The fourth dimension

96 THE FOURTH DIMENSION

This fourth dimension is supposed to run at right angles to any of the three space dimensions, as the third space dimension runs at right angles to the two dimensions of a plane, and thus it gives us the opportunity of generating a new kind of volume. If the whole cube moves in this dimension, the solid itself traces out a path, each section of which, made at right angles to the direction in which it moves, is a solid, an exact repetition of the cube itself,

The cube as we see it is the beginning of a solid of such a kind. It represents a kind of tray, as the square face of the cube is a kind of tray against which the cube rests.

Suppose the cube to move in this fourth dimension in four stages, and let the hyper-solid region traced out in the first stage of its progress be characterised by this, that the terms of the syllogism are in the first figure, then we can represent in each of the three subsequent stages the remaining three figures. Thus the whole cube forms the basis from which we measure the variation in figure. The first figure holds good for the cube as we see it, and for that hyper-solid which lies within the first stage ; the second figure holds good in the second stage, and SO on.

Thus we measure from the whole cube as far as figures are concerned.

But we saw that when we measured in the cube itself having three variables, namely, the two premisses and the conclusion, we measured from three planes. The base from which we measured was in every case the same.

Hence, in measuring in this higher space we should have bases of the same kind to measure from, we should have solid bases.

The first solid base is easily seen, it is the cube itself. The other can be found from this consideration.

That solid from which we measure figure is that