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 Calcutta. Differential or Fluxionary Calculus. La Grange gave this name to his particular view of the subject. (Théorie 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 Differentialis, 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 Dérivations.)

The Calculus of Probabilities. This treats of everything connected with the Doctrine of Chances. The most valuable work on this subject is La Place's Théorie 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.