Fundamentals of Arithmetic; Fundamentals of Music

There will be, therefore, numbers that join the first elements, which consist of substance and always remain. For nothing can be made from non-subsisting things, and they even and odd are themselves dissimilar and have the power of composition. These, however, are the things of which number consists: even and odd, which, by a certain divine power, although they are disparate and contrary, flow from one generation and are joined into one composition and modulation.

Definition and division of number and definition of even and odd.

III. And first, what number is must be defined. Number is a collection of units, or a heap of quantity flowing from units. Its first division, therefore, is into odd and even. And even, indeed, is that which can be divided into two equal parts, with no middle unit intervening; odd, however, is that which no one divides into equal parts, because the aforementioned one intervenes in the middle. And this definition of this kind is common and well known.

Definition of even and odd number according to Pythagoras.

IIII. That one, however, according to the Pythagorean discipline, is as follows: an even number is one that can be divided under the same division into the greatest and the smallest—greatest in space magnitude, smallest in quantity—according to the contrary passions of these two kinds. An odd number, however, is one to which this cannot happen, but one for which a natural section into two unequal sums is the only possibility.