Nicomachus begins by teaching that number is of 3 kinds. He says it is a: “limited multitude or a combination of units or a flow of quantity made up of units”. E.g. the number 3, 4 plus 4, 5 in motion. He then provides an ordinary definition of even being: “that which can be divided into two equal parts without a unit intervening in the middle” E.g. 4 is divided by 2 two times. Concerning the odd number species he says: “that which cannot be divided into two equal parts because of the aforesaid intervention of a unit” E.g. 3 is divided by 2 one time with 1 remaining, i.e. two unequal parts.
He goes deeper with both species according to Pythagorean doctrine. Concerning an even number, we learn that it admits of division into the greatest and smallest parts at the same operation. I.e. ‘a half’ is the greatest magnitude, and the ‘number 2’ is the least multitude possible post-division. Indeed, Nicomachus says that this is: “in accordance with the natural contrariety of these two genera”. On the other hand, odd is that which doesn’t allow this to be done to it. An odd number is divided into two unequal parts. E.g. 3 is divided by 2 two times with 1 remaining.
At this point he introduces a more ancient definition of both respective species. The ancients considered even to be that which can be divided into two equal and two unequal parts. E.g. 4 is divided by 2 two times, but 8 is divided by 6 is one time with 2 remaining. However, concerning the number 2, he says: “… except that the dyad, which is its elementary form, admits but one division, that into equal parts’. Interestingly, he teaches that in any division with even numbers, brings to light only one species of number, however it may be divided, independent of the other. I.e. even divided by even equates to even in parts. E.g. 22 is divided by 6 three times with 4 remaining, both the 22 and the 6 are even numbers.
Juxtaposed to this, concerning the ancient definition of odd, he says: “a number which in any division whatsoever, which necessarily is a division into unequal parts, shows both the two species of number together”. He goes on to explain that there is never an instance where there is not some intermixture of even and odd together. E.g. 17 is divided by 6 two times with 5 remaining, the two instances of 6 being the even numbers, with the 5 being the odd number, to complete the sum of 17.
He concludes with an interesting image of the positioning of even and odd with respect to each other. He says: “the odd is that which differs by a unit from the even in either direction, that is, toward the greater or the less, and the even is that which differs by a unit in either direction from the odd, that is, is greater by a unit or less by a unit”.
EAR