The fourth dimension

RECAPITULATION AND EXTENSION 223

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 zy. 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 « to y and 2 to w, and the second being « 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 @ 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 wy plane. In this case the plane of rotation is in the three-dimensional space of wyz, and we have—what has