On the Institution of Arithmetic (1488) - Page 22

An exposition of the description pertaining to the nature of the evenly odd number in width and the evenly even number in length. Chap. 12.

The reasoning of the description arranged above is this: if you look to the width, where there is one mean of two terms, and you join the terms themselves, you will find their doubles to be the mean, as 36 and 20 make 56, the mean of which is 28, which is constituted as the middle term between them. And again, if you join 28 and 12, they make 40, of which 20 is the mean, the middle term of those is found. But where they have two means, both extremities joined together become equal to both means, as 12 and 36 when you join them become 48; if you apply their means to each other, that is 20 and 28, it will be the same, and in the other part of the width, the numbers that result are noted in the same order. Nor will the reason of either width differ in any way, and you will note the same in the same order in the remaining numbers, and this is done according to the form of the evenly odd number, in which it has already been written above that this property exists. Again, if you look to the length where two terms have one mean, which is done from the multiplied extremities, this happens if the middle term takes the increments of its own plurality. For twelve times 48 makes 576. If the middle term of those, that is 24, multiplies itself, it will procreate those same 576. And again, if 24 multiplies 96, they make 2304. Of which, if the middle term, that is 48, leads into itself, they create the same 2304. Where, however, two terms include two means, which is done by the multiplied extremities, this same result is returned when the means are multiplied into one or the other sum. For twelve times 96 multiplied procreate 1152; if their two means, that is 24 and 48, multiply themselves, they will restore those same 1152. And this is by the imitation and kinship of the evenly even number, from which this property is recognized as generated, having drawn participation from it. And on the other side of the length, the same ratio and description is noted. Therefore, it is manifest that this number was procreated from the two previous ones, since it retains their properties.

On the odd number and its division. Chap. 13.

The odd number is also one that is disjoined from the nature and substance of an even number. For while that one can be divided into two equal members, this one is prevented from being able to be cut by the intervention of the unit. It has three similar subdivisions, of which one part is that number which is called prime and incompositus incomposite/indivisible. The second, however, is that which is secondary and compositus composite/divisible. And the third is that which is joined by a certain mean of these, and draws something naturally from the kinship of both, which is indeed secondary and composite by itself, but compared to others, is found to be prime and incomposite.