. of the Angle fubiended by Two 'Objects^ See, 405 
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 MIFR : then the tri- 
angles DEL, FBR, being equal, DL will be equal to BR ;• 
moreover, the triangles ENT, IRB, being equal, the arcs EN, 
RB, will be equal : from whence it follows, that EN == DL, or 
the perpendicular difiances of the points E and D from the arc 
of a great circle which paffes through the points I and F, are 
equal. It appears alfo, from the fame conflrudlion, 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 2IR — NR, and 2RF 
=2 LR : whence, by fubtradHng equals from equals, 2RI— 2RF 
= NR — LR, or 2IF ~NL, which, was the equality to be de- 
monfirated. 
V • - _ j 
9. From this laft conftrudbion and demonft 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 arcsNH, LH, will be qua- 
drants, and the triangle EHD ifofceles, which, from a property of 
fpherics 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 2PI 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. 
10. From the laft article it appears, that the fine of half 
the angle between the obferved objedls, or the fine of half ED, 
Vol, LXXL H h h is 
