De institutione musica II.15: Quemadmodum ab aequalitate supradictae processerint medietates.
But we must say for a little how these proportionalities are begotten from equality. For it was said before that what the unit is worth in number, equality is worth in proportions, and as the unit is the head of number, so equality is the beginning of proportions.
Therefore the arithmetic mean will be born from equality in this way. Three equal terms being set, there are these two ways by which this proportionality is produced. Let the first be set equal to the first, the second to the first and the second, the third to the first, second and third. Which is shown by this example. Let there be three units. Let the first be set equal to the first, that is one, the second to the first and second, that is 2, the third to the first, second and third, that is 3; and the arrangement will be this:
Again let there be three twos constituted in equality: 2, 2, 2. Let the first be set equal to the first, that is 2, the second to the first and second, that is 4, the third to the first, second and third, that is 6; and the arrangement will be this:
Again the same with three:
But in these it must be observed that if the unit is constituted as the beginning of equality, the unit will also be in the differences of the numbers, but the numbers themselves leave none between them. If two holds the equality, two is the difference and one number is always passed over between the terms. If three, the same is the difference, and between the numbers two naturally placed are passed over, and so on in this manner. There is also another way of begetting arithmetic proportionality. Let three equal terms be set, and let the first be set equal to the first and second, the second to the first and two seconds, the third to the first, two seconds and the third. As if there are three units. Let the first be equal to the first and second, that is 2, the second to the first and two seconds, that is 3, the third to the first, two seconds and the third, that is 4. Here the unit holds the difference of the terms. For between two and the unit and between three and two the difference is a unit. No natural number is passed over. For after the unit comes two, and after two, three, naturally placed. Let the same be done again with two, and let there be three twos, and let the first be equal to the first and second, that is four, the second to the first and two seconds, that is six, the third to the first, two seconds and the third, that is eight.
Here also two holds the difference of the terms, with one passed over naturally between them. For between 4 and 6 five is naturally placed, between 6 and 8 seven. But if three is the beginning of equality, three will be the difference, with always one fewer of the numbers passed over. And the same is seen in four and five. And what we pass over in silence for brevity the careful reader will find from the same rules by himself.
We showed how the geometric proportionality can be found from equality when we showed how all inequality flows from equality. Yet, unless it is wearisome, it must now be briefly repeated. Three equal terms being set, let the first be set equal to the first, the second to the first and second, the third to the first, two seconds and the third. And let the same be done continuously. And so from equality geometric proportionality takes its beginning. But of the properties of these proportions we have spoken most carefully in the Arithmetic. If the reader comes to these instructed by those, he will be disturbed by no error of doubt.
The harmonic mean, of which we must now treat a little more fully, is begotten by this reason. Let it be constituted, if we care to make doubles, three equal terms being set: the first equal to the first and two seconds, the second to two firsts and two seconds, the third to once the first, twice the second and three times the third. And in this manner let there be units:
Let the first, then, be made equal to the first and two seconds, that is three, the second to two firsts and two seconds, that is 4, the third to the first, two seconds and three thirds, that is 6. And if equality is constituted in twos or in threes, the same reason of the mean appears, the terms and differences standing apart by double, as the descriptions below show.
But if a triple proportion is to be made in the extremities, three equal terms being set, the first is to be made from the first and second, the second from the first and two seconds, the third from the first, two seconds and three thirds, as is the description below: