DEFINITIONS AND EDUCATION. 119 



No doubt they are not themselves fully aware of 

 what they require and could not formulate their 

 desire, but if they do not obtain satisfaction, they 

 feel vaguely that something is wanting. Then what 

 happens? At first they still perceive the evidences 

 that are placed before their eyes, but, as they are 

 connected by too attenuated a thread with those that 

 precede and those that follow, they pass without 

 leaving a trace in their brains, and are immediately 

 forgotten ; illuminated for a moment, they relapse 

 at once into an eternal night. As they advance 

 further, they will no longer see even this ephemeral 

 light, because the theorems depend one upon another, 

 and those they require have been forgotten. Thus 

 it is that they become incapable of understanding 

 mathematics. 



It is not always the fault of their instructor. Often 

 their intellect, which requires to perceive the connect- 

 ing thread, is too sluggish to seek it and find it. But 

 in order to come to their assistance, we must first of 

 all thoroughly understand what it is that stops them. 



Others will always ask themselves what use it is. 

 They will not have understood, unless they find 

 around them, in practice or in nature, the object of 

 such and such a mathematical notion. Under each 

 word they wish to put a sensible image ; the definition 

 must call up this image, and at each stage of the 

 demonstration they must see it being transformed 

 and evolved. On this condition only will they under- 

 stand and retain what they have understood. These 

 often deceive themselves : they do not listen to the 

 reasoning, they look at the figures ; they imagine that 

 they have understood when they have only seen. 



