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

NON-GEOMETRIC INTERPRETATION 99
(I) Ax +By +C =O into any given system (2) A'x +B'y +
C'==0 and, at the same time, any given dyad of the
former into any given dyad of the latter; to show this
possibility, transform (I) as above indicated, then equate
the two ratios of the coefficients in the resulting equation
to the corresponding ratios taken from (2); these two
equations (two conditions on a, b and 0) insure that
(I) has (2) for its transform; but our three parameters
can satisfy a third condition; notice what it is; let
di(xi, yi) be the given dyad of (I), and dl'(xi', yl') the
A C A' C'
given dyad of (2); then yi = - x- and y'=-x '-B-;
di is to be converted into di' and this gives the third condition, which is that xi=xi' cos 0+ Bx '+ ) sin O+a
or an equivalent one obtained from the second equation
of (t). The writing out of the three conditions and
solving them for a, b and 0 involves a little finger work
but no logical difficulty. You may wish to perform the
task as an exercise. Again, any one of our dyad-to-dyad
transformations converts any given segment into a
congruent segment. I say " any one of our dyad-to-dyad
transformations," for we have many, infinitely many,
of them, depending on the values we assign to the parameters a, b and 0. To prove the property in question
let the segment be determined by d,(x,, y,) and d2(x2, y2);
in (xi -x2)2+(yl -y2)2 replace Xi, yi, X2, y2 by their
transforms xi cos 0-y, sin O+a, xi sin O+y, cos o+b,
X2 cos O-y2 sin 0+a, x2 sin 6+y2 cos 0+b, simplify and
then note that the radical expression has suffered no
change. Finally, any one of our transformations leaves
the order of dyads unchanged; that is, if di, d2 and ds are
converted respectively into di', d2' and da', then, if dg