JULES HENRI POINCARfi. vir 



The theory of algebraic functions of one variable has as its most 

 striking auxiliary the manifolds of two dimensions known as Rie- 

 mann surfaces, and the theory of the connectivity of Riemann sur- 

 faces is the main object of the analysis situs of two dimensions. A 

 generalization of this theory to manifolds of any number of dimen- 

 sions was foreseen to some extent by Riemann himself and to a. 

 larger degree by Betti, who discovered a set of invariants of 7z-dimen- 

 sional manifolds which are known as the Betti numbers. Little real 

 progress, however, had been made till Poincare took the question up, 

 modified the Betti definitions, showed how the modified Betti num- 

 bers satisfy a generalization of Euler's theorem for polyhedra, and 

 introduced an entirely new set of constants, the coefficients of torsion. 

 This work is contained in a series of memoirs of which the first 

 appeared in 1892 and the last in 1904. They were accompanied and" 

 followed by a number of papers in which analysis situs is applied 

 to the theory of algebraic functions of two and more variables. 



I have now mentioned what appear to me the most important 

 achievements of Poincare, grouping them together in four classes 

 which, as I said, correspond very roughly to a chronological order, 

 The fouir sections of his work which I have signalized are his con- 

 tributions ( I ) to the classical theory of differential eciuations and 

 the theory of automorphic functions, (2) to the theory of stability 

 and of the dift'erential equations of celestial mechanics, (3) to physics 

 and (4) to analysis situs and the theory of functions of several 

 variables. 



Another side of Poincare's intellectual activity which has attracted 

 more general attention than any of his capital achievements in pure 

 science is represented by his semi-philosophical books, " Science and 

 Hypothesis," " The Value of Science," and " Science and Method."' 

 These books have been translated into most of the modern languages,, 

 including English, and have received much attention and praise. 

 They are characterized by a clarity and hard-headed " common 

 sense " which is more often sought than found in this class of 

 literature. 



After noticing the works for which Poincare was chiefly famous, 

 there still remains a host of mathematical papers which would be 



