​NICOMACHUS, a Neo-Pythagorean philosopher and mathematician, born at Gerasa in Arabia Petraea, flourished about a. d. 100. In his musical treatise he mentions Thrasyllus (d. 36), the astrologer and conhdant of Tiberius, and his Arithmetic was translated by Apuleius, who wrote under Antoninus Pius and Marcus Aurelius. He is the author of two extant treatises: (1) 'Apt0;.u7rt/vi; elowywyh (Introduction to Arithmetic), a metaphysical account of the theory and properties of numbers, and the first work in which arithmetic was treated quite independently of geometry. It was extremely popular, was the subject of commentaries by Iamblichus (ed. H. Pistelli, 1894) and others, was translated into Latin by Apuleius (according to Cassiodorus, the translation itself being lost) and Boétius, and used as a schoolbook down to the Renaissance. (2) 'E'yXeLpl6Lo1/ (l.P]J.01/LK'l'I$ (Manual of Harmony), complete in one book, to which are erroneously appended as a second book some fragments probably belonging to a larger treatise On Music now lost. It is the oldest authority on the Pythagorean theory of music. Photius (cod. 187) also mentions a work by Nicomachus called 'Apt0/.tn-ruca -0eo)o'yoi9/.teva (The Theology of Arithmetic), written in a spirit of Pythagorean mysticism and Oriental superstition, and setting forth the application of arithmetic, or rather of the first ten numbers, to the origin and attributes of the gods. -But the extracts in Photius are now generally attributed to Iamblichus. Other works of Nicomachus were: a Life of Pythagoras and a Collection of Pythagorean Doctrines, the chief source of the life of Pythagoras and the account of his philosophy by Iamblichus.

Editions.-Introd. to Arith., by R. Hoche (1866); Manual of Harmony, by C. de Ian in Mnsici scriptures Graeci (1895), with account of Nicomachus and his works, and French translation, with bibliography and notes, by C. E. Ruelle (1881); Theology of Arithmetic, by F. Ast (1817); see W. Christ, Geschichte der griechischen Literatur (1898); M. Cantor, Vorlesnngen über Geschichte der Mathematik, i. (1894) p. 400, and J. Gow, A Short History of Greek Mathematics (1884), p. 88, both of whom give summaries of the Arithmetic.