Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

HYPER6PACES 325
line is a point-space of two or more dimensions. Ail this
is obvious. I have stressed it because it is essential to a
good understanding of any one of the meanings of hyperspace. Let us turn to analogous considerations in the
case of a plane.
In Lecture V we saw that, by means of a pair of
rectangular axes and a unit of length, a one-to-one correspondence can be established between the points P of a
plane and the pairs (x, y) of real numbers. The pair
represents the point, and the point the pair; the x and y
of a pair are at once the coordinates of the pair and of the
corresponding point. There are as many points in the
plane as there are pairs of real numbers in the system of
such pairs. Hence a plane contains oo2 points; a point
that is free to move in a plane and is confined thereto
has two and but two degrees of freedom; a plane, regarded
as a space of points, is a two-dimensional space; and you
see why it is so called,-it is because the points of a plane
match, in one-to-one fashion as we have seen, the pairs of
real numbers in the two-dimensional system of such pairs.
And now what shall we say of ordinary space? What,
I mean, shall we say of that immense region or room in
which we are immersed and, with us, our floating world
and the stars? Let us think of it as a field, or a plenum,
of points. What, then, is its dimensionality? It is easy
to ascertain.
Choose three mutually perpendicular planes; they
have a common point 0, called the origin; they determine
three lines, OX, OY, OZ, called axes; agree that a distance
measured parallel to an axis shall be positive or negative
according as it is reckoned in the sense of the arrow or in
the opposite sense; note that the three planes divide the
whole of space into eight compartments; choose a unit