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

172

NON-EUCLIDEAN GEOMETRY

[CH. VIII.

Then A, A', B, B' are concyclic, and the circle which passes
through them cuts the fundamental circle diametrally
(Fig. 116). There is one, and only one, circle cutting the
fundamental circle diametrally, which passes through two
different points within the fundamental circle.
AFIG. 11
FIG. 116.
In discussing the properties of the family of circles cutting
the fundamental circle diametrally, two methods can be
followed. We can restrict the nominal points of the geometry
to the points within and upon the fundamental circle. In
this case we regard the points on the circumference at the
extremities of a diameter as one and the same nominal point.
In the other case, we extend the field of nominal points outside
the circle to infinity, and the points on the circumference do
not require special treatment.
These two alternatives, we shall see below, correspond to
the two forms of the Elliptic Geometry, in one of which every
straight line intersects every other straight line in one point,
while in the other form, straight lines have always two points
of intersection. The nominal lines are the circles which cut
the fundamental circle diametrally.
When the field of nominal points is restricted to points within
or upon the fundamental circle, any two different nominal
points A, B determine a nominal line AB. Also any two nominal
lines must intersect at a single nominal point.