Page:Popular Science Monthly Volume 83.djvu/393

Rh The equation

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

The equation

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

So, by analogy, the equation

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.