The Second Mode of Multiplying Proportions

It occurring that in multiplications there be need to prefix the proportions one to another, we shall proceed in this manner: We shall first multiply each term of the first proportion — beginning from the lesser [of the first] — by the lesser term of the second proportion placed on the left side; and then the greater of the one by the greater of the other together; and from such multiplication we shall have three terms containing such proportions. Then, multiplying these products by the greater term of the third proportion, and the greater of them by the greater, we shall have our purpose.

If, then, we take the lesser term of the Sesquiquarta set down in the previous chapter — which is 4 — and multiply it by the 5 and the 6, the terms of the Sesquiquinta, there will result 20 and 24, which we shall place, as we did above, beneath a straight line. Then, the 5, the greater term of the said Sesquiquarta, being multiplied by 6, the greater term of the Sesquiquinta, there will come 30, which, set beside the 24, will give three terms, 30, 24, 20, which contain the multiplied proportions.

But to multiply the Sesquiterza with these, we shall take its lesser term, which is 3, and multiply it by the three products, beginning from the right and coming toward the left side; and we shall have 90, 72, 60, arranging them one after another beneath their producers, which are 30, 24, 20. And again, multiplying the 4, the greater term of the Sesquiterza, by the 30, there will come 120, which, after we have added it to the three aforesaid, will give this order: 120, 90, 72, 60 — containing the Sesquiquinta, the Sesquiquarta, and the Sesquiterza proportion.

But wishing to multiply the Sesquialtera with these, we shall take its lesser term, and multiply it, in the manner stated, by the four products, and we shall have 240, 180, 144, 120. We shall then multiply the 3, its greater term, by the 120, the greater term of the products, and there will arise 360; which, accompanied with the four products, will give the whole multiplication between these terms: 360, 240, 180, 144, 120 — which contain the four named proportions, as is seen in the example set below, similar to that which we showed in the previous chapter.

[Editorial note: Here Zarlino’s original sets the example in a two-part table, like that of the previous chapter but built from the right. The upper part, “Proportioni da moltiplicare,” shows the four superparticulars in their radical terms — Sesquialtera (3/2), Sesquiterza (4/3), Sesquiquarta (5/4), Sesquiquinta (6/5) — with the diagonal rules now linking the rightmost pair, where the operation begins. The lower part, “Proportioni moltiplicate,” shows the successive products growing leftward on three lines: 30 . 24 . 20; then 120 . 90 . 72 . 60; and finally 360 . 240 . 180 . 144 . 120.]