Arithmetic, Geometry, and Music

On the multiple and its species and their generations

Chapter 23.

The multiple is the first part of the greater inequality, older in nature and more eminent than all others, as we shall demonstrate a little later. This number is of such a kind that, when compared with another, it contains that against which it is compared more than once. This will become evident in the arrangement of the natural number. For all that follow correspond to the unit, and they maintain the varieties and order of all multiples. For against the first, that is the unit, 2 is the double, 3 the triple, 4 the quadruple, and thus progressing in order, all multiple quantities are carried forth. But what has been said, "more than once," takes its beginning from the binary number, and it progresses to infinity through the ternary, the quaternary, and the sequence of the others. Contrasted against this is that which is called the submultiple, and this is also the first species of the lesser quantity. This number is of such a kind that, when produced in comparison to another, it counts the sum of the greater more than once, beginning equally with its own quantity and determining it equally. I say "counts" in the same sense as "measures." If, therefore, the lesser number measures the greater number only twice, it will be called a subdouble; if three times, a subtriple; if four times, a subquadruple; and through these, the progression continues to infinity. You will always name them with the preposition "sub," such as 1 being the subdouble of 2, the subtriple of 3, the subquadruple of 4, and consequently.

Since multiplicity and submultiplicity are naturally infinite, their species also are turned over in infinite consideration through their own generations. For if, having placed numbers in their natural constitution, you choose individual even numbers through their consequences, they will be the doubles of all the even and odd numbers following one another, and the limit of this speculation does not fail. For let the natural number be set in this way: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. If you take the first even number, which is 2, it will be the double of the first, that is, the unit. If you take the following even number, which is 4, it is the double of the second, that is, of 2. If you take the third even number, which is 6, it is the double of the third number in the natural constitution, that is, of 3. If you look at the fourth even number, which is 8, it is the double of the fourth number, that is, of 4. And the same proceeds in the others taking them to infinity without any impediment. Triples, however, are born if, in the same natural arrangement, two are always skipped, and those that are after the two are compared to the natural number, except for the ternary, which skips only the binary so that it may be the triple of the unit. After 1 and 2, there is 3, which is the triple of 1. Again, after 4, 5, and 6, there is 7, which is the triple of the second number, that is, 2. Again, after 6, there are 7 and 8, and after these, 9, which is the triple of the third number, that is, 3. And if anyone were to do this, the same thing proceeds to infinity without any offense. The generation of quadruples, however, begins if one skips three numbers. After the unit, indeed, there are 2 and 3, and then 4, which is the quadruple of the first, that is, 1. Again, if I skip the ternary, the senary, and the septenary, the octonary occurs to me as the fourth, having skipped three, which is the quadruple of the binary, that is, the second number. But if, after eight, I skip three terms, that is, 9, 10, and 11, the duodenary that follows is the quadruple of the ternary number. It is necessary for this same thing to happen as they progress to infinity, and you will wonder to find the ordered turns of the multiple number if you always increase the addition by one term of intermission. For if you skip 4, the quintuple is found; if 5, the sextuple; if 6, the septuple; and they are always produced by the name of the multiplicity itself, being one less in the term of omission. For the double skips one, the triple skips 2, the quadruple skips 3, the quintuple skips 4. And subsequently, the sequence is to the same order. And all doubles according to their own sequences of even numbers are even. Triples, however, are always found to be one even term and one odd term. Quadruples, however, again always keep the

even quantity. And they are constituted from the fourth number, by having one of the prior even numbers skipped in order: first the even binary, after this 8, having skipped the senary, after this 12, having skipped the denary. And this same thing holds true for the others. The proposition of quintuples, however, is ordered alternately with even and odd numbers placed according to the similarity of the triple.

On the superparticular and its species and their generations.

Chapter 24.

A superparticularis superparticular number, when compared to another, is one that contains the whole of the lesser number and some part of it. If it contains the half of the lesser, it is called sesqualter one and a half; if it contains a third part, it is called sesquitertius one and a third; if it contains a fourth, it is called sesquiquartus one and a fourth; and if a fifth, it is called sesquiquintus one and a fifth. And with these names led to infinity, the form of the superparticular also progresses to infinity. And the larger numbers are called by these names, while the smaller numbers, which possess the whole and a part of them, are called sub-sesqualter, sub-sesquitertius, and another sub-sesquiquartus, and another sub-sesquiquintus; and the same extends according to the norm and multitude of the larger ones. I call the larger numbers the leaders and the smaller ones the followers. There is also an infinite multitude of superparticulars for the reason that their species are joined in interminable progression. For the sesqualter will have all the leaders after the ternary number as naturally triples. All the followers after the binary will be naturally even, in this way: that the first is compared to the first, the second to the second, the third to the third, and so on. For let the longest verses of the triples and doubles be described, and let it be in this way:

The first verse contains the natural number, the second its triple, and the third the double. And in this, if the ternary is compared to the binary, or if the senary to the quaternary, or the nonary to the senary, or if all the triples are opposed to the following double numbers, the emiolia sesquialteral proportion will be born. For three contains within itself two and their middle part, that is, 1. Six also contains within itself 4 and their half, that is, 2. And nine encloses within itself the senary and its middle part, that is, 3. And in the same way with the others. It must be said, if anyone wishes to consider the second species of the superparticular number, that is, the sesquitertial, by what reason he may find it; and indeed the definition of this comparison is such: a sesquitertius is that which, compared to a lesser, holds it once and its third part. But these are found if, with all numbers from the quaternary constituted as quadruples, the triples are compared from the ternary number, and the leaders will be the quadruples, and the followers the triples. Let there be in order in this way the natural number, so that under it there are quadruples and under that the triples. Let the first triple be placed under the first quadruple, the second under the second, the third under the third, and in the same way let all the triples of the same first verse be directed in order.

Therefore, if you compare the first to the first, the sesquitertial ratio will be contained. For if you compare 4 to 3, they will contain within themselves the whole ternary and its third part, that is, 1. And if you compare the second to the second, that is, 8 to 6, you will find the same thing: for 8 will have the whole senary and its third part, that is, 2. And by the same sequence, one must proceed to infinity. It is to be noted that 3 are the followers, 4 the leaders. Again, 6 are the followers, 8 the leaders, and in the same order the others. They are called leaders of the sesquitertius and the followers sub-sesquitertius, and it is fitting to keep these terms for all things placed according to this mode.