Arithmetic, Geometry, and Music

...lesser. And these are indeed of this kind: so that that former one, which exceeds its own parts, might appear as such: just as if someone were born with many hands beyond nature, like a hundred-handed giant, or joined with a triple body, like the three-bodied Geryon, or anything monstrous that nature has snatched up in the multiplication of parts. But that other one, as if a necessary part were naturally removed, or if he were born with a missing eye, as was the disgrace of the Cyclops’ forehead, or curtailed by some other limb, would receive a natural expenditure of his entire fullness. Between these, however, as if between equal intemperances, that number has obtained the temperament of a middle limit which is called perfectus perfect: that is, an emulator of virtue: which is neither stretched by superfluous progression, nor relaxed by contracted diminution, but holding the limit of a middle ground, it is neither thickened by abundance nor in need due to deficiency in its equal parts: such as six or 28. For the number six has a middle part, that is, 3, and a third, that is, 2, and sixths, that is, 1, which, when brought into one sum, are found to be equal to the whole body of the number by its parts. 28, however, has a half, 14, and a seventh, 4, and it does not lack a fourth, that is, 7, and it possesses a fourteenth, 2, and you will find in it a twenty-eighth, 1, which, when brought into one, will equal the whole body by its parts. For the parts of 28 joined together will result in 28.

On the generation of the perfect number. Chapter 20.

There is, moreover, in these things also a great similarity of virtue and vice. For you will rarely find perfect numbers, and they are easily countable, inasmuch as they are few and created in an excessively constant order; but indeed, you will find many and infinite superfluous and deficient numbers, neither disposed here and there in any order, nor generated by any certain end. Perfect numbers, however, are: within the number ten, 6; within the hundred, 28; within the thousand, 496; within ten thousand, 8128. And these numbers always end with two even numbers: 6 and 8. And they always arrive at the end of their sums alternately in these numbers. For the first is six, then 28. After these, 496, the same six that was the first; after that 8128, the same eight that was the second. The generation and procreation of them, however, is fixed and firm; nor can they be made in any other way, nor, if they are made in this way, could any other thing in any way be created. For you will arrange all even-even numbers in order as far as you wish: you will first add them according to the sequence, and if the first number created from that accumulation is prime and incompositus indivisible/prime, you will multiply the whole sum by that which you had added last. But if, once the accumulation is made, the first is not found to be prime and indivisible, but is composite and secondary, pass over this and you will add another that follows. If, however, it is still not prime and indivisible, add another again and see what happens. And if you find a prime indivisible number, then you will multiply the sum of the accumulation by the final multitude. For let all even-even numbers be arranged in this way: 1, 2, 4, 8, 16, 32, 64, 128. You will therefore do thus: you will place 1, and to it you will add 2. Then you will look at what number has been made from this aggregation: it is 3, which is, namely, prime and indivisible, and you had added the last binary number after the unit. If, therefore, you multiply the ternary—that is, the one which was collected from the accumulation—by the binary which is the last added, a perfect number will be born without any doubt. For twice 3 makes 6, which has one part named after itself, that is, a sixth; three indeed have a half according to quality; but two according to the accumulation, that is, according to the ternary, because three were added and multiplied. Twenty-eight, however, are born in the same way. For if upon one and two, which are three, you add the following even-even number, that is, 4, you will make the sum of seven; but you had consecutively added the last number, four: therefore, if you multiply that accumulation by this, a perfect number is created. For seven times 4 is 28, which is equal to its parts, having one named after itself,

that is, the twenty-eighth; the half, however, according to the binary, 14; according to the quaternary, 7; the seventh, however, according to the septenary, 4; according to the collection of all, the fourteenth is two, which is opposed by the name of half. Therefore, when these have been found, if you seek to find others, it is necessary that you investigate by the same logic. For you will place one, if you wish, and after this 2 and 4, which are accumulated into seven; but this has already existed. 28 is a perfect number. To this, therefore, let the following even-even number, that is, 8, be added as an accession, which, coming upon the previous ones, restores 15. But this is not prime and indivisible. For it has a part of a different kind beyond that which is named after itself: namely, the fifteenth unit. Since this is therefore secondary and composite, pass it over and add to the previous ones the continuing even-even number, that is, 16. Which, when joined with 15, will complete 31. But this, in turn, is prime and indivisible. Therefore, multiply this with the sum of the extreme aggregate: so that sixteen times 31 are made, which explain 496. This, moreover, is the numerosity within the thousand number that is perfect and equal to its parts. Therefore, the first unit is perfect in virtue and potentiality, though not yet in act or in reality. For if I take the first one from the proposed order of numbers, I see it is prime and indivisible, which, if I multiply by itself, the same unit is created for me. For once one makes only a unit, which is equal to its parts only in potentiality, whereas the others are perfect also in act and in work. Rightly, therefore, the unit is perfect by its own virtue, because it is both prime and indivisible, and multiplied by itself, it preserves itself. But since it has been said concerning that quantity which is made by itself, let us transfer the sequence of the work to that which is related to something.

On quantity related to something. Chapter 21.

The division of quantity related to something is twofold. For everything is either equal or unequal, which measures itself by comparison to another. And indeed, that is equal which, when compared to something, neither falls below it by a smaller sum nor exceeds it by a greater, such as ten to ten, or three to three, or a cubit to a cubit, or a foot to a foot, and things like these. This part of quantity related to something, that is, equality, is naturally indivisible. For no one can say that of equality, this is indeed such, and that is of this kind. For every equality preserves one measure in its own moderation. Also, the quantity which is compared to it is not called by a different name than that to which it is compared. For just as a friend is a friend to a friend, and a neighbor to a neighbor, so an equal is said to be equal to an equal. Unequal quantity, however, has a twofold division. For that which is unequal is cut into greater and lesser, which perform their roles by mutually contrary denominations. For the greater is greater than the lesser, and the lesser is lesser than the greater, and both are not marked by the same names, as is the case according to equality, but by diverse and distant ones, after the manner of one learning and one teaching, or one striking and one being struck, or whatever things related to something are compared by otherwise denominated contraries.

On the species of greater and lesser inequality. Chapter 22.

There are five parts of greater inequality. For there is one which is called multiple, another superparticular, the third superpartient, the fourth multiple-superparticular, the fifth multiple-superpartient. To these five parts of the greater, there are opposed five other parts of the lesser, just as the greater itself is always opposed to the lesser, which species of the lesser are thus individually opposed to those five species of the greater which were mentioned above, so that they are called by the same names, differing only by the preposition placed before them. For they are called submultiple, subsuperparticular, subsuperpartient, multiple-subsuperparticular, and multiple-subsuperpartient.