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

80

NON-EUCLIDEAN GEOMETRY

[-CH. III.

Therefore L 2PR + RP =two right angles, which is impossible, since PR would make equal alternate angles with P2
and RQ2, and these two parallels would have a common perpendicular.
3. If P corresponds to Q on the parallels (i) and (ii), and Q
to R on the parallels (ii) and (iii), then P corresponds to R on the
parallels (i) and (iii).
P
R
FIG. 55.
This follows from the concurrence of the perpendicular
bisectors of the sides of a triangle (~ 39).
The perpendicular bisector of PQ is parallel to the given lines;
the same holds of the perpendicular bisector of QR.
It follows that the line bisecting PR at right angles is parallel
to the other two bisectors, and to (i) and (iii).
Therefore P and R correspond.
~ 48. The Limiting-Curve or Horocycle.*
We now come to one of the most important curves in the
Hyperbolic Geometry.
The locus of the corresponding points on a pencil of parallel
lines is a curve called the Limiting-Curve or Horocycle.
It is clear that this is the circle of infinite radius, and from
~ 47 (2) it follows that it is not a straight line.
* Lobatschewsky uses the terms grenzkreis, courbe-limite, and horicycle; Bolyai speaks of the linea-L.