Fundamentals of Arithmetic; Fundamentals of Music

And this is the exemplar: if any given even number is divided, a greater one regarding the spaces of division will not be found than the split half, and in quantity, it is no smaller than the partition made into two; as if the even number which is 8 is divided into 4 and another 4, there will be no other division that produces greater parts; furthermore, there will be no other division that divides the whole number into a smaller quantity. For in a division into two parts, there is nothing smaller. For when one has parted the whole into a triple division, the sum of space is indeed diminished, but the number of the division is increased. But what was said, "according to the contrary passions of two kinds," is of this sort: for we have taught that quantity increases into infinite pluralities, but spaces, that is, magnitudes, are diminished into the most infinite smallnesses, and therefore here the opposite happens. For this division of the even is greatest in space, but smallest in quantity.

Another division of even and odd according to the older method.

V. According to the older method, however, there is another definition of an even number. An even number is that which admits a partition into two equal parts and into two unequal parts, but in such a way that in neither division is evenness mixed with oddness or oddness with evenness, except for the principle of evenness alone, the number two, which does not admit a section into an equal part, because it consists of two units and is, in a way, the first evenness of two. What I say, however, is this: for if an even number is posited, it can be divided into two equal parts, as ten is divided into fives; furthermore, also into unequal parts, as the same ten into 3 and 7, but in this way, that when...