On the Institution of Arithmetic

Let the first unit be posited: 1, and after this, one that differs from it by two, that is, three; and after this, one that again differs from the superior by two, that is, 5; and so to infinity; and let the arrangement be of this kind: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. These, therefore, are the naturally following odd numbers, which no even number distinguishes in between. If you multiply these by the number two, you will effect it in this way: twice one, that is, 2, which indeed is divided, but its parts are found to be indivisible, on account of the nature of the unsectable unit. Twice 3, twice 5, twice 7, twice 9, twice 11, and so on, from which are born these: 2, 6, 10, 14, 18, 22. If you divide these, they receive one section, rejecting the other, because the second division is excluded by the mean of the odd part. In these numbers, however, the only distance among themselves is the quaternary. For between the numbers two and six, there are 4. Again, between 6 and 10, and between 10 and 14, and between 14 and 18, the same quaternary makes the difference. For all these transcend each other by a quaternary number. This happens because the first ones that were posited, that is, their foundations, preceded each other by the number two, which, because we multiplied them by the number two, that progression grew into the number four. For two multiplied by twice two make the sum of the quaternary. Therefore, in the arrangement of the natural number, the evenly odd numbers differ from each other in the fifth place, preceding each other only by 4, passing through 3 in the middle, procreated by multiplying the odds by the number two. But these species of numbers are said to be contrary, that is, the evenly even and the evenly odd, because in the evenly odd number, only the greater extremity receives division, while in the other, only the lesser term is released by section. And because in the form of the evenly even number, starting from the extremities and proceeding up to the middle, what is contained under the extreme terms is the same as that which is contained under the sums placed within them. And this is the same until one arrives at two means in the even arrangements. But if the arrangements are odd, what is accomplished by one mean, this same thing is procreated under the parts placed on the other side. And this is until the progression reaches the extremities. For in that arrangement which is 2, 4, 8, 16, the same is rendered: 2 multiplied by 16 is the same as 4 multiplied by the number eight. For in both ways, 32 will be made. But if the order is odd, as is 2, 4, 8, the extremes will do the same as the mean. For twice 8 are 16; four times four are 16, which number is perfected by the quaternary multiplied by itself. In the evenly odd number, however, if there is one term in the middle, if the terms placed around it are reduced into one, it is a mean. And the same is the mean of those terms that are above these. And this is until the extreme of all the terms, as in that order of evenly odd numbers: 2, 6, 10; the two joined with ten completes 12, of which the six is found to be the mean. If, however, there are two means joined, they will both be equal.