by E. T. Bell
Not many of the great mathematicians have been the absentminded dreamers that popular fancy likes to picture them. Poincaré was one of the exceptions, and then only in comparative trifles, such as carrying off hotel linen in his baggage. But many persons who are anything but absentminded do the same, and some of the most alert mortals living have even been known to slip restaurant silver into their pockets and get away with it.
One phase of Poincaré’s absentmindedness resembles something quite different. Thus (Darboux does not tell the story, but it should be told, as it illustrates a certain brusqueness of Poincaré’s later years), when a distinguished mathematician had come all the way from Finland to Paris to confer with Poincaré on scientific matters, Poincaré did not leave his study to greet his caller when the maid notified him, but continued to pace back and forth—as was his custom when mathematicizing—for three solid hours. All this time the diffident caller sat quietly in the adjoining room, barred from the master only by flimsy portières. At last the drapes parted and Poincaré’s buffalo head was thrust for an instant into the room. “Fous me dérangez beaucoup” (You are disturbing me greatly) the head exploded, and disappeared. The caller departed without an interview, which was exactly what the “absentminded” professor wanted.
Poincaré’s elementary school career was brilliant, although he did not at first show any marked interest in mathematics. His earliest passion was for natural history, and all his life he remained a great lover of animals. The first time he tried out a rifle he accidentally shot a bird at which he had not aimed. This mishap affected him so deeply that thereafter nothing (except compulsory military drill) could induce him to touch firearms. At the age of nine he showed the first promise of what was to be one of his major successes. The teacher of French composition declared that a short exercise, original in both form and substance, which young Poincaré had handed in, was “a little masterpiece,” and kept it as one of his treasures. But he also advised his pupil to be more conventional—stupider—if he wished to make a good impression on the school examiners.
Being out of the more boisterous games of his schoolfellows, Poincaré invented his own. He also became an indefatigable dancer. As all his lessons came to him as easily as breathing he spent most of his time on amusements and helping his mother about the house. Even at this early stage of his career Poincaré exhibited some of the more suspicious features of his mature “absentmindedness”: he frequently forgot his meals and almost never remembered whether or not he had breakfasted. Perhaps he did not care to stuff himself as most boys do.
The passion for mathematics seized him at adolescence or shortly before (when he was about fifteen). From the first he exhibited a lifelong peculiarity: his mathematics was done in his head as he paced restlessly about, and was committed to paper only when all had been thought through. Talking or other noise never disturbed him while he was working. In later life he wrote his mathematical memoirs at one dash without looking back to see what he had written and limiting himself to but a very few erasures as he wrote. Cayley also composed in this way, and probably Euler, too. Some of Poincaré’s work shows the marks of hasty composition, and he said himself that he never finished a paper without regretting either its form or its substance. More than one man who has written well has felt the same. Poincaré’s flair for classical studies, in which he excelled at school, taught him the importance of both form and substance.
The Franco-Prussian war broke over France in 1870 when Poincaré was sixteen. Although he was too young and too frail for active service, Poincaré nevertheless got his full share of the horrors, for Nancy, where he lived, was submerged by the full tide of the invasion, and the young boy accompanied his physician-father on his rounds of the ambulances. Later he went with his mother and sister, under terrible difficulties, to Arrancy to see what had happened to his maternal grandparents, in whose spacious country garden the happiest days of his childhood had been spent during the long school vacations. Arrancy lay near the battlefield of Saint-Privat. To reach the town the three had to pass “in glacial cold” through burned and deserted villages. At last they reached their destination, only to find that the house had been thoroughly pillaged, “not only of things of value but of things of no value,” and in addition had been defiled in the bestial manner made familiar to the French by the 1914 sequel to 1870. The grandparents had been left nothing; their evening meal on the day they viewed the great purging was supplied by a poor woman who had refused to abandon the ruins of her cottage and who insisted on sharing her meager supper with them.
Poincaré never forgot this, nor did he ever forget the long occupation of Nancy by the enemy. It was during the war that he mastered German. Unable to get any French news, and eager to learn what the Germans had to say of France and for themselves, Poincaré taught himself the language. What he had seen and what he learned from the official accounts of the invaders themselves made him a flaming patriot for life but, like Hermite, he never confused the mathematics of his country’s enemies with their more practical activities. Cousin Raymond, on the other hand, could never say anything about les Allemands (the Germans) without an accompanying scream of hate. In the bookkeeping of hell which balances the hate of one patriot against that of another, Poincaré may be checked off against Kummer, Hermite against Gauss, thus producing that perfect zero implied in the scriptural contract “an eye for an eye and a tooth for a tooth.”
