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

106

NON-EUCLDEAN GEOMETRY

[CH. IV.

We proceed to obtain this relation:
Consider the function f(oc.) defined b)y the equation
tanh a = cosf(u).
and let us write a=A(A().
When uc j, a =0,tanha=0,csf(o)==0; i.e.!
When =, 0a = co, tanh a = 1, cosf(o) = 1; i.e. f(O) =0.
Further, as a increases from 0 to cc, fzx) diminishes con
tinuously from Z to 0.
2
Next consider a triangle ABC-not right-angled-and let
the perpendicular from B cut the base AC at D. Let the
elements of the triangle ABD be denoted by AB=c, BD=a,
DA=b, L ABD= I, / BAD=X. Also let the elements of th6s
triangle BDC be denoted by BC=c,, CD=b1, DB=a,,
LBCD=X1, LDBC=Ma.
As the side BD is common, a = a,.
B
A b D b, C
FIG. 76.
Then, from the Cosine Formula, ~ 60, we have
tanh cn = cosh c cosh c1 - cosh (b + bl)
sinh c sinh c1
* The method of this and the preceding sections is due to Liebmann,
"Elementare Ableitung der nichteuklidischen Trigonometrie," Ber.
d. k. siichs. Ges. d. Wiss. Math. Phys. Kiasse, vol. lix. p. 187 (1907), and
Nichteulelidische Geometrie, 2nd ed. p. 71. Another method, also independent of the geometry of space, is to be found in Gerard's work, and
in the paper by Young referred to below, p. 136.