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

NON-GEOMETRIC INTERPRETATION 103
arises from assigning to the variables in HAF' the following
values, or meanings: vt is to denote a triad of N'; v2, the
system of triads common to a pair of equations, Ax +By +
Cz+D = O, A'x+B'y+C'z+D = O; v3, the system of triads
satisfying one such equation; v4 is to mean N'; R1 is to
mean between in the sense that if the triads, tî(xi, yi, zi),
t2(X2, y2, 2), t3(x3, y3, Z3), belong to a system of the
former kind, then t2 will be between tl and t3 when and
only when x2 is between xi and x3 or y2 is between yl and
y3 or Z2 is between zi and z3; and R2 is to mean congruent,
or equal, in the sense that the segment t1t2 will be congruent to segment t3t4 when and only when,
/(X -X2)2 +(y -y2)2 + (Z -2)2 =
/(X3 -X4)2 +(y3 -y4)2 + (Z3 - Z4)2,
and, for angles, in the sense analogous to that given in
the preceding interpretation.
The interpretation is worked out with some detail in
a very interesting and enlightening way in the Elementare
Geometrie cited in the preceding lecture. The doctrine,
D41, arising thus from HàF' is, as you see, non-spatial and
non-geometric; it is a purely algebraic three-dimensional
theory of triads and systems of triads of real numbers and
is, of course, isomorphic with ordinary Euclidean geometry of space.