AS PRACTISED BY THE ARABS. m 



(4o+ 1) r — - 4 r 2 2 a — 8 a 2 



becomes — and is the greatest denomina- 



(4a-f 1/ 16a 2 +8a -f- 1 



tion of the negative deficiency, and consequently as in Par. 66,) this expres- 



8* a 2 — 2 a 8 « 2 2 a 

 sion with its signs changed, that is — ■ — — _ 



16 a 3 + 8 a -j- 1 (4a-j-l) 2 (4a+l) 2 



is the maximum of the positive error of excess. 



70). Now this expression increases by the increase of a. For let 



8 a 2 

 a be any other value greater than the present, and let — p, and 



(4 a -|- I) 2 



8 a 2 



z= X. And then by the very same reasoning that was employed 



(4 a 4- 1) 2 ia 2 o . 2 . 2 



v ■ ' 16 a a~ -j-8«a~ -\-ar 



in Par. 67) with z, ^, x andg, it will be found thatp — t x ■ 



16a 2 a" -f-8a 2 a -{-a 2 



and that 16 a 2 a 2 -j- 8 a a 2 -J- a 2 Z_ 16 a 2 a 2 -f- 8 a 2 « -j- a 2 , and consequently 



that jo- = <x multiplied by a proper fraction, that is, jo Z_ sr. And hence 



8 a 2 2 a 2 a 



increases by the increase of a. Again — ■ 



(4a + l) 2 (4«-fl) 2 16a 2 -f- 8a -|-1, 



and since a 2 increases faster than a, so 16 a 2 -|- 8 a -f- 1 increases faster 



than 2 a. That is, 2 a 



■ — • diminishes by the increase of a. And 



16a 2 -|- 8a -|-1 



consequently by the same reasoning as in Par. 67,) the whole expression 



8 a 2 2 a 



■ increases by the increase of a, and is by supposi- 



(4a + 1) 2 (4a-(-l) 2 



tion positive and real. And by a continuation of the reasoning of the 



8 a 2 2 a 8 a 2 



same paragraph, it will be seen that ■ ■ Z_ — or 



(4a +1) 2 (4a + l) 2 (4a+l) 2 



