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

84 MATHEMATICAL PHILOSOPHY
center C, then two points, P and P', on a line through C,
are inverses of each other provided the distance CP times
the distance CP' =r2. You easily see that to each point
there corresponds one and but one point, except that C
has no correspondent (in the Euclidean space). To annul
the exception we assume one " ideal " point, or point at
infinity, to serve as the transform of the inversion center
C. The new space'thus obtained is called inversion space.
The lines and planes through C are transformed into themselves. All lines and planes not through C are converted
respectively into circles and spheres through C.
And now for the field of our new interpretation. You
probably guess what it is to be. Let S be an inversion
space; O a chosen point in it. The ensemble of all the
spheres (including planes as spheres of infinite radius)
that go through O may be called the O-cluster of spheres.
Now remove the point O from S; the cluster is now the
O-cluster of pathospheres; and the cluster of circles bereft
of O will be called the O-cluster of pathocircles. Our
field,-let us denote it by K',-is composed of the points
(except O) of S, the pathospheres and pathocircles of the
O-clusters.
I need hardly say,-for you doubtless foresee,-that
our new interpretation of H/lF' springs from agreeing that
vi shall mean a point of K'; V2 shall mean a pathocircle;
V3 shall mean a pathosphere; Ri shall mean between in
the sense explained for the field K; and R2 shall mean
congruent in the sense that the transforms of segments or
angles congruent in the familiar sense of D1' shall be
congruent in the new sense.
Call the new doctrine thus arising D3'. It evidently
is a three-dimensional geometry of the points, pathocircles and pathospheres of the field K' and matches