of the Angle fubt ended by Two Objeffis, &c. ' 415 
or the value of ED may be obtained from the folution of two 
triangles KQF and DFL, with the proportion demonftrated in 
art. 9. 
21. To proceed with the computation, through D draw 
the arc DL perpendicular to FI, and let the fine of QKF:=jft, 
being the fine of half the arc OP, the meafure of IKF : put the 
fine of KF = r, the fine of the fine of DFK — w, ra- 
dius = 1. In the right-angled fpherical triangle KQF, the pro- 
perties of fpherics give this proportion : as radius to the fine 
of KF fo is the fine of QKF to fine of QF; wherefore 
fin. QF~j/; cof. QF—y/i -s 2 p"; and fin. FI (FI being 
double to QF) ~ 2 sp x ^/i — s z p z • Moreover, becaufe as rad. 
to cof. QK fo is cof QF to cof KF, we have cof KQ== 
V ' 1 .7 - - ; and fin. KQ ~ V -LlZlA . And fince as rad. : fin. QFK. 
v/i 
fo is fin. KF to fin. KQ ; this proportion gives fin. QFK 
; and becaufe the fine of the angle LFD is the fine of 
the difference (or fum) of the angles QFK, DFK, of which 
juft found, and fin. DFKss# 
the fines are,fin.QFK 
'A'-p 1 
s/ 
■sy 
by the data, we have from the rules of trigonometry, 
flft # TTFT ~ A —p~x A —n L z+i \/p z n z — n z s l p L 
Vi-. s y * 
and fince in the right-angled triangle LDF, as rad, : fin. DF V: fin. 
DFL : fin. DL, and by the problem fin. DF ~m it appears, that 
* If the points P and QJ>e on different * fides of the point O as they are repre- 
fented in the conftru£tion, the laft term will be affected with the fign — : if P 
and Q.be on the fame fide of O, the fign of the laft term will be -j-. It may be 
here obferved, concerning the geometrical conftr action (fig. 2. and 3.) that when 
P and Q^are on different fides of O, the angle obferved ED will be greater than 
when thofe points are on the fame fide of the initial point O, the arcs OP, Oj>, 
being equal. 
* Compare fig. 2. 
I i i 2 fin. 
