On the Institution of Arithmetic

...is astronomy, but the doctrine of immobiles is geometry, or because the motion of the stars itself is celebrated by harmonic modulations. Therefore, it is also certain that the power of music precedes the course of the stars in antiquity, which it is not doubtful that arithmetic exceeds by nature, since it seems older than those things which are prior to it. Yet, properly, the very nature of numbers is what every course of the stars and every astronomical reason is constituted upon. For so we gather the risings and settings, so we keep the slownesses and velocities of the wandering stars, so we recognize the eclipses and the manifold variations of the moon. Therefore, since it has become clear that the power of arithmetic is prior, let us take the beginning of our discussion from here.

On the substance of number. Chapter 2.

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All things that have been constructed by the primeval nature of things seem to be formed by the reason of numbers. For this was the principal exemplar in the mind of the Creator. From this, the multitude of the four elements was borrowed; from this, the turns of the times; from this, the motion of the stars and the revolution of the heavens. Since these things are so, and the state of all things functions by the connection of numbers, it is also necessary that number itself, always having itself in its own way, remains in substance equally, and is composed not from diverse things. For what would join the substance of number, when its exemplar had joined all things? But it seems to be composed from itself. Furthermore, nothing seems to be composed from similar things, nor from those which are joined by no proportion of reason, and are distinct from one another in every substance and nature. It is certain, therefore, since number is joined, that it is not joined from similar things, nor from those which, toward one another, adhere by no reason of proportion. Therefore, there will be the first things of numbers which join toward a substance, indeed, which consist of parts and always remain. For nothing can be made from things that do not exist, and they are themselves dissimilar and have the power of being composed. These, however, are the things of which number consists: even and odd. These, through a certain divine power, although they are unequal and contrary, nevertheless flow from one generation and are joined into one composition and modulation.

On the definition of number and the division of even and odd. Chapter 3.

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And first, what number is must be defined. A number is a collection of unities, or a heap of quantity flowed from unities. Its first division, therefore, is into odd and even. And an even number is that which can be divided into two equals, with no intermediate unit standing in the middle; an odd number is that which no number divides into equals, because one, as previously mentioned, stands in the middle. And this definition of this sort is vulgar and known.