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

386 MATHEMATICAL PHILOSOPHY
the reading of which requires no more preparatory knowledge of geometry and geometric method than can be
acquired in a good high school.
Another of the ways is the deliberate and patient way
of postulate procedure. I am not going to take the time
that would be necessary to spread before you here a
system of postulates for the geometry of a point-space of
n dimensions. To do so would be wasteful, for I may
assume that you are now pretty familiar with one or
more postulate systems for a space of three dimensions,as that of Hilbert or that of Veblen or that of Veblen
and Young (for 3-dimensional projective geometry),and only slight alteration of any such system is needed to
convert it into a system available for 4, 5 or n dimensions.
I will content myself with referring you to page 24 of
Veblen and Young's Projective Geometry, for example, for
a clear indication, if you require to be shown, of the
simple sort of alteration that will suffice.
The two foregoing ways of working into and working in
the worlds of hyperspace are, as you observe, the ways of
pure, or synthetic, geometry as distinguished from analytic, or algebraic, geometry (which latter, let me remind
you, is only a geometric method). The " pure " ways are
followed with especial frequency by the Italian geometricians, though, of course, the latter often employ the
analytic method also.
Of the latter method, I have already said enough for
the stated purpose of this lecture. An excellent way for a
beginner is found in a somewhat rough mixture of the
C" pure " and the algebraic ways, guided by the chief of all
intellectual guides- analogy-as follows:
Let L be a line (say a projective line) of points; it is
a space of one dimension, Si; if you like, you may say