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SCIENCE 



[N. S. Vol. XXXVI. No. 927 



parable mathematician was a strong supporter of 

 good writing, of those humanities which for so 

 long have guided the French genius along a right 

 and safe road. One might hear him, when the dic- 

 tionary was under discussion, ask about the origin, 

 and, as it were, the titles of nobility of words. 

 This modern, who stimulated contemporary life by 

 his discoveries and his calculations, defended with 

 boldness the heritage of our ancestors. He knew 

 that the French language is itself a country, and, 

 against every perilous invasion, this soldier of 

 sound speech stood firmly at the frontier. 



While it would be futile to try to give an 

 account of the mathematical work of Poin- 

 care in such a hasty sketch, yet it seema 

 hardly appropriate to omit entirely the things 

 which were dearest to him. One of the 

 earliest problems which he attacked was the 

 study of linear differential equations with ra- 

 tional or algebraic coefficients. This study 

 led him to the discovery of new functions 

 which may be regarded as generalizations of 

 elliptic and of modular functions. These 

 functions were characterized by the property 

 that they are invariant under certain linear 

 transformations. He was thus led to the 

 study of various groups of transformations, 

 and in his " Notice," to which we referred 

 above, he remarks that there is a theory which 

 has been equally useful to him in all his re- 

 searches, namely, that of the groups formed 

 by linear substitutions. In fact, these substi- 

 tutions play a preponderant role in the study 

 of linear equations and in that of arithmetic 

 forms. It is to this circumstance that one 

 ought to attribute the interrelations, often 

 unexpected, between the theory of numbers 

 and the theory of Fuchsian functions, theories 

 which, moreover, do not at first appear to 

 have any point of contact. 



He pointed out relations between the theory 

 of complex numbers and the theory of con- 

 tinuous groups, and thus he threw new light 

 on these far-reaching subjects. The theory of 

 the solution of systems of an infinite number 

 of linear equations with an infinite number 

 of unknowns is largely due to Poincare. He 

 was the first to establish definite criteria of 

 convergence in reference to the infinite de- 

 terminants employed by the American astron- 



omer, G. W. Hill, with so much success. It 

 should, however, be observed that infinite de- 

 terminants had been studied earlier by E. 

 Fiirstenau and T. Kotteritzsch, and that these 

 determinants should not be accredited to G. 

 W. Hill, as is sometimes done. 



Poincare wrote a number of books espe- 

 cially on mathematical physics, but the three 

 books which are perhaps the most commonly 

 known deal with philosophical questions and 

 bear the following titles, respectively : " La 

 Science et I'Hypothese," " La Valeur de la 

 Science," and " Science et Method." In re- 

 gard to the first of these, the Director of the 

 French Academy said at the time when Poin- 

 care entered this academy : " By the sale of 

 16,000 copies of ' La Science et I'Hypothese ' 

 you have increased your personality (person- 

 nel) ten-fold." 



He was fond of traveling and many Amer- 

 icans recall his visit to the St. Louis Exposi- 

 tion, in 1904, where he delivered aii address 

 entitled, " The Present and the Future of 

 Mathematical Physics." This address was 

 translated for the Bulletin of the American 

 Mathematical Society by J. W. Young, and 

 published in the February, 1906, number of 

 this journal. Poincare visited all the coun- 

 tries of Europe and also some of the coun- 

 tries of Africa. He was married and had 

 four children — three daughters and a son. 



The great mainspring of Poincare's activ- 

 ity was seeking the truth. This made his life 

 both simple and beautiful. Seeking the truth 

 implies an open acknowledgment of igno- 

 rance. In fact, one of the strongest mathe- 

 matical methods consists of putting an x for 

 the unknown; but how could we put an x for 

 the unknown unless we were willing to admit 

 that we are ignorant in regard to this fact. 

 Every one who has worked in elementary 

 algebra knows that much is frequently gained 

 by admitting our ignorance and by calling 

 some particular ignorance x and another par- 

 ticular ignorance y, etc. Even in his mature 

 years Poincare could honestly ask the question. 

 " La terre tourne-t-elle ? " Things that are 

 commonly accepted as true but have not been 

 fully established, frequently offer the most 



