lo INTRODUCTION. 



reckon with errors due to imperfections of our senses 

 and of our instruments. Happily we may admit that, 

 under certain conditions, there is a partial compensa- 

 tion of these errors, so that they disappear in averages. 

 This compensation is due to chance. But what is 

 chance? It is a notion which is difficult of justifica- 

 tion, and even of definition ; and yet what I have just 

 said with regard to errors of observation, shows that 

 the scientist cannot get on without it. It is necessary, 

 therefore, to give as accurate a definition as possible 

 of this notion, at once so indispensable and so elusive. 



These are generalities which apply in the main to 

 all sciences. For instance, there is no appreciable 

 difference between the mechanism of mathematical 

 discovery and the mechanism of discovery in general. 

 Further on I approach questions more particularly 

 concerned with certain special sciences, beginning with 

 pure mathematics. 



In the chapters devoted to them, I am obliged to 

 treat of somewhat more abstract subjects, and, to begin 

 with, I have to speak of the notion of space. Every one 

 knows that space is relative, or rather every one says 

 so, but how many people think still as if they con- 

 sidered it absolute. Nevertheless, a little reflection 

 will show to what contradictions they are exposed. 



Questions concerning methods of instruction are of 

 importance, firstly, on their own account, and secondly, 

 because one cannot reflect on the best method of 

 imbuing virgin brains with new notions without, at 

 the same time, reflecting on the manner in which 

 these notions have been acquired by our ancestors, 

 and consequently on their true origin — that is, in 

 reality, on their true nature. Why is it that, in most 



