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, _x_ to _y_ and _z_ 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 _x_ 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, _x_, _y_, _z_; in the other the axes _x_, _w_, and _z_. In fig. 13 we have the view of a four-dimensional sphere in the space of _xyz_. Fig. 13 shows all that we can see of the four sphere in the space of _xyz_, 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_ take the place of the _y_ axis. Here, in fig. 14, we have the space of _xzw_. In this space we have to take all the points which are at the same distance from the centre, consequently we have another sphere. If we had a three-dimensional sphere, as has been shown before, we should have merely a circle in the _xzw_ space, the _xz_ circle seen in the space of _xzw_. But now, taking the view in the space of _xzw_, we have a sphere in that space also. In a similar manner, whichever set of three axes we take, we obtain a sphere. [Illustration: _Showing axes xyz_ Fig. 13 (141).] [Illustration: _Showing axes xwz_ Fig. 14 (142).] In fig. 13, let us imagine the rotation in the direction _xy_ to be taking place. The point _x_ will turn to _y_, and _p_ to _p´_. The axis _zz´_ remains stationary, and this axis is all of the plane _zw_ which we can see in the space section exhibited in the figure. In fig. 14, imagine the rotation from _z_ to _w_ to be taking place. The _w_ axis now occupies the position previously occupied by the _y_ axis. This does not mean that the _w_ axis can coincide with the _y_ axis. It indicates that we are looking at the four-dimensional sphere from a different point of view. Any three-space view will show us three axes, and in fig. 14 we are looking at _xzw_.