On the Institution of Arithmetic

There are three species of even number. For one is called pariter par evenly even, another is pariter impar evenly odd, and the third is impariter par unevenly even. The contrary ones, which occupy the places of the summits, seem to be pariter par and pariter impar. A certain mean, however, which participates in both, is the number that is called impariter par.

An even-even number is one that can be divided into two even numbers, and its part into two other even numbers, and the part of the part into two other even numbers, and this is done so many times until the division of the parts naturally arrives at the indivisible unit. For example, the number sixty-four has a mean of thirty-two; this, in turn, has a mean of sixteen; this one, eight; this is divided by the number four into equal parts, which is the double of two; but the two is divided by the mean of the unit, which unit, being naturally singular, does not admit of section. It seems to happen to this number that whatever its part may be, it is found with the same name and title of evenly even, both in name and in quantity. But it seems to me that this number is called evenly even because all its parts are found to be even both in name and in quantity. How this number has parts that are even both in name and quantity, we shall say later. The generation of these is as follows: from one, you may note any numbers in double proportion; they will always produce evenly even species. Aside from this generation, it is impossible for others to be born. An example of this matter described in order appears to be such. Let these be all the doubles from one: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, and if an infinite progression is made from here, you will find all such numbers. They are made from one in double proportion, and all are evenly even. That, however, is not worthy of the least consideration: that every part of it is named from whichever part that is within that number, and it contains such a sum of quantity as the part is of the other number of that evenly even set that contains its quantity. Thus it happens that the parts correspond to each other: just as one is a certain part, the other has that same amount of quantity, and as this part is of that, so it is necessary that the same multitude be found in the previous sum. And first it happens, if the arrangements are even, that two mean parts correspond to each other; after that, those that are above them are converted to each other, and this is done until each term reaches the extremities. For let the order of the evenly even be posited from one up to 128 in this way: 1, 2, 4, 8, 16, 32, 64, 128, and let that be the maximum sum. In this, therefore, since the arrangements are even, one mean cannot be found. There are therefore two: that is, eight and sixteen, which are to be considered in the way they correspond to each other. For sixteen is the eighth part of the whole sum, that is, 128; and eight is the sixteenth. Again, those that are above these parts will correspond to each other: