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

096^ MATHEMATICAL PHILOSOPHY
to bé y m'x+b'; then, since d1d2 and d1'd2' are congruett I+m2(xl-x2) = v/I+m'2(xi'-x2'); so, too,
/I +m2(X2 -X3) = /i -m'2( 2' —x3'); whence, by addition, vi+m2(x -X3) = V/I -m'2(xl'-X3'); but this last
equation tells us that d1d3 and di'd3' are congruent; and
so, we see, the postulate is verified.
Before attacking postulate (I7) let us make due
preparation for it in the way of a simple theorerti, some
definitions and a little acquaintance with a very ftndamental kind of algebraic transformation.
Theorem.-Every system s separates the remaining
dyads of N into two classes such that, if dl and d2 be any
two dyads the segment d1d2 contains or does not contain
a dyad of s according as the given dyads belong, one of
them to the one class and one of them to the other, or both
of the dyads belong to the same class. [The theorem is
the correspondent of Hilbert's theorem 5.}
The proof is not difficult. The given system s is of the
form (I) x=k or of the form (2) y==mx+b. If s be of
fôrm (I), it is clear that the classes required are respectively composed of dyads for which x> k and of those for
which x<k. Next suppose s to be y=mx+b. Let
di(x, yi) anid d2(x2, y2) be any two given dyads not
belonging to s. It is plain that there is a system si,
y =mx+bi, containing dl, and a system s2, y=mx+b2,
containing d2; so that yi=mxi+bi and y2=mx2+b2.
The dyads di and d2 determine a system s', namely,
x-xi y-yi
x-X2 y-y2
now let d(x, y) be the dyad common to s and s'; then
x-xi m(x-xi) +b-b
X-X2 m(x-X2)+b-b2