DEFINITIONS AND EDUCATION. 121 



3. Since the \V'ord understand has several meanings, 

 the definitions that will be best understood by some 

 are not those that will be best suited to others. We 

 have those who seek to create an image, and those 

 who restrict themselves to combining empty forms, 

 perfectly intelligible, but purely intelligible, and de- 

 prived by abstraction of all matter. 



I do not know whether it is necessary to quote 

 any examples, but I will quote some nevertheless, 

 and, first, the definition of fractions will furnish us with 

 an extreme example. In the primary schools, when 

 they want to define a fraction, they cut up an apple or 

 a pie. Of course this is done only in imagination and 

 not in reality, for I do not suppose the budget of primary 

 education would allow such an extravagance. In the 

 higher normal school, on the contrary, or in the 

 universities, they say : a fraction is the combination 

 of two whole numbers separated by a horizontal line. 

 By conventions they define the operations that these 

 symbols can undergo ; they demonstrate that the rules 

 of these operations are the same as in the calculation 

 of whole numbers ; and, lastly, they establish that 

 multiplication of the fraction by the denominator, 

 in accordance with these rules, gives the numerator. 

 This is very well, because it is addressed to young 

 people long since familiarized with the notion of 

 fractions by dint of cutting up apples and other 

 objects, so that their mind, refined by a considerable 

 mathematical education, has, little by little, come to 

 desire a purely logical definition. But what would 

 be the consternation of the beginner to whom we 

 attempted to offer it } 



Such, also, are the definitions to be found in a 



