The fourth dimension

52 THE FOURTH DIMENSION

This relation always holds. Look at fig. 29. Shear square on hypothenuse7 internal 8 » 7

4 atcorners , - 1 . > 8 Fig. 29. Square on one side—which the reader can draw for p- himself—

° 4 internal

8 onsides . 4 at corners

ole pm

.K, and the square on the other side is 1. Hence in this ease again the difference is

. equal to the shear square on

. the hypothenuse, 9—1=8.

. Thus ina world of shear

. the square on the hypothen-

. use would be equal to the

. difference of the squares on

Fig. 29 Bis. the sides of a right-angled triangle.

In fig. 29 b¢s 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.