CALCULUS, Tin INFINITESIMAL, otherwise sometimes called the transcendental analysis, is a branch of mathematical science•which commands, by one general method, the most difficult problems in geometry and physics. The merit of the invention of this powerful mathematical instrument has been claimed for Leibnitz, but is undoubtedly due with equal justice to Newton, who laid the foundations for it in that celebrated section of his Principia, in which he demonstrates the chief theorems regarding the ulti mate values or limits of the ratios of variable quantities. The view of one class of writers is, that these distinguished men invented the C. simultaneously and independ ently; and it is the fact that Leibnitz's system is unfolded from premises differing some what from those of Newton. See PLuxtoxs. Another class of writers hold that New ton is the real inventor, and that to Leibnitz no more can be conceded than that he was the first who, using the suggestions of Newton's genius, gave a systematic statement to the principle of the transcendental analysis, and invented its appropriate symbolic lan guage. He had the doctrine of limits before him when he wrote, and did little more than upfold more fully the logic of the processes therein suggested, and exhibit them in algebraical forms.
The infinitesimal C., both in its pure and applied forms, whether of geometry or mechanics, is a branch of the science of number; its symbols are of the same kind, are operated on according to the same laws, and lead to analogous results. It differs from the other branches of the science of number, such as arithmetic and algebra, in regard ing number as continuous—i.e., as being capable of gradual growth and of infinitesimal increase, whereas they deal with finite and discontinuous numbers. It differs from ordinary algebra in another respect. In the latter, the values of unknown quantities, and their relations with each other, are detected by aid of equations established between these quantities directly; in the C., on the other hand, the equations between the quanti ties are not directly established, but are obtained by means of other equations primarily established, not between them, but certain derivatives from them, or elements of them. This artifice is most fertile, for it can be shown that in the great majority of cases the relations of quantities concerned in any problem may more easily be inferred from equations between these their derivatives or elements than between themselves.
It will be seen that the C. created a new notion of number—as continuous or growing. It is now necessary, in order to a proper conception of it, that a precise idea should be formed of a differential. The simplest idea of a differential is unquestionably that got by considering number as made up of infinitesimal elements, and a differential or " infinitesimal" as being the value of the difference between a num ber at one stage of its growth and as another very near it. Every finite number being—in the view of the C. as first conceived by Leibnitz—composed of an infinite number of these infinitesimal elements, certain axioms at once present themselves regarding infiuitesimals; as, for instance, " that a finite number of them has no value at all when added to a finite quantity." Many other such axioms readily follow, from which, on this view, the whole theory of the infinitesimal C. may be constructed. But there are logical objections to this mode of forming the theory of the transcendental analysis, and of three views that have been propounded, that now universally accepted as the most logical, and as being capable of the easiest application, is that founded, on the method of limits, already referred to as the invention of Newton. The meaning pf a differential on this view will now be explained.
It is clear that the C. can be applied only where numbers may have the continuous character—i.e., where they are or may be conceived as being variable. If two unknown quantities are connected by a single equation only, we clearly have the condition satis fied, as where y and x are connected by the equation Y = .Rx), where F is a sign denoting some function. of x, as tan. x, cos. x, e, etc. This equality may be satisfied by innumerable values of y and x. One question which the C. solves is, how does y vary when x varies? To solve it, and, at the same time, show how the doctrine of limits affects the definition of a differential, suppose z, y, and x+Dx, y+ Dy, to be two pairs of values of the variables which satisfy the above equation; then (2) y = F(x), and (3) Dy =F(x+Dx).