Now N— a2 is the remainder after n +1 figures of the
root, represented by a, have been found; and 2a is the
divisor at the same stage of the work. We see from (1)
that N — a2 divided by 2a gives x, the rest of the quotient
required, increased bv {x2}{2a}. We shall show that {x2}{2a} is a
proper fraction, so that by neglecting the remainder arising from the division, we obtain x, the rest of the root.
For x contains n figures, and therefore x2 contains 2n figures at most ; also a is a number of 2n + 1 figures (the last n of which are ciphers) and thus 2a contains 2n + 1 figures at least ; and therefore {x2}{2a} is a proper fraction.
From the above investigation, by putting a = 1, we see that two at least of the figures of a square root must have been obtained in order that the method of division, used to obtain the next figure of the square root, may give that figure correctly.
Ex. Find the square root of 290 to five places of decimals.
290(17.02 1 27)190 189 3402 ) 10000 6804 3196
Here we have obtained four figures in the square root by the ordinary method. Three more may be obtained by division only. using 2 1702, that is 3404, for divisor, and 3196 as remainder. Thus
3404)31960(938 30636 13240 10212 30280 27232 3048
