MENELAUS’S SFHÂERICA 
271 
Mene- 
lelaus’s 
tor the 
mietry, 
3m the 
ly con- 
lelaus’s 
icluded 
) much 
letrical 
1) of 
is were 
eorems 
¡.s, 1 and 
ssumed 
3gy in 
herical 
BG in 
n AC; 
irnally 
D and 
drawn 
dangle 
III. 9 and III. 10 sh^vv, for a spherical triangle, that (1) the 
great circles bisecting the three angles, (2) the great circles 
through the angular points meeting the opposite sides at 
right angles meet in a point. 
The remaining propositions, III. 11-15, return to the same 
sort of astronomical problem as those dealt with in Euclid’s 
Phaenomena, Theodosius’s Sphaerica and Book II of Mene- 
laus’s own work. Props. 11-14 amount to theorems in 
spherical trigonometry such as the following. 
Given arcs a x , a 2 , a 3 , a 4 , /3 l , (3 2 , (3 3 , /3 4 , such that 
90°^ a x > a 2 > a 3 > cx 4 , 
90°>/3 i >/3 2 > /?3>0 4 , 
and also oc 1 >(3 1 , a 2 >/3 2 , a 3 >(3 3 , a 4 >/3 4 , 
(1) If sin a x : sin a 2 
then 
sin a 3 : sin a 4 = sin ß x ; sin ß 2 : sin ß 3 : sin /3 4 , 
^ ßi 
Äg a 4 ß 3 ß i 
sin (oh+ft) _ sin (« 2 + ß 2 ) _ sin (« 3 + ß 3 ) 
sin (a 1 - ßj sin (a 2 - ß 2 ) ~ sin (a 3 - ß 3 ) 
_ sín («4 + ßj) 
~ sin (oc 4 — /3 4 ) ’ 
then 
(3) If 
then 
^2 ßi ßi m 
«s-«* ßz~ß± 
sinf^-QCg) sin iß x — ß^ 
sin (a s — a 4 ) sin (ß 3 - ß A ) 
^2 ^ ß\ ßi 
a 3 a 4 ß‘i ßi 
Again, given three series of three ares such that 
a 1 >a 2 >a 3 , ß l >ß 2 >ß 3 , 90° > y x > y 2 > y 3 , 
and sin («J - y x ) : sin (a 2 — y 2 ) : sin (a 3 —y 3 ) 
= sin (ft - y x ) : sin (ft - y 2 ) : sin {ß 3 - y 3 ) 
pus as 
ry of 
ar to 
= sin y x : sin y 2 : sin y 3