In fig. 94, in which the cube is turned about the yellow line, we have a certain difficulty, for the plane being will find that the position his squares are to be placed in will lie below that which they first occupied. They will come where the support was on which he stood his first set of squares. He will get over this difficulty by moving his support. Then, since the cubes come upon his plane by the light yellow face, he will have, taking the null cube as before for an example, null, light yellow face; null, red section, because the section is perpendicular to the red line; and finally, as the null cube leaves the plane, null, light yellow face. Then, in this case red following on null, he will have the same series of views of the red as he had of the null cube. [Illustration: Fig. 95.] There is another set of considerations which we will briefly allude to. Suppose there is a hollow cube, and a string is stretched across it from null to null, _r_, _y_, _wh_, as we may call the far diagonal point, how will this string appear to the plane being as the cube moves transverse to his plane? Let us represent the cube as a number of sections, say 5, corresponding to 4 equal divisions made along the white line perpendicular to it. We number these sections 0, 1, 2, 3, 4, corresponding to the distances along the white line at which they are taken, and imagine each section to come in successively, taking the place of the preceding one. These sections appear to the plane being, counting from the first, to exactly coincide each with the preceding one. But the section of the string occupies a different place in each to that which it does in the preceding section. The section of the string appears in the position marked by the dots. Hence the slant of the string appears as a motion in the frame work marked out by the cube sides. If we suppose the motion of the cube not to be recognised, then the string appears to the plane being as a moving point. Hence extension on the unknown dimension appears as duration. Extension sloping in the unknown direction appears as continuous movement. CHAPTER XII THE SIMPLEST FOUR-DIMENSIONAL SOLID A plane being, in learning to apprehend solid existence, must first of all realise that there is a sense of direction altogether wanting to him. That which we call right and left does not exist in his perception. He must assume a movement in a direction, and a distinction of positive and negative in that direction, which has no reality corresponding to it in the movements he can make. This direction, this new dimension, he can only make sensible to himself by bringing in time, and supposing that changes, which take place in time, are due to objects of a definite configuration in three dimensions passing transverse to his plane, and the different sections of it being apprehended as changes of one and the same plane figure.