The fourth dimension

222 THE FOURTH DIMENSION

diagrams. Now, in fig. 14 let the zw rotation take place, the z axis will turn toward the point w of the w axis, and the point p will move in a circle about the point a.

Thus in fig. 13 the point p moves in a circle parallel to the zy 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 fourdimensional 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, 2, w. The rotation of @ to y can be accompanied with the rotation of 2 to w. Call this the A kind.

But also the rotation of a 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, 2 to w rotation would be y to 2, wtoz. The semi-negative is x to y and w to z.