POINCAR, HENRI

born April 29, 1854, Nancy, Fr. died July 17, 1912, Paris in full Jules-henri Poincar French mathematician, theoretical astronomer, and philosopher of science who influenced cosmogony, relativity, and topology and was a gifted interpreter of science to a wide public. Poincar was from a family distinguished by its contributions to government and administration. His first cousin was Raymond Poincar, president of the French Republic during World War I. Poincar was ambidextrous and was nearsighted; during his childhood he had poor muscular coordination and was seriously ill for a time with diphtheria. He received special instruction from his gifted mother and excelled in written composition while still in elementary school. Becoming deeply interested in mathematics during adolescence, he attended in 187275 the cole Polytchnique in Paris, where he easily won top honours in mathematics, but was undistinguished in physical exercise and in art. Poincar had an unusually retentive memory for everything he read; moreover, he could visualize what he heard, a useful faculty because he could not clearly see at a distance the mathematical symbols that were on the blackboard. Throughout his life he was able to perform complex mathematical calculations in his head and could quickly write a paper without extensive revisions. He received, in 1879, a doctorate from the cole Nationale Suprieure des Mines with a thesis on differential equations. Following a brief appointment in mathematical analysis at the University of Caen, in 1881 Poincar joined the University of Paris, where, during the rest of his life, he lectured and wrote prolificallyalmost 500 paperson mechanics and experimental physics, in all branches of pure and applied mathematics, and in theoretical astronomy. Changing his lectures every year, he would review optics, electricity, the equilibrium of fluid masses, the mathematics of electricity, astronomy, thermodynamics, light, and probability. Many of these lectures appeared in print shortly after they were delivered at the university. In his extensive writings on probability, Poincar anticipated the concept of ergodicity that is basic to statistical mechanics. Applying his mastery of analysis to the question of the solvability of algebraic equations, Poincar developed before age 30 the idea of the automorphic functionone that is invariant under a group of transformations that are characterized algebraically by ratios of linear terms. He showed how these functions can be used to integrate linear differential equations with rational algebraic coefficients and how to express the coordinates of any point on an algebraic curve as uniform functions of a single algebraic variable (or parameter). Some of these automorphic functions he called Fuchsian, after the German mathematician Immanuel Lazarus Fuchs, who was one of the founders of the theory of differential equations; he found that they were associated with transformations arising in non-Euclidean geometry. In recognition of his fundamental contributions to mathematics, he was elected, in 1887, to membership in the Acadmie des Sciences in Paris. In celestial mechanics, Poincar made substantial contributions to the theory of orbits, particularly the classical three-body problem (for example, the system involving the Sun, Moon, and Earth). This was part of the problem of n bodies (planets, stars, etc.), set for a prize by King Oscar II of Sweden: given the present masses, velocities, motions, and mutual distances of n bodies, how long will they remain stable in their present spatial relationships, or will their orbits change at some future date? In his solution, Poincar developed powerful new mathematical techniques, including the theories of asymptotic expansions and integral invariants, and made fundamental discoveries on the behaviour of the integral curves of differential equations near singularities. He was awarded the prize in 1889, even though his solution to the problem was only partially correct; in the same year he was also made a knight of the French Legion of Honour. Poincar summarized his new mathematical methods in astronomy in Les Mthodes nouvelles de la mcanique cleste, 3 vol. (1892, 1893, 1899; The New Methods of Celestial Mechanics). Another result of this work, his Analysis situs (Positional Analysis) in 1895, was an early systematic treatment of topology, which deals with properties of a system that endure when metric distortion occursthat is, topology deals with the qualitative characteristics of spatial configurations that do not vary during cumulative transformations. He also contributed to the theory of numbers by demonstrating how the conception of binary quadratic forms, which was developed by the German mathematician Carl Friedrich Gauss, could be cast in geometric form. In 1904 he lectured at the St. Louis Exposition. In mathematical analysis, Poincar made important contributions to the theory of equilibrium of rotating fluid masses. In particular, he described the conditions of stability of the pear-shaped figures that played so prominent a part in the researches of later cosmogony, with reference to the evolution of celestial bodies. He attempted an application of these ideas to the stability of Saturn's rings and to the origin of binary stars. In 1906, in a paper on the dynamics of the electron, he obtained, independently of Albert Einstein, many of the results of the special theory of relativity. The principal difference was that Einstein developed the theory from elementary considerations concerning light signaling, whereas Poincar's treatment was based on the full theory of electromagnetism and was restricted to phenomena associated with the concept of a universal ether that functioned as the means of transmitting light. After Poincar achieved prominence as a mathematician, he turned his superb literary gifts to the challenge of describing for the general public the meaning and importance of science and mathematics. Always deeply interested in the philosophy of science, he wrote La Science et l'hypothse (1903; Science and Hypothesis), La Valeur de la science (1905; The Value of Science), and Science et mthode (1908; Science and Method), all of which reached a wide public of nonprofessionals. His works were translated into English, German, Hungarian, Japanese, Spanish, and Swedish. He emphasized the subconscious, while probing the psychology of mathematical discovery and invention. He was a forerunner of the modern intuitionist school in that he believed that some mathematical induction is a priori and independent of logic. In his view, sudden illumination, following long subconscious work, is a prelude to mathematical creation. But his greatest contribution to philosophy was to emphasize the role played in scientific method by conventioni.e., by the arbitrary choice of concepts. Poincar's prestige and influence increased in French science, and in 1906 he was elected president of the Acadmie des Sciences and in 1908 to membership in the Acadmie Franaise, the highest honour accorded a French writer. Gerald James Whitrow