AS PRACTISED BY THE ARABS. 69 



Then 3 is the number which fulfils these conditions. For since 2 is 

 the first found figure of the Root, if 3 is the sought number, the above 

 expression becomes — 



6'20 5 '3 - 63,200,000- 3 = 57,600,000 



+ 15'20 4 '3 2 = 15- 160,000- 9 = 21,600,000 



•$- 20-20 3 -3 3 =: 20* 8,000- 27 = 4,320,000 



-|- 15-20 2 -3 4 = 15- 400- 81 rr 486,000 



+ 6-20 -3 5 = 6- 20243 = 29,160 



-f 3 6 729 = 729 



84,035,889, which is 

 Z. 102,571,800. And 3 is the greatest number which will answer these 

 conditions. For if 4 be substituted in. this expression for the sought 

 number, it will become — 



6-20 5 '4 = 6-3,200,000- 4 == 76,800,000 



_j_ 1520* 4 2 = 15- 160,000- 16 — 38,400,000 



+ 20-20 3 -4 3 5= 20- 8,000- 64 = 10,240,000 



+ 15-20 2 -4 4 = 15- 400- 256 = 1,536,000 



+ 6-20 -4 5 = 6- 20-1024 = 122,880 



4 6 = 4096 = 4,096 



127,102,976, which is 

 > 102,571,800, and consequently would not answer. 



(b'). Place 3, the number now found, to the right hand of 2, the first 

 found figure of the Root, in the semicircle Article b.) so as to make it 23. 

 Call 3 the second found figure of the Root, and 23 the found figures of the 

 Root. 



