Arithmetica

Every evenly
odd number by the
ratio of all
groups of four,
which if...
or half doubled
among the means,
produced by
twos.

the nature of the unbreakable unit. Twice 3, twice 5, twice 7, twice 9, twice 11, and so on. From these are born: 2, 6, 10, 14, 18, 22. If you divide these, they receive only one cut, rejecting any others, because the second division is excluded by the oddness of the half part. Between these numbers, there is only a distance of four. For between the numbers 2 and 6, there are 4. Again, between 6 and 10, and between 10 and 14, and between 14 and 18, the same four makes the difference. For all these transcend each other by a quantity of four. This happens because the first numbers placed, which are their foundations, preceded each other by the number two. Since we multiplied those by two, that progression grew into the number four. For two multiplied by two makes the sum of four.

Therefore, in the arrangement of natural numbers, evenly odd numbers are distant from each other by five places. They precede each other by 4, passing over 3 in the middle, created by odd numbers multiplied by two. These species of numbers, namely evenly even and evenly odd, are said to be contrary because in the evenly odd number, only the greater extreme receives a division. In the former, only the smaller term is free from cutting. Also, in the form of an evenly even number, starting from the extremes and progressing toward the middle, the product of the extreme terms is the same as the product of the smaller sums placed within. This remains true until two means are reached in even arrangements.

But if the arrangements are odd, the product produced by one mean is the same as that produced by the parts placed on either side. Let this progression continue until the extremes. For in the arrangement 2, 4, 8, 16: 2 multiplied by 16 gives the same result as 4 led by the number eight. Both ways, they make 32. If the order is odd, as in 2, 4, 8: the extremes make the same as the mean. For twice 8 is 16, and four times four is 16. This number is perfected by the four led into itself.

In an evenly odd number, however, if there is one term in the middle, it is the mediety middle or mean of the terms placed around it if they are combined into one. And the same is true for the terms above those, all the way to the extremes of all terms. For example, in the order of evenly odd numbers 2, 6, 10: two joined with ten completes 12, of which six is found to be the half. But if there are two means joined, they will both be equal to the terms established above them. As in this order: 2, 6, 10, 14. For 2 and 14 joined grow into 16, which six coupled with ten will produce. This happens in arrangements with more terms, starting from the means until the extremes are reached.

COMMENTARY ON CHAPTER SEVEN.

Aristotle in
book of manuscripts?
of the system of two and
the world in demonstrations

An ornamental woodcut initial I with floral motifs. IN this chapter, he first brings out the substance of the evenly odd number by definition, and finally its following nature by five properties. This is the understanding of the definition: an evenly odd number is one that, since it is even, is divisible into two equals, namely into two means. But these means suffer no further division into two equals. This property of the evenly odd is first

accessible to us in the number six. For six is cut into 3 and 3, its means, but neither of these is subject to division into two equals. The same happens to 10, which is cut into 5 and 5, as these means reject the same cutting.

Furthermore, he adds that evenly odd numbers oppose the nature of evenly even numbers, and this is in a contrary partition. Every evenly even number suffers division into two equals all the way to unity. Thus unity, the smallest of all parts, is entirely odd, or it is the only one that does not receive this cut. But in the evenly odd, on the contrary, there is only one number, the greatest, that has received that cut. Therefore they oppose each other in division. Here, there is only one thing that is divisible; there, on the contrary, only one thing is exempt from division. Here, there is one division; there, the division is manifold. He shows this later.

For this reason, he concludes that it is appropriate for an evenly odd number to have an odd mean. This is not even difficult to know from the definition. For by definition, the half is not divisible into two equals. Now, a number that cannot be divided into two equals is necessarily odd. This is from the definition of an odd number. Therefore, every half of an evenly odd number is odd.

It is also easy to conclude that those numbers are far more formal than material. For things that suffer only one division pertain to evenness and matter. But a division that is immediately stopped and finished pertains to oddness and form. Therefore, it is not without reason that the Pythagoreans philosophize through the evenly odd about the highest things, even if they are primarily sensible. Surely, they have the most form and the least matter. Division follows matter. Thus, by rising away from matter, celestial spirits are found to be less subject to division than other things. Likewise, the immense simplicity of God becomes known. For if division comes from matter, those things that recede from matter must also recede from division.

Hence, fire is less divisible than earth. A human is less divisible than a beast. A beast less than a tree. In the number of beings in the sensible world, animals are by far the least subject to division. Divide an animal, and the name of the animal immediately falls away and perishes. For they are anhomiomeria composed of unlike parts, such as organs and limbs. Inanimate things, on the contrary, are subject to the greatest division. The denomination is not taken away by this division. Indeed, they are called homiomeria having parts of the same kind, such as water or stone because the parts are of the same genre and name as the whole.

Therefore, things that recede further from matter are proven to be more fully absent from division. And if they recede infinitely from the mass of matter, they recede infinitely from division. But what is it to recede from division, other than to approach indivisibility and simplicity? Therefore, angels, who among created things are furthest from matter, are the most indivisible of them. God, however, who recedes infinitely from matter, is infinitely simple and indivisible.

In the second place, he adds: in an evenly odd number, the parts do not agree in quantity and denomination. If the denomination is even, the quantity is odd, and vice versa. This should be understood only regarding the numerative part. For example, the numerative parts of 30 are 15, 10, 6, 5, 3, 2, 1. Now, 15, which cannot be divided into two equals, is odd in quantity, but in the name of the total sum, it is even. For it is called the "second" part, from the number two, which is even. On the contrary, 10 is even in quantity but odd in denomination, for it is the "third" part. Similarly, 6 is even in quantity but odd in denomination, since it is called the "fifth" part. Conversely, 5 is odd in quantity but even in name, since it is the "sixth" part. Two is even in quantity but odd in name, for it is the "fifteenth" part. Finally, unity is odd in quantity and even in denomination.

Therefore, he rightly asserts that it can in no way happen that any part has a name and number of the same genre. This means that on both sides of evenly odd numbers, there cannot be a numerative part that aligns in both name and quantity, such as being both even or both odd. But when one part is even, the other is odd.

Furthermore, this is an argument for us that the sum of beings does not consist of parts that are of the same nature, but of parts that differ in their natural fellowship. Therefore, they are and are held to be anhomiomeria and dissimilar. Thus, even in the parts, one cannot see similar operations. Instead, according to the differing nature of the parts and their different temperament, different operations become known. For vision is in the eye, hearing in the ears, and smelling in the nostrils. In inanimate things, this is certainly not seen. However, plants, which are in the middle, proceed in a middle way. This will be shown more fully in what follows.

If you notice that passibility the capacity to be acted upon or suffer change is connected to evenness and matter, and that operative power is connected to oddness and form (for it is the nature of form to act), an ascent to the remarkable operations of things presents itself. Because the reason for suffering is connected to the mass of matter, to approach matter is to approach passibility. To recede from matter is to recede from passibility. Therefore, those things that are absent from matter are also absent from passion and passibility. But what else is it to recede from the nature of passion and passibility than to approach the nature of action and active power? For to recede from one of two contraries is to approach the other. Therefore, things that are further from matter are also further from passion and passibility.