328 Mr . Horner's new method of solving numerical * 
root-figure will be obtained and verified at an expence of 
\ (n . n f) - j- 2 new lines of calculation, containing in all 
somewhat above f(n.?i — i.ft-j-4) digits more than the 
preceding root-figure cost. But as the necessity for unlimited 
continuity can rarely, if ever, occur, we may consider our- 
selves at liberty to check the advance of the resolvend, as 
soon as it contains one or tw r o figures more than the number 
we yet propose to annex to the root. This will happen, 
generally speaking, when d. th of the numeral places of the 
root are determined. 
By arresting the advance of the resolvend, we diminish it 
in the first instance by an optional number (/>) of places, and 
by n places more at each succeeding step. Neglecting at the 
same time an equal number of figures in the right hand 
places of each closing addend and its derivatives, as contribu- 
ting nothing to the diminished resolvend, we thus cause the 
effective units’ place of each derivee to retrograde* in the first 
instance p-\-m,—n places, and at every subsequent step, a 
number of places (m) equal to the index of the derivee. 
In the mean time, while these amputations are diminishing 
the derivees and addends on the right hand, a uniform ave- 
rage diminution of one digit on the left hand is taking place 
in the successive classes of addends in each column. The 
obvious consequence is, that after about th of the root- 
* As the advance of the closing addend is prepared by annexing dots to the superior 
preparatory addend, so its retrogradation may be prepared by a perpendicular line 
beginning before the proper place (the m th or m -) - p — n‘ b ) of tire said preparatory 
addend, and continued indefinitely downward. One digit, or two, of those which fall 
to the right of this line in the next succeeding sum and preparatory addends, must 
be retained, for the sake ultimately of correctly adjusting the effective units of the 
subtrahend. 
