XXIV. A new demonstration of the binomial theorem. By 
Thomas Knight, Esq. Communicated by W. H. Wollaston, 
M. D. Sec. R. S. 
Read July 4, 1816. 
It is somewhat remarkable that, amongst the various and far- 
fetched methods and artifices by which the binomial theorem 
has been obtained, no one should once have thought of the 
only course which seems obvious and natural. The equation 
{a -f- x) m x ( a -f- y) m = j ( a -f- x) ( a +y) j m expresses the 
general property of powers, whether m be positive or nega- 
tive, whole or fractional ; and from this equation, without the 
help of any artifice, the series in question is deduced. 
Some investigations have been found fault with, as drawn 
from principles allied to the method of fluxions ; whilst, on 
the other hand, a demonstration, taken from the “ Thkorie 
des Fonctionsf has been represented as perfect: but I cannot 
help thinking that it is as much connected with the fluxional 
calculus as any of the rest ; for it seems to make no difference 
whether, in (a s)'% T »ve substitutes u for x, and take the 
coefficient of u, or substitute x **J— x, and take the coefficient 
of x. The former substitution was made because it was 
known to be equivalent to the other, and has so little appa- 
rent connection with the subject, that a student would hardly 
understand why it was made. The demonstration of Mr. 
La Croix in the Introduction to his “ Calcul Differentier is 
MDCCCXVI. X X 
