On the assumption of a fourth dimension we have to suppose a fourth axis, which we will call AW. It must be supposed to be at right angles to each and every one of the three axes AX, AY, AZ. Just as the two axes, AX, AZ, determine a plane which is similar to the original plane on which we supposed the plane being to exist, but which runs off from it, and only meets it in a line; so in our space if we take any three axes such as AX, AY, and AW, they determine a space like our space world. This space runs off from our space, and if we were transferred to it we should find ourselves in a space exactly similar to our own. We must give up any attempt to picture this space in its relation to ours, just as a plane being would have to give up any attempt to picture a plane at right angles to his plane. Such a space and ours run in different directions from the plane of AX and AY. They meet in this plane but have nothing else in common, just as the plane space of AX and AY and that of AX and AZ run in different directions and have but the line AX in common. Omitting all discussion of the manner on which a plane being might be conceived to form a theory of a three-dimensional existence, let us examine how, with the means at his disposal, he could represent the properties of three-dimensional objects. There are two ways in which the plane being can think of one of our solid bodies. He can think of the cube, fig. 8, as composed of a number of sections parallel to his plane, each lying in the third dimension a little further off from his plane than the preceding one. These sections he can represent as a series of plane figures lying in his plane, but in so representing them he destroys the coherence of them in the higher figure. The set of squares, A, B, C, D, represents the section parallel to the plane of the cube shown in figure, but they are not in their proper relative positions. [Illustration: Fig. 8.] The plane being can trace out a movement in the third dimension by assuming discontinuous leaps from one section to another. Thus, a motion along the edge of the cube from left to right would be represented in the set of sections in the plane as the succession of the corners of the sections A, B, C, D. A point moving from A through BCD in our space must be represented in the plane as appearing in A, then in B, and so on, without passing through the intervening plane space. In these sections the plane being leaves out, of course, the extension in the third dimension; the distance between any two sections is not represented. In order to realise this distance the conception of motion can be employed. [Illustration: Fig. 9.]