Monthly Archives: April 2021

A mathematical person: the last universalist: Henri Poincare

Reference: E. T. Bell “Men of Mathematics”.

PS: Of course, some one might say that Prof John von Neumann was the last universalist. I leave you the dear reader to think…

Below I want to share the words of Prof. E. T. Bell in his classic “Men of Mathematics”. I am doing it for my own learning more rather than even sharing it with the general public. Perhaps, these are good times to be introverted 🙂

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Quote: (Henri Poincare): A scientist worthy of the name, above all a mathematician, experiences in his work, the same impression as an artist; his pleasure is as great and of the same nature.

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In The History of his Life and Times, the astrologer William Lilly (1602-1681) records an amusing — if incredible — account of the meeting between John Napier (1559-1617), of Merchiaton, the inventor of logarithms, and Henry Briggs (1561-1631) of Gresham College, London, who computed the first table of common logarithms. One John Marr, “an excellent mathematician and geometrician,” had gone ‘into Scotland before Mr Briggs, purposely to be there when these two so learned persons should meet.Mr Briggs appoints a certain day when to meet in Edinburgh, but failing thereof, the Lord Napier was doubtful he would not come. It happened one day as John Marr and Lord Napier were speaking of Mr Briggs, “Ah John (said Merchiaton), Mr Briggs will not now come.” At the very moment, one knocks at the gate; John Marr hastens down, and it proved Mr. Briggs to his great contentment. He brings Mr. Briggs up into my lord’s chamber, where almost one quarter of an hour was spent, each beholding the other with admiration, before one word was spoke.”

Recalling this legend, Sylvester tells how he himself went after Briggs’ world record for flabbergasted admiration when, in 1885, he called on the author of numerous astonishingly mature and marvelously original papers on a new branch of analysis which had been swamping the editors of mathematical journals since the early 1880’s.

“I quite entered into Briggs’ feelings at his interview with Napier,” Sylvester confesses, “when I recently paid a visit to Poincare (1854-1912) in his airy perch in the Rue Gay-Lussac…In the presence of that mighty reservoir of pent-up intellectual force my tongue at first refused its office, and it was not until I had taken some time (it may be two or three minutes) to peruse and absorb as it were the idea of his external youthful lineaments that I found myself in a condition to speak.”

Elsewhere Sylvester records his bewilderment when, after having toiled up the three flights of narrow stairs leading to Poincare’s “airy perch,” he paused, mopping his magnificent bald head, in astonishment at beholding a mere boy, “so blond, so young,” as the author of the deluge of papers which had heralded the advent of a successor to Cauchy.

A second anecdote may give some idea of the respect in which Poincare’s work is held by those in a position to appreciate its scope. Asked by some patriotic British brass hat in the rabidly nationalistic days of the World War — when it was obligatory on all academic patriots to exalt their esthetic allies and debase their boorish enemies — who was the greatest man France had produced in modern (recent) times, Bertrand Russell answered instantly, “Poincare.” “What ! That man! ” his uninformed interlocutor exclaimed, believing Russell meant Raymond Poincare, President of the French Republic. “Oh,” Russell explained when he understood the other’s dismay, “I was thinking of Raymond’s cousin, Henri Poincare.”

Poincare was the last man to take practically all mathematics, both pure and applied, as his province. It is generally believed that it would be impossible for any human being starting today to understand comprehensively, much less do creative work of high quality in more than two of the four main divisions of mathematics — arithmetic, algebra, geometry, analysis, to say nothing of astronomy and mathematical physics. However, even in the 1880’s. when Poincare’s great career opened, it was commonly thought that Gauss was the last of the mathematical universalists, so it may not prove impossible for some future Poincare once more to cover the entire field.

As mathematics evolves, it both expands and contracts, somewhat like one of Lemaitre’s models of the universe. At present, the phase is one of explosive expansion, and it is quite impossible for any man to familiarize himself with the entire inchoate mass of mathematics that has been dumped on the world since the year 1900. But already to certain important sectors a most welcome tendency toward contraction is plainly apparent. This is so, for example, in algebra, where the wholesale introduction of postulational methods is making the subject at once more abstract. more general, and less disconnected. Unexpected similarities — in some instances amounting to disguised identity— are being disclosed by the modern attack, and it is conceivable that the next generation of algebraists will not need to know much that is now considered valuable, as many of these particular, difficut things will have been subsumed under simpler general principles of wider scope. Something of this sort happened in classical mathematical physics when relativity put the complicated mathematics of the ether on the shelf.

Another example of this contraction in the midst of expansion is the rapidly growing use of the tensor calculus in preference to that of numerous special brands of vector analysis. Such generalizations and condensations are often hard for older men to grasp at first and frequently have a severe struggle to survive, but in the end it is usually realized that general methods are essentially simpler and easier to handle than miscellaneous collections of ingenious tricks devised for special problems. When mathematicians assert that such a thing as the tensor calculus is easy — at least in comparison with some of the algorithms that preceded it — they are not trying to appear superior or mysterious but are stating a valuable truth which any student can verify for himself. This quality of inclusive generality was a distinguishing trait of Poincare’s vast output.

If abstractness and generality have obvious advantages of the kind indicated, it is also true that they sometimes have serious drawbacks for those who must be interested in details. Of what immediate use is it to a working physicist to know that a particular differential equation occuring in his work is solvable, because some pure mathematician has proved that it is, when neither he nor the mathematician can perform the Herculean labour demanded by a numerical solution capable of application to specific problems?

To take an example from the field in which Poincare did some of his most original work, consider a homogeneous, incompressible fluid mass held together by the gravitation of its particles and rotating about an axis. Under what conditions will the motion be stable and what will be the possible shapes of such a stably rotating fluid? MacLaurin, Jacobi, and others proved that certain ellipsoids will be stable; Poincare, using more intuitive, “less arithmetical” methods than his predecessors, once thought he had determined the criteria for the stability of a pear-shaped body. But he had made a slip. His methods were not adapted to numerical computations and later workers, including G. H. Darwin, son of the famous Charles, undeterred by the horrific jungles of algebra and arithmetic that must be cleared out of the way before a definite conclusion be reached, undertook a decisive solution.

The man interested in the evolution of binary stars is more comfortable if the findings of the mathematicians are presented to him in a form to which he can apply a calculating machine. And since Kronecker’s fiat of “no construction, no existence”, some pure mathematicians themselves have been less enthusiastic than they were in Poincare’s day for existence theorems which are not constructive. Poincare’s scorn for this kind of detail that users of mathematics demand and must have before they can get on with their work was one of the most important contributory causes to his universality. Another was his extraordinarily comprehensive grasp of all the machinery of the theory of functions of a complex variable. In this he had no equal. And it may be noted that Poincare turned his universality to magnificent use in disclosing hitherto unsuspected connections between distant branches of mathematics, for example, between (continuous groups) and linear algebra.

One more characteristic of Poincare’s outlook must be recalled for completeness before we go on to his life: few mathematicians have led the breadth of philosophical vision that Poincare had, and none is his superior in the gift or clear exposition. Probably, he had always been deeply interested in the implications of science and mathematics, but it was only in 1904, when his greatness as a technical mathematician was established beyond all cavil, that he turned as a side-interest in what may be called the popular appeal of mathematics and let himself go in a sincere enthusiasm to share with the non-professionals the meaning and human importance of his subject. Here his liking for the general in preference to the particular aided him in telling intelligent outsiders what is of more than technical importance in mathematics without talking down to his audience. Twenty or thirty years ago workmen and shopgirls could be seen in the parks and cafes of Paris avidly reading one or other of Poincare’s popular masterpieces in its cheap print and shabby paper cover. The same works in a richer format could also be found — well thumbed and evidently read — on the tables of the professionally cultured. These books were translated into English, German, Spanish, Hungarian, Swedish and Japanese. Poincare spoke the universal languages of mathematics and science to all in accents which they recognized. His style, peculiarly his own, loses much by translation.

