I have said that geometry is not intelligible to children; but it is our own fault. We do not observe that their method is different from ours, and that what is to us the art of reasoning should be to them only the art of seeing. Instead of giving them our method, we should do better to take theirs. For in our way of learning geometry, imagination really does as much as reason. When a proposition is stated, we have to imagine the demonstration; that is, we have to find upon what proposition already known the new one depends, and from all the consequences of this known principle select just the one required. According to this method the most exact reasoner, if not naturally inventive, must be at fault. And the result is that the teacher, instead of making us discover demonstrations, dictates them to us; instead of teaching us to reason, he reasons for us, and exercises only our memory. Make the diagrams accurate; combine them, place them one upon another, examine their relations, and you will discover the whole of elementary geometry by proceeding from one observation to another, without using either definitions or problems, or any form of demonstration than simple superposition. For my part, I do not even pretend to teach Émile geometry; he shall teach it to me. I will look for relations, and he shall discover them. I will look for them in a way that will lead him to discover them. In drawing a circle, for instance, I will not use a compass, but a point at the end of a cord which turns on a pivot. Afterward, when I want to compare the radii of a semi-circle, Émile will laugh at me and tell me that the same cord, held with the same tension, cannot describe unequal distances. When I want to measure an angle of sixty degrees, I will describe from the apex of the angle not an arc only, but an entire circle; for with children nothing must be taken for granted. I find that the portion intercepted by the two sides of the angle is one-sixth of the whole circumference. Afterward, from the same centre, I describe another and a larger circle, and find that this second arc is one-sixth of the new circumference. Describing a third concentric circle, I test it in the same way, and continue the process with other concentric circles, until Émile, vexed at my stupidity, informs me that every arc, great or small, intercepted by the sides of this angle, will be one-sixth of the circumference to which it belongs. You see we are almost ready to use the instruments intelligently. In order to prove the angles of a triangle equal to two right angles, a circle is usually drawn. I, on the contrary, will call Émile's attention to this in the circle, and then ask him, "Now, if the circle were taken away, and the straight lines were left, would the size of the angles be changed?"