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

NON-GEOMETRIC INTERPRETATION 91
by definition, x2 is between X3 and xi, or y2 is between
y3 and yi, and so d2 is between d3 and d1. Hence the
postulate is satisfied.
Postulate (9).-It is evident that in determining a
dyad of any given system we can assign the x (or y) at
will. Now let di(xi, yi) and d3(x3, y3) be two given
dyads of any given system s; let d2(x2, y2) be a dyad of
s such that xi <X2<x3; then d2 is between d1 and d3;
next let d4(x4, y4) be such that Xi <X3 <x4; then d3 is
between di and d4. Hence the postulate is satisfied.
Postulate (Io).-We need consider only four possibilities: (a) A=O, and s is By+C=O; (e) B=O, and s
is x +C =; (y) A s0, B0O, C=O, and s isx +By =0;
(6) A 0, B 0, CsO, and s is Ax+By+C=O.
You know that of three numbers one and only one is
between the other two. In (a) any three dyads of s are
of the form (X,-), (x2, -), (Xa -); hence one
and only one of the x's is between the other two, and so,
too, of the dyads; in (f) like reasoning leads to the same
conclusion; in (y) let di(xi, yi), d2(x2, y2), d3(x3, y3) be
any three dyads of s; then - — Y -;2 y hence
B Xi X2 X3
no two x's (or y's) are equal for, if they were, the corresponding y's or (x's) would be equal and we should not
have three distinct dyads; hence one and only one of the
x's (and also one and only one of the y's) is between the
other two; hence so, too, the dyads; finally, in (ô) we
have
i yi I y2 I y3.,4 I y2 I y3 y- I yl
C xi yi X2 y2 2 X3 Y3
X2 2 x2 X 3 Xi y i