For the literary excellence of his popular writings Poincare was awarded the highest honour a French writer can get, membership in the literary section of the Institut. He has been somewhat spitefully said by envious novelists that Poincare achieved this distinction, unique for a man of science, because one of the functions of the literary Academy is the constant compilation of a dictionary of the French language, and the universal Poincare was obviously the man to help out the poets and grammarians in their struggle to tell the world what automorphic functions are. Impartial opinion, based on a study of Poincare’s writings, agrees that the mathematician deserved no less than he got.

Closely allied to his interest in the philosophy of mathematics was Poincare’s preoccupation with the psychology of mathematical creation. How do mathematicians make their discoveries? Poincare will tell us later his own observations on this mystery in one of the most interesting narratives of personal discovery that was ever written. The upshot seems to be that mathematical discoveries more or less make themselves after a long spell of hard labour on the part of the mathematician. As in literature — according to Dante Gabriel Rossetti — “a certain amount of fundamental brainwork” is necessary before a poem can mature, so in mathematics there is no discovery without preliminary drudgery, but this is by no means the whole story. All “explanations” of creativeness that fail to provide a recipe whereby a gifted human being can create are open to suspicion. Poincare’s excursion into practical psychology, like some others in the same direction, failed to bring back the Golden Fleece, but it did at least suggest that such a thing is not wholly mythical and may some day be found when human beings grow intelligent enough to understand their own bodies.

Poincare’s intellectual heredity on both sides was good. We shall notgo 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: Leon Poincare, 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. Leon’s son Henri, born on April 29, 1854, at Nancy, Lorraine, became the leading mathematician of the early 20th 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, and just seeing that each of his playmates got his or her full share of officeholding. This perhaps is conclusive evidence that “child is father to the man” and that Poincare was constitutionally incapable of understanding the simplest principle of administration, which his cousin Raymond applied intuitively.

Poincare’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 Poincare’s death). Something may have escaped the present writer but it seems that Darboux, after having stated that Poincare’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 “the three big K’s” —- noted in connection with Dedekind — from their late instructors after the kultural drives of Germany and 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 wife of Emile Boutroux and the mother of a mathematician (who died young).

Due partly to his mother’s constant care, Poincare’s mental development as a child was extremely rapid, 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 ambidexterous and that he could write or draw as badly as with the left hand as with the right hand. Poincare never outgrew this physical awkwardness. As an item of some interest in this connection it may be recalled that when Poincare 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 the famous mathematician that he was, he would have been rated — by the tests — as an imbecile.

At the age of five, Henri suffered a setback from diptheria 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 of 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 occured. He retained this powerful memory all his life. This rare faculty, which Poincare 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 the eye; with Poincare it was mostly by the 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 skillfully of course handicapped 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 Poincare 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 asbsentminded dreamers that popular fancy likes to picture them. Poincare 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 Poincare’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 Poincare’s later years), when a distinguished mathematician had come all the way from Finland to Paris to confer with Poincare on scientific matters> Poincare 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 portieres. At last the drapes parted and Poincare’s buffalo head was thrust for an instant in to the room. “vous me derangen 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.

Poincare’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. The mishap affected his 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 execise, original in both form and substance, which young Poincare had handed in, was a “little masterpiece,” and kept it as one of his treasures. But he also advised his pupil to be conventional — stupider — if he wished to make a good impression on the school examiners.

Being out of the more boisterous games of his schoolfellows, Poincare 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 age of his career Poincare 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 paced restlessly about, and was committed to paper only when all had been thought through. Talking or other noise never disturbed his 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 Poincare’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. Poincare’s flair for classical studies, in which he excelled at school, taught him the importance of both form and substance.

The Franco-Prusssian war broke over France in 1870 when Poincare was sixteen. Although he was too young and too frail for active service, Poincare 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 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 battle-field 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 pillages, “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 refined to abandon the ruins of her cottage and who insisted on sharing her meagre supper with them.

Poincare 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, Poincare 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, Poincare 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 Poincare took the examinations for his first degree (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 Poincare would have been plucked,” the head examiner declared.

He next prepared for the entrance examination 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, Poincare 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 Poincare’s career. He never seemed to think when a mathematical difficulty was submitted to him by a colleague. “The reply came like an arrow.”

At the end of his year, he passed first into the Ecole Polytechnique. Several legends of his unique examination survive. One tells how a certain examiner forewarned that young Poincare was a mathematical genius, superseded the examination for three quarters of an hour in order to devise a “nice” question — a refined torture. But Poincare got the better of him and the inquisitor “congratulated the examinee warmly, telling him he had won the highest grade.” Poincare’s experiences with his tormentors would seem to indicate that the French mathematical examiners have learned something since they ruined Galois and came within an ace of doing the like by Hermite.

At the Polytechnique Poincare 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 and 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 embarassed 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 Poincare 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 Poincare 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 Poincare’s inability to draw also had its serious side when he came to geometry, and he lost his first place, passing out of the school second in rank.

On leaving the Polytechnique in 1875 at the age of twenty one Poincare 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 simply 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 Poincare worked is wanted, many points called for corrections or explanations. Poincare 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 *(“There is no royal road to Geometry,” as Menaechinus is said to have told Alexander the Great when the latter wished to conquer geometry in a hurry. ) destined to lead more easily to the end. He willingly enough made the corrections and tidying up which seemed necessary to me. 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 solutions he was to give us.”

Thus, young Poincare, like young 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 fruit of his labour 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.

Poincare 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 27 years) to the University of Paris where, in 1886, he was again promoted, taking charge of the course in mecnanics and experimental physics (the last seems rather strange, in view of Poincare’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, Poincare spent the rest of his life in Paris as the ruler of French mathematics.

Poincare’s creative period opened with the thesis of 1878 and closed with his death in 1919 — when he was 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 those 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 labour one would have to be a second Poincare, 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.

Poincare’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 his leaning towards 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 life 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 Poincare 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 molding analysis itself and discovering those elements in it which it is essential to know and which science might always to conserve. These fundamental elements are those which recur in all natural phenomena.” To which some might retort: No doubt, but what about the arithmetic in the sense of Gauss? However, Poincare 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 Poincare 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, but to bring out what Poincare did, we may repeat the essentials. The trigonometric function \sin{x} has the period 2\pi, namely, \sin{x+2\pi}=\sin{x}, that is, when the variable x is increased by 2\pi, the sine function of x returns to the initial value. For an elliptic function, say, E(x), there are two distinct periods, say p_{1} and p_{2} such that E(x+p_{1})=E(x) and E(x+p_{2})=E(x). Poincare found this 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 transformationsof itself, and all these transformations form a group. A few symbols will clarify this statement.

Let x be replaced by \frac{ax+b}{cx+d}. Then, for a denumerable infinity of sets of values a, b, c, d, there are uniform functions of x, say F(x) is one of them, such that

F(\frac{ax+b}{cx+d})=F(x)

Further, if a_{1}, b_{1}, c_{1}, d_{1} and a_{2}, b_{2}, c_{2}, d_{2} are two sets of values of a, b, c, d, and if x be replaced first by \frac{a_{1}x+b_{1}}{c_{1}x+d_{1}} and in this x, be replaced by \frac{a_{2}x+b_{2}}{c_{2}x+d_{2}}, giving say, \frac{Ax+B}{Cx+D} then not only do we have

F(\frac{a_{1}x+b_{1}}{c_{1}x+d_{1}})=F(x), and F(\frac{a_{2}x+b_{2}}{c_{2}x+d_{2}}) = F(x),

but also F(\frac{Ax+B}{Cx+D}) = F(x)

Further the set of all substitutions

x \rightarrow \frac{ax+b}{cx+d}

which leave the value of F(x) unchanged as just explained form a group; the result of the successive performance of two substitutions in the set

x \rightarrow \frac{a_{1}x+b_{1}}{c_{1}x+d_{1}} and x \rightarrow \frac{a_{2}x+b_{2}}{c_{2}x+d_{2}}

is in the set; there is an “identity substitution” in the set, namely x \rightarrow x where a=1, b=0, c=0, d=1 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 this terminology, we see that F(x) 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 as the points on a line. Poincare 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 be made here to indicate what Poincare 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, Poincare found two memorable propositions which “gave him the keys of the algebraic cosmos.”

