The fourth dimension

RECAPITULATION AND EXTENSION 215

the face of ABFE, we see that any line in the face can take the place of the vertical and horizontal lines we have examined. Take the diagonal line a¥ and the section through it toGu. The portions of matter which were on one side of aF in this section in fig. 3 are on the opposite side of it in fig. 8. They have gone round the ine ar. Thus the rotation round a face can be considered as a number of rotations of sections round parallel lines in it.

The turning about two different lines is impossible in three-dimensional space. To take another illustration, suppose A and B are two parallel lines in the zy plane, and let cD and EF be two rods crossing them. Now, in the space of wyz if the rods turn round the lines a and B

in the same direction they

will make two independent circles,

When the end F is goin, down the end c will be coming up. They will meet and con-

a flict.

But if we rotate the rods about the plane of aB by the z to w rotation these movements will not conflict. Suppose all the figure removed

with the exception of the plane wz, and from this plane draw the axis of w, so that we are looking at the space of xew.

Here, fig. 10, we cannot see the lines A and 8B, We see the points @ and u, in which a and B intercept the x axis, but we cannot see the lines themselves, for they run in the y direction, and that is not in our drawing.

Now, if the rods move with the z to w rotation they will

Pig. 9 (137).