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

94 MATHEMATICAL PHILOSOPHY
contains a d of segment d2d3 or segment dids. Hence,
you see, our postulate is verified.
Postulate (I3).-That this postulate of parallels is
verified in our new interpretation may be quickly seen as
follows. Let Ax +By+C=O be any given system s, and
let d(x', y') be any dyad not belonging to s. Then any
system s' containing d is A'(x-x')+B'(y-y') =0, or
A'x+By'+C' = where C' =-(A'x' +B'y'). Solving s
and s' for x and y, we get
C B A C
C' B' IA' Ci
x I A B ' A B j
A B' A' B'
the two terms of neither fraction can be zero, for, if they
A B C
were, then -'= B= C and s and s' would coincide,
Al Y C'
contrary to hypothesis; hence x and y have definite
finite values and accordingly s and s' have a common
dyad (x,y), except when the denominator is zero, but
A' A
this can happen when and only when B-=, and hence
there is one and only one s' having no dyad in common
with s, this unique s' being parallel to s. And, as you
see, the postulate is satisfied.
Before examining postulate (I4) we require a
DEFINITION.-If d'(x', y') be a given dyad of a system
s, any dyad d(x, y) will be said to be on the one side or on the
opposite side of d' according as x> x' or <x', except when s
is of the form Ax +C=0 and then the distinction of sides
will depend on whether y> y' or y <y'.
Postulate (14).-Let di(xi, yi) and d2(x2, y2) be any
segment did2; let d'(x',y') be any given dyad of any