Etymologiae, Book III, chapters 13-15: De numeris geometriae to De musica et eius nomine
Chapter XIII. On the numbers of geometry According to Geometry you seek numbers thus. Its extremes multiplied make as much as the means doubled: for example 6 and 12 multiplied make seventy-two, the means 8 and 9 multiplied make the same.
Chapter XIV. Exposition of the figures written below Another reckoning in the motion of the stars is likewise gathered in eight figures: whether they are diametrical or quadrate or trigonal or hexagonal or unconnected or together or circumferent, that is, overtaking or overtaken. They are diametrical when five signs lie between. Tetragonal, when two. Hexagonal, when one. Unconnected (asyndeton), when none. Together, when they are in the same small part. Overtaking, when it comes upon or makes the act. Overtaken, when it precedes. Trigonal, when three lie between. Likewise, according to another reckoning, there are eight differences, that is: sign, parts, limits, conjunction, whether retrograde or on a direct course, latitude and longitude. The reckoning of the inner form. A question of this kind could arise at this place. Since in the order of number VIII come first, he has put IX first here, since in the reckoning of arithmetic or geometry VIII are more than IX. For VIII is a cube or solid, that is, a body, which can find no more. IX, however, are surfaces, that is, a thing which is not full, but needs perfection.
Here two cubes, that is, two solidities, are found in this way. Six is the first perfect number; for it is divided by even numbers thus: a sixth by one; in a third by two; three twos, six; in a half, that is, twice three, six. Another which you divide thus by even numbers you will find, that suits the proposal. Among the first in order, that is, X, which on account of the first perfect number, multiplying with the first row, six times nine, 54; nine times six, 54. And the matter is known to have had so many parts, not undeservedly, of two, of which it has one in such an order: 1, 2, 3, 4, 9, 8, the others together 27.
Chapter XV. On music and its name Music is skill in modulation, consisting in sound and song. It is called Music by derivation from the Muses. The Muses are so named from apo tou masthai, that is, from seeking, because through them, as the ancients wished, the force of songs and the modulation of the voice were sought. Their sound, since it is a sensible thing, flows past into past time and is impressed on the memory. Hence it was feigned by the poets that the Muses are the daughters of Jove and Memory. For unless sounds are held in memory by man, they perish, because they cannot be written.