The fourth dimension

THE ANALOGY OF A PLANE WORLD 9

to any other, by means of two straight lines drawn at right angles to each other.

Let ax and ay be two such axes. He can accomplish the translation from a to B by going along ax to c, and then from c along cs parallel to ay.

The same result can of course be obtained by moving to D along ay and then parallel to ax from pD to B, or of course by any diagonal movement compounded by these axial movements.

By means of movements parallel to these two axes he can proceed (except for material obstacles) from any one point of his space to any other.

Fig. 5.

If now we suppose a third line drawn out from 4 at right angles to the plane it is evident that no motion in either of the two dimensions he knows will carry him in the least degree in the

Z direction represented by A Z.

Fig. 6. The lines az and ax determine a plane. If he could be taken off his plane, and transferred to the plane 4xZ, he would be in a world exactly like his own. From every line in his world there goes off a space world exactly like his own.

From every point in his world a line can be drawn parallel to az in the direction unknown to him. If we suppose the square in fig. 7 to be a geometrical square from every point of it, inside as well as on the contour, a straight line can be drawn parallel to az. The assemblage of these lines constitute a solid figure, of which the square in the plane is the base. If we consider the square to represent an object in the plane

Fig. 7.