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

ANOTHER GEOMETRIC INTERPRETATION 77
What is the transform, or inverse, of a line not through
the center? Let Ax+By+C=O be such a line; replace
the coordinates (x, y) of any point in it by their values
taken from (I), simplify the result and (if you like) drop
the primes; we thus get
d B
(3) x2+y2+r2X+ry =0.
FIG. I3.
This, you note, is a circle through the inversion center,
which is here the origin, for the coordinates (0, 0) of the
origin satisfy the equation. Hence every line not through
the center has for its transform, or inverse, a circle through
the inversion center.
With these simple ideas held in reserve for use as we
need them, let us proceed to our third geometric interpretation. It will be advantageous to deal first with HAF.
Denote by wr an inversion plane. Let O be a chosen point
of 7r. The ensemble of all circles through 0 is called a