In How Many Ways One Quantity Is Compared to Another

The containing of one [number] by another, and the being contained, is not always taken simply, but indeed in another manner as well. Whence, this comparison being considered more minutely, from each of these genera [of inequality] there arise another five; for the greater number can be compared to the lesser in five ways and no more, and likewise, conversely, the lesser to the greater.

For in the proportion of greater inequality, the greater number contains the lesser within itself: more than once and entirely; or once only, and besides a part of that lesser, called an Aliquot part; or once only, and besides a part of it, called a Non-aliquot part. The greater number also contains the lesser more than once, and besides an aliquot part of it; or else it contains it several times, and besides a non-aliquot part of it.

From the first mode arises that genus of proportion called Multiple; from the second, that called Superparticular; and from the third, that named Superpartient. And these are called simple genera; for from the fourth mode is generated another, called Multiple superparticular, and from the fifth and last is born that which is termed Multiple superpartient — which genera are compounded from the first and from the two following it, as is understood from the name of each by itself; and these are called Composite.

In the proportion of Lesser inequality, then, the lesser number is likewise contained by the greater in five ways, and no more; and thus one has another five genera, called of lesser inequality. These are denominated by the proper names of the aforesaid, with only the particle “Sub” — which signifies “Under” — added to mark their difference; and they are named Submultiple, Subsuperparticular, Subsuperpartient, Submultiple superparticular, and Submultiple superpartient: of which the first three are likewise called simple, but the other two composite. And since these last five genera are not apt for the generation of musical consonances, as we shall see in the second part, I shall therefore reason no further about them.