Two automorphic functions invariant 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).

An algebraic curve is one whose equation is of the type P(x,y)=0, where P(x,y) is a polynomial in x and y. As a simple example, the equation of the circle whose centre is at the origin —- (0,0) — and whose radius is a is x^{2}+y^{2}=a^{2}. According to the second of Poincare’s “keys”, it must be possible to express x, y as automorphic functions of a single parameter, say t. It is, for if x=a\cos{t}, y=a\sin{t}, then squaring and adding, we get rid of t (since \sin^{2}{t}+\cos^{2}{t}=1) and find x^{2}+y^{2}=a^{2}. But, the trigonometric functions \cos{t} and $latex\sin{t}$ are special cases of elliptic functions, which in turn are special cases of automorphic functions.

The creation of this vast theory of automorphic functions was but one of many astounding things in analysis which Poincare did before he was thirty. Nor was all his time devoted to analysis; the theory of numbers, parts of algebra, and mathematical astronomy also shared his attention. In the first he recast the Gaussian theory of binary quadratic forms in a geometrical shape which appeals particularly to those who, like Poincare, prefer the intuitive approach. This of course was not all that he did in the higher arithmetic, but limitations of space forbid further details.

Work of this calibre did not pass unappreciated. At the unusually early age of thirty-two (in 1887) Poincare was elected to the Academy. His proposer said some pretty strong things, but most mathematicians will subscribe to their truth. “(Poincare’s) work is above ordinary praise and reminds us inevitably of what Jacobi wrote of Abel — that he had settled questions which before him were unimagined. It must indeed be recognized that we are witnessing a revolution in Mathematics comparable in every way to that which manifested itself, half a century ago, by the accession of elliptic functions.”

To leave Poincare’s work in pure mathematics here is like rising from a banquet table after having just sat down, but we must turn to another side of his universaility.

Since the time of Newton and his immediate successors astronomy has generously supplied mathematicians with more problems than they can solve. Until the late nineteenth century the weapons used by mathematicians in their attack on astronomy were practically all immediate improvements of those invented by Newton himself, Euler, Lagrange and Laplace. But all through the nineteenth century, particularly since Cauchy’s development of the theory of functions of a complex variable and the investigations of himself and others on the convergence of infinite series, a huge arsenal of untried weapons had been accumulating from the labours of pure mathematicians. To Poincare, to whom analysis came as naturally as thinking, this vast pile of unused mathematics seemed the most natural thing in the world to use in a new offensive on the outstanding problems of celestial mechanics and planetary evolution. He picked and chose what he liked out of the heap, improved it, invented new weapons of his own, and assaulted theoretical astronomy in a general fashion it had not been assaulted in for a century. He modernized the attack; indeed his campaign was so extremely modern to the majority of experts in celestial mechanics that even today, forty years or more after Poincare opened his offensive, few have mastered his weapons and some, unable to open his bow, insinuate that it is worthless in a practical attack. Nevertheless Poincare is not without forceful champions whose conquests would have been impossible to the men of the pre-Poincare era.

Poincare’s first (1889) great success in mathematical astronomy grew out of an unsuccessful attack on “the problem of n bodies.” For n=2, the problem was completely solved by Newton; the famous problem of “three bodies” will be noticed later, when n exceeds 3 some of the reductions applicable to the case n=3 can be carried over.

According to the Newtonian law of gravitation, two particles of masses, m and M at a distance D apart attract one another with a force proportional to \frac{m \times M}{D^{2}}. Imagine n material particles distributed in space; the masses, the initial motions, and the mutual distances of all the particles are assumed known at a given instant. If they attract one another according to the Newtonian law, what will be their positions and motions (velocities) after any stated lapse of time? For the purposes of mathematical astronomy, the stars in a cluster, or in a galaxy, on in a cluster of galaxies, may be thought of as material particles attracting one another according to the Newtonian law. The “problem of n bodies” thus amounts to — in one of its applications — to asking what will be the aspect of the heavens a year from now, or a billion years hence, it being assumed that we have sufficient astronomical data to describe the general configuration now. The problem of course is tremendously complicated by radiation — the masses of stars do not remain constant for millions of years, but a complete, calculable solution of the problem of n bodies in its Newtonian form would probably give results of an accuracy sufficient for all human purposes — the human race will likely be extinct long before radiation can introduce observable inaccuracies.

This was substantially the problem proposed for the prize offered by King Oscar II of Sweden in 1887. Poincare did not solve the problem, but in 1889 he was awarded the prize by a jury consisting of Weierstrass, Hermite, and Mittag-Leffler for his general discussion of the differential equations of dynamics and an attack on the problem of three bodies. The last is usually considered the most important case of the n body problem, as the Earth, the Moon and the Sun furnish an example of three body problem. In his report to Mittag-Leffler, Weierstrass wrote, ” You may tell your Sovereign that this work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that the publication will inaugurate a new era in the history of Celestial Mechanics. The end which his Majesty had in view in opening the competition may therefore be considered as having been attained.” Not to be outdone by the King of Sweden, the French Government followed up the prize by making Poincare a Knight of the Legion of Honour — a much less expensive acknowledgement of the young mathematician’s genius than the King’s 9500 crowns and gold medal.

As we have mentioned the problem of three bodies we may now report one item from its fairly recent history; since the time of Euler it has been considered one of the most difficult problems in the whole range of mathematics. Stated mathematically, the problem boils down to a system of nine differential equations(all linear, each of the second order). Lagrange succeeded in reducing this system to a simpler. As in the majority of physical problems, the solution is not be expected in finite terms; if a solution exists at all, it will be given by an infinite series. The solution will “exist” if these series satisfy the equations (formally) and moreover converge for certain values of the variables. The central difficulty is to prove the convergence. Up till 1905 various special solutions had been found, but the existence of anything that could be called general had not been proved.

In 1906 and 1908 a considerable advance came from a rather unexpected quarter — Finland, a country which sophisticated Europeans even today consider barely civilized, especially for its queer customs or paying its debts, and which few Americans thought advanced beyong the Stone Age till Paavo Nurmi ran the legs off the United States. Excepting only the rare case when all three bodies collide simultaneously, Karl Frithiof Sundman of Helsingfors, utlizing analytical methods due to the Italian Levi-Civita and the French Painleve, and making an ingenious transformation of his own, proved the existence of a solution in the sense described above. Sundman’s solution is not adapted to numerical computation, nor does it give much information regarding the actual motion, but that is not the point of interest here: a problem which had not been known to be solvable was proved to be so. Many had struggled desperately to prove this much; when the proof was forthcoming, some, humanly enough, hastened to point out that Sundman had done nothing much because he had not solved some problem other than the one he had. This kind of criticism is as common in mathematics as it is in literature and art, showing once more that mathematicians are as human as anybody.

Poincare’s most original work in astronomy was summed up in his great treatise Les methodes nouvelles de la mecanique celeste (New methods of celestial mechanics; three volumes 1892, 1893, 1899). This was followed by another three volume work in 1905-1910 of a more immediately practical nature, Lecons de mecanique celeste, and a little later by the publication of his course of lectures Sur les figures d’equilibre d’une masse fluids (on the figures of equilibrium of a fluid mass), and a historical critical book Sur les hypotheses cosmogoniques (On cosmological hypotheses).

