What a Denominator Is, and in What Manner It Is Found; and How, of Two Proposed Proportions, One May Know the Greater or the Lesser

We must note that the Denominator (as Euclid would have it) is that number according to which the part is taken in its whole; it is properly called by some an Aliquot part, and by others the Quotient, since it denotes how many times the greater term of the proportion contains the lesser. And it is that which is produced by the division of the greater term, made by the lesser, of any proposed proportion of whatsoever genus. As, for example, dividing the greater term of the Duple — which is found to be the first in the Multiple genus, and is 2 — by Unity, which is the lesser, there will come 2, which I say to be the Denominator of that proportion; for the Binary contains that unity twice, and this divides it entirely into two parts.

Likewise we shall say that 3 is the denominator of the Triple, and 4 the denominator of the Quadruple; for 3 contains Unity three times, and 4 four times; and so of all the others in turn. And such denominations are called Simple, because they are denominated by simple numbers, which are 2, 3, 4, and the like.

But if in the Superparticular genus we divide the terms of the Sesquialtera in the manner stated — that is, the greater by the lesser — there will come 1½, which I say to be the denominator of the Sesquialtera; for its greater term, 3, contains the term 2 once, with one half-part, which, according to the custom of the mathematicians, is written in this manner, 1½. And such a denomination is called Composite, because it is composed of unity and of a part of it.

It is indeed true that the parts which arise in this manner are sometimes called Aliquot, and sometimes Non-aliquot, of the lesser term that the proportion contains; but the number set above the line is called the Numerator of that part, and the one set below, the Denominator.

Whence this particle “Sesqui” derives, and what it signifies, is no easy thing to know — unless it be that which Augustine would have, who (reading “Sesque,” and not “Sesqui”) thinks it to be said as though from “Se absque,” that is, from “Absque se,” which signifies “Without itself”; because (if I do not deceive myself) it takes the denomination of the proportions from the part of the greater number by which the lesser is exceeded, in the terms, or numbers, of the proportions of the Superparticular genus, which he names “Sesquati” numbers, and those of the Multiple, “Complicati.”

And although there have been some who held the opinion that it is a syllabic addition signifying nothing, but was devised only that the said species might be uttered more conveniently, this seems to me to be said with little consideration; and better have those spoken who said that “Sesqui” means “Whole,” and that “Sesquialtera” is so called from that word, which is Latin, and from “Altera,” likewise a Latin word, which is used when one speaks of two only and signifies “Other” — as though a proportion whose greater term contains the whole of the lesser one entire time, with one of the two parts. And this is well said; for were it otherwise (as certain ones would have it, that “Sesqui” signifies “as much again, and the half”), such a word could not be fitted to the others, as to the Sesquiterza, the Sesquiquarta, and the like.

Nonetheless, it is to be noted that the Denominator of any proportion is found in two ways: that is, either in pure numbers, or by adding parts to these. And we shall be able to find this second way in four manners: for sometimes we shall find Unity and some part; and sometimes Unity and several parts; or else we shall find some number and one part; or some number joined to several parts.

If we find simple numbers, we ought to denominate the proportion simply, according as has been shown in the species of the Multiple; and if we find unity joined to some part, we ought to denominate it according as those of the Superparticular were denominated above. When then unity is found with several parts, then, leaving aside unity, one sets before the Numerator of the parts this particle “Super,” and to the Denominator this other, “Partiente”; and the denomination of the proportion is composed of the said two particles and of the terms of the parts. As, for example, may be seen in the first species of the Superpartient genus, where the proportion called Superbipartienteterza is denominated by 1⅔, its denominator; for, the greater term of that proportion, which is 5, being divided by 3, which is the lesser, there results 1⅔. Whence, taking the numerator of the parts, which is 2, and adding to it the particle “Super,” one says “Superbi”; then taking the 3, the denominator, with the second particle “Partiente,” one says “Partienteterza”; and thus joined together one says “Superbipartienteterza” — which is done in the others also, according to its denominator.

But when the denominator is composed of some number and of one part only, the proportion is denominated first from the number, as was said of the Multiple, and then the part is added, in the manner I have declared in the Superparticular; for such a proportion is found necessarily in the first composite genus, called Multiple-superparticular. As may be seen in the Duplasesquialtera, which is denominated by 2½; for its greater term, which is 5, contains the 2, which is the lesser, twice, and one half-part of the lesser — so that from the 2 it takes the denomination of the Duple, and from the part, which is ½, it takes that of the Sesquialtera.

When then the denominator is contained by a whole number and by several parts, then the proportion is denominated first from the number, in the manner shown in the Multiple, and then the parts are added, denominating them as we did in the Superpartient genus; for such a proportion falls necessarily in the second composite genus, called Multiple-superpartient. We have the example of this in the Duplasuperbipartienteterza, which is the first species of that genus — as we shall see — denominated, for the reasons stated, by 2⅔, its denominator.

It would be long, were I to set down the examples of each species; but because many of them will be seen in their place, I shall not extend myself further on this now. I shall only say this for a conclusion: that each proportion is greater than another (as Euclid advises us) by as much as its denominator makes it, and this in every genus of proportion — which is manifest, since the Duple is without any doubt greater than the Sesquialtera, for its Denominator, 2, is greater than 1½, the Denominator of the Sesquialtera; and so one may say also of the others.