AS PRACTISED BY THE ARABS. 61 



6a + 1, and since R is an integer, so Maximum of R is 6 a 5 -j- 15 a 4 -j- 

 20 a 3 -|- 15 a 2 -f- 6 a, and Maximum of R p 6 is 6 a 5 p 6 4- 15 a 4 p 6 + 20 a 3 

 p 6 4- 15 a 2 p 6 4- 6 a p 6 . Again by Par. 17). B contains 6 figures and 

 hence by Lem 4). Maximum of B is p 6 — 1. Hence Maximum of R p 6 

 -f B is 6 a 5 f -f 15 a 4 p 6 + 20 a 3 p 6 -f- 15 a 2 p 6 4 6«/ + p 6 — 1. 

 But 6 a 5 ^ 6 -f- 15 a 4 £ 6 +20 a 3 p 6 + 15 a 2 £ 6 -f 6 a p 6 -f p 6 exceeds 6 a s 

 ^e _ji |5 ft 4 pa _j_ 20 « 3 pe 4. 15 a 2 <?, 6 -f- 6 a £ 6 -f p 6 — 1, by Unit. That 

 is, if b — p, then G a s p s b + 15 a 4 f 4 5 2 4 20 a 3 <p 3 Z> 3 + 15 a" p°- b* + 6 a 

 p b s -j- & 6 ,must exceed Rip 6 4- B at least by 1, and yet is also Z. R p 6 + 

 B which is absurd. A fortiori b cannot be > p. 



(21.) Then let A p 6 4 B be put =: P and a p -f- b zz p, then since /> 

 is the approximate Root of P, so j p 6 Z_and Q> -j- l) e >P. Let P — p 6 zz R'. 

 Then is R' Z. 6 p 5 -f 15 jt? 4 -f 20 jo 3 4 15 p 2 -f 6 j» 4- 1. If not R' = or 

 > 6p s -j- I5p* -f 20jo 3 -f 15/> 2 -f &p + 1. 



First. Let it be equal. ThenP— p 6 =6/+ 15 p i 4 20 />* _j_ 15^5 

 4- 6p -f- 1, and P r: / + 6jo s + 15p 4 -f 20 p 3 + 15/> 2 + 6p + 1 = 

 (j» 4- l) 6 - And yet by supposition P Z. (jo 4 l) 6 which is absurd. Second. 

 A fortiori it cannot be greater. 



(22.) Since P—p 6 = R'andP=p 6 -f-R'soPp 6 -f C — |A p« 4- B|<p 6 



4 C = (p 6 4 R') p° 4 C = p 6 p 6 4 R'ip 6 4C = j»-p| 6 -f (R' <p 6 -j- C). 



jj 6 



Now since 6 | P p j ' — p p so by Par. 13). p pwili here represent the S and 



R'$ 6 -f C the £ of that Paragraph, and if c be the greatest possible integer 

 such that 6- pp j 5 c 4 15- /> p \ 4 c~ 4 20* p p j 3 c l 4 15" p <p | 2 c 4 + G p 

 p c s 4c 6 that is 6 jo 5 p 5 c -\- 15jt> 4 p 4 c 2 4- 20 jo 3 <p 3 r 3 -f 15 jo 2 <p 2 c 4 4 6p 

 p c 5 4 c 6 should be less than R' <p 6 4 C then is p p 4 c the highest ap- 

 proximate 6th Root of p p\ 6 4 R' p 6 4 C or P p 6 4- C or \Ap 6 -f- B| p 6 

 4- C and c will represent the u of Par. 13). 



