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

96

NON-EUCLIDEAN GEOMETRY

[CH. IV.

the tangent at Q is parallel to the axis through P, in the
opposite sense to that in which the axis is drawn (Fig. 68).
Q
4
f _
P
FIG. 68.
We shall for the present denote the length of this arc by S.*
Let B be a point on the Limiting-Curve through A, such that
the are AB is less than S (Fig. 69).
B
A, t
S-s
C,' G\ \ )n
C, C
FIG. 69.
It follows that the tangent at B must intersect the axis
through A.
Let it cut 2A in D, and let the segments AD and BD be u
and t. It is easy to show that u < t.
Produce the are BA to the point C, such that the are
BC =S.
On 2D produced take the point A,, such that DA = DB =t.
Then the perpendicular through A1 to the axis is parallel to
BD, and therefore to CQ2'.
Let the Limiting-Curve through A1 meet CO2' in C1.
Since the tangent at A1 is parallel to C2', the are A1C1 =S.
*Cf. p. 119.