Chapter 23: On the numbers of music

1. You inquire into the numbers according to music as follows: setting down the extremes, as for example, VI and XII, you see by how many units VI is surpassed by XII, and it is by VI units; you square it; six times six make XXXVI. You add those first-mentioned extremes, VI and XII; together they make XVIII; you divide XXXVI by XVIII; two is the result. This you add to the smaller amount, VI namely; the result will be VIII and it will be the mean between VI and XII. Because VIII surpasses VI by two units, that is by a third of six, and VIII is surpassed by XII by four units, a third part [of twelve]. By what part, then, the mean surpasses, by the same is it surpassed.

2. Just as this proportion exists in the universe, being constituted by the revolving circles, so also in the microcosm—not to speak of the voice—it has such great power that man does not exist without harmony.[263]

[263] The general sense of the passage: “ut sine ipsius
perfectione etiam homo symphoniis carens non consistat.” 3, 23,
2. See p. 65.

ON ASTRONOMY