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. 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. [Illustration: Fig. 35.] 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. Take the simplest four-dimensional body—one which begins as a cube, fig. 36, in our space, and consists of sections, each a cube like fig. 36, lying away from our space. If we turn the cube which is its base in our space about a line, if, _e.g._, in fig. 36 we turn the cube about the line AB, not only it but each of the parallel cubes moves about a line. The cube we see moves about the line AB, the cube beyond it about a line parallel to AB and so on. Hence the whole four-dimensional body moves about a plane, for the assemblage of these lines is our way of thinking about the plane which, starting from the line AB in our space, runs off in the unknown direction. [Illustration: Fig. 36.] In this case all that we see of the plane about which the turning takes place is the line AB.