De institutione musica II.7: Omnem inaequalitatem ex aequalitate procedere eiusque demonstratio.
As the unit is the beginning of plurality and number, so is equality the beginning of proportions. For by three rules, as was said in the Arithmetic, we produce the multiple proportions from equality, and from the multiples inverted we beget the superparticular relations. Likewise from the superparticulars inverted we make the superpartient comparisons. Let three units be set, or three twos or three threes, or any equal terms, and let the first be equal to the first, placed in the following order, the second equal to the first and the second, the third equal to the first, two seconds and the third. For by this progress of number the double is made, the first proportion of multiplicity, as the description shows:
For the unit placed in the second rank is equal to the first unit placed in the upper place. Likewise two is equal to the first unit and the second unit; likewise four is equal to the first unit and two second units and the third unit, and 1, 2, 4 is a double proportion. If you do the same with these, the triple comparison will be begotten, and from the triple the quadruple, from the quadruple the quintuple, and so on runs the begetting of relations. Again, by the same three rules the superparticulars will be made, as we prove by one example. Let us now invert and set the greater number first: 4, 2, 1. Let the first be equal to the first, that is 4, the second to the first and the second, that is 6, the third to the first, two seconds and the third, that is 9; and these being set out, the proportion is seen to be sesquialter.
And if this is done with triples, the sesquitertian arises; if with quadruples, the sesquiquartan; and with like names on either side the proportionality will be born from multiplicity. From the superparticular inverted, however, the superpartient relation is drawn. Let the sesquialter comparison be set out inverted: 9, 6, 4. Let the first be equal to the first, that is 9, the second to the first and second, that is 15, the third to the first, two seconds and the third, that is 25. And let them be set in order in this manner:
The superbipartient relation is thus produced from inverted sesquialters. If a careful investigator comes to this speculation, he will wonder that from inverted sesquitertians the supertripartient is produced, and, with like names matched, that all the kinds of superpartients are begotten from superparticularity. From superparticulars not inverted, but remaining as they were begotten from the multiple, multiple superparticulars must be created. From superpartients remaining as they came forth from superparticulars, none other than multiple superpartients will be begotten. And so much for these; for this comparison has been discussed more carefully in the books of the Arithmetic.