294 HISTORICAL VIEW OF THE

the hints supplied in the inventions of his prede-cessors.

The serieses which he deduced, as applying generallyto the quadrature of curves, were, as he tells us, foundedon Wallis’s principle ; and, from comparing the formsof those already obtained, he discovered a law whichwas common to them all. In such of these forms asinvolved fractional exponents, he found the series in-finite ; though, wherever he could render it convergent,it would of course give a value to any requisite degreeof approximation. This, however, was closely connectedwith an object of more extensive importance and utility,a theorem for expressing any root of a quantity composedof two terms in a series involving certain combinationsof those terms, according to a given law, and whichwould be convergent. To such a theorem he was almostimmediately led; and, as he instantly perceived thatroots were comprised under the same algebraic formulasas powers, by the mere adoption of the idea of fractionalindices, he saw that this theorem was no other than themost general form of that for raising a binomial quan-tity to any given power, in a series of terms involvingthe index and the whole powers of the two terms in acertain regular progression. It hence acquired the nameof the binomial theorem.

This theorem, besides its other numberless applica-tions, was directly employed in the quadrature of curves,as supplying a more direct method than that originallyadopted, and out of which it had in fact originated.The whole course of the discovery is related in detail,in a letter to Oldenburgh ; one of a valuable series towhich we shall often have occasion to refer, collectedunder the name of “ Commercium Epistolicum.” Theapplication, then, consisted mainly in this; that he re-duced the value of the ordinate of a curve into an in-finite series of the integer powers of the abscissa by thebinomial theorem, or by any simpler algebraical pro-cess, where the case admitted it. After this, other con-siderations, like 'those employed in the method of in-