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ON THE NON-EUCLIDIAN GEOMETRY.
[528
The distance BC, or say a, of any two points B, G is by definition as follows
Radius of circle = 1:
In A ABC, sides are a, b, c:
angles „ A, B, C:
OA , OB , OC are = sin p, sin q, sin r :
OA' , OB' , OG' „ „ sin a, sin b, sin c:
$B0C, CO A, AOB „ „ a, ¡3, y.
_ . . BI.GJ
a ~ 2 ° g BJf. Cl ’
(where I, J are the intersections of the line BC with the circle); that is,
- - 0 /BI.CJ, /BJ.CI
s -(- 6 j oi .j cosh cl BJ Cl BI CJ ’
where the numerator is
BI .CJ+BJ.CI
•Jbi . BJ V CI. CJ ’
BI (BJ - BC) + CI (BC + CJ), = BI. BJ +CI.CJ+BC(CI-BI),
= BI.BJ+CI. CJ+BCi
Hence taking a for the distance BC, and sing, sinr, for the distances OB, OC respec
tively, we have BI. BJ = cos 2 q, CI. CJ = cos 2 r; and the formula is
, _ cos 2 q -f cos 2 r + a 2
cosh a = ¿r 3 —- ,
l cos q cos r
or, what is the same thing, taking a for the angle BOC, and therefore
a 2 = sin 2 q + sin 2 r — 2 sin q sin r cos a,
we have
, _ 1 — sin q sin r cos a
cosh a = 1 .
cos q cos r
In a similar manner, if sin a is the perpendicular distance from 0 on the line BC
(that is, a sin a = sin q sin r sin a) it can be shown that
. , _ a cos a
smh a ,
cos qcos r
the equivalence of the two formulae appearing from the identity
cos 2 q cos 2 r = (1 — sin q sin r cos a) 2 — a 2 + a? sin 2 a,
which is at once verified.
Next for an angle; we have by definition
j _ 1 , sin BAI. sin CAJ
21 8 sin CAI. sin BAJ’