54 SCIENCE AND METHOD. 



away to spend a few days at the seaside, and 

 thought of entirely different things. One day, as I 

 was walking on the cliff, the idea came to me, again 

 with the same characteristics of conciseness, sudden- 

 ness, and immediate certainty, that arithmetical trans- 

 formations of indefinite ternary quadratic forms are 

 identical with those of non-Euclidian geometry. 



Returning to Caen, I reflected on this result and 

 deduced its consequences. The example of quadratic 

 forms showed me that there are Fuchsian groups 

 other than those which correspond with the hyper- 

 geometric series ; I saw that I could apply to them 

 the theory of the Theta-Fuchsian series, and that, 

 consequently, there are Fuchsian functions other than 

 those which are derived from the hypergeometric series, 

 the only ones I knew up to that time. Naturall}^, I 

 proposed to form all these functions. I laid siege 

 to them systematically and captured all the outworks 

 one after the other. There was one, however, which 

 still held out, whose fall would carry with it that of the 

 central fortress. But all my efforts were of no avail at 

 first, except to make me better understand the difficulty, 

 which was already something. All this work was per- 

 fectly conscious. 



Thereupon I left for Mont-Valerien, where I had 

 to serve my time in the army, and so my mind was 

 preoccupied with very different matters. One day, afe 

 I was crossing the street, the solution of the difficulty 

 which had brought me to a standstill came to me 

 all at once. I did not try to fathom it immediately, 

 and it was only after my service was finished that 

 I returned to the question. I had all the elements, 

 and had only to asscmlole and arrange them. Accord- 



