., Leipzig and Berlin: Teubner (1928). 6. A. Pringsheim, Zur Theorie der Taylor'schen Reihe und der analytischen Funktionen mit beschr/inktem Existenzbereich, Math. Ann. 42 (1893), 153-184. 7. H. Salzmann and K. Zeller, Singularit/iten unendlich oft differenzierbarer Funktionen, Math. Z. 62 (1955), 354-367. o 8. I. Vrkoc, Holomorphic extension of a function whose odd derivatives are summable, Czechoslovak Math. J. 35(110) (1985), 59-65. 9. Z. Zahorski, Sur l'ensemble des points singuliers d'une fonction d'une variable r6elle admettant les d6riv6es de tousles ordres, Fund. Math. 34 (1947), 183-245; supplement, ibid. 36 (1949), 319-320.

Department of Mathematics Northwestern University Evanston, IL 60208 USA T H E M A T H E M A T I C A L I N T E L L I G E N C E R VOL. 11, N O . 4, 1989

37

The catapult that Archimedes built, the gambling-houses that Descartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions--not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the caf~ where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? 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 the European Editor, Ian Stewart.

The Mathematical Miller of Nottingham Susan Friedlander and Anton Powell

A l t h o u g h the existence of Green's t h e o r e m and Green's functions have ensured that Green is a household name in the mathematical world, George Green died in 1841, at the age of 47, in relative obscurity. An obituary in a local Nottingham paper admitted that it knew little: "We believe he was the son of a m i l l e r . . . had his life been prolonged, he might have stood eminently high as a mathematician." There are few reliable details of the life of George Green. But now, for the first time, Green has been the subject of a full-length scholarly biography, by Mary Cannell. She has most generously allowed us to use here findings of her illuminating work, in advance of its formal publication. Green, whose family roots lay in the rural Trent Valley, east of Nottingham, was enrolled at the age of eight at the private school of Robert Goodacre, the author of a popular arithmetic textbook. At nine he left school; so ended, it seems, the formal education of the child. The next firm record comes from 1823 when, at the age of 29, Green is recorded as having joined the Nottingham Subscription Library. This was a kind of proto-university, owning scientific i n s t r u m e n t s and h o l d i n g lectures and meetings of learned societies. At this time Green lived in the village of Sneinton (now a Nottingham suburb), where his father had established a successful milling business. The mill, which can now be visited by the mathematical tourist, was reportedly used as Green's study. Here, it seems, he worked in isolation on math38

emafical ideas that he developed in advance of those of his English contemporaries. In the long period between leaving school at the age of nine and publishing his first work, Green worked as a miller at the Sneinton windmill. A document survives in which he signs himself " G e o r g e Green, miller, Sneinton." A cousin of Green, William Tomlin, was to state in a memoir after the mathematician's death that Green found work at the mill irksome. Oral tradition among Green's descendants tells of the mathematician in 1831 having to defend the building with a musket against a crowd of rioters agitating for the right to vote in parliamentary elections. Green's first and most important work, A n Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, was published privately by

subscription in 1828. The preface concludes with a request that the work be read with indulgence in view of the limitations of the author's education. Sadly this seminal work, in which Green's theorem and Green's functions were introduced and the term 'potential' coined, appears to have been almost entirely ignored at the time of publication. It was neglected until William Thomson (Lord Kelvin) discovered it and recognised its great value. Thomson was responsible for republishing the work, with an introduction, in the Journal far die reine und angewandte Mathematik (18501854). In 1928, to mark the centenary of the Essay, Albert Einstein sent a congratulatory telegram to the city

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4 9 1989 Springer-Verlag New York

of Nottingham. (On another occasion Einstein remarked that Green had anticipated the work of later mathematicians, including Gauss.) In 1832 and 1833 two further works of Green were read at the Cambridge Philosophical Society. In October 1833, at the age of 40, he became an undergraduate at Gonville and Caius College, Cambridge, where, following further significant publications, he was made a Fellow six years later. In 1840 Green returned to Nottingham in poor health. He died in the next year at the home of Jane Smith, his mistress and the mother of his seven children. What brought Green into contact with the contemporary French mathematics on which his theories were to build? As regards the new continental analysis, the Nottingham of Green's day might seem to be a backwater of a backwater. Even at Cambridge, m a t h e m a t i c i a n s p e r s i s t e d w i t h the N e w t o n i a n "fluxions" rather than using the continental notation. Copies of new French works would be expensive and hard to obtain, not least because Britain and France had been long at war. As a possible link between Green and the new French mathematics, Mary Cannell points to John Toplis who, as well as being the translator of Laplace's M~chanique C~leste (cited in Green's Essay) and a Fellow of Queens', Cambridge, was also a near neighbour of Green in Nottingham as well as being a member of the Subscription Library there. The Subscription Library still functions in its original Georgian building by the town square. No longer a proto-university, the library is a peaceful place where visitors are welcomed. Green is now remembered with honour; there is anguish about the decision in the early 20th century to sell the copy of his Essay that the mathematician had presented to the library. Copies of the Essay are now extremely rare. Green's windmill is a delight, restored in the early 1980s and now open to visitors--after decades as a ruinous shell amid brambles and weeds. In the mill yard is a small science museum for young people, indicating the significance of Green's work for the development of modern physics. Flour is again milled and sold there along with such a m u s e m e n t s as green pencils decorated with Green's theorem. From the upper windows you can see, much as Green did, both the rural Vale of Belvoir and the grime of industrial South Nottingham. In 1937 the Mathematical Association met in Nottingham; members visited Green's grave and made a protest about its decrepit state. N o t t i n g h a m city council responded by restoring both it and the adjacent grave of his mistress, Jane Smith. In the northeastern corner of St. Stephen's churchyard, Sneinton, 200 yards from the windmill, lies the man who "had his life been prolonged, might have stood eminently high as a mathematician."

The windmill where George Green, of Green's functions and Green's theorem, ground corn for a living.

~lanchoster

Notting Coventry 9 ~.1 Cambridge

Oxford London

George Green's most important work was done in Nottingham, England. THE MATHEMATICAL tNTELLIGENCER VOL. 11, NO. 4, 1989

39

Tourist addresses: Green's Mill and Science Museum Belvoir Hill Sneinton, Nottingham Tel: [0] 602 503 635 Nottingham Subscription Library Bromley House Angel Row Nottingham Tel: [0] 602 473 134 [The library holds a draft copy of Cannell's biographical study of George Green.]

40

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

Library Hours: Wednesday to Sunday (and Bank Holidays) 10.00 am-5.00 pm; closed Monday and Tuesday, and Christmas Day. An 80-page booklet, with 63 black-and-white illustrations, describes the history of Green's Mill. George Green--Miller and Mathematician 1793-1841 by D. M. Cannell is available from the address above (Sneinton) for s + 50p. postage. Susan Friedlander Department of Mathematics University of Illinois at Chicago Chicago, IL 60680 USA

Anton Powell 20 Hound Road Nottingham England

Some Recent Developments in Differential Geometry* Brian White

The last few years have brought m a n y exciting developments in differential geometry. In this article I will describe several that can be readily appreciated by J nonspecialists.

Evolution of Curves and Surfaces Until recently differential geometry was the study of fixed curves or surfaces in space and of abstract manifolds with fixed Riemannian metrics. N o w geometers have begun to study curves and surfaces that are subjected to various forces and that flow or evolve with time in response to those forces. Perhaps the simplest example (but already a very subtle one) is the curve-shortening flow. Consider a simple closed curve in the plane, and suppose that it moves so that the velocity at each point on the curve is equal to the curvature vector of the curve at that point. Thus on a convex curve, every point moves into the region b o u n d e d by the curve, whereas on a general curve, points on the portions that b e n d inward will move outward. This motion of curves arises naturally from thinking of the space of all smooth e m b e d d e d curves as an infinite dimensional manifold M: a curve moves so that its velocity is minus the gradient of the length function on M. What h a p p e n s to a curve as it flows in this way? Some facts are rather straightforward to establish:

(1) Disjoint curves remain disjoint. This fact is an example of the m a x i m u m principle for parabolic partial differential equations. To see w h y it is true, consider two closed curves, one inside the other, that are initially disjoint. Suppose that at some later time they intersect each other. At the first such time, the curves m u s t be tangent at the point p where they meet, and the curvature of the inner curve at p must be greater than or equal to the curvature of the outer curve at p. If strict inequality holds, t h e n (at p) the inner curve is moving inward faster than the outer curve. But that implies that a m o m e n t earlier a portion of the "inner" curve was outside of the " o u t e r " curve, a contradiction. If the curvatures at p are equal, a more subtle argument is required (Cf. [21]).

* This p a p e r w a s m a d e possible in part by g r a n t s from t h e National Science F o u n d a t i o n (DMS-8611574 a n d DMS-8553231) a n d from t h e Arco Corporation. It is b a s e d o n a n a d d r e s s delivered at the 90th S u m m e r M e e t i n g of t h e A m e r i c a n Mathematical Society. The a u t h o r w o u l d like to t h a n k Rob K u s n e r for s u g g e s t i n g a n u m b e r of imp r o v e m e n t s to t h e paper. THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 4 9 1989Springer-VerlagNew York 41

(2) The enclosed area decreases at a constant rate. In fact, if a closed curve C is moving so that (at a certain time) the point x is m o v i n g with velocity v(x), then --

d (area inside C) = fc v o n,

dt

where n(x) is the outward unit normal to C at x. This equation holds for any flow, not just for the curvature flow. In our case, v(x) 9 n(x) is the curvature of C at x, so

d dt (area inside C) = total curvature of C = - 2~r, because the total curvature of a n y simple closed curve is - 2"rr. Consider for example a curve that is initially a circle of radius R. Clearly it must remain circular (by symmetry), a n d its area at time t is "rrR2 - 2~rt. Thus its lifespan is 1/2R2, for at that time it has already shrunk to a point. (3) A circle shortens more slowly than other curves of the same length. To see this, we use the "first variation formula" which says that for a n y flow of curves, if L(t) is the length of the curve at time t, then --L = -

v.K,

dt

where K(x) is the curvature vector of C at x. In our case v(x) = K(x), so d L = - - f K 2.

Jr

dt

Thus by the Cauchy-Schwarz inequality ~L~

dt

--

L

K

1

L (2~r)2' with equality if a n d only if K(x) is constant (independent of x), i.e., if and only if the curve is a circle. (4) Curves last for a limited time. Given a n y initial curve C, we can d r a w a large circle, say of radius R, around it. As the curve and the circle evolve, the curve always remains inside the circle by the m a x i m u m principle. Since the circle collapses in time 1/2R2, the inner curve can flow for at most that time. 42

T H E M A T H E M A T I C A L INTELLIGENCER VOL. 11, N O . 4, 1989

Figure 1. Can curvature flow make this curve convex? Other properties of the flow are m u c h more subtle: (1) M u s t every curve collapse to a point? Property (4) above suggests that they do. The difficulty is that the curve might conceivably develop a singularity (such as a sharp corner) before collapsing to a point. Consider for example a long thin ellipse centered at the origin with the major axis horizontal a n d the minor axis vertical. Does it collapse to a point or to a segment? Of course the two ends are moving inward much faster than the top and bottom are, which suggests that it should collapse to a point. But then the top a n d bottom have m u c h less distance to travel; if they reach the origin before the two e n d s do, then the curve will collapse to a segment. (2) If a curve collapses to a point, must it collapse to a r o u n d point? It m a y surprise the reader that points come in all shapes (though only one size). To see w h a t it means to collapse to a round point, let C(t) be the curve at time t. Dilate C(t) to get a n e w curve C(0 centered at the origin and enclosing area "rr. We say that C(t) converges to a r o u n d point provided the dilated curves converge to a circle. In 1986, Michael Gage a n d Richard H a m i l t o n [6] s h o w e d t h a t convex curves do collapse to r o u n d points. Then in 1987 Matt Grayson [7] s h o w e d that a n y e m b e d d e d curve m u s t eventually become convex

9 9

9 Figure2.

Curvature flow for varifolds.

and thus (by the Gage-Hamilton result) collapse to a round point. The proofs are too complicated to be described here, but the following example shows w h y Grayson's result is surprising. Consider an annulus of inner radius r and outer radius R. Starting at any point in the annulus, draw a spiral by going around the annulus N times. N o w turn around (by tracing out a tiny semicircle) and spiral back around N times to get back to the starting point and thus form a simple closed curve C (see Figure 1). Note that at most of its points C has curvature between r - i and R -~ and hence is not moving very fast initially. Note also that the curve can flow for time at most ~/2R2, independent of N. Thus if we choose a very large N, the curve will have to do a lot of unwinding in a short time if it is to collapse to a r o u n d point. Unlikely as it seems, according to Grayson's work that is exactly what happens. More generally, one can consider smooth n-dimensional surfaces in R n+ 1 and let each point move with velocity equal to the mean curvature of the surface at that point. (This corresponds to flowing by minus the gradient of n-dimensional area in the relevant infinite dimensional manifold.) This flow has been studied by Gerhardt Huisken, who was able to prove that convex surfaces collapse to round points [10]. Huisken proved his result at about the same time that Gage and Hamilton proved theirs, ,and, oddly enough, Huisken's proof works for surfaces of every dimension except one, the case handled by Gage and Hamilton. In fact Huisken shows that the dilated surfaces flow to a surface with the property that at each point x, the prin-

cipal curvatures are all equal (though their common value is allowed to depend on x.) Such surfaces are called "'totally umbilic" and in dimensions/>2 must be round spheres. But for curves the condition is vacuous. Incidentally the higher dimensional analogue of Grayson's theorem is definitely false: ff the initial surface is two spheres joined by a thin tube (like a dumbbell), then the tube pinches closed (forming a singularity with infinite curvature) before the spheres have time to collapse. Classification of the singularities that occur in the mean curvature flow is a challenging open problem. Several years before Huisken's work, Ken Brakke [3] studied other aspects of mean curvature flow. Brakke was interested in the problem in part because grain boundaries in annealing metals seem to flow with velocity proportional to mean curvature. Because grain b o u n d a r i e s are like curved polyhedral complexes, Brakke needed to define the flow on a class of shapes (so-called rectifiable varifolds) much more general than the smooth hypersurfaces studied by Huisken. The initial curve formed by two circles of different sizes joined by a line segment (see Figure 2) provides a good example of mean curvature flow for varifolds. According to Brakke, the two circles instantly deform so that at each of the nodes the three curves meet at equal angles. The two deformed circles proceed to shrink until the smaller has collapsed to a point. The segment then vanishes instantly, so that all that remains is a single convex curve, which must then shrink to a round point. Recently Stanley Osher and James Sethian [19] (using earlier work of Sethian) have devised effective techniques for simulating mean curvature flow on computers. Though not general enough for the surfaces Brakke studied, their methods do allow one to follow the evolution of an initally smooth surface even after singularities have formed. Their methods also apply to other curvature-dependent flows, such as the propagation of flame fronts. Sethian describes the ideas for non-experts in [22].

