[ 439 ] 
In the next place, the arch A D being drawn, in the 
rectangular triangle AED, the radius is to the cofine 
ofDE, as the cofine of A E to the cofine of AD j and 
in the rectangular triangle AF D, the cofine of D F is 
to the radius, as the cofine of AD to the cofine of AF ; 
therefore, by equality, the cofine of D F is to the 
cofine of D E as the cofine of A E to the cofine of 
AF [<3], the arch AF counted according to the 
order of the figns being to be taken fimilar in 
fpecies to A E : For when AE is lefs than a qua- 
drant (as in fig. I ), AF will be lefs than a quadrant;, 
and when AE fhall be greater than i, 2, or 3 qua- 
drants, AF counted according to the order of the 
fig-ns, fhall exceed the fame number of quadrants. 
For, fince DE and DF are each lefs than quadrants, 
when A E in the triangle DEA is alfo lefs than a 
quadrant, the hypothenufe A D is lefs than a quadrant, 
when in the triangle DF A the legs DF and F A. 
are fimilar, that is, F A will be lefs than a quadrant ; 
(as in fig, 1.) but if AE is greater than a quadrant; 
(as in fig. 2.) that is, diflimilar to DE, the hypothe- 
nufe DA will be greater than a quadrant, and the 
arches DF, FA likewife diffimilar, and AF greater 
than a quadrant ; alfo in fig. 3 and 4, the arches AE, 
AF counted from A^ in confequence, will be the 
complements to a circle of the arches AE, AF in the 
triangles A DE, ADF. 
For an exampje, let the cafe be taken in Dr. Halley’s 
aftronomical tables, where an occultation of the moon i 
with a fixed fiar is propofed to be computed, the lati- 
\ a~\ The fame may be concluded from the f. HD being to . 
f.TDas f. HID. to f. IHD, 
