154 ARITHMETIC. 



3. Phcc the double of the root, already found, on the 

 left hand of the dividend for a divisor, 



4. Consider what figure must be annexed to the divisor, 

 so that if the result be r^ultiplied by it, the product inaf 

 be equal to, or the next less than, the dividend, and it will 

 be the second figure of the root. 



5. Subtract the said product from the dividend, and to 

 the remainder bring dovim the next period for a new divi- 

 dend. 



6. Find 



«t- ■ ■■ - • ' ^ - 



Suppose N to consist of two periods, and let the figures in the 

 root be represented by a and b. 



Then a-^-b zza^ '\-2ah-\'h'=N=::: given number j and to find 

 the root of N is the same as finding the root of a^ •^zab-^-b'^^ the 

 Tijethod of doing which is as follows : 



ist divisor fl}^* + 2^^-f ^^ (a -f*=: root. 



zd divisor 2a'{-b)2ah-\rh^ 

 tal-^-b' 



Again, suppose N to cbnsist Of 3 periods, and let the figures of 

 the root be represented by a, b and c. 



Then a-{-b-\-c =:^* + 2^3-f 3* -f-2^<r-f 23c- -fr*, and the man- 

 ner of finding Oj b and r, will be as before ; thus, 



ist divisor a)a'^ '^-zab-^-b'^ •^2ac-\-2bc'\-c'^ {a'\-b'\'Ci=, root. 



2d divisor 2a'if-b)2ab'\-b* 

 2ah^b* 



3d divisor 2a-\-2b'\-c)2ac-Jf2hc-{'c''' 

 2ac-\'2bc'\^c^ 



Now, the operation,, in each of these cases, exactly agrees with 

 the rule, and the same will be found to be true when N consists 

 of any number of periods whatever. 



