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Nicomachus of Gerasa
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Nicomachus of Gerasa

The following summary of Nicomachus of Gerasa is sourced from a Stanford University article on pythagoreanism:

Nicomachus of Gerasa (modern Jerash in Jordan) was probably active a little before Numenius, in the first half of the second century AD. Unlike Neopythagoreans such as Eudorus, Moderatus and Numenius, whose works only survive in fragments, two complete works of Nicomachus survive, Introduction to Arithmetic and Handbook of Music. More than anyone else in antiquity he was responsible for popularizing supposed Pythagorean achievements in mathematics and the sciences. The Handbook of Music gives the canonical but scientifically impossible story of Pythagoras' discovery of the whole number ratios, which correspond to the basic concordant intervals in music: the octave (2:1), fifth (3:2), and fourth (4:3); he supposedly heard the concords in the sounds produced by hammers of varying weights in a blacksmith's shop, which he happened to be passing (Chapter 6 — translation in Barker 1989, 256 ff.). In the next century, Iamblichus took this chapter over virtually verbatim and without acknowledgement in his On the Pythagorean Life (Chapter 26) and it was repeated in many later authors. The harmonic theory presented by Nicomachus in the Handbook is not original and is, in fact, somewhat retrograde. It is tied to the diatonic scale used by Plato in the Timaeus (35b-36b), which was previously used by the Pythagorean Philolaus in the fifth-century (Fr. 6a Huffman) and shows no awareness of or interest in the more sophisticated analysis of Archytas in the fourth century BC. Nicomachus also relies heavily and without acknowledgement on a non-Pythagorean treatment of music, Aristoxenus' Elementa Harmonica, many of the ideas of which he assigns to the Pythagoreans (e.g. in Chapter 2; see Barker 1989, 245 ff.).

The Handbook was influential because it put forth an accessible version of Pythagorean harmonics. Nicomachus may have written a more detailed treatment of Pythagorean harmonics in a lost Introduction to Music, which was probably the basis of Books I-III of Boethius' De Institutione Musica (see Bower in Boethius 1989, xxviii). Even more influential than his work on harmonics was Nicomachus' Introduction to Arithmetic. Again Nicomachus was not an original or particularly talented mathematician, but this popularizing textbook was widely influential. There were a series of commentaries on it by Iamblichus (3rd AD), Asclepius of Tralles (6th AD), and Philoponus (6th AD) and it was translated into Latin already in the second half of the second century by Apuleius. Most importantly, Boethius (5th-6th c. AD) provides what is virtually a translation of it in his De Institutione Arithmetica, which became the standard work on arithmetic in the middle ages.

In the Introduction to Arithmetic, Nicomachus assigns to Pythagoras the Platonic division between the intelligible and sensible world, quoting the Timaeus as if it were a Pythagorean text (I 2). He also assigns Aristotelian ideas to Pythagoras, in particular a doctrine of immaterial attributes with similarities to the Aristotelian categories (I 1). Nicomachus divides reality into two forms, magnitude and multitude. Wisdom is then knowledge of these two forms, which are studied by the four sciences, which will later be known as the quadrivium: arithmetic, music, geometry and astronomy. He quotes a genuine fragment of Archytas (Fr. 1 Huffman) in support of the special position of these four sciences. Nicomachus presents arithmetic as the most important of the four, because it existed in the mind of the creating god (the demiurge) as the plan which he followed in ordering the cosmos, so that numbers thus appear to have replaced the Platonic forms as the model of creation (I 4). It is striking that, along with this Platonization of Pythagoreanism, Nicomachus does give an accurate presentation of Philolaus' basic metaphysical principles, limiters and unlimiteds, before attempting to equate them with the Platonic monad and dyad (II 18).

Another work by Nicomachus, The Theology of Arithmetic, which can be reconstructed from a summary by Photius and an anonymous work sometimes ascribed to Iamblichus and known as the Theologoumena Arithmeticae (Dillon 1977, 352-353), suggests that he largely returned to the system of principles found in Plato's unwritten doctrines and did not follow Eudorus and Moderatus in attempts to place a supreme god above the demiurge. Nicomachus apparently presents the monad as the first principle and demiurge, which then generates the dyad, but much is unclear (Dillon 1977, 353-358). The Theology of Arithmetic may have been most influential in its attempt to set up an equivalence between the pagan gods and the numbers in the decad, which was picked up later by Iamblichus and Proclus (Kahn 2001, 116). Nicomachus also wrote a Life of Pythagoras, which has not survived but which Porphyry (e.g. VP 59) and Iamblichus used.





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