Now, it is evident that we have twenty-seven positions, each of them named. If the reader will follow this nomenclature in respect of the positions marked in the figures he will have no difficulty in assigning names to each one of the twenty-seven positions. A is _oi_, _oj_, _ok_. It is at the distance 0 along _i_, 0 along _j_, 0 along _k_, and _io_ can be written in short 000, where the _ijk_ symbols are omitted. The point immediately above is 001, for it is no distance in the _i_ direction, and a distance of 1 in the _k_ direction. Again, looking at B, it is at a distance of 2 from A, or from the plane ADC, in the _i_ direction, 0 in the _j_ direction from the plane ABD, and 0 in the _k_ direction, measured from the plane ABC. Hence it is 200 written for 2_i_, 0_j_, 0_k_. Now, out of these twenty-seven “things” or compounds of position and dimension, select those which are given by the rule, every one of one kind with every other of every other kind. Take 2 of the _i_ kind. With this we must have a 1 of the _j_ kind, and then by the rule we can only have a 0 of the _k_ kind, for if we had any other of the _k_ kind we should repeat one of the kinds we already had. In 2_i_, 1_j_, 1_k_, for instance, 1 is repeated. The point we obtain is that marked 210, fig. 66. [Illustration: Fig. 66.] Proceeding in this way, we pick out the following cluster of points, fig. 67. They are joined by lines, dotted where they are hidden by the body of the cube, and we see that they form a figure—a hexagon which could be taken out of the cube and placed on a plane. It is a figure which will fill a plane by equal repetitions of itself. The plane being representing this construction in his plane would take three squares to represent the cube. Let us suppose that he takes the _ij_ axes in his space and _k_ represents the axis running out of his space, fig. 68. In each of the three squares shown here as drawn separately he could select the points given by the rule, and he would then have to try to discover the figure determined by the three lines drawn. The line from 210 to 120 is given in the figure, but the line from 201 to 102 or GK is not given. He can determine GK by making another set of drawings and discovering in them what the relation between these two extremities is. [Illustration: Fig. 67.] [Illustration: Fig. 68.] [Illustration: Fig. 69.] Let him draw the _i_ and _k_ axes in his plane, fig. 69. The _j_ axis then runs out and he has the accompanying figure. In the first of these three squares, fig. 69, he can pick out by the rule the two points 201, 102—G, and K. Here they occur in one plane and he can measure the distance between them. In his first representation they occur at G and K in separate figures.