[Illustration: Fig. 42.] To bring out this point more clearly let us take two parallel lines, A and B, in the space of _xyz_, and let CD and EF be two rods running above and below the plane of _xy_, from these lines. If we turn these rods in our space about the lines A and B, as the upper end of one, F, is going down, the lower end of the other, C, will be coming up. They will meet and conflict. But it is quite possible for these two rods each of them to turn about the two lines without altering their relative distances. To see this suppose the _y_ axis to go, and let the _w_ axis take its place. We shall see the lines A and B no longer, for they run in the _y_ direction from the points G and H. [Illustration: Fig. 43.] Fig. 43 is a picture of the two rods seen in the space of _xzw_. If they rotate in the direction shown by the arrows—in the _z_ to _w_ direction—they move parallel to one another, keeping their relative distances. Each will rotate about its own line, but their rotation will not be inconsistent with their forming part of a rigid body. Now we have but to suppose a central plane with rods crossing it at every point, like CD and EF cross the plane of _xy_, to have an image of a mass of matter extending equal distances on each side of a diametral plane. As two of these rods can rotate round, so can all, and the whole mass of matter can rotate round its diametral plane. This rotation round a plane corresponds, in four dimensions, to the rotation round an axis in three dimensions. Rotation of a body round a plane is the analogue of rotation of a rod round an axis. In a plane we have rotation round a point, in three-space rotation round an axis line, in four-space rotation round an axis plane. The four-dimensional being’s shaft by which he transmits power is a disk rotating round its central plane—the whole contour corresponds to the ends of an axis of rotation in our space. He can impart the rotation at any point and take it off at any other point on the contour, just as rotation round a line can in three-space be imparted at one end of a rod and taken off at the other end. A four-dimensional wheel can easily be described from the analogy of the representation which a plane being would form for himself of one of our wheels. Suppose a wheel to move transverse to a plane, so that the whole disk, which I will consider to be solid and without spokes, came at the same time into contact with the plane. It would appear as a circular portion of plane matter completely enclosing another and smaller portion—the axle.