Square on one side—which the reader can draw for himself— 4 internal 4 8 on sides 4 4 at corners 1 — 9 and the square on the other side is 1. Hence in this case again the difference is equal to the shear square on the hypothenuse, 9 - 1 = 8. Thus in a world of shear the square on the hypothenuse would be equal to the difference of the squares on the sides of a right-angled triangle. In fig. 29 _bis_ another shear square is drawn on which the above relation can be tested. What now would be the position a line on turning by shear would take up? We must settle this in the same way as previously with our turning. Since a body sheared remains the same, we must find two equal bodies, one in the straight way, one in the slanting way, which have the same volume. Then the side of one will by turning become the side of the other, for the two figures are each what the other becomes by a shear turning. We can solve the problem in a particular case— [Illustration: Fig. 30.] In the figure ACDE (fig. 30) there are— 15 inside 15 4 at corners 1 a total of 16. Now in the square ABGF, there are 16— 9 inside 9 12 on sides 6 4 at corners 1 — 16 Hence the square on AB would, by the shear turning, become the shear square ACDE. And hence the inhabitant of this world would say that the line AB turned into the line AC. These two lines would be to him two lines of equal length, one turned a little way round from the other. That is, putting shear in place of rotation, we get a different kind of figure, as the result of the shear rotation, from what we got with our ordinary rotation. And as a consequence we get a position for the end of a line of invariable length when it turns by the shear rotation, different from the position which it would assume on turning by our rotation. A real material rod in the shear world would, on turning about A, pass from the position AB to the position AC. We say that its length alters when it becomes AC, but this transformation of AB would seem to an inhabitant of the shear world like a turning of AB without altering in length. If now we suppose a communication of ideas that takes place between one of ourselves and an inhabitant of the shear world, there would evidently be a difference between his views of distance and ours. We should say that his line AB increased in length in turning to AC. He would say that our line AF (fig. 23) decreased in length in turning to AC. He would think that what we called an equal line was in reality a shorter one. We should say that a rod turning round would have its extremities in the positions we call at equal distances. So would he—but the positions would be different. He could, like us, appeal to the properties of matter. His rod to him alters as little as ours does to us.