The question naturally occurs, looking at these numbers 2, 2², 2³, by what figure shall we represent 2⁴, or 2 × 2 × 2 × 2. We know that in the figure there must be sixteen units, or twice as many units as in the cube. But the unit also itself must be different. And it must not differ from a cube simply in shape. It must differ from a cube as a cube differs from a square. No number of squares will make up a cube, because each square has no thickness. In the same way, no number of cubes must be able to make up this new unit. And here, instead of trying to find something already known, to which the idea of a figure corresponding to the fourth power can be affixed, let us simply reason out what the properties of such a figure must be. In this attempt we have to rely, not on a process of touching or vision, such as informs us of the properties of bodies in the space we know, but on a process of thought. Each fact concerning this unknown figure has to be reasoned out; and it is only after a number of steps have been gone through, that any consistent familiarity with its properties is obtained. Of all applications of the reason, this exploration is perhaps the one which requires, for the simplicity of the data involved, the greatest exercise of the abstract imagination, and on this account is well worth patient attention. The first steps are very simple. We must imagine a finite straight line to generate a square by moving on the plane of the paper, and this square in its turn to generate a cube by moving vertically upwards. Fig. 1 represents a straight line; Fig. 2 represents a square formed by the motion of that straight line; Fig. 3 represents perspectively a cube formed by the motion of that square A B C D upwards. It would be well, instead of using figure 3, to place a cube on the paper. Its base would be A B C D, its upper surface E F G H. [Illustration: Generating a line, square, and cube.] The straight line A B gives rise to the square A B C D by a movement at right angles to itself. If motion be confined to the straight line A B, a backward and forward motion is the only one possible. No sideway motion is admissible. And if we suppose a being to exist which could only move in the straight line A B, it would have no idea of any other movement than to and fro. The square A B C D is formed from the straight line by a movement in a direction entirely different from the direction which exists in A B. This motion is not expressible by means of any possible motion in A B. A being which existed in A B, and whose experience was limited to what could occur in A B, would not be able to understand the instructions we should give to make A B trace out the figure A B C D.