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

39, 40, 41] THE PARALLEL CONSTRUCTIONS

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Therefore they are parallel to one another in the same sense
and intersect in an improper point-a point at infinity.
If we take these three cases together, it will be seen that the
theorem is established.
~40. The Parallel Constructions.
In Hilbert's Parallel Axiom the assumption is made that
from any point outside any straight line two parallels can
always be drawn to the line. In other words, it is assumed that
to any segment p there corresponds an angle of parallelism
1(p).
The fundamental problems of construction with regard to
parallels are the following:
1. To draw the parallel to a given straight line from a given
point towards one end.
2. To draw a straight line which shall be parallel to one
given straight line, and perpendicular to another given straight
line which intersects the former.
In other words:
1. Given p, to find HI(p).
2. Given II(p), to find p.
For both of these problems Bolyai gave solutions; and
one was discussed by Lobatschewsky. In both cases the
argument, in one form or other, makes use of the Principle of
Continuity.
In the treatment followed in this book the Hyperbolic
Geometry is being built up independently of the Principle of
Continuity. For that reason neither Bolyai's argument
(Appendix, ~~ 34, 35), nor Lobatschewsky's discussion * of the
second problem, will be inserted.
~41. To draw the Parallel to a given Line from
a Point outside it. Bolyai's Classical Construction
(Appendix, ~34).
To draw the parallel to the stragqht line AN from a given point D,
Bolyai proceeds as follows:
Draw the perpendiculars DB and EA to AN (Fig. 48), and the
perpendicular DE to the line AE.
* Cf. Lobatschewsky, Geometrische Untersuchungen zur Theorie der
Parallelinien, ~ 23 (Halsted's translation, p. 135). Also New Principles
of Geometry, ~ 102 (Engel's translation).