On the Institution of Music (De institutione musica)

Why multiplicity excels the others. 105.

118

But in these matters, one must consider that the "multiple" genus of inequality original: "multiplicis inequalitatis genus"; this refers to ratios where the larger number is a clean multiple of the smaller, like 2:1 or 3:1 seems far older and more fundamental than the other two types. For the natural arrangement of numbers is perfected in multiples when compared to unity (which is first). In contrast, the "superpartient" ratio a ratio like 5:3, where the larger number contains the smaller plus several parts of it is not formed by a simple sequence, but is created through an unlimited scattering of those smaller numbers arranged after unity, such as three and the rest, in a certain way. The superpartient ratio actually occupies a much more "backward" or secondary position. It is not formed by continuous numbers, but by skipped ones; it must necessarily be created by skips: here by skipping one number, there by two, here by three, here by four, and so it grows into infinity. Furthermore, multiplicity begins from one; "superparticularity" ratios like 3:2 or 4:3, where the larger number contains the smaller plus one part of it begins from two; but the superpartient genus takes its start only from three. But enough of these things for now. Now it will be necessary to present certain things that the Greeks call axioms original: "axiomata"; fundamental truths that require no proof. We will understand what they are pointing toward only when we deal with the demonstration of each individual thing.

106.

What square numbers are, and an observation about them

A square number is one produced by a double dimension into equal parts, such as twice two, three times three, four times four, five times five, or six times six. Here is a description of them:

Thus, the natural number arranged in the top row is the side side: what we would today call the "square root" of the squares described below them. Continuous natural squares are those that follow one another in order, such as 4, 9, 16, and so on. If I subtract a smaller

continuous square from a larger continuous square, the remainder will be exactly equal to the sum of the sides of both squares. For example, if I take four from nine, five remains; this five is the sum of two and three, which are the sides of those two squares. Likewise, if nine is taken from the number marked 16, seven remains; this is the sum of three and four, which are the sides of the aforementioned squares. The same pattern holds for the rest. But if the squares are not continuous i.e., they are not next to each other in the sequence but one square is skipped between them, then half of the remainder is equal to the sum of both sides. For instance, if I take the square 4 from the square 16, 12 remains; the half of 12 is 6, which is the sum of the two sides (2 and 4). The same method applies to the others. If indeed two squares are skipped, then the sum of the sides will be the third part of the remainder. For example, if I take 4 from 25 (skipping the squares 9 and 16), the remainder is 21. Their sides are 2 and 5, which make 7; and 7 is the third part of the remainder, 21. This is the rule: if three are skipped, the sum of the sides is the fourth part of the remainder left by subtracting the smaller from the larger. If four are skipped, it is the fifth part. The name of the fractional part is always one number higher than the number of skipped squares.

That all inequality proceeds from equality, and its demonstration. 107.

Just as unity the number 1 is the principle of the plurality of numbers, so equality the ratio 1:1 is the principle of proportions. For by three rules, as has been said in Arithmetic, we may produce a multiple proportion from equality. From the inversion of multiples, we have all the superparticulars ready for us. Likewise, from the inversion of the superparti—