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

CHAPTER VI.

THE ELLIPTIC PLANE GEOMETRY.
~ 74. In Hilbert's Parallel Postulate, through any point A
outside any line b, two parallels a, and a2 can be drawn to the
line, and these separate the lines in the plane of the parallels
which cut b from the lines which do not cut it.
On the Euclidean Hypothesis, the two rays a, and a2
together form one and the same line, and there is but one
parallel to any line from a point outside it.
There is still another case to be examined, namely that in
which all the rays through A cut the line b. In this case there
is no parallel through a point outside a line to that line.
We shall see that this corresponds to the Hypothesis of the
Obtuse Angle of Saccheri, in accordance with which the sum of
the angles of a triangle exceeds two right angles. Saccheri and
Legendre were able to rule this case out as untrue; but their
argument depended upon the assumption that a straight line
was infinite in length. Riemann was the first to recognise that
a system of geometry compatible with the Hypothesis of the
Obtuse Angle became possible when, for the hypothesis that
the straight line is infinite, was substituted the more general
one that it is endless or unbounded. (Cf. ~~ 19, 20.)
The geometry built up on the assumption that a straight
line is unbounded, but not infinite, and that no parallel can be
drawn to a straight line from a point outside it will now be
treated in the same manner in which the Hyperbolic Geometry
was discussed.
~ 75. We proceed to the development of Plane Geometry
when the assumptions
(i) All straight lines intersect each other,
(ii) The straight line is not infinite,