AS PRACTISED BY THE ARABS. 131 



evident that the present operation from Art. 1' to 10'.) is equivalent to seek- 

 ing the c of Par. 23) and since 4 by Art. g\ et seq. is the found Digit, so 4 

 also expounds the c of Par. 23) and then — ■ 



g\ Since by Art. |') 138 — 6 p, and since 4 contains one figure, so 

 1384 — 6 p <p -f c by Lem. 6) and is the upper number in the Rank of the 

 Latus. 



a'. Then 1,384 X 4 — 5,536 = (6 jo p -f- c) X c =: 6 p p c X c\ and is 

 the product written in the Rank of the Square. 



/. Then since by Art. pf). 7,935 = 15 jo*, with its units put under 

 the 3d place of 5,536 so by Lem. 7) their sum in this situation — 5,536 

 + 7,935 X 10 2 — 5,536 + 793,500 — 799,036 - (6 p <p c + c~) 4- 15j» 2 X 0* 

 = \& p* <p* x 6 p <p c -\- c", and is the upper number in the Rank of the 

 Square. 



v. Then 799,036 X 4 = 3,196,144 — (15 pf f- x 6 p <p c X c c ) X c = 

 15 p- <p° c -f 6p <p C* -J- c 3 , and is the product written in the Rank of the 

 Cube. 



<ff. Then since by Art. g), 243,340 — 20 p 3 , with its units put under 

 the 4th place of 3,196,144 so by Lem. 7) their sum in this situation — 

 3,196,144 + 243,340 X 10 3 — 3,196,144 -j- 243,340,000 — 246,536,144 = 

 (15p 2 <p" c 4- 6 p <p c 2 4- c 3 ) -f 20 p 3 X f — 20 p 3 <p 3 -f 15 p- <p° c 4- 6 p <p c" -f 

 c 3 , and is the upper number in the Rank of the Cube. 



yj. Then 246,536,144 X 4 - 986,144,576 — (20 p 3 <p 3 4- 15 f £ 2 c 4- 

 6 p <p c" + e 3 ) x c •=. 20 p 3 p 3 c -f 15 p 2 <p" c" -|- 6 /> <p c 3 -j- <?*, and is the pro- 

 duct written in the Rank of the Biquadrate. 



l 1 