The Optimal Isoperimetric Inequality What is the greatest possible area that can be surrounded by a closed curve of a specified length? The Greeks knew that the answer is realized by a circle, t h o u g h t h e y did not have a p r o o f acceptable b y modern standards. In the nineteenth century mathematicians such as Jacob Steiner [24] gave proofs that if a solution curve exists, then it must be a circle, but they did not prove the existence of a solution. N o w m a n y complete, rigorous proofs are known. (See Robert Osserman's article [20] for the history of this and related problems.) The result is known as the isoperimetric inequality: the area bounded by a closed THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

43

curve is less than or equal to the length squared divided by 4~r, with equality if and only if the curve is a circle: 1 Area in C ~< - - (length of C) 2. 4"rr More generally, for any closed hypersurface S in R n + 1,

ment for the special case of two-dimensional surfaces in R 3. (Other proofs were known in this special case, but they do not generalize.) In this case the theorem may be stated as: THEOREM. If S is an area-minimizing surface in R 3 with area ~r, then the length of OS is greater than or equal to 2~r, with equality if and only if S is a round planar disk.

n+l

Volume in S ~< cn (area of S)-n- , with equality if and only if S is a round sphere (where cn is the c o n s t a n t that gives e q u a l i t y for r o u n d spheres). As we have stated it, the isoperimetric inequality makes sense only for hypersurfaces. But in 1960, Herbert Federer and Wendell Fleming [5] discovered a beautiful generalization to surfaces of arbitrary dimension and codimension.

Federer-Fleming Isoperimetric Inequality.

Let M be a

closed m dimensional surface in R x. Then there is an m + 1 dimensional surface ~ in R N such that the boundary of ~ is M and such that

I~1 ~ C~,~IMI ~-~ ,

PROOF: Among all area-minimizing surfaces with area ~r, let S be the one whose boundary is as short as possible. It follows that S minimizes the ratio

10sl2 Isl among all area-minimizing surfaces (because the ratio is invariant with respect to dilations). Of course since a disk of area "rr and circumference 2"rr is area-minimizing, OS must have length ~<2"rr; our goal is to show that it has length exactly 2~r. Let Ct be a one-parameter family of curves with Co = OS, and for each Ct let St be a surface of least area having boundary OSt = Ct. Because ]CII2/ISt] attains its minimum at t = 0 we have

(where Cm,N is a constant that depends only on m and N). They did not, however, determine the best possible constant Cm,~ (that is, the smallest value of Cm,N that makes the theorem true). In 1973 Jim Michael and Leon Simon [18] showed that the best constant Cm,N is bounded by a function of m, but otherwise no progress was made for many years. Finally in 1986 Fred Almgren [2] showed that the best constant is attained precisely by round spheres. Note that in the statement of the Federer-Fleming theorem, since we are allowed to choose any surface with boundary M, we may as well choose the surface of least possible area. Thus Almgren's optimal version of the Federer-Fleming isoperimetric inequality may be stated as follows.

THEOREM. (Almgren 1986). If S is an m+l

Isl ~ c~lasl ~

with equality if and only if S is congruent to a round ball in R m+l.

(A surface S is said to be area-minimizing if its area is less than or equal to the area of every other surface having the same boundary.) Although the proof involves some complicated technicalities, the basic idea is very elegant and not hard to understand, so I would like to sketch the arguTHE MATHEMATICAL 1NTELLIGENCER VOL. 11, NO. 4, 1989

(;t)

IGl~/Is,I

t=0

and therefore 0 = 21Col

ISol -~ -

dlGl ~ - l

= 2[col-~-

dlS,I

ICol 2 ~

-ICol ~

~

15ol -2

~r-2.

(*)

N o w if in the one-parameter family C t of curves, the initial velocity of each point x ~ Co = C is v(x), then

(;t)

t=O

(m + 1)-dimen-

sional area minimizing surface in R N, then

44

o =

Sv

t=0

(where n(x) is the unit vector at x ~ C that is normal to C, tangent to S, and that points away from S). The first of these formulas was mentioned above in connection with evolution equations. The second is less wellknown because it is true in general only if So is a minimal surface [23, w Combining these formulas with

(9 gives O = 2~r fcV " K + lCl fc v " n = [ v" (2"rrK + Jc

loin).

Because this holds for every vector field v, it follows that 2"rrK + ICIn must be identically 0. That is, the curve C has constant curvature [CI/(2w). Recall that ICI 2"rr (we are trying to prove equality), so C has curvature everywhere less than or equal to 1. The theorem then follows from the following proposition.

Proposition. (Almgren) If C is a closed curve in R 3 with curvature everywhere less than or equal to 1, then C has length >~2~r, with equality if and only if C is a circle. (Proofs of this were known before Almgren's work, but those proofs do not imply the higher-dimensional result, namely that if S is a closed k-dimensional surface in R n with mean curvature everywhere less than or equal to k, then the area of S is greater than or equal to the area of the unit k-sphere, with equality and only if S is congruent to the unit k-sphere.) PROOF: Let K be the convex hull of C. Let v : OK~ S2 be the Gauss map, which assigns to each x E 3K the outward unit normal v(x) to K at x. (The range of v is the set of unit vectors, i.e., the unit 2-sphere.) Note that at a corner of K there are many outward unit normals so the map v is multivalued. (To be more precise, v(x) is the set of unit vectors v such that the function x --~ v 9x attains its maximum value on K at x.) The first observation is that v maps 3K',C to a set of zero area in S 2. To see this, consider for example the case where C consists of two congruent circles (one above the other) so that K is a cylinder. Then v maps the sides of the cylinder to a great circle and the top and bottom to a pair of points. On the other hand, the image of OK under v is all of S 2, so the image of C (more precisely, of C N OK) under v must have area 4~r. It is not too hard to see that an infinitesimal piece of C at x of length ds is mapped to a set of area at most 2 9 (the curvature of C at x)ds, which is less than or equal to 2 ds (because the curvature is at most 1). Thus area of v(C)

Icl

~2.

But the area of v(C) is 4"rr, so IC[ t> 2~r. 9 It is not known whether in Almgren's theorem it suffices to assume that S is merely minimal instead of area-minimizing. (A surface is said to be minimal if it satisfies the first derivative test for being an area-minimizing surface.) Minimality does suffice for two-dimensional surfaces with only one boundary component, for surfaces that are topologically annuli, and for surfaces in R 3 that have only two boundary components. (See [19, w and [16].)

Soap Bubbles A soap bubble is affected primarily by two forces: the pressure exerted by the enclosed air, and the surface tension in the soap film itself. When these forces are in equilibrium, the mean curvature of the surface is the same at every point. For this reason I will use the terms "constant-mean-curvature surface" and "soap bubble" interchangeably. (Mean curvature is defined as follows. If cartesian coordinates for R 3 are chosen so that the surface passes through the origin and is tangent to the xyplane there, then near the origin the surface is the graph of a function z = f(x,y) with 0 = f(0) = Of/Ox(O) = 3f/Oy(O). The mean curvature of the surface at the origin is then O2f/Ox2(O) + 02g/Oy2(O).) The question I would like to discuss is: must soap bubbles be round? Certainly the soap bubbles we see are round, and of course spheres do have constant mean curvature, but could there be other constantmean-curvature surfaces? In some cases the answer has been known for years. In 1951 Heinz Hopf ([8] or [9, p. 138]) showed that if a constant-mean-curvature surface is topologically a sphere (embedded or immersed), then it must be a r o u n d sphere. And in 1958 A. D. Alexandrov [1] showed that among compact embedded surfaces, all soap bubbles are round. That left open the possibility of immersed constant-mean-curvature surfaces of higher genus. Until 1984, most people who were aware of the problem probably thought that no other constantmean-curvature surfaces should exist. But Henry Wente [25] surprised them by constructing a constant m e a n curvature i m m e r s e d torus. It was already known that solutions of a certain partial differential equation in the plane could be used to construct constant-mean-curvature surfaces in R 3. Wente focused on doubly periodic solutions to the partial differential equation. In general, such a solution gives an infinite surface in R3; two regions of the plane that differ by translation through a period give rise to two portions of the surface that are congruent but that do not necessarily coincide. Wente worked with rectangular period lattices in the plane so that one of the periods corres p o n d e d to a rotation in R 3 and the other corresponded to a translation in R 3. By carefully choosing the periods (or lattice points) in the plane, Wente could make rotation be through a rational angle and the translation distance be zero, so that the potentially infinite surface closed up to form an immersed torus. More recently, in 1987 Nikolaos Kapouleas [14] was able to construct constant-mean-curvature surfaces of every genus >13. Before describing his construction, I must describe the surfaces studied by C. Delaunay in 1841. So far, I have been speaking only of compact surfaces, and the theorems of Alexandrov and Hopf THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

45

do not rule out any noncompact surfaces. Delaunay was interested in all noncompact constant-mean-curvature surfaces that are rotationally symmetric about an axis. The symmetry reduces the problem to an ordinary differential equation that Delaunay was able to analyze. If we fix the mean curvature, then the resulting surfaces form a one-parameter family. The first example is just a cylinder. As we change the parameter, the cylinder develops periodic bulges so that it looks like a row of spheres joined by necks (see Figure 3). If we keep changing the parameter, the necks become smaller and smaller until they pinch off and we have a row of spheres. If we keep going, then the spheres start to overlap so that the necks are inside out and the surface is immersed. (See [4] for more about Delaunay surfaces.) Now consider seven spheres of the same size at the corners and at the center of a large hexagon. Kapouleas's idea was to join the spheres by pieces of Delaunay surfaces (see Figure 4). The resulting surface has constant mean curvature everywhere except at the places where the Delaunay surfaces were attached to the spheres (there it is necessary to modify the Delaunay surfaces slightly so that t h e y fit together smoothly with the spheres). But the mean curvature is so close to being constant that Kapouleas was able to show that a small perturbation of the surface would have mean curvature that is exactly constant. Figure 3. Surfaces of revolution that have constant mean Clearly one can create many other examples by curvature. starting with a different arrangement of spheres. For example, by starting with n + 1 spheres at the corners and the center of an n-gon, one gets a constant-mean- pact embedded constant-mean-curvature surfaces. In curvature surface of genus n. But there are some re- particular, they show that if such a surface has finitely strictions. In particular, each Delaunay surface that we many ends, then each end must converge rapidly to a attach to a sphere exerts a force on that sphere. If the Delaunay surface (as one moves away from the orDelaunay surface is embedded, it pulls the sphere, igin). If the surface has only two ends, then they show and if the Delaunay surface is immersed, it pushes it. that it must be a Delaunay surface. (Bill Meeks [17] The amount of pulling or pushing is determined by had showed earlier that any embedded example with the neck size of the Delaunay surface. Thus in Figure 4 two ends must be contained in a cylinder.) On the we could not have joined the spheres by Delaunay other hand, Wente's construction can produce imsurfaces all of which were embedded. For if we had, mersed examples with two ends that are not Dethen all three surfaces attached to a vertex sphere launay. would be pulling on it, and the configuration would The reader may wonder about higher-dimensional not be in equilibrium. Likewise it is not possible to join soap bubbles. Alexandrov's proof works in any dispheres at the corners of a square by Delaunay sur- mension, but Hopf's is strictly for two-dimensional faces (immersed or embedded) along the edges of the soap b u b b l e s - - i n 1982 Wu-Yi Hsiang [11] constructed square; since at each vertex one Delaunay surface immersed soap bubbles in R 4 that are topologically would be exerting a horizontal force and the other a spheres. vertical force, the two forces could not cancel each other. For this reason Kapouleas was able to construct References neither tori nor surfaces of genus 2. In [13] Kapouleas also constructed many noncom1. A. D. Alexandrov, Uniqueness theorems for surfaces in the large. Vestik Leningrad Univ. 13 (1958), 5-8; English pact examples, such as a sphere with three halves of trans., Amer. Math. Soc. Transl. (2) 21 (1962), 412-16. Delaunay surfaces growing out of it (Figure 5). Such 2. F. J. Almgren, Jr., Optimal isoperimetric inequalities. Inexamples may be immersed or embedded. diana University Math. J. 35 (1986), 451-547. Nick Korevaar, Rob Kusner, and Bruce Solomon [15] 3. K. Brakke, The motion of a surface by its mean curvahave done some work toward classifying all noncomhare, Math. Notes No. 20, (1978) Princeton Univ. Press.

0

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THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

I

Figure 4. Seven spheres connected by pieces of Delaunay surfaces,

Figure 5. A new noncompact embedded surface of constant mean curvature.

