INTEGRAL CALCULUS, a branch of infinitesimal calculus, treating of the methods of deducing relations between finite values of variables from given relations between con temporaneous infinitesimal elements of those variables. It is thus the inverse of the differ ential calculus. Its object is to discover the primitive function from which a given differen tial coefficient has been derived. This primi tive function is called the integral of the pro posed differential coefficient, and is obtained by the application of the different princi ples established in finding differential coeffi cients and by various transformations. To illustrate—with the integral calculus one may discover the relations connecting finite values of variables, as x and y, from the relation connecting their differentials, as dx and dy. In tegral calculus is thus the doctrine of the limit of the sum of infinitesimals of which the num ber increases, while the magnitude decreases, both without limit, yet according to some law. The more proper concern of the integral calcu lus is besides the finding and discussion of inte grals, with such matters as the theory, of spherical harmonics, the theory of residuation, and parts of the theory of functions. The sign of integration is "f," which is a form derived from the old or long "s." It is the initial of the word "sum," and came into use owing to the conception that integration is the process of summing an infinite series of infinitesimals. Before even the notation of the differential cal culus and its rules were discovered by Gottfried Wilhelm von Leibnitz (1646-1716), he had in vented the notation and had found some of the rules of the integral calculus. Leibnitz first used the now well-known sign "fp or long as short for the "sum of" when considering the sum of an infinity of infinitesimals as is done in the method of indivisibles. Leibnitz himself at tributed all his mathematical discoveries to his improvements in notation; and the fact that we still use and appreciate Leibnitz's (and one must add his "cP') even though our views as to the principles are very different from those of his school, is perhaps the best testimony to the question of notation. The integral cal
culus may be distinguished from the differential calculus by another feature than inversion; namely, by the greater importance in it of imag inaries. The integral calculus is frequently called the indirect or inverse method of fluxions, the analytic processes by which a function may be found such that being differentiated it shall produce the given differential. By writers on "fluxions" this function is called the "fluent" or the flowing quantity. By writers on the infini tesimal calculus it is called the integral of the proposed differential. The origin and constitu tion of quantities is called "fluxions" in the scheme of Sir Isaac Newton (1642-1727), be cause conceived to express the manner of gen eration of quantities by the motion of other quantities. In the scheme of Leibnitz the lan guage employed is "infinitesimals" or 'differen ces"; because they are conceived to express the constant addition of one indefinitely small quantity to another. In the scheme of Newton, or "fluxions," the finding of the sum of all dif ferences is called the "inverse" or the 'indirect method." It is thus that this method of obtain ing the quantity generated is identical with the method of finding the "fluents." And this same method in the language of Leibnitz is called "integration" or the finding of the "integrals." The two systems are therefore in no respect whatever different except in their origin and in their language; their rules, principles, applica tions and results are for all practical purposes the same. In Newton's system an ar or f is used to express the "fluent." To find the integral of a differential is to integrate that differential, and this process by which this integral is found is called integra tion. With the integral calculus a mathemati cian endeavors to transform the given expres sions into others which are differentials of known functions, and thus deduce formulas which may be applied to all similar forms. The number of such formulas is said to be unlim ited. The collection of Meyer Hirsch is a well arranged and though originally published long ago at Berlin is still useful.