PHYSICAL AND MATHEMATICAL SCIENCES. 2<)7

to the independence of Leibnitz ’s discovery, and ex-pressed a favourable opinion on its merits. Leibnitz ,moreover, seemed equally willing to admit the claim ofNewton. Leibnitz , indeed, enjoyed the advantage ofseeing his calculus rapidly improving, and extending itsapplications in the hands of his friends and disciples,whilst that of Newton remained in comparative ob-scurity. But still nothing like rivalry or hostility ap-peared between the parties. A few years later, we shallfind this tranquillity strangely disturbed : for the pre-sent we must briefly recur to some general view of thenature of the discoveries in question, and then proceedto other subjects of discussion, of which so many, andall of such high interest, crowd upon us at this mosteventful period of scientific history.

General Idea of the Fluxional Calculus.

What we have before said of the methods of infinit-esimals, and the attempts at the problems of tangentsand quadratures, will have sufficed to show how nearan approach had been made to general principles. Themethod of serieses is in itself, perhaps, at once the mostsatisfactory and the most easily applicable, in caseswhere an exact solution is impossible: the essential ideais one which is practically familiar to the mind of everyone who has even gone so far as to calculate by de-cimal fractions. The notion of a value carried on toany number of places of decimals, but which, strictlyspeaking, never terminates, though each successive figurewe obtain brings us, in a tenfold degree, nearer thetruth, and though we may approach nearer to the exactvalue than to be able to assign or even conceive thedifference, — this notion, we repeat, is practically fa-miliar to the mind of every computor. There is, then,nothing more than this in the principle of those appli-cations of serieses to which we have referred. Theobject and nature of the series is simply to afford themeans of expressing a succession of terms, each formed