132 



term now used to represent all the cases I have mentioned. 

 The Binomial Theorem extends to all of them with that aene- 

 rality which could never be attained by going through the 

 arithmetical operations denoted by the Indices or Exponents. 



Thus if ;=",:. then, '^/> + x)(p+x xp + xxScc . . to 7i terms 



I 



n -1 



or p + x' and v^ + ^x/i + xx^ + a;x &c. to m terms or p+x' 

 will be represented by the following General Theorem ; 



p^x^ —p' +"xp'' +'".(t-^)^/'' +&c. a Demonstration of this 



1 . 2 

 Theorem, which is the subject of the following Essay, was 

 the result, a few years ago, of pursuing the excellent mathe- 

 matical course, which is delivered by the Rev. Dr. Magee, in 

 the University of Dublin. 



Although Sir Isaac Newton discovered, so early as the year 

 1669, this Theorem for the extraction of Roots of powers by 

 the method of Infinite Series, yet it does not appear that he 

 had ever discovered a proof of the truth of the Theorem, and 

 iiotwitlistanding the Fluxional Demonstrations which later 

 mathematicians had given, it was long observed, that an 

 algebraical rule might more justly be proved by the principles 

 of algebra. Of the above series particular cases only, have, 

 as yet, been algebraically demonstrated, and the other cases 

 have been usually inferred by Induction. Sir Isaac Newton 

 first considered Roots as powers. In all cases he represented 



the 



