The Arithmetic of Boethius

The prince of parity. it receives division into two unequal parts: but so that in neither division is parity mixed with the odd, nor oddness with parity. Except only for the number two, the prince of parity, which does not receive an unequal section because it consists of two units and is, in a way, the first parity of two. And what I say is this: If an even number is posited, it can be divided into two equal parts, as ten is divided into fives. Furthermore, it can also be divided into unequal parts, as that same ten into three and seven; but in such a way that when one part of the division is even, the other is also found to be even; and if one is odd, the other does not differ from its oddness, as in the same number, which is ten. For when it is divided into fives, or into three and seven, both parts in each division were odd. If, however, it or another even number is divided into equals, as eight into four and four, and also into unequals, as that same eight into five and three, in that former division both parts were made even, and in the latter, both were odd. Nor can it ever happen that when one part of a division is even, the other could be found to be odd; or when one is odd, the other could be understood as even. The odd number, however, is one that, at any division of that kind, is always divided into unequals, so that both species of number always appear. Nor is one ever without the other, but one part is assigned to parity, and the other to oddness; for example, if you divide seven into three and four, one portion is even and the other is odd. And this same thing is found in all odd numbers. Nor can these twin species, which naturally compose the power and substance of number, ever exist outside of themselves in the division of an odd number.
¶ But if these must also be defined by the other species, an odd number is said to be one that differs from an even number by a unit, either by increase or decrease. Likewise, an even number is one that differs from an odd number by a unit, either by increase or decrease. For if you take one away from an even number or add one, it becomes odd; or if you do the same to an odd number, an even one is immediately produced.

COMMENTARY ON CHAPTER THREE.

A decorative woodcut initial 'Q' features a floral motif.Since the parts of number were explained a little earlier, he now proceeds to clarify its substance more fully by definition. And since quantity is twofold, continuous and discrete, number is ascribed to the discrete, not the continuous. Yet there is a certain number of continuous things that is also a measure, as in parts of both time and magnitude. Hence time, as well as the continuous, is called number. And many things are borrowed from numbers, yet it is a numbered thing, and not a numbering thing. For number that is also a numbering thing, which we have touched upon somewhat, is subjectively in the soul. And it has such a proximity to the soul that some philosophers have said that number is the soul itself; however, it is more correct to say it is the principal numbering thing, but not number itself, since number is the sole instrument of discretion. And the homonymy of the term regarding the numbering and the numbered perhaps provided the moderns with an opportunity for error. For what Aristotle, and likewise Boethius, expresses in some places—that it exists in things and also is distinguished from them—that moved the Nominalists, who cast aside the things themselves more than is proper and favored the reasoning of names (and the name by which they are distinguished indicates this well), to make the same thing for the things themselves. This is true of the numbered, which is all they notice, but it is proper to distinguish the real from the things; and that, indeed, is proper regarding the numbering thing. But that they assert it is immersed in things, Aristotle does not prove, and Boethius even less so. And it is not difficult to detect this, especially from Aristotle’s First Philosophy, in the 13th and 14th books, in which he refutes the various opinions of the ancients concerning numbers. And that indeed fits well, for the measure of continuous things...