AS PRACTISED BY THE ARABS. 125 



r'. By Art. ^). Since 2,937,321 - 15 « 4 <p* -f 20 a 3 p 3 ft 4 15 a 2 p 2 ft 2 

 4 6 a <p ft 3 + ft 4 , so 598,044 + 2,937,321 - 3,535,365 = (20 a 3 p 3 6 + 

 30 a 2 p 2 ft 2 + 18 a p V 4 4 ft 4 ) 4 (15 a 4 <p 4 + 20 a 3 f ft -f 15 a 2 <p 2 ft 2 4 

 6 a p ft 3 -f ft 4 ) — 15 a 4 p 4 + 4 a 3 p 3 b + 45 « 2 <p 2 ft 2 -j- 24 a 9 ft 3 + 5 ft 4 , 

 and is the upper number in the Rank of the Biquadrate. 



s'. Then 3,535,365 x3= 10,606,095 - (15 a 4 p* -f 40 a 3 p 3 b 4 45 

 a 2 p s ft 2 4 24 a <p ft 3 4 5 ft 4 ) X 5 = 15 a 4 <p 4 ft 4 40 a 3 <p 3 ft 2 4 45 a* f ft 3 

 4 24 a p ft 4 4 5 ft 5 , and is the product written in the Rank of the 

 Quadratus Cubi. 



I'. By Art. \). Since 28,011,963 — 6 a 5 p 5 4 15 a 4 p 4 ft -f 20 a 3 <p 3 6 s 

 4 15 a" p % ft 3 -f 6 a p ft 4 4 ft 5 , so 10,606,095 -f 28,011,963 — 38,618,058 = 

 (15 a 4 p 4 ft 4. 40 a 3 <p 3 ft 2 4 45 a 2 <p 2 ft 3 + 24 a p ft 4 4 5 ft 5 ) 4 (6 a 5 p 5 4 15 

 a 4 p* ft 4 20 a 3 <p 3 ft 2 4 15 a 2 f ft 3 -f 6 a p ¥ 4- ft 5 ) zr 6 a 5 p 5 4 30 a 4 p 4 ft 4 

 60 a 3 p 3 ft 2 4 60 a 2 p~ ft 3 4 30 a p ft 4 4 6 ft 5 — 6 ( a 5 p 5 4 5 a 4 p 4 ft 4 10 a 3 <p 3 

 ft 2 4 10 a" p" ft 3 4 5 a p ft 4 4 ft 5 ) — 6 (a p 4- ft) 5 , and since a p 4 ft is = p 

 by Par. 21), so 6 (a 9 4 ft) 5 = 6p 5 , and is the sum written in the Rank of 

 the Quadratus Cubi. 



u'. By the transference of 38,618,058, its units are put under the 6th 

 place of the third period, and hence 6 p 5 thus transferred is the upper 

 number in the Rank of the Quadratus Cubi. 



v f . Then by Art. k'.) Since 126 = 6 a p 4 2 ft, so 3 4 126 — 129 = 

 ft4(6«(p-f2ft) = 6 a <p 4 3 ft, and is the upper number in the Rank of 

 the Latus. 



iv'. Then 129 x 3 == 387 = (6 a p 4. 3 ft) X ft = 6 a p ft 4 3 ft', and 

 is the product written in the Rank of the Square. 



