A history of Greek mathematics - Wilbourhall.org

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MENELAUS'S SPHAERICA 271
III. 9 and III. 10 show, 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 Menelaus's
own work. Props. 11-14 amount to theorems in
spherical trigonometry such as the following.
Given arcs a 1}
a 2
, 0f 3
, a 4
, fi x , p
2
, /3 3 , /2 4
,
such that
90° ^oc x
> a 2
>a 3
>a 4 ,
90° >t3 1 >t3 2
>(3 3 >t3 4 ,
and also oc l
>/3 1
, a 2
>/3 2
, a 3 >/3 3
,
a 4
>/? 4
,
then
(1) If sin a x
: sin oc 2
: sin a 3
: sin cx 4
= sin /? x
: sin /? 2
: sin /3 3
: sin/? 4 ,
*=*>%=£'.
(2) If
siD ("i + ft) _ shi(a 2 + ft2 ) _ 8in(« 3 + /8 3 )
sin (o^-ft) " sin (a 2
-0 2 )
"" sin(a3 -/? 3 )
sin fa 4 + ft4 )
'
sin(a 4
-/3 4
)'
then
•
^
? <
a i~ a 2
a3-«4
^ ft-ft
P3-P4
If
sin (o^-o^)
< sin _(ft-£ 2 )
sin (a 3
— a 4
) sin (£3 - /2 4 )
then
*^
ol 2
>ol 33 @ 1 >P 2 >P B
,
90°>y 1 >y 2 >y 3
,
and sin (otj — y x ) : sin (a 2 — y 2 ) : sin (a 3—
= sin (ft
- y T
) : sin (0 2 - y 2 ) : sin (03 - y3 )
3)
= siny 1
:siny 2
:siny 3