On the Principles of Arithmetic and On the Principles of Music

PRINCIPLES OF ARITHMETIC, BOOK I, CHAPTERS.

III. The definition and division of number, and the definition of even and odd.
IIII. The definition of even and odd number according to Pythagoras.
V. Another division of even and odd according to an older method.
5 VI. The definition of even and odd through each other.
VII. On the primacy of unity Boethius follows the ancient tradition where "unity" (the number 1) is not a number itself, but the seed or source from which all numbers grow..
VIII. The division of the even number.
VIIII. On the evenly-even number and its properties In modern terms, these are powers of two ($2^n$, such as 4, 8, 16) which can be repeatedly halved until reaching unity..
10 X. On the evenly-odd number and its properties These are numbers like 6, 10, or 14—even numbers that, when halved, immediately result in an odd number..
XI. On the oddly-even number and its properties, and its relationship to the evenly-even and evenly-odd.
XII. An explanation of the diagram pertaining to the nature of the oddly-even number.
15 XIII. On the odd number and its division.
XIIII. On the first and incomposite number The "first and incomposite" refers to a prime number, which cannot be broken down into other factors besides itself and unity..
XV. On the second and composite number.
XVI. On that which is secondary and composite in itself, yet prime and incomposite in relation to another.
20 XVII. On the creation of the prime and incomposite, the secondary and composite, and the number secondary and composite in itself but prime and incomposite to another.
XVIII. On the discovery of those numbers that are secondary and composite in themselves, but prime and incomposite when compared to others.
25 XVIIII. Another classification of the even number into perfect, diminished original: inperfectos. These are "deficient" numbers, where the sum of the factors is less than the number itself., and more than perfect original: ultra quam perfectos. These are "abundant" numbers, where the sum of the factors exceeds the number itself..
XX. On the generation of the perfect number.
XXI. On relative quantity.

The following section contains technical variations found in the original Latin manuscripts used by the editor, Gottfried Friedlein, to reconstruct the text.

1. "diffinitio" [definition] appears here and throughout the book except in manuscript d || "imparis" [odd] throughout the greatest part of the book. 2. the word "numeri" [of number] is omitted in manuscripts c and r. 6. "super" [above] in manuscript b.
,im
9. "pari" [even] in manuscript f. 10. This entire verse is omitted, and the following chapters are numbered one less in manuscripts d, f, and l. 12. "cognomine" [by name] in manuscript r. 13. "pariter" [evenly] in manuscript a. 16. "De proprio" [Of its own] in manuscript a. 18. "compositus et" [composite and] in manuscript s. 20. "et secundi et compositi" [and the second and composite] omitted in manuscripts a, c, d, l, r, and s. 21. "incompositi et alterum" [incomposite and the other] in manuscript c || "et alterum" [and the other] in manuscript l. 23. "ad secundi" [to the second] in manuscript b. 24. "vero" [indeed] omitted in manuscript a. 26. "Alio" [In another] in manuscript c || "partio" [division] in manuscript a; "participio" [participation] in manuscript d; "participatio" [participation] in manuscript l. 27. "et utraque" [and both] in manuscript a || "et" [and] omitted in d, l, and r.