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? 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. [Illustration: Fig. 44. _Axis of x running towards the observer._] 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. [Illustration: Fig. 45.] Now take the section determined by the _zy_ plane. This will be a circle as shown in fig. 45. If we let drop the _x_ axis, this circle is all we have of the sphere. Letting the _w_ axis now run in the place of the old _x_ axis we have the space _yzw_, and in this space all that we have of the sphere is the circle. Fig. 45 then represents all that there is of the sphere in the space of _yzw_. In this space it is evident that the rods CD and EF can turn round the circumference as an axis. If the matter of the spherical shell is sufficiently extensible to allow the particles C and E to become as widely separated as they would be in the positions D and F, then the strip of matter represented by CD 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 it—a 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.