of adfebied Equations . 471 
39° 5'; and, having taken out the natural tangent, and loga- 
1 —c z 
rithmic fine of this arc, add the logarithm of to the lat- 
ter, and find the number correfponding to the fum, which will 
be lefs than the natural tangent of 39 0 f by 2869. As this 
affumption is fo near, take 39' 6" for the next, repeat the ope- 
ration, and the refult will be 1935 too great. Then 4804 
(2869 + 1935) : 60" :: 2869 : 36"; which being added to 39°5 , 
gives 39 0 f 36", for the co-latitude of the place fought, and 
the natural fine of this arc, or .6305856 is the value of x in 
this equation. 
example v. 
Let the equation 9c 1 + 
b z —2a 2 
2 2 , 2 a z -b z ab z , . 
x + — - — x — — =0, be taken, 
4 a 24 
which refults from a folution of one cafe of the problem de 
inclinationibus of Apollonius ; but which, as it naturally rifes 
to a folid problem, was not, I conceive, confidered by that cele- 
brated author. The refult of the analyfis, before any reduc- 
tion takes place, is this proportion, x + a : x — a :: 2 s/ ax : b ; 
and hence, - — - \/ax—\b . But it is here manifefh that if a 
be taken for the tangent of an arc, of which the radius is 
\Z ax, x will be the cotangent of it, and - — - \/ ax( — \b ) the co- 
+ a v ' 
fine of twice that arc. Confequently, we have to find an arc, 
the tangent of which is to the cofine of twice that arc as a is 
to 1 b; and this being done, the natural co- tangent of that arc, 
to the proper radius, will be the value of x. 
Vol. LXXI. Q q q 
Thus, 
