HIS'tORr of the SOCIETr. 21 



2. Take P'= N — n"- for a dlvifor, and 2n — R° = 2n for a 

 dividend. Let the quotient be j«,', and remainder R'. 



3. T A KE P* = P ° — (R ° — R') /*', for a divifor, and 2» — R' for 

 a dividend- Let the quotient be f^f and the remainder R^ 



4. Take P'" = P' — (R'— R''')Xp* for a divifor, and iti — K" 

 for a dividend. Let the quotient be /x'", and the remain- 

 der R"'. 



5. These operations may be continued without endj 

 the divifor P*" being found from the formula P'' zz: 

 V?—'^ — (R/^^ — R''"*) X f(/'~'' : the correfponding dividend 

 being 2« — R''"^ ; and the quotient of the divifion beiiig 

 denoted by (jif, and the remainder by Re But it will only be 

 neceflary to continue thefe operations till we arrive at a value 

 P'z: P° = I, which will always neceflarily be the cafe. After 



this, the feries of numbers, ft^"*"^, (/f^^, i^^'^^t will neceffarily be 

 the fame as the numbers (/.% [if^ ftl" , &e. ; iff^ continually re- 

 peated in their order. 



The rule may be fliortly exprefled in algebraic language, 



thus : 



p° — I ; f(,r:«X i+R°=:« ; 



F =:N — «'; 2h— R° = 2«=P'X//-'+R'j 



■?" = I — iCcYR ° — R') = I +A^'R'J 2K — R' :=: V Xf^' + R'^ ; 



P'" = F — /(R' — R'O ; 2«— R''' = P'" X /'+ R'"; 



P.v = r — [^"' (R" — R"') ; 2« — R'" = P" X ^"+R '"j,. 



and fo on. 



Having thus found the numbers, /*, /*', i^", p, we fliall have 

 the continued frac5tion fought, 



,«"'+&c. 



I 



And the fraction may be continued indefinitely, by repeating 



the denominators jm.', jM-", • . . /«•'', continually in their order. 



Example 



