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

82

NON-EUCLIDEAN GEOMETRY

[CH. III.

rays passing through its ends, as Q'R' does with the rays
through its ends.
We have thus shown that between the two Limiting-Curves
there is a one-one correspondence of the nature stated, and in
this case we say that the two curves are congruent.
Further, it is clear that it is immaterial at which line of
the pencil we begin our Limiting-Curve.
It is convenient to speak of the point at infinity, through
which all the parallel lines of the pencil pass, as the centre of
the Limiting-Curve; also to call the lines of the pencil the
axes of the curve. Concentric Limiting-Curves will be LimitingCurves with the same centre.
We can now state the following properties of these curves:
(a) The Limiting-Curve in the Hyperbolic Geometry corresponds to the circle with infinite radius in the Euclidean
Geometry.
(b) Any two Limiting-Curves are congruent with each other.
(c) In one and the same Limiting-Curve, or in any two
Limiting-Curves, equal chords subtend equal arcs, and equal
arcs subtend equal chords.
(d) The Limiting-Curve cuts all its axes at right angles, and
its curvature is the same at all its points.
~49. The Equidistant-Curve.
There remains the pencil of lines through an ideal point:
the set of lines all perpendicular to the same line.
1. If two given lines have a
common perpendicular, to any
point P on the one corresponds p -
one and only one point Q on the
other.
Let MN be the common perpendicular to the given lines,
and P any point on one of them.
From the other line cut off
NQ= MP, Q being on the same M N
side of the common perpen- FIG. 57.
dicular as P.
Then PM NQ is one of Saccheri's Quadrilaterals, and the angles
at P and Q are equal.