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.