On the Institution of Arithmetic (1488) - Page 12

Definition of even and odd number according to Pythagoras. Chapter 4.

Ornamental woodcut initial 'I' with foliage and floral motifs.
That, however, according to the Pythagorean discipline, is as follows. An even number is that which, under the same division, can be divided into the greatest and the smallest—greatest in space, smallest in quantity—according to the contrary passions of these two kinds. But an odd number is that to which this cannot happen, but which has a natural section into two unequal sums. This is the exemplar: if any given even number is divided, one will not find a greater division as far as the spaces of division pertain than a discrete half. But in quantity, there is none smaller than that which is made into a double partition. For instance, if the even number which is 8 is divided into 4 and another 4, there will be no other division that makes greater parts. Furthermore, there will be no other division that divides the whole number by a smaller quantity. For there is nothing less than a division into two parts. For when one has divided the whole into a triple division, the sum of the space is diminished, but the number of the division is increased. What was said "according to the contrary passions of two kinds" is of this sort: we previously taught that quantity grows into infinite pluralities, but spaces—that is, magnitudes—are diminished into most infinite smallnesses, and therefore the opposite happens here; for this division of the even is greatest in space and smallest in quantity.

Another division of even and odd according to the older method. Chapter 5.

Ornamental woodcut initial 'S' with swirling foliage and floral designs.
According to the older method, however, there is another definition of an even number. An even number is that which receives a partition into two equal parts and into two unequal parts, but so that in neither division is evenness mixed with oddness, or oddness with evenness, except only for the binarius the number 2, the prince of evenness, which does not receive an unequal section because it consists of two unities and is, in a way, the first evenness of two. What I say is of this sort. For if an even number is placed, it can be divided into two equal parts, just as the denarius the number 10 is divided into fives. Furthermore, it can be divided into unequals, as the same ten into 3 and 7. But in this way: 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. For example, in the same number which is ten. For when it is divided into fives, or when into 3 and 7, in both portions, both parts turned out to be odd. But if that or another even number is divided into equals, such as eight into 4 and 4, and likewise into unequals, such as the same eight into 5 and 3, in the former division both parts were made even; 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 can be found, or that when one is odd, another even can be understood. An odd number, however, is...