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

HYPERSPACES 339
6N values determine a point in a space of 6N dimensions.
Thus there subsists a correspondence between such points
and the varying gas states. As the state of the gas
changes (owing to the motions of the molecules) the
corresponding point generates a path, or locus, in the
space of 6N dimensions; and so the behavior or history of
the gas (as a whole) gets geometrically represented by
loci in the mentioned hyperspace. That will suffice as a
hint at what has become a recondite mathematical theory
-the kinetic theory of gases.
I have said that I cannot here exhibit the wonders to
be found in the worlds of hyperspace. To do so in any
fair measure would require many lectures as long as this
one. I can not refrain, however, from leading you, if
you be willing, to see one of the minor wonders met with
on the very threshold of 4-dimensional space. We can
find it in the "mixed way " we were following a little
while ago, guiding ourselves by analogy, and at the same
time you will see how you can yourselves discover further
wonders. Note the facts carefully and note their analogies as we start at the bottom and ascend the scale.
Observe, to begin, that in a line (Si) an equation ax+b =0
of first degree in one variable (x) represents a point (So);
in a plane (S2) an equation ax +by +c = 0 of first degree in
two variables (x, y) represents a line (S1), and that two
such equations taken as simultaneous represent a point
(So)-the common point of the two lines; in an ordinary
space (S3) an equation ax+by+cz +d = of first degree
in three variables (x, y, z) represents a plane (S2), two
such equations taken as simultaneous represent a line
(S1),-the line common to the two planes,-and that
three such equations (if independent) together represent a
point (So)-the common point of the three planes, You