DEFINITIONS AND EDUCATION. 125 



We used to possess a vague notion, formed of in- 

 congruous elements, some a priori and others derived 

 from more or less digested experiences, and we im- 

 agined we knew its principal properties by intuition. 

 To-day we reject the empirical element and preserve 

 only the a priori ones. One of the properties 

 serves as definition, and all the others are de- 

 duced from it by exact reasoning. This is very well, 

 but it still remains to prove that this property, which 

 has become a definition, belongs to the real objects 

 taught us by experience, from which we had drawn 

 our vague intuitive notion. In order to prove it we 

 shall certainly have to appeal to experience or make 

 an effort of intuition ; and if we cannot prove it, our 

 theorems will be perfectly exact but perfectly useless. 



Logic sometimes breeds monsters. For half a 

 century there has been springing up a host of weird 

 functions, which seem to strive to have as little resem- 

 blance as possible to honest functions that are of some 

 use. No more continuity, or else continuity but no 

 derivatives, etc. More than this, from the point of 

 view of logic, it is these strange functions that are 

 the most general ; those that are met without being 

 looked for no longer appear as more than a particular 

 case, and they have only quite a little corner left them. 



Formerly, when a new function was invented, it 

 was in view of some practical end. To-day they are 

 invented on purpose to show our ancestors' reasonings 

 at fault, and we shall never get anything more than 

 that out of them. 



If logic were the teacher's only guide, he would 

 have to begin with the most general, that is to say, 

 with the most weird, functions. He would have to 



