De institutione musica II.17: Quemadmodum inter duos terminos supradictae medietates vicissim locentur.
It is customary for two terms to be given and proposed, so that between them we may place now an arithmetic mean, now a geometric, now a harmonic. We have spoken of these in the arithmetical books also; yet let us now briefly set out the matter itself once more. If the arithmetic mean is sought, the difference of the given terms must be observed, divided, and added to the lesser term. Let the terms 10 and 40 be set at either end, and let their mean be sought according to arithmetic proportionality. First I look at the difference between the two, which is 30. I divide this; it makes 15. I add this to the lesser term, that is, to ten; it makes 25. If therefore this number is placed between 40 and 10 as the middle, an arithmetic proportionality arises in this way: 10, 25, 40. Likewise let us place a geometric mean between the same terms. I multiply the extremes by their own numbers, as 10 by 40; it makes 400. Of this I take the square root; it makes 20, for twenty times 20 make 400. If I place these 20 as the middle between 10 and 40, the geometric mean arises as set out below. But if we seek the harmonic mean, we join the extremes together, as 10 and 40; they make 50. Their difference, which is 30, we multiply by the lesser term, that is, by ten, so that ten times 30, which is 300, results. This we divide by 50; it makes 6. When we have added this to the lesser term, 16 results. If therefore we place this number as the middle between 10 and 40, the harmonic proportionality is obtained: 10, 16, 40.