AS PRACTISED BY THE ARABS. 149 



very tedious to demonstrate, and excessively laborious to exemplify. I 

 shall not therefore, by entering upon this task, render this very long paper 

 yet unnecessarily longer, but as a proof of this assertion I refer to the great 

 deficiency in the 2d and 3d example of Par. 44). This imperfection, the 

 Arabians seem to have been fully sensible of, and anxious to remedy ■ and 

 I shall conclude with an account of their attempts for this purpose in the 

 extraction of the Square Root. Of these I have not been able to obtain 

 the Arabic original, but their detail is as follows. 



55). Let on the principles of Par. 28). A, be a surd to the 2d 

 Power, of which a, is the approximate integral Square Root, so that a" Z_ 

 A and (a -f 1)~ or a" -f 2 a -f 1 is > A. Then let A — a" sd r and ar -|- r 

 = A. Then evidently r Z. 2 a -\- 1 and the Root, to be assumed is 



r r 



a 4 == a -) Then the deficiency arising from this assump- 



(a + If—a 1 2 a + I. 



J r y I 2ar r! v 



tion is evidently A < — ( a -\ ] = (a 2 + r) — ■[or -j -\ \ 



\ 2 a + 1/ \ 2 a + 1 (2 a + I) 2 / 

 (2 a + 1) r — r* 



— ■ . Now since r Z. 2 a -f 1 so r 2 or r r Z_ (2 a -f- 1)»% and 



(2 a + 1)* 



hence this can never be a negative expression, but must be always posi- 

 tive and real. 



56). Then the Arabian Arithmeticians observe that the deficiency 

 incurred by employing this assumed Root as the true Root, must always 

 be less than \. To prove this, if to a be assigned any constant value, 

 then r may be considered as a variable. For the only known properties 

 of r are that it should be real, and Z_ 2 a -j- 1. Hence if a be put — 1, 

 then 2 a -j- 1 — 3, and r is expoundable by 1 and 2. If a be put 

 ~ 2, then r is expoundable by 1, 2, 3, 4. If a be put = 3, then r 

 is expoundable by 1, 2, 3, 4, 5 or 6, and so on. The shortest and most 



