AS PRACTISED BY THE ARABS. 121 



of Art. y, the units of the upper number in the Rank of the Biquadrate, 

 are put under the 5th place of the second period, so they are also put 

 under the 5th place of this product. 



7. Then if this product and that upper number be in this situation ad- 

 ded together since by Art. y) that upper number — 15 a 4 so by Lem. 7, the 

 sum = 20 a 3 p 3 X that Digit -j- 15a 2 <p 2 X that Digit | 2 -f 6 a p x thatDigit| 3 

 4- that Digit | 4 -f 15 a 4 X <p* = 15a 4 p* -f- 20 a? p* X thatDigit -f- 15 a 2 p* 

 X that Digit | " -J- 6 a p X that Digit) 3 -f- that Digit | 4 . 



8. Then if that Digit be multiplied into this sum, the product will be- 

 come 15a 4 p 4 X thatDigit-)- 20a 3 p 3 x that Digit | 2 -f 15 cff X tnatDigiFj 3 

 -\-6ap X that Digit | 4 -f tliat Digit | s . Thenif this product bewrittenin 

 the Rank of the Quadratus Cubi, opposite to the second period, then since 

 by the transference of Art. u, the units of the upper number in the Rank 

 of the Quadratus Cubi are put under the 6th place of the second period, 

 so they are also put under the 6th place of this product. 



9. Then if this product and that upper number be in this situation 

 added together since by Art. u) that upper number — 6 a 5 , so by Lem. 7, 



the sum = 15 a 4 p 4 X that Digit -f 20 a 3 e 3 X thatDigitj* -j- 15 a~ p" 



X that Digit| 3 -f Qap X that Digit| 4 -{- thatDigit| s + 6 a 5 X p 5 = 6 a 5 <p s 



+ 15 a 4 p 4 X that Digit -j- 20 a 3 <p 3 X thatDigit| 2 -f 15 a 2 f X thatDigitj : 



-f 6 a p X that Digit| 4 -f thatDigit| s . 



10. Then if that Digit be multiplied into this sum, the product willbe- 

 come0a s £ 5 x thatDigit -f 15a 4 p 4 x that Digit | 2 -|- 20a 3 p 3 x thatDigit} 3 

 -j- 15 a" p" X thatDigit] * -f 6 a p X that Digit | 5 -f that Digit \ 6 , which is 

 required to be not greater thanR p 6 -J- B by Art. j). Noav this is evidently 

 the same as the expression of Par. 20). 6 a 5 p 5 -f 15 a 4 p l b" -j- 20 a 3 p 3 b 3 



