BRAIN WORK AND MANUAL WORK. 195 



elementary theorem two years after having left the 

 school, will be unable to do it, unless mathematics is 

 their speciality. They will forget which auxiliary lines 

 to draw, and they never have been taught to discover 

 the proofs by themselves. No wonder that later on 

 they find such difficulties in applying geometry to phy- 

 sics, that their progress is despairingly sluggish, and 

 that so few master higher mathematics. There is, how- 

 ever, the other method which permits progress, as a 

 whole, at a much speedier rate, and under which he 

 who once has learned geometry will know it all his 

 life long. Under this system, each theorem is put as 

 a problem ; its solution is never given beforehand, 

 and the pupil is induced to find it by himself. Thus, 

 if some preliminary exercises with the rule and the 

 compass have been made, there is not one boy or girl, 

 out of twenty or more, who will not be able to find the 

 means of drawing an angle which is equal to a given 

 angle, and to prove their equality, after a few sugges- 

 tions from the teacher ; and if the subsequent problems 

 are given in a systematic succession (there are excel- 

 lent text-books for the purpose), and the teacher does 

 not press his pupils to go faster than they can go at the 

 beginning, they advance from one problem to the next 

 with an astonishing facility, the only difficulty being 

 to bring the pupil to solve the first problem, and thus 

 to acquire confidence in his own reasoning. 



Moreover, each abstract geometrical truth must be 

 impressed on the mind in its concrete form as well. As 

 soon as the pupils have solved a few problems on paper, 

 they must solve them in the playing-ground with a few 

 sticks and a string, and they must apply their knowledge 

 in the workshop. Only then will the geometrical lines 

 acquire a concrete meaning in the children's minds ; 

 only then will they see that the teacher is playing no 

 tricks when he asks them to solve problems with the 

 rule and the compass without resorting to the protractor : 