Of the first of these works, Darboux (seconded by many others) declares that it did indeed start a new era in celestial mechanics and that it is comparable to the Mecanique celeste of Laplace and the earlier work of D’Alembert on the precession of the equinoxes. “Following the road in analytical mechanics opened up by Lagrange,” Darboux says, “…Jacobi had established a theory which appeared to be one of the most complete in dynamics. For fifty years, we lived on the theorems of the illustrious German mathematician, applying them and studying them from all angles, but without adding anything essential. It was Poincare who first shattered these rigid frames in which the theory seemed to be encased and contrived for it vistas and new windows on the external world. He introduced or used, in the study of dynamical problems, different notions: the first, which had been given before and which, moreover, is applicable not solely to mechanics, is that of variational equations, namely, linear differential equations that determine solutions of a problem infinitely near to a given solution; the second, that of integral invariants, which belong entirely to him and play a capital part in these researches. Further fundamental notions were added to these, notably those concerning so-called periodic solutions, for which the bodies whose motion is studied return after a certain time to their initial positions and original relative velocities.

The last started a whole department of mathematics, the investigation of periodic orbits: given a system of planets, or of stars, say, with a complete specification of the initial position and relative velocities of all members of the system at a stated epoch, it is required to determine under what conditions the system will return to its initial state at some later epoch, and hence, continue to repeat the cycle of motions indefinitely. For example, is the solar system of this recurrent type, or if not, would it be were it isolated and not subject to perturbations by external bodies ? Needless to say the general problem has not yet been solved completely.

Much of Poincare’s work in his astronomical researches was qualitative rather than quantitative, as befitted an intuitionist, and this characteristic led him, as it had Riemann, to the study of analysis situs. On this he published six famous memoirs which revolutionized the subject as it existed in his day. The work on analysis situs in its turn was freely applied to the mathematics of astronomy.

We have already alluded to Poincare’s work on the problemof rotating fluid bodies — of obvious importance in cosmogony, one brand of which assumes that the planets were once sufficiently like such bodies to be treated as if they actually were without patent absurdity. Whether they were or not is of no importance for the mathematics of the situation, which is of interest in itself. A few extracts from Poincare’s own summary will indicate more clearly than any paraphrase the nature of what he mathematicized about in this difficult subject.

“Let us imagine a (rotating) fluid body contracting by cooling, but slowly enough to remain homogeneous and for the rotation to be the same in all its parts.

“At first, very approximately a sphere, the figure of this mass will become an ellipsoid of revolution which will flatten more and more, then, at a certain moment, it will be transformed into an elllipsoid with three unequal axes. Later, the figure will cease to be an ellipsoid and will become pear-shaped until at last the mass, hollowing out more and more at its “waist,” will separate into two distinct and unequal bodies.

“The preceding hypothesis certainly can not be applied to the solar system. Some astronomers have thought that it might be true for certain double stars and that double stars of the type of Beta Lyrae might present transitional forms analogous to those we have spoken of.”

He then goes on to suggest an application to Saturn’s rings, and he claims to have proved that the rings can be stable only if their density exceeds 1/16 that of Saturn. It may be remarked that these questions were not considered as fully settled as late as 1935. In particular a more drastic mathematical attack on poor old Saturn seemed to show that he had not been completely vanquished by the great mathematicians, including Clerk Maxwell, who have been firing away at him off and on for the past seventy years.

Once more we must leave the banquest having barely tasted anything and pass on to Poincare’s voluminous work in mathematical physics. Here his luck was not so good. To have cashed in on his magnificent talents he should have been thirty years later or have lived twenty years longer. He had the misfortune to be in his prime just when physics had reached one of its recurrent periods of senility, and he was so thoroughly saturated with nineteenth century theories when physics began to recover its youth — after Planck, in 1900, and Einstein in 1905, had performed the difficult and delicate operation of endowing the decrepit roue with its first pair of new glands — that he had barely time to digest the miracle before his death in 1912. All his mature life Poincare seemed to absorb knowledge through his pores without a conscious effort. Like Cayley, he was not only a prolific creator but also a profoundly erudite scholar. His range was probably wider than ever Cayley’s, for Cayley never professed to be able to understand everything that was going on in applied mathematics. This unique erudition may have been a disadvantage when it came to a question of living science as opposed to classical.

Everything that boiled up in the melting pots of physics was grasped instantly as it appeared to Poincare and made the topic of several purely mathematical investigations. When wireless telegraphy was invented he seized the new thing and worked out its mathematics. While others were either ignoring Einstein’s early work on the (special) theory of relativity or passing it by as a mere curiosity, Poincare was already busy with its mathematics, he was the first scientific man of high standing to tell the world what had arrived and urge it to watch Einstein as probably the most significant phenomenon of the new era which he foresaw but could not himself usher in. It was the same with Planck’s early form of the quantum theory. Opinions differ, of course; but, at this distance it is beginning to look as if mathematical physics did for Poincare what Ceres did for Gauss; and although Poincare accomplished enough in mathematical physics to make half a dozen great reputations, it was not the trade to which he had been born and science would have got more out of him if he had stuck to pure mathematics — his astronomical work was nothing else. But science got enough, and a man of Poincare’s genius is entitled to his hobbies.

We pass on now to the last phase of Poincare’s universality for which we have space: his interest in the rationale of mathematical creation. In 1902 and 1904 the Swiss mathematical periodical L’Enseignement Mathematique undertook an enquiry into the working habits of mathematicians. Questionnaires were issued to a number of mathematicians,of whom over a hundred replied. The answers to the questions and an analysis of general trends were published in final form in 1912. Anyone wishing to look into the psychology of mathematics will find much of interest in this unique work and many confirmations of the views at which Poincare had arrived independently before he saw the results of the questionnaire. A few points of general interest may be noted before we quote from Poincare.

The early interest in mathematics of those who were to become great mathematicians has been frequently exemplified. To the question “At what period…and under what circumstances did mathematics seize you?” 93 replies to the first part were received; 35 said before the age of ten; 43 said eleven to fifteen; 11 said sixteen to eighteen; 3 said nineteen to twenty, and the lone laggard said twenty six.

Again, anyone with mathematical friends will have noticed that some of them like to work early in the morning (one distinguished mathematician began work at the inhuman hour of 5a.m. daily) while others do nothing till after dark. The replies on this point indicated a curious trend — possibly significant, although there are numerous exceptions: mathematicians of the northern races prefer to work in the night, while Latins favour the morning. Among night-workers prolonged concentration often brings on insomnia as they grow older and they change — reluctantly — to the morning. Felix Klein, who worked day and night as a young man, once indicated a possible way out of this difficulty. One of his American students complained that he could not sleep for thinking of his mathematics. “Can’t sleep, eh?” Klein snorted. “What choral for?” However, this remedy is not to be recommended indiscriminately; it probably had something to do with Klein’s own tragic breakdown.

Probably the most significant of the replies were those received on the topic of inspiration versus drudgery as the source of mathematical discoveries. The conclusion is that “mathematical discoveries small or great…are never born of spontaneous generation. They always presuppose a soil seeded with preliminary knowledge and will prepared by labour, both conscious and subconscious.”

Those who, like Thomas Alva Edison, have declared that genius is ninety nine percent perspiration, one percent inspiration are contradicted by those who reverse the figures. Both are right. One man remembers the drudgery while another forgets it all in the apparently discovery but both, when they analyze their impressions, admit that without drudgery and a flash of “inspiration” discoveries are not made. If drudgery alone sufficed, how is it that many gluttons for hard work who seem to know everything about some branch of science, while excellent critics and commentators, never themselves make even a small discovery! On the other hand, those who believe in “inspiration” as the sole factor in discovery or invention — scientific or literary — may find it instructive to look at an early draft of any of Shelley’s “completely spontaneous” poems (so far as these have been preserved and reproduced), or the successive versions of any of the greater levels that Balzacx inflicted on his maddened printer.

