De institutione musica II.12: De arithmetica geometrica armonica medietate.
Since we have said beforehand what was meanwhile to be treated of proportions, we must now speak of the means. A proportion is a certain comparison of two terms with each other. By terms I mean the sums of numbers. Proportionality is a gathering of equal proportions. Proportionality consists in three terms at the least. For when the first term keeps to the second the same proportion that the second keeps to the third, this is called proportionality, and among the three terms the middle is the second. There is a threefold division of the middle term joining these proportions. For either the difference of the lesser term to the middle and of the middle to the greatest is equal, but the proportion not equal, as in these numbers 1, 2, 3. Between 1 and 2 and between 2 and 3 a unit alone holds the difference; but the proportion is not equal; for 2 to 1 is double, 3 to 2 sesquialter. Or the proportion is equal in both, but not set in equal differences, as in these numbers 1, 2, 4. For 2 to 1 is double, just as four to two. But between four and two the difference is two, between two and the unit the difference is one. The third kind of mean is that which stands neither in the same proportions nor in the same differences, but as the greatest term stands to the least, so stands the difference of the greater terms to the difference of the lesser terms, as in these numbers 3, 4, 6. For 6 to 3 is double; between 6 and 4 the difference is two, between four and three one. But two compared to the unit is again double. So as the greatest term is to the least, so is the difference of the greater terms to the difference of the lesser. That mean in which the differences are equal is called arithmetic, that in which the proportions are equal geometric, and that which we described third harmonic. Let us set under them these examples:
We are not unaware that there are also other means of proportions, which we have mentioned in the Arithmetic. But for the present treatise these are meanwhile necessary. Among these three means, proportionality is properly and chiefly called geometric because it is woven wholly of equal proportions. Yet we shall use the same word without distinction, calling the others also proportionalities.