Dr. hutton on Cubic Equations 
123. Again, 
v 7 c+b — v' 7 c—b 
ih l 2 . C 2 . t ■. 8 . II b* 0 
-xr* x : — + — ~ — r + ■ — r * 2 —2 See* 
^c 2, 3 3.6.9c 2, 3 . 6 . 9 . 1 2 . 1 5 c 4 
Is the leaft root of the equation a? 3 + 3 \ / c ! '-b z . x — ib. 
2 ^ IP 
Then, by taking = 1 , and - — g z , this becomes 
— v l ~g_ 1 2 -5 
■ — ■ — • — ■ * — — — ■ + 
3.6.9 
8cc. = the leaft root of the 
'equation x 3 + 
3 ^ -r 
4 g z 
X 
4 -g 
l —. And when f or c 1 is ne 
V Q 
gative, this becomes 
v' 1 + gV~ l — \/i-g>/-L r 2 . ;/ _ tin. 
• 7— = — — r-¥- + &c. = the leaft root 
2g^-i 3 3.6.9 
•of the equation x l 
the infinite feries 
sf* +/ 
x = — r* So that in general 
2 • j£ + ± 2 - 5.8. 11 . 14, 17/ ^ c 
3.O.9 3.6.9,12.15 3.6.9.12,15.18.21 
IS 
^ = the leaft root of the equa- 
tion x z =t 
3 ^ r / 
4/ 
x = 
Of the Roots by another Clafs of Series. 
1 24. But there are yet other feries, converging much, 
falter than thofe in the foregoing clafs, by the help of 
which, and cardan’s rule conjointly, may always be 
found 
