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Jeremy Gray is professor of the history of mathematics at the Open University, and an honorary professor at the University of Warwick. His most recent book is Plato's Ghost: The Modernist Transformation of Mathematics (Princeton).

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"Poincaré was much more than a mathematician: he was a public intellectual, and a rare scientist who enthusiastically rose to the challenge of explaining and interpreting science for the public. With amazingly lucid explanations of Poincaré's ideas, this book is one that any reader who wants to understand the context and content of Poincaré's work will want to have on hand."--Dana Mackenzie, author ofThe Universe in Zero Words

"This engaging book recounts the achievements of Henri Poincaré, covering his mathematics, physics, and philosophy, and his activities as a public intellectual. He is an eminently worthy subject for an intellectual biography of this kind."--Benjamin Wardhaugh, University of Oxford

"This comprehensive scientific biography of Poincaré situates the scientist's life and work in the sociopolitical context of his era. Covering his varied and wide-spanning work--from the most philosophical to the most technical--this book gives the general reader a clear historical sense of the man's voluminous accomplishments."--Jimena Canales, Harvard University

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"Poincaré was much more than a mathematician: he was a public intellectual, and a rare scientist who enthusiastically rose to the challenge of explaining and interpreting science for the public. With amazingly lucid explanations of Poincaré's ideas, this book is one that any reader who wants to understand the context and content of Poincaré's work will want to have on hand."--Dana Mackenzie, author ofThe Universe in Zero Words

"This engaging book recounts the achievements of Henri Poincaré, covering his mathematics, physics, and philosophy, and his activities as a public intellectual. He is an eminently worthy subject for an intellectual biography of this kind."--Benjamin Wardhaugh, University of Oxford

"This comprehensive scientific biography of Poincaré situates the scientist's life and work in the sociopolitical context of his era. Covering his varied and wide-spanning work--from the most philosophical to the most technical--this book gives the general reader a clear historical sense of the man's voluminous accomplishments."--Jimena Canales, Harvard University

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HENRI POINCARÉ

PRINCETON UNIVERSITY PRESS

Contents

List of Figures....................................................ixPreface............................................................xiIntroduction.......................................................1Views of Poincaré.............................................3Poincaré's Way of Thinking....................................61 The Essayist.....................................................272 Poincaré's Career...........................................1533 The Prize Competition of 1880....................................2074 The Three Body Problem...........................................2535 Cosmogony........................................................3006 Physics..........................................................3187 Theory of Functions and Mathematical Physics.....................3828 Topology.........................................................4279 Interventions in Pure Mathematics................................46710 Poincaré as a Professional Physicist.......................50911 Poincaré and the Philosophy of Science.....................52512 Appendixes......................................................543References.........................................................553Articles and Books by Poincaré................................554Other Authors......................................................564Name Index.........................................................585Subject Index......................................................589

Chapter One

POINCARÉ AND THE THREE BODY PROBLEM

On 20 January 1889 the ambitious Swedish mathematician Gösta Mittag-Leffler went to the Court of King Oscar II of Sweden to announce the judges' decision concerning the prize competition the King had announced in 1885. Mittag-Leffler had administered it, and would have been feeling very pleased, for the competition had been a success: the result would surely play well with the King and would add, as intended, to the celebrations for the King's 60th birthday. Moreover, it would be popular with professional mathematicians; none of this could do other than advance Mittag-Leffler's own career.

King Oscar II was an enlightened monarch who proposed prize competitions from time to time. He had studied mathematics at Uppsala University and retained an interest in it all his life. He had given financial support to Mittag-Leffler's new journal, Acta Mathematica, occasionally sponsored individual mathematicians, and in 1884 had asked Mittag-Leffler to run the prize competition whose result he was shortly to discover (it is not known if he had had the original idea or if it was a suggestion of Mittag-Leffler's). Together they took the risky decision of calling for essays on specific topics, rather than merely awarding a fine recent piece of work. The four topics they chose, and which were published in Acta Mathematica, some German and French journals, and in English translation in Nature (30 July 1885) were all substantial, but the winning one addressed in suitably technical language the most publicly accessible topic:

A system being given of a number whatever of particles attracting one another mutually according to Newton's law, it is proposed, on the assumption that there never takes place an impact of two particles to expand the coordinates of each particle in a series proceeding according to some known functions of time and converging uniformly for any space of time.

As every mathematician and astronomer could see, a successful resolution of this problem would enable one to describe the position of every planet in the solar system at every past and future moment of time. For this reason the problem is often referred to as asking for a proof of the stability of the solar system, although an alarming range of life-destroying orbits for the earth are compatible with the stated requirements.

Mittag-Leffler had ensured that the entries, all of which had to be received by 1 June 1888, would be judged by a very distinguished panel: Weierstrass, Hermite, and, in a lesser capacity, Mittag-Leffler himself. Karl Weierstrass in Berlin and Charles Hermite in Paris were the leading mathematicians in each country in their generation, the most powerful influences behind the scenes, and, not coincidentally, people whom Mittag-Leffler had already used to help him in his career. All three worked chiefly or exclusively in an area of mathematics called complex function theory, which was one of the dominant fields of mathematics at the time and taught at least to the better students in all leading universities. Entries were to be submitted anonymously, tied to their authors by a system of sealed envelopes, although it is hard to believe that the panel would not recognize the handwriting of most of the people capable of entering the competition. And there was a prize: a gold medal bearing his Majesty's image and having a value of one thousand francs, and a sum of 2,500 crowns (about 2,600 francs, equivalent to a professor's salary for half a year).

