THE FUTURE OF MATHEMATICS. 29 



by putting together pebbles. By teaching children the 

 multiplication table we save them later on countless 

 operations with pebbles. Some one once recognized, 

 whether by pebbles or otherwise, that 6 times 7 

 are 42, and had the idea of recording the result, and 

 that is the reason why we do not need to repeat the 

 operation. His time was not wasted even if he was 

 only calculating for his own amusement. His opera- 

 tion only took him two minutes, but it would have 

 taken two million, if a million people had had to 

 repeat it after him. 



Thus the importance of a fact is measured by the 

 return it gives — that is, by the amount of thought it 

 enables us to economize. 



In physics, the facts which give a large return are 

 those which take their place in a very general law, 

 because they enable us to foresee a very large number 

 of others, and it is exactly the same in mathematics. 

 Suppose I apply myself to a complicated calculation 

 and with much difficulty arrive at a result, I shall 

 have gained nothing by my trouble if it has not 

 enabled me to foresee the results of other analogous 

 calculations, and to direct them with certainty, avoid- 

 ing the blind groping with which I had to be con- 

 tented the first time. On the contrary, my time will 

 not have been lost if this very groping has succeeded 

 in revealing to me the profound analogy between the 

 problem just dealt with and a much more extensive 

 class of other problems ; if it has shown me at once 

 their resemblances and their differences ; if, in a word, 

 it has enabled me to perceive the possibility of a 

 generalization. Then it will not be merely a new 

 result that I have acquired, but a new force. 



