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Classical Arithmetic, Lesson 03. Book I, Chapter 3

The following tasks are required to complete this lesson:

  1. Study the Lesson carefully
  2. Watch the lesson video, if available.
  3. Complete Comprehension Questions

Lesson

In this lesson, we study the natural species (or divisions) of number–the even and the odd–and their various definitions.

The first division of number therefore is into the even and the odd.

1. (First definition) And the even number indeed is that which may be divided into two equal parts, without the intervention of unity in the middle. But the odd number is that which cannot be divided into equal parts, without unity intervening in the middle. And these indeed are the common and known definitions of the even and the odd.

2. (Second definition) But the definition of them according to the Pythagoreans is as follows: The even number is that which under the same division may be divided into the greatest and the least; the greatest in space, and the least in quantity, according to the contrary passions of these two genera. But the odd number is that to which this cannot happen, but the natural division of it is into two unequal parts. Thus, for instance, if any given even number is divided, there is not any section greater than half, so far as pertains to the space of division, but so far as pertains to quantity, there is no division less than that which is into two parts. Thus, if the even number 8, is divided into 4 and 4, there will be no other division, which will produce greater parts, viz. in which both the parts will be greater. But also there will be no other division which will divide the whole number into a less quantity; for no division is less than a section into two parts. For when a whole is separated by a triple division, the sum of the space is diminished, but the number of the division is increased. As discrete quantity however, beginning from one term, receives an infinite increase of progression, but continued quantity may be diminished infinitely, the contrary to this takes place in the division of the even number; for here the division is greatest in space, but least in quantity. In other words, the portions of continued quantity are the greatest, but the discrete quantity is the least possible.

3. (Third definition) According to a more ancient mode likewise, there is another definition of the even number, which is as follows:  The even number is that which may be divided into two equal, and into two unequal parts; yet so that in neither division, either parity will be mingled with imparity, or imparity with parity; except the binary number alone, the principle of parity, which does npt receive an unequal section, because it consists of two unities.  Thus for instance, an even number as 10 may be divided iuto 5 and 5 which are two equals; and it may also be divided into unequal parts, as into 3 and 7. It is however, with this condition, that when one part of the division is even, the other also is found to be even; and if one part is odd, the other part will be odd also, as is evident in the same number 10. For when it is divided into 5 and 5, or into 3 and 7, both the parts in each division are odd. But if it, or any other even number, is divided into equal parts, as 8 into 4 and 4, and also into unequal parts, as the same 8 into 5 and 3, in the former division both the parts are even, and in the latter, both are odd. Nor can it ever be possible, that when one part of the division is even, the other will be found to be odd; or that when one part is odd, the other will be even. But the odd number is that which in every division is always divided into unequal parts, so as always to exhibit both species of number. Nor is the one species ever without the other; but one belongs to parity, and the other to imparity. Thus if 7 is divided into 3 and 4, or into 5 and 2, the one portion is even, and the other odd. And the same thing is found to take place in all odd numbers. Nor in the division of the odd number can these two species, which naturally constitute the power and essence of number, be without each other.

4. (Fourth definition) It may also be said, that the odd number is that which by unity differs from the even, either by increase, or diminution. And also that the even number is that which by unity differs from the odd, either by increase or diminution. For if unity is taken away or added to the even number, it becomes odd; or if the same thing is done to the odd number, if immediately becomes even

5. Some of the ancients also said that the monad is the first of odd numbers. For the even is contrary to the odd; but the monad is either even or odd. It cannot however be even; for it cannot be divided into equal parts, nor in short does it admit of any division. The monad therefore is odd.  If also the even is added to the even, the whole becomes even; but if the monad is added to an even number, it makes the whole to be odd. And hence the monad is not even, but odd.

6. Aristotle however, in his treatise called Pythagoric says, that the one or unity participates of both these natures; for being added to the odd it makes the even, and to the even the odd; which it would not be able to effect if it did not participate of both these natures. And hence the one is called “evenly-odd”. Archytas likewise is of the same opinion.  The monad therefore is the first idea of the odd number, just as the Pythagoreans adapt the odd number to that which is definite and orderly in the world. But the indefinite duad is the first idea of the even number; and hence the Pythagoreans attribute the even number to that which is indefinite, unknown, and inordinate in the world. Hence also the duad is called “indefinite”, because it is not definite like the monad. The terms, however, which follow these in a continued series from unity, are increased by an equal excess; for each of them surpasses the former number by the monad. But being increased, their ratios to each other are diminished. Thus, in the numbers 1, 2, 3, 4, 5, 6, the ratio of 2 to 1 is double; but of 3 to 2 sesquialter; of 4 to 3 sesquitertian; of 5 to 4 sesquiquartan; and of 6 to 5 sesquiquintan. This last ratio, however, is less than the sesquiquartan, the sesquiquartan is less than the sesquitertian, the sesquitertian than the sesquialter, and the sesquialter than the double. And the like takes place in the remaining numbers. The odd and the even numbers also surveyed about unity alternately succeed each other.

Assessment

  1. Into what species may number be divided?
  2. What is the “even” and the “odd” according to the first, or common, definition?
  3. What is the “even” and the “odd” according to the second, or Pythagorean, definition?
  4. What is the “even” and the “odd” according to the third, or pre-Pythagorean, definition?
  5. What is the “even” and the “odd” according to the fourth definition?
  6. Is the number one even or odd?
  7. Briefly summarize the content of this lesson.
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