Nicomachus: Number

“Number is limited multitude or a combination of units or a flow of quantity made up of units; and the first division of number is even and odd.”

Nicomachus

Context: Nicomachus of Gerasa: Introduction to Arithmetic (1926), Book I, Chapter VII

“Every number is at once half the sum of the two on either side of itself…”

Nicomachus

Nicomachus of Gerasa: Introduction to Arithmetic (1926)

“Scientific number, being set over such things as these, should be harmoniously constituted, in accordance with itself; not by any other but by itself.”

Nicomachus

Book I, Chapter VI
Nicomachus of Gerasa: Introduction to Arithmetic (1926)

“Some… agreeing with Philolaus, believe that the proportion is called harmonic because it attends upon all geometric harmony, and they say that 'geometric harmony' is the cube because it is harmonized in all three dimensions, being the product of a number thrice multiplied together. For in every cube this proportion is mirrored; there are in every cube 12 sides, 8 angles and 6 faces; hence 8, the [harmonic] mean between 6 and 12, is according to harmonic proportion…”

Nicomachus

this harmonic proportion may be expressed as <math>\frac{12}{6}=\frac{12-8}{8-6}</math> or inversely.
Nicomachus of Gerasa: Introduction to Arithmetic (1926)

“Plato, too, at the end of the thirteenth book of the Laws, to which some give the title The Philosopher… adds: "Every diagram, system of numbers, every scheme of harmony, and every law of the movement of the stars, ought to appear one to him who studies rightly; and what we say will properly appear if one studies all things looking to one principle, for there will be seen to be one bond for all these things, and if anyone attempts philosophy in any other way he must call on Fortune to assist him. For there is never a path without these… The one who has attained all these things in the way I describe, him I for my part call wisest, and this I maintain through thick and thin." For it is clear that these studies are like ladders and bridges that carry our minds from things apprehended by sense and opinion to those comprehended by the mind and understanding, and from those material, physical things, our foster-brethren known to us from childhood, to the things with which we are unacquainted, foreign to our senses, but in their immateriality and eternity more akin to our souls, and above all to the reason which is in our souls.”

Nicomachus

Footnote: The Epinomis, from which Nicomachus here quotes 991 D ff., is now recognized as not genuinely Platonic. Nicomachus doubtless cited the passage from memory, for he does not give it exactly...
Nicomachus of Gerasa: Introduction to Arithmetic (1926)

Books , Law , Way , Soul

“Without the aid of these, then, it is not possible to deal accurately with the forms of being nor to discover the truth in things, knowledge of which is wisdom, and evidently not even to philosophize properly, for "just as painting contributes to the menial arts toward correctness of theory, so in truth lines, numbers, harmonic intervals, and the revolutions of circles bear aid to the learning of the doctrines of wisdom," says the Pythagorean Androcydes. Likewise Archytas of Tarentum, at the beginning of… On Harmony, says… in about these words: "It seems to me that they do well to study mathematics, and it is not at all strange that they have correct knowledge about each thing, what it is. For if they knew rightly the nature of the whole, they were also likely to see well what is the nature of the parts. About geometry, indeed, and arithmetic and astronomy, they have handed down to us a clear understanding, and not least also about music. For these seem to be sister sciences; for they deal with sister subjects, the first two forms of being."”

Nicomachus

Nicomachus of Gerasa: Introduction to Arithmetic (1926)

Wisdom , Truth , Art , Music