[ 38 * 3 
XVI. Newton's Binomial Theorem legally demonstrated by Al- 
gebra. By the Rev. William Sewell, A. M. Communicated 
by Sir Joseph Banks, Bart. K. B. P. R. S. 
Read May 12, 1 79b. 
J—/ET m and n be any whole positive numbers ; and 1 + 
a binomial to be expanded into a series, as 1 Ax -f- Bx a -{- 
Cx 3 -j-, &c. where A, B, C, D, &c. are the coefficients to be 
determined. 
m 
Assume v m =i-\-x\ n = 1 -f- Ajc Bx* -f Cx' -f- Djc 4 -J-> &c. 
And z m =i -f- yl” = 1 -f Ay -f By 1 + Cy s + Dy 4 &c. 
Then will v” = 1 -j- %■> and z n = 1 -j- y . • . V — z n = x — y. 
And v m — z m — A x x — y -f B x / — y 1 - j- C x x 3 — y 5 + 
D x x 4 — y 4 -f-j &c. 
Consequently =-A-{-Bxx+y-\-Cxx x +xy-\-y* 
-j- D x i 1 1 jc y -f- x y z + y J +, &c. Now p* — z m = v — % 
x : 1 + v m ~ 2 z -f- v m -*z 1 -f » &c • Also v n — z n — 
z -f v n ~ 3%* -f"> &c... • 1 • Therefore 
3 z* -f- , ftc....Z w — 1 
V — Zx\V n 
v m — z' 
+ V” 
reduces to, and becomes 
l +v a - 
Z + V"— 3 Z 2 + , ■XC....Z 1 '- 
= A-f B XX +y-f C x/-f xy -j-y*-f Dxf-f .z 2 y + vry^y 3 
4-, &c. 
The law is manifest ; and it is likewise evident that the 
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