VOL. LXXXV.] PHILOSOPHICAL TRANSACTIONS. 573 



XVl. The Binomial Theorem demonstrated by the Principles of Multiplication. 

 By the Rev. A. Robertson, A. M., of Christ Church, Oxford, F. R. S. p. 298. 

 A consideration of the very high importance and extensive utility of the bino- 

 mial theorem, having induced me to enter on an examination of the methods in 

 which, at different times, it has been demonstrated; and having frequently re- 

 viewed them, and deliberated on the subject, I was convinced that a demonstra- 

 tion begun and conducted on the obvious principles of multiplication was still 

 wanted, much to be desired, and also attainable. For to these principles invo- 

 lution must be ultimately referred, in whatever form it may be presented; and it 

 therefore appeared, that an investigation of the theorem effected by them only, 

 was likely to be as simple and perspicuous as the subject will permit. 



I think it needless to enter into a minute account of the demonstrations here- 

 tofore published, or to enumerate the objections which have been or may be 

 made to them. It is well known to mathematicians that they are effected either 

 by induction, by the summation of figurative numbers, by the doctrine of com- 

 binations, by assumed series, or by fluxions: but that multiplication is a more 

 direct way to the establishment of the theorem than any of these, cannot I sup- 

 pose 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 assertions, and the certainty and extent of our con- 

 clusions. 



1. The product arising from the multiplication of any number of quantities 

 into each other, continues the same in value, in every variation which may be 

 made in the arrangement of the quantities which compose it. Thus/? X 9 X r X 

 s = bars = spar = psqr = pqsr = any other arrangement of the same quantities. 



2. It is evident that each of the quantities a, b, c, &c. will be found the same 

 number of times in the compound product arising from {x + a) X {x + b) X 

 (x+ c) X {x + d) X {x + e) &c. For this product is equal to pqrsi = pgrs 

 X {x + e) = pqrt X {x + d) = pqst X {x -{■ c) = prst X {x + b) = qrst X 

 (x -\- a), by substituting for the compound quantities, x -^ a, x + b, &c. their 

 equals p, q, &c. Therefore, in the compound product, each of the quantities, 

 a, b, c, &c. will be found multiplied into the products of all the others. 



3. These things being premised, we may proceed to the multiplication of the 

 compound quantities x -{• a, x -{- b, x + c, &c. into each other; and in order 

 to be as clear as possible in what follows, let us consider the sum of the quan- 

 tities, a, b, c, &c. or the sum of any number of them multiplied into each other, 

 as co-efficients to the several powers of x, which arise in the multiplication. By 

 considering products which contain the same number of the quantities a, b, c, 

 &c. as homologous, the multiplication will appear as follows, and equations of 

 various dimensions will arise, according to the powers of x. Mr. R. here sets 