Poincare stated his views on mathematical discovery in an essay first published in 1908 and reproduced in Science et Methode. The genesis of mathematical discovery, he says, is a problem which should interest psychologists immensely, for it is the activity in which human mind seems to borrow least from the external world, and by understanding the process of mathematical thinking we may hope to reach what is most essential in the human mind.

How does it happen, Poincare asks, that there are persons who do not understand mathematics? “This should surprise us, or rather it would surprise us if we were not so accustomed to it.” If mathematics is based only on the rules of logic, such as all normal minds accept, and which only a lunatic would deny (according to Poincare), how is it that so many are mathematically impermeable? To which it may be answered that no exhaustive set of experiments substantatiating mathematical incompetence as the normal human mode has yet been published. “And, further,” he adds. “how is error possible in mathematics?” Ask Alexander Pope: “To err is human,” which is as unsatisfactory a solution as any other. The chemistry of the digestive system may have something to do with it, but Poincare prefers a more subtle explanation — one which could not be tested by feeding the “vile” body hashish and alcohol.

“The answer seems to me evident,” he declares. Logic has little to do with discovery or invention, and memory plays tricks. Memory however is not importtant as it might be. His own memory, he says without blush is bad:”Why then does not it desert me in a difficult piece of mathematical reasoning where most chess players (whose memories he assumes excellent) would be lost? Evidently because it is guided by the general course of reasoning. A mathematical proof is not a mere juxtaposition of syllogisms; it is syllogisms arranged in a certain order, and the order is more important than the elements themselves.” If he has the “intuition” of this order, memory is at a discount, for each syllogism will take its place automatically in the sequence.

Mathematical creation however does not consist merely in making new combinations of things already known, “anyone could do that, but the combinations thus made would be infinite in number and most of them entirely devoid of interest. To create consists precisely in avoiding useless combinations and in making those which are useful and which constitute only a small minority. Invention is discernment, selection.” But has not all this been said thousands of times before? What artist does not know that selection — an intangible — is one of the secrets of success? We are exactly where we were before the investigations began.

To conclude this part of Poincare’s observations it may be pointed out that much of what he says is based on an assumption which may indeed, be true but for which there is not a particle of scientific evidence. To put it bluntly he assumes that majority of human beings are mathematical imbeciles. Granting him this, we need not even accept his purely romantic theories. They belong to inspirational literature not to science. Passing to something less controversial, we shall now quote the famous passage in which Poincare describes how one of his own “greatest inspirations” came to him. It is meant to substantiate his theory of mathematical creation. Whether it does or not is left to the reader.

He first points out that technical terms need not be understood in order to follow his narrative. “What is of interest to the pyschologist is not the theorem but the circumstances.

“For fifteen days I struggled to prove that no functions analogous to those I have since called Fuchsian functions could exist. I was then very ignorant. Every day I sat down at the work table where I spent an hour or two, I tried a great number of combinations and arrived at no result.One evening, contrary to my custom, I took black coffee, I could not go to sleep, ideas swarmed up in clouds: I sensed them clashing until, to put it so, a pair would hook together to form a stable combination. By morning I had established the existence of a class of Fuchsian functions, those derived from the hypergeometric series. I had only to write up the results, which took me a few hours.

“Next I wished to represent these functions by the quotient of two series; this idea was perfectly conscious and thought out, analogy with elliptic functions guided me. I asked myself what must be the property of these series if they existed, and without difficulty I constructed the series which I called the thetafuchsian.

“I then left Caen, where I was living at the time, to participate in a geological trip sponsored by the School of Mines. The exigencies of travel made me forget my mathematical labours; reaching Coutances we took a bus for some excursion or another. The instant I put my foot on the step of the bus the idea came to me, apparently with nothing whatsoever in my previous thoughts having prepared me for it, that the transformations which I had used to define Fuchsian functions were identical with those of non-Euclidean geometry. I did not make the verifications; I should have not had the time, because once in the bus I resumed an interrupted conversation, but I felt an instant and complete certainty. On returning to Caen, I verifed the result at my leisure to satisfy my conscience.

“I then undertook the study of certain arithmetical questions without much apparent success and without suspecting that such matters should have the slightest connection with my previous studies. Disgusted at my lack of success, I went to spend a few days at the seaside and thought of something else. One day, while walking along the cliffs, the idea came to me, again with the same characterisitcs of brevity, suddenness, and immediate certainty, that the transformations of indefinite ternary quadratic forms were identical with those of non-Euclidean geometry.

“On returning to Caen, I reflected on this result and deduced its consequences; the examples of quadratic forms showed me that there were Fuchsian groups other than those corresponding to the hypergeometric series: I saw that I could apply to them the theory of thetafuchsian functions, and hence that there existed thetafuchsian functions other than those derived from the hypergeometric series, the only ones I had known up till then. Naturally, I set myself the task of constructing all these functions. I conducted a systematic siege and, one after another, carried all the outworks; there was however one which still held out and whose fall would bring about that of the whole position. But all my efforts were served only to make me better acquainted with the difficulty, which in itself was something. All this work was perfectly conscious.

“At this point, I left for Mont-Valerien, where I was to discharge my military service. I had therefore very different preoccupations. One day, while crossing the boulevard, the solution of the difficulty which had stopped me appeared to me all of a sudden. I did not seek to go into it immediately, and it was only after my service that I resumed the question. I had all the elements, and had only to assemble and order them. So I wrote out my definitive memoir at ohe stroke and with no difficulty.”

Many other examples of this sort of thing could be given from his own work, he says, and from that of other mathematicians as reported in L’Enseignement Mathematique. From his experiences, he believes that this semblance of “sudden illumination {is} a manifest sign of previous long subconscious work,” and he proceeds to elaborate his theory of the subconscious mind and its part in mathematical creation. Conscious work is necessary as a trigger to fire off the cumulated dynamite which the subconscious has been emitting — he does not put it so, but what he says amounts to the same. But what is gained in the way of rational explanation if, following Poincare, we foist off the “subconscious mind,” or the “subliminal self,” the very activities which it is our object to understand? Instead of endowing this mysterious agent with a hypothetical tact enabling it to discriminate between the “exceedingly numerous” possible combinations presented (how, Poincare does not say) for its inspection, and calmly saying that the “subconscious” rejects all but the “useful” combinations because it has a feeling for symmetry and beauty, sounds suspciously like solving the initial value problem by giving it a more impressive name. Perhaps this is exactly what Poincare intended, for he once defined mathematics as the art of giving the same name to different things; so here he may be rounding out the symmetry of his view by giving different names to the same thing. It seems strange that a man who could have been satisfied with such a “psychology” of mathematical invention was the complete skeptic in religious matters that Poincare was. AFter Poincare’s brilliant lapse into psychology skeptics may well despair of ever disbelieving anything.

During the first decade of the twentieth century Poincare’s fame increased rapidly and he came to be looked upon, especially in France, as an oracle on all things mathematical. His pronouncements on all manner of questions, from politics to ethics, were usually direct and brief, and were as accepted as final by the majority. As almost invariably happens after a great man’s extinction, Poincare’s dazzling reputation during his lifetime passed throught a period of partial eclipse in the decade following his death. But his intuition for what was likely to be of interest to a later generation is already justifying itself. To take but one instance of many, Poincare was a vigorous of opponent of the theory that all mathematics can be rewritten in terms of the most elementary notions of classical logic; something more than logic, he believed, makes mathematics what it is. Although he did not go so far as the current intuitionist school he seems to have believed, as that school does, that at least some mathematical notions precede logic, and if one is to be derived from the other it is logic which came out of mathematics, not the other way about. Whether this is to the ultimate creed remains to be seen, but at present it appears as if the theory which Poincare assailed with all the irony at his command is not the final one, whatever may be its merits.

