**Nicomachus** was one of the leading members of the Pythagorean School whose *Introduction to Arithmetic* was the standard arithmetic text for more than 1000 years.

- Nicomachus himself refers to Thrasyllus who died in 36 AD so this gives lower limits on his dates.
- On the other hand Apuleius, the Platonic philosopher, rhetorician and author whose dates are 124 AD to about 175 AD, translated Nicomachus's Introduction to Arithmetic into Latin so this gives an upper limit on his dates.
- His argument is based on the fact that Marinus claimed that Proclus believed that he was the reincarnation of Nicomachus.
- Let us move from conjectures to more certain ground, and record that Nicomachus was a Pythagorean.
- Nicomachus wrote Arithmetike eisagoge (Introduction to Arithmetic) which was the first work to treat arithmetic as a separate topic from geometry.
- Unlike Euclid, Nicomachus gave no abstract proofs of his theorems, merely stating theorems and illustrating them with numerical examples.
- However Introduction to Arithmetic does contain quite elementary errors which show that Nicomachus chose not to give proofs of his results because he did not in general have such proofs.
- Many of the results were known by Nicomachus to be true since they appeared with proofs in Euclid, although in a geometrical formulation.
- Sometimes Nicomachus stated a result which is simply false and then illustrated it with an example which happens to have the properties described in the result.
- An example of this we look more closely at the results which Nicomachus quotes on perfect numbers.
- These statements must be merely false deductions from the fact that there were four perfect numbers known to Nicomachus, namely 6, 28, 496 and 8128.
- To illustrate Nicomachus's rather strange approach to numbers, giving the moral properties, we look at his description of abundant numbers and deficient numbers.
- She concludes that most Arabic texts on number theory written by mathematicians were influenced by both Euclid and Nicomachus, but were mainly influenced by Euclid.
- However, texts by non-mathematicians were most strongly influenced by Nicomachus.
- Nicomachus also wrote two volumes Theologoumena arithmetikes (The Theology of Numbers) which was completely concerned with mystic properties of numbers.
- Another work by Nicomachus which has survived is Manual of Harmonics which is a work on music.
- Again Nicomachus shows the influence of Pythagoras but also Aristotle's theories of music.
- But, unlike Euclid, who attempts to prove musical propositions through mathematical theorems, Nicomachus seeks to show their validity by measurement of the lengths of strings.
- Both Porphyry and Iamblichus wrote biographies of Pythagoras which quote from Nicomachus.
- From this evidence some historians have conjectured that Nicomachus also wrote a biography of Pythagoras and, although there is no direct evidence, it is indeed quite possible.

Born about 60, Gerasa, Roman Syria (possibly now Jarash, Jordan). Died about 120.

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Ancient Greek, Origin Jordan, Number Theory, Special Numbers And Numerals

Epochs: 1

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive