Men of Mathematics
Page 68
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 center is at the origin—(0, 0)—and whose radius is a, is x2 + y2 = a2. According to the second of Poincaré’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 and y = a sin t, then, squaring and adding, we get rid of t (since cos2 t sin2 t = l), and find x2 + y2 = a2. But the trigonometric functions cos t, 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 astonishing things in analysis which Poincaré 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 (see chapter on Gauss) in a geometrical shape which appeals particularly to those who, like Poincaré, 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 caliber did not pass unappreciated. At the unusually early age of thirty two (in 1887) Poincaré was elected to the Academy. His proposer said some pretty strong things, but most mathematicians will subscribe to their truth: “[Poincaré’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 Poincaré’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 universality.
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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 labors of pure mathematicians. To Poincaré, 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 grand 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 Poincaré opened his offensive, few have mastered his weapons and some, unable to bend his bow, insinuate that it is worthless in a practical attack. Nevertheless Poincaré is not without forceful champions whose conquests would have been impossible to the men of the pre-Poincaré era.
Poincaré’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” (n = 3) 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, M at a distance D apart attract one another with a force proportional to Imagine n material particles distributed in any manner in space; the masses, 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, or 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—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 observational data to describe the general configuration now. The problem of course is tremendously complicated by radiation—the masses of the 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. Poincaré did not solve the problem, but in 1889 he was awarded the prize anyhow 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, Moon, and Sun furnish an instance of the case n = 3. 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 its 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 Poincaré a Knight of the Legion of Honor—a much less expensive acknowledgment of the young mathematician’s genius than the King’s 2500 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 solving a system of nine simultaneous 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 to be expected infinite terms; if a solution exists at all it will be given by 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 1909 a considerable advance came from a rather unexpected quarter—Finland, a country which sophisticated Europeans even today consider barely civilized, especially for its queer custom of paying its debts, and which few Americans thought advanced beyond 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, utilizing analytical methods due to the Italian Levi-Civita and the French Painlevé, 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.
Poincaré’s most original work in mathematical astronomy was summed up in h
is great treatise Les méthodes nouvelles de la mécanique céleste (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, Leçons de mécanique céleste, and a little later by the publication of his course of lectures Sur les figures d’équilibre d’une masse fluide (On the figures of equilibrium of a fluid mass), and a historical-critical book Sur les hypothèses 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 Mécanique céleste 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 Poincaré 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 positions 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 its 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 Poincaré’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 Poincaré’s work on the problem of 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 Poincaré’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 ellipsoid 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.
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Once more we must leave the banquet having barely tasted anything and pass on to Poincaré’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 born 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 roué 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 Poincaré 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 by Poincaré and made the topic of several purely mathematical investigations. When wireless telegraphy was invented he seized on 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, Poincaré was already busy with its mathematics, and 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 Poincaré what Ceres did for Gauss; and although Poincaré 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 Poincaré’s genius is entitled to his hobbies.
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We pass on now to the last phase of Poincaré’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 Mathématique 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.IV Anyone wishing to look into the “psychology” of mathematicians will find much of interest in this unique work and many confirmations of the views at which Poincaré had arrived independently before he saw the results of the questionnaire. A few points of general interest may be noted before we quote from Poincaré.
The early interest in mathematics of those who were to become great mathematicians has been frequently exemplified in preceding chapters. 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 (I know one very distinguished mathematician who begins his day’s work at the inhuman hour of five a.m.), 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 at night, while the Latins favor 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’s chloral 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 well prepared by labor, both conscious and subconscious.”
Those who, like Thomas Alva Edison, have declared that genius is ninety nine per cent perspiration and only one per cent inspiration, are not contradicted by those who would reverse the figures. Both are right; one man remembers the drudgery while another forgets it all in the thrill of apparently sudden 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 novels that Balzac inflicted on his maddened printer.