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

103]

THIRD SYSTEM OF CIRCLES

173

When the domain of the nominal points is both within and
without the fundamental circle, two nominal points do not
always determine uniquely a nominal line. If the points
A and B are upon the circumference of the circle at opposite
ends of a diameter, a pencil of nominal lines passes through
A and B. Again, if the points A and B lie on a line through
O and OA. OB= -k2; the same remark holds true.
Further, with the same choice of nominal points; every
nominal line intersects every other nominal line in two nominal
points.
The simplest way of discussing the properties of the system
of circles with which we are dealing, is to make use of the
fact that they can be obtained by projecting the great circles
of a sphere stereographically from a point on the surface of
the sphere on the tangent plane at the point diametrally
opposite. If the centre of projection is a pole of the sphere,
the equator projects into the fundamental circle, and one
hemisphere projects into points outside this circle, the other
into points within it. This projection is a conformal one,
and the angle at which two great circles intersect is the same
as the angle at which the corresponding circles in the plane
cut each other.
We define the angle between two nominal lines as the angle
between the circles with which they coincide.
We are now able to prove some of the theorems of this
Nominal Geometry.
Since all the great circles perpendicular to a given great
circle intersect at the poles of that circle, it follows that all
the nominal lines perpendicular to a given nominal line intersect
at one point, in the case when the nominal points are within or
upon the circumference of the fundamental circle; in two points,
when this field is both within and without. (Cf. ~~ 75-77.)
The point of intersection is spoken of as a pole, or the pole,
of the line.
Again, in a right-angled spherical triangle ABC, in which
C is the right angle, the angle at A - a right angle, according
as the pole of AC lies on CB produced, or coincides with B,
or lies between C and B.
When translated into the language of the nominal geometry,
we have the theorem which corresponds to ~ 78 (1).
Further, the sum of the angles of a spherical triangle is
greater than two right angles. It follows, since the projection