MATHEMATICAL DISCOVERY. 51 



be done by any one, and t'ne combinations that could 

 be so formed would be infinite in number, and the 

 greater part of them would be absolutely devoid of 

 interest. Discovery consists precisely in not con- 

 structing useless combinations, but in constructing 

 those that are useful, which are an infinitely small 

 minority. Discovery is discernment, selection. 



How this selection is to be made I have explained 

 above. Mathematical facts worthy of being studied 

 are those which, by their analogy with other facts, 

 are capable of conducting us to the knowledge of a 

 mathematical law, in the same way that experimental 

 facts conduct us to the knowledge of a physical law. 

 They are those which reveal unsuspected relations 

 between other facts, long since known, but wrongly 

 believ^ed to be unrelated to each other. 



Among the combinations we choose, the most fruit- 

 ful are often those which are formed of elements 

 borrowed from widely separated domains. I do not 

 mean to say that for discovery it is sufficient to bring 

 together objects that are as incongruous as possible. 

 The greater part of the combinations so formed would 

 be entirely fruitless, but some among them, though 

 very rare, are the most fruitful of all. 



Discovery, as I have said, is selection. But this is 

 perhaps not quite the right word. It suggests a pur- 

 chaser who has been shown a large number of samples, 

 and examines them one after the other in order to 

 make his selection. In our case the samples would be 

 so numerous that a whole life would not give sufficient 

 time to examine them. Things do not happen in this 

 way. Unfruitful coinbinations do not so much as 

 present themselves to the mind of the discoverer. In 



