AS PRACTISED BY THE ARABS. 129 



ten in the Rank of the Square opposite to the third period, then since by 

 the transference of Art. /, the units of the upper number in the Rank of 

 the Square are put under the 3d place of the third period, so they are 

 also put under the 3d place of this product. 



3'. Then' if this product, and that upper number be in the situation 

 added together, since by Art. v') that upper number = 15 p 2 so by Lem 7). 



the sum == 5pp X that Digit -J- that Digitj 2 -}- 15 p" x <p 2 ■=. 15 p" p 2 -f 6 

 p p X that Digit -{- tiiat Digit. | 2 



4'. Then if that Digit be multiplied into this sum, the product will 



become 15p z f~ X that Digit -f p p X that Digit |* -f- that Digit | 3 . Then if 

 this product be written in the Rank of the Cube, opposite to the third 

 period, since by the transference of Art. /, the units of the upper number 

 in the Rank of the Cube are put under the 4th place of the third period, 

 so they are also put under the 4th place of this product. 



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

 added together, since by Art. /) that upper number =z 20 a 3 so by Lem. 7). 

 the sum = 



15 p*p* X that Digit -{- 6p p X thatDigiii 2 + that Digit, 3 -f 20 p 3 X p 5 =z 



20p 3 p 3 + 15p 2 £ 2 x that Digit -}- 6p px that Digit | • x that Digit | 3 . 



\ 

 6'. Then if that Digit be multiplied into this sum the product will 



become 20 p 3 p 3 X that Digit + 15p 2 p~ X that Digit | 2 -j- 6 pp X that Digit | s 

 4- that Digit |*. Then if this product be written in the Rank of the 

 Biquadrate opposite the third period, since by the transference of Art. </) 

 the units of the upper number in the Rank of the Biquadrate are put un- 

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

 place of this product. 



