322 Mr. Horner’s new method of solving numerical 
be binomial or adfected, as soon as the transformation in R 
has been accomplished. The following description, there- 
fore, of a familiar process in arithmetic, will convey an accu- 
rate general idea of our more extensive calculus, and obviate 
the necessity of any formal precepts. 
In Evolution, the first step is unique, and if not assisted 
by an effort of memory, could only be tentative. The whole 
subsequent process may be defined, division by a variable divi- 
sor. For an accurate illustration of this idea, as discoverable 
in the existing practice of arithmeticians, we cannot however 
refer to the mode of extracting any root, except that of the 
square ; and to this, only in its most recently improved state. 
Here, in passing from one divisor to another, two additive 
corrections are introduced ; the first depending on the last 
correction of the root, the second on the correction actually 
making. And this new quotient correction of the root, since it 
must exist previously to the completion of the divisor by 
which it is to be verified, is required to be found by means of the 
incomplete divisor ; and may be taken out, either to one digit 
only, as is most usual, or to a number of digits equal to that 
which the complete and incomplete divisors possess in com- 
mon. And farther, as these divisors may not, in the first in- 
stance, agree accurately even in a siqgle digit, it is necessary 
at that stage of the operation, mentally to anticipate the effect 
of the new quotient, so as to obtain a sufficiently correct idea 
of the magnitude of the new divisor. 
24. This is an accurate statement of the relation which the 
column headed by the first derivee bears to the analysis. The 
remaining columns contribute their aid, as successively sub- 
sidiary to each other ; the contributions commencing with 
