AS PRACTISED BY THE ARABS. 137 



Then by the method of Par. 28. 

 First. Let there be sought the approximate 6th Root of 65.- 



Here, since 2 6 == 64, and which is Z. and 3 6 — 729, which is > 65, so M 

 = 65, m = 2, and (m -f l) 6 — m 6 = 729 — 64 = 665, and r — M — m 6 



r 



— 65 — ■ 64 = 1. And hence the approximate Root, or m -J- 



(m 4- l) 6 — m 6 



= 2-^-. Then2 T |-| G =: 



l l l ill 



26 + 6-2 5 - {- 15-2 4 - J- 20-2 3 . [■ 152 s fr- 6-2- . + or 



665 665 2 665 3 665 + 665 s 665 5 



6- 2 5 - 665 s + 15- 2 4 - 665 4 - + 20- 2 3 - 665 3 + 15- 2 s 665 2 + 6. 2. 665 + 1 



&4- 



665 5 



equal to 

 24,969,477,535,800,000 — 6'2 5 665 5 



46,935,108,150,000 = 15.2*.665 4 



47,052,740,000 = 20.2 3 .665 3 



26,533,500 = 15.2 2 .665 2 



7,980 = 6.2.665 

 1 



665 6 = 86,482,825,840, 140,625)25,016,459,723,231,481, (0 + 64 = 64 



25,016,459,723,231,481 



Hence the deficiency in this case is 65 — 64- : 



86,482,825,840,140,625 

 61,460,366,116,909,144 



86,482,825,840,140,625 



Second. Let there be sought the approximate 6th Root of 396 



