165 

 E S S A Y II. 



CHAP. I. 



PROP. I. 

 As a principle in the proof of the Binomial Theorem, it has 



th I th 



been already demonstrated, that the n differences of the n 

 powers of numbers in arithmetical progression will be con- 

 stant and equal to the common difference of the Roots mul- 

 tiplied by the product of the digits from unity to the 

 Index of the power. But, as this shall be a principle m the 

 following Propositions, I shall demonstrate it here in more 

 particular terms than could be admitted in the proof of the 

 more general Proposition from which it was deduced as a 

 Corollary. 



Let^, y> r, *, if, &c. be the terms of an arithmetical series, 

 whose common difference is d, then q-\d, r+d, s+d, t+d, v+d, 

 &c- will be respectively equal to the corresponding terms of 

 the former series, />, q, r, s, /, &c. since each Binomial has 

 for its first member the next lesser term of the series, and for 

 its second member, the common difference ; and the first 



members of the « powers of those Binomes being the n 



» 2 powers 



