[Illustration: Fig. 40.] [Illustration: Fig. 41.] In fig. 36 suppose the _z_ axis to go. We have then simply the plane of _xy_ and the square base of the cube ACEG, fig. 39, is all that could be seen of it. Let now the _w_ axis take the place of the _z_ axis and we have, in fig. 39 again, a representation of the space of _xyw_, in which all that exists of the cube is its square base. Now, by a turning of _x_ to _w_, this base can rotate around the line AE, it is shown on its way in fig. 40, and finally it will, after half a revolution, lie on the other side of the _y_ axis. In a similar way we may rotate sections parallel to the base of the _xw_ rotation, and each of them comes to run in the opposite direction from that which they occupied at first. Thus again the cube comes from the position of fig. 36. to that of fig. 41. In this _x_ to _w_ turning, we see that it takes place by the rotations of sections parallel to the front face about lines parallel to AB, or else we may consider it as consisting of the rotation of sections parallel to the base about lines parallel to AE. It is a rotation of the whole cube about the plane ABEF. Two separate sections could not rotate about two separate lines in our space without conflicting, but their motion is consistent when we consider another dimension. Just, then, as a plane being can think of rotation about a line as a rotation about a number of points, these rotations not interfering as they would if they took place in his two-dimensional space, so we can think of a rotation about a plane as the rotation of a number of sections of a body about a number of lines in a plane, these rotations not being inconsistent in a four-dimensional space as they are in three-dimensional space. We are not limited to any particular direction for the lines in the plane about which we suppose the rotation of the particular sections to take place. Let us draw the section of the cube, fig. 36, through A, F, C, H, forming a sloping plane. Now since the fourth dimension is at right angles to every line in our space it is at right angles to this section also. We can represent our space by drawing an axis at right angles to the plane ACEG, our space is then determined by the plane ACEG, and the perpendicular axis. If we let this axis drop and suppose the fourth axis, _w_, to take its place, we have a representation of the space which runs off in the fourth dimension from the plane ACEG. In this space we shall see simply the section ACEG of the cube, and nothing else, for one cube does not extend to any distance in the fourth dimension. If, keeping this plane, we bring in the fourth dimension, we shall have a space in which simply this section of the cube exists and nothing else. The section can turn about the line AF, and parallel sections can turn about parallel lines. Thus in considering the rotation about a plane we can draw any lines we like and consider the rotation as taking place in sections about them.