346 Account of certain Improvements 
Rule. 
Divide the proposed number into as many periods as possible, 
consisting of three figures each, proceeding from right to left ; sub- 
tract the highest cube that can be taken out of the remaining 
figure or figures on the left, placing that cube under those figures, 
and the remainder under the cube with a line between them; to 
the remainder annex the next period,which is called the resolvend, 
and place the root of the cube taken in the quotient. 
Then any resolvend and the quotient being given, a new figure 
of the quotient will be obtained by annexing a cipher to the 
quotient figure ; and calling the quotient thus increased, the in- 
creased quotient. Subtract the sum of three times the square 
of the increased quotient into the new figure, three times the in- 
creased quotient into the square of the new figure, and the cube 
of the new figure from the resolvend, and place the new figure 
in the quotient instead of the cipher, and annex the next period 
to the remainder for the new resolvend, 
It is obvious that the new figure must be such that the sum 
to be subtracted must be as nearly equal as possible, but less than 
the given resolvend. 
This rule as generally given in books of arithmetic is far from 
being explicit, as they do not show clearly how it is derived from 
the polynomial. In obtaining a new figure in the root, most 
authors direct the student to multiply the square of the quotient 
by 300, and this again by the new figure: then to multiply the 
quotient by 30, and this product by the square of the new figure ; 
and lastly, to take the cube of the new figure, and add these three 
products together, and subtract as above: but this does not fol- 
low from the principle, though the effect must be the same. 
Example. 
Find a number whose cube shall be the nearest less number to 
160634321856. 
160°634°321°S56(5436 
@=125 
35634, Ist resolvend 
ab= 30000=3(50)* x4 
al?’= 2400=3(50) x4? 
= pee vy noe eae 
32464 
3170321, 2d resol. 
(a+ 
