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 acquisition 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 mathematical 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 integral 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, you think the following demonstration worthy the attention of mathematicians, you will much oblige me by presenting it to the Royal Society.