Thus in [fig. 13] the point p moves in a circle parallel to the xy plane; in [fig. 14] it moves in a circle parallel to the zw plane, indicated by the arrow.
Now, suppose both of these independent rotations compounded, the point p will move in a circle, but this circle will coincide with neither of the circles in which either one of the rotations will take it. The circle the point p will move in will depend on its position on the surface of the four sphere.
In this double rotation, possible in four-dimensional space, there is a kind of movement totally unlike any with which we are familiar in three-dimensional space. It is a requisite preliminary to the discussion of the behaviour of the small particles of matter, with a view to determining whether they show the characteristics of four-dimensional movements, to become familiar with the main characteristics of this double rotation. And here I must rely on a formal and logical assent rather than on the intuitive apprehension, which can only be obtained by a more detailed study.
In the first place this double rotation consists in two varieties or kinds, which we will call the A and B kinds. Consider four axes, x, y, z, w. The rotation of x to y can be accompanied with the rotation of z to w. Call this the A kind.
But also the rotation of x to y can be accompanied by the rotation, of not z to w, but w to z. Call this the B kind.
They differ in only one of the component rotations. One is not the negative of the other. It is the semi-negative. The opposite of an x to y, z to w rotation would be y to x, w to z. The semi-negative is x to y and w to z.
If four dimensions exist and we cannot perceive them, because the extension of matter is so small in the fourth dimension that all movements are withheld from direct observation except those which are three-dimensional, we should not observe these double rotations, but only the effects of them in three-dimensional movements of the type with which we are familiar.
If matter in its small particles is four-dimensional, we should expect this double rotation to be a universal characteristic of the atoms and molecules, for no portion of matter is at rest. The consequences of this corpuscular motion can be perceived, but only under the form of ordinary rotation or displacement. Thus, if the theory of four dimensions is true, we have in the corpuscles of matter a whole world of movement, which we can never study directly, but only by means of inference.
The rotation A, as I have defined it, consists of two equal rotations—one about the plane of zw, the other about the plane of xy. It is evident that these rotations are not necessarily equal. A body may be moving with a double rotation, in which these two independent components are not equal; but in such a case we can consider the body to be moving with a composite rotation—a rotation of the A or B kind and, in addition, a rotation about a plane.
If we combine an A and a B movement, we obtain a rotation about a plane; for, the first being x to y and z to w, and the second being x to y and w to z, when they are put together the z to w and w to z rotations neutralise each other, and we obtain an x to y rotation only, which is a rotation about the plane of zw. Similarly, if we take a B rotation, y to x and z to w, we get, on combining this with the A rotation, a rotation of z to w about the xy plane. In this case the plane of rotation is in the three-dimensional space of xyz, and we have—what has been described before—a twisting about a plane in our space.