On the Principles of Arithmetic and On the Principles of Music

...the total sums is a natural division. This is the pattern: if any given even number is divided, no part will be found larger in terms of the extent original: spatia. In this context, it refers to the size or magnitude of the resulting parts. of the division than the separated half; yet, in terms of quantity original: quantitate. Here, it refers to the number of parts created by the division., no division is smaller 5 than that made into a pair. For example, if the even number 8 is divided into 4 and another 4, there will be no other division that produces larger parts; furthermore, there will be no other division that divides the whole number into a smaller quantity of parts. For there is nothing less than a division into two parts. 10 For when one has partitioned the whole by a threefold division, the sum of the extent of each part is diminished, but the quantity of the divisions increases. As for what was said—"according to the contrary properties of two kinds"—it is of this sort: we previously taught that quantity grows toward infinite pluralities, 15 but extents (that is, magnitudes) decrease toward the most infinite smallness, and therefore here the opposite occurs. For this division of an even number is greatest in its extent, but smallest in its quantity.

Another division of the even and the odd according to an older method.

V. According to an older method, there is another 20 definition of an even number. An even number is that which allows for partition into two equals and into two unequals, but in such a way that in neither division is evenness mixed with oddness, nor oddness with evenness—excepting only the ruler of evenness, the binary original: binarium; the number 2, which does not admit an 25 unequal cutting, because it consists of two units and is, in a sense, the first evenness of two. What I am saying is this: if an even number is set down, it can be divided into two equals, as 10 is divided into two 5s; furthermore, it can also be divided into unequals, as the same 10 into 3 and 7, but in such a way that when...

1 "This however is", with the word "pattern" omitted in manuscripts d and l. "This however is the pattern" in r. 2 "anyone" in a and s. 4 "let it be" in a, d, and l. 6 "four" instead of the Roman numeral "IIII" in d; "four and into IIII" in s. || "others" in c. 12 "of two" corrected to a different form of "of two" in a. 15 "infinite" in f. 18 The title of the chapter is omitted in d. 22 "or oddness to evenness or evenness to oddness" in f. || "or oddness to evenness" in the upper margin of d. 25 "consists of units" in f. 27 "equal" in b, d, and l. 28 "by" omitted in d. 29 "three and seven" in r; "three and into VII" in f.