AS PRACTISED BY THE ARABS. 159 



causes to increase, and to diminish, so it must evi- 



{2a z + l) 2 2a: -f 1 



r" z" r 



dently cause their difference — to increase also. 



(2«2+l/ 2az + 1 



Now r by Par. 55) is Z. 2 a -{- 1, that is, r is not greater than 2 



«. 



r- z* r 



Substitute this value of r and the expression — • becomes 



(2az+iy 2a: + l 



4a 2 2 2 2 a 



4 « s f -f 4 « s -f 1 2az -\-l 



4 a 2 z* 4 a 2 s 2 



Now since 4 a" z" -\- 4 a z -\- 1 > 4 a 2 .s 2 so • Z. 



4 a" z" -f- 4 a z -\- 1 4a 2 2 ! 



4a 2 f 2 a 4a s f 



or 1. And evidently Z. 



4 r s 2 -(- 4 a : -f- 1 2a z -j- 1 4a 2 f -j-4«s-j-l, 

 and consequently is, a fortiori, also Z_ 1 ; and since it is also by supposi- 

 tion, positive and real, it must be a proper fraction. That is, though the 



r z 



error of excess committed by assuming a -j as the true Root of 



2 az -f 1 



a" -f r continually increases both as r and z are taken greater and greater, 



yet, although r be taken as great as possible, and though z be taken as 



great as we please, yet this error must always be less than unit, which is 



the limit to which it continually tends, but cannot pass. 



68). This may be more directly, I will not say more satisfactorily, 



proved, in the method of modern Geometers, by considering Infinity as 



r z~ r 



a positive Idea. In this case, since the expression — 



(2as -f- l) s 2az -j- 1 



s 1 



