21 9 
the Quadrature of the Circle. 
5. With respect to the indices m and n, as they are here sup- 
posed to be affirmative whole numbers, and will be so in the 
use I am about to make of them, the reader need not be de- 
tained with any observations on the cases in which these fluents 
will fail, when the indices have contrary signs. 
6 ■ lt ma y be Proper further to remark, that by putting 
= z, and calling the first, second, third, &c. terms of the 
series — — - , n . 2 n x m + z » 
m.i—x" m. m + n. 1 — x n y m . m + n. m -f 2 n. 1 — xtf ' C ‘ 
A, B, C, &c. respectively, the series will be expressed in the 
concise and elegant notation of Sir Isaac Newton; viz. 
-- fZ nz A I znzB 3 nzC |C . . . 
m, i — x n m + n * m -f-2 n m + 3 n * Wllicll IS W6ll 
ted to arithmetical calculation. 
7. I come now to the transformation proposed, which will 
appear very easy, as soon as the common series, expressing the 
length of an arch in terms of its tangent, is properly arranged. 
If the radius of a circle be i, and the tangent of an arch of 
it be called t, it is well known that the length of that arch will 
series be written in one line, and the second, fourth, sixth, &c. 
m another, the same arch will be expressed thus : 
* i 7 ' 9 11 1 ^vv, il 
affirmative terms of this series be written in one line, and 
the negative ones in another, the arch will be 
Ff a 
