MATHEMATICAL DISCOVERY. 55 



ingly I composed my definitive treatise at a sitting 

 and without any difficulty. 



It is useless to multiply examples, and I will con- 

 tent myself with this one alone. As regards my other 

 researches, the accounts I should give would be exactly 

 similar, and the observations related by other mathe- 

 maticians in the enquiry of V Enseignement Mathe- 

 matique would only confirm them. 



One is at once struck by these appearances of 

 sudden illumination, obvious indications of a long 

 course of previous unconscious work. The part played 

 by this unconscious work in mathematical discovery 

 seems to me indisputable, and we shall find traces 

 of it in other cases where it is less evident. Often 

 when a man is working at a difficult question, he 

 accomplishes nothing the first time he sets to work. 

 Then he takes more or less of a rest, and sits down 

 again at his table. During the first half-hour he still 

 finds nothing, and then all at once the decisive idea 

 presents itself to his mind. We might say that the 

 conscious work proved more fruitful because it was 

 interrupted and the rest restored force and freshness 

 to the mind. But it is more probable that the rest 

 was occupied with unconscious work, and that the 

 result of this work was afterwards revealed to the 

 geometrician exactly as in the cases I have quoted, 

 except that the revelation, instead of coming to light 

 during a walk or a journey, came during a period 

 of conscious work, but independently of that work, 

 which at most only performs the unlocking process, 

 as if it were the spur that excited into conscious form 

 the results alread)' acquired during the rest, which till 

 then remained unconscious. 