4. J. Eells, The surfaces of Delaunay, Math. Intelligencer 9 (1987), 53-57. 5. H. Federer and W. H. Fleming, Normal and integral currents. Ann. of Math. 72 (1960), 458-520. 6. M. Gage a n d R. S. H a m i l t o n , The h e a t e q u a t i o n shrinking convex plane curves; J. Differential Geom. 23 (1986), 69-96. 7. M. Grayson, The heat equation shrinks embedded plane curves to round points. J. Differential Geom. 26 (1987), 285-314. 8. H. Hopf, Uber Fl/ichen mit einer Relation zwischen den Hauptkriimmungen, Math. Nachr. 4 (1951), 232-249. 9. H. Hopf, Differential geometry in the large, Lecture Notes in Mathematics No. 1000, (1983) Springer-Verlag. 10. G. Huisken, Flow by mean curvature of convex surfaces into spheres. J. Differential Geom. 20 (1984), 237-266. 11. W. Y. Hsiang, Generalized rotational hypersurfaces of constant mean curvature in the euclidean spaces, J. Differential Geom. 17 (1982), 337-356. 12. N. Kapouleas, Constant mean curvature surfaces in euclidean three space, (research announcement), Bull. Amer. Math. Soc. 17 (1987), 318-320. 13. - - , Complete constant mean curvature surfaces in euclidean three space, (preprint). 14. - - , Compact constant mean curvature surfaces in euclidean three space, (preprint). 15. N. Korevaar, R. Kusner, and B. Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), 465-503. 16. P. Li, R. Schoen, and S. T. Yau, On the isoperimetric inequality for minimal surfaces, Annali Scuola Norm. Sup.

Pisa 21 (1984), 237-244. 17. W. H. Meeks, HI, The topology and geometry of embedded surfaces of constant mean curvature, J. Differential Geom. 27 (1988), 539-552. 18. J. H. Michael and L. Simon, Sobolev and mean-value inequalities on generalized submanifolds of R n, Comm. Pure Appl. Math. 26 (1973), 361-379. 19. S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Computational Phys. (1987). 20. R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), 1182-1238. 21. M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, (1984) Springer-Veflag, p. 168. 22. J. Sethian, Hypersurfaces moving with curvature-dependent speed: Hamilton-Jacobi equations, conservation laws and numerical algorithms, preprint. 23. L. Simon, Lectures in geometric measure theory, Centre for Mathematical Analysis, Australian National University, Canberra, Australia 1983. 24. J. Steiner, Sur le maximum et le minimum des figures dans le plan, sur la sphere et dans l'espace en g@n@ral,J. Reine Angew. Math. 24 (1842), 93-152. 25. H. Wente, Counterexample to a conjecture of H. Hopf, Pacific J. Math. 121 (1986), 193-243.

Mathematics Department Stanford University Stanford, CA 94305 USA THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

47

Steven H. Weintraub* For the general philosophy of this section, see Volume 9, No. 1 (1987). A bullet (o) placed beside a problem indicates a submission without solution; a dagger (~-) indicates that it is not new. Contributors to this column who wish an acknowledgment of their contribution should enclose a self-addressed postcard. Problem solutions should be received by 1 February 1990.

Problems The square root of a homeomorphism: Problem 89-70 by the Column Editor Let f:R---~R be an orientation-preserving h o m e o m o r p h i s m . M u s t there exist an orientation-preserving hom e o m o r p h i s m g such that f(x) = g(g(x))?

Chess queens: Problem 89-80 by Bernardo Recam~n (Instituto Alberto Merani, Colombia) W h a t is the m a x i m u m n u m b e r of c h e s s q u e e n s of three different colors that can be a r r a n g e d on a chessb o a r d so that no t w o q u e e n s of the s a m e color are attacking each other? It is k n o w n that the a n s w e r is at least 44. (For one color the a n s w e r is well k n o w n to be 8, for two colors the a n s w e r is easily seen to be 32, a n d for four or m o r e colors the a n s w e r is trivially 64.)

* Column editor's address: Department of M a t h e m a t i c s , L oui s i a na State University, Baton R o u g e LA 70803-4918 USA

48 THE MATHEMATICALINTELLIGENCERVOL. 1I, NO. 4 9 1989Springer-VerlagNew York

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

49

The Calculus of Variations Today t Stefan Hildebrandt Translated by A. Shenitzer*

The calculus of variations is of ancient origin. The Greeks knew the isoperimetric properties of the circle and the sphere, and Heron derived the law of reflection for light rays from a minimum principle. Similarly, in 1662, Pierre de Fermat derived the law of refraction for light rays from the principle that light minimizes the time of passage from source to observer. In 1686 Isaac Newton investigated the shape of a solid of revolution that experiences least resistance in a resisting medium. The development of the calculus of variations as an independent mathematical discipline

began with the papers of the Bernoulli b r o t h e r s - Jacob I and Johannes I - - i n s p i r e d by the brachistochrone (curve of quickest descent) competition. In the eighteenth century, largely as a result of the contributions of Leonhard Euler and Joseph Louis Lagrange, the calculus of variations became an effective mathematical tool. The calculus of variations was initially known as the "isoperimetric method." We can fix the date when the new name came into being. The minutes of meeting No. 441 of the Berlin Academy of Thursday, 16 December 1756 contain the phrase: Mr. Euler a l~ Elementa calculi variationum. During the meetings of 9 September and 16 September 1756, Euler lectured on his two papers (not published until 1766) bearing on Lagrange's 8-calculus. They are related to Lagrange's letter to Euler, dated 12 August 1755, in which the 19year-old French mathematician set forth his method of variations--stiU in use t o d a y - - o f calculating with the symbols ~x, 8y. . . . . At first both Euler and Lagrange regarded the calculus of variations as a kind of higher infinitesimal calculus. But in 1771 Euler discovered the well-known device for reducing the calculus of variations to the ordinary calculus. The name "calculus of variations" was generally adopted only at the beginning of the nineteenth century.

~"This article is an edited translation of Variationsrechnung heute, which was published by the Rheinisch-Westf/ilische Akademie der Wissenschaften in Natur-, Ingenieur- und Wirtschaflswissenschaflen 345 (1986). The original paper in German was based upon a lecture intended for non-mathematicians. * The translator wishes to thank the author and his friends George Booth and Hardy Grant for reading the translation and suggesting a number of improvements. 50 THEMATHEMATICALINTELLIGENCERVOL.11, NO. 4 9 1989Springer-VerlagNew York

What is the fundamental problem of the calculus of variations? To answer this question with a measure of precision we require a few mathematical concepts. Consider functions (mappings) u : F~ ~ M that associate with every point x of some (open and bounded) set f~ in n-dimensional space R * an image point u(x) belonging to a given manifold M. We assume that the mappings u are differentiable, that is, that they each have a tangent mapping Du(x) at every x. N o w let F(x,z,p) be a real-valued function, which m a y be thought of as a "density" function. If u : f~ ~ M is a "sufficiently regular" function, then we can form the integral

I(u) = fa F(x,u(x),Du(x))dx.

(1)

We consider next a class C of admissible functions described by a number of subsidiary conditions. Examples of subsidiary conditions--about to be illustrated--are boundary conditions, content conditions (isoperimetric conditions), and obstacle conditions. The integral (1) associates with every function u in the class C a value I(u). To visualize this association we think of the class C as a plane, each of whose points corresponds to an admissible function u. We lay off the number I(u) above u and obtain in this way a kind of mountain landscape. The fundamental problem of the

calculus of variations is to determine the highest and lowest spots, that is, the minima and maxima of this integral mountain range. We look first for a geometric property that characterizes peaks and valleys under the assumption--similar to that made about a geologically old mountain landscape--that the integral mountain range is sufficiently round and smooth. Every hiker knows such a criterion: at each peak and valley the ground is horizontal, slantless. In other words, the integral mountain

tions. If M = ~N and u = (ul(x) . . . . equations are

uN(x), then these

F,i(x,u(x),Du(x))- ~ D~Fpi(x,u(x),Du(x))= 0, a~=l

i = 1. . . . .

N.

(2)

For a long time the calculus of variations meant the study of equations (2); those at a remove from mathematics are likely to think that the calculus of variations had essentially been completed with the papers of Euler and Lagrange. This is a basically mistaken impression. The truth is that the development of the calculus of variations really began to gather speed only in the present century. We shall now discuss this (possibly excessive-sounding) claim and the p r e s e n t problems of the calculus of variations. We note that eighteenth-century mathematicians regarded the existence of extremal points as obvious, and that, n o w and again, even Gauss and Riemann were victims of this false assumption. They also took for granted the differentiability of the extremal function u--a property clearly essential for setting up equations (2). In fact, to write down equations (2) we require objects that are twice differentiable, that is, of second order of smoothness. The now central questions of existence and regularity of the solutions of extremaI problems were ignored by mathematicians of the Baroque period and even today are viewed by some as redundant or, at best, overly clever. Why worry about, say, extremal objects that cannot be described by smooth functions? This question can be answered in many ways, but one answer is b o u n d to impress all: nature produces such objects. In fact, many natural phenomena cannot be described other than by nonsmooth functions, and, the closer we look, the more singularities we discover. To shed some light on these problems we consider a few relevant examples.

range has a horizontal tangent plane at each extremum.

Minimal Surfaces

Other points in the mountains with horizontal tangent planes are the saddle points between two peaks and two valleys. The points u at which the integral mountain range has a horizontal tangent plane are called stationary (or critical) points of the integral in the class C. Lagrange's g-calculus is the mathematical technique for expressing in formulas the geometric property of horizontality of the tangent plane: a stationary point u is characterized by the vanishing of the first variation

An old problem of the calculus of variations is to fit a surface of least area in a given boundary configuration. The simplest cases of this problem involve surfaces of minimal area fitted in closed curves; Figure 1 shows a few relevant examples. In the language of the calculus of variations, the surfaces are described by mappings u : fl ~ ~3 of a 2-dimensional region f~ ; the area I(u) is given by the integral

~I(u) = I_ {G(x,u(x),Du(x))Su + Fm(x,u(x),Du(x))DSu}dx of the integral I at u for every variation ~u. If F and u are sufficiently regular, then one can derive from the condition ~I(u) = 0, which characterizes the stationary points of I, a system of partial differential equations for u, usually referred to as the Euler-Lagrange equa-

I(u) = f~ lux /h uytdxdy.

(3)

The class C of admissible objects consists of functions that describe surfaces fitted in the prescribed curve F. If we consider the integral mountain range associated with the integral (3), then the counterparts of its "valley points" (minima) are just the surfaces of least area fitted in F. They belong to the stationary points of THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

51

Figure 1. Surfaces of minimal area fitted in closed curves.

Figure 2. Solutions for some semifree boundary value problems.

the integral mountain range. It turns out that their mean curvature H is everywhere zero, that is, they satisfy the equation H = 0.

(4)

This is the Euler-Lagrange equation that goes with (3). The surfaces described by (4) are called minimal surfaces. The minimal surfaces that correspond to the valley points of the integral mountain range of (3) can be realized by means of a physical experiment. If we 52

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

make a thin-wire replica of the curve F, then the stable soap films in the wire ring correspond to the surfaces of least area in F. The subsidiary condition defining the class is the boundary condition that requires an admissible surface to fit the curve F. Already in the nineteenth century the Belgian physicist J. A. Plateau guessed that every closed curve F bounds a minimal surface. This is true but was proved only around 1930. For a long time it was thought that just as there is only one shortest link between two points in space (the straight-line link), so, too, a curve F can b o u n d at most one minimal surface. This is not so. Thus the curve in Figure 1 bounds three minimal surfaces. In fact, for every natural number k there is a curve P with total curvature greater than 4"rr in which one can fit at least k minimal surfaces. It is not known if there are curves that bound infinitely many minimal surfaces of the same topological type. The minimal surfaces la and lb can be realized by means of stable soap films. But then the saddle-point lemma implies the existence of a third, unstable, minimal surface corr e s p o n d i n g to a saddle point b e t w e e n two valley points in the integral mountain range; this surface is shown in Figure lc. Such an unstable surface is difficult to get and, so far, no one has managed to obtain its general numerical representation. The reason for this is that a numerical procedure based on the variational method is necessarily unstable, for approximate solutions tend to the two stable solutions rather than to the unstable o n e - - m u c h as a skier ends up in a valley soon after moving s o m e w h a t away from a saddle point. Different boundary conditions for minimal surfaces are obtained with boundary configurations consisting wholly or partly of support surfaces. For example, choose as support surface a half plane (in the experiment, a thin plate of plexiglass) with which the curve F (a wire) has in common just its two end points. The curve begins at the u p p e r side of the plate, goes around its edge, and bends at its lower side. If we fit a minimal surface (a soap film) in this configuration, then part of its boundary, namely P, is fixed and part

Figure 3. S o a p f i l m bounded by a tetrahedron,

F i g u r e 4. S o a p f i l m bounded by an octahedron.

is a free boundary component ~ resting on the support surface (the plexiglass plate). Then there is an obstacle, the edge of the plate. The soap film must hang from this edge. Far from being arbitrarily smooth, the curve may have a cusp. Figure 2 s h o w s the solutions for some semifree boundary value problems. It turns out that the surfaces of least area (the soap films) make right angles with the support surfaces at their interior points. This is a natural boundary condition that likewise follows from the equation 8I(u) = 0 if one takes into consideration the free mobility of the minimal surface on the support surface. This is another condition that must be satisfied by the stationary surfaces in addition to the equation H = 0. There are boundary configurations R for which the fitting surfaces of least area are invariably objects with singularities. For instance, if R consists of the edges of a tetrahedron (see Figure 3), then, in a soap film experiment, the solution is a system of six soap films with four liquid edges. Any two films meeting along the same edge form an angle of 120~. Moreover, the four liquid edges have exactly one vertex in common, and each pair of liquid edges forms an angle o~ of approximately 109~ ' (exactly: coscx = -1/3). In the case of an octahedron (Figure 4) the solutions are significantly more complicated; in this case there are many noncongruent solutions. The angles of 120 ~ and 109~ ' occur very frequently in nature, in particular in crystals. In the experiment depicted in Figure 5 there is, in addition to the boundary condition, an isoperimetric condition: the system of surfaces fitted in R is required to span a certain volume (a certain amount of air). Accordingly, an air bubble, bounded by four surfaces of constant mean curvature H # 0, is located on the inside of the six minimal surfaces that issue from the edges. A similar solution would result if one replaced the rectilinear edges of the tetrahedron by circular arcs (see Figure 6). The protozoan Callimitra, whose siliceous skeleton is shown in Figure 7, has similar shape. (It belongs to the order Radiolaria, classified and drawn by Ernst H/ickel in the nineteenth century.)

