154 



power of a Binomial quantity in the case of a Fractional or 

 a negative Index ; and until we shall have by some means 



m 

 — -^ r" 



discovered the law of the series which is equal to 1 + x, we 



//( th 



shall call it the r Root of the 7» power of 1 + x, instead of 



ih 



calling it the " power of 1 + x, or a power of 1 + ^:' whose 

 Index is the Fraction y. And even considering the quantity 



^ th m 



l+,v in this point of view as the r root of l+^r, it has 

 been usual to assume an infinite series of the above Form, 



m 



]+^'V+c,r + rfx-f&c. = l+.r; but it should first be made to 

 appear why no Fractional Index shall be found in the series, 



or why in the r power of the infinite series, the powers of x 

 which are higher than m will break off. In deducing the 

 law of the indices of the powers of a:, I shall not attempt to 



m 



y 



express a finite quantity l + .r by a series having an infinite 

 number of terms, which attempt must appear to be impos- 

 sible, nor in truth shall I assume a polynomial expression for 



Y 



any part of the quantity 1+x, until I shall prove that such 

 expression will have arisen from a previous extraction ot Roots. 

 As to the CoeflScients of .r, it will be sulFicienf to sliew from- 

 the extraction of Roots that tlie: Coefficient of the second. 



: term 



