I have said already that geometry is beyond the child's reach; but that is our own fault. We fail to perceive that their method is not ours, that what is for us the art of reasoning, should be for them the art of seeing. Instead of teaching them our way, we should do better to adopt theirs, for our way of learning geometry is quite as much a matter of imagination as of reasoning. When a proposition is enunciated you must imagine the proof; that is, you must discover on what proposition already learnt it depends, and of all the possible deductions from that proposition you must choose just the one required. In this way the closest reasoner, if he is not inventive, may find himself at a loss. What is the result? Instead of making us discover proofs, they are dictated to us; instead of teaching us to reason, our memory only is employed. Draw accurate figures, combine them together, put them one upon another, examine their relations, and you will discover the whole of elementary geometry in passing from one observation to another, without a word of definitions, problems, or any other form of demonstration but super-position. I do not profess to teach Emile geometry; he will teach me; I shall seek for relations, he will find them, for I shall seek in such a fashion as to make him find. For instance, instead of using a pair of compasses to draw a circle, I shall draw it with a pencil at the end of bit of string attached to a pivot. After that, when I want to compare the radii one with another, Emile will laugh at me and show me that the same thread at full stretch cannot have given distances of unequal length. If I wish to measure an angle of 60 degrees I describe from the apex of the angle, not an arc, but a complete circle, for with children nothing must be taken for granted. I find that the part of the circle contained between the two lines of the angle is the sixth part of a circle. Then I describe another and larger circle from the same centre, and I find the second arc is again the sixth part of its circle. I describe a third concentric circle with a similar result, and I continue with more and more circles till Emile, shocked at my stupidity, shows me that every arc, large or small, contained by the same angle will always be the sixth part of its circle. Now we are ready to use the protractor. To prove that two adjacent angles are equal to two right angles people describe a circle. On the contrary I would have Emile observe the fact in a circle, and then I should say, "If we took away the circle and left the straight lines, would the angles have changed their size, etc.?"