AS PRACTISED BY THE ARABS. 71 



Then 4 is the number which fulfils these conditions. For since 23 

 are the two first found figures of the Root, if 4 is the sought number, the 

 above expression becomes — ■ 



6-230 5 -4 = 6-643,634,300,000' 4 s 15,447,223,200,000 



+ 15-230 4 '4 2 — 15- 2,798,410,000- 16 = 671,618,400,000 



-f 20-230 3 -4 3 = 20- 12,167,000- 64 = 15,573,760,000 



-f 15-230 2 -4 4 — 15- 52,900- 256 = 203,136,000 



~'-f 6-230 -4 5 = 6- 230- 1024 — 1,413,120 



+ 4 6 = 4096 = 4,096 



16,134,619,913,216, which 

 is Z. 18,535,911,758,593. And 4 is the greatest number which will 

 answer these conditions. For if 5 be substituted in this expression for the 

 sought number, it will become— 



6-230 5 *5 ±s 6-643,634,300,000- 5 = 19,309,029,000,000 



-f 15-230 4 -5* = 15- 2,798,410,000- 25 — 1,049,403,750,000 

 + 20-230 3 -5 3 — 20- 12,167,000- 125 — 30,417,500,000 



+ 15'230 a -5 4 — 15- 52,900- 625 == 495,937,500 



-{- 6-230- 5 5 — 6- 230- 3,125 = 4,312,500 



5 6 = 15,625 = 15,625 



20,389,350,515,625, which 

 is > 18,535,911,758,593, and consequently would not answer. 



V . Place 4, the number now found, to the right hand of 23, the two 

 first found figures of the Root in the semi-circle Art. b.) so as to make 

 it 234. Call 4 the third found figure of the Root, and 234 the found figures 

 of the Root. 



u 



