52 SCIENCE AND METHOD. 



the field of his consciousness there never appear any 

 but really useful combinations, and some that he- 

 rejects, which, however, partake to some extent of 

 the character of useful combinations. Everything 

 happens as if the discoverer were a secondary examiner 

 who had only to interrogate candidates declared eli- 

 gible after passing a preliminary test. 



But what I have said up to now is only what can 

 be observed or inferred by reading the works of 

 geometricians, provided they are read with some 

 reflection. 



It is time to penetrate further, and to see what 

 happens in the very soul of the mathematician. For 

 this purpose I think I cannot do better than recount 

 my personal recollections. Only I am going to confine 

 myself to relating how I wrote my first treatise on 

 Fuchsian functions. I must apologize, for I am going 

 to introduce some technical expressions, but they need 

 not alarm the reader, for he has no need to under- 

 stand them. I shall say, for instance, that I found the 

 demonstration of such and such a theorem under such 

 and such circumstances ; the theorem will have a 

 barbarous name that many will not know, but that 

 is of no importance. What is interesting for the 

 psychologist is not the theorem but the circumstances. 



For a fortnight I had been attempting to prove 

 that there could not be any function analogous to 

 what I have since called Fuchsian functions. I was at 

 that time very ignorant. Every day I sat down at my 

 table and spent an hour or two trying a great number 

 of combinations, and I arrived at no result. One 

 night I took some black coffee, contrary to my custom, 

 and was unable to sleep. A host of ideas kept surging 



