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 observation, 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 apprehend. 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 ABCD 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 ABCD 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 of a higher dimensionality.
Fig. 35.
Let him now, adopting the conception of a three-dimensional 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 square which forms its base, all rigidly connected together.
If now he turns the square about the point A in the plane of xy, each parallel section turns with the square he moves. In each of the sections there is a point at rest, that vertically over A. Hence he would conclude that in the turning of a three-dimensional body there is one line which is at rest. That is a three-dimensional turning in a turning about a line.
In a similar way let us regard ourselves as limited to a three-dimensional world by a physical condition. Let us imagine that there is a direction at right angles to every direction in which we can move, and that we are prevented from passing in this direction by a vast solid, that against which in every movement we make we slip as the plane being slips against his plane sheet.
We can then consider a four-dimensional body as consisting of a series of sections, each parallel to our space, and each a little farther off than the preceding on the unknown dimension.