Calculus, in mathematics, a mode of calculating. In this broad signification we may speak of common arithmetic and algebra as forms of a calculus. Thus also trigonometry is called the calculus of sines, and the doctrine of chances is spoken of as the calculus of probabilities. The branches of mathematics to which the term is more especially applied are the differential calculus, integral calculus, calculus of variations, to which we may add the calculus of imaginaries, that of residuals, and that of quaternions. - The Imaginary Calculus investigates the nature of quantities which are required to fulfil apparently impossible conditions. It has been discovered by means of this calculus that every absurdity in geometry can be reduced to an attempt to measure a straight line in a direction different from that of its length; and that every algebraic absurdity can be represented by one symbol, always capable of this one geometrical interpretation. This extensively useful calculus has been chiefly developed by M. Cauchy. - The Residual Calculus investigates cases of apparent impossibility, arising from the attempt to measure a quantity which has become immeasurably great.

Imaginaries and residuals are chiefly employed as subsidiary to the operations of the higher species of calculus. - The Differential Calculus, called by the English fluxions, is the most valuable of mathematical modes, from the great variety of subjects to which it is applicable, and from the strength of its solvent power. Its discovery is justly assigned to the latter part of the 17th century, although there were doubtless some hints of it among earlier writers. Archimedes had demonstrated the area of a parabola to be two thirds of its circumscribing rectangle, and also the truth of his celebrated propositions concerning the sphere and the cylinder. Kepler, seizing the spirit of his method, introduced the words infinite and infinitesimal into geometry. Cavalieri, Roberval, and Term at enlarged the application of his mode. In the mean while Vieta, Cardan, Harriot, and others had improved algebra, and Descartes had applied it to geometry by his invaluable system of variable coordinates. Tims the way was prepared for Leibnitz and Newton, who, independently of each other, invented the differential calculus, although differing in the form in which they conceived of and expressed the same truths. Newton's discovery or invention was made in 1665, and that of Leibnitz several years later.

The notation of the latter was so convenient, and his mode of attacking the subject has such a practical superiority for the learner, that Newton's method of fluxions has now gone completely out of use; although in a metaphysical point of view Newton's mode is not open to the objections which may be brought against that of Leibnitz. The discovery of this method originated in the investigation of curved lines, but is extended to the consideration of every species of magnitude. Newton conceived of a curved line as generated by the motion of a point; and the spirit of his method consists in determining the velocity with which the point, at each instant, is moving in a given direction different from that of the line; that is, e. g., if the point be moving in a general southwesterly direction, in determining the velocity with which it souths compared with that with which it wests. The spirit of Leibnitz's method consists in supposing the curve to be composed of infinitely short straight lines, and in determining the direction of each of these lines.

Lagrange in his Theorie des fonctiom endeavored to treat the calculus from a purely algebraic point of view, and invented a new notation, but in his other works he always made use of the notation of Leibnitz. - The Integral Calculus is the reverse of the differential, and seeks to find from a known ratio between the changes of two quantities mutually dependent on each other what the relation or law of dependence between the quantities themselves must be; or, in the language of the calculus, the integral of a given function (i. e., law of dependence) is a required new function of which the given function is the differential. - The Calculus of Variations investigates the changes produced by gradually altering the laws of dependence which bind the variable quantities together. This invention of Lagrange crowns the calculus of functions, which by means of these five branches is capable, under a master's hand, of tracing out very complicated and intricate chains of inter-dependence in every part of the domain of quantity. And yet there is not one of these calculi that can answer all the questions which the physical sciences ask of it.

More powerful engines of analysis may yet be invented by future mathematicians. - The Calculus of Quaternions, published by Sir W. R. Hamilton in 1853, promises to do something toward supplying this defect. By combining in one notation the direction as well as the length of a line, he is able to express in a single symbolical sentence an amount of geometrical truth which in ordinary analytical geometry would require at least four sentences. No other writer has yet mastered this powerful instrument sufficiently to use it with ease; but the verdict of mathematicians is unanimous in praise of its ingenuity and probable future utility. - The difference between the powers of the principal calculi may be familiarly illustrated by the cycloid, a curve described by a nail head in the tire of a wheel rolling on a straight level road. The differential calculus would investigate the direction in which the nail head moves at each instant of its motion, and show the proportion between its rise, its fall, its horizontal motion, its motion through space, the curvature of its real path, and the revolution of the wheel at each instant.

The integral calculus would, from these elements, discover how far the nail head travelled in one revolution of the wheel, how much space is enclosed between its path and the ground, etc. The calculus of variations would consider the change made by the wheel rolling over a hill, or would show how the cycloid differs in its properties from similar curves. - The calculus is too difficult and abstruse for any popular exposition. The reader may find general views upon the subject in Davies's "Logic of Mathematics," and Comte's " Philosophy of Mathematics," translated by Prof. Gillespie, or in French in Carnot's Reflexions. For gaining a practical acquaintance with the science there are numerous accessible treatises, among which Church's and Courtenay's are well adapted to ordinary students, but Peirce's conducts much more rapidly into the highest walks. Of English treatises, Price's holds a high rank; but the most extensive treatise in the English language is that by Augustus De Morgan, published by the society for the diffusion of useful knowledge. The treatise of I. Todhunter is highly esteemed as a practical work.

Among the best German works is that of Dr. Martin Ohm. The French have been prolific writers upon the subject; among them Duhamel holds a high rank, and the treatise of Lacroix (3 vols. 4to, 1810-'19) is the most elaborate that has yet appeared in any language.