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.?"
Exactness in the construction of figures is neglected; it is taken for granted and stress is laid on the proof. With us, on the other hand, there will be no question of proof. Our chief business will be to draw very straight, accurate, and even lines, a perfect square, a really round circle. To verify the exactness of a figure we will test it by each of its sensible properties, and that will give us a chance to discover fresh properties day by day. We will fold the two semi-circles along the diameter, the two halves of the square by the diagonal; he will compare our two figures to see who has got the edges to fit most exactly, i.e., who has done it best; we should argue whether this equal division would always be possible in parallelograms, trapezes, etc. We shall sometimes try to forecast the result of an experiment, to find reasons, etc.
Geometry means to my scholar the successful use of the rule and compass; he must not confuse it with drawing, in which these instruments are not used. The rule and compass will be locked up, so that he will not get into the way of messing about with them, but we may sometimes take our figures with us when we go for a walk, and talk over what we have done, or what we mean to do.
I shall never forget seeing a young man at Turin, who had learnt as a child the relations of contours and surfaces by having to choose every day isoperimetric cakes among cakes of every geometrical figure. The greedy little fellow had exhausted the art of Archimedes to find which were the biggest.
When the child flies a kite he is training eye and hand to accuracy; when he whips a top, he is increasing his strength by using it, but without learning anything. I have sometimes asked why children are not given the same games of skill as men; tennis, mall, billiards, archery, football, and musical instruments. I was told that some of these are beyond their strength, that the child's senses are not sufficiently developed for others. These do not strike me as valid reasons; a child is not as tall as a man, but he wears the same sort of coat; I do not want him to play with our cues at a billiard-table three feet high; I do not want him knocking about among our games, nor carrying one of our racquets in his little hand; but let him play in a room whose windows have been protected; at first let him only use soft balls, let his first racquets be of wood, then of parchment, and lastly of gut, according to his progress. You prefer the kite because it is less tiring and there is no danger. You are doubly wrong. Kite-flying is a sport for women, but every woman will run away from a swift ball. Their white skins were not meant to be hardened by blows and their faces were not made for bruises. But we men are made for strength; do you think we can attain it without hardship, and what defence shall we be able to make if we are attacked? People always play carelessly in games where there is no danger. A falling kite hurts nobody, but nothing makes the arm so supple as protecting the head, nothing makes the sight so accurate as having to guard the eye. To dash from one end of the room to another, to judge the rebound of a ball before it touches the ground, to return it with strength and accuracy, such games are not so much sports fit for a man, as sports fit to make a man of him.
The child's limbs, you say, are too tender. They are not so strong as those of a man, but they are more supple. His arm is weak, still it is an arm, and it should be used with due consideration as we use other tools. Children have no skill in the use of their hands. That is just why I want them to acquire skill; a man with as little practice would be just as clumsy. We can only learn the use of our limbs by using them. It is only by long experience that we learn to make the best of ourselves, and this experience is the real object of study to which we cannot apply ourselves too early.
What is done can be done. Now there is nothing commoner than to find nimble and skilful children whose limbs are as active as those of a man. They may be seen at any fair, swinging, walking on their hands, jumping, dancing on the tight rope. For many years past, troops of children have attracted spectators to the ballets at the Italian Comedy House. Who is there in Germany and Italy who has not heard of the famous pantomime company of Nicolini? Has it ever occurred to any one that the movements of these children were less finished, their postures less graceful, their ears less true, their dancing more clumsy than those of grown-up dancers? If at first the fingers are thick, short, and awkward, the dimpled hands unable to grasp anything, does this prevent many children from learning to read and write at an age when others cannot even hold a pen or pencil? All Paris still recalls the little English girl of ten who did wonders on the harpsichord. I once saw a little fellow of eight, the son of a magistrate, who was set like a statuette on the table among the dishes, to play on a fiddle almost as big as himself, and even artists were surprised at his execution.
To my mind, these and many more examples prove that the supposed incapacity of children for our games is imaginary, and that if they are unsuccessful in some of them, it is for want of practice.
You will tell me that with regard to the body I am falling into the same mistake of precocious development which I found fault with for the mind. The cases are very different: in the one, progress is apparent only; in the other it is real. I have shown that children have not the mental development they appear to have, while they really do what they seem to do. Besides, we must never forget that all this should be play, the easy and voluntary control of the movements which nature demands of them, the art of varying their games to make them pleasanter, without the least bit of constraint to transform them into work; for what games do they play in which I cannot find material for instruction for them? And even if I could not do so, so long as they are amusing themselves harmlessly and passing the time pleasantly, their progress in learning is not yet of such great importance. But if one must be teaching them this or that at every opportunity, it cannot be done without constraint, vexation, or tedium.
What I have said about the use of the two senses whose use is most constant and most important, may serve as an example of how to train the rest. Sight and touch are applied to bodies at rest and bodies in motion, but as hearing is only affected by vibrations of the air, only a body in motion can make a noise or sound; if everything were at rest we should never hear. At night, when we ourselves only move as we choose, we have nothing to fear but moving bodies; hence we need a quick ear, and power to judge from the sensations experienced whether the body which causes them is large or small, far off or near, whether its movements are gentle or violent. When once the air is set in motion, it is subject to repercussions which produce echoes, these renew the sensations and make us hear a loud or penetrating sound in another quarter. If you put your