The fourth dimension

THE ANALOGY OF A PLANE WORLD 11

Again, just as the plane being can represent any motion in his space by two axes, so we can represent any motion in our three-dimensional space by means of three axes. There is no point in our space to which we cannot move by some combination of movements on the directions marked out by these axes.

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 4x in common.

Omitting all discussion of the manner on which a plane being might be conceived to form a theory of a threedimensional existence, let us examine how, with the means at his disposal, he could represent the properties of threedimensional 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