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

62 MATHEMATICAL PHILOSOPHY
recall, I have not assumed on your part a knowledge of
Projective Geometry. It will be sufficient for our purpose to introduce them in the rough traditional way
instead of the very refined way employed by Veblen and
Young, for example, in their Projective Geometry, which is
based upon a postulate system appropriate for projective
geometry.
L: ----_._.
FIG. I.
Let the figure be in a Euclidean plane-the kind of
plane belonging to Di. All lines of the plane that contain a given point P constitute a pencil of lines; P is the
pencil's vertex. All the points of a line L constitute a
range of points; L is the range's base. It is plain that
each point of range L is on one line of pencil P; and
that, reciprocally, each line of P has one point of L, with
a single exception,-L', parallel to L, contains no point
of L. To remove this exception to the one-to-one correspondence, otherwise perfect, there is made in projective
geometry an agreement or convention: namely, that
each line has (at an infinite distance) a so-called " ideal "
point, or point at infinity, and that the "ideal" points
of any two parallel lines are coincident. We thus get,
as you see, a new sort of straight line and of plane and of