The fourth dimension

—e

THE SECOND OHAPTER IN THE HISTORY OF FOUR SPAOE 51

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.

Fig. 27 represents a square unsheared. Fig. 28

Fig. 27.

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 acpz, we find—

Qinside = 2

4 atcorners= 1 or a total of 3.

Now, the square on the side asp has 4 points, that on Bo, 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. |