136 AN ESSAY ON THE ROOTS OF INTEGERS, 



J. 253,125,396,471,245,260, the upper number transferred in the 

 Rank of the Cube = 20 f l — 20 m 3 by Par. 25). 



j. 825,325,162,335, the upper number transferred in the Rank of the 



Square = \b f- — 15 m" by Par. 25). 



x. 1,407,402, the upper number transferred in the Rank of the Latus 

 zz 6p iv — 6m by Par. 25). 



i^. Hence then the sum with the additional Unit = 4,260,793,105, 



941,366,382,119,977,455 = 6 m 5 + 15 m 4 -f 20 m? -f 15 nf + 6 m -f 1 — 



( m _)_ i)6 _ m «, and since by Art. cl> 1v ). 987, 654, 321 = r of Par. 28) and 



r 



by Par. 34). 234,567 =r m so m -| — the mixed number 



(m -f. I) 6 — m 6 



987,654,321, 



234,567 and is by Par. 28). the approximate 



4,260,793,105,944,366,382,119,977,455, 



6th Root of the given number M, or 166,571,800,758,593,887,308,296,025, 



335,490. 



(44.) To prove by tentation that this is the Case, would require the 

 actual involution of the above mixed number, which is the approximate 

 Root, to the sixth Power, a task of vast labour, which, after so much cal- 

 culation, I willingly decline, as it could serve little purpose except the 

 mere gratification of curiosity, and therefore to illustrate this part of the 

 subject, I shall chuse the following examples in simpler numbers, but 

 which, in all probability, will be thought sufficiently complicated. Besides 

 their present use, they will afterwards be satisfactory for reference in a 

 future part of this paper. 