Figure 5. Soap film determined by an isoperimetric condition.

Figure 6. Soap film determined by circular arcs.

Figure 7. The protozoan Callimitra.

Still other conditions can be obtained by using as parts of the boundary configuration inextensible but free-to-move threads. Take a flat soap film. Place on it a thread in the form of a loop attached to the tip of a wire that terminates in the soap film. N o w burst the bubble inside the loop with a pin. The remaining outer soap film then pulls the thread taut and shapes it into a circle (Figure 8). This shows that of all plane curves of given length the circle encloses the largest area (a fact known in antiquity but first proved rigorously in THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

53

Figure 8. Solution of the isoperimetric problem.

takes on the shape of a space curve of constant curvature (Figure 9). The sphere has a property analogous to the previously stated isoperimetric property of the circle: among all closed surfaces of given surface area, the sphere encloses a solid of largest volume. A minimum property equivalent to this maximum property is that, of all solids of given volume, the sphere has the least surface area. This exemplifies the following phenomenon: If one can formulate for a mathematical object a minimum principle, then it is possible to state a dual variational problem for which the object in question has a maximum property. We shall soon come across another instance of this dualism that can be investigated by elementary means. We note in passing that the first rigorous proof of the isoperimetric property of the sphere was given in 1884 by H. A. Schwarz.

Shortest Pathways

Figure 9. A space of constant curvature.

Figure 10. The law of reflection. the last century by Karl Weierstrass). But then the circle is the solution of the isoperimetric problem, for, by the principle of virtual work, the outer soap film tends to shrink so as to minimize its potential energy and this is equivalent to its attaining minimal area. If we move the wire downward and thus pull thread and soap film out of their plane position, then, other than at the point where~it is attached to the wire, the thread 54

THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 4, 1989

The problem of the shortest pathway linking a finite number of points in the plane is even o l d e r - - a n d in a sense simpler--than the problem of surfaces of minimal area that fit in a given contour. This problem has interesting as well as surprising solutions that can often be obtained by elementary geometric means. Everyone knows that the shortest path linking two points is the straight-line segment that joins them. A s o m e w h a t more difficult question is that of the shortest path from a point P to a line G to a point Q, where P, Q and G are coplanar and P and Q are on the same side of G. There is just one shortest p a t h - - c o n sisting of two segments (see Figure 10)--that forms with the line G equal incidence and reflection angles (~ = 13). This is the well-known law of reflection, obeyed by a light ray striking a plane mirror and by an elastic point mass (ball or billiard ball) hitting a fiat wall. We have just stated a minimality principle for a polygonal path obeying the law of reflection, or as one often says, we have determined such a path by means of a variational principle (or an extremal principle). The variational principle just stated was known to the Greek mathematician Heron (first century A.D.) and is a modest forerunner of the important variational principles employed in modern physics. We treat next the problem of the shortest pathway joining three points. This problem and its generalization (to find the shortest pathway linking 4, 5, 6 . . . . coplanar points) is known as the Steiner problem [after the Swiss mathematician J. Steiner (1796-1863), who taught at the University of Berlin], although it was posed earlier by Fermat and was already solved by Galileo's student Torricelli, and by Cavalieri. Thus, let A, B, C be three points. We may think of A, B, C as vertices of a triangle A. If one of the interior angles of A, say the angle at B, is greater than or equal to 120~ t h e n (as was first n o t e d by H e i n e n in 1834) the

shortest pathway linking A, B, and C consists of the segments AB and BC. On the other hand, if each interior angle of A is less than 120~ then there is just one point P - - w h i c h lies in the interior of & - - s u c h that the three segments AP, BP, and CP yield the required shortest pathway. Torricelli knew that the circumcircles of the outward equilateral triangles on the sides of & intersect at P, and Cavalieri found that each side of A is seen from P at an angle of 120~ Thomas Simpson (1750) realized that the lines joining the outer vertices of Torricelli's equilateral triangles to the opposite vertices of A intersect at P (Figure 11). The three Simpson linking segments are equal to one another as well as to the least value of the sum of the distances AP + BP + CP. In 1846 Fasbender discovered the following maximum property of P associated with its minimum__ p r o p e ~ The least value of the sum of the distances AP + BP + CP in the triangle & is equal to the maximum of the altitudes of all equilateral triangles circumscribed about the triangle A. This is another instance of the duality between the maximum and minimum property that we encountered earlier in connection with the isoperimetric property of circle and sphere. We note that the Steiner figure can be obtained in a so~ip film experiment. To this end we take two glass plates kept parallel by three perpendicular pins of equal length. If we immerse this configuration in a soap bath and take it out again, then we obtain a system of three soap films perpendicular to each of the plates. These soap laminae touch each plate in three segments that yield the shortest pathway linking the three pin ends at either plate (Figure 12). As noted, for two and three points the minimal p a t h w a y is uniquely determined. For four or more points, however, we must generally expect more than one minimal pathway. We must even distinguish bet w e e n stationary and stable pathways. The stable pathways yield absolute or merely relative minima. If w e choose four points that are vertices of a square, then we obtain two different but congruent minimal pathways (Figure 13). If we stretch the square slightly into a rectangle, then we obtain two minimal pathways of different (path) length, one of which is an absolute and the other a relative minimum. C o n s i d e r , m o r e generally, a stable (minimal) pathway for N given points A t, A 2. . . . . A N with n "free" vertices P1, P2. . . . . Pn. If we use the results on minimal p a t h w a y s for three vertices, then we can easily conclude that the number of branches issuing from each vertex Pj is two or three. At a two-branch vertex the branches must form an angle ~ with 120 ~ ~ 180~ and at a three-branch vertex the branches must form equal angles of 120~ each. We see that as the number of vertices A1, A2. . . . . A N increases it is not only increasingly difficult to guess the form of the minimal pathways but it is also not possible to estimate the number of minima and the

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ll i ii ,+11rFi~_ i Figure 12. The shortest pathway linking three pin ends.

Figure 13. Minimal pathways on the vertices of a square. THEMATHEMATICAL INTELUGENCERVOL.11,NO.4, 1989 55

Figure 14. Minimal pathways with many vertices.

number of merely stationary systems (Figure 14). The problem of the shortest connecting network, of significance for, say, the location of new factories, turns out to be a nasty one. We note that there is no good algorithm for producing all minimal solutions associated with an arbitrary set of vertices. 56

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

Nonsmooth Solutions The examples just described show that there are easily formulated variational problems that have no smooth s o l u t i o n s a n d that n o n s m o o t h s o l u t i o n s arise throughout nature. To save the honor of the great

mathematicians of the eighteenth and nineteenth centuries we note that the phenomenon of nonsmooth solutions was discovered by the 72-year-old Euler, not on the basis of a physical observation but in connection with the study of the minima of the integral

f V x X/dx 2 + du 2.

(5)

This discovery, made in 1779, surprised him so much that he spoke of a paradox in the calculus of variations (De

insigni paradoxo quod in analysi maximorum et minimorum occurrit. Opera omnia, Ser. I, vol. 25). Euler showed that under suitable boundary conditions there can exist a smooth extremal (that is, a solution of the Euler-Lagrange equation) and, at the same time, the absolute minimum is attained at a "broken" (polygonal) line, and thus at a curve that is not smooth and not a true extremal. Euler's paper did not appear until 1811, and it was only in 1831 that Goldschmidt "explained" the Euler paradox in a paper in which he answered a prize question of the G6ttingen Academy proposed by Gauss. In this paper Goldschmidt considered minimal surfaces of rotation, that is, surfaces of zero mean curvature that arise by rotating a curve u = u(x) about the x-axis. These surfaces are the stationary points of the integral

f uX/dx + du2. The twice-differentiable stationary points of this integral are the familiar catenaries, given by the hyperbolic cosine, whose surfaces of revolution are the catenoids. But there are also broken minima of the kind shown in Figure 15, that is, consisting of a segment of the x-axis and two segments perpendicular to it. If this polygonal line is rotated about the x-axis, then we obtain two disks joined by the segment of the x-axis as if by an umbilical cord; the latter can be dispensed with for it has zero area.

Existence of Solutions We saw that the appearance of nonsmooth solutions is very natural and can sometimes be connected with an important geometric phenomenon. For example, the catenoid falls apart if we move the two boundary disks sufficiently far apart and goes over into a disconnected solution--the two disks. It may be surmised that the appearance of singularities in the case of solutions of variational problems in elasticity theory has similar significance. However, these phenomena have not yet been thoroughly investigated. The question of existence of solutions of an extremal problem is an important and eminently nontrivial

problem. In this connection we observe that virtually all physical phenomena can be formulated as variational problems or, to put it differently, physical theories use variational principles to describe progress of m o t i o n s or states of equilibrium. But a physical theory, which, after all, is supposed to supply a model for a class of phenomena, would be in a sad plight indeed if it were not possible to ascertain that the theory is free of contradictions and that there exist objects described by the theory. At the very least, the variational principles associated with the theory must admit solutions. Now it may seem obvious that an integral mountain range must have stationary points or even minima. That this is not at all the case is shown by the following example due to Oskar Perron. Assume that the set of natural numbers contains a largest number. Denote this number by n. If n > 1, then n2 > n. But then n2, which is also a natural number, exceeds the maximum n. The remaining possibility is that n = 1. Thus the assumption of the existence of a largest natural number n leads to the untenable conclusion that n = 1. It can be argued that, while logically correct, this example is somewhat artificial and that, surely, "geometrically sensible" problems must always have solutions. Apart from the vagueness of the term "geometrically sensible," this article of faith is easily shown to be baseless. Consider all surfaces above a plane E that are b o u n d e d by a circle K in E and pass through a point P on the perpendicular to E through the center M of K and one unit above M. We shall show that the class C of these surfaces has no element with least area. To see this, it suffices to realize that the area of each admissible surface is greater than the area A of the disk B bounded by the circle K and that we can find a sequence of admissible surfaces F whose areas are arbitrarily close to A. To obtain the required surfaces we need only remove from B a disk B' that is concentric with B and has small radius and replace it with the right circular cone with base B' and vertex P. The following is a truly baffling example. Take a needle of given length and consider all open sets M in which it can be rotated once through 360~. We wish to find an M of least area. One can certainly claim that

Figure 15. Broken minima. THE MATHEMATICAL 1NTELLIGENCER VOL. 11, NO. 4, 1989 5 7

this problem, posed by the Japanese mathematician S6ichi Kakeya, is geometrically reasonable. For a long time it was believed that a certain three-cornered hypocycloid yields its solution. But in 1927 Abram Besicovitch showed that there are plane figures of arbitrarily small area in which the needle can be rotated through 360~ in other words, the Kakeya problem has no solution. These examples show that the question of existence of solutions of extremal problems is anything but the trivial issue that it may at first seem to be. During the International Congress of Mathematicians at Paris in 1900, David Hilbert delivered a lecture in which he listed 23 problems whose solution he regarded as essential for the future d e v e l o p m e n t of mathematics. Three of these problems deal with the calculus of variations and two of the three, problems 19 and 20, pertain to the existence and regularity of the solutions of extremal problems. Problem 19. Are the solutions of regular problems in the calculus of variations necessarily analytic? Problem 209 The general problem of boundary values9 It is my conviction that it will be possible to prove these existence theorems by means of a general principle whose nature is indicated by Dirichlet's principle. This general principle will then perhaps enable us to approach the question: Has not every regular variational problem a solution, provided certain assumptions . . . are satisfied . . . . and provided also, if need be, that the notion of a solution shall be suitably extended? 9

.

.

With these two problems Hilbert charted the course of development of a considerable part of mathematical analysis. Commenting on problem 20 Hilbert says, with good reason, that it is not at all clear in what class of objects one should look for the solution of an extremal problem. This being so, one should water down the concept of "solution" to the point where objects of greatest possible generality are admissible as solutions (or extremal points). It should then not be difficult to construct "'generalized solutions." One should then show that in the case of reasonable extremal problems--Hilbert had in mind integrals with real analytic elliptic i n t e g r a n d s - - t h e solutions are well-behaved (analytic). Richard Courant compared this proposal with Mephisto's introduction of paper money in Goethe's Faust (part two): Ein solch Papier, an Gold und Perlen Statt, Ist so bequem, man weiss doch, was man hat; Man braucht nicht erst zu markten noch zu tauschen, Kann sich nach Lust in Lieb' und Wein berauschen; Will man Metall, ein Wechsler ist bereit . . . . (Such paper, in place of gold and pearls Is so very convenient, one knows what one has! One need not first deal in the market or exchange, One can, if so inclined, get intoxicated with love and wine; If one wants metal, the moneychanger is ready . . . . ) Marshal, army master, king--all are delighted. All 58

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

debts are balanced, the army is paid, the economy flourishes, the king alone asks in amazement: Und meinen Leuten gilt's ftir gutes Gold? (And my people take it for good gold?) Mephisto allays all fears. After all, everything is covered, there is enough gold in the country, one need only auction off the jewelry or, at worst, dig a bit for gold. Then all w h o wish can exchange their paper money for gold.

