146 



which I have deduced the Law of the Indices in the alge- 

 braical operations, and not for the purpose of giving a general 

 rule for numerical extraction of Roots. For, I have supposed 



certain arrangements in 1 + x which will not always apply to 



m 



l+_i_, since the dispositions by decimal places in numbers, 

 and those of the letters in algebra are usually different. In 

 the latter case the characters or letters are indeterminate, and 

 therefore no powers but those which are homologous are ca- 

 pable of coalescing under the same coefficient. In the Invo- 

 lution and Evolution of numbers, the members are multiples 

 of terms in the decimal series of powers, the Indices of which 

 powers are indeed in arithmetical progression like the Indices 

 of A' in the algebraical formula, but the multiples are denoted 

 by the digits, although, if the algebraical arrangements were 

 followed, they would be denoted by numbers or coefficients, 

 which may themselves contain ^)owers of 10. Hence the 

 powers of the second member of the Root, and the coefficients 

 of those powers, which are kept distinct in the algebraical ar- 

 rangement, will coalesce into one number in the numerical no- 

 tation, and the necessary preservation of places and distances 

 will prevent the members of a power of a Binomial number 

 from being arranged according to the Indices of the second 



5 



member of the Root. Thus 1 + -jC according to the natural 

 places, or the numerical arrangements of the powers in the 



deci mal series, is 1 .6105 1 , or ] + t'- + -viz + —Vo + t oi-o + to oV— • 



But 



