The fourth dimension, page 72

64 THE FOURTH DIMENSION

To see the whole he must relinquish part of that which he has, and take the whole portion by portion.

Consider now a plane being in front of a square, fig. 34.

The square can turn about any point in the plane—say the point a. But it cannot turn about a line, as AB. For, in order to turn about the line AB, the square must leave the plane and _ move in the third dimension. This motion is out of his range of observa-

tion, and is therefore, except for a process of reasoning, inconceivable to him.

Rotation will therefore be to him rotation about a point. Rotation about a line will be inconceivable to him.

The result of rotation about a line he can appprehend. He can see the first and last positions occupied in a half revolution about the line ac. The result of such a half revolution is to place the square aBcD on the left hand instead of on the right hand of the line ac. It would correspond to a pulling of the whole body azncp through the line ac, or to the production of a solid body which was the exact reflection of it in the line ac. It would be as if the square AgcD turned into its image, the line aB acting as a mirror. Such a reversal of the positions of the parts of the square would be impossible in his space. The occurrence of it would be a proof of the existence ofa higher dimensionality.

Let him now, adopting the conception of a threedimensional body as a series of sections lying, each removed a little farther than the preceding one, in direction at right angles to his plane, regard a cube, fig. 36, as a series of sections, each like the A 8 * square which forms its base, all

Fig. 35. rigidly connected together.

Fig. 34.

P 4

8,