We can also take a section parallel to the face ABCD, and then letting drop all of our space except the plane of that section, introduce the w axis, running in the old y direction. This section can be represented by the same drawing, [fig. 38], and we see that it can rotate about the line on its left until it swings half way round and runs in the opposite direction to that which it ran in before. These turnings of the different sections are not inconsistent, and taken all together they will bring the cube from the position shown in [fig. 36] to that shown in [fig. 41].
Since we have three axes at our disposal in our space, we are not obliged to represent the w axis by any particular one. We may let any axis we like disappear, and let the fourth axis take its place.
Fig. 39.
Fig. 40.
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.