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

248 MATHEMATICAL PHILOSOPHY
that are strictly geometric and some that are neither
geometric nor numerical.
(8) Consider the class 2 of the spheres having a given
point C for center and any radius greater than zero.
Each of the spheres has many points, many tangent lines
and many tangent planes. Each sphere has an interior,the region or room bounded by the sphere,-and an
exterior-the region outside of it. If you let S, R, R', r,
P, L, ir, l, A, ', P', P" respectively represent any one of
the spheres, any one of the interiors, any one of the
exteriors, any one of the radii (taken as line segments),
any one of the points on any one of the spheres, any one
of the tangent lines, any one of the tangent planes, any
one of the radial lengths, any one of the sphere areas,
any one of the sphere volumes, any one of the points
common to all the interiors, any one of the points common
to all the exteriors, then the symbols S, R, R', r, P, L, ir,
P', P" are geometric variables, and 1, A, V are arithmetic,
or numerical, variables. Of the former ones the respective
ranges are: the class S, the class of all the interiors of the
spheres in 2; the class of all the sphere exteriors; the
class of all the radii; the class of all the points of our
space except the point C; the class of all the lines of space
except the lines of the line-sheaf having C for vertex;
the class of all the planes of space except those of the
bundled vertexed at C; the class whose sole member is
the point C (P' being thus a constant); and the null
(empty) class of points (the range of P" containing no
terms); the ranges of 1, A and F are the same, namely,
the class of all the positive real numbers (zero not
included).
Let me remind you of something and then submit a
few questions that should give you a happy hour or so of