Following the usual French custom Poincaré took the examinations for his first degrees (bachelor of letters, and of science) before specializing. These he passed in 1871 at the age of seventeen—after almost failing in mathematics! He had arrived late and flustered at the examination and had fallen down on the extremely simple proof of the formula giving the sum of a convergent geometrical progression. But his fame had preceded him. “Any student other than Poincaré would have been plucked,” the head examiner declared.
He next prepared for the entrance examinations to the School of Forestry, where he astonished his companions by capturing the first prize in mathematics without having bothered to take any lecture notes. His classmates had previously tested him out, believing him to be a trifler, by delegating a fourth-year student to quiz him on a mathematical difficulty which had seemed particularly tough. Without apparent thought, Poincaré gave the solution immediately and walked off, leaving his crestfallen baiters asking “How does he do it?” Others were to ask the same question all through Poincaré’s career. He never seemed to think when a mathematical difficulty was submitted to him by his colleagues: “The reply came like an arrow.”
At the end of this year he passed first into the École Polytechnique. Several legends of his unique examination survive. One tells how a certain examiner, forewarned that young Poincaré was a mathematical genius, suspended the examination for three quarters of an hour in order to devise “a ‘nice’ question”—a refined torture. But Poincaré got the better of him and the inquisitor “congratulated the examinee warmly, telling him he had won the highest grade.” Poincaré’s experiences with his tormentors would seem to indicate that French mathematical examiners have learned something since they ruined Galois and came within an ace of doing the like by Hermite.
At the Polytechnique Poincaré was distinguished for his brilliance in mathematics, his superb incompetence in all physical exercises, including gymnastics and military drill, and his utter inability to make drawings that resembled anything in heaven or earth. The last was more than a joke; his score of zero in the entrance examination in drawing had almost kept him out of the school. This had greatly embarrassed his examiners: “. . . a zero is eliminatory. In everything else [but drawing] he is absolutely without an equal. If he is admitted, it will be as first; but can he be admitted?” As Poincaré was admitted the good examiners probably put a decimal point before the zero and placed a 1 after it.
In spite of his ineptitude for physical exercises Poincaré was extremely popular with his classmates. At the end of the year they organized a
public exhibition of his artistic masterpieces, carefully labelling them in Greek, “this is a horse,” and so on—not always accurately. But Poincaré’s inability to draw also had its serious side when he came to geometry, and he lost first place, passing out of the school second in rank.
On leaving the Polytechnique in 1875 at the age of twenty one Poincaré entered the School of Mines with the intention of becoming an engineer. His technical studies, although faithfully carried out, left him some leisure to do mathematics, and he showed what was in him by attacking a general problem in differential equations. Three years later he presented a thesis, on the same subject, but concerning a more difficult and yet more general question, to the Faculty of Sciences at Paris for the degree of doctor of mathematical sciences. “At the first glance,” says Darboux, who had been asked to examine the work, “it was clear to me that the thesis was out of the ordinary and amply merited acceptance. Certainly it contained results enough to supply material for several good theses. But, I must not be afraid to say, if an accurate idea of the way Poincaré worked is wanted, many points called for corrections or explanations. Poincaré was an intuitionist. Having once arrived at the summit he never retraced his steps. He was satisfied to have crashed through the difficulties and left to others the pains of mapping the royal roadsII destined to lead more easily to the end. He willingly enough made the corrections and tidying-up which seemed to me necessary. But he explained to me when I asked him to do it that he had many other ideas in his head; he was already occupied with some of the great problems whose solution he was to give us.”
Thus young Poincaré, like Gauss, was overwhelmed by the host of ideas which besieged his mind but, unlike Gauss, his motto was not “Few, but ripe.” It is an open question whether a creative scientist who hoards the fruits of his labor so long that some of them go stale does more for the advancement of science than the more impetuous man who scatters broadcast everything he gathers, green or ripe, to fall where it may to ripen or rot as wind and weather take it. Some believe one way, some another. As a decision is beyond the reach of objective criteria everyone is entitled to his own purely subjective opinion.
Poincaré was not destined to become a mining engineer, but during his apprenticeship he showed that he had at least the courage of a real engineer. After a mine explosion and fire which had claimed sixteen victims he went down at once with the rescue crew. But the calling was uncongenial and he welcomed the opportunity to become a professional mathematician which his thesis and other early work opened up to him. His first academic appointment was at Caen on December 1, 1879, as Professor of Mathematical Analysis. Two years later he was promoted (at the age of twenty seven) to the University of Paris where, in 1886, he was again promoted, taking charge of the course in mechanics and experimental physics (the last seems rather strange, in view of Poincaré’s exploits as a student in the laboratory). Except for trips to scientific congresses in Europe and a visit to the United States in 1904 as an invited lecturer at the St. Louis Exposition, Poincaré spent the rest of his life in Paris as the ruler of French mathematics.
