156 AN ESSAY ON THE ROOTS OF INTEGERS, 



64). It may be presumed the Arabians would be anxious to correct 

 or diminish such important deficiencies as these. The method they have 

 employed for this purpose in the Square is as follows : 



r 



The assumed Root of a" -j- r by Par. 55), is a -J that is, 



2a + l 



a -f- . Instead of 1 here employed as the multiplier of r, and 2a, 



2a xl+1 



let there be substituted the general real integer z, and this expression will 



r z 



become a A in which z may be taken any integer at pleasure. 



2 az+ l 



r z 



Then if this expression a -j- be assumed the approximate Square 



2az + 1 



Root of a" -f r, the deficiency in this case will evidently be or -f r — . 



1 rz \ 2 (2 a z -\- 1) r — j- 2 z- 



( a _| j — ,'. Let any constant value be given 



\ 2az + 1/ (2az + If 



to z and put this expression into Fluxions as in Par. 56), and then 



2 a z + 1 



2 a z r -f r — 2 z " r r rr o and r — . Substitute this value of r 



2z° 

 (2 a z + 1) r—r- z~ 



and the expression becomes 



(2 a z + I) 2 

 (2 az + 1 (2 as + l) 2 



(2«* + 1) 



2 2 2 4 z 4 



1 



(2a:+ 1) J 4z 2 



1 



65). Now as evidently becomes less, as z becomes greater, so 



4 z" 

 it might at first be supposed that if z were taken very large, the error 

 would be very inconsiderable. But then it must be observed that since z 2 

 increases faster than z, so if z be taken very great, the numerator 

 (2 a z -f 1) r — r- z 2 becomes negative, and since the denominator 



