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

NON-GEOMETRIC INTERPRETATION 89
three cases are possible and only tnree: (a) xl =x2, yl $y2;
(3) xi OX2, yi =y2; (y) XiX2, y1 #y2. In (a) B =0 and
C C
=-xi=-x2; in (3) A=0, and B~y =-y2; in
(y) plainly A 0, B 0, and if C=0, then
A Yi Y2
B Xi X2
but if C 0, then
I yi X 1
A I y2 B x2 I
and - - _
C xI y, C Xi Yi
X2 2 X2 y2
so that in all cases s is determined, and postulate (I) is
verified.
Postulate (2).-The question is: do every two dyads of
a system determine it? Let Ax+By+C=O be a system
s; (xl, yi) (x2, y2), any two dyads, di, d~, of s. These
dyads, by postulate (I), do determine a system s', say
A'x+B'y+C'=O. We are to show that s and s' are the
same system.
If A=0, then A'=0, for di is (x,,) and d2 is
l C\
X2, -B; hence
f B'x - B'C+BC' = O,
(I) { BA'x2-B'C+ BC'=,
since d1 and d2 belong to s'; as xilx2, A'=0; so s' is
B'y+C'=O; hence B'yl+C'=O, and as di belongs to s,