On Geometric Division, or Proportionality

Geometric Division is made when the Divisor is so placed between the extremes of some proportion that it preserves the conditions touched upon in the preceding chapter. Whence it is to be known that every other Proportionality is of such a nature that it divides the proposed proportion only into two unequal parts; but the property of the Geometric is always to divide it into two equal parts — from which effect it is properly called Proportionality, since among its greater terms, and among the lesser also, the proportions are equal; and the product of the Divisor multiplied by itself is equal to the product of the extreme terms of the said Proportionality, multiplied together.

But to find such a Divisor we shall observe this rule: having proposed whatsoever Proportion to be divided, contained in its radical terms — to avoid the length of the operation, the labor, and the many errors that occur — we shall first multiply those terms one with the other; then we shall extract the square Root of the product, which will be a number that, multiplied by itself, will render exactly such a product; and such a Root will be the sought Divisor.

And that I may more easily be understood, I shall come to the example. Let us set the Quadrupla proportion, contained in its radical terms, 4 and 1; wishing to divide it Geometrically, we must first multiply the said terms one by the other, and thus we shall have 4; then, its square Root being taken, which will be 2, we shall say that number to be the geometric Divisor of such proportion; for the product that comes from the multiplication of it by itself is equal to that which arises from the multiplication of the proposed terms multiplied together — since 4 multiplied by the unity renders as much as 2 multiplied by itself, as is seen in the figure.

[Editorial note: Here Zarlino’s original contains a figure headed “Proportione da dividere secondo la Geometrica proportionalità.” It shows the Quadrupla in its radical terms (4 . 1); the label “Proportione divisa in due parti equali” (proportion divided into two equal parts); an arc marked “Quadrupla” and “Divisore” indicating the division by the mean term 2; and the result, 4 . Dupla . 2 . Dupla . 1. Beneath, labelled “Differenze inequali di i termini delle proportioni” (unequal differences of the terms), stand the two unequal differences, 2 and 1.]

The Quadrupla, then, is divided into two equal parts by such Divisor — that is, into two Duple, the one of which is found to be between 4 and 2, and the other between 2 and 1. But one must observe that, although the property of the Geometric proportionality is to divide whatsoever proportion into two equal parts, this is done universally in continuous quantity; for in the discrete quantity not all proportions are divisible in such a manner, since the numbers do not suffer the division of the unity.

Whence, just as it is impossible to divide rationally into two equal parts any proportion contained in the Superparticular genus — as Boethius affirms in his Music, and Giordano [Nemorarius] in his Arithmetic — there falling between its terms no other number than the unity, which cannot be divided; so it will be impossible to divide those of the other genera that come after this one. For those which can be divided are contained in the Multiple genus, and have in one of their extremes a square number, and in the other the Unity, and thus are capable (as the same Giordano affirms) of such division.

So that from the Geometric proportionality we may have two divisions, namely the Rational and the Irrational. The Rational, I say, is that which is made by way of rational numbers, so that its Divisor is exactly the square Root of the product of the multiplication of the terms of some proportion multiplied together; and the parts of such division may be denominated, as is the one shown, contained between these terms, 4, 2, 1. But the Irrational is that which is made by way of measures, and also of numbers, which are called Surd and Irrational; for such division can in no way be made, nor even circumscribed, with rational numbers or like measures; and this happens when from the product we cannot have its Root exactly.

As, for example, would happen when we wished to divide in such manner a Sesquialtera; for then, the terms being multiplied together, which are 3 and 2, from the 6 that will be the product, one cannot extract such a root — that is, one cannot have a number that, multiplied by itself, makes 6. It is indeed true that such a number may be denominated, according to the custom of the Mathematicians, in this manner, saying “Root of 6” — that is, the square Root that could be extracted from this number, if it were possible; and this would be its Divisor. But such a Root, or number, that is seen in the example below, for the reason stated will always be named Surd and Irrational. And because one cannot have the rational root of such a number, the parts of this division cannot be denominated or described, even though its extremes are comprised by rational numbers. Whence such division, for the reasons stated, will always be called Surd and Irrational, and is not considered by the Musician.

[Editorial note: Here Zarlino’s original contains a second figure, headed as before, showing the Sesquialtera in its radical terms (3 . 2) with the label “Proportione divisa irrationalmente in due parti equali” (proportion divided irrationally into two equal parts); an arc marked “Sesquialtera” and “Divisore” indicates the division, whose mean term is given beneath as ”℞. 6” — the Root of 6, an irrational number — standing between 3 and 2.]