To take the special case with which we are already familiar. The line AB, is bounded by the points A and B. (Fig. 10). The square, A B C D, is bounded by four lines AB, BC, CD, DA. The cube, A B C D E F G H, is bounded by six surfaces, namely, ABCD, CDEF, EFGH, GHAB, ADEH, BCFG. Similarly we must conclude that a tesseract is bounded by cubes. We shall see later that there are eight of them. (4) We may put (3) in a slightly different way, by saying that: "Two adjacent portions of space, of any dimensionality, are separated by a space of the next lower dimensionality." The portions AB and BC of the line AC are separated by the point B. (Fig. 11.) The portions ABEF and BCDE of the fig. ACDF are separated by the line EB. The portions ABEFGHIM and BCDEMIKL of the solid ACDFGHKL are separated by the surface BIME. [Illustration: _Fig. 11_] Similarly we must suppose that any two adjacent portions of four space are separated by a three space figure. Or, again, to alter it slightly, "any space is no more than a boundary between two adjacent portions of the next higher space." Whence it follows that the whole of our three space is but the boundary between two adjacent portions of four space. (5) "A tesseract, which is the four-dimensional analogue of the cube, is bounded by Eight cubes. It has Twenty-four plane square faces, Thirty-two linear edges, and Sixteen corner points." This may at first sight seem difficult to grasp. In reality however, it is quite simple. We have only to remember that the tesseract is generated by the movement of a cube, in a direction at right angles to every direction that can be drawn in the cube, and that whenever a figure of a given dimensionality moves thus it generates a figure of the next higher dimensionality. Thus every point in the cube will trace out a line, every line a surface, and every surface a solid, and, since the distance moved is equal to the length of the side of the cube, these surfaces will be squares and the solids will be cubes. But let us first consider the analogous case of the generation of the cube by the movement of a square. Let A B C D represent the original position of the square. It moves, a distance equal to one of its sides, in a direction at right angles to every direction that can be drawn within itself--at right angles, _i.e._, to every one of its sides--and finally comes to rest in the position E F G H. [Illustration: _Fig. 12_] Every side has traced out another square and we have, in addition, the old square ABCD, with which we started and the new square EFGH, with which we end. Thus even if we had no idea how many sides, edges, and corners a cube had we could deduce them. We should say:-- Every side of the original square has traced out a new square--that makes 4--and we also have the original square and the "final" square making a total of 6. A cube, therefore, must be bounded by 6 square surfaces.