Page:Popular Science Monthly Volume 83.djvu/393

The equation

$ax+by=c$

represents a line in $2$-space, but has no meaning in $1$-space.

The equation

$ax+by+cz=d$

represents a plane in $3$-space, but has no meaning in $2$-space or $1$-space.

So, by analogy, the equation

$ax+by+cz+dw=e$

would have a meaning in $4$-space,—say a $3$-space section of $4$-space—but has no meaning in $3$-space.

In general, an algebraic equation of $k$ variables has no meaning in a space of lower dimension than $k$, but has a meaning in $n$-space, where $n>=k$.

Discarding experience and reasoning wholly from analogy, we introduce some properties of the fourth dimension as follows.

Four-dimensional measure depends upon length, breadth, height and a fourth dimension all multiplied together. In the graphical representation of $3$-space, points are referred to three mutually perpendicular planes formed by three lines mutually at right angles. In a similar way, to represent $4$-space we must assume another axis at right angles to each of the other three. In the present development of human thought, this is purely subjective, a mere mental conception, and it is upon this conception that the theory of hyperspace is built.

The position of a point in a plane may be determined, as we have seen, by its distance from each of two perpendicular right lines; in $3$-space, by its distance from each of three mutually perpendicular planes; and in $4$-space, by its distance from each of four mutually perpendicular $3$-spaces, for there are four arrangements of the four axes taken three at a time, and each independent set of three perpendicular axes define a $3$-space, for example, $wxy,wxz,wyz,xyz$. Just as in our space it requires at least three points to determine a plane ($2$-space), so in $4$-space four points are necessary to determine a $3$-space.

As portions of our space are bounded by surfaces, plane or curved, so portions of $4$-space are bounded by hyperspace (three-dimensional).

In our space, a point moving in an unchanging direction generates a straight line.

This straight line (say of $a$ units in length), moving perpendicular to its initial position through the distance a, generates a square.

This square, moving perpendicular to its initial position through the distance $a$, generates a cube.

This cube, we will suppose, moving perpendicular to our space for a distance equal to one of its sides (that is, equal to $a$), will generate a hypercube.

Now the line contains $a$ units, the square $a^{2}$ units, the cube $a^{3}$ units, the hypercube $a^{4}$ units.