On the Principles of Arithmetic and On the Principles of Music

if one part of the division is even, the other is also found to be even; and if one is odd, the remaining part does not differ from its oddness—as in the case of the same number, which is ten original: denarius. For when it is divided into fives, or when into three and seven, both parts in
5 each portion turned out to be odd. If, however, the number itself, or another even number, is divided into equal parts—such as eight original: octonarius into 4 and 4—and likewise through unequal parts—as the same eight into 5 and 3—in that first division both parts are made even, while in this one both turned out to be odd;
10 nor can it ever happen that, when one part of the division is even, another can be found to be odd; or, when one is odd, another can be understood as even. An odd number, however, is that which, regardless of the division, is always divided into unequal parts, so that it always shows both kinds original: species; referring to the two types of numbers: even and odd of number, and
15 never is one without the other; instead, one part is assigned to evenness, the other to oddness—as, if you divide 7 into 3 and 4, one portion is even and the other is odd. And this same thing is found in all odd numbers; they can never be anything other than themselves in the division of an odd number. These are the twin kinds,
20 which naturally compose the power and substance of number.

VI. But if these must also be defined by means of each other's types, it will be said that an odd number is that which differs by a single unit original: unitas; the fundamental "one" from which all numbers are built from an even number, either by increase or by decrease. Likewise, an even number is that which differs by a unit from an odd number, either by
25 increase or by decrease. For if you take one away from an even number, or add one, it becomes odd; or if you do the same to an odd number, an even number is immediately produced.

The following section contains variant readings from different historical manuscripts of this text.

1 the word "was" is omitted in manuscript c. 4 "three or in seven" in a, "3 and 7" in c and l, "three and 7" in f. 6 "and eight" in f. 8 "five and three" in d. 11 "that" in c. 14 "as" added by a second hand in d. || "it shows" in c. 16 "seven" in d || "3 and 4" in a, c, f, "three and four" in d, "3 and in 4" in r and s. 18 the word "numbers" is omitted in f. 19 "These" (spelling variant) in a. 21 The heading is omitted in d. || "and odd" is omitted in f. 24 "increase" in f. || "Likewise even" in c, f, s, "Likewise even again" in a. 25 "or increase" written above the line as a correction in d. 26 "you take away" (spelling variant) in a, b, s.

Technical vocabulary: even, odd, division, the number ten, the number eight, unit, increase, decrease.