On the other side, we all know how children like to make toys themselves, how they gladly imitate the work of full-grown people if they see them at work in the workshop or the building-yard. But the parents either stupidly paralyse that passion, or do not know how to utilise it. Most of them despise manual work and prefer sending their children to the study of Roman history, or of Franklin’s teachings about saving money, to seeing them at a work which is good for the “lower classes only.” They thus do their best to render subsequent learning the more difficult. And then come the school years, and time is wasted again to an incredible extent. Take, for instance, mathematics, which every one ought to know, because it is the basis of all subsequent education, and which so few really learn in our schools. In geometry, time is foolishly wasted by using a method which merely consists in committing geometry to memory. In most cases, the boy reads again and again the proof of a theorem till his memory has retained the succession of reasonings. Therefore, nine boys out of ten, if asked to prove an 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 physics, that their progress is despairingly sluggish, and that so few master higher mathematics. There is, however, the other method which permits 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 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.