Men of Mathematics (103 page)

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Poincaré's intellectual heredity on both sides was good. We shall not go farther back than his paternal grandfather. During the Napoleonic campaign of 1814 this grandfather, at the early age of twenty, was attached to the military hospital at Saint-Quentin. On settling in 1817 at Rouen he married and had two sons: Léon Poincaré, born in 1828, who became a first-rate physician and a member of a medical faculty; and Antoine, who rose to the inspector-generalship of the department of roads and bridges. Léon's son Henri, born on April 29, 1854, at Nancy, Lorraine, became the leading mathematician of the early twentieth century; one of Antoine's two sons, Raymond, went in for law and rose to the presidency of the French Republic during the World War; Antoine's other son became director of secondary education. A great-uncle who had followed Napoleon into Russia disappeared and was never heard of after the Moscow fiasco.

From this distinguished list it might be thought that Henri would have exhibited some administrative ability, but he did not, except in his early childhood when he freely invented political games for his sister and young friends to play. In these games he was always fair and scrupulously just, seeing that each of his playmates got his or her full share of officeholding. This perhaps is conclusive evidence that “the child is father to the man” and that Poincaré was constitutionally incapable of understanding the simplest principle of administration, which his cousin Raymond applied intuitively.

Poincaré's biography was written in great detail by his fellow countryman Gaston Darboux (1842-1917), one of the leading geometers of modern times, in 1913 (the year following Poincaré's death). Something may have escaped the present writer, but it seems that Darboux, after having stated that Poincaré's mother “coming from a family in the Meuse district whose [the mother's] parents lived in Arrancy, was a very good person, very active and very intelligent,” blandly omits to mention her maiden name. Can it be possible that the French took over the doctrine of “the three big K's”—noted in connection with Dedekind—from their late instructors after the kultural drives of Germany into France in 1870 and 1914? However, it can be deduced from an anecdote told later by Darboux that the family name
may
have been Lannois. We learn that the mother devoted
her entire attention to the education of her two young children, Henri and his younger sister (name not mentioned). The sister was to become the wife of Émile Boutroux and the mother of a mathematician (who died young).

Due partly to his mother's constant care, Poincaré's mental development as a child was extremely rapid. He learned to talk very early, but also very badly at first because he thought more rapidly than he could get the words out. From infancy his motor coordination was poor. When he learned to write it was discovered that he was ambidextrous and that he could write or draw as badly with his left hand as with his right. Poincaré never outgrew this physical awkwardness. As an item of some interest in this connection it may be recalled that when Poincaré was acknowledged as the foremost mathematician and leading popularizer of science of his time he submitted to the Binet tests and made such a disgraceful showing that, had he been judged as a child instead of as the famous mathematician he was, he would have been rated—by the tests—as an imbecile.

At the age of five Henri suffered a bad setback from diphtheria which left him for nine months with a paralyzed larynx. This misfortune made him for long delicate and timid, but it also turned him back on his own resources as he was forced to shun the rougher games of children his own age.

His principal diversion was reading, where his unusual talents first showed up. A book once read—at incredible speed—became a permanent possession, and he could always state the page and line where a particular thing occurred. He retained this powerful memory all his life. This rare faculty, which Poincaré shared with Euler who had it in a lesser degree, might be called visual or spatial memory. In temporal memory—the ability to recall with uncanny precision a sequence of events long passed—he was also unusually strong. Yet he unblushingly describes his memory as “bad.” His poor eyesight perhaps contributed to a third peculiarity of his memory. The majority of mathematicians appear to remember theorems and formulas mostly by eye; with Poincaré it was almost wholly by ear. Unable to see the board distinctly when he became a student of advanced mathematics, he sat back and listened, following and remembering perfectly without taking notes—an easy feat for him, but one incomprehensible to most mathematicians. Yet he must have had a vivid memory of the “inner eye” as well, for much of his work, like a good deal of Riemann's,
was of the kind that goes with facile space-intuition and acute visualization. His inability to use his fingers skilfully of course handicapped him in laboratory exercises, which seems a pity, as some of his own work in mathematical physics might have been closer to reality had he mastered the art of experiment. Had Poincaré been as strong in practical science as he was in theoretical he might have made a fourth with the incomparable three, Archimedes, Newton, and Gauss.

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 roads
^II
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.”