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

24, 25, 26] SOME THEOREMS ON PARALLELS

47

Then it must cut (1), and on being produced must also cut
(3).
Since this holds for every line such as AD, (2) is parallel to
(3).
Case II. Let the line (1) be outside both (2) and (3), and
let (2) lie between (1) and (3). (Fig. 20.)
2
FIo. 20.
If (2) is not parallel to (3), through any point chosen at
random upon (3), a line different from (3) can be drawn which
is parallel to (2).
This, by Case I., is also parallel to (1), which is absurd.*
~26. We shall now consider the properties of the figure
[cf. Fig. 21] formed by two parallel rays through two given
points and the segment of which these two points are the ends.
A
FIG. 21.
It is convenient to speak of two parallel lines as meeting at
infinity. In the Hyperbolic Geometry each straight line will
have two points at infinity, one for each direction of parallelism.
With this notation the parallels through A, B may be said to
meet at Q2, the common point at infinity on these lines.
* The proof in the text is due to Gauss, and is taken from Bonola,
loc. cit. p. 72.