The fourth dimension

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,