The development of the calculus of variations really began to gather speed only in the present century. Hilbert's proposal is similar: One should use paper money--generalized functions--to solve a problem. The generalized (or weak) solutions can be handled as conveniently as paper money. If one wants g o o d - that is, sufficently smooth--solutions then at worst one may have to do some digging--that is, give proof of regularity. Then one can exchange the paper money for gold, the generalized solution for a regular solution. This proposal turned out to be extremely successful and has greatly advanced the development of mathematics. Concepts like "Sobolev spaces of functions" and "distribution spaces" have been indispensable aids. Functional analysis has developed a variety of methods for finding generalized solutions of extremal problems and of differential equations. There has also arisen a regularity theory demonstrating in many important instances the regularity of weak solutions. In the case of linear Euler-Lagrange equations of elliptic type, the regularity theory is entirely satisfactory: all weak solutions prove to be regular. In the case of nonlinear Euler equations it turns out to be far more difficult to demonstrate the regularity of weak solutions. While Charles B. Morrey (1938) and John Nash and Ennio De Giorgi (1957) obtained important partial results, the general case baffled all efforts. Nevertheless, in the early 1960s there was a widespread feeling that a regularity proof covering all cases was in the offing and that Hilbert's Problem 19 would thereby be completely solved. To everybody's surprise, in 1968 Ennio De Giorgi, Enrico Giusti, and Mario Miranda published examples of variational integrals satisfying Hilbert's basic assumption w h o s e stationary points were not only not smooth but, in fact, discontinuous. This p h e n o m e n o n can be convincingly clarified by means of a simple example found ten years ago. Consider mappings u : ~ --~ R" of a bounded open region F/in Euclidean space ~n and take as the integral to be minimized the Dirichlet integral

f lDul2dx.

(6)

Its Euler-Lagrange equation is the potential equation Au=0.

(7)

If we attempt to minimize the integral (6) in the class of functions with prescribed smooth boundary values on the boundary of 1~, then it turns out that this minimum problem has exactly one solution. This solution is arbitrarily smooth (in fact, real-analytic), it satisfies equation (7), and it takes on the prescribed boundary values in a continuous and arbitrarily smooth manner. If we modify this example by replacing the mappings

u : f~---> ~" with the mappings

u : l~--, S', that is, if we replace the linear range space R n with the n-dimensional (curved) sphere /

s n = {u ~ R~+I : lu I = 1},

But the "Leibniz dogma" fails even in the sense in which Hilbert formulated it in his Problem 19: It is not true that every regular (in m o d e r n terminology: elliptic) problem of the calculus of variations--however smooth the data and side conditions--has a smooth, let alone analytic, solution. Questions

We have seen that, after many successes, the Hilbert program formulated in Problems 19 and 20 ran into difficulties due not to insufficiently developed mathematical techniques but to "the nature of things." However effective the method of first constructing weak solutions by means of fttnctional-analytic, measuretheoretic, and topological procedures, it is a sobering experience to realize that the stationary points, and even the extrema, of regular (elliptic) variational problems and, similarly, the solutions of nonlinear systems of elliptic partial differential equations can have singularities. Hence the following questions: (i) How can one characterize variational problems whose solutions are free of singularities? Is it at least possible to single out large classes of extremal problems of physical and geometric interest whose solutions are free of singularities?

then the corresponding Euler-Lagrange equation is

(ii) How can one estimate the "'size" of the set of singularities by using suitable measures? Is it true that, at least for - A u = u IDul2. (8) certain well-defined classes of variational problems, the sets If we take f~ to be the n-dimensional unit ball f~ = of singularities have some definite geometric, analytic, or {x ~ ~" : Ixl < 1} and attempt to minimize the Dirichlet perhaps even algebraic structure? What is the behavior of integral (6) in the class of mappings that take on the weak solutions in the vicinity of singularities? Which singuboundary values u(x) = (x,0) on the boundary 01~ of larities are removable? How do the singularities of a variafl, then it turns out that for n/> 7 we do have a sin- tional problem vary as a result of variation of its data? gular solution of the minimum problem, namely We have only preliminary and provisional answers to all of these questions (see the bibliography at the u(x) = ,0 , x = (x 1 . . . . . xn), end of this article). Without progress in this area it is difficult to see how to make headway in answering the but no smooth solution. In fact, there is not even a following group of questions: smooth stationary point. Moreover, the nonlinear el- Off) How many solutions does a variational problem have? liptic system (8) has no smooth solution in 1~ with the More generally, can one describe the dependence of the prescribed boundary values on aft. structure of the space of solutions on the data of a problem so Let us once more exploit the Mephistopheles ex- that it is possible to tell how solutions change as a result of ample. The devil has a few surprises up his sleeve-variation of the data? (Of special interest are changes in the not every item of paper money has gold backing. The topological character of solutions.) presence of singularities (in our example the irregular In principle, Marston Morse's Morse theory is a point x = 0) cannot be ruled out. The Leibniz notion of prestabflized harmony of the world must not be taken means of answering the questions in (iii). Unfortuto mean t h a t - - a s Leibniz thought--all natural phe- nately, this theory can so far be used without restricnomena are describable by means of analytic func- tion only in the case of the problem of geodesics in tions. Already Euler recognized that this is false for manifolds (the stationary points of the length functional, or the one-dimensional Dirichiet integral), bethe hyperbolic wave equation cause, from the analytic standpoint, one-dimensional problems are relatively harmless. But in the meantime utt - &u = o. THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989 5 9

the Morse theory has been applied with some success to minimal surfaces a n d related geometric objects and one can only hope that a complete Morse theory can be developed for at least these geometric problems. I hope that these few limited vistas over the calculus of variations s h o w that, in spite of its advanced age, it is a vigorous area of mathematics with an abundance of challenging, unsolved problems. I would even venture to say, with w h a t I think is a measure of justification, that, all its successes notwithstanding, the calculus of variations has just emerged from its infancy.

8. S. Hildebrandt and A. Tromba, Mathematics and Optimal Form, Scientif. Amer. Library, New York: W. H. Freeman and Co. (1985). 9. O. A. Ladyzenskaja and N. N. Uralzeva, Linear and quasilinear elliptic equations, New York and London: Acad. Press (1968). 10. C. B. Morrey, Multiple integrals in the calculus of variations, Berlin-Heidelberg-New York: Springer-Veflag (1966). 11. J. C. C. Nitsche, Vorlesungen iiber Minimalfldchen, BerlinHeidelberg-New York: Springer-Verlag (1975). 12. L. Simon, Lectures on Geometric Measure Theory, Centre for Math. Anal., Austral. Nat. Univ. Canberra Vol. 3 (1983).

Picture Credits Bibliography 1. F. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Memoirs of the Amer. Math. Soc. 4, No. 165 (1976). 2. ]. M. Ball, editor, Systems of nonlinear partial differential equations. N A T 9 ASI series, Series C, No. 111, Proc. of N A T 9 Adv. Study Inst. Oxford 1982, Reidel Publ., Dordrecht-Boston-Lancaster (1983). 3. M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Math. Studies, Princeton Univ. Press (1983). 4. D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Heidelberg-New York: SpringerVerlag (1977). 5. D. Hilbert, Mathematische Probleme, Archiv f. Math. und Phys. 3. Reihe, Bd. 1, 44-63, 213-237 (1901), und: Ges. Abhandl. Bd. 3, Springer-Verlag, 290-329 (1935). 6. S. Hildebrandt, Nonlinear elliptic systems and harmonic mappings, Proc. 1980 Beijing Symp. on Diff. Geom. and Diff. Equ., Vol. 1, Beijing: Science Press (1982) 481-615. 7. S. Hildebrandt, Calculus of Variations Today, reflected in the Oberwolfach meetings, Perspectives in mathematics, W. J/iger, J. Moser, R. Remmert, (ed.), Basel-BostonStuttgart: Birkh/iuser Vedag (1983), 321-336.

Figures 1, 6, 10: S. Hildebrandt and A. Tromba, Mathematics and Optimal Form, New York: W. H. Freeman & Co. (1985). Figure 2: S. Hildebrandt and J. C. C. Nitsche, A uniqueness theorem for surfaces of least area with partially free boundaries on obstacles, Archive for Rational Mechanics and Analysis 79, 189-218 (1982). Figures 3, 5, 8, 9: Bildarchiv. Inst. fiir Leichte F1/ichentragwerke, Universit/it Stuttgart. Figure 7: E. H/ickel, Reports of the Scientific Results of H.M.S. Challenger, London, 1881-89. Figure 14: B. Winkel und J. Kron, Minimalwegenetze mit vielen Knoten, Studienarbeit 2/1985, Inst. fiir Leichte F1/ichentragwerke, TU Stuttgart.

Stefan Hildebrandt Mathematisches Institut Universitiit Bonn D-5300 Bonn, Federal Republic of Germany Abe Shenitzer Department of Mathematics York University Downsview, Ontario M3J 1P3 Canada

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THE MATHEMATICALINTELLIGENCER VOL. 11, NO. 4,

1989

Chandler Davis*

Darstellung und Begriindung einiger neuerer Ergebnisse der Funktionentheorie (dritte, erweiterte Auflage) by Edmund Landau and Dieter Gaier Springer-Veflag, 1986, 201 pp.

Reviewed by Lawrence Zalcman Great books deserve great readers. Edmund Landau's

Darstellung und Begriindung einiger neuerer Ergebnisse der Funktionentheorie is one of the great books of twentieth-century function theory, 1 and in Dieter Gaier it has found its ideal reader. In this third edition of Landau's classic, Gaier has added some eighty pages of commentary, supplementary material, and bibliography, augmenting the text of the second edition by two thirds. The result is, as they say, a volume that belongs on the bookshelf of every friend of complex analysis. It is n o w just over half a century since Landau passed from the mathematical scene. A very satisfactory sense of the man and his work can be got from the obituary [2] written by Hardy and Heilbronn and from Littlewood's comments [6, pp. 125-6], which only strengthen the impressions one gets from reading Hardy. It would seem that Landau's strong personality and uncompromising commitment to the highest standards of mathematical excellence occasionally * C o l u m n editor's address: Mathematics Department, University of Toronto, Toronto, Ontario M5S 1A1 Canada 1 I imagine a consensus can be secured as to the top three. They are P61ya and Szeg6's Aufgaben und Lehrs~tze aus der Analysis (1924), translated as Problems and Theorems in Analysis I and II (SpringerVerlag, 1972 and 1976); Hermann Weyl's Die Idee der Riemannschen FMche (1913), translated as' The Concept of a Riemann Surface (Addison-Wesley, 1955); and Rolf Nevanlinna's Eindeutige Analytische Funktionen (1936), translated as Analytic Functions (Springer-Verlag, 1970). The DarsteUung has not been as influential as any of these books, but it is no less beautiful than the most beautiful of them. I would list it right after the "big three."

gave rise to friction in personal relationships. However, there was never any disagreement as to his stature as a scientist. Even the anti-Semitic Bieberbach, in defending the Nazi-inspired boycott of Landau's lectures at G6ttingen in 1933, felt compelled to concede that he was a great mathematician. 2 He certainly was enormously industrious. All told, Landau wrote over 250 papers and seven books. Hardy opined that the books might prove his most enduring monument. Whatever the merits of the claim, there is no denying that Landau's books have been extremely influential. The Handbuch der Lehre vonder Verteilung der Primzahlen marks the birth of analytic number theory as a subject in its own right; for an astonishing account of the state of ignorance in the preHandbuch years, see Littlewood's confession [6, p. 89]. The massive three-volume Vorlesungen fiber Zahlentheorie was the standard reference of its day and is still well w o r t h reading. Generations of students have learned the construction of the real number system (not to m e n t i o n the r u d i m e n t s of m a t h e m a t i c a l German) from Grundlagen der Analysis. And Landau's classic text Einfiihrung in die Differentialrechnung und Integralrechnung (translated as Differential and Integral Calculus, Chelsea, 1950) is, quite simpIy, one of the t w o best calculus b o o k s w r i t t e n this century, 3 a volume which no one teaching honors calculus can afford to keep outside arm's reach. The "jewel in the crown" is the Darstellung. Hardy described it as "'Landau's most beautiful book . . . a collection of elegant, significant, and entertaining theorems of modern function theory." These include a 2 In an article that w a s to become notorious, Bieberbach lauded "the manly rejection of a great mathematician, Edmund Landau, by the students in G6ttingen. The un-German style of this man in teaching and research proved intolerable to German sensibilities." For precise references and background see [9]. 3 The other, a very different sort of book, is Courant's Differential and Integral Calculus, Interscience, 1934-6.

THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 4 9 1989Springer-VerlagNew York 61

selection of results on analytic functions satisfying ~(z)l 1 on the unit disc, the high point of which is Landau's precise bound Isn]

1 +

+

+

.

.

.

