Arithmetica

❦An evenly even pariter par a number divisible into even parts until reaching the unit number is one that can be divided into two even parts, and each of those parts into two other even parts, and the parts of those parts into two other even parts, so that this process continues until the division of the parts reaches an indivisible unit by nature. For example, the number 64 has a half of 32. This also has a half of 16. This one indeed has 8. The number 4 also divides this into equal parts, which is double the 2, but the 2 is divided by the half of the unit; this unit, being naturally singular, does not admit division. It seems to happen to this number that whatever its part may be, it is found to be evenly even both in its name and in its vocabulary, as well as in its quantity. But for this reason, 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. We will say later how this number has parts that are even in both name and quantity.

❦The generation of these, however, is as follows. For from one, whatever numbers you mark in a double proportion, evenly even numbers are always produced. Furthermore, it is impossible for them to be born otherwise than through this generation. An example of this matter, described in order, seems to be of such a kind. Let all be doubles from one: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, and from here, if an infinite progression is made, you will find all such numbers. They are made from one in a double proportion, and all are evenly even.

That which is worthy of no small consideration is that every part of it, which is denominated from any part whatsoever within the number itself, includes as much in the sum of its quantity as the other number, which contains it, is a part of that evenly even number. Therefore, the parts respond to one another in such a way that as great a fraction as one part is, the other part holds that much quantity; and as great a fraction as that part is, it is necessary that an equal multitude be found in the former sum.

❦And first, if the arrangements are even, let the middle parts correspond to each other. Afterwards, let those that are above them be turned toward one another, and let this same thing be done until both terms reach the extremities. For let the order of the evenly even be placed from one to 128 in this manner: 1, 2, 4, 8, 16, 32, 64, 128, and let that be the maximum sum. In this, therefore, since the arrangements are even, one middle cannot be found. There are, therefore, two: that is, 8 and 16, which must be considered in how they respond to each other. For 16 is the eighth part of the whole sum, which is 128; 8 is the sixteenth. Again, those parts that are above these respond to each other in turn, that is, 32 and 4. For 32 is the fourth part of the total sum, while 4 is the thirty-second. Again, above these parts, 64 is the second part, while 2 is the sixty-fourth. This continues until the extremities form the limit, which without doubt enjoy the same correspondence. For the whole sum is 128 once, while one is the one hundred twenty-eighth part.

But if we place odd terms, that is, sums—for I name the terms as sums—according to the nature of the odd, one middle can be found, and one is to respond to itself. For if this order is placed: 1, 2, 4, 8, 16, 32, 64, there will be only one middle, that is, 8. This 8 is the eighth part of the total sum and turns toward itself in denomination and quantity. And in the same way, just as the terms above it were treated, they give each other mutual names according to their own quantities and exchange their vocabulary. For 4 is the sixteenth part of the total sum, while 16 is the fourth. And again, above these terms, 32 is the second part of the total sum, while 2 is the thirty-second; and the total sum 64 is once, while the unit is found as the sixty-fourth. This, therefore, is what was said: all its parts are found to be evenly even both in name and in quantity.

This also is perfected with much consideration and much constancy of the divine: that the lesser sums, if arranged in order in this number and piled upon themselves, are always equal to the following sum minus one. For if you join one to the two that follow, they become 3, that is, one less than the number four. And if you add 4 to the previous ones, they are 7, which are surpassed by only one unit by the following number eight. But if you add the same 8 to the aforementioned, they will become 15, which would be an even quantity for the number 16

were it not that the smaller unit hinders it. This, however, the first offspring of the number also preserves and guards. For the unit, which is the first, is a contraction by only one unit from the two following; hence it is not a wonder that the entire growth of the sum agrees with the primary principle. This consideration will profit us most in knowing those numbers which we will show to be superfluous, deficient, or perfect.

❦There, the piled-up quantity of the parts is compared to the whole term of the number. We also cannot pass over with any forgetfulness that in this number, when the parts responding to each other are multiplied, the greater extremity of that same number and the sum are produced. And first, if the arrangements are even, the middles are multiplied, and so too are those that are above them, up to the aforementioned extremities. ❦For if the arrangements are even, according to the nature of the even, it will contain two terms in the middle, as in that arrangement of numbers in which the extreme term ends at 128. In this number, the halves are 8 and 16, which, multiplied among themselves, will produce the sum of the greater plurality. For if you multiply eight times 16 or sixteen times 8, the sum 128 is formed. And these numbers that are above them, if they are multiplied, do the same thing. For if you multiply 4 and 32 among themselves, they will produce the aforementioned extremity. For 4 times 32, or 4 times 32 original: "quater 32" duct, will complete 128 by immutable necessity. And this falls down to the extreme terms, that is, 1 and 128. For the extreme term 128 is once; if the unit is multiplied one hundred twenty-eight times, nothing will be changed from the prior quantity. ❦If, however, the arrangements are odd, one middle term is found, and it responds to itself by its own multiplication. In that order of numbers where the extreme term is concluded by the plurality of 64, only one middle is found, that is, 8. If you multiply this eight times, that is, by itself, it will explain 64. And those that are above this middle return the same result, just as those placed above the two did before. For 4 times 16 is 64, and 16 times 4 complete the same. Again, 2 times 32 do not depart from 64, and 32 times 2 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.

❦An evenly odd pariter impar a number that is even but has odd factors number is that which itself has received the nature and substance of parity, but in the opposite division, it is opposed to the nature of the evenly even number. For it will be taught here that it is divided by a far different reason. For because it is even, it admits section into equal parts, but its parts soon remain indivisible and inseparable, as are 6, 10, 14, 18, 22, and similar to these. For if you have divided these numbers into a double division, you immediately run into an odd number which you cannot cut. It happens twice that they have all parts denominated conversely, which are the quantities of those parts that are denominated. And it can never 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 the medium, which is a name of parity, but 9 is an odd quantity. The third part, which is an odd denomination, is 6, to which an even plurality belongs. Again, if you reverse it, the sixth part, which is an even denomination, is 3, but the number three is odd. And the ninth part, which is an odd name, is 2, which is an even number. And the same is found in all others that are evenly odd. It can never happen that the name and number of any part are of the same genus. The procreation of these numbers happens if those which differ by two are arranged from one, that is, all the odd numbers established in a natural sequence and order. For if through the binary number...