analytical and geometrical Methods of Investigation. pp 
already stated, according to the conditions* of the formula de- 
manded by mathematicians. It enables us, however, imme- 
diately to ascertain that the roots are possible, and to calculate 
■their approximate value; for, when s/.i — or y = s~ l 
{ f ‘ :r - + r-'~ }, s- =/— 
=—{? + -£- + -£ + &+& c. > 
when % = o y = j e° + s~ 3 j = 1 
K-i + tt~ + ~hr “f ? 
-f &C. 
■7T. 
3.2 ■ 5.8 ' 7.16 
Hence, we may numerically approximate to the value of % from 
the expression z = it — { y + -|/- + -|£ + &c. } when y is 
given, and < 1. Nov/, in the case of the cubic equation, 
a 37V3 1 • r* q 3 
y~ — === . = - ■ ■ N ; and, since ™ < -t 
• V Va' + tn zqVq 4 27 
3 *V 3 - 
“ is < 1, conse- 
a 
quently the value of z may be obtained ; suppose it t , then the 
roots are to be approximated to, by means of the series that result 
from the developements of the forms by which they are repre- 
sented ; to wit, 
e 
-l/i — 
v/l/ 1 - 
V 3 | i.z. 
± J 1 
r 
! 1 . 
t* 
I.- 2 . 3 2 
1 1.2. 3-4 ' 
3 + 
(0+0* 
| d 
(9+g4 
I.2. 3 i 
• 1.2.34* 
■ 3 + 
(20 +0* 
| J 
(20 + 0 
1.2.3 1 
1 1.2.34 * 
3 4 
} 
&C. 
&c. | 
&c 
Now these series converge; for, since t is finite, we must at 
length arrive at a term A , in which [n— 1) n is > {-jf ; and* 
since (n+i)th term ( A ] is = A x (— Y x - — — -,\ A is 
1 ! 1 \-»+i / n \ 3 I n + ipz+z) n+i 
* The conditions of the formula are, that it should be finitfe in regard to the num- 
ber of terms, free from imaginary quantities, and containing only the coefficients 
q and -r. See Mem. de TAcad. 1738. 
