AS PRACTISED BY THE ARABS. 127 



£'. Then by Art. i/.) Since 129 — 6 a p -f .3 b, so 3 -J- 129 = 132 = & 

 -f(6«(p-f3Z>) = 6 « p -)- 4 i, and is the upper number in the Rank 

 of the Latus. 



s'. Then 132 X 3 = 396 — (6 a p -j- 4 5) X b =z 6 a p b -f 4 6 2 , and 

 is the product written in the Rank of the Square. 



£'. Then by Art. x.) Since 7,134 = 15 a" p 2 + 18 a p 5 4- 6 &*, so 396 

 -j- 7,134 = 7,530 — (Qap b + 4 i 2 ) + (15 a~ p 2 4- 18 a p 6 -j- 6 6 2 ) = 

 15 a 2 <z> 2 -f. 24 a ip & 4- 10 6 2 , and is the upper number in the Rank of the 

 Square. 



rf. Then 7,530 X 3 = 22,590 = (15 a" p* + 24 a p b 4- 10 6 s ) X 6 

 r= 15 a 2 <p 2 b -\- 24 a p b % -j- 10 6 3 , and is the product written in the Rank 

 of the Cube. 



&. Then by Art. z'.) Since 220,750 — 20 a 3 p 3 -f 45 a* p 2 b + 36 a p ¥ 

 4- 10 ¥ so 22,590 4- 220,750 — 243,340 = (15 a" p- b -j- 24 a 9 ¥ -j- 10 6 3 ) 

 4- (20 a 3 <p 3 4- 45 a 2 <p 2 & 4- 36 a p ¥ 4- 10 ¥) = 20 a 3 <p 3 4- 60 a 2 <p 2 & + 60a p ¥ 

 -j- 20 6 3 = 20 (a 3 <p 3 4- 3 a- <p 2 & + 3 a p ¥ 4- b 3 ) =z 20 (a p 4. bf, and since 

 a p 4- £ is = p by Par. 21). so 20 (a p 4- Z») 3 == 20 j» 3 , and is the sum 

 written in the Rank of the Cube. 



/. By the transference of 243,340, its units are put under the 4th place 

 of the third period, and hence 20/> 3 , thus transferred, is the upper number 

 in the Rank of the Cube. 



%.'. Then by Art. h'). Since 132 = 6 a p 4. 4 b so 3 4- 132 = 135 = b 

 •\-(6ap + £b)— 6ap + 5b, and is the upper number in the Rank of 

 the Latus. 



K 1 



