1€^ Application of the Fluxional Ratio, &fc. 



in equation (6.), and is what we designate by — ; and that the coef- 



ficient of -^ in the third term, which is [Sx''-^, is the same with the 



fluxional coefficient n(w— 1)^" '■' in equation (7.) designated by"-^=^ ;. 



and that the coefficient of ^"o" ^^ the fourth term, which is yx"'^, 

 is the same with the fluxional coefficient n{n — l){n—2)x"~^ in 



bV 



equation (8.) designated by —7^, and so on. These coefficients are 

 represented in equation (4.) hy f'x,f"x,f"'x, &;c. hence bysubsti- 



tui\onf{x-{-hr=y+-h+-^ y+-^ 2:3+^^ 2:3:4 + ^^" 

 (9.). It is in this manner, that, without making use of the Differen- 

 tial Calculus, we arrive at the formula of Taylor, which is in fact the 

 binomial series accommodated to fluxions. 



h^ h^ h' . , . ^ 



Withdraw the divisors from "2"?"^3"? 9 3 4 ' ^^* '" series of 



Taylor, and we have an expression of the several orders of 

 fluxions. Thus it is demonstrated that a relation exists between 

 the binomial theorem and the several orders of fluxions. And 



since -^ must indicate the coefficient of h in the second term of 



X'- 



y 



the series, it follows that — =7ia:"~', and y=nx'' ^x\ Hence it 



' X' T J 



appears that Lagrange had a sufficient reason for assuming the second 

 term of the binomial series for the first fluxion. 



It would be no uncommon occurrence, if, when the steps leading 

 to a discovery or improvement are once laid, by those who have 

 gone before us, the same improvement should be made by several 

 individuals. This happened in the separate and nearly cotemporane- 

 ous invention of fluxions by Sir Isaac Newton and Leibnitz. And 

 this may possibly be the case in regard to the views of that science 

 here exhibited. No such thing, however, has come to my knowl- 

 edge. But it is what appears by the VII. Article in No. 47 of the 

 Journal of Science, to have taken place in respect to the invention 

 of a universal method of computing the area of an irregular pol- 

 ygon. While Doct. Stiles was President of Yale College, which 

 must have been previous to the year 1795, my method of solving 



