On the Multiplying of Proportions

Having sufficiently shown how the proportions arise, and their denominations, we shall give beginning to reasoning of their operations, which are five: namely, to Multiply, to Add, to Subtract, to Divide, and to Find their roots. As to the first, we must know that there have been some who held the opinion that Multiplying and Adding were one and the same thing, and some who held the opposite, that they were two separate operations; and the same they held of Subtracting and of Dividing. But, leaving the disputes aside, I shall demonstrate by example that such operations are not one and the same thing, but separate operations — a thing very useful and necessary to the present business. Coming, then, to the purpose, I say that Multiplying is a disposition of several proportions in a continued order, placed one after another in such a way that the lesser term of the one is the greater of the other, and so conversely. But Adding, I say, is an aggregation of several proportions gathered together under a single denomination.

Multiplying may be done in two ways. The first is when to one proportion another, or more, is multiplied, beginning from the left side and coming toward the right — which way we shall name Adjoining (soggiungere). The second is when we proceed contrariwise, that is, from the right toward the left — which way we shall call Prefixing (preporre). And because these two ways are both necessary and serve well, we shall therefore show the operation of the one way and of the other.

Beginning, then, with the first, I say that if we had to multiply together two or more proportions, of one same genus or of different ones (which matters not), we shall first dispose the proportions — contained in their radical terms — one after another in order, according as we intend to multiply them; and, taking the greater term of the second proportion to be multiplied (placed on the left side), we shall multiply it by the greater and by the lesser term of the first; and this latter [the lesser term of the first] we shall then multiply by the lesser term of the second; and we shall have three numbers, containing two continuous proportions. Now we shall multiply these by the greater term of the third proportion that is to be multiplied — which is third in the aforesaid order — beginning from the left and coming, one after another, toward the right side. Which done, taking again the lesser term of that proportion, we shall multiply it by the lesser of the products; and there will result four terms, or numbers, in which the multiplied proportions will be contained. And whenever it should be needful to adjoin to these proportions yet another, we shall always multiply the product-numbers by the greater term of the proportion we wish to adjoin, and the lesser of the products by its lesser; and from such multiplication we shall always have that which we seek.

But because examples move the intellect more to the understanding of a thing than do words — and most of all in operations with numbers — therefore, desiring to be understood, I shall come to the example. Let us suppose, then, that we have to multiply together four proportions, contained in the Superparticular genus, and let these be a Sesquialtera, a Sesquiterza, a Sesquiquarta, and a Sesquiquinta. First we shall place them one after another, in the order in which they are to be multiplied — that is, contained in their radical terms — in this manner: 3/2, 4/3, 5/4, 6/5. And then we shall multiply the greater term of the Sesquiterza, which is 4, by the 3 and the 2, the terms of the Sesquialtera; and from such multiplication we shall have 12 and 8, which likewise will contain the Sesquialtera. For the terms of any proportion, multiplied by whatsoever number, make no variation of quantity — as is manifest by trial, and by the 18th proposition of Book 7 of the Principles of Euclid, and by what Boethius says in chapter 29 of Book 2 of his Music. And such numbers we shall place beneath a straight, level line, which shall divide these from the proposed proportions.

This done, we shall multiply together the lesser terms of these two proportions, and there will come 6, which we shall place on the right side beside the 8; and we shall have multiplied the said proportions together — that is, adjoined to the Sesquialtera the Sesquiterza, between these terms, 12, 8, 6. Now, to adjoin to these the Sesquiquarta, we shall multiply these terms by its greater term, which is 5, beginning from the left side and coming toward the right, and we shall have 60, 40, 30. Which done, we shall multiply the lesser term of the first three, which is 6, by the lesser term of the Sesquiquarta, which is 4, and there will arise 24; which, set with the others, will give this order: 60, 40, 30, 24 — containing the Sesquialtera, the Sesquiterza, and the Sesquiquarta proportion.

The same we shall do when we wish to multiply to these the Sesquiquinta; for, multiplying first the aforesaid four terms by its greater, which is 6, there will come 360, 240, 180, 144; and then, the lesser of those shown, which is 24, being multiplied by the lesser term of that proportion, which is 5, it will give 120; which, set in its place — from such multiplication we shall have five numbers, or terms, that is, proportions: as between 360 and 240 the Sesquialtera; the Sesquiterza between 240 and 180; between 180 and 144 the Sesquiquarta; and between 144 and 120 the Sesquiquinta — although they are not found to be in their radical terms, as is seen here in the example set below.

[Editorial note: Here Zarlino’s original sets the worked example in a two-part table. The upper part, “Proportioni da moltiplicare” (proportions to be multiplied), shows the four superparticulars in their radical terms — Sesquialtera (3/2), Sesquiterza (4/3), Sesquiquarta (5/4), and Sesquiquinta (6/5) — with diagonal rules linking the shared terms. The lower part, “Proportioni moltiplicate” (multiplied proportions), shows the successive products on three lines: 12 . 8 . 6; then 60 . 40 . 30 . 24; and finally 360 . 240 . 180 . 144 . 120.]

When, therefore, we have to multiply and adjoin together many proportions, operating in the manner we have demonstrated, we shall always be able to attain our intent.