[ 2 78 ] 
the feries aftiimed by the number two ; or by finding 
the powers of the feries above defcribed (which ex- 
hibits the length of the tangent of the double arc) 
multiplying each power by its proper coefficient, 
ranging the produdts under each other (according to 
the powers of the tangent of the fingle arc) and find- 
ing their fum. 
Now, fince the length of the double arc may be 
thus exprefled, by two infinite feries, each conftituted 
of the tangent of the fingle arc, and its powers; 
therefore the coefficients of the like powers of that 
tangent, in each feries, will be equal between them- 
felves ; and confequently the values of the unknown 
coefficients may be obtained by fimple equations. 
Laftly, fince the feries, which gives the length of 
the tangent of the double arc confifts only of the odd 
powers of the tangent of the fingle arc, therefore 
none of the even powers thereof can range there- 
with : now thefe will not occur in the odd powers 
of that feries ; and therefore the feries, aflumed to 
exprefs the length of the fingle arc, whofe double is 
to be compared with the fum of the former, muff 
confift only of the odd powers of that tangent ; and 
then the feries firft mentioned refults from the opera- 
tion, as will appear by examining the fame, as hereto 
annexed. 
I am thoroughly fenfible, that, to the learned, 
who are already m afters of the inveftigation of thefe 
feries by other methods, this will appear to be an 
eftay of more curiofity than ufe; but, with regard 
to ftudents, I hnmbly conceive it will have its ad- 
vantages ; becaufe by thofe, who are acquainted with 
the notation of algebra, and the manner of folving 
fimple 
