The fourth dimension

THE EVIDENCES FOR A FOURTH DIMENSION 81

the strip of matter represented by cp and EF and a multitude of rods like them can turn round the circular circumference.

Thus this particular section of the sphere can turn inside out, and what holds for any one section holds for all. Hence in four dimensions the whole sphere can, if extensible turn inside out. Moreover, any part of ita bowl-shaped portion, for instance—can turn inside out, and so on round and round.

This is really no more than we had before in the rotation about a plane, except that we see that the plane can, in the case of extensible matter, be curved, and still play the part of an axis,

If we suppose the spherical shell to be of four-dimensional matter, our representation will be a little different. Let us suppose there to be a small thickness to the matter in the fourth dimension. This would make no difference in fig. 44, for that merely shows the view in the xyz space. But when the « axis is let drop, and the w axis comes in, then the rods cp and EF which represent the matter of the shell, will have a certain thickness perpendicular to the plane of the paper on which they are drawn. If they have a thickness in the fourth dimension they will show this thickness when looked at from the direction of the w axis.

Supposing these rods, then, to be small slabs strung on the circumference of the circle in fig. 45, we see that there will not be in this case either any obstacle to their turning round the circumference. We can have a shell of extensible material or of fluid material turning inside out in four dimensions,

And we must remember that in four dimensions there is no such thing as rotation round an axis. If we want to investigate the motion of fluids in four dimensions we must take a movement about an axis in our space, and