cj' adfedicd Equations, 
and, becaufe of the fimilar triangles ^ 
ABC and BFC, HEC and dG C, 
x 2 :: a : 
4 73 
r v:: £ 
r ... 
ax 
r 
CE, r : s/ r 
b 
DF ; r : y : : b : — = CG, and 
y r 
z 
-~dG; con- 
- r = ( 31 ; and 
b vfi 
\ G 
ji 
~ _ /XT 
B /\ 
r 
* r 
C 
v \ s/r % — y 1 ".b \ 
fequently, ~ -r — FI 
becaufe the angles DIF, ^IG, are equal 
by the nature of the problem, and the 
angles DFI and dGI both right angles, the triangles DE I and 
° a /A — " 
dGl are alfo fimilar, and confequently - — -- 
ax 
r 
r 
i' / S-y\.h_r. and - 
r y r 
x 
y 
r 
. a 
2 A 
G 
-y 
bV i 
or 
y 
bV 
a i 
r -y 
aVf — x* ^r z —y" y r -- x ~ 
arc HI -A b — co-fecant of AT -r- a = the co-tangent HI -f-r — co- 
tangent Al-f-r; or, laftly, the co-tangent of HI - co-tangent 
of AI = co-fecant of HI x 7 — co-fecant AI x - . Confequently , 
we have to find two arcs, the fum of which is given, and fucn 
that the difference of their co-tangents may be equal to the dif- 
ference of the products of their co-fecants into given quan- 
tities. 
To do this affume the angle DCF as near as poffibie ; and, 
becaufe the fum of the two angles is given, the angle dCGwill 
be known alfo. Take the difference of the logarithms of r and 
<7, r and b , which will be conftant, alfo the difference of the 
co-tangents of the two affumed arcs, and having taken oul the 
log. co-fecants, add to them refpe&ively the- two logarithmic 
Q^q q 2 differences. 
or, the co-fecant of the 