+ ( 1 " 32" ' ' ( 2 n__'2; - 4 1)) for the nth partial sum of the Taylor series of such a function; an excursus on summability, culminating in the Knopp-Schnee theorem on the equivalence of Cesaro and H61der summability of order k; the basic Tauberian theorems, including those of Tauber, Fej~r, and Hardy-Littlewood; some striking examples concerning the c o n v e r g e n c e of p o w e r series on (the boundary of) the circle of convergence; results about the analyticity of power series on the boundary (the Fatou-Riesz theorem on convergence of a power series at points of analyticity on the circle of convergence, the Fabry gap theorem); theorems on the logarithmic convexity of the m a x i m u m modulus and integral means of analytic functions (Hadamard's three-circle theorem, Hardy's theorem); a treatment of the Picard theorems, and the theorems of Landau and Schottky based on Bloch's theorem; and, finally, the elementary theory of univalent functions. All this was transmitted in the famous Landau style of which Hardy had this to say: "There are no mistakes--for Landau took endless trouble, and was one of the most accurate thinkers of his d a y - - n o ambiguities, and no omissions . . . . L a n d a u w o u l d not, or could not, think or write vaguely, and a reader . . . will be astonished to find h o w often Landau has given him the shortest, the simplest, and in the long run the most illuminating proof." Sixty years have passed since the publication of the second edition of the Darstellung, and its contents, though grown classical, have (with the sole exception of the material on summability) lost none of their punch. Gaier has now added thirty-six pages of detailed commentary and a sixteen-page bibliography numbering 368 items. The effort involved in collecting and organizing this material must have been enormous, and even the expert is likely to find something new here (as well as much that has been forgotten). Gaier has also contributed a selection of new material chosen in accordance with Landau's own criteria of high elegance and easy accessibility. The choice of results is impeccable. In the classical direction, there are function-theoretic proofs (due, respectively, to Jurkat and Hahisz) of a generalization of Littlewood's O(1/n) Tauberian theorem and of the Hardy-Littlewood "high indices" theorem, as well as a proof of the Fabry gap theorem based on Turan's lemma, which is developed ab initio. The last of these, especially, provides an al62 THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 4, 1989

ternative, and more satisfactory, approach to a result discussed in the original Darstellung. Some modern themes are also sounded. A section devoted to the W e r m e r maximality t h e o r e m contains both Paul Cohen's elementary proof of Wermer's theorem and Gunter Lumer's elegant deduction of the result via the Hahn-Banach theorem, with the characterization of maximum modulus algebras as an application. The algebra theme is continued in the final section, which concerns the theorem of Bers that the algebraic structure of the ring of analytic functions on a plane domain determines the domain up to (anti)conformal equivalence. The temptation to include more must have been almost overwhelming, but Gaier has laudably resisted it and limited the new material to just under 30 pages. Landau would undoubtedly have approved. This is not to place the book above all criticism. A number of important references have been omitted. S u r e l y Szdsz's b e a u t i f u l g e n e r a l i z a t i o n [10] of Schwarz's lemma deserves at least an honorable mention in the bibliography, if not more extended discussion. Nor should Ian Richards' elegant survey on analyticity [8] have gone unmentioned. The section on the high indices theorem demands reference to Ingham's elementary proof of that result [3]; see also [4]. Another article by Ingham on a related theme actually appears in a memorial volume for Landau [11]. I wish Gaier had found room to include the beautiful Karamata-Wielandt proof of the Hardy-Littlewood Tauberian theorem ([358] of the bibliography) and Hironaka's extension [156a] of Bers's theorem to fields of meromorphic functions (and Riemann surfaces); but I can also imagine his reasons for not doing so. In a subject in which "less is more," it is difficult to argue with an approach based on minimalism. Landau has not been forgotten. He is remembered not only for his books, but also for his students (the most famous of w h o m was Carl Ludwig Siegel) and, of course, for his results. L a n d a u ' s t h e o r e m , 4 a striking finite version of the Little Picard Theorem, is still part of the standard graduate curriculum in complex variables, and the evaluation of the Landau constant 5 remains one of the central open problems in 4 L a n d a u ' s Theorem: Let F(z) = E~= 0 a~zn, al # 0, be analytic on ~R = (]Z[ < R} a n d s u p p o s e that F(z) # 0, I for zed~R. Then R ~ C(ao, al). Picard's Theorem is an instant consequence. Indeed, s u p p o s e F is entire a n d omits the values 0 a n d 1. If F is nonconstant, w e m a y choose c o o r d i n a t e s so that F'(0) # 0 a n d t h e n a p p l y L a n d a u ' s T h e o r e m to obtain a contradiction to the analyticity of F on arbitrarily large discs about the origin. 5 According to the Bloch-Landau Theorem, w h e n e v e r F(z) = z + ~ = 2 a~z" is analytic on the unit disc A = {Iz[ < 1}, F(~) will contain a disc of radius e, an absolute constant. The best (i.e., largest) value of is the Landau constant L. It is k n o w n that .5 < L ~ F (%) F (u F (1/6) < .5432588 . . . . For an up-to-date a n d detailed discussion of these a n d related matters, see [7].

geometric function theory. It is Landau's proof of (his version of) Bloch's theorem that we teach our students; and such is the vitality of his ideas that the seemingly innocent trick which makes that proof work is really all that is needed to solve a number of other problems that have baffled and beguiled mathematicians over the years [13], [1], (cf. [12, p. 95]). Recently Landau's name was in the news again. This past February marked the opening of the Landau Center for Research in Mathematical Analysis, funded by the West German government, at the Hebrew University of Jerusalem. It is not widely k n o w n , but Landau played a central role in establishing the study of mathematics in Israel. 6 Along with Hadamard, Einstein, Freud, and other notables, he served on the first Curatorium (Board of Governors) of the Hebrew University. It was Landau who arranged for the purchase of Felix Klein's mathematical library by the Hebrew University, and it was he who drafted the plan for establishing the Institute of Mathematics there. In fact, Landau was actually the first professor of mathematics in Israel. He spent the winter semester of the academic year 1927-28 in Jerusalem as visiting professor at the n e w l y o p e n e d Institute of Mathematics, where he gave a course of lectures--in Hebrew. (Reference to these lectures a p p e a r s - - i n Hebrew characters--at the top of page 620 of his important article [5].) For a time, Landau was even under serious consideration for the post of rector of the new university; but, in the end, he returned to G6ttingen. The opening festivities for the Landau Center stretched over the better part of a week and included a dozen lectures by distinguished Israeli and German mathematicians, including Agmon, Furstenberg, Grauert, Hirzebruch, and Piatetski-Shapiro. A bibliography of Landau's work, compiled by I. J. Schoenberg (his son-in-law), can be f o u n d in [11]. Gaier's new edition of the Darstellung und Begrfindung is one more worthy m o n u m e n t to the memory of this hero of twentieth-century mathematics.

References 1. Robert Brody, Compact manifolds and hyperbolicity, 6 Landau was the scion of one of Berlin's leading Jewish families. His father, LeopoldLandau(1848-1920), Professorof Gynecologyat the Universityof Berlin, ran together with his brother Theodor the most famouswomen's clinic in Germany.His mother, Johannan~e Jacoby, came from a wealthy familyof bankers. Leopold Landau also found time to be activein publicand communalaffairs, serving as an alderman, helping to found the BerlinAkademiefiir die Wissenschaft des Judentums, and taking an active part in Zionist causes. A more remote ancestor, whose Hebrew name Edmund bore, was EzekielLandau(1713-1793) of Prague, ChiefRabbi of Bohemia and one of the leadingrabbinicalauthoritiesof the eighteenth century. Landau'sfather-in-lawwas the distinguishedchemistPaul Ehrlich, pioneer of modem hematology, histology, immunology, and chemotherapy,and winner of the 1908NobelPrize in Medicine.

Trans. Amer. Math. Soc. 235 (1978), 213-219. 2. G. H. Hardy and H. Heilbronn, Edmund Landau, J. London Math. Soc. 13 (1938), 302-310. 3. A. E. Ingham, On the 'high-indices theorem' of Hardy and Littlewood, Quart. J. Math. 8 (1937), 1-7. 4. A. E. Ingham, On Tauberian theorems, Proc. London Math. Soc. (3) 14A (1965), 157-173. 5. Edmund Landau, Uber die Blochsche Konstante und zwei verwandte Weltkonstanten, Math. Z. 30 (1929), 608-634. 6. J. E. Littlewood, Littlewood's Miscellany, Cambridge: Cambridge University Press (1986). 7. C. David Minda, Bloch constants, J. d'Analyse Math. 41 (1982), 54-84. 8. Ian Richards, Axioms for analytic functions, Adv. in Math 5 (1970), 311-338. 9. Allen Shields, Klein and Bieberbach: Mathematics, race, and biology, The Mathematical Intelligencer, vol. 10, no. 3 (1988), 7-11. 10. Otto Sz~isz, Ungleichheitsbeziehungen ffir die Ableitungen einer Potenzreihe, die eine im Einheitskreise beschr/inkte Funktionen darstellt, Math. Z. 8 (1920), 303-309. 11. Paul Tur~in (ed.), Number Theory and Analysis, A collection of papers in honor of Edmund Landau (1877-1938), New York: Plenum Press (1968). 12. H. Wu, Some theorems on projective hyperbolicity, J. Math. Soc. Japan 33 (1981), 79-104. 13. Lawrence Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), 813-817. Department of Mathematics and Computer Science Bar-Uan University 52100 Ramat-Gan, Israel

Geometries and Groups by V. V. Nikulin and I. R. Shafarevich (Translated by Miles Reid) Berlin-Heidelberg: Springer-Verlag, 1988 viii + 251 pp.

Reviewed by John Stillwell If geometry is a good t h i n g - - a n d most of us agree that it i s - - t h e n w h y is so little taught in our schools and universities? I believe it is because we mathematicians have killed off natural interest in the subject by being too boring, complicated, and abstract. Let me begin with a confession: I find Euclidean geometry boring. Euclid's Elements was a landmark in the history of mathematics, but it is surprising that it remained influential until the twentieth century. It was already superseded in the seventeenth century by the more powerful techniques of analytic geometry and calculus. But not only does Euclidean geometry live on, m o d e r n mathematicians have found a way to make it more b o r i n g - - b y taking out all the pictures and expressing everything in terms of linear algebra! And what of the seventeenth-century developments that superseded Euclid? Analytic geometry and calTHE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989 6 3

culus open our eyes to exciting new vistas of curves and surfaces, b e y o n d the bland straight lines and planes of Euclid. (Admittedly Euclid studied the circle, but being without calculus he could not answer one of the simplest questions about i t - - w h a t is its length?) Students get occasional glimpses of this new territory in calculus courses but if, at some late stage, they opt for a course in algebraic or differential geometry they are in for a big shock. The basic concept of differential g e o m e t r y - - t h e differential manifold--takes an hour to define, and things generally get worse after that. (A colleague told me that students dubbed his differential geometry course "notation on manifolds.") As for algebraic g e o m e t r y - - n o w a d a y s you can't begin until you know all of commutative algebra and hefty amounts of analysis and algebraic topology. And these are just the technical requirements. To see the point of the subject, the student should be fluent in number theory, complex analysis, and projective geometry. Projective geometry? What the heck is that? This is the problem with modern geometry. In order to understand anything, you are supposed to know everything. We withhold the glories of the subject until students have all the technical prerequisites, then obliterate them by excessive abstraction and generality. Fortunately, not all mathematicians have succumbed to this tendency. Hans Rademacher wrote a lovely little book called Higher Mathematics from an Elementary Point of View [4] that tackled some deep (and currently fashionable) ideas like modular forms using only elementary methods. Now, in the book under review, Nikulin and Shafarevich have presented some beautiful geometry with a minimum of prerequisites instead of the maximum. Geometries and Groups could be called "Euclid on Manifolds," except that Nikulin and Shafarevich never use the term " m a n i f o l d . " Instead, they invite the reader to think about spaces in which Euclidean geometry holds within a certain radius of each point, as seems to be the case for the three-dimensional space in which we live. This raises an interesting question, even in two dimensions: If a surface S has this "locally Euclidean" property, must S be the Euclidean plane? The short answer is no (as is quickly shown by examples such as the cylinder) and a complete answer is given by a splendid theorem of Killing and Hopf which is the centrepiece of the book: A surface is locally

Euclidean if and only if it is the quotient of the plane by a discontinuous group of isometries, acting without fixed points. Before proving the Killing-Hopf theorem, which is rather subtle because it involves covering spaces, Nikulin and Shafarevich treat the easier problem of finding the discontinuous groups of Euclidean isometries. This enables them to find the four surfaces that are nontrivial quotients of the plane: the cylinder, the twisted cylinder, the toms, and the Klein bottle. Each 64

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

nmmmmnmmd imllllmnnml imnummllmlUl immummmmu immminnmmu immmlllllmnl immmmmmmm immmmmmmm rmmmmmmmmma immmmmmmm immmmmmmma immmmmmilmm immmmmmmmm immmmmmmmm immmmmmmmm ~mmmmmmmm immmmnmmmi ~mmmmmmmmm tmmmmmmmm immmmmmma immmmmmmmma

ii) 1

Figure 1. Quotients of the plane.

of these surfaces is constructed from a fundamental region of the corresponding discontinuous group. For example, the cylinder results from joining opposite sides of a strip bounded by parallel lines, the torus from joining opposite sides of a parallelogram. There is a degree of abstraction in the torus construction, because it cannot be carried out in ordinary space without distorting lengths, as is suggested by Figure 1 (from Geometries and Groups, p. 31). One either has to think of the torus as the ordinary doughnut shape with a nonstandard metric on it (the "length before the surface was distorted") or, better, as an abstract surface given by coordinate patches that are discs in the plane.

The basic concept of differential geometrym the differential manifold~takes an hour to define, and things generally get worse after that. This paves the way for the precise definition of a locally Euclidean surface and a proof of the KillingHopf theorem. Readers familiar with the definition of a differentiable manifold will notice that Nikulin and Shafarevich are essentially defining a two-dimensional Euclidean manifold--the transition maps between the charts being Euclidean isometries--except that they make the stronger-than-usual assumption that the charts are all discs of the same radius. This assumption enables them to prove the KillingHopf theorem without using any facts about open and closed sets or compactness. One wonders whether it is wise to be quite so fundamentalist at this point since their proof, which uses only high school geometry, is considerably longer than the proof of Hopf [2], which uses a very modest amount of topology. Nevertheless, it completes a remarkable achievement. Simply by "localising" Euclidean geometry, Nikulin and Shafarevich have launched the reader into the modern geometric world of manifolds, group actions, and quotient spaces.

