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

162

NON-EUCLIDEAN GEOMETRY [OH. VIII.

This property is what is assumed in the Parallel Postulate
on which the Hyperbolic Geometry is based. We are therefore
led to define parallel nominal lines in the plane geometry we
are investigating as follows:
The nominal lines through a nominal point parallel to a
nominal line are the two circles of the system passing through
the given point which touch the circle with which the given nominal
line coincides at the points where it cuts the fundamental circle.
Thus we have in this geometry two parallels-a right-handed
parallel and a left-handed parallel-and these separate the
lines of the pencil which intersect the given line from those
which do not intersect it.
~ 98. At this stage we can say that any of the theorems of
the Hyperbolic Geometry which involve only angle properties
will hold in the geometry of the circles, and vice versa. Those
involving metrical properties of lines we cannot discuss until
the nominal length of a nominal segment has been defined.
Fra. 109.
For example, it is obvious that there are nominal triangles
whose angles are all zero (Fig. 109). The sides of these triangles
are parallel in pairs, and we regard parallel lines as containing
an angle zero.
Further, we can prove that the sum of the angles in any nominal
triangle is less than two right angles, by inversion, as follows:
Let C1, C2, C3, be three circles of the system-i.e. three
nominal lines forming a nominal triangle, say PQR. We
suppose these circles completed, and we deal with the whole