102 Application of the Fluxional Ratio, &c. 
in equation (6.), and is what we designate by - ; and that the coef- 
he ‘SE 
ficient of @ in the third term, which is 8z"~?, is the same with the 
fluxional coefficient n(n—1)2"~* in equation (7.) designated by“ 
hs : 
and that the coefficient of 5-3 in the fourth term, which is yz"-*, 
is the same with the fluxional coefficient n(n— 1)(n—2)a"-8 i in 
Ly]? 
lesa 
represented in equation (4.) by f’z,f"2, f’2, &c. hence by substi- 
2 2 3 3 4 4 
tution fle+h)y=y+ 5 v,4.Wl : ered merit ee 3.4 + Xe. 
(9.). It isin 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. 
3 
heh 
Withdraw the divisors from SOPoaP Xe. in the series of 
equation (8.) designated by and so on. ‘These coefficients are 
‘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 & in the second term of 
the series, it follows that =n", and yr=ne"~ 2, Hence it 
appears that Lagrange had a sufficient reason for _— the second 
term of the binomial series for the first fluxion. 
It would be no uncommon occurrence, if, when the pans 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. Butit 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 
