Nicomachus (Greek: Νικόμαχος; c. 60 – c. 120) was an important mathematician in the ancient world and is best known for his works Introduction to Arithmetic (Arithmetike eisagoge) and Manual of Harmonics in Greek. He was born in Gerasa, in the Roman province of Syria (now Jerash, Jordan), and was strongly influenced by Aristotle. He was a Pythagorean.



Little is known about the life of Nicomachus except that he was a Pythagorean and that he came from Gerasa. The age in which he lived (c. 100 AD) is only known because he mentions Thrasyllus in his Manual of Harmonics, and because his Introduction to Arithmetic was apparently translated into Latin in the mid 2nd century by Apuleius.[1] His Manual of Harmonics was addressed to lady of noble birth, at whose request Nicomachus wrote the book, which suggests that he was a respected scholar of some status.[1] He mentions his intent to write a more advanced work, and how the journeys he frequently undertakes leaves him short of time.[1]


Introduction to Arithmetic

Introduction to Arithmetic (Greek: Ἀριθμητικὴ εἰσαγωγή), the lesser work on arithmetic. As a Neo-Pythagorean, Nicomachus was often more interested in the mystical properties of numbers rather than their mathematical properties. He distinguishes between the wholly conceptual immaterial number, which he regards as the 'divine number', and the numbers which measure material things, the 'scientific' number.[1] He writes extensively on numbers, especially on the significance of prime numbers and perfect numbers and argues that arithmetic is ontologically prior to the other mathematical sciences (music, geometry, and astronomy), and is their cause. Boethius' De institutione arithmetica is in large part a Latin translation of this work.

Manual of Harmonics

Manual of Harmonics (Greek: Ἐγχειρίδιον ἁρμονικῆς). This is the first important music theory treatise since the time of Aristoxenus and Euclid. It provides the earliest surviving record of the story of Pythagoras's epiphany outside a smithy that pitch is determined by numeric ratios. Nicomachus also gives the first in depth account of the relationship between music and the ordering of the universe via the "music of the spheres." Nicomachus's discussion of the governance of the ear and voice in understanding music unites Aristoxenian and Pythagorean concerns, normally regarded as antitheses.[2] In the midst of theoretical discussions, Nicomachus also describes the instruments of his time, also providing a valuable resource. In addition to the Manual, ten extracts survive from what appear to have originally been a more substantial work on music.

Lost Works

The works which are lost are:

  • Art of Arithmetic (Greek: Τέχνη ἀριθμητική), the larger work on arithmetic, mentioned by Photius.
  • A larger work on music, promised by Nicomachus himself, and apparently referred to by Eutocius in his comment on the sphere and cylinder of Archimedes.
  • An Introduction to Geometry, referred to by Nicomachus,[3] although whether it was his work in unclear.
  • Theology of Arithmetic (Greek: Θεολογούμενα ἀριθμητικῆς), on the Pythagorean mystical properties of numbers in two books mentioned by Photius. There is an extant work sometimes attributed to Iamblichus under this title written two centuries later which contains a great deal of material thought to have been copied or paraphrased from Nicomachus' work.
  • A Life of Pythagoras, one of the main sources used by Porphyry and Iamblichus, for their (extant) Lives of Pythagoras.
  • A collection of Pythagorean dogmata, referred to by Iamblichus.
  • On Egyptian festivals (Greek: Περὶ ἑορτῶν Αἰγυπτίων), is mentioned by Athenaeus, but whether by this Nicomachus is uncertain.

See also

  • Monad
  • Dyad
  • Triad
  • Pentad
  • Tetrad
  • Decad
  • Nicomachus's theorem


  1. ^ a b c d Henrietta Midonick, (1965), The Treasury of Mathematics, Volume 2, pages 15-16. Penguin Books.
  2. ^ Levin, "Nicomachus of Gerasa," Grove Music Online.
  3. ^ Nicomachus, Arithmetica, ii. 6. 1.


  • Flora R. Levin, "Nicomachus of Gerasa," Grove Music Online, visited 3 June 2007.
  • Andrew Barker, editor, Greek Musical Writings vol 2: Harmonic and Acoustic Theory (Cambridge: Cambridge University Press, 1989), pp. 245–69.

External links

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