380 BRAIN WORK AND 



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 apply- 

 ing geometry to physics, that their progress is 

 despairingly sluggish, and that so few master 

 higher mathematics. 



There is, however, the other method which per- 

 mits the pupil to 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 suggestions from 

 the teacher ; and if the subsequent problems 

 are given in a systematic succession (there are 

 excellent text-books for the purpose), and the 

 teacher does not press his pupils to go faster 

 than they can go at the beginning, they ad- 

 vance from one problem to the next with an 

 astonishing facility, the only difficulty being 

 to bring the pupil to solve the first problem, 



