Let us dwell a moment longer on the rotation of a rigid body. Looking at the cube in fig. 3, which turns about the face of ABFE, we see that any line in the face can take the place of the vertical and horizontal lines we have examined. Take the diagonal line AF and the section through it to GH. The portions of matter which were on one side of AF in this section in fig. 3 are on the opposite side of it in fig. 8. They have gone round the line AF. Thus the rotation round a face can be considered as a number of rotations of sections round parallel lines in it. The turning about two different lines is impossible in three-dimensional space. To take another illustration, suppose A and B are two parallel lines in the _xy_ plane, and let CD and EF be two rods crossing them. Now, in the space of _xyz_ if the rods turn round the lines A and B in the same direction they will make two independent circles. When the end F is going down the end C will be coming up. They will meet and conflict. [Illustration: Fig. 9 (137).] But if we rotate the rods about the plane of AB by the _z_ to _w_ rotation these movements will not conflict. Suppose all the figure removed with the exception of the plane _xz_, and from this plane draw the axis of _w_, so that we are looking at the space of _xzw_. Here, fig. 10, we cannot see the lines A and B. We see the points G and H, in which A and B intercept the _x_ axis, but we cannot see the lines themselves, for they run in the _y_ direction, and that is not in our drawing. Now, if the rods move with the _z_ to _w_ rotation they will turn in parallel planes, keeping their relative positions. The point D, for instance, will describe a circle. At one time it will be above the line A, at another time below it. Hence it rotates round A. [Illustration: Fig. 10 (138).] Not only two rods but any number of rods crossing the plane will move round it harmoniously. We can think of this rotation by supposing the rods standing up from one line to move round that line and remembering that it is not inconsistent with this rotation for the rods standing up along another line also to move round it, the relative positions of all the rods being preserved. Now, if the rods are thick together, they may represent a disk of matter, and we see that a disk of matter can rotate round a central plane. Rotation round a plane is exactly analogous to rotation round an axis in three dimensions. If we want a rod to turn round, the ends must be free; so if we want a disk of matter to turn round its central plane by a four-dimensional turning, all the contour must be free. The whole contour corresponds to the ends of the rod. Each point of the contour can be looked on as the extremity of an axis in the body, round each point of which there is a rotation of the matter in the disk.