466 Mr. wales on the Refolution 
It would be endlefs were I to undertake to enumerate all the 
various circumftances and cafes in which this method of bring- 
ing out the unknown quantity may be applied w r ith fuccefs ; 
what has already been laid will be fufficient to explain the 
nature of it, and to enable the analyft to apply it in other in- 
fiances as they occur to him, I lhall therefore only add a few 
examples to illuflrate it. 
example i» 
Let it be required to find the value of x in an equation of 
the form # 3 — r' 1 x — a. 
If r' 1 be expounded by 50, and a by 120 (fee Phil. Tranf. 
vol. LXV 1 II. p. 937-) the equation may be reduced to 
x — 50 = ^120 ; and, confequently, by tab. III. if x 
be confidered as the fecant of an arc, of which the radius is 
V 50, \/ x z — 50 will be the tangent of it, and we fhall have 
to find an arc, fuch that the tangent multiplied by the fquare 
root of the fecant may be equal to \/i2o ; or, which amounts 
to the fame thing, fuch an arc that the log. tang, together 
with half the log. fecant may be equal to half the log. of 1 20. 
But becaufe the tangent and fecant, here required, are to the 
radius of the \/ 50, the log. tangents and fecants in the tables 
muft be increafed by the logarithm of that number, and there- 
fore log. tang. 4- I log. 50 + f log. fecant -f | log. 50 = \ log. 1 20 : 
or log. tang, -f | log. fecant = | log. 120 — | log. of 50. Hence, 
having taken J the log. of 50 from § the log. of 120, run 
the eye along the tables of logarithmic tangents and fecants 
until an arc be found of which the fum of the log. tangent 
and half the log, fecant is equal to 19*7653631, the remainder. 
In 
