456 Mr * wales ; on the Refolution 
The firft thought of extending the ufe of the tables of fines, 
tangents, and fecants, farther than to the cafes which have 
been already mentioned, occurred to me while I was confidering 
the problem which produced the equation given in this paper as 
the fourth example. And it is remarkable, that the very fame 
thought occurred to Dr. hutton about the fame time, and in 
the refolution of the fame problem ; and we were not a little 
furprized, on comparing opr folutions together, to find that 
our ideas had taken fo exactly the fame turn ; and that both 
Ihould have {fumbled on a thought, which, as far as either of 
us knew, had never prefented itfelf to any one before. Hav- 
ing fince examined farther into the matter, I have the fatisfac- 
tion to find, that the principle is very extenfive, and that a great 
number of equations, efpecially fuch as arife in the practice of 
geometry, aftronomy, and optics, may be refolved by it with 
great eafe and expedition. 
Tut befide the facility with which the value of the unknown 
quantity is brought out by means of the tables of fines, tan- 
gents, and fecants, this method of refolution has another con- 
{referable advantage over moft others which have been propofed, 
inafmuch as the fir If ftate of the equation, without any previous 
reduction, is generally the beff it can be in for refolution ; and 
from which it may molt readily be difeovered, how to feparate 
it into fuch parts as exprefs the fine, or the tangent, or the 
fecant of the arc of a circle ; or into the fine, tangent, or fe- 
cant of fome multiple of that arc, or of a part of it; and in 
the doing of which confifts the principal part of the bufinefs in 
queftion. It will alfo be of fome advantage to preferve the 
original fubftitutions as diftindt as poffible, by ufing only' the 
figns of the feveral operations which it may be necefifary to go 
’ through 
