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

THE PSYCHOLOGY OF MATHEMATICS 419
The former is the conceptual range's so-called image or
mental picture, commonly represented to the physical
eye by a row or series of dots; the latter is the conceptual
pencil's so-called image or mental picture, commonly
represented to the physical eye by a set of physical lines
(or dot rows) having a physical point (or dot) in common.
You are familiar with the sensuous figures.
Still keeping n = 2, let us take a pair of equations like
(i), writing them, for short, (2) E'(u, x) =0, (3)
E"(u, x)=0; and consider (4) E'+XE"=O where X
is a parameter. Denote the ranges represented by (2) and
(3) by R' and R", and denote the pencils represented by
them by P' and P". What are T(u) and T(x) of (4)?
The former is a conceptual variable range of the pencil
(of ranges) determined by R' and R"; the latter is a
conceptual variable point (or pencil) determined by P' and
P"; it is plain that I(u) and I(x) are respectively the
so-called images of the variable range and variable point
(or pencil) just mentioned.
Advancing to the case where n =3, we have for field of
operation the space Sa of ordinary geometry. Consider
(5) uixl+U2X2+uax3+I =0. T(u) of (5) is obviously a
conceptual plane of points, while T(x) is a conceptual
bundle of planes (a point, that is, enveloped by the planes
containing it); and, of course, I(u) and I(x) are respectively the " images " of the conceptual plane and bundle
(or point as the bundle's vertex or carrier).
Let us now take a pair of equations like (5), namely,
(6) E'=O, (7) E"=O; and consider (8) E'+xE"=O.
Let 7r' and r" be the planes, and B' and B" the bundles
(or points), represented by (6) and (7). T(u) of (8) is a
conceptual variable plane of the axial pencil (of planes)
determined by r' and 7r"; T(x) is a conceptual variable