Shear can be considered as produced in another way. Take the square ABCD (fig. 25), and suppose that it is pulled out from along one of its diagonals both ways, and proportionately compressed along the other diagonal. It will assume the shape in fig. 26. This compression and expansion along two lines at right angles is what is called shear; it is equivalent to the sliding illustrated above, combined with a turning round. [Illustration: Fig. 25.] [Illustration: Fig. 26.] In pure shear a body is compressed and extended in two directions at right angles to each other, so that its volume remains unchanged. Now we know that our material bodies resist shear—shear does violence to the internal arrangement of their particles, but they turn as wholes without such internal resistance. But there is an exception. In a liquid shear and rotation take place equally easily, there is no more resistance against a shear than there is against a rotation. Now, suppose all bodies were to be reduced to the liquid state, in which they yield to shear and to rotation equally easily, and then were to be reconstructed as solids, but in such a way that shear and rotation had interchanged places. That is to say, let us suppose that when they had become solids again they would shear without offering any internal resistance, but a rotation would do violence to their internal arrangement. That is, we should have a world in which shear would have taken the place of rotation. A shear does not alter the volume of a body: thus an inhabitant living in such a world would look on a body sheared as we look on a body rotated. He would say that it was of the same shape, but had turned a bit round. Let us imagine a Pythagoras in this world going to work to investigate, as is his wont. [Illustration: Fig. 27.] [Illustration: Fig. 28.] Fig. 27 represents a square unsheared. Fig. 28 represents a square sheared. It is not the figure into which the square in fig. 27 would turn, but the result of shear on some square not drawn. It is a simple slanting placed figure, taken now as we took a simple slanting placed square before. Now, since bodies in this world of shear offer no internal resistance to shearing, and keep their volume when sheared, an inhabitant accustomed to them would not consider that they altered their shape under shear. He would call ACDE as much a square as the square in fig. 27. We will call such figures shear squares. Counting the dots in ACDE, we find— 2 inside = 2 4 at corners = 1 or a total of 3. Now, the square on the side AB has 4 points, that on BC has 1 point. Here the shear square on the hypothenuse has not 5 points but 3; it is not the sum of the squares on the sides, but the difference. This relation always holds. Look at fig. 29. [Illustration: Fig. 29.] Shear square on hypothenuse— 7 internal 7 4 at corners 1 — 8 [Illustration: Fig. 29 _bis_.]