On Arithmetic Proportionality, or Division

One will be able, then, to divide whatsoever proportion according to the arithmetic proportionality, when we shall have found a Divisor which, placed in the mean of the terms of the proportion to be divided, will divide it in such a manner that, the differences of the terms (as has been said) being equal, its proportions will be unequal — so that among the greater numbers the lesser proportions will be found, and among the lesser the greater: a thing which belongs only to the arithmetic proportionality.

This we shall be able to find easily when, the terms of the proposed proportion being added together, we divide the product into two equal parts; for that number which will arise from such division will be the sought Divisor, which will divide the said proportion, according to the aforesaid conditions, into two parts.

Nevertheless, one must observe that, the proposed proportion being in its radical terms, the aforesaid mode cannot be observed; for it will necessarily be contained by numbers prime to one another (Contraseprimi), which, added together, will give an odd number, that cannot be divided into two equal parts — that is, into two whole numbers. Whence, wishing to find such a divisor, and to avoid broken numbers, which are not received by the arithmetician, we shall always double the said terms, and there will come two even numbers, which will not vary the first proportion.

Now, this done, the said even numbers being added together, and the product divided into two equal parts, that which will come will be the sought Divisor. And let it be, for example, that we wished to divide the Sesquialtera proportion — contained between these radical terms, 3 and 2 — according to the arithmetic division; these numbers being prime to one another, they must be doubled; which done, we shall have 6 and 4, containing the Sesquialtera; which, added together, will give 10, that, divided into two equal parts, will give 5.

Whence I say that 5 will be the Divisor of the proposed proportion; for, besides constituting in such proportionality the equal differences, it also divides the proportion (as is proper to such proportionality) into two unequal proportions, in such a manner that among the greater numbers the lesser proportion is found, and, conversely, among the lesser the greater: as between 6 and 5 the Sesquiquinta, and between 5 and 4 the Sesquiquarta, as is seen here.

[Editorial note: Here Zarlino’s original contains a figure headed “Proportioni da dividere secondo l’Arithmetica proportionalità” (proportions to be divided according to the arithmetic proportionality). It shows the Sesquialtera first in its radical terms (3 . 2), then doubled (6 . 4); an arc marked “Sesquialtera” and “Divisore” indicates the division by the mean term 5; and the result is given as 6 . Sesquiquinta . 5 . Sesquiquarta . 4. Beneath, labelled “Differenze equali di i termini delle proportioni” (equal differences of the terms of the proportions), stand the two equal differences, 1 and 1.]