The fourth dimension

70 THE FOURTH DIMENSION

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 4, 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 acc, 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 ar, 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.

To bring out this point more clearly let us take two y : B parallel lines, A and B, in

the space of xyz, and let cp and EF be two rods running above and below the plane of zy, from these lines. If we

Fig. 42.