On the Institution of Arithmetic

that extends to any division of this kind: for it always divides into unequals, so that it always displays both species of number, and one is never without the other. Instead, one part is assigned to evenness, and the other to oddness. For example, if you divide seven into three and four, one portion is even and the other is odd. This same phenomenon is found in all odd numbers. Nor can these twin species, which naturally compose the power and substance of a number, ever exist in the division of an odd number apart from themselves.

The definition of even and odd through one another. Chapter 6.

A decorative woodcut initial 'Q' features a stylized floral vine motif inside a square border.But if these even and odd numbers are also to be defined through one another, it shall be said that an odd number is one that differs from an even number by a unit, either by an increase or a decrease. Likewise, an even number is one that differs from an odd number by a unit, either by an increase or a decrease. For if you take one away from an even number or add one to it, an odd number is created; or if you do the same to an odd number, an even one is immediately produced.

On the primacy of the unit. Chapter 7.

A decorative woodcut initial 'O' contains a central five-petaled rosette surrounded by scrolling foliage.Every number is also a mean of those numbers placed around it in their natural order. And if those two numbers that are joined to the middle are added together, the aforementioned number is also the mean part of them. Furthermore, those that are joined in the second place above them, since they are also combined, the number prior to them is in the place of the mean, and this continues until the unitas the unit/one is encountered and creates the boundary. For instance, if someone posits the number five, around it on one side are four, and below it are six. If these are joined, they make ten, of which the number five is the mean. Furthermore, those that are around them, that is, around six and four, are three and seven; if these are joined, the number five is their mean. Again, if those that are placed on the other side are joined, these are also doubles of the number five. For above three are two, and above seven are eight. If these are joined, they make ten, of which five is again the mean. This same thing occurs in all numbers, until one can reach the limit of the unit. For only the unit does not have two terms around it, and therefore it is the mean of only that which is near it. For next to one, only the binarius the number two is naturally established, of which the unit is the mean part. Therefore, it is clear that the unit is the first of all those in the natural order of numbers, and it is rightly recognized as the progenitor of the entire, however extensive, plurality.

The division of an even number. Chapter 8.