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

338 MATHEMATICAL PHILOSOPHY
where the matter is handled systematically, elementally
and many-sidedly. Among the aims of the course in
Modern Theories of Geometry, which I have given at
Columbia University for many years, is that of helping
students to acquire a working knowledge of n-dimensional
geometry. The importance of such a knowledge is by no
means restricted to students of so-called pure mathematics.
Indeed, a few years ago, I had the honor to give a series
of lectures on n-dimensional geometry to a group of
physicists who had found that without some knowledge
of hyperspace methods, they could not read the literature
of their own subject, especially that of the kinetic theory
of gases. That was before the present relativity rage,
which, as we saw in a previous lecture, avails itself of the
idea of four-dimensional space. It will take but a minute
and it will be instructive to show why students of gas
theory are now obliged to acquire some knowledge of
n-dimensional geometry. It is because some of the foremost writers on the theory,-J. H. Jeans, for example,have adopted and elaborated the following considerations.
Suppose we have a closed vessel, say a sphere, filled with
gas. Let us suppose the gas is composed of N molecules.
These are flying about hither and thither, all of them in
motion. Think of one of them; at a given instant it is at
a point (x, y, z); at the same time it is moving so that the
components of its velocity along the axes of reference
are (say) u, v and w; if and only if we know the six
coordinates of the molecule at an instant, we know where
it is and the direction and rate of its going. The N
molecules constituting the gas thus depend, you see, upon
6N coordinates. At any instant these have definite values.
Together these values define the " state" of the gas at
that instant. Now, say the writers in question, the