Fortunately the answer to Masson's question is to be found in Poincare's own writings, and it becomes the more interesting when taken in connection with his further thesis that the method of research in mathematics is precisely that of all pure science. This method I desire to consider at some length, for I conceive that such a consideration will be entirely appropriate in this place.
The first research mentioned by Rados in the report of the committee to the Hungarian Academy in 1905, when Poincaré 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 rational algebraic coefficients, just as elliptic functions and abelian functions enable us to integrate certain algebraic differentials. With regard to these researches, Poincaré tells us that for a fortnight he had tried without success to demonstrate their non-existence. He investigated a large number of formulæ 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 combinations. 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. Returning 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 problem of the linear differential equations and the fuchsian functions and reached result after result, save one thing which seemed to be the key
