Thus the plane being would find that the ends of each of the lines was distant by the diagonal of a unit square from the corresponding end of the last and he could then place the three lines in their right relative position. Joining them he would have the figure of a hexagon. [Illustration: Fig. 70.] We may also notice that the plane being could make a representation of the whole cube simultaneously. The three squares, shown in perspective in fig. 70, all lie in one plane, and on these the plane being could pick out any selection of points just as well as on three separate squares. He would obtain a hexagon by joining the points marked. This hexagon, as drawn, is of the right shape, but it would not be so if actual squares were used instead of perspective, because the relation between the separate squares as they lie in the plane figure is not their real relation. The figure, however, as thus constructed, would give him an idea of the correct figure, and he could determine it accurately by remembering that distances in each square were correct, but in passing from one square to another their distance in the third dimension had to be taken into account. Coming now to the figure made by selecting according to our rule from the whole mass of points given by four axes and four positions in each, we must first draw a catalogue figure in which the whole assemblage is shown. We can represent this assemblage of points by four solid figures. The first giving all those positions which are at a distance O from our space in the fourth dimension, the second showing all those that are at a distance 1, and so on. These figures will each be cubes. The first two are drawn showing the front faces, the second two the rear faces. We will mark the points 0, 1, 2, 3, putting points at those distances along each of these axes, and suppose all the points thus determined to be contained in solid models of which our drawings in fig. 71 are representatives. Here we notice that as on the plane 0_i_ meant the whole line from which the distances in the _i_ direction was measured, and as in space 0_i_ means the whole plane from which distances in the _i_ direction are measured, so now 0_h_ means the whole space in which the first cube stands—measuring away from that space by a distance of one we come to the second cube represented. [Illustration: Fig. 71.] Now selecting according to the rule every one of one kind with every other of every other kind, we must take, for instance, 3_i_, 2_j_, 1_k_, 0_h_. This point is marked 3210 at the lower star in the figure. It is 3 in the _i_ direction, 2 in the _j_ direction, 1 in the _k_ direction, 0 in the _h_ direction. With 3_i_ we must also take 1_j_, 2_k_, 0_h_. This point is shown by the second star in the cube 0_h_. [Illustration: Fig. 72.] In the first cube, since all the points are 0_h_ points, we can only have varieties in which _i_, _j_, _k_, are accompanied by 3, 2, 1.