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XVI. The Binomial Theorem demonstrated by the Principles of 
Multiplication. By Abram Robertson, A. M. of Christ 
Church, Oxford, F. R. S. In a Letter to the Rev. Dr. Mas- 
kelyne, F. R. S. and Astronomer Royal. 
Read May 21, 1795. 
REV. SIR, Christ Church, Oxford, Oct. 27th, 1794. 
A consideration of the very high importance and exten- 
sive utility of the binomial theorem, having induced me to 
enter upon an examination of the methods in which, at dif- 
ferent times, it has been demonstrated; and having frequently 
reviewed them, and deliberated with myself upon the subject, 
I was convinced that a demonstration begun and conducted 
upon the obvious principles of multiplication was still wanted, 
much to be desired, and also attainable. For to these prin- 
ciples involution must be ultimately referred, in whatever form 
it may be presented ; and it therefore appeared, that an in- 
vestigation 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 heretofore published, or to enumerate the ob- 
jections 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 figurate numbers, by the 
