On the Principles of Arithmetic and On the Principles of Music

...are connected. Therefore, those things which first join numbers together must exist in their substance and remain forever. For nothing can be made from things that do not exist; these elements are both different from one another and possess the power of composition. These elements, of which number consists, are the even and the odd 5 which, by a certain divine power, although they are unequal and opposites, nevertheless flow from a single origin and are joined into one composition and harmony original: modulationem. In Boethius's worldview, mathematical "modulation" refers to the harmonious proportion found in both music and the structure of the universe..

Definition and division of number, and the definition of even and odd.

III. And first, it must be defined what a number is. 10 A number is a collection of units, or a heap of quantity pouring forth from units. Therefore, the primary division of number is into the odd and the even. An even number is that which can be divided into two equal parts without a single middle unit coming between them; an odd number, however, is that which no one can divide into equal parts 15 because that aforementioned single unit comes between them in the middle. And this definition is common and well-known.

The definition of even and odd number according to Pythagoras.

IIII. But that definition according to the Pythagorean school is as follows: an even number is one which, under the same division, can be 20 divided into the greatest and the smallest—the greatest in space, but the smallest in quantity—according to the opposing properties of these two types The Pythagoreans viewed "even" as the most perfect division because it splits a number into the largest possible parts (two halves) but the smallest possible number of parts (the number two).. An odd number, however, is one to which this cannot happen, but which is divided into two unequal...

The following are variant readings from different historical manuscripts of this text:

1 Therefore they will be (in manuscript l). || Above the line "of numbers" (in manuscript b), and "of numbers" in the margin (in manuscript r). || that (in manuscript f). || they join together (in manuscript f). 2 they consist (in manuscripts a, c, s). 7 however from one (in manuscripts c, f, s). || origin in the margin (in manuscript a). 9 A second hand added the heading in manuscript d || "Definition and" omitted (in manuscript f). "and definition" omitted (in manuscripts c, r, s) || the difference of the even (in manuscript f). 11 For a number (in manuscript f). 13 is indeed (in manuscript f). 14 falling between (in manuscript f). 16 falls between (in manuscripts b, f, l, s). 18 Heading omitted (in manuscript d). || of number omitted (in manuscripts f, l, r). 21 divided... quantity in the upper margin (in manuscript d). and the smallest quantity (in manuscript f). 22 of those two (in manuscript s) || corrected "of two" to "of both" (in manuscript a). 23 and 24 They place the definition of an odd number at the end of this chapter (in manuscripts c, d, f, l, r, s). || these (in manuscript f). || the same (in manuscripts d, l, s). 24 to happen (in manuscript a).