AS PRACTISED BY THE ARABS. 153 



In the 2d set of Cases. r 



When A = 5 = 2* -f- 1, and when A = 8 — 3* — 1., the deficiency is 

 the same, viz. -^-. 



When A — 6 — 2° -f 2, and when A = 7 3 2 — 2, the deficiency is the 

 same, viz; -^-. 



In the 3d set of Cases. 



When A rr 10 — 3 2 -f- 1, and when A — 15 zr 4 a — 1, the deficiency 

 is the same, viz. ^\. 



When A — 11 — 3 2 -f 2, and when A = 14 — 4 2 — 2, the deficiency 

 is the same, viz. ±%. 



When A — 12 = 3 2 -f 3, and when A = 13 — 4 2 — 3, the deficiency 

 is the same, viz. -£-§-. 



This is easily proved generally, for since by Par. 57) the excess of 



(2 a -f 1) r r" 



a" -\- r over the Square of its assumed Root, is let the surd 



power whose Root is required be (« -}- 1) " — r. This is =: a" -j- 2 a -\- 1 — r, 

 and hence the remainder is in this case 2 a -f 1— r. This being the 

 numerator, and 2 a -f- 1 still being the denominator, the assumed Root 



2 a -\- 1 — r r 



is in this case a -f- — a -f- 1 and hence the deficien- 



2 a + 1 2 a + 1 



r \ 2 ( 2 a + 1 ) ?• r- 



cy is (a -}- 1)- — r — (« -f- 1 ) =: the same expres- 



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



sion as before. 



