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

170 NON-EUCLIDEAN GEOMETRY [cH. vII.
(ii) the limiting-curve, with its centre at infinity, or at a point
where two parallels meet; and (iii) the equidistant-curve,
with its centre at the ideal * point of intersection of two lines
which have a common perpendicular.
All these curves are ordinary circles, but they do not
belong to the system of circles orthogonal to the fundamental
circle.
As to the first, the nominal lines through a point A are all
cut orthogonally by the circles of the coaxal system with A
and its inverse point A' as Limiting Points. Thus these
circles are the circles of this nominal geometry with A as their
centre. They would be traced out by the end of a nominal
segment through A, when it is reflected in the nominal lines
of the pencil.
As to the second, the circles which touch the fundamental
circle at a point U cut all the circles of the system which
pass through U orthogonally. They are orthogonal to the
pencil of parallel nominal lines meeting at infinity in U.
Thus these circles are the circles of this nominal geometry
with their centre at the point at infinity common to a pencil
of parallel nominal lines. They would be obtained when the
reflection takes place in the lines of this pencil.
As to the third, all circles through U, V cut all the nominal
lines perpendicular to the line AB (cf. Fig. 111) orthogonally.
Thus these circles are the circles of the nominal geometry with
their centre at the ideal point common to this pencil of notintersecting nominal lines. They would be obtained when
the reflection takes place in the lines of this pencil.
These three circles correspond to the ordinary circle, the
Limiting-Curve and the Equidistant-Curve of the Hyperbolic
Geometry.
~ 102. The Impossibility of proving Euclid's Parallel
Postulate.
We can now assert that it is impossible for any inconsistency
to exist in this Hyperbolic Geometry. If such a contradiction
entered into this plane geometry, it would also occur in the
interpretation of the result in the nominal geometry. Thus
a contradiction would also be found in the Euclidean Geometry.
We can therefore state that it is impossible that any logical
*Cf. ~37.