Of course, this is not the first time Euclidean geometry has been used as a point of departure to new geometric worlds, but it is probably the first time beginning geometers have been transported so far. The Killing-Hopf theorem, in particular, has previously appeared only in advanced monographs such as Wolf [7]. The last chapter of Nikulin and Shafarevich takes readers even further: into the moduli space of Euclidean geometries on the torus, which is made up of the different "shapes" a fundamental parallelogram can have. (Euclidean geometries are classified by shape, i.e., "up to similarity," because there is no preferred unit of Euclidean length.) One reason for the belated appearance of this circle of ideas at the elementary level is its formidable historical b a c k g r o u n d - - i n quadratic forms, elliptic functions, hyperbolic geometry, and Riemann surfaces. This is where coverings, discontinuous groups, and moduli first emerged and where mathematicians have come to think t h e y s h o u l d stay. H o w e v e r , they emerged as simplifying and unifying ideas that can stand alone like the group concept itself. Just as one now teaches group theory before the Galois theory, which was its historical precedent, so one can put locally Euclidean surfaces before quadratic forms or hyperbolic geometry. In fact, Nikulin and Shafarevich do just that in the book. They are well aware of the history, but they turn it upside down so as to start at the easy end. My only disappointment is that they say too little about the tangled background of their subject for the reader to appreciate h o w well they have untangled it. The climax of the book is the following theorem stated on p. 242. The set of all geometries on the torus (or of lattices on the plane) Up to similarity is itself a geometry, that of a triangle in the Lobachevsky plane with one vertex at infinity and angles of ~ and ~ at the two finite vertices. The triangle in question is the standard fundamental region for the action of the (extended) modular group on the upper half plane, concerning which Nikulin and Shafarevich add, "A further result of our study is the fact that we have constructed an example of a discrete group of motions of the Lobachevsky plane, the modular group." The book ends only a few lines after this, perhaps before the magic of these results has had time to sink in. The authors add some historical remarks, which may help readers to appreciate the story behind the theorem. However, they are rather brief and, in my opinion, they miss some of the main episodes. I would therefore like' to add a little more history, although a full explanation would require a book in itself. The t w o t y p e s of object i n v o l v e d in the final theorem, the Euclidean lattices and the hyperbolic tri-

Figure 2. Independent periods.

angle, lived parallel lives for nearly a century before being definitively linked by group theory and geometry. With hindsight, we can see that the group GL(2,Z) (and its close relatives SL(2,77) and PSL(2,Z)) appears at nearly every turn in the story. The slow emergence of the final theorem is no doubt due to the fact that the role of groups in geometry was not recognised until the 1870s. GL(2,7/), the group of 2 x 2 integer matrices with determinant _ 1, can be viewed as the group of isomorphisms of a two-dimensional lattice. It arose from Lagrange's study of quadratic forms in 1773 (see [5], p. 35). Lagrange s h o w e d , in effect, that two forms ax 2 + 2bxy + cy 2 and A X 2 + 2BXY + CY 2 are equivalent if one can be converted to the other by a substitution X = c~x + BY

Y = ~/x + 8y Of course, Lagrange did not use the language of matrices, or groups, as it had not yet been invented. SL(2,Z) is the subgroup of matrices with determinant + 1 (corresponding to orientation-preserving transformations of the lattice, or proper equivalence in the language of quadratic forms). The modular group PSL(2,Z) is the quotient SL(2,Z)/{ +_-I}. The hyperbolic triangle emerges when the modular group (or its extension) is viewed as acting, not on lattices, but on points of the complex plane. Each lattice can be encoded up to similarity by a single complex n u m b e r 9 by expressing the generating vectors as complex numbers ~i, (~2 and setting ~ = ~ol/o~2 (see THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

65

Figure 2). Then the extended modular group is the group of linear fractional transformations

Z

~z + ~ w h e r e ~ , ~ , ~ , S E Z , ad - bc = +-1. 8z +

~.-> m

In geometric terms, it is the group of the famous modular tessellation below, consisting of the images of any one region u n d e r the transformations in the group. Thus any region can serve as a fundamental region for the extended modular group. One such is the triangle described by Nikulin and Shafarevich shown in black in Figure 3. A fundamental region for the modular group was first discovered by Gauss in his research on imaginary quadratic forms. For further information on this see [5], p. 92. A stronger link b e t w e e n lattices and the fundamental region for the modular group was forged in the theory of elliptic functions and elliptic curves, which flourished b e t w e e n the 1820s and 1870s and is enjoying a revival today. As early as 1797, Gauss discovered that the inverse function of the elliptic integral

l§

I 1

|

~

I

$25

1

435

2

5 3

4

A = {moo1 + nta2: m,n ~ Z},

where 001, 002 are independent complex numbers. Jacobi noticed in 1834 that the curve y2 = p(x) is parametrized by x = f(z), y = f'(z), with distinct points on the curve corresponding to distinct equivalence classes {z + m(91+ nco2}modulo A. For this reason, y2 = p(x) is called an elliptic curve. We would say that the elliptic function f defines an analytic isomorphism between the curve and the quotient C/A of the plane by the lattice. Thus, with the intervention of elliptic functions, the problem of classifying lattices received additional motivation from the problem of classifying cubic curves y2 = p(x). The latter problem had been of interest ever since N e w t o n had shown that every cubic curve is projectively equivalent to one of this special type. H o w e v e r , the illumination of geometry by elliptic functions did not actually happen for some decades. It was probably not even realized that y2 = p(x) was a torus until Riemann introduced Riemann surfaces in 1851, enabling the topological form of all algebraic curves to be easily understood. Indeed, it was Riemann w h o first made it obvious w h y elliptic functions have two periods. He pointed out that the independent periods (91 and (92 correspond to integrals around topologically independent paths on the torus (Figure 4). As for the classification of cubic curves, this also was done first without elliptic functions. Again in 1851, Salmon discovered that cubic curves are classified up

$

Figure 3. The modular tessellation. 66

f~) dt/X/-i -Z t 4 had two periods, 6 and i&, where 6 is a certain real number. It turned out to be a general property of elliptic integrals, rediscovered by Abel and Jacobi in the 1820s, that their inverse functions had two independent periods, (91 and o02, over the complex numbers. If for example f - l(z) = f~ d t / V ~ , p(t) cubic without repeated roots, then f(z + m(91 + n(92) = f(z) for all z ~ C and m,n ~ Z. Such an f is called an elliptic function, and there is a field of elliptic functions associated with each lattice

T H E M A T H E M A T I C A L I N T E L L I G E N C E R VOL. 11, N O . 4, 1989

Figure 4. Independent paths on the torus.

to projective equivalence by a single complex number "r. Salmon defined ~" geometrically, as a cross ratio of four tangents, but it was later found that ~- is nothing but to~/to2. In other words, if the curve is viewed as a quotient C/A of C by a lattice A, then its projective class "r is simply the shape of the lattice. For reasons connected with the history of elliptic integrals, "r became known as the modulus.

We withhold the glories of the subject until students have all the technical prerequisites, then obliterate them by excessive abstraction and generality. To summarise the story so far: by 1851 it had been discovered that cubic curves, which were tori, corresponded to lattices, and that geometrically distinct (at this stage, projectively distinct) curves corresponded to distinct lattice shapes. What was to stop mathematicians from defining tori directly as quotient spaces C/A at this stage? J For one thing, the role of groups, let alone that of quotient spaces, had not yet been recognised in geometry. Without the formal justification of the quotient construction, it was probably difficult to believe in the torus C/A. C/A is not just a topological toms, it is fiat (locally like a plane) and cannot exist in ordinary three-dimensional space. At this point a key contribution was made by Clifford, which probably accounts for the considerable credit Nikulin and Shafarevich give to Clifford in their historical remarks. In 1873 he discovered a fiat torus, now called the Clifford surface, in three-dimensional projective space. In fact, once one realises where to look for it, such a torus is obvious. A projective line is closed and finite, hence its cylindrical neighbourhoods are tori and, being locally like an ordinary cylinder, they are flat. Clifford emphasised the flatness by saying "the geometry of this surface is the same as that of a finite parallelogram whose opposite sides are regarded as identical." The discovery of a "concrete" fiat torus removed a psychological barrier to the consideration of quotient surfaces. H o w e v e r , Clifford died in 1877, before the scope of the quotient construction was realised. In the 1870s, the only other known examples of groups acting on surfaces were the modular group on C and a few other groups of linear fractional transformations originating from differential equations. (For more on the input of' differential equations to the story, see [1].) All this changed in 1880 when Poincar6 realised that these transformations were identical with those of Lobachevsky (hyperbolic) geometry. This completed the proof of the final theorem by showing

that the tessellation for the action of the modular group was a regular tessellation of hyperbolic geometry, and in particular that its fundamental region is a hyperbolic triangle. It also created an environment in which the theorem was the centrepiece of a broad and exciting theory. Poincar6 showed that regular tessellations of the hyperbolic plane were infinitely more varied than those of the Euclidean plane, and this led to an interplay between geometry, topology, and analysis far richer than in the Euclidean case (see [3]). But was the Euclidean case properly understood after all this? It was known how to construct certain locally Euclidean surfaces as quotients of the plane, but did all locally Euclidean surfaces arise in this way? In 1890 Klein raised this q u e s t i o n as part of the problem of determining "space f o r m s " - - t h e manifolds which have constant curvature and which are therefore locally spherical, Euclidean, or hyperbolic. Killing in 1893 dubbed this the "Clifford-Klein space form problem" and attempted a proof that all such space forms are quotients of the corresponding n-dimensional spherical, Euclidean, or hyperbolic space (5 n, R", or H") by a discontinuous group of isometries. His proof does not meet modern standards of rigour, partly for want of a good definition of space form. This became possible with the appearance of Weyl's 1913 book on Riemann surfaces [6], which introduced the apparatus of charts now used to define geometric or differentiable structures on manifolds. Using this and some other ideas of Weyl, Hopf [2] gave the first rigorous proof of Killing's theorem in 1925. Thus the starting point of Geometries and Groups is, in a sense, the culmination of 150 years of mathematics. It doesn't quite allow us to dear our library shelves of collected works from Lagrange to Weyl, but it certainly makes the road less rocky for those who want to grasp some of their key ideas.

References

1. J. Gray, Linear Differential Equations and Group Theory from Riemann to PoincarE, Basel: Birkh/iuser (1986). 2. H. Hopf, Zum Clifford-Kleinschen Raumformproblem, Math. Ann. 95 (1925), 313-339. 3. H. Poincar6, Papers on Fuchsian Functions, New York: Springer-Verlag (1985). 4. H. Rademacher, Higher Mathematics from an Elementary Point of View, Basel: Birkh/iuser (1983). 5. W. Scharlau & H. Opolka, From Fermat to Minkowski, New York: Springer-Verlag (1985). 6. H. Weyl, The Concept of a Riemann Surface, Cambridge: Addison-Wesley (1955). 7. H. Wolf, Spaces of Constant Curvature, New York: McGraw-Hill (1967).

Department of Mathematics Monash University Clayton, Victoria 3168 Australia THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

67

A Source Book in Mathematics, 1200-1800 by Dirk J. Struik Princeton: Princeton University Press, 1986 iv + 427 pp.

Reviewed by Craig G. Fraser With the re-issue in paperback in 1986 of Dirk J. Struik's 1969 Source Book, we now possess a readily available collection of original readings, translated, edited, and annotated, that will be useful both to the professional interested in historical background and to students of the history of mathematics. I have used the Source Book in a third-year undergraduate historyof-mathematics course, supplementing it with historical monographs and English-language articles from such journals as Historia Mathematica, Archive for History of Exact Sciences, and Centaurus. These articles have appeared since the book was first published in 1969 and represent the contributions of professional historians to the study of seventeenth- and eighteenth-century mathematics. In the present review I share my impressions of Struik's book with readers of the Mathematical Intelligencer, providing a possibly idiosyncratic selection of references to recent historical work and discussing how this work has altered our understanding of the history. [For a broader look at sources and histories of mathematics see J. V. Grabiner's review of H o w a r d Eves, Great Moments in Mathematics (after 1650) (1981), in American Mathematical Monthly 93 (1986), 491-494.] Over half of the Source Book is devoted to the history of the calculus, beginning with Stevin's study of centers of gravity and following through to the work of Lagrange on the algebraic foundations of analysis. This emphasis accurately reflects the development of mathematics as it assumed its present form in the early m o d e m period. The calculus has attracted and continues to attract intense historical interest as an example of the development of a new and major branch of mathematics and as an extremely useful tool in the mathematical investigation of nature. Two good surveys of the subject that have appeared since 1969 are M. E. Baron's The Origins of the Infinitesimal Calculus (1969) and C. H. Edwards's The Historical Development of the Calculus (1979). One of the fascinations of the period prior to Newton and Leibniz is to contemplate the range and diversity of techniques devised to determine areas and tangents. Galileo's principle of composition of motions and its employment by Roberval to study the cycloid are presented by Struik in two interesting extracts. The sections on Cavalieri are usefully supplemented by K. A n d e r s o n ' s "Cavalieri's Method of Indivisibles," Archive for History of Exact Sciences 31 (1985), 291-367, a study that focuses on the conceptual fea68

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 4, 1989

MENSISOCTOBRISA. MDC LXXXIV. 467 2VOYtl METHODV$ PRO M A X I M I $ Z T MI. nira~ ~ i:ernque t4ngentihu~ ~ ~tt~ necfra~t~;, nec irrationAkl quaatila/et maratur, ~'jTngulArepro i#~ calculigr G.G. L. ltaxisAX,&curvzplures, ut VV, "~'\~7, YY, ZZ, quarum ordi- ran. xm

S

nattr, ad axem normales, VX, \v/X, YX, ZX, qua vocentur refFedive, v, vv, yj z; & ipfa AX ab ftiffa ab axe, voeetur x. Tangcmes tint VB, W C , YD, ZE axi occurrentes refpe~ive in pun~qis B~ C, D~ E. Jamre~a aliquaproarbitrioaffumta vocetur dx, & recta qua fit ad dx, utv(velw,,vely, velz)ef~adVg (velXY/G~ vel YD, vel ZE) vocetur d v fvel d vv, vel dy vel dz) five differentia ipl~rum ~ (vel ipfa. turn vv, auty, aut O Hispofitis calculi regulz crunt tales : Sira quantitas data conf~ans~ erit da ~quahs o, & d ax erk ~qu" a dx: fi fit y ~clu. v (feu ordi,ata quzvis eurvae YY, Zclualis cuivis or. dinat~erefpondenticurv.-eVV)eritdyzqu, dg. Jam Additio ff st4btratti,..~ fit z -y~ vv'~ x .xqu. v, erit dz--y~vv'~x feu d r , ~qu. dz - d y ' ~ d v v t dx. ~lulti?licatlo, " d ~ x v x q u . x d v t v d x , feu pol~to y ~qu. xv, act d y ~qu. x d ~-Ip d x In ;,rbitrio enim e~ vel forrnulam~ ut xz', vel compendio pro ca literam~ ut y, adhibere. Notandum & x & d x eodem modo in hoc calculo tra~ari,ut y & dy,vel alia m litera m indeterminatam cure fua differenfiali. Notandum etiarn non dari femper regreffuln a diffcrentiali ~quatione, nifi cum quadam cautione, de quo alibi. '~dy~-yd~ -

Porro Di~;f;o, d--vr y

@ofitoz zqu.

