C 471 ] 
following paper : the defign of which is to point out a; 
method of folving feveral of the moh ufeful queftions 
in fpherical trigonometry in a manner fomewhat limilar 
to that ufed m approximating to the roots of algebraic 
equations. This method is founded on the following 
L E M M A. 
If the radius is fuppofed equal to 
unity, the fine of the fum of Uvo arcs, ol 
and /?, is equal to fin. + cof. oc x fin.,/S- 
fin. a X verf. fin. (3. And its cofine=cof.a- 
fin.-cc X fin. j6-cof. a x-verf. fin, 
D E M O N S T R A* T I o N.. 
Let the arc echo, ra, and the arc /S be ab, their fines ■ 
Ka, BD, refpeeftively ; then Bb being drawn perpendicu- 
lar to the radius cr will be the fine of ol + ^. Draw Dp 
and An, parallel to cr. Then, by fimilar triangles, 
ca : c^ :: bd : b/), and ca : a^ :: ad \ np. Therefore, 
-&b( -Aa^^p~pn)-Aa-^r 
CflXBI? AtfXAD 
; .that is, fine it + /S, 
CA CA 
:fin. oL -}- cof. OL X fin. j6-fin. a, x verf. fin. 
In the fame manner, drawing Dq parallel. to ha w^e. 
AflXBD CflXAD ^ 
-,orcofi 
may prove cb (-za-bq-aq)-Qh- 
fit + /S = cof. Of, — fin., it X fin. cof. « x verf. /S.. 
CA 
In 