To minimize the risk that no one would enter, or that no entries of sufficient quality would be received, the four questions were carefully tailored to current interests, specifically the work of Lazarus Fuchs, a German mathematician, the French mathematicians Charles Briot and Jean-Claude Bouquet, both professors at the Sorbonne and known for their work on complex function theory and differential equations, and Henri Poincaré. In fact, as the judges knew very well, Poincaré's name could have been mentioned in connection with the work of Fuchs, for he had astounded the mathematical community with his extension of it in the years from 1880 to 1884 and made the reputation of Acta Mathematica in so doing. It could also have been mentioned in connection with the problem of the stability of the solar system, because Poincaré had also written extensively and with remarkable success on a simpler system of equations but always with an eye on the problem of planetary motion.

It was, of course, Poincaré who had won the competition. The King could be assured that his generosity had brought forth new work from the most talked about new member of the mathematical community. Mittag-Leffler could feel pleased that the man whose career he had encouraged by opening the pages of his journal to him would now repay him with a new work of great importance, one that was also going to be published in Acta Mathematica. All in all, a most satisfactory outcome. And indeed, the news played very well. Hermite was able to persuade the editors of the scientific pages of Le Temps and Le journal des débats to publicize the news, and they were followed by the Journal officiel and Le correspondant. Since a second prize had gone to another French mathematician, Paul Appell, the whole thing was turned into something of a French celebration. Both men were made Knights of the Légion d'honneur for their work and, as one of Poincaré's obituarists was to note (Darboux 1916, xxxi–xxxii), once the prize was announced:

The name of Henri Poincaré became known to the public, who then became accustomed to regarding our colleague no longer as a mathematician of particular promise, but as a great scholar of whom France has the right to be proud.

It helped, of course, that astronomy was a popular subject and celestial mechanics had a significant place in French scientific life, marked, for example, by Le Verrier's discovery of Neptune in 1846, an achievement commemorated by his being one of the 72 mathematicians, scientists and engineers to have their names inscribed on the Eiffel Tower in 1889.

Poincaré's account of what he had done in tackling the problem of the stability of the solar system formed the content of his first essay for the general public, his (1891d) in the Revue générale des sciences pures et appliquées. The Revue générale had been founded by Louis Olivier. It was published twice a month, and its first issue had come out on 15 January 1890. From the start, Olivier aimed to make it preeminent among the many journals which proliferated in France at this time, and he filled its pages with articles on every kind of science from agriculture to mathematics. He intended to speak to working scientists—his own training was in microbiology and he had studied with Pasteur—as well as educated general readers, and to convey a progressive message about the benefits of science and rational thought in solving problems of national importance while educating its citizens in the virtues of honesty and integrity. He edited his journal vigorously, and worked hard to bring in the best authors, and he seems to have succeeded: when he died in 1910, at the age of only 56, Nobel laureates were among those who wrote obituaries of him.

Poincaré was sympathetic to the cause of the small journals, and wrote a preface (1911b) to an otherwise almost valueless account of the Revue bleue and the Revue scientifique by Jacques Lux in 1911/12. Lux observed that the two revues had been founded in 1863 to revive French intellectual life, and had dedicated themselves anew to this task after the defeat in the Franco-Prussian War. The Revue scientifique in particular aimed not so much to popularize science as to make the methods of science better known—this was a call that Poincaré was to answer with more skill than most. In his preface, Poincaré noted the role these journals had had since their founding in breaking down the gulf between "those charmed by Letters and those passionate to research into scientific truths." He ascribed this gulf to the university system, and to the greatly varied nature of scientific work, and he dismissed the idea that some preestablished harmony could be relied upon to bring many muddled accounts into something "solidly established, free of error, and rich in beauty." Instead, he said, it was possible to know the whole world only by train, and "the revues are our trains. No doubt," he went on, "the trips are rather quick, and one can only see the essential things—but, as with good guides, one does see the essentials."

In his essay (1891d), Poincaré carefully explained that Newton's laws permit the completely accurate solution of two bodies orbiting around their common center of gravity: as Kepler had been the first to suggest, each body moves in an ellipse. But no one had been able to solve the problem of three or more bodies exactly. The best that could be done was to approximate the motion by expressing the coordinates of the separate bodies in polynomials of arbitrary degree. But this was not without its problems, he said. Suppose, he went on, that the motion is actually described by a periodic function with a very long period, and that the masses of the various bodies enter this function as parameters and are very small. Even so, increasing powers of these masses appear in the approximation, and these powers will grow indefinitely although the function being approximated does not (being periodic). For example, to approximate the function sin mt one can use the expression mt - [m³t³/6 . This is a very good approximation for small values of t but it increases indefinitely with t while sin mt remains between -1 and +1.

