The fourth dimension

dimensions. His limitation to a plane must be the result of physical conditions.

We will therefore think of him as of a figure cut out of paper placed on a smooth plane. Sliding over this plane, and coming into contact with other figures equally thin as he in the third dimension, he will apprehend them only by their edges. To him they will be completely bounded by lines. A“ solid” body will be to him a two-dimensional extent, the interior of which can only be reached by penetrating through the bounding lines.

Now such a plane being can think of our threedimensional existence in two ways.

First, he can think of it as a series of sections, each like the solid he knows of extending in a direction unknown to him, which stretches transverse to his tangible universe, which lies in a direction at right angles to every motion which he made.

Secondly, relinquishing the attempt to think of the three-dimensional solid body in its entirety he can regard it as consisting of a number of plane sections, each of them in itself exactly like the two-dimensional bodies he knows, but extending away from his two-dimensional space.

A square lying in his space he regards as a solid bounded by four lines, each of which lies in his space.

A square standing at right angles to his plane appears to him as simply a line in his plane, for all of it except the line stretches in the third dimension.

He can think of a three-dimensional body as consisting of a number of such sections, each of which starts from a line in his space.

Now, since in his world he can make any drawing or model which inyolves only two dimensions, he can represent each such upright section as it actually is, and can represent a turning from a known into the unknown dimension as a turning from one to another of his known dimensions,

THE HIGHER WORLD 63