93 § 
'Extenfion of Cardan’s Rule to the 
and - is = 3600, we fhall have ss = J — - - = 4629. 629, 
629, 629, See. — 3600= 1029. 629, 629, 629, 8cc. 
which is confiderably lefs than 3600, or ee; and confe- 
quently the feries will converge. 
39. We fhall therefore have - = 4 ° 29 ' 6291 629, 629 - Sec. 
^ 3000 
J 4 i 6 
= .286,00; and- = .081,796; and - 6 = .023,393; an ^ 
confequently — = = '511 — .063,55 5 and 
2 or 
9^ 
243 * 
20x^081^796 _ = >qo6 73 . and 3^1 = 121x^,393 _ 
243 243 7 / ^ 7 6c6l<T 6^6l 
20i 4 308^ 
« IS 
243 243 " ~ ' 6361*? 
= .001,098. Therefore 2 + — — 4 . , , 6 
6561 7 9^ 243^ 63OK? 
= 2 + .063, 55— .00 6,7 3 + .00 1,0 9 = 2. 064,64— .006,73 
And or */?*, is = n/ 3 [60 = 3.914,867. 
= 2.0 
57>9 
1. 
Therefore e 1 x the feries 2+— — 
9^ 243^ 
20 ^ 3o8r 
— X + - — 
6561/ 
- 8cc. is = 
3.914,867 x 2.057,91=8.0564; that is, the root of the 
propofed equation x 3 - 50#= 120 is 8.0564; which is 
true in three places of figures, the error being in the 
fourth place of figures, or third place of decimal frac- 
tions, where the figure ought to be a 5 inflead of a 6, 
the more accurate value of x in that equation being 
8*055)8io,345, 702, as may eafily be found by Mr. 
raphson’s method of approximation. But 8.0564, 
the value of x found by the foregoing procefs, is fuf- 
ficiently near to its more accurate value 8.055,810, 8cc, 
to 
