3g2 Mr. Horner’s new method of solving numerical 
Consequently the root is 1.356895867. This example was 
selected by Lagrange for its difficulty. 
Of its three roots, that which we have now found is the 
most difficult to obtain ; yet the whole work, including the 
preparatory portion in Art. 22, may be performed without 
one subsidiary figure, in less than a quarter of an hour. 
A little attention and practice will render the mental ag- 
gregation of positive and negative numbers as familiar as the 
addition of either sort separately. The introduction of r small 
negative resolvend, instead of a large positive one, where the 
opportunity occurs, will then greatly abridge the operations. 
For example, the cube root of 2, or 1.26.. 
— 0078950105 
1 . 259921049895 
was determined to this extent, true to the 12th decimal place, 
within as small a compass and as short a time as the result 
of Example III. 
Ex. V. As an example of a finite equation of a higher 
order, let the equation x s -{- 2X 4 + 3 X * + 4,-z* -f- $x = 32 1 be 
proposed. The root appears to be >2, <3; and the equa- 
59 , , 
7 5 6 
66 5b. 
7 92 
828 
8 64 , , 
- 4509 
978509 
45/ S 
452 7 
tion in % = x — 2 is 
207 = 201% + 150%* + 59% 3 + 122 4 -f % 
Hence, 
750 , 20/ , . . . 
39 936 7/396/6 
4 
53 6 
20 
933 
29 
. . 3 
60 
94 
78993 6 
44688 
49656 . , . 
2755527 
287035527 
276908/ 
2782662 
746632 
293333 9 
74 7 5 
7485 
5 
64 
2 9488 
■ // 
96 
02 
92 
50 
64 
28 
3/49 6/6 
7407744 . , . . 
867/0 658/ 
464347 O 6 5 8 / 
8694/3 82 
234667/2 
7 6 
^75387 8752 
2352 6 5 /\9 
77693/ 
4779/7459 
/ 7 69 
7 4 
48095900 
/ 4 
66 . 
52 
4 
0 6 
6\8 
75 
23 
6 / 
23 
207.00000 /, 638 6 058033 2 
788 97696 
78 0230400000 
73 9 3 0^/79743 
4 0 9 2 62 80257 
3 803 l 03002/ 
2895 250236 
286 75 04 7 54 
27745482 
23 904 7 95 
38400 87 
3 8 2478/ 
7590 6 
74343 
7563 
7434 
7 29 
96_ 
33 
