9 26 Extenfion of Cardan’s Rule to the 
— 1.976,959. But e is - 1054. Confequently, ej, or 
y/ 3 e, is = s/ 3 io54=io.I 768. Therefore e\x the feries 
3081 - &c. is = xo. 1768 x 1.976,959 
2 fX 
20 / 
~ gee 
243 
6561 
= 20.119,116. Therefore the root of the equation 
x 3 - 300 x — 2108 is . = 20.1 19,1 16. Q. E. I. 
2,7. This value of at is true to five places of figures, the 
more accurate value of it being 20.1 1 9,0 5 3, as will eafily 
appear by profecuting it to three or four more places of 
figures by Mr. raphson’s method of approximation. 
28. That 20.x 19 is very nearly equal to, but fome- 
what lefs than, the true value of x in the equation 
x 3 — 300 2 1 08, will appear by fubftituting it inftead 
of x in the left-hand fide of that equation. Fox’, if we 
take x = 20.119, we ftxall have xx — 404.774,161, 
and a 3 = 8143.651,345,159, and 300a; = 6035.700; 
and confequently, x 3 - 300a; = 8143.65 1,345,1 59, 
-6035.700 = 2107.95 1,345,1 59, which is fomewhat 
lefs than 2108, or the accurate value of x 3 - 300 a; in 
the propofed equation a; 3 - 300 a; = 2108. Therefore, 
20.119 tnuft be nearly equal to, but fomewhat lefs 
than, the accurate value of x in that equation. 
29. It appears therefore from this example, that this 
exprelfion, e\ x the infinite feries 2- — - 
6 gee 243 e 4 6561 e 
— &c. does truly exhibit the root of the equation 
4 
a; 3 
