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

148 NON-EUCLIDEAN GEOMETRY [CH. VI'.
From ~ 83, we have
b'Q.
9(Q KL
Now, when Bb tends to zero, LQ and Kb' tend to BA, and
from ~ 84, O(LQ) and O(Kb') tend to /(AB).
b'Q
Thus, Lt KL b(AB).......... (iv)
In the same way we have
L t (B C )................~......... ~ ~~~ ~~~ \
Bb/
There remains the limit of '
MN
Let s be the point at which Bc" meets AC.
We know from ~ 81 (I.), that s lies between C and c, and
we have Bs> BC.
Then, since Cb' = Be", we have Bb' > sc".
Bb' SC//
Therefore B c (Nc").
MN Mn> >
Produce BC till it meets a"c" in R.
We have BR> Bc", so that BR> cb'.
From BR cut off Bc' =cb'.
Draw c'P perpendicular to MN.
Then we have
BYBY Cc'
Bb B C,'< p(PC') < 0 (cm - PM - Cce).
MN MP MP
Thus S(CM - PM Bcc Y > > 4(Nc").
MN
Proceeding to the limit,
~lar\-Tt fBb'\
p(AC) Lt (...........).... (vi)
From (iii)-(vi), it follows that / (AB) q(BC) 4 (CA), or with
the usual notation from ~ 86,
C a b
COS = COS -COS-c
k k k'