DEFINITIONS AND EDUCATION. 133 



and prepare it. The general statement will be led up 

 to by the study of some particular examples. 



One word more. The aim of each part of the 

 statement of a definition is to distinguish the object 

 to be defined from a class of other neighbouring 

 objects. The definition will not be understood until 

 you have shown not only the object defined, but the 

 neighbouring objects from which it has to be dis- 

 tinguished, until you have made it possible to grasp 

 the difference, and have added explicitly your reason 

 for saying this or that in stating the definition. 

 • But it is time to leave generalities and to enquire 

 how the somewhat abstract principles I have been 

 expounding can be applied in arithmetic, in geometry, 

 in analysis, and in mechanics. 



Arithmetic. 



12. We do not have to define the whole number. 

 On the other hand, operations on whole numbers are 

 generally defined, and I think the pupils learn these 

 definitions by heart and attach no meaning to them. 

 For this there are two reasons : first, they are taught 

 them too early, while their mind still feels no need of 

 them ; and then these definitions are not satisfactory 

 from the logical point of view. For addition, wc 

 cannot find a good one, simply because we must 

 stop somewhere, and cannot define everything. The 

 definition of addition is to say that it consists in adding. 

 All that we can do is to start with a certain number 

 of concrete examples and say, the operation that has 

 just been performed is called addition. 



For subtraction it is another matter. It can be 

 defined logically as the inverse operation of addition. 



