MANUAL WORK. 381 



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 play ing-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 ; only then will they 

 know geometry. 



" Through the eyes and the hand to the brain " 

 this is the true principle of economy of time in 

 teaching. I remember, as if it were yesterday, 

 how geometry suddenly acquired for me a new 

 meaning, and how this new meaning facilitated 

 all ulterior studies. It was as we were mastering 

 at school a Montgolfier balloon, and I remarked 

 that the angles at the summits of each of the 

 twenty strips of paper out of which we were 

 going to make the balloon must cover less than 

 the fifth part of a right angle each. I remember, 

 next, how the sinuses and the tangents ceased 

 to be mere cabalistic signs when they permitted 

 us to calculate the length of a stick in a working 



