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; 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 profile of a fortification; and how geometry in space became plain when we began to make on a small scale a bastion with embrasures and barbettes--an occupation which obviously was soon prohibited on account of the state into which we brought our clothes. “You look like navvies,” was the reproach addressed to us by our intelligent educators, while we were proud precisely of being navvies, and of discovering the use of geometry. By compelling our children to study real things from mere graphical representations, instead of _making_ those things themselves, we compel them to waste the most precious time; we uselessly worry their minds; we accustom them to the worst methods of learning; we kill independent thought in the bud; and very seldom we succeed in conveying a real knowledge of what we are teaching. Superficiality, parrot-like repetition, slavishness and inertia of mind are the results of our method of education. We do not teach our children how to learn.