The system will be clear, if we look at a representation in the plane of a tesseract with three, and one with four divisions in its side. The tesseract made up of three tesseracts each way corresponds to the cube made up of three cubes each way, and will give us a complete nomenclature. In this diagram, fig. 101, 1 represents a cube of 27 cubes, each of which is the beginning of a tesseract. These cubes are represented simply by their lowest squares, the solid content must be understood. 2 represents the 27 cubes which are the beginnings of the 27 tesseracts one inch on in the fourth dimension. These tesseracts are represented as a block of cubes put side by side with the first block, but in their proper positions they could not be in space with the first set. 3 represents 27 cubes (forming a larger cube) which are the beginnings of the tesseracts, which begin two inches in the fourth direction from our space and continue another inch. [Illustration: Fig. 101.] [Illustration: Fig. 102[4]] [4] The coloured plate, figs. 1, 2, 3, shows these relations more conspicuously. In fig. 102, we have the representation of a block of 4 × 4 × 4 × 4 or 256 tesseracts. They are given in four consecutive sections, each supposed to be taken one inch apart in the fourth dimension, and so giving four blocks of cubes, 64 in each block. Here we see, comparing it with the figure of 81 tesseracts, that the number of the different regions show a different tendency of increase. By taking five blocks of five divisions each way this would become even more clear. We see, fig. 102, that starting from the point at any corner, the white coloured regions only extend out in a line. The same is true for the yellow, red, and blue. With regard to the latter it should be noticed that the line of blues does not consist in regions next to each other in the drawing, but in portions which come in in different cubes. The portions which lie next to one another in the fourth dimension must always be represented so, when we have a three-dimensional representation. Again, those regions such as the pink one, go on increasing in two dimensions. About the pink region this is seen without going out of the cube itself, the pink regions increase in length and height, but in no other dimension. In examining these regions it is sufficient to take one as a sample. The purple increases in the same manner, for it comes in in a succession from below to above in block 2, and in a succession from block to block in 2 and 3. Now, a succession from below to above represents a continuous extension upwards, and a succession from block to block represents a continuous extension in the fourth dimension. Thus the purple regions increase in two dimensions, the upward and the fourth, so when we take a very great many divisions, and let each become very small, the purple region forms a two-dimensional extension.