cardan’s Rule to the fecond Cafe , &c. 89 
wit, \/ 3 fe + zl + is algebraick, and the latter 
part, to wit, \Z } e x the feries — + + 
Pz '♦ Tz'* 
+ “itt — r r-4- 
&c. is tranfcendental. Q. E. I. 
Of the convergency of the Series obtained in the preceding 
Article, 
Art*. 6. This feries — + — + —~ 
Pz‘< , TV 
M 
+ 8cc. 
evidently converges fafter than the feries 2A + 
ee 
2E2 4 aOz 6 
- 8cc. or 2 +—- 
20 z 
4 308^ 
e e ye 243c 
fequently the expreflion V 
the feries 
+ 
e + z 
, , s - Sec : and con- 
6$6i<r 
+ a/ 5 ji? — + 4\A? x 
Czz Gz 6 Lz'° Pz 1 * Tz' 8 or 
— +-T-+—+— 5-+ -rr-+ &c. feems rather fitter 
ee e £ 4 e 
(a) N. B. I have been informed that both this mixed expreflion of the root of 
the equation — in the fecond cafe of it, and the merely tranfcendental 
expreflion of it publiflied in the former paper, and from which this expreflion is 
derived, were invented by Monfieur nicole, and publiflied in the memoirs of the 
French Academy of Sciences fo long ago as the year 1738 ; and the latter of them, 
to wit, the tranfcendental expreflion the feries 2 + — — ififl -f- . — 
()ee 243^ 6561*° 
&c. I had myfelf feen many years ago in Monfieur clairaut’s. algebra, in the 
place cited in the 50th Article of my former paper, to wit, in pages 286, 287, 288. 
But it was obtained by the intervention of negative quantities, and the roots of 
negative quantities, which gave it, in my opinion, an air of great obfcurity. And 
therefore I thought an inveftigation of the fame feries, by a method that keeps 
clear of thofe difficulties, might not be unacceptable to the lovers of thefe fciences, 
nor unworthy of a place in the Tranfa&ions of this learned body, 
Vol. LXX. N to 
