Arithmetic is the science that treats of numbers (Gr. arithmos). It is sometimes divided into theoretical and practical; the former investigating the properties of numbers and their combinations, the latter applying the principles so established, in the form of rules, to actual calculations. Some restrict the term arithmetic to this art of reckoning, assigning the investigation of the principles to analysis. Anciently the science of reckoning was usually called logistic; while arithmetic dealt with forms of numbers, primary numbers, &c.
Arithmetic is said to have been first developed into a science in India; the Egyptians reckoned the god Thoth the first teacher of numeration. Among the ancient Greeks and Romans, arithmetic made little progress, owing to their clumsy modes of notation. Few of their writings on the subject have come down to us; the most important are those of Euclid, Archimedes, Diophantus, and Nicomachus. After the introduction of the decimal system and the Arabic or Hindu numerals (see NUMERALS), about the 11th century, arithmetic began to assume a new form. Early in the 13th century, Sacro Bosco wrote his Algorithmus seu Arithmetice Introductio; Pacioli wrote in the 15th century; and in the 16th, Adam Ries and Apianus. It was not till the 16th century that the Double Rule of Three, or Compound Proportion, was discovered, and decimal fractions were introduced. The invention of Logarithms in the 17th century is the last great step in advance that the art has made. The elementary operations in Arithmetic are Addition, Subtraction, Multiplication, and Division. The subjects of Fractions, Decimals, Practice, Proportion, Logarithms, Interest, Discount, Involution and Evolution, will be noticed in their proper places. Annuities, Averages, Insurance, Mensuration, and Partnership give rise to branches or special applications of arithmetic. The various methods of Notation and the forms of the numerals employed by the Greeks, Romans, Chinese, and other nations, are separately treated at Notation and Numerals. The theory of Num- bers (q.v.) is a subject cognate to arithmetic. See also ALGEBRA and ANALYSIS. Contrivances such as the Abacus, the Calculating Machine, and Napier's Bones, are treated under their own heads.