Arithmetic, Geometry, and Music

if all evenly odd pariter impares even numbers that can be divided only once by an odd number, not reaching unity numbers are multiplied, the plurality is rightly measured. For let the first unit be set: 1. And after this, one that differs from it by two, that is, three: 3. After this, one that again differs from the one above by two, that is, 5. And this to infinity; let the arrangement be of this kind: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. These, therefore, are naturally succeeding odd numbers, which no even number separates in between. If you multiply these by the number two, you will accomplish it in this way: twice one, that is, 2, which is indeed divided, but its parts are found to be indivisible because of the nature of the indivisible unit. Twice 3, twice 5, twice 7, twice 9, twice 11, and subsequently, from which these are born: 2, 6, 10, 14, 18, 22. If you divide these, they admit one section, rejecting the other, because the second division is excluded by the half-part of an odd number. But between these numbers, there is a distance of only a quaternary four to each other. For between the numbers two and six, there are 4. Again, between 6 and 10, 4; between 10 and 14, and between 14 and 18, the same quaternary makes the difference. For these all surpass each other by a quaternary numerosity. This happens because the first ones that were set, that is, their foundations, preceded each other by the number two; because we multiplied these by two, that progression grew into the number four. For two multiplied by twice two make the sum of four. Therefore, in the arrangement of natural numbers, the evenly odd numbers, by how much they are distant from each other in place, precede each other only by 4, passing through 3 in the middle, created by multiplying the odd numbers by the number two.

These species of numbers are said to be contrary, that is, the evenly even and the evenly odd, because in an evenly odd number only the greater extremity receives division, while in the other, only the smaller term is solved by a section. And because in the form of an evenly even number, starting from the extremities and proceeding toward the middle, that which is contained under the extreme terms is the same as that which is contained under the sums placed within them. And this same thing will be reached even to two halves in even arrangements, of course. But if the arrangements are odd, that which is produced from one half is produced under the parts placed on the other side. And this continues until the process reaches the extremities. For in that arrangement which is 2, 4, 8, 16, they yield the same; 2 multiplied by 16 is that which 4 multiplied by the number eight is. 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 half. For twice 8 is 16; four times four is 16, which number is perfected from the quaternary into itself by eight. But in an evenly odd number, if there is one term in the middle, it is the half of the terms placed around it if they are reduced into one. And the same is the half of the terms that are above these. And this reaches up to the extremes of all terms, as in that order which is of evenly odd numbers, 2, 6, 10; joined together, the binary with the denary ten completes 12, of which six is found to be the half. But if two halves have been joined, both will be equal to the terms established above them, as in this order: 2, 6, 10, 14. For 2 and 14 joined grow into 16, which the six joined with the ten will effect. And this happens in more numerous terms, having taken the beginning from the middle, until the extremes are reached.

On the evenly odd The text here shifts to the definition of "impariter par" which is traditionally translated as "unevenly even" or "oddly even" number and its properties.

Chapter II.

i

The impariter par unevenly even number is constructed from both; in the place of a half, it is concluded by a double extremity, so that by whatever it differs from one, it is joined to the other by the same relationship. This, however, is such a one that divides into equal parts, and each part can be divided into other equal parts, and also sometimes the parts of the parts are divided, but not so that that equal distinction progresses all the way to unity; as are 24, 7, 28 the numbers 7 and 28 here may be placeholders or examples of non-divisibility in the specific Boethian sequence. For these can be divided into halves, and their parts again are solved into other halves without any doubt. There are also

certain other numbers whose parts receive other divisions, but that division itself does not reach up to unity. Therefore, in the fact that it admits more than one section, it has a similarity to the evenly even, but it is separated from the evenly odd. In the fact, however, that that section is not led up to one, it does not refute the evenly odd, but is disjoined from the evenly even. It happens, however, to this number to have both things which the former do not have, and to obtain both things which they receive. And it indeed has what both do not have: that whereas in one only the greater term was divided, and in the other only the smaller term was not divided, in this one, neither the greater term alone receives division, nor does the smaller term alone join itself from division. For the parts are solved, and yet that section does not reach to unity; if before unity a term is found which you cannot cut. It obtains also what those others receive: that certain parts of it respond, and are denominated according to their genus to their own quantity, that is, to the similarity of the evenly even number. Other parts, however, take a contrary denomination of their own quantity, to the form of the evenly odd, of course. For in the number 24, the quantity of the part is even, denominated from an even number. For the fourth is 6, the second is 12, the sixth is 4, the twelfth is 2, which names of parts do not differ from the parity of quantity. But they are rightly denominated when the third part is eight, the eighth is 3, the twenty-fourth is 1, which denominations, since they are from pairs, are found to be odd quantities. And when the sums are even, the denominations are odd. Such numbers are born in such a way that they designate their substance and nature even in their own proper generation, created from evenly even and evenly odd numbers. For all evenly odd numbers were born from the odd numbers placed in order. But the evenly even from a double progression. Let all odd numbers therefore be disposed in order naturally, and under them, starting from four, all the doubles, and let them be in this way:

These being placed in this way, if the first grows together by the multiplication of the first—that is, if the ternary three of the quaternary—or if the same first of the second—that is, the ternary of the octonary eight—or if the same first of the third—that is, the ternary of 16—and the same up to the last; or if the second of the first and second, or if the second of the third, and the same multiplication is brought forth up to the extreme; or if the third, starting from the first, passes up to the extreme. And thus the fourth, and all the higher ones in order, multiply those which are placed below in the arrangement; they will create all the unevenly even numbers. Let us take an example of this matter: if you multiply 3 by 4, 12 will be made; or if 5 by 4, the number 20 will grow; or if the same 5 by 7 The text likely implies multiplication of the arranged values, 28 will grow, and this up to the end. Again, if 8 multiplies 3, 24 will be born. If 8 by 5, 40 are made; if 8 by 7, 56 will be collected. And in this way, if all the lower doubles are multiplied by the upper ones, or if the upper ones multiply those same lower ones, you will find all that have been born are unevenly even. And this is the admirable form of this number: that when that arrangement and description has been perceived by number, the property is found to be as if to the width of the evenly odd, and to the length of the evenly even numbers. For two extremities are equal to two halves, or two double extremities to one half. In length, however, it designates the property of the evenly even number. For what is contained under two halves is equal to that which is concluded under the extremes, or what is born from one half is equal to that which is contained under both extremities. The description, however, which is supposed, was made in this way. As many as the number three multiplied in the order of evenly even numbers, whichever were created from it