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HISTORICAL VIEW OP THE

according to the same rule or law, but each (like thesuccessive decimal figures), having a value rapidly de-creasing ; so that, by taking the sum of a certain num-ber of them, we obtain the same kind of approximatevalue as that given by taking a certain number of placesof decimals.

The labours of Wallis and Mercator , and the firstefforts of Newton and Leibnitz , had been directed tothe discovery of a variety of such serieses, for the dif-ferent purposes of expressing those values which theycould not obtain in simple and finite terms, but whichgave the areas and other results of which they were insearch. The grand point, however, was to connect allsuch values, whether exact or approximate, and themethod of arriving at them, by a common principle anda common rule.

Now we have seen several methods successively startedby Kepler, Cavalieri, and others, which were devisedwith the view of passing from elementary problems re-ferring to finite quantities, to those more recondite pro-perties which essentially involved the principle of limits;and, at the same time, contrived to obviate the excessivetediousness of the ancient methods: some of thesemethods approached very nearly to certain cases offluxions, or differentials. But what distinguishes theinvention of the calculus (to give it at once its genericdesignation), is the generalisation of such methods, andreducing the elementary idea to such a form, as to sup-ply rules applying to all the objects before referred to, andconnect together, by a common principle, the processes hvwhich these objects were attained ; and thus, not merelyestablishing those particular truths, hut putting us in pos-session of a method of investigating all questions of thesame kind, and opening the way to the solution of a vastnumber of others as yet unthought of.

This grand step was taken by the two great inventorsof the calculus in very nearly the same manner, thoughin the mode of describing it a different language wasemployed.