"The even is that which can be divided into two equal parts without a unit intervening in the middle; and the odd is that which cannot..."

“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.”

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

“Nicomachus of Gerusa: Introduction to Arithmetic . Tr. Martin Luther D'Ooge with studies in Greek arithmetic by Frank Egleston Robbins and.”

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

“If geometry exists, arithmetic must also needs be implied... But on the contrary 3, 4, and the rest might be 5 without the figures existing to which they give names. Hence arithmetic abolishes geometry along with itself, but is not abolished by it, and while it is implied by geometry, it does not itself imply geometry.”

“And once more is this true in the case of music; not only because the absolute is prior to the relative, as 'great' to 'greater' and 'rich' to 'richer' and 'man' to 'father,' but also because the musical harmonies, diatessaron, diapente, and diapason, are named for numbers; similarly all of their harmonic ratios are arithmetical ones, for the diatessaron is the ratio of 4 : 3, the diapente that of 3 : 2, and the diapason the double ratio ; and the most perfect, the di-diapason , is the quadruple ratio .”