410 
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’