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

LECTURE V
Another Geometric Interpretation
BRIEF INTRODUCTION TO THE METHOD OF DESCARTES AND
FERMAT-INVERSION GEOMETRY AND INVERSION
TRANSFORMATION-THE INFINITE REGION OF INVERSION SPACE A POINT-BUNDLES OF CIRCLES AND
CLUSTERS OF SPHERES-PATHOCIRCLES AND PATHOSPHERES-ONE-TO-ONE CORRELATION.
IN presenting a third interpretation of our two doctrinal functions, it will be convenient to borrow a few
ideas from Cartesian Analytical Geometry and Inversion
Geometry. It will be advantageous to explain them in
advance.
The perpendicular lines OX and OY, Fig. 9, are called
coordinate axes; O is the origin of distances, which, if
measured upward or rightward, are regarded positive, but,
if downward or leftward, negative. I am supposing the
figure to be in a Euclidean plane. Choose some unit of
length; then any point has a pair of numbers (x, y),
P's distances from the axes and called its coordinates.
Conversely, to any such a pair belongs a point. Let (I),
Fig. Io, be any line through 0; then (2), parallel to (I),
is any line of the plane. Let P(x, y) be any point of (I);
let m =tan 0; then y=mx; this equation is the equation
of (I); it is so called because to any pair (x, y) satisfying
it belongs a P of (I) and any P of (I) has a pair satisfying
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