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

78 MATHEMATICAL PHILOSOPHY
bundle of circles. The bundle includes, as infinite circles
(i.e., circles of infinite radius), the straight lines through
O. Now, in thought, let us, once for all, remove the
point O from ir. Each circle of the bundle now lacks a
point; we may call them pathocircles, and speak of the
O-bundle of pathocircles. Our field of operation-which
may be denoted by K-is composed of the pathocircles of the O-bundle and the points (except O,
of course) of -r. We are going to give the doctrinal
function HaF an interpretation in the field K; it will be
a geometry of certain elements of K. The interpretation
arises from assigning to the variable-symbols in the
postulates of HaF definite meanings as follows: vi will
mean a point of K; v2, a pathocircle; R1 will mean
between in the sense that, if A, B, C be three points of a
pathocircle, B will be said to be between A and C, if A
(or C) must go through B in moving on the pathocircle
to C (or A); and R2 will mean congruent in the sense
that, if two segments or angles be congruent in the
ordinary sense (interpretation D1), their transforms, or
inverses, with respect to a given circle with O as center,
will be called congruent in the field of K.
We have now to show that the postulates are verified
by the meanings assigned. Before giving a proof valid
for all of the postulates, it will be instructive to deal with
a selected few of them singly by means of simple figures,
as in the preceding lecture. Postulate (I) is evidently
satisfied. In Fig. 14 the two points A and B determine
the pathocircle a of the O-bundle.
Fig. I5 exhibits the fact that postulate (8) is verified.
Point B is between A and C and between C and A; neither
A nor C is between the other two of the three points;
of course, no point can move through the absent 0.