54 AN ESSAY ON THE ROOTS OF INTEGERS, 



perhaps be difficult to refer to a book in which this is detailed so minutely 

 as is required here, I shall endeavour to give one, and shall, accordingly, 

 arrange the present Essay in the following order. There shall be given — 



I. A general demonstration of the extraction of the Hoots of all 

 powers. 



II. An example of this operation after the common European 

 method, exhibiting its conformity to the demonstration. 



III. A similar example after the Arabian method, with a similar 

 exhibition. 



IV. An extract from the original Ayoun-ul-Hisab, containing the 

 Rule, together with a translation and remarks. 



(9.) Then to begin orderly — 



I. In this demonstration, the symbol Z. shall be used to express 

 less than > to express greater than, and then the following Lemmas must 

 be premised. 



Lemma 1. The « th power of 10 is 1 with n cyphers to its right hand. 



Thus the first power of 10 is 10, the second is 100, the third is 1,000, 

 the fourth is 10,000, &c. that is 1 with 1, 2, 3, 4, &c. cyphers to its right 

 hand. 



Lem. 2. Hence the n th power of 10 contains « + 1 figures, and is 

 the least possible number which can do so. 



