On the Institution of Arithmetic

For if you multiply four and thirty-two among themselves, they will make the aforementioned extremity. For four taken thirty-two times, or four times thirty-two, will complete 128 by an immutable necessity. And this falls down to the extreme terms, that is, one, two, 128. For the extreme term is once 128. With the unit multiplied one hundred and twenty-eight times, it will not change anything from the previous quantity. But if the arrangements are odd, one middle term is found, and it responds to itself by its own multiplication. For in that order of numbers, where the extreme term concludes in the plurality of 64, only one mean is found, that is, eight. If you multiply this eight times, that is, into itself, it will unfold 64. And those that are above this mean render the same as those that were formerly placed above the two made. For four times sixteen is 64, and sixteen times four complete the same. Again, twice 32 does not depart from 64, and thirty-two times two accumulate the same, and once 64, or the unit multiplied sixty-four times, will restore the same number without any variation.

On the evenly odd number and its properties. Chapter 10.

A decorative woodcut initial 'P' features intertwined floral vines and foliage within a square frame.
The pariter impar evenly odd number is that which has itself obtained the nature and substance of evenness, but in a contrary division, it is opposed to the nature of the evenly even number. It will be taught how far this is divided by a dissimilar ratio. For since it is even, it admits of section into equal parts, but its parts soon remain indivisible and unsectable, as are 6, 10, 14, 18, 22, and those similar to them. For soon, if you have parted these numbers into a double division, you fall into an odd number that you cannot cut. It happens to these that they have all parts denominated contrarily, which are the quantities of the parts themselves that are denominated. Nor can it ever happen that any part of this number receives the denomination and quantity of the same genus. For always, if the denomination is even, the quantity of the part will be odd, and if the denomination is odd, the quantity will be even; as in 18, its second part is 9, which is an odd quantity—that is, the mean, which is the name of evenness. The third, however, which is an odd denomination, is six, to which there is an even plurality. Again, if you invert it, the sixth part, which is an even denomination, is three; but the ternary is odd. And the ninth part, which is an odd vocabulary, is 2, which is an even number. And the same is found in all others that are evenly odd. Nor can it ever happen that for any part, the name and the number are of the same genus. But the procreation of these numbers happens if they are arranged from one, those which differ by two, that is, all the odd numbers established in a natural sequence and order. For if these are multiplied by the number two, they will all make the evenly odd plurality measured.