138 



altering tlio value of x, and the highest powers of ,r being the 

 same in both those expressions, if we divide by those powers, 

 then the common first members are the only members in both 

 Avhich do not contain negative powers of x, and therefore 

 the common first members in both expressions are identical. 



Newton, in his letter of the 13th of June, I676, to Mr. 

 Oldenburg, the Secretary to the Royal Society of London, 

 has expressed the Binomial Theorem in the following form, 



m — Sn 



viz. P+PQ^" = P"+;:AQ+^rBQ+-^CQ + -i;rDQ + &c. 

 where P-j-PQ signifies a quantity of which some Root, or some 

 dimension, or some Root of a dimension is to be investigated, 

 also P denotes the first term of that quantity, Q the remain- 

 ing terms divided by the first, and 7, the numeral Index of the 

 dimension of P-{-PQ, Avhether that dimension be a whole or 

 a broken quantity, affirmative or negative, and A, B, C, D, 

 &c. are used for the terms found in the progress of the 



operation, that is A for the first term P-;, and B for the se- 

 cond term ^AQ, &c. See Commercuim Epistolicum, No. 

 XLVIII. 



In the 45th Proposition of the First Book of the Principia, 

 Newton gives the same Theorem in the following Terms, viz. 



'r=Xl"=T"-^XT"~'+!^»X'r"~Vc. and also in the 93d 

 Proposition of the same Book, he gives the following expres- 



sion, viz. A+0^ =A +^0A +^^^0A &c. 



