DEFINITIONS AND EDUCATION. 123 



at once, a circle is a round, and we should have 

 understood." No doubt it is the teacher who is 

 right. The pupils' definition would have been of no 

 value, because it could not have been used for any 

 demonstration, and chiefly because it could not have 

 given them the salutary habit of analyzing their con- 

 ceptions. But they should be made to see that they 

 do not understand what they think they understand, 

 and brought to realize the roughness of their primitive 

 concept, and to be anxious themselves that it should 

 be purified and refined. 



4. I shall return to these examples ; I only wished 

 to show the two opposite conceptions. There is a 

 violent contrast between them, and this contrast is 

 explained by the history of the science. If we read 

 a book written fifty years ago, the greater part of the 

 arguments appear to us devoid of exactness. 



At that period they assumed that a continuous func- 

 tion cannot change its sign without passing through 

 zero, but to-day we prove it. They assumed that the 

 ordinary rules of calculus are applicable to incommen- 

 surable numbers ; to-day we prove it. They assumed 

 many other things that were sometimes untrue. 



They trusted to intuition, but intuition cannot give 

 us exactness, nor even certainty, and this has been 

 recognized more and more. It teaches us, for instance, 

 that every curve has a tangent — that is to say, that 

 every continuous function has a derivative — and that 

 is untrue. As certainty was required, it has been 

 necessary to give less and less place to intuition. 



How has this necessary evolution come about? It 

 was not long before it was recognized that exactness 



