DEFINITIONS AND EDUCATION. 135 



commutative, associative, and distributive, making it 

 quite clear to the listeners that the verification has 

 been made in order to justify the definition. 



We see what part is played in all this by geometrical 

 figures, and this part is justified by the philosophy and 

 the history of the science. If arithmetic had remained 

 free from all intermixture with geometry, it would 

 never have known anything but the whole number. 

 It was in order to adapt itself to the requiremeiTts of 

 geometry that it discovered something else. 



Geometry. 



In geometry we meet at once the notion of the 

 straight line. Is it possible to define the straight 

 line ? The common definition, the shortest path from 

 one point to another, does not satisfy me at all. I 

 should start simply with the ruler, and I should first 

 show the pupil how we can verify a ruler by revolving 

 it. This verification is the true definition of a straight 

 line, for a straight line is an axis of rotation. We 

 should then show him how to verify the ruler by 

 sliding it, and we should have one of the most im- 

 portant properties of a straight line. As for that 

 other property, that of being the shortest path from 

 one point to another, it is a theorem that can be 

 demonstrated apodeictically, but the demonstration is 

 too advanced to find a place in secondary education. 

 It will be better to show that a ruler previously veri- 

 fied can be applied to a taut thread. We must not 

 hesitate, in the pre.sence of difficulties of this kind, 

 to multiply the axioms, justifying them by rough 

 examples. 



Some axioms \vc must admit ; and if wc admit a 



