The foregoing examples make it clear that beings can be conceived as living in a more limited space than ours. Is there a similar limitation in the space we know? At the very threshold of arithmetic an indication of such a limitation meets us. If there is a straight line before us two inches long, its length is expressed by the number 2. Suppose a square to be described on the line, the number of square inches in this figure is expressed by the number 4, _i.e._, 2 × 2. This 2 × 2 is generally written 2², and named “2 square.” Now, of course, the arithmetical process of multiplication is in no sense identical with that process by which a square is generated from the motion of a straight line, or a cube from the motion of a square. But it has been observed that the units resulting in each case, though different in kind, are the same in number. If we touch two things twice over, the act of touching has been performed four times. Arithmetically, 2 × 2 = 4. If a square is generated by the motion of a line two inches in length, this square contains four square inches. So it has come to pass that the second and third powers of numbers are called “square” and “cube.” We have now a straight line two inches long. On this a square has been constructed containing four square inches. If on the same line a cube be constructed, the number of cubic inches in the figure so made is 8, _i.e._, 2 × 2 × 2 or 2³. Here, corresponding to the numbers 2, 2², 2³, we have a series of figures. Each figure contains more units than the last, and in each the unit is of a different kind. In the first figure a straight line is the unit, viz., one linear inch; it is said to be of one dimension. In the second a square is the unit, viz., one square inch. The square is a figure of two dimensions. In the third case a cube is the unit, and the cube is of three dimensions. The straight line is said to be of one dimension because it can be measured only in one way. Its length can be taken, but it has no breadth or thickness. The square is said to be of two dimensions because it has both length and breadth. The cube is said to have three dimensions, because it can be measured in three ways.