Except for a distressing illness during his last four years Poincare’s busy life was tranquil and happy. Honors were showered upon him by all the leading learned societies of the world, and in 1906, at the age of 52, he achieved the highest distinction possible to a French scientist, the Presidency of the Academy of Sciences. None of all this inflated his ego, for Poincare was truly humble and unaffectedly simple. He knew of course that he was without a close rival in the days of his maturity, but he could also say without a trace of affection that he knew nothing compared to what it is to be known. He was happily married and had a son and three daughters in whom he took much pleasure, especially when they were children. His wife was a great-granddaughter of Etienne Geoffroy Saint-Hilaire, remembered as the antagonist of that pugnacious comparative anatomist Cuvier. One of Poincare’s passions was symphonic music.

At the International Mathematical Congress of 1908, held at Rome, Poincare was prevented by illness from reading his stimulating (if premature) address on The Future of Mathematical Physics. His trouble was hypertrophy of the prostate, which was relieved by Italian surgeons, and it was thought that he was permanently cured. On his return to Paris he resumed his work as energetically as ever. But in 1911 he began to have presentiments that he might not live long, and on December 9 wrote asking the editor of a mathematical journal iwhether he would accept an unpublished memoir — contrary to the usual custom —- on a problem which Poincare considered of the highest importance:”….at my age, I may not be able to solve it, and the results obtained, susceptible or putting researchers on a new and unexpected path, seem to me full of promise, in spite of the deceptions they have caused me, that I should resign myself to sacrificing them…” He had spent the better part of two fruitless years trying to overcome the difficulties.

A proof of the theorem which he conjectured would have enabled him to make a striking advance in the problem of three bodies; in particular it would have permitted him to give the existence of an infinity of periodic solutions in cases more general than those hitherto considered. The desired proof was given shortly after the publication of Poincare’s “unfinished symphony” by a young American mathematician, George David Birkhoff.

In the spring of 1912, Poincare fell ill again and underwent a second operation on July 9. The operation was successful, but on July 17 he died very suddenly from an embolism while dressing. He was in the fifty ninth year of his age and at the height of his powers —- “the living brain of the rational sciences,” in the words of Painleve.

Regards,

Nalin Pithwa

Derivatives part 14: IITJEE maths tutorial problems for practice

This is part 14 of the series

Question 1:

Let f(x)= \sqrt{x-1} + \sqrt{x+24-10\sqrt{x-1}} for x<26 be a real valued function. Then, find f^{'}(x) for 1<x<26:

Answer 1:

Consider (\sqrt{x-1}-5)^{2} = x-1+25-10\sqrt{x-1} = x+24 -10\sqrt{x-1} so that we have

\sqrt{x+24-10\sqrt{x-1}}=\sqrt{x-1}-5

Hence, f(x) = \sqrt{x-1} + \sqrt{x-1}-5 = 2\sqrt{x-1}-5 when 1<x<26

Hence, f^{'}{x} = \frac{-2}{2\sqrt{x-1}} = -\frac{1}{\sqrt{x-1}}

Question 2:

Let 3f(x) - 2 f(\frac{1}{x})=x, then find f^{'}(2).

Answer 2:

Given that 3f(x) - 2 f(\frac{1}{x})=x….call this I.

Also, from above, we get 3f(\frac{1}{x}) - 2 f(x)= \frac{1}{x}…call this II.

so we get 6f(x)-4f(\frac{1}{x})=2x….call this I’

and 9f(\frac{1}{x})-6f(x) = 9/x…call this II’.

5f(\frac{1}{x})=2x+ \frac{9}{x} and hence, f^{'}(1/x) = \frac{2x}{5} + \frac{9}{5x}

Also, again 3f(x)-2f(1/x)=x….A

3f(1/x)-2f(x)=1/x…B

So, we now we get the following two equations:

9f(x)-6f(1/x)=3x…..A’

6f(1/x)-4f(x)=2/x….B’

so, now we have 5f(x) = 3x + \frac{2}{x} so that we get f(x) = \frac{3}{5}x+\frac{2}{5x} andf(1/x) = \frac{2}{5}x+\frac{9}{5x}

so f^{'}(x) = \frac{3}{5} + \frac{2}{5}\frac{-1}{x^{2}} = \frac{3}{5} - \frac{1}{10}=\frac{1}{2}

Question 3:

If x= \frac{a(1-t^{2})}{1+t^{2}} and y = \frac{2bt}{1+t^{2}}, then find \frac{dy}{dx}.

Answer 3:

Given that x = \frac{a(1-t^{2})}{1+t^{2}} where a is a parameter (constant) and t is a variable.

Let t=\tan{\theta} so that x = \frac{a(1-\tan^{2}{\theta})}{1+\tan^{2}{\theta}} = a \cos{2\theta}

so that y = \frac{2bt}{1+t^{2}}=b \sin{2\theta}

\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{2b\cos{2\theta}}{-2a\sin{2\theta}}=- \frac{b}{a\tan{2\theta}}

so that we have

\frac{dy}{dx} = - \frac{b(1-t^{2})}{a}

Question 4:

If y = \arccos{\frac{x-x^{-1}}{x+x^{-1}}} then find \frac{dy}{dx}

Answer 4:

Given that \arccos{\frac{x^{2}-1}{x^{2}+1}} = \arccos{\frac{1-x^{2}}{1+x^{2}}} and put x=\tan{\theta}

\frac{1-\tan^{2}{\theta}}{1+\tan^{2}{\theta}}=\cos{2\theta} so that y = \arccos {\cos{2\theta}}=2\theta

\frac{dy}{dx} = 2\frac{d}{dx}(\arctan{x})=\frac{2}{1+x^{2}} which is the required answer.

Question 5:

If \arcsin{(\frac{x^{2}-y^{2}}{x^{2}+y^{2}})}=a, where a is a parameter, then find \frac{dy}{dx}.

Answer 5:

Given that a=\arcsin{(\frac{x^{2}-y^{2}}{x^{2}+y^{2}})} so that \sin{a} = \frac{x^{2}-y^{2}}{x^{2}+y^{2}}

(x^{2}+y^{2})\sin{a} = x^{2}-y^{2}

Differentiating both sides w.r.t. x, we get

2x\sin{a} + \sin{a} ,2y.\frac{dy}{dx}=2x-2y\frac{dy}{dx}

\frac{dy}{dx}(2y\sin{a}+2y)=2x-2x\sin{2a}

\frac{dy}{dx} = \frac{2x(1-\sin{a})}{1+\sin{a}}=\frac{x}{y}\times \frac{2y^{2}}{2x^{2}}=\frac{y}{x}

Question 6:

If y = cot^{-1}{(\sqrt{\frac{1+x}{1-x}})} then find \frac{dy}{d(\arccos{x})}.

Answer 6:

Given that y = cot^{-1}(\sqrt{(\frac{1+x}{1-x})}) so that y = \arctan{(\sqrt{(\frac{1-x}{1+x})})} = \arctan{(cot {(2\theta)})} where x=\tan^{2}{\theta} so that \frac{d\theta}{dx} = \frac{1}{1+x^{2}}

and \sec^{2}{y}.\frac{dy}{dx} = - cosec^{2}{(2\theta)}.2\frac{d\theta}{dx}

Let f=\arccos{x} so that \frac{df}{dx} = - \frac{1}{\sqrt{1-x^{2}}}

Now, note that \sec^{2}{y} = cosec^{2}{2\theta} so we get the following simplification:

\frac{dy}{dx} = - \frac{2}{1+x^{2}}

Now, \frac{dy}{df} = -\frac{\frac{dy}{dx}}{\frac{df}{dx}}= \frac{2\sqrt{1-x^{2}}}{1+x^{2}}

Cheers,

Nalin Pithwa

Derivatives: part 13: IITJEE Math tutorial problems for practice

Question 1:

Let f(x) be a differentiable function w.r.t. x at x=1 and \lim_{h \rightarrow 0} \frac{1}{h}f(1+h)=5, then evaluate f^{'}(1)

Solution 1:

By definition, derivative of a function f(x) is f^{'}(x) = \lim_{t \rightarrow x}\frac{f(t)-f(x)}{t-x}, where let us substitute t-x=h, t=x+h, as t \rightarrow x, then h \rightarrow 0

So that above expression is equal to \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}

f^{'}(1) = \lim_{h \rightarrow 0} \frac{f(1+h)-f(1)}{h} exists and can be evaluated if we know the value of the function f(x) at x=1.

