320 Mr. Horner’s new method of solving numerical 
Ex. 1. Has the equation x * — 4X 3 -fi 8.r 2 - — i6.r-f-20 = o 
any real root? — See Euler, C. D. p. 678. 
0 
1 
1 2 
20 
9 
4 
—1 6 
—8 
0 
8 
2 
8 
— 4 
0 
4 
1 
1 
1 
Here the first column consists of the given co-efficients 
taken in reverse order. In the second, g is = the sum of the 
first column, — 8 is = — 16 + 2 (8) + 3 (— 4) -f 4 ( 1 ), 
2 is = 8 + 3 ( —4) + 6(1), &c. The third column is 
formed from the second, by the same easy process. We need 
proceed no farther; for the sequences 2, o, 1 in the second 
column, and 4, o, 8 in the third, show that the equation has 
two pairs of imaginary roots. Consequently it has no real 
root. 
Ex. 2. To determine the nearest distinct limits of the posi- 
tive roots of x 3 — 7# -fi 7 = o. See Lagrange, Res. des E. N. 
§ 27, and note 4. § 8. 
Operating as in the former example, we have 
0 
1 
2 
7 
1 
1 
“7 
—4 
5 
0 
3 
6 
1 
1 
1 
Since all the signs are now positive, 2 is greater than any 
of the positive roots. Again, between — 4 and -f- 5, it is 
manifest, that 0 will occur as a value of the first derivee, and 
