INFINITESIMALS

, are certain infinitely or indefinitely small parts; as also the method of computing by them.

In the method of Infinitesimals, the element by which any quantity increases or decreases, is supposed to be infinitely small, and is generally expressed by two or more terms, some of which are infinitely less than the rest, which being neglected as of no importance, the remaining terms form what is called the difference of the proposed quantity. The terms that are neglected in this manner, as infinitely less than the other terms of the element, are the very same which arise in consequence of the acceleration, or retardation, of the generating motion, during the infinitely small time in which the element is generated; so that the remaining terms express the element that would have been produced in that time, if the generating motion had continued uniform. Therefore, those differences are accurately in the same ratio to each other, as the generating motions or fluxions. And hence, though in this method, Infinitesimal parts of the elements are neglected, the conclusions are accurately true, without| even an infinitely small error, and agree precisely with those that are deduced by the method of fluxions.

But however safe and convenient this method may be, some will always scruple to admit infinitely little quantities, and insinite orders of Infinitesimals, into a science that boasts of the most evident and accurate principles, as well as of the most rigid demonstrations. In order to avoid such suppositions, Newton considers the simultaneous increments of the flowing quantities as finite, and then investigates the ratio which is the limit of the various proportions which those increments bear to each other, while he supposes them to decrease together till they vanish; which ratio is the same with the ratio of the fluxions. See Maclaurin's Treatise of Fluxions, in the Introduc. p. 39 &c, also art. 495 to 502.