and Infinite Series. 
4 r S 
f = j + ^ = v / io + 5\/— i +^io- 5 \/ — i = the firft 
>t 
root; and s ~ d V-x =s - + ^ I0 ~ ^ 
sfe v 10 + . sy~ I ~ v 10 ~~ JAAl \Z —% = the other two roots. 
2 ^ 
66. Hence it appears, that cardan’s rule s + d brings 
out fometimes the greatest root, fometimes the middle 
root, and fometimes the leaft root. 
Of the Roots by Infinite Series. 
67. Another way of affigning the roots of a cubic 
equation, may be by infinite feries, derived from the fore- 
going formulae, namely,/ + d and - s -~ =t s -~ 'S- 3, or. 
s/b + c-vs/b-c and 
— \ x. -fib + c + fib — v / — 3 x -V b + c — fib — c. 
For by expanding d/ b ± c in an infinite feries, we fhall 
evidently have all the roots exprefied in fuch feries. 
2 '* ■ 2 - 5i3 &c.. 
68. Now / - -fib + / = 3 /b x : 1 + 
3 b 3 * 3.6. 9 b* 
and d -fib-c-fib x : 1 -4 - 
2 c 
2 . 5 £ 
3 b 3 • 3 • 6 • 9 b * 
8cc. 
Hence s + d = 2 -fib x : 1 
1C 
_ 2 - ^ gee 
3, 6 6 Z 3.6.9.12^ 
for the firft root, as it was found by Mr. nicole, in the 
MemoireS: 
