MATHEMATICAL DISCOVERY. 53 



in my head ; I could almost feel then jostling one 

 another, until two of them coalesced, so to speak, to 

 form a stable combination. When morning came, I 

 had established the existence of one class of Fuchsian 

 functions, those that are derived from the hyper- 

 geometric series. I had only to verify the results, 

 which only took a few hours. 



Then I wished to represent these functions by the 

 quotient of two series. This idea was perfectly con- 

 scious and deliberate ; I was guided by the analogy 

 with elliptical functions. I asked myself what must 

 be the properties of these series, if they existed, and 

 I succeeded without difficulty in forming the series 

 that I have called Theta-Fuchsian. 



At this moment I left Caen, where I was then living, 

 to take part in a geological conference arranged by 

 the School of Mines. The incidents of the journey 

 made me forget my mathematical work. When we 

 arrived at Coutances, we got into a break to go 

 for a drive, and, just as I put my foot on the 

 step, the idea came to me, though nothing in my 

 former thoughts seemed to have prepared me for it, 

 that the transformations I had used to define Fuchsian 

 functions were identical with those of non-Euclidian 

 geometry. I made no verification, and had no time to 

 do so, since I took up the conversation again as soon 

 as I had sat down in the break, but I felt absolute 

 certainty at once. When I got back to Caen I verified 

 the result at my leisure to satisfy my conscience. 



I thcMi l)cc;an to study arithmetical questions without 

 any great apparent result, and without suspecting that 

 they could have the least connexion with my previous 

 researches. Disgusted at mj' want of success, I went 



