'NICOMACHUS, a Neo-pythagorean philosopher and mathematician, born at Gerasa in Arabia Petraea, flourished about A.D. Ioo. In his musical treatise he mentions Thrasyllus (d. 36), the astrologer and confidant 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) ApLOgn7TtK1) Eiaaywy17 (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 Boetius, and used as a schoolbook down to the Renaissance. (2) E-yxapiScov apµovt4s (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 ApLOj.o ruca. 1 Matthias may have been a cousin of Richard Nicolls; his family were of Islip, Oxford; he was secretary of the province, held various judicial positions, and was mayor of New York City in 1672. Matthias's son William (1657-1723), a lawyer, was a member of the New York Assembly from 1702 until his death and was speaker in 1702-1718; he received a royal patent for what is now the town of Islip on Long Island. Descendants of Richard and of Matthias Nicolls spell the name "Nicoll." OeoXc youµcva (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.
Introd. to Arith., .by R. Hoche (1866); Manual of Harmony, by C. de Jan in Musici scriptores 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, Vorlesungen'fiber 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_
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