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Poincaré’s creative period opened with the thesis of 1878 and closed with his death in 1912—when he was at the apex of his powers. Into this comparatively brief span of thirty four years he crowded a mass of work that is sheerly incredible when we consider the difficulty of most of it. His record is nearly five hundred papers on new mathematics, many of them extensive memoirs, and more than thirty books covering practically all branches of mathematical physics, theoretical physics, and theoretical astronomy as they existed in his day. This leaves out of account his classics on the philosophy of science and his popular essays. To give an adequate idea of this immense labor one would have to be a second Poincaré, so we shall presently select two or three, of his most celebrated works for brief description, apologizing here once for all for the necessary inadequacy.
Poincaré’s first successes were in the theory of differential equations, to which he applied all the resources of the analysis of which he was absolute master. This early choice for a major effort already indicates Poincaré’s leaning toward the applications of mathematics, for differential equations have attracted swarms of workers since the time of Newton chiefly because they are of great importance in the exploration of the physical universe. “Pure” mathematicians sometimes like to imagine that all their activities are dictated by their own tastes and that the applications of science suggest nothing of interest to them. Nevertheless some of the purest of the pure drudge away their lives over differential equations that first appeared in the translation of physical situations into mathematical symbolism, and it is precisely these practically suggested equations which are the heart of the theory. A particular equation suggested by science may be generalized by the mathematicians and then be turned back to the scientists (frequently without a solution in any form that they can use) to be applied to new physical problems, but first and last the motive is scientific. Fourier summed up this thesis in a famous passage which irritates one type of mathematician, but which Poincaré endorsed and followed in much of his work.
“The profound study of nature,” Fourier declared, “is the most fecund source of mathematical discoveries. Not only does this study, by offering a definite goal to research, have the advantage of excluding vague questions and futile calculations, but it is also a sure means of molding analysis itself and discovering those elements in it which it is essential to know and which science ought always to conserve. These fundamental elements are those which recur in all natural phenomena.” To which some might retort: No doubt, but what about arithmetic in the sense of Gauss? However, Poincaré followed Fourier’s advice whether he believed in it or not—even his researches in the theory of numbers were more or less remotely inspired by others closer to the mathematics of physical science.
The investigations on differential equations led out in 1880, when Poincaré was twenty six, to one of his most brilliant discoveries, a generalization of the elliptic functions (and of some others). The nature of a (uniform) periodic function of a single variable has frequently been described in preceding chapters, but to bring out what Poincaré did, we may repeat the essentials. The trigonometric function sin z has the period 2π, namely, sin (z + 2Π) = sin z; that is, when the variable z is increased by 2π, the sine function of z returns to its initial value. For an elliptic function, say E(z), there are two distinct periods, say pi and p2, such that E(z + p1) = E(z), E(z + p2) = E(z). Poincaré found that periodicity is merely a special case of a more general property: the value of certain functions is restored when the variable is replaced by any one of a denumerable infinity of linear fractional transformations of itself, and all these transformations form a group. A few symbols will clarify this statement.
Let z be replaced by Then, for a denumerable infinity of sets of values of a, b, c, d, there are uniform functions of z, say F(z) is one of them, such that
Further, if a1, b1, c1, d1, and a2, b2, c2, d2 are any two of the sets of values of a, b, c, d, and if z be replaced first by and then, in this, z be replaced by giving, say, then not only do we have
but also
Further the set of all the substitutions
(the arrow is read “is replaced by”) which leave the value of F(z) unchanged as just explained form a group: the result of the successive performance of two substitutions in the set,
is in the set; there is an “identity substitution” in the set, namely z → z (here a = 1, b = 0, c = 0, d = l); and finally each substitution has a unique “inverse”—that is, for each substitution in the set there is a single other one which, if applied to the first, will produce the identity substitution. In summary, using the terminology of previous chapters, we see that F(z) is a function which is invariant under an infinite group of linear fractional transformations. Note that the infinity of substitutions is a denumerable infinity, as first stated: the substitutions can be counted off 1, 2, 3, . . . , and are not as numerous a
s the points on a line. Poincaré actually constructed such functions and developed their most important properties in a series of papers in the 1880’s. Such functions are called automorphic.
Only two remarks need be made here to indicate what Poincaré achieved by this wonderful creation. First, his theory includes that of the elliptic functions as a detail. Second, as the distinguished French mathematician Georges Humbert said, Poincaré found two memorable propositions which “gave him the keys of the algebraic cosmos”:
Two automorphic functionsIIIinvariant under the same group are connected by an algebraic equation;
Conversely, the coordinates of a point on any algebraic curve can be expressed in terms of automorphic functions, and hence by uniform functions of a single parameter (variable).
