CALCULUS primarily denotes a little stone or pebble, anciently used in making computations, taking suffrages, playing at tables, and the like. In after times, pieces of ivory, and counters struck of silver, gold, and other
matterials,
were used in lieu of the calculi, but still retained the ancient name. Computists were by the lawyers called calculones, when they were either slaves or newly freed men; those of a better condition were named calculatores or numerarii ; there was ordinarily one of these in each family of distinction. The Roman judges anciently gave their opinions by calculi, which were white for absolution, and black for condemnation. Hence calculus albus, in ancient writers, denotes a favourable vote, either in the case of a person to be absolved and acquitted of a charge, or elected to some dignity or post; and calculus niger had a contrary signification. This usage is said to have been borrowed from the Thracians, who marked their happy or prosperous days by white, and their unhappy by black pebbles, which were put each night into an urn.

Besides the diversity of colour, there were some calculi also which had figures or characters engraven on them, as those which were in use in taking the suffrages both in the senate and in assemblies of the people. These calculi were made of thin wood, polished and covered over with wax. Their form is still seen in some medals of the Cassian family; and the manner of casting them into the urns is represented in the medals of the Licinian family. The letters marked upon these calculi were U. R. for uti rogas, and A. for antiquo; the first of which expressed an approbation of the law, the latter a rejection of it. Afterwards the judges who sat in capital causes used calculi marked with the letter A. for absolvo ; C. for condemno ; and-N. L. for non liquet, signifying that a more full information was required.

Calculus is also used by ancient writers for a kind of weight equal to two grains of cicer. Some make it equivalent to the siliqua, which is equal to three grains of barley. Two calculi composed the ceratium.

Calculus, in Mathematics, is a general name given to various ways of investigating or establishing the truths of that science by the aid of conventional symbols or characters which represent the things treated of, also the operations to be performed on them, and the relations in which they stand to one another. Thus we have the common Arithmetical Calculus, and the Algebraic Calculus. The term is applied to a considerable number of distinct mathematical theories, the principal of which are these:

The Differential Calculus and the Integral Calculus. The invention of these is claimed for Leibnitz. They are identical with the Fluxionary Calculus, the invention of Newton.

The Calculus of Partial Differences, which is a branch of the Differential and Integral Calculus.

The Calculus of Variations, another branch of the same theory. Its principal object is to determine when mathematical quantities, subject to certain conditions, are the greatest or least possible. This theory, first broached by James and John Bernoulli, was perfected by Euler and La Grange, who have discussed it in their writings. There is a distinct treatise on this subject by the late Mr Woodhouse of Cambridge.

The Calculus of Exponentials, or Exponential Calculus. This may include the doctrine of logarithms; but the name is commonly applied to the method of finding the differentials or fluxions of exponential and logarithmic quantities. John Bernoulli was the first who treated of this subject as a distinct calculus. (Bernoulli Opera, tom. i. page 179.)

The Calculus of Functions, the same in effect with the [7:5:788]Differential or Fluxionary Calculus. La Grange gave this name to his particular view of the subject. ( Theorie des Fonctions Analytiques, also Leçons sur le Calcul des Fonctions.)

The Calculus of Finite Differences. This investigates the properties of quantities by means of their differences; it is of great value in the summation of infinite series. Brooke Taylor’s Methodus Incrementorum, Stirling’s Methodus Differentiatis, and Emerson’s Method of Increments, also his Differential Method, all treat of this subject. There are also various treatises in works on the Differential Calculus, as Lacroix, &c

The Calculus of Derivations. This is applicable to the doctrine of series, and is due to a continental mathematician, Arbogast, who has composed a treatise on the subject. (Arbogast Du Calcul des Derivations.)

The Calculus of Probabilities. This treats of every thing connected with the Doctrine of Chances. The most valuable work on this subject is La Place’s Theorie Analytique des Probabilités.

The Calculus of Sines. This branch of mathematical science was embodied in a distinct form by Euler. See his various writings, particularly his Analysis Infinitorum. We have explained this calculus in our article Algebra.

There are some other mathematical theories which have been distinguished each as a separate Calculus, as Landen’s Residual Analysis, Glenie’s Antecedental Calculus (Edin. Phil. Trans, vol. iv.), &c.

Calculus Minervoe, among the ancient lawyers, denoted the decision of a cause in regard to which the judges were equally divided. The expression is taken from the history of Orestes, represented by Aeschylus and Euripides; at whose trial before the Areopagites, for the murder of his mother, the votes being equally divided for and against him, Minerva interposed, and gave the casting vote or calculus in his behalf.