Hence it is not from any phenomenon explained by mathematics that we can derive a proof of four dimensions. Every phenomenon that has been explained is explained as three-dimensional. And, moreover, since in the region of the very minute we do not find rigid bodies acting on each other at a distance, but elastic substances and continuous fluids such as ether, we shall have a double task.
We must form the conceptions of the possible movements of elastic and liquid four-dimensional matter, before we can begin to observe. Let us, therefore, take the four-dimensional rotation about a plane, and enquire what it becomes in the case of extensible fluid substances. If four-dimensional movements exist, this kind of rotation must exist, and the finer portions of matter must exhibit it.
Consider for a moment a rod of flexible and extensible material. It can turn about an axis, even if not straight; a ring of india rubber can turn inside out.
What would this be in the case of four dimensions?
Fig. 44.
Axis of x running towards the observer.
Let us consider a sphere of our three-dimensional matter having a definite thickness. To represent this thickness let us suppose that from every point of the sphere in [fig. 44] rods project both ways, in and out, like D and F. We can only see the external portion, because the internal parts are hidden by the sphere.
In this sphere the axis of x is supposed to come towards the observer, the axis of z to run up, the axis of y to go to the right.
Fig. 45.