Arithmetic, Geometry, and Music

iThere is also the odd number: which differs from the nature and substance of an even number. For while the latter can be divided into twin equal parts, the former cannot be cut because the intervention of unity prevents it. It has likewise three subdivisions, of which one is a part and is the number that is called first primus prime and incomposite incompositus indivisible/prime. The second indeed is that which is second secundus and composite compositus composite. And the third is that which is joined by a certain intermediary of these, and naturally draws something from the relationship of both, which is indeed in itself second and composite, but when compared to others is found to be first and incomposite.

¶ Concerning the first and incomposite. Chapter 14.
eThe first and incomposite is indeed that which has no other part except that which is denominated from the whole quantity of the number, so that this part is nothing but unity, as are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. In these, therefore, no other part will ever be found in any of them, except that which is denominated by them, and that unity alone, as has already been said above. For in three there is only one part, that is, the third, which is denominated from three, and this third part is unity. In the same way, the only part of five is the fifth, and this is unity; and the same will be found following in each one. It is said to be first and incomposite because no other number measures it, except unity alone, which is the mother of all things. Furthermore, they do not count two as a number, for the reason that if you compare only two against three, they are fewer. If, however, you make two twice, it grows into four, which is more than three. Rightly, a number measures a number, whenever, once or twice or three times or however many times a number compared to a number arrives at the limit of the compared number without the sum being diminished or increased, as if you compare two to 6, the number two will measure six three times. Therefore, no number will measure the first and incomposite numbers, except unity alone, since they are composed of no other numbers but are produced only by units added and multiplied among themselves. For three times one makes 3, and five times one makes 5, and seven times one makes 7. And the others which we have described above are born in the same way. These, however, multiplied into themselves make other numbers just like the first; and having obtained the first substance and power of things, you will find them to be, as it were, elements of all things created from them, which are clearly incomposite and formed by a simple generation, and into them are resolved all those numbers which are made from them, but they themselves are neither produced from others, nor are they reduced into others.

¶ Concerning the second and composite. Chapter 15.
fThe second and composite is indeed odd because it is formed by the same property of the odd, but it retains no principle of substance in itself and is composed of other numbers. It has parts denominated both by itself and by an alien term, but you will always find in these only unity as the part denominated by itself. From an alien term, however, it has either one part, or as many others as there are numbers by which that composite is produced, as are 9, 15, 21, 25, 27, 33, 39. Therefore, each of those has indeed its own proper parts denominated by itself, namely units; as 9 has the ninth, that is 1; 15 has the fifteenth, the same unit again; and the same holds true in the others we have described above. They also have a part from an alien term, as 9 has a third, that is three; and 15 has a third, that is 5, and a fifth, that is 3. 21 indeed has thirds, that is 7, and a seventh, that is 3, and in all others the same consequence is found. This number is called second because it is not measured only by unity, but also by another number from which it is joined. Nor does it have in itself any principle of understanding. For it is produced from other numbers: 9 indeed from three, 15 from three and 5, 21 from three and 7, and the rest in the same way. It is called composite because it can be resolved into those very things from which it is said to be composed, namely into those that measure the composite number. Nothing, however, which can be dissolved is incomposite, but is composed by the necessity of things.

¶ Concerning that which is in itself second and composite but in relation to another is first and incomposite. Chapter 16.
bBut with these placed against each other, that is, the first and incomposite and the second and composite, and separated by natural diversity, another is considered in the middle. He himself is indeed composite and second, and receives measurement from another, and therefore is capable of a part from an alien term, but when he is compared to another number of the same kind, he is not joined with any common measure, nor will they have equivalent parts; as 9 is to 25. No common measure of numbers measures these, unless perhaps unity, which is the common measure of all numbers. And these indeed do not have equivalent parts. For what is the third in 9 is not in 25, and what is the fifth in 25 is not in 9. Therefore, they are by nature both second and composite, but compared to each other they are rendered first and incomposite, because no other measure measures both of them except unity, which is denominated by both. For in nine there is the ninth, and in 25 there is the twenty-fifth.

¶ Concerning the generation of the first and incomposite, and the second and composite; being second and composite in relation to themselves, but first and incomposite in relation to one another. Chapter 17.
gThe generation and origin of these is gathered by a method of investigation which Eratosthenes called a sieve cribrum sieve, by which, with all odd numbers placed in the middle, it is distinguished through the art we are about to convey which are of the first, which of the second, and which of the third kind. For let all odd numbers be arranged in orders starting from the number three, in which there is the longest extension: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49. With these thus arranged, it must be considered whether the first number of those placed in the order can measure them. But after two are passed, it measures the one that is placed after them. And if, after the same one that it measured, two others are passed, it again measures the one that is after those two. And in the same way, if one leaves two, the one after them is to be measured by the first number. In the same way, always leaving two, those proceeding from the first to infinity will measure. But this is not done commonly or confusedly. For the first number measures the one that is after two placed after it, by its own quantity. For the number three measures the number 9, which is the third. If, however, after nine I leave two, the one that occurs to me after them is to be measured by the first number through the quantity of the second odd, that is, through five. Soon after 9, if I leave two, that is 11 and 13, the number three measures 15 by the quantity of the second number, that is, by five, since three measures 15 five times. Again, if starting from fifteen I interpose two, the one that is placed after is measured by the first number through the plurality of the third odd. For if I interpose 17 and 19 after 15, 21 occurs, which the number three measures according to seven. For three is the seventh part of the number 21. And doing this to infinity, I find the first number, if I interpose two, measures all those following after it according to the quantity of the odd numbers placed in the order. If, however, one wishes to find for the number five, which is established in the second place, what is its first and subsequent measure, by passing over four odd numbers, the fifth one occurs to it that it can measure. For let four odd numbers be interposed, that is 7, 9, 11, 13; after these is 15, which five measures, according to the quantity of the first, that is three; for five measures 15 three times. And subsequently, if one interposes four, the one that is placed after them is measured by the second, that is five, by its own quantity. For after fifteen, with 17, 19, 21, and 23 interposed, I find 25 after them, which the number five measures by its own plurality. For five times five multiplied is 25.