Page:Popular Science Monthly Volume 83.djvu/394

390 Again, to repeat in words slightly different from the foregoing (Fig. 11) considering the $$a$$ units as $$a$$ points (an indefinite number), the square $$ABCD$$ is derived from the line $$AB$$, which for convenience suppose to be one foot in length, by letting $$AB$$ with its a points move through a distance of one foot in a direction perpendicular to itself, that is, perpendicular to the one dimension of $$AB$$, every point of $$AB$$ describes a line, and $$ABCD$$ contains therefore $$a$$ lines and $$a^2$$ points.

The cube $$ABCD-G$$ is derived from the square $$ABCD$$ which moves one foot in a direction perpendicular to its two dimensions, its $$a$$ lines and $$a^2$$ points describing $$a$$ squares and $$a^2$$ lines respectively. The cube $$ABCD-G$$ therefore contains $$a$$ squares, $$a^2$$ lines and $$a^3$$ points.

Similarly, the four-dimensional unit is derived from the cube, $$ABCD-G$$, by letting that cube move one foot in a direction perpendicular to each of its three dimensions, that is, in the direction of the fourth dimension; its $$a$$ squares, $$a^2$$ lines, and $$a^3$$ points describing respectively $$a$$ cubes, $$a^2$$ squares, $$a^3$$ lines. The hypercube, therefore, contains $$a$$ cubes, $$a^2$$ squares, $$a^3$$ lines and $$a^4$$ points.

Now, as to the boundaries of the units, $$AB$$ has two bounding points, $$ABCD$$ has four, two each from the initial and the final position of the moving line, $$ABCD-G$$ has eight,—four each from the initial and the final position of the moving square,—and the hypercube has sixteen,—eight each from the initial and the final position of the moving cube.

Bounding Lines.—Of bounding lines, $$AB$$ has one (or is itself one), $$ABCD$$ has 4, one each from the initial and the final position of the moving line and 2 generated by the 2 bounding points of that line; $$ABCD-G$$ has 12,—4 each from the initial and the final position of the moving square and 4 generated by the 4 bounding points of that square; and the hypercube has 32,—12 each from the initial and the final position of the moving cube and 8 generated by the 8 bounding points of that cube.

Bounding Squares.—Of bounding squares, $$ABCD$$ has one (itself); $$ABCD-G$$ has 6,—one each from the initial and the final position of