The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

156 NON-EUCLIDEAN GEOMETRY [CIo. vm.
proving the Parallel Postulate and of the logical consistency
both of the Hyperbolic and Elliptic Geometries. In this
discussion the " dictionary method " of ~ 29 will be more fully
explained.
We shall consider three families of circles in a planeextending the argument to spheres later. These are the
family of circles passing through a fixed point; the family
of circles cutting a fixed circle orthogonally; and the family
of circles cutting a fixed circle diametrally (i.e. the common
chord of the fixed circle and any of the variable circles is to
be a diameter of the fixed circle). Denoting the fixed point
by 0, and taking the fixed circle as a circle with centre O and
radius k, the first family of circles has power zero with regard
to 0; the second, power k2; and the third, powzer -k2. We
shall see that the geometries of these three families of circles
agree with the Euclidean, Hyperbolic, and Elliptic Geometries,
respectively.
~ 94. The System of Circles through a Fixed Point.
If we invert from a point 0 the lines lying in a plane through
O we obtain a set of circles passing through that point. To
every circle there corresponds a straight line, and to every
straight line a circle. The circles intersect at the same angles
as the corresponding lines. The properties of the family of
circles could be deduced from the properties of the set of lines,
and every proposition concerning points and lines in the one
system could be interpreted as a proposition concerning points
and circles in the other.
There is another method of dealing with the geometry of
this family of circles. We shall describe it briefly, as it will
make the argument in the case of the other families, which
represent the Non-Euclidean Geometries, easier.
If two points A and B are given, these, with the point 0,
fully determine a circle passing through the point O. We
shall call these circles nominal lines.* We shall refer to the
points in the plane of the circles as nominal points, the point O
being supposed excluded from the domain of the nominal
* In another place, cf. Bonola, loc. cit., English translation, Appendix
V., and Proc. Edin. Math. Soc., Vol. 28, p. 95 (1910), I have used the
terms ideal points, ideal lines, etc. For these I now substitute nominal
points, nominal lines, etc., owing to possible confusion with the ideal
points, ideal lines, etc., of ~~ 37, 38.