INFINITESIMALS, among mathema ticians, are defined to be infinitely small quantities. In the method of infinitesi mals, the element, by which any quantity increases or decreases, is supposed to be infinitely small, and is generally express ed by two or more terms, some of which are infinitely less than the rest, which be neglected as of no importance, the re maining terms form what s called the dif ference 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 motion, during the infinitely small time in which the element is generated : so that the re maining terms express the elements 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 infini tesimal parts of the elements are neglect ed, the conclusions are accurately true, without even an infinitely small error, and precisely with those that are de duced by the method by fluxions.
• In order to render the application of this method easy, some analogous princi ples are admitted, as that the infinitely small elements of a curve are right lines,
.or that a curve is a polygon of an infinite number of sides, which, being produced, give the tangents of the curve ; and by their inclination to each other measure the curvature. This is as if we should stip pose, when the base flows uniformly, the ordinate flows with a motion which is uni form for every infinitely small part of time, and increases or decreases infi nitely small differences at the end of every such time.
But however convenient this principle may be, it must be applied with caution and art on various occasions. It is usual, therefore, in many cases, to resolve the element of the curve into two or more in finitely small right lines ; and sometimes it is necessary, if we would avoid error, to resolve it into an infinite number of such right lines, which are infinitesimals of the second order. In general, it is a postulatum in this method, that we may descend to the infinitesimals of any order whatever, as we find it necessary ; by which means, any error that might arise in the application of it may be discovered and corrected by a proper use of this method itself. See Maclaurin's Fluxions.