132 SCIENCE AND METHOD. 



^ it involves no contradiction either in its terms or with 

 the truths previously admitted. 



But that is not enough. A definition is stated as 

 a convention, but the majority of minds will revolt 

 if you try to impose it upon them as an arbitraiy 

 convention. They will have no rest until you have 

 answered a great number of questions. 



Mathematical definitions are most frequently, as 

 M. Liard has shown, actual constructions built up 

 throughout of simpler notions. But why should these 

 elements have been assembled in this manner, when 

 a thousand other assemblages were possible ? Is it 

 simply caprice? If not, why had this combination 

 more right to existence than any of the others? What 

 need does it fill ? How was it foreseen that it would 

 play an important part in the development of the 

 science, that it would shorten our reasoning and our 

 calculations? Is there any familiar object in nature 

 that is, so to speak, its indistinct and rough image ? 



That is not all. If you give a satisfactory answer 

 to all these questions, we shall realize that the new- 

 comer had the right to be baptized. But the choice of 

 a name is not arbitrary either ; we must explain what 

 analogies have guided us, and that if we have given 

 analogous names to different things, these things at 

 least differ only in matter, and have some resemblance 

 in form, that their properties are analogous and, so to 

 speak, parallel. 



It is on these terms that we shall satisfy all propen- 

 sities. If the statement is sufficiently exact to please 

 the logician, the justification will satisfy the intui- 

 tionist. But we can do better still. Whenever it is 

 possible, the justification will precede the statement 



