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

64 MATHEMATICAL PHILOSOPHY
assigning to the v's and the R's in the postulates of HF the
following meanings, or constant values: vl is to mean a
line of q>; v2, a pathopencil of,; Ri is to mean " between"
in the sense that, if A, B, C be three lines of a pathopencil,
B will be considered to be between A and C, if A (or C)
must rotate through the position of B to coincide with C
(or A) (for, of course, a line of a pathopencil must not
be supposed to rotate into the position left vacant by
the absent O-line); and R2 is to mean "congruent" in a
sense to be given later.
A B
FIG. 2.
We have to show that the indicated meanings satisfy,
or verify, the postulates of HAF. That some of them are
thus satisfied may be made evident by simple figures;
and it will be interesting and enlightening to exhibit
such evidence before giving the proof for all the postulates.
At the same time, we will lay bare, by means of figures,
the significance of one or two theorems of the new doctrine. I shall not here repeat the postulates, but will
suppose you to have them in hand.
Postulate (I) is plainly satisfied, for any two lines
A and B of q determine, as in Fig. 2, a pathopencil a,
which consists of all the lines through a except the 0-line
Oa.