432 Demonstration of the Binomial Theorem. 
or more cylinders; and to construct the furnace with two 
tuyeres ; and, of course, to have two air-pipes. 
There are reasons, however, which render it better to em- 
ploy two or more cylinders even when the furnace has but 
one tuyere; and when the quantity of blast required per mi- 
nute, is not great. These are—lIst, The blast will be more 
uniform; and, 2nd, A smaller receiver will be sufficient. 
; A. B. Quinney. 
August 11, 1826. 
Art. XVIII.—New Demonstration of the Binomial Theo- 
rem; by Prof. THEopoRE StronG, of Hamilton Col- 
lege. 
CLINTON, Jan, 30, 1827. 
TO THE EDITOR. 
Dear Sir,— 
SHOULD you think the following demonstration of the 
Binomial theorem worthy of a place in your valuable Jour- 
Assume the identical equation (a+2)'=a+e2 =a'+7%x 
a}-! x! (For the first power of a quantity is the same as the 
quantity itself; therefore (a+<)* =a+<« and a’ =a, also a’~* 
=a° =: =1) to this add the expression 
1x(1—1) j1-2,2 af. 1x (1—1)x (1 —2) 1-343 _ 
1x = 1x2xs 
1x (I—1) x (1-2) x (1-3) 91 
1x2x3x4 
Every term of this expression equals nothing, for it has the 
factor 1—1=0, therefore 21+} xa)"*2" will not be in- 
creased or decreased by the addition of this expression. The 
addition gives (a+2)* =a'+}xa'"'2' =a'+7 x@tal+l 
x(I—1), 1-22 x (1-1) x (1—2) es . 
as a 2 1-3,3 . ad fin. 
1x2 x2x3 eae. oe 
= 
~4 24+ &e. ad infinitum. 
