Remarks on Professor Wallace^s reply to B. 207 



is also made 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 hitn Le Gendre's work, 

 and not the original paper of Euler. 



It is not necessary for B. to notice the many irrelevant sub- 

 jects which Professor Wallace has brought into notice in his 

 reply. The remarks hereafter to be made will therefore be 

 confined chiefly to the proof of the identity of Euler's 

 method with that republished by Professor Wallace. For 

 this purpose the detail of Euler'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 Eulei first proposes the Binomial 

 Theorem in page 103, vol. XIX., Nov. Comm., is 



n n n — 1 



b ' h 



and by putting a; = -, he reduces it to the form a" (I -j--)" — 



«"(] -fjc)" , and the question is thus reduced to the more 

 simple case of (l+o;)". But it may be observed that this 

 form, though more simple, is equally extensive with the 

 former, and the one may be deduced from the other. Euler 

 then observes that the development of this, when w is a posi- 

 tive integer, is well known to be of the following form ; 



n nn—\ n n—\ n — 2 



(1+ar)" = 1 + ^.0; + --^-. x^'+Y'^l-'-i-'^'+jetc.* 



(A) 

 He also remarks, that if n is not a positive integer, the value of 

 this series may be considered as an unknown quantity which 

 may be expressed by the symbol [n], or as La Croix more 

 simply denotes it by fn, so that generally for all values of«,f 

 positive, negative, integer, fractional, or surd, 



* The demonstration of the binomial theorem when n is a positive inte- 

 ger, is easily obtained by several methods. For the convenience of 

 reference, a demonstration is given at the end of this paper. 



f The words of Euler, in vol xix. p. 107, J^^ov. Comm., are "verum si 

 n nonfuerit numerus integer positivus valorem hujus seriei tanquam in- 

 cognitum spegtemus, ejus loco hoc signo [n,] utamur." La Croix uses 

 Vol. IX.— No. 2. 3B 



