DEFINITIONS AND EDUCATION. 127 



tion. To begin with, it is only a perceptible image, 

 a line drawn with chalk on a blackboard. Little by- 

 little it is purified ; it is used for constructing a com- 

 plicated system of inequalities which reproduces all 

 the lines of the original image ; when the work is 

 quite finished, the centering is removed, as it is after 

 the construction of an arch ; this crude representation 

 is henceforth a useless support, and disappears, and 

 there remains only the edifice itself, irreproachable in 

 the eyes of the logician. And yet, if the instructor 

 did not recall the original image, if he did not replace 

 the centering for a moment, how would the pupil 

 guess by what caprice all these inequalities had been 

 scaffolded in this way one upon another? The defini- 

 tion would be logically correct, but it would not show 

 him the true reality. 



7. And so we are obliged to make a step back- 

 wards. No doubt it is hard for a master to teach 

 what does not satisfy him entirely, but the satisfaction 

 of the master is not the sole object of education. We 

 have first to concern ourselves with the pupil's state 

 of mind, and what we want it to become. 



Zoologists declare that the embrj^onic development 

 of an animal repeats in a very short period of time 

 the whole history of its ancestors of the geological 

 ages. It seems to be the same with the development 

 of minds. The educator must make the child pass 

 through all that his fathers have passed through, more 

 rapidly, but without missing a stage. On this account, 

 the history of any science must be our first guide. 



Our fathers imagined they knew what a fraction 

 was, or continuity, or the area of a curved surface ; it 