That said, he went on, this method of approximations is reliable, and it is already used to calculate tables of ephemerides accurate for navigational purposes or for tracking asteroids several years ahead. However, the final aim of mechanics is altogether more grand: it is to see if Newton's laws describe all astronomical phenomena. One can imagine, said Poincaré, any amount of improvement in these approximative methods, and indeed many people had invented new ones, but no one had been able to show that the infinite series that expressed these solutions converged and so could be used to establish the behavior of the solar system at all future times. This is what the Prize problem had called for when it asked that the solutions be expressed as "known functions of time ... converging uniformly for any space of time."

But, said Poincaré, could not complete rigor be achieved in some new way, say for the problem of three bodies moving under their mutual gravitational attraction? It had long been known that there were some extremely special cases that could be treated this way. There were results about three bodies constrained to lie in a plane. The American astronomer George William Hill had found a periodic solution in which the three orbits are all nearly circular (Hill 1877) and had used it to give a new and improved theory of the motion of the moon. Other special cases allowed for the bodies to move in different planes. There were solutions in which the planets slowly spiraled outwards to a closed orbit, and others in which, far back in the past, they had departed from such a closed orbit and were now spiraling into another; Poincaré called these the singly and doubly asymptotic solutions. But the general case seemed very far from being amenable to any kind of analysis.

In his work, he said, he had exploited another of Hill's insights: the actual motion, although not periodic, can sometimes be very well described by an approximation in terms of periodic functions. Laplace had shown this a century before in analyzing the motion of the moons of Jupiter. However, the existence of the asymptotic solutions already mentioned raised the worrying possibility that the power series solutions of the equations of motion would in fact diverge for large enough values of the time variable t (and maybe for all values of t). This was not, after all, a fatal objection, but it suggested a need to look for other ways of tackling the problem. And in fact he had been able to show that as far as the stability of the solar system was concerned, it was possible to give new results. He had looked at a special case of the three body problem, in which a small body of negligible mass orbits two larger bodies (such as the sun and Jupiter) which are going round in circles about their center of gravity in a common plane (it is further assumed that the small body moves in this common plane). In this case, he said, he had been able to show that under certain conditions the small body will return infinitely often indefinitely close to any position it has ever occupied, and in this sense its orbit is stable. (This form of stability was called Poisson stability by Poincaré in the prize-winning memoir (1890c, para. 8).) The asymptotic solutions mentioned before are not stable in this sense, so there are both unstable and stable solutions, and, said Poincaré, he had been able to show that the initial conditions that lead to an asymptotic orbit are exceptional and so stability holds in general.

This short account concealed as much as it revealed; there was a much more complicated and mathematically deeper story going on behind the scenes (see chap. 4 below). It is nonetheless informative. The problem at stake is to establish the stability of the solar system, not so much because anyone doubts it as because it would help confirm the insight that Newton's laws apply to the entire physical universe. Poincaré gave a reasonable hint as to the methods used, and a good indication of their imperfections. Even when the answer is very simple, say when the orbit is periodic, the approximations may lack that simplicity. The solutions will generally be given as infinite power series, but there is no proof that these make numerical sense because convergence may not be established, or even true. Nor were these problems a reflection of how poor had been the attempts to solve the problem: they were intrinsic to the problem as the variety of orbits, especially the asymptotic ones, showed. Then, rather modestly, Poincaré indicated that where everyone had been blocked he had managed to find a way forward, and show that there were good reasons to believe, at least in the first mathematically nontrivial case, that most orbits were stable. It was both an explanation of why the problem is so difficult and an indication that Newton's laws do indeed lead to plausible conclusions.

POINCARÉ'S POPULAR ESSAYS

Controversies over a New Geometry

In 1891 Poincaré also published another substantial essay in the second volume of the Revue générale des sciences pures et appliquées. This was his "Les géométries non euclidiennes" (1891b). Poincaré's success with the three body problem is what brought him into the public eye, but the essay on non-Euclidean geometry proved to be the start of a controversy that was to keep him there. It drew on what was not only one of his earliest but also one of his lifelong interests, it spoke to a contemporary concern about the nature of space and geometry, and Poincaré had something important and unexpected to say.

Geometry in 19th-century France was dominated by the legacy of two mathematicians, Gaspard Monge and Adrien-Marie Legendre, and was perpetuated through the influential École polytechnique. Monge, who had played a major role in establishing this École and its sister establishment the École normale supérieure during the French Revolution, had created the discipline of descriptive geometry. This was an ingenious way of coupling the depiction of objects in three-dimensional space on two planes (plan and elevation) with some simple algebra. This made it possible to calculate lengths and angles for masonry and it became an essential tool for military and civil engineers. Since in principle the job of the École polytechnique was to provide two years basic training for candidates to the Écoles d'application (such as Ponts et Chaussées, Mines, Artillerie, Arts et Métiers) this gave descriptive geometry a permanent place in the syllabus there, which indeed it retained until after the First World War.

(Continues...)

Excerpted from HENRI POINCARÉby JEREMY GRAY Copyright © 2013 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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