Remarks on Professor Wallace’s reply to B. = 29% 
is also mats in exactly the same manner by Professor Wal- 
lace. /These circumstances render it highly probable that 
in this part of his reply he had before him Le Gendre’s work, 
‘and not the original paper of Euler. 
It is not necessary for B. to notice the many irrelevant sub- 
e 
Potued chiefly to the proof of the identity of Euler’s 
method with that republished by Professor Wallace. For 
this purpose the detail of Kuler’s demonstration will be given 
and compared with that of Professor Wallace. This will 
also serve to bring into more general notice one of the best 
demonstrations ever given of that important theorem. 
The manner in which Euler first proposes “es Binomial! 
Theorem in page 103, vol. XIX. a Comm., 
(a+b) =" +70” bo a an-2b? +, &e. 
: sa b b 
and by putting c=7, he reduces it to the form a” (1 += 
a” (i+cx)", and the question is thus reduced to the more 
simple case of (1-++-x)". But it may be observed that this 
form, though more simple, is equally extensive with the 
ormer, and the one may be deduced from the other. Euler 
then observes that the development of this, when n is a posi- 
tive integer, is well gpowa to be of the following form: 
eS 
*, g-po on de Ma we. ts 
1 ta)"= =i bee 
He also remarks, that if n is not a positive integer, th 
this series may be considered as an * pabicie quantiee 
may be expressed by the symbol [n], or a 
simply denotes it by fn, so that generally for all en of fit 
positive, negative, integer, fractional, or sur 
* The demonstration of the binomial theorem when n is a positive inte- 
ger, is easily obtained by several methods. For the convenience of 
reference, 2 demonstration is given at the end of this paper. 
+ The words of Euler, in eh KIX. p- 107 Nov. Comm., are “verum si 
er positt rem Gewe setiel tanquam i in- 
cognitum spectemus, ejus loco hoc signo [rn] utamur.” La Croix uses 
Vou. “3B 
Tk =No, 2. 
