Fundamentals of Arithmetic; Fundamentals of Music

one part of the division will be even, and the other will also be found to be even; and if one is odd, the remaining part will 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, in both cases, the parts in each portion turned out to be odd. But if that number, or another even number, is divided into equal parts, as eight into 4 and 4, and likewise into unequal parts, as that same eight into 5 and 3, in the former division, both parts were made even, while in the latter, both turned out to be odd; nor can it ever happen that, when one part of the division is even, another odd part could be found, or when one is odd, another even could be understood. An odd number, however, is that which is always divided into unequal parts for any given division, so that it always exhibits both species of number, and one is never without the other, but one part is assigned to parity, the other to oddness, as if you divide 7 into 3 and 4, the one portion is even, the other odd. And this same thing is found in all odd numbers, and they can never exist in a division of an odd number except by being unequal. These twin species naturally compose the force and substance of number. 5, 10, 15, 20

Definitio paris et inparibus per alterutrum Definition of even and odd through one another

VI. But if these are also to be defined through one another, it shall be said that an odd number is that which differs from an even number by a unit, either by an increase or a decrease. Likewise, an even number is that which differs from an odd number by a unit, either by an increase or a decrease. For if you subtract one from an even number, or add one to it, an odd number is created, or if you do the same to an odd number, an even number is immediately brought forth. 25