AN ESSAY ON THE ROOTS OF INTEGERS, 



r r 



Then since x > so m -f- x > m + — and (m-\-x) 6 



(m + 1 ) 5 — m 6 . (m + 1 ) 5 — m « . 



r 

 or M also > m -J-- 



(m-j-1) 6 — wi< 



as above. 



(29.) Here it is evident, that if any constant value be assigned to 



m, then if r be small, that is in case the integral Root differ but little from 



r 



the true Root, then the fraction is also small, but if r be 



( m + 1 ) a — i» ft 



large, that is in case the integral Root, differ much from the true Root, 



r 



then the fraction is large. That is, the compensation made 



( m + 1 ) n — m a 



by the fraction is proportionate to the error of the integral Root. 



(30.) I need scarely add, that whenever it is required to extend the 



above demonstration to any other index than 6, then for the individual 



numbers 6, 15, 20, 15, 6, 1, there are to be substituted the general co-effi- 



iv n — 1 iv n — 1- n — -2 rv n — 1- n — 2' n — 3 



cients of the binomial theorem n, , , . 



2 23 2 3-4 



&c. and the same reasoning applied as that given above. 



(31.) I now proceed in order : — ■ 



II. To exemplify the above demonstration, by the actual extraction 

 of the 6th Root of a given number, according to the directions contained 

 in European books of Arithmetic. I therefore chuse to extract the 6th 

 Root of the number 



166, 571, 800, 758, 593, 887, 308, 296, 025, 335, 490. 

 which consists as in par. 14) of 33 figures. And the operation is thus 

 exhibited. 



