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

92 MATHEMATICAL PHILOSOPHY
since any two of the dyads determine s; if two of the y's
Wefe equal, then A =0, contrary to hypothesis, unless the
corresponding x's were also equal, but then we should hot
have three distinct dyads. Hence ohe and only one of
the y's (or x's) is between the other two, and, the same
being consequently true of the dyads, the postulate is
verified.
The next postulate in HaF is the beautiful postulate
(12). First, however, we must have a
DEFINITION.-A pair of dyads, d1(xl, yi) and d2 (x2, y2)
of an s, is a segment d\d2 or d2d1; d1 and dz are its ends;
all dyads between the ends are the segment's dyads.
Postulate (ii).-Let us notice, in the first place, that,
taken two at a time, three dyads, dl(xl, yl), d2(x2, y2),
d3(x3, y3), nôt belonging to a same system, determine three
systems, si, J a, as follows:
x-x2 Y-y2
x -X3 y -Y3
x-x3 y-y1
S2: ` - = ^,
X-Xi y-Yi
X-Xi y-yi
S3 ~= =X3;
X-X2 y-y2
it is plain that there is but one restriction on the X's,
namely, Xi I, X2 # I, X3 - I; for, except for the inequalities, the given dyads would not be distinct. Looking at
jl, for example, you see that, when the variable dyad
d(x, y) is between d2 and da (i.e., when it belongs to the
segment d2d3), Xi is negative; and that, if Xi is negative
(and neither zéro nor oo), d is in the segment d2d3.
Clearly the same statetnent, mutatis mutandis, is valid for
X2 and X3.