XVII. Note respecting the demonstration of the binomial theorem inserted in the last volume of the Philosophical Transactions

In looking into Mr. Spence’s ingenious “Essay on Loga­rithmic Transcendents,” a work published in 1809, but which I have been so unfortunate as never to have seen till within the last fortnight, I was not a little surprised to find that a demonstration of the binomial theorem, similar to the one I had the honour to present to the Royal Society, had been already given by that writer. The same may be said of the first proposition of the preceding Paper on the construction of Logarithms. Having made this acknowledgment, I shall perhaps be pardoned for observing, that Mr. Spence is not particularly happy in the manner of developing the kind of functions he treats of in his preface. I shall endeavour to give the solu­tion of a class of equations of which he (Pref. p. vii.) has considered a particular case: with this we will begin.