Arithmetic, Geometry, and Music

...you may add: it is allowed to multiply their roots by four—that is, 9, 7, 5, 3—and those that are brought forth by that multiplication, again by four, and you will find the same proportion to be made, clearly, in an unoffended reason. And other species arise with roots growing always by one plus the multiplication. But the roots of the proportions are the eight numbers described in the upper arrangement, as if everything is based on the sum of the aforementioned comparison. In this, also, it must be seen that since two parts of the lesser are more in the greater, the name of the third is always implied. As the superbipartiens, which is said because it has two third-parts of the lesser number, should be called superbipartiens tertias superbipartiens of thirds. And when I say supertripartiens, it is necessary that it be implied as supertripartiens quartas supertripartiens of fourths, since it exceeds by three on fourths. And for superquadripartiens, one implies superquadripartiens quintas superquadripartiens of fifths, and in the same mode in the others, with one always added, the parts of the implied must be made. So that their kindred and fitting names are twice these: so that what is called superbipartiens is also called superbitertius over-by-two-thirds. He who is called supertripartiens, let him be supertriquartus over-by-three-fourths; and he who is called superquadripartiens, the same be called superquadriquintus over-by-four-fifths. And by the same similarity, let names be produced even to infinity.

On the multiplici superparticulari multiple superparticular. Chapter 29.

Therefore, these are the simple and first species of quantity related to something. But two others are composed from these—that is, as if from some principles—such as multiplices superparticulares multiple superparticulars, and multiplices superpartientes multiple superpartients. And their companions, the submultiplices superparticulares and submultiplices superpartientes. For in these, as in the aforesaid proportions, smaller numbers and all their species also are said with the preposition added. Whose definition can be rendered as such: a multiple superparticular is as often as a number compared to a number has it more than once and one part of it—that is, it has it either double, or triple, or quadruple, or however many times, and any part of it, whether a half, or a third, or a fourth, or whatever other exuberance of parts it may happen to be. This, therefore, consists of both the multiple and the superparticular. For that which has a compared number more than once is of the multiple. But that which transcends the lesser in having a part is superparticular. Thus, it is a word made from both names. And its species are made to the image of those proportions from which the number itself draws its origin. Thus, the first part of this word, which is possessed by the name of multiple, is to be noted by the name of the species of the multiple number. But that which is superparticular will be named by the same word by which the species of the superparticular number were called. For he will be called a duplex sesqualter double-three-halves who has another number doubled and its half part. He who has a third is duplex sesquitertius double-four-thirds; he who has a fourth is duplex sesquiquartus double-five-fourths, and so forth. If, however, it contains him three times whole and a half part, or third, or fourth, it is called triplex sesqualter, triplex sesquitertius, triplex sesquiquartus, and in the same mode in the others. And he will be called quadruplus sesqualter, quadruplus sesquitertius, quadruplus sesquiquartus; and as often as he contains the whole number in himself, he is called by the species of the multiple number. But which part of the compared number he has closed is called according to the superparticular comparison and relationship. Examples of these are of this kind: duplex sesqualter is as 5 to 2. For 5 has the binary number twice and its half, that is, 1. But the duplex sesquitertius is the septenary compared to the ternary. But 9 to 4 is duplex sesquiquartus. If, however, 11 to 5, it is duplex sesquiquintus. And these will always be born if all natural even and odd numbers are arranged in order from the binary number, if against them all odd numbers starting from five...

number are compared, as if you carefully and diligently place the first to the first, the second to the second, the third to the third, so that the arrangement is such:

If, from the two even numbers all arranged in terms, those starting from the number five, skipping over by the quaternary number, are compared against them, they create all the duplices sesqualteros, as is the description below:

If the arrangements start from three, and skip over by three, and there are fitted to them those starting from the septenary, which exceed each other by the septenary number, all duplices sesquitertii are born by having carefully observed the comparison, as the description below advises:

If all quadruples are arranged in order, those which are quadruples of the natural number, such as the quadruple of unity, and of two, three, and four, and of five, and of the others following them, so that there are fitted to them those starting from the number nine, always preceding each other by nine, then the form of the duplex sesquiquarta proportion will be fashioned:

But the species of this number which is triplex sesqualtera is generated in this mode: if all even numbers are arranged in order from the binary number, and to them are fitted, in the usual mode of comparison one to another, those starting from the septenary number, exceeding each other by the septenary:

If those having entered from the ternary number arrange all the triples of the natural number, and we compare to them those starting from the denary, exceeding each other by the denary in order, all triplices sesquitertii will result in that continuation of terms:

On their examples to be found in the superior formula. Chapter 30.

We can note fully and plainly the examples of these and of those that follow if we wish to turn our acumen diligently to the prior description which we made when we were speaking of the superparticular and the multiple, where the sum of multiplications grew from one up to the denary. For all that follow, collated to the first verse, will render the ordered and suitable species of the multiple. If, however, you fit all those that are of the third order to the second, you will recognize the ordered species of the superparticular. But if you compare whosoever are in the fifth verse to the third order, you will conveniently see the species of the superpartient number positioned. But the multiple superparticular is shown when all those that are in the series of the fifth verse are compared to the second verse, or those that are in the seventh, or in the ninth, and thus if this description were to infinity, the species of this proportion will be generated to infinity. But it is also manifest that their companions are always said with the preposition sub added: as subduplex sesqualter, subduplex sesquitertius, subduplex sesquiquartus, and others indeed in this mode.

On the multiplici superpartiente multiple superpartient. Chapter 31.

A multiple superpartient is as often as a number compared to a number has another number twice in itself...