Mr . Robertson's Demonstration, &c. 299 
doctrine of combinations, by assumed series, or by fluxions : 
but that multiplication is a more direct way to the establish- 
ment of the theorem than any of these, cannot, I suppose, be 
doubted. Proceeding by it, we have always an evident first 
principle in view, to which, without the aid of any doctrine 
foreign to the subject, we can appeal for the truth of our as- 
sertions, and the certainty and extent of our conclusions. 
The following demonstration, which owes its origin to the 
abovementioned train of thinking, might be divided into two 
parts ; but I thought it more advisable to divide it into ar- 
ticles, and number them for the sake of references. That 
which might be called the first part, extends from the first to 
the end of the twelfth article, and contains the investigation 
of the theorem, as far as it relates to the raising of integral 
powers. The remaining articles constitute the second part, 
which contains the demonstration of the theorem as applicable 
to the extraction of roots, or the raising of powers, when the 
exponents are vulgar fractions. If the assumption of the series, 
in which the theorem is usually expressed, be allowed, the first 
part might be inferred as a corollary from the demonstration 
of the second. For having proved that x z ] ~ = x~ -f 
jzx r + y x d— — z x r -}-, &c. it follows, that when, r 
is equal to 1, then x -f- % n — x n n z x n ~ 1 + n x x 
z* x”'-* &c. I could not, however, think of suppressing 
the first part, as the binomial series is so easily investigated in 
it from first principles. 
Upon examining the Philosophical Transactions, I found a 
demonstration of this important theorem by Castillioneus, 
