On the Adding of Proportions

The adding of proportions (as I have said) is nothing other than to reduce as many of them as one wishes — of one or of diverse genera — under a single denomination, which is also found in the extreme numbers, or terms, of those proportions when they are multiplied together; with this difference, that these extremes are mediated by other proportions, but those that arise from adding are immediate, as we shall see.

If, then, we had to add together two or more proportions, of one or of diverse genera, one must proceed in this manner: that is, to place first the greater and radical terms of the proportions that are to be added one under the other, or one facing the other, and likewise the lesser; then to multiply the greater ones one by the other, beginning from the first two, and the product of these by the third, and what arises by the fourth, and so on, one after another; and the product of such multiplication will be the greater term, containing the proportion that is to arise. Which done, one must likewise multiply the lesser ones one by the other; and the product will be the lesser term, which together with the greater contains the sought proportion.

Thus, if we had to add together the multiplied proportions, we shall first arrange them, as they are seen in the example; and, beginning from the greater terms of those, we shall multiply the first two — that is, 3 and 4 — one with the other, and we shall have 12. This, then, multiplied by 5, will give 60, which, multiplied by 6, will produce 360; and this number will be the greater term that is to arise from such a sum. In the same way we shall then multiply the lesser terms — that is, the 2 by the 3, and there will come 6, which, multiplied by the 4, will give 24; with this, then, the 5 shall be multiplied, and it will give 120; which number will be the lesser term that, together with the greater, contains the produced proportion — the same that is found in the extreme terms of the proportions multiplied above, as may be seen.

Having thus reduced such proportions under a single denominator — which is 3 — and under a single proportion — which is the Tripla — one may now see the difference that is found between adding and multiplying: that the one is found mediated by intervening proportions, and the other is without any mean in its extreme terms, as may be seen in the examples set below.

[Editorial note: Here Zarlino’s original sets the operation out as a table in two arrangements. In the first, marked “Primo,” the four proportions are listed vertically — Sesquialtera, Sesquiterza, Sesquiquarta, Sesquiquinta — with their greater terms (3, 4, 5, 6) in a left-hand column and their lesser terms (2, 3, 4, 5) in a right-hand column; beneath, the products give the sum: 360 . Tripla . 120. In the second, marked “Secondo modo,” the same proportions are arranged horizontally — the greater terms 3, 4, 5, 6 in an upper row and the lesser 2, 3, 4, 5 in a lower row — yielding the same result, the Tripla, 360 : 120.]