AS PRACTISED BY THE ARABS. 



63 



First. p'" = P" <P + e 



a <p -f b <p -f c <p + d <p+e, the great 



est approximate 6th Root of P'" or P" <p 6 -f E, or , 



tf -f E. And V'"—p"> will be = R' lV . 



<p 6 + D 



SeconcL p^—p'''^, ~^-f— a<p,+ b <p -\- c \ tp + d <p + e <p + f 



the greatest approximate 6th Root of P iv or V" <p 6 -f- F or 















^ + D 





A £ 6 + B J tf 4- c 



+ E 



p 5 + F, which is the original 



number M proposed in par. 16). and P' lV — p lV = R v . That is jp iv = m of 

 par. 14). 



(26.) And evidently similar reasoning may be continued to any 

 length. Let it now be proposed to pursue this operation so as to find a 

 certain number of decimal places in the Root, and to avoid as before 

 par. 10), the trouble of a general demonstration let it be proposed to find 

 the Root to 3 decimal places. Then multiply M by cf X 3 or p 18 , and by 

 the principles of the series in par. 1). M <p 18 will contain 33 -J- 18, or 51 

 figures as by par. 15), And of these, as will easily be perceived, the 18 

 lowest places are Cyphers. That is by the reasoning of par. 15. 



M^-A^-i-B^ + C^ + D^ + E^ + F^+O^ + O^ + O- 

 and, consequently, by operating according to the principles given above, 

 the approximate Root will be found of this form. 



a <p 8 + b ip 1 + c <p 6 -f d £ s + e p* +fp 3 + g <p 2 + h <p + 1c. 

 Let this Root be put = p so that p 6 Z. and (> -f- Vf > M <p 18 . Then 



s 



