330 Mr. Horner's new method of solving numerical 
half of the root will be found by division simply. Meantime, 
the number of figures employed in verification of each suc- 
cessive root-digit, instead of increasing, is rapidly diminishing. 
3. The process with which we commence, need not be of 
a higher order than is indicated by the number of places to 
which we would extend the root ; and may be even reduced 
to an order as much lower as we please, by means of an in- 
troductory approximation. 
Ex. III. Let the root of the equation in Ex. II. be deter- 
mined to the tenth place of decimals. 
Arranging the derivees as before, we proceed thus : 
6 . . 
609- 
184 
62 74 
. 8 . 
62 
82 
10 ... . 
54 § i 
105481 " 
5562 . . 
25096 
11129396/ 
2511 
314 
2 
12 
1115764 
3 M 
3 i 
[A 
L 4 
11161 
1 C 
3 1 
>4 
L 
11161 
4 ' 
L 
• • • 
1 .oooooo( .094551 48 1 5 
^949329 
50671000 
-44517584 
6153416 
5578825 
574591 
558055 
1,1111611)16536(14815 
11161 
5375 
44 6 o 
910 
893 
17 
11 
6 
6 
