2i 4 THE POPULAR SCIENCE MONTHLY 



the occasion to see how mathematical genius manifests itself, whether it is the 

 result of atavism, or the product of a special culture, at what moment and 

 under what conditions it sees light, at what epoch of life it is most active and 

 brilliant ? 



Fortunately the answer to Masson's question is to be found in Poin- 

 care's own writings, and it becomes the more interesting when taken in 

 connection with his further thesis that the method of research in mathe- 

 matics is precisely that of all pure science. This method I desire to con- 

 sider at some length, for I conceive that such a consideration will be 

 entirely appropriate in this place. 



The first research mentioned by Eados in the report of the com- 

 mittee to the Hungarian Academy in 1905, when Poincare was awarded 

 the first Bolyai prize as the most eminent mathematician in the world, 

 is the series of investigations relating to automorphic functions. These 

 functions enable us to integrate linear differential equations with ra- 

 tional algebraic coefficients, just as elliptic functions and abelian func- 

 tions enable us to integrate certain algebraic differentials. With regard 

 to these researches, Poincare tells us that for a fortnight he had tried 

 without success to demonstrate their non-existence. He investigated a 

 large number of formulae with no results. One evening, however, he was 

 restless and got to sleep with difficulty ; ideas surged out in crowds and 

 seemed to crash violently together in the endeavor to form stable combi- 

 nations. The next morning he was in possession of the particular set of 

 automorphic functions derived from the hypergeometric series; he had 

 only to verify the calculations. Having thus found that functions did 

 exist of this kind, he conceived the idea of representing these functions 

 as the quotients of two series, analogous to the theta series in elliptic 

 functions. This he did purely by the analogy, and arrived at theta- 

 fuchsian functions. Having occasion to take a journey, mathematics 

 was laid aside for a time, but in stepping into an omnibus at Coutances, 

 the idea flashed over him that the transformations which he had used to 

 define these automorphic functions were identical with certain others he 

 had used in some researches in non-euclidean geometry. Eeturning 

 home he took up some questions in the theory of arithmetic forms, and 

 with no suspicion that they were related to the fuchsian functions or 

 the geometric transformation, he worked for some time with no success. 

 But one day while taking a walk, the idea suddenly came to him that the 

 arithmetic transformations he was using were essentially the same as 

 those of his study in non-euclidean geometry. From this fact he saw at 

 once by the connections with the arithmetic forms that the fuchsian 

 functions he had discovered were only particular cases of a more general 

 class of functions. He laid siege now systematically to the whole prob- 

 lem of the linear differential equations and the fuchsian functions and 

 reached result after result, save one thing which seemed to be the key- 



