134 SCIENCE AND METHOD. 



But is that how we should begin ? Here, again, we 

 should start with examples, and show by these 

 examples the relation of the two operations. Thus 

 the definition will be prepared and justified. 



In the same way for multiplication. We shall take 

 a particular problem ; we shall show that it can be 

 solved by adding several equal numbers together ; 

 we shall then point out that we arrive at the result 

 quicker by multiplication, the operation the pupils 

 perform already by rote, and the logical definition will 

 spring from this quite naturally. 



We shall define division as the inverse operation 

 of multiplication ; but we shall begin with an example 

 drawn from the familiar notion of sharing, and we 

 shall show by this example that multiplication 

 reproduces the dividend. 



There remain the operations on fractions. There is 

 no difficulty except in the case of multiplication. The 

 best way is first to expound the theory of proportions, 

 as it is from it alone that the logical definition can 

 spring. But, in order to gain acceptance for the 

 definitions that are met with at the start in this theory, 

 we must prepare them by numerous examples drawn 

 from classical problems of the rule of three, and we 

 shall be careful to introduce fractional data. We shall 

 not hesitate, either, to familiarize the pupils with the 

 notion of proportion by geometrical figures ; either 

 appealing to their recollection if they have already 

 done any geometry, or having recourse to direct 

 intuition if they have not, which, moreover, will prepare 

 them to do it. I would add, in conclusion, that after 

 having defined the multiplication of fractions, we must 

 justify this definition by demonstration that it is 