) dzxqu, Y

Y~ Q~oad gignalaoc probe notandum, tuna in calculo pro llteca ~'ubl~ituitur fimplicitcr ejus differentially/fervari quidem eadem ~igna~ ~lepre r t~rYoi~f dz, pro..z s ut 9 additione & fubtraO.ionepaulo ante llol?ra apparel; fed quando ad exege~n valot'~a ]enkur~ I'eu cure confideratur ipl~us z rdatio ad x , mnc~ apFarere, an v.alor iplqus dz llt quahtitas afl~rmativa, an nihilo minor l~unegadva : uod poRerins cure fit~ tune tangera ZE ducimra pund~o Z non vers ^~ led in partes contrarias fee infra X, id e~ tmac c-urn ipf~ ordina tat lq n n ~ z dr

Figure 1. The opening page of Leibniz's Acta Eruditorum 3 (1684), 467, containing the first publication of the differential calculus. (Reproduced from the Source Book, p. 273). tures that distinguish his approach from modern integration. An important omission in the Source Book is Descartes's method of normals, a technique that was refined by the Leiden school of Cartesians at the middle of the century. This subject is examined in detail along with problems of rectification, also not treated by Struik, in J. Van Maanen's "Hendrik van H e u r a e t (1634-1660?) His Life and Mathematical Work," Centaurus 27 (1984), 218-279. The content and meaning of the early calculus was to a c o n s i d e r a b l e d e g r e e e x p r e s s e d in the n e w branches of mechanics, particle dynamics, theory of rigid bodies, fluid mechanics, and the theory of elasticity. Struik states in the preface that he will consider applied m a t h e m a t i c s " o n l y w h e n it had a direct bearing on the development of pure mathematics." Even interpreted strictly this condition would permit a larger scope for applications than he allows, and a weakness of the Source Book remains its failure to include more topics from mathematical mechanics. The selections from Newton, for example, do not convey a sense for how his calculus functioned in the actual derivations of the new dynamics. This subject is discussed in D. T. Whiteside's "The Mathematical Prin-

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Aprds cela prenant vn point a difcretion dans la courbe, comme C, fur lequel ie fuppofe que l'inftrument'qui fert a la defcnre eft app tque, ie tire de ce point C- la ligne C B parallele a G A, & pourceque C B & B A font deux quantates mdetermmees & inconnu~s , ie les nomme l~ & rautre x. mais attln de trouuer le rapport de rvne ~.l'autre; ie confidere auffy le's quantitds connu~s qui determinent h defcription de cite ligne courbe, comme G A que ie nomme a, K L que ie nomme b, & N LparaUele'a G A queieno~me c. puis ie dis, comme N L eft ~ L K; ou c.~b, ainfi C B, ouy, eft ~lB K, qui eft 9

9

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par confequent ~ y : & B L eft -7 Y -- b, & ~ L'eft x-l-b "" b. de plus comme C B eft k L B, ouy/t 7Y'", b b ainti 7Y b a, ou GA, eft~ L A , oux q - T y --b, de fa~on que mulSf tipliant

53 Figure 2. Descartes's tracing device for the hyperbola. From La G~om~trie(1637), p. 321. (Presented as a diagram with translation, SourceBook, p. 156.) ciples Underlying Newton's Principia Mathematica," Journal for the History of Astronomy 1 (1979), 116-138; readers with a knowledge of French should consult F. De Gandt's "Le Style Mathdmatique des Principia de N e w t o n , " Revue d'Histoire des Sciences, 39 (1986), 195-222. R. Westfall's Never at Rest: A Biography of Isaac Newton (1980) contains chapters on mathematics and dynamics. Huygens's Horologium oscillatorium is now available (1988) in a full English translation with notes by R. J. BlackweU and introduction by H. J. M. Bos. Joella Yoder's recent book (1988) on Huygens should also be consulted. Eighteenth-century applications of the calculus in m e c h a n i c s are described in J. T. Cannon and S. Dostrov'sky's The Evolution of Dynamics, Vibration Theoryfrom 1687 to 1742 (1981) and C. Fraser's "D'Alembert's Principle: The Original Formulation and Application in Jean D'Alembert's Traitd de Dynamique (1743)," Centaurus 28 (1985), 31-61, 145-159.

Struik presents several selections to illustrate the Leibnizian calculus in the early eighteenth century. Recent historical studies that have significantly revised our understanding of this subject are Bos's "Differentials, Higher-Order Differentials and the Derivative in the Leibnizian Calculus," Archive for History of Exact Sciences 14 (1974), 1-90, and S. B. Engelsman's Families of Curves and the Origins of Partial Differentiation (1984). These authors develop a naturalistic picture of an early calculus that differs conceptually and technically from modern analysis. The extracts from the Source Book, in which the context is necessarily suppressed and the technical idiom has been to some extent modernized, do not always fully convey the style and content of the historical subject. The "otherness" of past mathematics, indicated impressionistically in the original reproductions that appear throughout the book, is documented by Bos and Engelsman for the case of the early Leibnizian calculus in full and interesting detail. A specific contribution of these studies is to have clarified eighteenth-century conceptions of rigor. In an article on the history of analysis in the nineteenth century, I. Grattan-Guinness notes that "a prominent feature o f . . . [modern] mathematical analysis is its apparent autonomy of algebra and geometry. Although algebraic formulae are manipulated and geometrical diagrams drawn, they are not essential to the rigour or justification of the subject" (From the Calculus to Set Theory, 1630-1910 (1980), p. 95). In the early eighteenth-century, by contrast, the primary notions of calculus were the algebraic variable and geometric curve. The "metaphysics" of the subject was rooted in an implicit understanding concerning the relation of geometrical and analytical conceptions, not in any particular logical attitude towards the infinitely small. This viewpoint assumed a faith in formalism and a global approach to the validity of expressions that might mistakenly be termed "naive," but which in fact was based on definite convictions about the nature of mathematics 9 Another question clarified by these histories concerns the relevance of non-standard analysis for interpretation of pre-Cauchy calculus. Bos (pp. 82-83) rejects Abraham Robinson's suggestion that the early calculus needs to be reappraised in the light of his discoveries: I do not think that the appraisal of a mathematical theory, such as Leibniz's calculus, should be influenced by the fact that two and three-quarters centuries later the theory is "vindicated" in the sense that it is shown that the theory can be incorporated in a theory which is acceptable by present-day mathematical standards . . . . If the Leibnizian calculus needs a rehabilitation... I feel that the legitimate grounds for such a rehabilitation are to be found in the Leibnizian theory itself. He further observes (p. 84):

THEMATHEMATICAL INTELLIGENCER VOL.11,NO.4,198969

Leibnizian infinitesimal analysis deals with geometrical quantities, variables and differentials, while non-standard analysis, as well as modem real analysis in general, deals with real numbers, functions and (notwithstanding its acceptance of differentials) derivatives. Bos and Engelsman note the increasing tendency of the calculus in the first half of the eighteenth century to favor analytical over geometrical methods. Research that continued in this tradition is described in J. L. Greenberg's "Alexis Fontaine's Route to the Calculus of Several Variables," Historia Mathematica 11 (!984), 22-38, J. V. Grabiner's The Origins of Cauchy"s Rigorous Calculus (1981), Engelsman's "Lagrange's Early Contributions to the Theory of First Order Partial Differential Equations," Historia Mathematica 7 (1980), 7-23, and Fraser's "Joseph Louis Lagrange's Algebraic Vision of the Calculus," Historia Mathematica 14 (1987), 38-53. Struik's illustration of the British fluxional calculus, a selection from Brook Taylor, is profitably read in conjunction with L. Feigenbaum's "Brook Taylor and the Method of Increments," Archive for History of Exact Sciences 34 (1985), 1-140. (A facsimile of Taylor's work of 1715 with English translation is presented in Feigenbaum's 1981 Yale dissertation, available by order from Ann Arbor.) The passage to the limit used by Taylor to obtain his original series remains interesting as a remarkable example of mathematical reasoning. Background to the extract from Euler on trigonometry will be found in V. J. Katz's "The Calculus of the Trigonometric Functions," Historia Mathematica 14 (1987), 311-324. The selections from Johann and Jakob Bernoulli, Euler, and Lagrange on the calculus of variations are complemented by H. Goldstine's A History of the Calculus of Variations from the 17th Through the 19th Century (1980) a n d F r a s e r ' s "J. L. L a g r a n g e ' s Changing Approach to the Foundations of the Calculus of Variations," Archive for History of Exact Sciences 32 (1985), 151-191. Next to the calculus, the most prominent subject of the Source Book is the invention and development of symbolic algebra and analytic geometry in the writings of Vi6te, Descartes, and Fermat. This chapter in the history of mathematics has exerted a special fascination for scholars in recent years. Historical study of the analytic art reveals interesting "meta-mathematical" aspects of algebra that have been by-and-large overlooked in m o d e r n mathematical philosophy. Descartes' original program of research, an investigation of curves that could be used in the construction of geometrical problems, indicates how traditional concerns often dominated mathematics during the Scientific Revolution. Historical studies are M. S. Mahoney's The Mathematical Career of Pierre de Fermat (1974) and Bos's "On the Representation of Curves in Descartes' G~omdtrie," Archive for History of Exact Sciences 24 (1981), 295-338. The 1980 collection Descartes" Philos70

THE MATHEMATICAL INTELLIGENCER VOL. I1, NO. 4, 1989

ophy, Mathematics and Physics, ed. S. Gaugkroger, contains notable essays by J. A. Schuster, S. Gaugkroger, Mahoney and E. Grosholz. Schuster provides a detailed investigation of Descartes' conception of "universal mathematics" and its role in the evolution of his scientific philosophy during the 1620s. Reaction to analytical mathematics in the writings of Isaac Barrow is the subject of C. Sasaki's "The Acceptance of the Theory of Proportion in the Sixteenth and Seventeenth Centuries," Historia Scientiarum 29 (1985), 83-116. Subjects not dealt with in the Source Book, or considered very incompletely, include the development of numerical analysis, the foundations of Euclidean geometry, probability and statistics, and the philosophy of mathematics. Recent monographs or articles on these topics are Goldstine's A History of Numerical Analysis from the 17th through the 19th Century (1977); J. L. Greenberg's "Breaking a 'Vicious Circle': Unscrambling A.-C. Clairaut's Iterative Method of 1743,'" Historia Mathematica 15 (1988), 228-239; J. Gray's Ideas of Space (1979); I. Hacking's The Emergenceof Probability (1975); S. M. Stigler's The History of Statistics: The Measurement of Uncertainty before 1900 (1986); G. Buchdahl's Metaphysics and the Philosophy of Science: The Classical Origins Descartes to Kant (1969); and G. G. Brittan's Kant's Theory of Science (1978). Although topics in social and institutional history are less easily integrated into a programme of original readings, let me refer to the background presented in M. C. Jacob's The Newtonians and the English Revolution, 1689-1720 (1976); R. Hahn's The Anatomy of a Scientific Institution: The Paris Academy of Sciences, 1666-1803 (1971); and J. E. McClellan's Science Reorganized: Scientific Societies in the 18th Century (1985). Institute for the History and Philosophy of Scienceand Technology University of Toronto Toronto, OntarioM5S 1K7 Canada

by Robin Wilson*

Mathematics Education The teaching of mathematics has been featured on stamps of several countries throughout the world. The teaching of mechanics appears on the German Democratic Republic stamp (1972). The Guinea stamp (1962) shows a boy at a blackboard, and the stamp from the Maldive Islands (1970) shows the teaching of geometry

by television. On the Soviet stamp (1961), some young adults are studying trigonometry, and arithmetical calculations are depicted on the audio-visual stamp from Haiti (1966). Finally, the Anguillan stamp (1965) shows what happens to those who do not learn their multiplication tables!

* C o l u m n editor's a d d r e s s : Faculty of M a t h e m a t i c s , T h e O p e n University, Milton K e y n e s MK7 6 A A E n g l a n d THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 4 9 1989Springer-VerlagNew York 71