Arithmetics of Boethius

And whatever part that is, it is necessary that a sum of the same magnitude be found in the prior quantity. First, let it be thus: if the arrangements are even, so that the two middle parts correspond to each other. Afterward, those which are above them should be turned toward each other in turn. And let this same thing be done until each term reaches the extremities. For let the order of an evenly even a number that can be divided by two, and its results divided by two, all the way down to one be placed from one up to 128 in this way: 1, 2, 4, 8, 16, 32, 64, 128. And let that be the greatest sum. In this series, since there are an even number of arrangements, a single middle cannot be found. There are, therefore, two, which are 8 and 16: which must be considered in how they respond to each other. For of the whole sum, which is 128, the eighth part is 16, and the sixteenth part is 8. Again, those parts which are above these will respond to each other in turn, namely 32 and 4. For 32 is the fourth part of the whole sum, and 4 is the thirty-second part. Again, above these parts, 64 is the second part, and 2 is the sixty-fourth part, until they reach the limit of the extremities, which no doubt enjoy the same correspondence. For the whole sum is 128 taken once, and one is the one-hundred-and-twenty-eighth part. If, however, we place an odd number of terms, that is, sums the author uses "terms" and "sums" interchangeably here, according to the nature of the odd, a single middle can be found, and it will respond to itself. For if this order is placed here: 1, 2, 4, 8, 16, 32, 64, there will be only one middle, which is 8. This 8 is the eighth part of the whole sum, and it is converted to itself in both name and quantity. In the same way as above, the terms that are around it give each other mutual names according to their own quantities and exchange their designations. For 4 is the sixteenth part of the whole sum, while 16 is the fourth part. And again above these terms, 32 is the second part of the whole sum, while 2 is the thirty-second part. And the whole sum taken once is 64, while unity is found to be the sixty-fourth part. This, therefore, is what was said: that all its parts are found to be evenly even in both name and quantity.

4 This also was perfected with much consideration and the great constancy of the divinity: that the smaller sums arranged in order in this number, when heaped together upon themselves, always equal one less than the following number. For if you join one to the two that follow, they become 3, which falls one short of the number four. And if you add 4 to the previous ones, they become 7, which is surpassed by the following eight by only a single unity. But if you join the same 8 to those mentioned above, they will become 15, which would exist as equal to the quantity of the number 16 if the smaller unity did not prevent it. The first offspring of number also keeps and guards this. For unity, which is first, is smaller by only a single unity than the two following numbers combined. Whence it is no wonder that the whole growth of the sum agrees with its own principle. This consideration will be of greatest use to us in recognizing those numbers which we will show to be superfluous abundant, diminished deficient, or perfect. For there, the accumulated quantity of the parts is compared to the limit of the whole number. We also can pass over nothing with forgetfulness regarding how, in this number, when the parts responding to each other are multiplied, the greater extremity and the sum of that same number will be produced. First, if the arrangements are even, the middles are multiplied, and from there those which are above them, up to the aforementioned

Book I, 19

extremities. For if there are even arrangements, they will contain two terms in the middle according to the nature of the even, as in that arrangement of numbers in which the last term ends at 128. For in this number, the middles are 8 and 16, which, when multiplied by each other, produce the sum of the greater increasing plurality. For if you multiply eight times 16 or sixteen times 8, the sum grows to 128. And these numbers which are above them, if they are multiplied, do the same. For if you multiply 4 and 32 by each other, they will produce the aforementioned extremity. For 4 taken thirty-two times, or 32 taken four times, will complete 128 by immutable necessity. And this holds all the way to the extreme terms, which are 1 and 128. For once the extreme term is 128. If unity is multiplied one hundred and twenty-eight times, nothing of the previous quantity will be changed. If, however, 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 last term is concluded with the plurality of 64, only one middle is found, which is 8. If you multiply this eight times, that is, into itself, it will explain 64. And those who are above this middle return the same, just as those placed above the two middles did before. For four times 16 are 64, and sixteen times 4 complete the same. Again, 2 taken 32 times does not depart from 64, and 32 taken twice heaps up the same. And 64 taken once, or unity multiplied sixty-four times, restores the same number without any variation.

COMMENTARY ON CHAPTER SIX

14 A decorative initial C depicts foliage and scrollwork. Consequently, he pursues the species of the even number, and primarily the evenly even, which he indeed defines and to which he ascribes and assigns five properties. He defines the evenly even number as that which both itself and all its parts admit a division into two equal portions. This must be understood regarding the part that is a number. For the unity of the evenly even number itself is indeed a part, but not one that can be divided into two, much less into two equal parts. In this matter, let us take this example: 64 is an evenly even number. For first it is cut into two equals, namely 32 and 32. Likewise, all its parts (I speak of those which are numerical, in which genus are 32, 16, 8, 4, 2) admit a section of equals. And that division finds its first end in unity. For 32 is divided into 16 and 16; sixteen into 8 and 8; 8 into 4 and 4; four into 2 and 2; and two into one and another unity, in which all division is ended and absolved. In this way, one might argue that human discernment of actions terminates in the unity of right reason. Likewise, the multiple discrepancy and division of truths terminate in that unity which serves consensus regarding the matter. For in this way all things love to be resolved into one and to be closed and absolved by one. Thus every composition is finally led to unity, and things that are divided demand unity, by which indeed individual things are preserved and consist. Hence, those things attributed to age are united in time, as if in a simple and regular measure. But those things of time and age are far more fully in Aevum unbounded time or the age of celestial beings, approaching unity, which unity alone measures, when it acknowledges only its beginning as its limit. Truly, all things are in eternity as in the simplest and super-immense unity, which all things desire to be assimilated to and individual things strive for. The same can be seen in magnitudes and in a certain paradigm. For surfaces stop and terminate the section of bodies, while lines terminate surfaces. But the limit and simplest measure of all is the point, in which all division in magnitude is finished. Beyond the point there is nothing, just as there is no division of bodies without a surface, or any surface without a line. Thus the section of each thing enjoys being absolved by its own principle, in which part every otherness and every composition is weighed, absorbed into the concord of unity. Furthermore,