CALCULUS, a branch of mathemat ical science. The lower or common anal ysis contains the rules necessary to calculate quantities of any definite mag nitude whatever. But quantities are sometimes considered as varying in mag nitude, or as having arrived at a given state of magnitude by successive varia tions. This gives rise to the higher analysis, which is of the greatest use in the physico-mathematical sciences. Two objects are here proposed: First, to descend from quantities to their ele ments. The method of effecting this is called the differential calculus. Second, to ascend from the elements of quanti ties to the quantities themselves. This method is called the integral calculus. Both of these methods are included un der the general name infinitesimal, or transcendental analysis. Those quanti ties which retain the same value are called constant; those whose values are varying are called variable. When vari able quantities are so connected that the value of one of them is determined by the value ascribed to the others, that variable quantity is said to be a func tion of the others. A quantity is infin itely great or infinitely small, with re gard to another, when it is not possible (io assign any quantity sufficiently large or sufficiently small to express the ratio of the two.
When we consider a variable quantity as increasing by infinitely small degrees, if we wish to know the value of those increments, the most natural mode is to determine the value of this quantity for any given period, as a second of time, and the value of the same for the period immediately following. This dif ference is called the differential of the quantity. The integral calculus, as has been already stated, is the reverse of the differential calculus. There is no vari able quantity expressed algebraically, of which we cannot find the differential; but there are differential quantities which we cannot integrate: some be cause they could not have resulted from differentiation; others because means have not yet been discovered of inte grating them. Newton was the first dis coverer of the principles of the infinites imal calculus, having pointed them out in a treatise written before 1669, but not published till many years after. Leib nitz, meanwhile, made the same dis covery, and published it before Newton, with a much better notation, which is now universally adopted.