64 AN ESSAY ON THE ROOTS OF INTEGERS, 



evidently — Z. that is Z_ M. And similarly > M. Hence 



P 18 



then 



or — Z_ M 6 and 



*' 



or > M 



> 18 p 3 



p a $* -\- b tf + c <p 6 -\- d <p s + e <p* -\- f (p 1, + g <p % -f h <p -f & 

 That is — or — — 



<P 3 P 3 



is the approximate 6th Root of M, and is equal to a <p 5 -f bp* -fcp 3 -{- e? p 2 -f- 



e< p 1 + ./V ~\~g<P~ x +^<P -2 -f-^p -3 which, according to the well known laws of 

 the series of par. 1,) contains 3 decimal places; viz. g <2> _1 -j-AqT 3 -J- kp~ 3 . 



And since > M 6 so is — the highest approximate Root with 3 de- 



cimal places. 



(27.) If for 3 decimal places, there be required any other number, 

 then let the number of decimal places required be put zz z, and then it is 



obvious that in this reasoning for M p 6 * 3 and there is to be substituted 



(A 

 M 6Z and -, and the very same process will give a Root with a z number 



V 



of decimal places. 



(28.) But I have not been able to find that the Arabs were acquaint- 

 ed with this method of approximating to the truth, and I therefore pro- 

 ceed to explain their contrivance for adding a fraction to the integral 

 approximate Root, such that the sum should of course be greater than 

 that integral Root, and yet less than the truth, and consequently should 

 approximate still more nearly to the truth. 



