CHAPTER IV. On the Multiple Kind.

When the smaller number is contained by the larger exactly twice, it will be called double; if three times, triple; if four times, quadruple; and so on to infinity. See, if you please, examples of some of these species original: "specierum"; in this context, "species" refers to the specific types of mathematical ratios within a broader category or "genus.".

In this way, infinite species of this kind can be formed, as will be clear to the experimenter.

CHAPTER V.

On the second Kind of Proportions.

Which is called superparticular A ratio where the larger number contains the smaller number once, plus one single part of the smaller number (e.g., 3:2 or 4:3). These ratios are the basis for the most important musical intervals like the fifth and fourth.. Before I approach the explanation of this kind, it must be explained what an aliquot part and an aliquant part are.

An aliquot part, which is also called multiplicative, is that which, when multiplied, always perfectly restores its whole. For example: 3 is an aliquot part of the numbers 6, 9, 12, 15, and 18, because 3 taken twice adequately makes 6, its whole; likewise, the same number three multiplied by itself The author uses "multiplied by itself" loosely here to mean "taken three times" to reach 9. produces 9, its whole; and so with the others.

An aliquant part, or additive original: "aggregativa", is that which, taken in any way, does not equal its whole; it must either exceed it or fall short of it. For example: 3 is an aliquant part of the number 5; because 5 contains 3 once, and in addition two of its parts—that is, the number two—which, taken once, is not enough to restore the three, and taken twice, exceeds it. This is what is required for the essence of an aliquant, or additive, part.

These things having been established, I say: the superparticular kind is a proportion whose larger number contains the smaller once, and in addition its...