On the Subtracting of Proportions

The third operation is called Subtracting, which is nothing other than the removing of a proportion, or lesser quantity, from a greater, in order to know the differences — that is, by how much quantity the one surpasses, or is surpassed by, the other; which operation is done in this manner.

First, one must dispose the radical terms of the proportions in the manner of a square figure, so that the terms of the greater be in the upper part, and those of the lesser in the lower, one beneath the other — taking care, however, that the greater terms of the one and of the other hold the left side, and the lesser the right. This done, we shall cross-multiply the terms; that is, the greater of the upper proportion by the lesser of the lower, and likewise the greater of the lower by the lesser of the one placed above; and the products we shall set perpendicularly beneath the multiplied terms placed above, dividing them from the proportions by a straight, level line; and then from such products one will have by how much the one proportion surpasses the other, and the difference that is found between the one and the other.

Wishing, then, to remove a Sesquiterza from a Sesquialtera, and to know by how much the Sesquialtera surpasses the Sesquiterza, and the difference that is found between them, we shall operate in this manner. We shall first order the terms of the proportions in the way they are seen in the example set below; then, having drawn a straight, level line beneath, under it we shall place the terms produced by the multiplication that is made of one term with another. Beginning from the 3, the greater term of the Sesquialtera, we shall multiply it by the 3, the lesser of the Sesquiterza, and the product, which will be 9, we shall set perpendicularly beneath the 3, the greater term of the Sesquialtera, under the line on the left side; and this will be the greater term of the proportion that is to arise, which will contain the difference we seek. Which done, we shall multiply the 4, which is the greater term of the Sesquiterza, by the 2, which is the lesser of the Sesquialtera; and the product, which will be 8, will come to be the lesser term of the proportion containing the aforesaid difference; for, set under the named line, perpendicularly beneath the 2, the lesser term of the Sesquialtera, we shall have the proportion Sesquiottava, contained between the 9 and the 8 — which I say is the difference by which the one is greater than the other, as is seen here.

[Editorial note: Here Zarlino’s original contains a square figure illustrating the subtraction. The greater proportion, the Sesquialtera, stands at the top with its terms — 3 on the left, 2 on the right; the lesser proportion, the Sesquiterza, at the bottom — 4 on the left, 3 on the right. The left column is labelled “Termini maggiori” (greater terms), the right “Termini minori” (lesser terms). Crossed diagonals mark the cross-multiplication: the greater term of the upper proportion by the lesser of the lower (3 × 3 = 9), and the greater of the lower by the lesser of the upper (4 × 2 = 8). Beneath, labelled “Differenza,” stands the result — the Sesquiottava, 9 : 8.]

We may now say that, a Sesquiterza being subtracted from a Sesquialtera, there remains a Sesquiottava; and that this is the difference found between the one and the other, and is that quantity by which the greater surpasses the lesser, and by which this lesser is surpassed by that greater. And that this is true may be proved: for, adding together (in the manner shown) the Sesquiterza with the Sesquiottava, we shall have from such sum the Sesquialtera — which was that proportion which surpassed the Sesquiterza by a Sesquiottava. Whence from this we may also see that the adding of proportions is the proof of Subtracting, and, conversely, subtracting the proof of adding.