8 president's address — SECTION A. 



combinations, and reach no result. One evening, contrary to his custom^ 

 he had taken a cup of black coffee, and could not sleep. The ideas 

 seemed to crowd and jostle one another in his head, and in the 

 morning he was able in a few hours to establish the existence of a class 

 of Fuchsian Functions, those which are derived from the Hypergeo- 

 metric Series. 



The next step was to represent these functions as the quotient 

 of two series. To this the analogy with the elliptic fimctions guided 

 him. He investigated what should be the properties of these series, 

 and established without difficulty the existence of those he called 

 Theta-Fuchsian Series. 



At this stage he had to leave for the country on some official work. 

 On the steps of an omnibus, the idea flashed into his mind that the 

 transformations of which he had made iise in defining the Fuchsian 

 Functions were identical with those of the Non-Euclidean Geometry. 

 He did not verify the conjecture, but resumed the interrupted con- 

 versation with his companion. Still, he felt perfectly certain that his 

 idea was correct ; on his return to his home he looked into the 

 matter, and found that what he had surmised was true. 



Then he took up the study of some arithmetical questions without 

 much success, far from suspecting that they would have anything to 

 do with his previous investigations. Annoyed at his comparative 

 failure with his new task, he went to the seaside to spend a few days 

 and turn his thoughts to other things. One day walking on the cliff, 

 the thought came to him, suddenly and surely as before, that the 

 arithmetical transformations of certain quadratic forms were identical 

 with those of the Non-Euclidean Geometry. 



On his return he reflected upon this result and the consequences 

 to be derived from it. The example of the quadratic forms showed 

 him that there were Fuchsian groups other than those which correspond 

 to the Hyper-geometric Series. He saw that he could apply to them 

 the theory of the Theta-Fuchsian Series, and that consequently there 

 must exist Fuchsian Functions other than those which were derived 

 from that series. He set himself to form such functions. He made 

 a systematic attack upon the position and carried all the outworks 

 save one. This he could not reduce, however hard his efforts. 



Again his work was interrupted ; this time that he might put in 

 his military service. One day, passing down the street, all of a sudden 

 the solution of the difficulty appeared to him. At the time he made 

 no attempt to go into the point in detail, but let it stand over till the 

 end of his service. When the opportunity came, all his material was 

 there. He had only to arrange it and the complete memoir was 

 written, practically, at a sitting. 



All this story, Poincare told in a fascinating lecture entitled — 

 L'Invention Mathematique ; his view of the matter being that after 



