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

98 MATHEMATICAL PHILOSOPHY
The use of those terms here is purely metaphorical and,
had we desired to dispense with their use, it would not
have been difficult, as you no doubt see, to do so.]
Let me now explain briefly a simple but exceedingly
important algebraic transformation which will be very
helpful in dealing with postulate (I7). Consider the pair
of equations
(t) { x =x' cos -y' sin O+a,
( y = x' sin +y' cos +b;
solving these for x' and y', we get the pair
() i x' =x cos ++y sin 6-a cos 8-b sin 0,
(') y'= -x sin e+y cos e-b cos e+a sin e;
you notice that (t) and (t') are equivalent, either pair
being obtainable from the other. Either pair, say (t)
defines a dyad-to-dyad transformation; that is, given a
dyad (x', y'), there corresponds to it, by virtue of (t)
a definite dyad (x, y), and conversely. Of two dyads
thus related, we say that each is the other's transform
or that each is converted or transformed into the other.
I wish to call your attention to four further properties
of the transformation. One of them is that the dyads
of a system are converted into the dyads of a system. To
prove this proposition, take any system Ax+By+C
=0, replace x and y by their values from (t), simplify,
and note that you then have the equation of a system.
Thus a dyad-to-dyad transformation is also a system-tosystem transformation: the transform of a system is a
system. Another property of the transformation, showing its power, is that, owing to the presence of three
undetermined quantities, or parameters as they are called
-a, b and O-, we can convert any given system