On the Institution of Arithmetic

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

An ornamental woodcut initial 'S' features swirling foliage and floral designs.

According to the older method, there is another definition of an even number. An even number is that which admits of 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 binarius the number two, the principle of evenness, which does not admit of an unequal section because it consists of two units and is, in a certain way, the first evenness of numbers.

What I mean is this: if an even number is posited, it can be divided into two equal parts, such as ten being divided into five and five. Furthermore, it can be divided into unequal parts, such as that same ten being divided into three and seven. But it occurs in such a way that 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.

Take, for example, the same number, ten. When it is divided into five and five, or when it is divided into three and seven, both parts in each instance of division turn out to be odd. However, if this number or another even number is divided into equals, such as eight into four and four, and likewise into unequals, such as the same eight into five and three, in the former division both parts are made even, while in the latter both turn out to be odd. It can never happen that when one part of the division is even, the other can be found to be odd, nor when one is odd can the other be understood to be even. An odd number, however, is...