27 2 
TRIGONOMETRY 
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(1) If 0f 1 >^ 1 >2y 1 , Oi 2 >/? 2 >2y 2 * cx 3 >/? 3 >2y 3 , 
then 
a 2 > ft 2 . 
a 3 Æ 2 -/V 
and 
(2j If /? 2 < cx 2 < y 2 , £ 3 •< oi 3 < y 3 , 
then 
a i-a 2 ftx-fto 
a 2 — a 3 /3 2 & 
III. 15, the last proposition, is in four parts. The first part 
is the proposition corresponding to Theodosius III. 11 above 
alluded to. Let BA, BC be two quadrants of great circles 
(in which we easily recognize the equator and the ecliptic), 
P the pole of the former, PA 1 , PA 3 quadrants of great circles 
meeting the other quadrants in .d^, A 3 and C 1 , C 3 respectively. 
Let B be the radius of the sphere, r, r x , r 3 the radii of the 
‘parallel circles’ (with pole P) through C, G 1 , C 3 respectively. 
Then shall 
sin A- i A 3 _ Er 
sin C l C 3 r x r 3 
In the triangles PGC 3 , BA 3 G 3 the angles at G, A 3 are right, 
and the angles at G 3 equal; therefore (III. 2) 
sin PG _ sin BA 3 
sin PG 3 sin BC 3