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

ANOTHER GEOMETRIC INTERPRETATION 81
tion, explained a little while ago. Let us suppose our
field K to be laid down upon a Euclidean plane w. Remember that O is absent from K but that, below the vacant
position, r has a point, which we may call 0'. In K take
a definite circle I for inversion circle having O for center.
Regard the transformation as converting the points of
K (or r) into the points of r (or K), noting that O' of 7
and the " ideal " point of K are each the other's transform; that the lines of 7 are converted into the pathocircles of K, and conversely; and that, if, in 7r, a point
B is between A and C on a line, then in K the transform
of B is between the transforms of A and C on a pathocircle, the transform of the line. You see that there is
thus established a one-to-one correspondence between the
points and lines of wr and the points and pathocircles of
K, in such a way that all postulated relations among the
elements of 7 hold equally among the corresponding
elements of K.
Though logically superfluous, it will be instructive to
illustrate the matter a little further by simple figures.
In Fig. I8, I is the inversion circle; a is a line in 7r; pathocircle a' is the transform of a; points A', B', C' are the
transforms of A, B, C; segments AB and BC are congruent in the familiar sense-in doctrine D1; their
transforms A'B' and B'C' are congruent in the new sense.
You see that the postulate of Archimedes, postulate (20),
is verified; for as congruent segments stretch upward in
endless succession along a, their congruent transforms
proceed on a' in endless succession towards 0, never
reachirig this vacant point-position.
Fig. I9 illustrates congruence of triangles in the new
interpretation. Triangles ABC and AlB1C1 are congruent in r-in D1; their transforms,-the new triangles