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(Read December 6, 1912.)

In a much quoted sentence, Klein has said "We shall have a picture of the development of mathematics if we imagine a chain of lofty mountains as representative of the men of the eighteenth century, terminating in a mighty outlying summit,—Gauss, and then a broader, hilly country of lower elevation; but teeming with new elements of life.” This was written in 1893 and would perhaps still be received as the truth by most observers. During the ensuing period it has, however, become more and more evident that two of the contemporary hills rise quite above the others, and it may be that when they are seen in perspective they will compare favorably with the more distant mountains.

Hilbert and Poincaré are associated and contrasted in the minds of all students of mathematics both by the brilliance of their achievements and the difference of their methods. To enter into a comparative study of these two great men would not be an appropriate exercise for this occasion, but to have suggested it may help to indicate the relation of Poincaré to the time in which he lived.

Aside from his intellectual triumphs, which one could adequately comprehend only by reading a series of his papers, the life of Poincaré presents little of interest. He was born at Nancy on the 29th of April, 1854. His unusual gifts were recognized early, so that he had an excellent education. He received the degrees of Bachelor of Letters and of Science in 1871 and that of Mining Engineer in 1879. He was attached in one capacity or another to the Department of Mines of the French government for the rest of his life, but not in such a way as to interfere with his scientific work.

In the year 1879 he also received his doctorate of Science from the University of Paris. He was immediately made a member of the faculty of science at Caen. From there he was called to the C'niversity of Paris in 1881. In 1886 he was appointed professor

of physics and of the calculus of probabilities in the University of Paris and in 1896 became professor of mathematical astronomy at the same university. In 1904 he was also made professor of general astronomy at the École Polytechnique. Since 1902 he occupied the chair of electricity at the École professionnelle supérieure des Postes et des Télégraphes. He died on the 17th of July, 1912,

The importance of his scientific contributions was recognized from the very beginning of his career. He received practically all the distinctions which are open to a mathematician. Among the most notable were: Election to the Academy of Science of France (Section of Geometry) in 1887, election to the French Academy in 1908, and the Bolyai Prize for excellence in all fields of mathematics in 1905. He was elected a member of the American Philosophical Society in 1899.

Poincaré was particularly distinguished among his contemporaries by the wide range of his creative power. He left behind enduring works not only in the several branches of pure mathematics but in astronomy, physics and philosophy. He has often been described as the last of the universals. Indeed in this respect as well as in the brilliance of his individual works, he is like those earlier heroes of science whom Klein compared to the chain of lofty mountains.

It goes without saying that one could not expect to give an adequate account of Poincaré's complete work in a short address like

I shall try, however, to mention certain main divisions of his work, taking them up in an order which is roughly chronological. Naturally, the periods to which I shall refer all overlap but I shall try to arrange them according to the dates of the central papers in each subject.

Poincaré's doctoral dissertation, which was his first published work of importance, appeared in 1879. Its title was "On the Properties of Functions Defined by Partial Differential Equations" and it supplies the existence theorem for solutions in the neighborhood of singular points of a very general type. This memoir initiated a long series of brilliant contributions to the theory of differential equations, especially to that of linear differential equations. Most of these papers appeared in the period before 1886.

this one.

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