Question 2:

If x\sqrt{1+y} + y \sqrt{1+x}=0, then find \frac{dy}{dx}.

Answer 2:

Given that x\sqrt{1+y} + y\sqrt{1+x}=0

Taking derivative of both sides w.r.t. x, we get the following equation:

\sqrt{1+y} \times 1 + \frac{x}{2\sqrt{1+y}}.\frac{dy}{dx} + \sqrt{1+x}\frac{dy}{dx}+\frac{y}{2\sqrt{1+x}}.1=0

\sqrt{1+y}+ \frac{y}{2\sqrt{1+x}}+\frac{dy}{dx} \times (\frac{x}{2\sqrt{1+y}}+\sqrt{1+x})=0

This further simplifies to :

\frac{dy}{dx}. (\frac{x+2\sqrt{(1+x)(1+y)}}{2\sqrt{1+y}}) = \frac{2\sqrt{(1+x)(1+y)}+y}{2\sqrt{1+x}}

\frac{}{} = 2 \times \sqrt{\frac{1+y}{1+x}} \times \frac{2\sqrt{(1+x)(1+y)+y}}{x+2\sqrt{(1+x)(1+y)}}

But, we already know that x\sqrt{1+y}=-y\sqrt{1+x} so that \sqrt{\frac{1+y}{1+x}} = - \frac{y}{x}

\frac{dy}{dx} = - \frac{2y}{x} \times \frac{2\sqrt{(1+x)(1+y)}+y}{x+2\sqrt{(1+x)(1+y)}}

\frac{dy}{dx}= -2 \times \frac{2y\sqrt{(1+x)(1+y)}+y^{2}}{x^{2}+2x\sqrt{(1+x)(1+y)}}

\frac{dy}{dx}=-2. \frac{2x(1+y)+y^{2}}{x^{2}-2y(1+y)} is the desired answer.

You can see how ugly it looks. Is there any way to simplify above? Let us give it one more shot. As follows:

Given that x\sqrt{1+y} + y\sqrt{1+x}=0 Hence, x^{2}(1+y)=y^{2}(1+x) so that

x^{2}-y^{2}=y^{2}x-x^{2}y

(x+y)(x-y) = y^{2}x-x^{2}y=xy(y-x). If x \neq y, then

x+y= -xy. Taking derivative of both sides w.r.t. x, we get:

1+\frac{dy}{dx} = y(-1)+(-x)\frac{dy}{dx}

(1+x)\frac{dy}{dx} = -1-y

\frac{dy}{dx} = - \frac{1+y}{1+x}= - \frac{y^{2}}{x^{2}} which is such an elegant answer 🙂

Question 3:

If x^{y}.y^{x}=c, where c is a parameter constant, then find \frac{dy}{dx} at (e,e).

Solution 3:

Let u=x^{y} and v=y^{x}.

Taking logarithm of both sides:

\log {u} = y \log {x} and \log {v} = x \log{y}.

Consider the LHS equation:

Taking derivative of both sides w.r.t.x, we get:

\frac{1}{u}\frac{du}{dx} = \frac{y}{x} + \frac{dy}{dx}. \log{x}

\frac{1}{u}. \frac{du}{dx} = \frac{y}{x} + \frac{dy}{dx} (\log{x})

\frac{du}{dx} = x^{y} \times (\frac{y}{x}+(\log{y}).\frac{dy}{dx})

\log {v} = x \log{y}

\frac{1}{v}\frac{dv}{dx} = \frac{x}{y}\frac{dy}{dx} + (\log{y})

\frac{dv}{dx} = y^{x} \times (\frac{x}{y}\frac{dy}{dx}+ \log{y}).

Also, x^{y}\frac{dv}{dx} + y^{x}\frac{du}{dx}=0

x^{y}y^{x} \times (\frac{x}{y}\frac{dy}{dx}+\log{y}) + y^{x}.x^{y}. (\frac{y}{x}+(\log{x}).\frac{dy}{dx}) =0

x^{y}y^{x} \times (\frac{dy}{dx}(\frac{x}{y}+\log{x})+\log{y}+\frac{y}{x})=0 Now substitute (x,y)=(e,e) and get the required answer.

x^{y}y^{x} \times (\frac{dy}{dx}(\frac{x}{y})+\log{x}) = - x^{y}y^{x}(\frac{y}{x}+\log{y})

\frac{dy}{dx} (\frac{x}{y}+\log{x})=-(\frac{y}{x}+\log{y})

Substituting (x,y) = (e,e)

Hence, then, (\frac{dy}{dx})_(e,e) = - \frac{e/e + log e}{e/e + \log {e}}=-1 is the desired answer.

Question 4:

Find \frac{d}{dx}(\tan(\arctan{x} + cot^{-1}(x+1))

Answer 4:

Consider \tan(A+B) = \frac{\tan{A}+\tan{B}}{1-\tan{A}\tan{B}}

Subsituting A= \arctan{x} and B=cot^{-1}(x+1), we get the following:

\tan{(A+B)} = \frac{tan(\arctan{x}+tan(cot^{-1}(x+1)))}{1-x. tan(cot^{-1}(x+1))} = \frac{x+\frac{1}{(x+1)}}{1-\frac{x}{x+1}}

which in turn equals \frac{x+\frac{1}{x+1}}{1-\frac{x}{x+1}}=\frac{x^{2}+x+1}{1} noting that \arctan{(\frac{1}{x+1})}=cot^{-1}(x+1)

Hence, the answer is \frac{d}{dx}(x^{2}+x+1)=2x+1

Question 5:

If y = \arctan{\sqrt{\frac{1+\sin{x}}{1-\sin{x}}}}, find \frac{dy}{dx}

Solution 5:

Given that \tan{y} = \sqrt{\frac{1+\sin{x}}{1-\sin{x}}}

\tan^{2}{y} = \frac{1+\sin{x}}{1-\sin{x}}. Taking derivative of both sides w.r.t. x,

2(\tan{y})/ \sec^{2}{y}\frac{dy}{dx} = \frac{(1-\sin{x})(\cos{x})}{(1-\sin{x})^{2}}

2\tan{y}\sec^{2}{y}\frac{dy}{dx} = \frac{\cos{x}-\sin{x}\cos{x}+\cos{x}+\sin{x}\cos{x}}{(1-\sin{x})^{2}} which in turn equals

\frac{2\cos{x}}{(1-\sin{x})^{2}}

But, \tan^{2}{y} = \frac{1+\sin{x}}{1-\sin{x}}

so that \sec^{2}{y}=1+\tan^{2}{y}=1+\frac{1+\sin{x}}{1-\sin{x}} = \frac{2}{1-\sin{x}}

Hence, 2. \sqrt{\frac{1+\sin{x}}{1-\sin{x}}} \times \frac{2}{1-\sin{x}} \times \frac{dy}{dx} = \frac{2\cos{x}}{(1-\sin{x})^{2}}

Hence, \frac{dy}{dx} = \frac{1}{2} \times \frac{\cos{x}}{\sqrt{(1+\sin{x})(1-\sin{x})}} = \frac{\cos{x}}{2\sqrt{1-\sin^{2}{x}}} = \frac{1}{2}

Question 6:

Find \frac{d}{dx}cot^{-1}(\frac{1+\sqrt{1+x^{2}}}{x})

Solution 6:

Let y = cot^{-1} (\frac{1+\sqrt{1+x^{2}}}{x})

Put x = \sin{\theta} so that \sqrt{1-x^{2}}=\sqrt{1-\sin^{2}{\theta}}=\cos{\theta}

\frac{1+\cos{\theta}}{\sin{\theta}} = \frac{2\cos^{2}(\theta/2)}{2\sin{\theta/2}\cos{\theta/2}} = cot (\theta/2)

cot^{-1}(cot{(\theta/2)}) = \theta/2

y=\theta/2

\frac{dy}{dx} = \frac{1}{2}\frac{d}{dx}( \arcsin {x} )=\frac{1}{2\sqrt{1-x^{2}}} where |x|<1

Question 7:

If y = \arcsin{(\frac{2x}{1+x^{2}})}+sec^{-1}(\frac{1+x^{2}}{1-x^{2}}). Find \frac{dy}{dx}.

