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

74

NON-EUCLIDEAN GEOMETRY

[CH. III.

These two bisectors have therefore an ideal point in common,
and the perpendicular bisector of SD must pass through the
same ideal point (cf. ~ 39).; i.e. it must also be perpendicular
to M2f'.
Suppose the parallel M'2' drawn through M' to A2'.
The bisector of the angle 2'M'f is perpendicular to 22',
and therefore to SD.
It follows that M'S bisects the angle f"M'12'.
But M'f" and M'2' are the parallels from M' to f"AQ'.
Therefore M'S is perpendicular to f2"A'.
And AS was made equal to FD in our construction.
The result to which we are brought can be put in the following words: Let the perpendicular AF be drawn from the point
A to the given line a (Ff2), and let the perpendicular Af2' be
drawn at A to AF. From any point D on the ray Ff2 drop the
perpendicular D B to A2'. This line D B cuts off from the
parallel Af2 a length equal to FD.
The parallel construction follows immediately. We need only
describe the arc of a circle of radius FD with A as centre. The
parallel A2 is got by joining A to the point at which this arc
cuts DB.
The existence of the parallel, given by Hilbert's Axiom,
allows us to state that the arc will cut the line once between
B and D, without invoking the Principle of Continuity.*
~ 44. Construction of a Common Parallel to two given
Intersecting Straight Lines.t
Let Of2 and O2' be the two rays a and b meeting at O and
containing an angle less than two right angles.
From these rays cut off any two equal segments OA and OB.
From A draw the parallel A2' to the ray 0f2', and from B
the parallel B2 to the ray O12.
Bisect the angles f2Af2' and Q2B2' by the rays a' and b'.
By ~ 26 (4), we know that
L OAf2'= LO BO2.
* In Euclid's Elements the fundamental problems of construction of
Book I. can be solved without the use of Postulate 3,: "To describe a
circle with any centre and distance." To draw the parallel from a
given point to a given line can be reduced to one of the problems of
~ 3. On the other hand, in the Hyperbolic Geometry, the parallelconstruction requires this postulate as to the possibility of drawing
a circle.
t Cf. Hilbert, loc. cit. p. 163.