The fourth dimension

220 THE FOURTH DIMENSION

and all parallel sections move in a similar manner, then the point will describe a circle. If, now, in addition to the rotation by which the x axis changes into the y axis the body has a rotation by which the z axis turns into the w axis, the point in question will have a double motion in consequence of the two turnings. The motions will compound, and the point will describe a circle, but not the same circle which it would describe in virtue of either rotation separately.

We know that if a body in three-dimensional space is given two movements of rotation they will combine into a single movement of rotation round a definite axis. It is in no different condition from that in which it is subjected to one movement of rotation. The direction of the axis changes ; that is all. The same is not true about a four-dimensional body. The two rotations, z to y and 2 to w, are independent. A body subject to the two is in a totally different condition to that which it is in when subject to one only. When subject to a rotation such as that of « to y, a whole plane in the body, as we have seen, is stationary. When subject to the double rotation no part of the body is stationary except the point common to the two planes of rotation.

If the two rotations are equal in velocity, every point in the body describes a circle. All points equally distant from the stationary point describe circles of equal size.

We can represent a four-dimensional sphere by means of two diagrams, in one of which we take the three axes, 2, y, 2; in the other the axes 2, w, apd z. In fig. 13 we have the view of a four-dimensional sphere in the space of ayz. Fig. 13 shows all that we can see of the four sphere in the space of ayz, for it represents all the points in that space, which are at an equal distance from the centre.

Let us now take the xz section, and let the axis of w