Solution 7:

Let x = \tan{\theta} so that \frac{2x}{1+x^{2}} = \frac{2\tan{\theta}}{1+\tan^{2}{\theta}} = \sin{2\theta}

so that \arcsin{(\frac{2x}{1+x^{2}})} = \arcsin{\sin{2\theta}}=2\theta

We now have \frac{1-x^{2}}{1+x^{2}} = \frac{1-\tan^{2}{\theta}}{1+\tan^{2}{\theta}} = \cos{2\theta}

so that sec^{-1}{(\frac{1+x^{2}}{1-x^{2}})} = sec^{-1}(sec {(2\theta)}) =2 \theta

so the desired answer is 4\frac{d\theta}{dx}=\frac{4}{1+x^{2}}

Question 8:

If y= \arcsin{(\frac{\sqrt{1+x}+\sqrt{1-x}}{2})} then find \frac{dy}{dx}

Solution 8:

Given that \sin{y} = \frac{\sqrt{1+x}+\sqrt{1-x}}{2}

2\cos{y}\frac{dy}{dx} = \frac{1}{2\sqrt{1+x}} \times 1 + \frac{1}{2\sqrt{1-x}} \times (-1)

2\cos{y} \frac{dy}{dx} = \frac{\sqrt{1-x}-\sqrt{1+x}}{2\sqrt{1-x^{2}}}

2\cos{y} \frac{dy}{dx} = \frac{1-x-(1+x)}{(2\sqrt{1-x^{2}})(\sqrt{1-x}+\sqrt{1+x})} = - \frac{2x}{(2\sqrt{1-x^{2}})(\sqrt{1-x}+\sqrt{1+x})}

2\cos{y} \frac{dy}{dx} = - \frac{x}{(\sqrt{1-x^{2}})(2\sin{y})}

\frac{dy}{dx} = - \frac{x}{2\sqrt{1-x^{2}} (\sin{2y})} but \sin{2y} = \frac{\sqrt{1+x}+\sqrt{1-x}}{2} and \cos{y} = \frac{\sqrt{1+x}-\sqrt{1-x}}{2}

so now we have \sin{2y} = \frac{2}{4} (1+x-(1-x)) = x

Hence, we get \frac{dy}{dx} = - \frac{x}{2\sqrt{1-x^{2}}(x)}=-\frac{1}{2\sqrt{1-x^{2}}}.

Question 9:

If g(x) = x^{2}+2x+3f(x) and f(0)=5 and \lim_{x \rightarrow 0} \frac{f(x)-5}{x}=4, then evaluate g^{'}(0).

Solution 9:

We have g^{'}{(x)} = 2x+2+3f^{'}(x) and hence, g^{'}{(0)}=2+3f^{'}{(0)}

By definition of derivative, we have f^{'}{(x)}= \lim_{t \rightarrow x} \frac{f(t)-f(x)}{t-x} where let us say t-x=h so that t \rightarrow x, and h  \rightarrow 0

f^{'}(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}

\lim_{h \rightarrow 0} \frac{f(h)-5}{h}=4 and f(0)=5 and hence, f^{'}(0)=4.

Hence, g^{'}(0)=2+3 \times 4=14

Question 10:

If \tan{y} = \frac{2t}{1-t^{2}}, and \sin{x}=\frac{2t}{1+t^{2}}, then find \frac{d^{2}y}{dx^{2}}.

Answer 10:

Let t = \tan{\theta} and hence, \frac{2t}{1-t^{2}} = \frac{2\tan{\theta}}{1-\tan^{2}{\theta}} = \tan{2\theta}

Hence, \tan{y} = \tan{2\theta}

so that \sec^{2}{y} \frac{dy}{dx} = \sec^{2}{(2\theta)}.2.\frac{d\theta}{dx}

Let t=\tan{\theta} so that \theta = \arctan{t} and \frac{d\theta}{dx}=\frac{1}{1+t^{2}} \times \frac{dt}{dx}

Hence, we get (\sec^{2}{y})\frac{dy}{dx} = \sec^{2}(2\theta) \times \frac{2}{(1+t^{2})}.\frac{dt}{dx} so that

\sec^{2}{(2\theta)}\frac{dy}{dx} = \sec^{2}{(2\theta)} \times \frac{2}{1+t^{2}} \times \frac{dt}{dx}

Hence, \frac{dy}{dx} = \frac{2}{(1+t^{2})}\frac{dt}{dx}

Hence, \sin{x}=\frac{2t}{1+t^{2}} = \frac{2\tan{\theta}}{1+\tan^{2}{(\theta)}}=2 \frac{\sin{\theta}}{\cos{\theta}} \frac{\cos^{2}{\theta}}{1}

Hence, \sin{x} = \sin{2\theta} and hence x=2\theta and so t=\tan{\theta} hence, \theta=\arctan{t}

x =\arctan{t} so that t=\tan{x}

\frac{dt}{dx} = \frac{1}{1+x^{2}}….call this A.

\frac{dy}{dx} = \frac{2}{1+t^{2}} \frac{dt}{dx}…call this B.

\frac{dy}{dx} = \frac{2}{(1+t^{2})(1+x^{2})}

\frac{d^{2}y}{dx^{2}} = \frac{d}{dx}(\frac{2}{(1+t^{2})(1+x^{2})}) = \frac{2}{(1+t^{2})} \frac{d}{dx}(\frac{1}{1+x^{2}}) + \frac{2}{(1+x^{2})} \frac{d}{dx}(\frac{1}{(1+t^{2})})

\frac{d^{2}y}{dx^{2}} = \frac{2}{(1+t^{2})}. \frac{-2x}{(1+x^{2})^{2}}+\frac{2}{(1+x^{2})}.\frac{-2t}{(1+t^{2})^{2}}.\frac{dt}{dx}

=\frac{-4x}{(1+t^{2})(1+x^{2})^{2}} - \frac{4t}{(1+x^{2})^{2}(1+t^{2})^{2}}

\frac{d^{2}y}{dx^{2}} = \frac{-4x(1+t^{2})-4t}{(1+x^{2})^{2}(1+t^{2})^{2}} = \frac{-4(t+x(1+t^{2}))}{(1+x^{2})^{2}(1+t^{2})^{2}} which in turn equals

\frac{-4(\tan{x}+x (1+\tan^{2}{x}))}{(1+x^{2})^{2}(1+\tan^{2}{x})^{2}} so where did we go wrong….quite clearly, practice alone can help us develop foresight…below is a cute proof:

\frac{dy}{dx}= \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{\frac{d}{dt}(\arctan{(\frac{2t}{1-t^{2}})})}{\frac{d}{dt}\arcsin(\frac{2t}{1+t^{2}})} and put t=\tan{\theta}

so that \frac{dy}{dx} = \frac{2\frac{d\theta}{dt}}{2\frac{d\theta}{dt}}=1 so we have bingo 🙂 an elegant answer

\frac{d^{2}y}{dx^{2}}=1

Cheers,

Nalin Pithwa.