324* Mr. Horner's new method of solving numerical 
the known arithmetical process for extracting the square 
root. 
3. At Cubic equations , the aberration of the old practice of 
evolution commences, and our theorem places us at once on 
new ground. We have here 
A = az -f- bz*- f- z 3 
and must proceed thus : 
1 b a A(r-f*r-J- 
r 
1 
2r 
y_ 
B’ 
&C. 
Br— B 
o 
B:+r*s B 
o 1 
B V 
A' 
&C. 
This ought to be the arithmetical practice of the cube root, 
as an example will prove. 
Ex. I. Extract the cube root of 4,8228544. 
Having distributed the number into tridigital periods as 
usual, we immediately perceive that the first figure of the 
root is 3 = R. Consequently, the first subtrahend is R 3 =27, 
the first derivee 3^=27, the second 311 = 9 ; the third (sal,) 
need not be written. Hence 
9- 
27.. 
48228544(364 
27 
6 
—-576 
— — . 
21228 
32 76—.—. 
19656 
12 . 
612 . . 
4 
_>433 6 
1572544 
— 
^i57 2 544 
1084' 
393136/" 
In this example the reader will perceive that no supple- 
mentary operations are concealed. The work before him is 
complete, and may be verified mentally. I need not intimate 
