of the Angle fubtended by fwo QbjeMs, &c. 40^ 
perpendicular diftances from this arc, which may be demon- 
ftrated thus. Through the points E, D, and B, draw the 
arcs EN, DL, and BR, perpendicular to NIFR : then the tri- 
angles DFL, FBR, being equal, DL will be equal to BR ; 
moreover, the triangles ENI, IRB, being equal, the arcs EN, 
RB, will be equal : from whence it follows, • that EN = DL, or 
the perpendicular diftances of the points E and D from the arc 
of a great circle which pafles through the points I and F, are 
equal. It appears alfo, from the fame con ftrufiion, that the 
arc NL, intercepted between the two perpendiculars EN, DL, 
is equal to twice IF : for becaufe the triangles EIN, RIB, are 
equal, as are the triangles DLF, RFB, it follows, that NI is 
equal to IR, and LF to FR, wherefore 2lR=r NR, and 2RF 
— LR : whence, by fribtradting equals from equals, 2KI — 2RF 
= NR — LR, or 2IF — NL, which was the equality to be de- 
mon ftra ted. 
9. From this laft conftruction and demon# ration the follow* 
ing proportion is inferred. As radius : cofine of DL or EN, 
fo is the fine of IF to the fine of half the arc ED, or of half the 
obferved angle : for if the arcs NE, LD (fig. 3.), be continued 
until they meet in the pole H, the arcs NH, LH, will be qua- 
drants, and the triangle EHD ifofceles, which, from a property of 
Ipherics too obvious to need demonftrating, gives this propor- 
tion : as the chord of NL to the chord of ED, fo is radius 
to the fine of DH, or cofine of DL ; but the chord of NL is 
equal to the chord of 2IF from art. 8. We have, therefore* 
as radius : cofine DL, fo is the chord of 2FI to the chord of 
ED, or, which is the fame proportion, as radius : cofine DL* 
fo is the fine of IF to the fine of half ED. 
I/O. From the laft article it appears, that the fine of half 
the angle between the obferved ebje<fts, or the fine of half ED* 
Vol, LXXh Hhh is 
