286 SCIENCE AND METHOD. 



their methods of discovery are very similar. In the 

 case of both they consist in rising from the fact to the 

 law, and in seeking the facts that are capable of 

 leading up to a law. 



In order to elucidate this point, I have exhibited 

 the mathematician's mind at work, and that under 

 three forms : the mind of the inventive and creative 

 mathematician ; the mind of the unconscious geome- 

 trician who, in the days of our far-off ancestors or in 

 the hazy years of our infancy, constructed for us our 

 instinctive notion of space ; and the mind of the youth 

 in a secondary school for whom the master unfolds the 

 first principles of the science, and seeks to make him 

 understand its fundamental definitions. Through- 

 out we have seen the part played by intuition and 

 the spirit of generalization, without which these 

 three grades of mathematicians, if I may venture 

 so to express myself, would be reduced to equal 

 impotence. 



And in demonstration itself logic is not all. The 

 true mathematical reasoning is a real induction, 

 differing in many respects from physical induction, 

 but, like it, [proceeding from the particular to the 

 universal. All the efforts that have been made to 

 upset this order, and to reduce mathematical induction 

 to the rules of logic, have ended in failure, but poorly 

 disguised by the use of a language inaccessible to the 

 uninitiated. 



The examples I have drawn from the physical 

 sciences have shown us a good variety of instances of 

 facts that give a large return. A single experiment of 

 Kaufmann's upon radium rays revolutionizes at once 

 Mechanics, Optics, and Astronomy. Why is this? It 



