26 SCIENCE AND METHOD. 



ployed only rule and compass ; later it became one 

 obtained by the extraction of radicals, then one in 

 which algebraical functions and radicals alone figured. 

 Thus the pessimists found themselves continually 

 passed over, continually forced to retreat, so that at 

 present I verily believe there are none left. 



My intention, therefore, is not to refute them, since 

 they are dead. We know very well that mathematics 

 will continue to develop, but we have to find out in 

 what direction. I shall be told "in all directions," 

 and that is partly true ; but if it were altogether true, 

 it would become somewhat alarming. Our riches 

 would soon become embarrassing, and their accumula- 

 tion would soon produce a mass just as impenetrable 

 as the unknown truth was to the ignorant. 



The historian and the physicist himself must make 

 a selection of facts. The scientist's brain, which is 

 only a corner of the universe, will never be able to 

 contain the whole universe ; whence it follows that, 

 of the innumerable facts offered by nature, we shall 

 leave some aside and retain others. The same is 

 true, a fortiori, in mathematics. The mathematician 

 similarly cannot retain pell-mell all the facts that are 

 presented to him, the more .so that it is himself — I was 

 almost going to say his own caprice — that creates these 

 facts. It is he who assembles the elements and con- 

 ' structs a new combination from top to bottom ; it is 

 generally not brought to him ready-made by nature. 



No doubt it is sometimes the case that a mathe- 

 matician attacks a problem to satisfy some require- 

 ment of physics, that the physicist or the engineer 

 asks him to make a calculation in view of some par- 

 ticular application. Will it be said that we geometri- 



