MENELAUS’S SP HAFRICA 
369 
It follows at once (Prop. 4) that, if AM, A'M' are great 
circles drawn perpendicular to the bases BO, B'G' of two 
spherical triangles ABO, A'B'O' in which B = B', 0 = O', 
sin BM _ sin MO 
sin B'M' ~ sin M'C 
^since both are equal to 
inn AM \ 
tan A'M') 
III. 5 proves that, if there are two spherical triangles ABO, 
A'B'O' right-angled at A, A' and such that 0=0', while h 
and h' are less than 90°, 
sin (a + h) _ sin (a' + h') 
sin (a — h) sin (a! — h') 
from which we may deduce 1 the formula 
sin (a + h) 1 + cos 0 
sin (a — h)~ 1 — cos 0 
which is equivalent to tan h = tan a cos C. 
(y) Anharmonic property of four great circles through- 
one point. 
But more important than the above result is the fact that 
the proof assumes as known the anhar 
monic property of four great circles 
drawn from a point on a sphere in rela 
tion to any great circle intersecting them 
all, viz. that, if ABCD, A'B'O'D' be two 
transversals, 
sin Al) sin BO sin A'D' sin B'G' 
sin DO sin AB sin D'G' sin A'B' 
1 Braunmiihl, op. cit, i, p. 18; Bjornbo, p. 96.