of adfeSled Equations, 
and, becaufe of the fimilar triangles 
473 
ABC and BFC, HEC and dGC, 
r a : — = CE, r : y/r^x 
r 
,vC^ 
= DF; r:y::£:^ = CG, and 
^ r 
tStS—L = dG; con- 
r \ •/ r z - y ‘ :: b : 
■F‘ 
\ r 1 
\ ' jr 
i/V 
B / N 
Ficr. i. 
fequently, ~ - r = FI,— — r — GI ; and 
becaufe the angles DIF, dlG, are equal j|H| 
by the nature of the problem, and the 
angles DFI and dGl both right angles, the triangles Db I and 
dG I are alfo fimilar, and confequently 
ax 
- r 
v 2 . by 
r ; and 
V r z — a 2 
y 
r V r‘ 
C, 
2 -/ 
r z —f 
or 
7 — ; or, the co-fecant of the 
bVF- 3 / ^ S -x 
arc HI -4- 3 - co-fecant of jAI -*■ = the co-tangent HI -f-r - co- 
tangent Al-r-r; or, laftly, the co-tangent of HI - co-tangent 
of AI = co-fecant of HI x r - - co-fecant Al* y Confequently, 
we have to find two arcs, the fum of which is given, and fucli 
that the difference of their co-tangents may be equal to the dif- 
ference of the produds of their co-fecauts into given quan- 
titles. 
To do this affume the angle DCF as near as poffible ; and, 
becaufe the fum of the two angles is given, the angle ^CG will 
be known alfo. Take the difference of the logarithms of r and 
r and b, which will be conftant, alfo the difference of tne 
co-tangents of the two affumed arcs, and having taken out the 
log. co-fecants, add to them refpedively the two logaiithmic 
Qq q 2 differences. 
