I may as well repeat here that I do not for a moment expect that the reader will have been able to visualise four-dimensional space. But I do hope that he will have seen the force of the analogy and will be prepared to admit that so far as we have gone at present four dimensional space is by no means inconceivable though it may not be distinctly imaginable. The foregoing is really all that is necessary on the mathematical or theoretical side for the understanding of the basic ideas with which I am dealing but for the benefit of those readers who like that sort of thing I have added a few simple propositions and extensions of the analogy in the form of an appendix. The only other question that need really concern us here is that of the phenomena of _change_ in a two-dimensional world. We have already seen that a cube laid on a flat surface will present to a plane being, in that surface, the appearance of a square. It is also clear that if it is pushed through the surface it will continue to present the same appearance until it has passed right through, when it will suddenly vanish away. He would be unconscious of any movement on the part of the cube unless there was some difference between the first and last sections which he perceived. If, for instance, the bottom face was red and the top face blue he would be conscious of a colour change on the part of the square which he perceived. It would start by being red and would pass through various shades of purple till, just before its final disappearance, it would be pure blue. But now suppose that it was pressed through his surface not "normally" but corner wise as indicated in Fig. 6--that is to say with one of its corners leading and one of its diagonals vertical. The plane being would then see quite a different set of figures. First would be a point; this would grow into a triangle which would increase in size until it reached a certain maximum when it would begin to develope three new sides at its corners which would grow, at the expense of the original sides, until a regular hexagon was produced when the reverse process would set in and the hexagon gradually change back into a triangle which in turn would dwindle away and disappear. It is easy to work out what would happen in the case of other solids, _e.g._, Sphere, Cone, Tetrahedron, etc. All such changes would appear very mysterious to the plane being if he had formed no conception of three-dimensional space or the shapes of bodies therein. [Illustration: _Fig. 6._] Let us now extend this idea rather further.