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

76, 77] THE TWO ELLIPTIC GEOMETRIES

131

region of the plane, I. 16 does hold, and theorems dependent
upon it are true in such a region.
The plane of this geometry has properties completely analogous to those possessed by the surface of a sphere. The great
circles of the sphere correspond to the straight lines of the
plane. Like the line, they are endless. Any two points on
the surface of the sphere determine a great circle, provided the
points are not the opposite ends of a diameter. The great
circles through any point on the sphere intersect all other
great circles.
We shall find that this analogy can be carried further. The
sum of the angles of a spherical triangle is greater than two
right angles. The sum of the angles of a triangle in this plane
is greater than two right angles. The Spherical Excess
measures the area of spherical triangles. With suitable units
the area of plane triangles is equal to their excess. Indeed
the formulae of this Plane Trigonometry, as we shall show
later, are identical with the formulae of ordinary Spherical
Trigonometry.*
~ 77. It must be remarked, however, that in the argument
of ~ 76 it is assumed that the point 01 is a different point from
0. If the two points coincide, the plane of this geometry has
a wholly different character. The length of a straight line is
now 2V instead of 4S. If two points P, Q are given on the
plane, and any arbitrary straight line, we can pass from P to
Q by a path which does not leave the plane, and does not cut
the line. In other words, the plane is not divided by its lines
into two parts.
The essential difference between the two planes is that in
the one the plane has the character of a two-sided surface, and
in the other it has the character of a one-sided surface.t The
first plane-that which we have been examining-is usually
called the spherical plane (or double elliptic plane); the second
plane is usually called the elliptic (or single elliptic) plane.
The geometries which can be developed on both of these
planes are referred to as Riemann's (Non-Euclidean) Geometries. It seems probable that the Spherical Plane was the only
* Spherical Geometry can be built up independently of the Parallel
Postulate, so it is not necessary to say ordinary Spherical Trigonometry
when referring to it.
t Cf. Bonola, loc. cit. ~ 75.