C30S] 
XV. A new Demonstration of the Binomial Theorem , when the 
Exponent is a positive or negative Fraction . By the Rev. 
Abram Robertson, A.M. F.R.S. Savilian Professor of 
Geometry in the University of Oxford. In a Letter to Davies 
Giddy, Esq. F. R. S. 
Read June 5, 1806, 
DEAR SIR, 
Being perfectly convinced of your love of mathematical 
science, and your extensive acquirements in it, I submit to 
your perusal a new demonstration of the binomial theorem, 
when the exponent is a positive or negative fraction. As I 
am a strenuous advocate for smoothing the way to the acqui- 
sition of useful knowledge, I deem the following articles of 
some importance ; and unless I were equally sincere in this 
persuasion, and in that of your desire to promote mathemati- 
cal studies, in requesting the perusal, I should accuse myself 
of an attempt to trifle with your valuable time. 
The following demonstration is new only to the extent 
above mentioned ; but in order that the reader may perceive 
the proof to be complete, a successive perusal of all the 
articles is necessary. As far as it relates to the raising of in- 
tegral powers, it is in substance the same with one which I 
drew up in the year 1794,, and which was honoured with a 
place in the Philosophical Transactions for 1795. If, therefore,, 
R r 2 
