The following tasks are required to complete this lesson:
- Study the Lesson carefully
- Watch the lesson video, if available.
- Complete Comprehension Questions
Lesson
Study the following lesson carefully, including all footnotes. For help studying, please see the article, “How to Study for Mastery”.
The Division of the Even Number
Of the even number however there are three species. For one species is that which is called the evenly-even, but another is denominated the evenly-odd, and the third is the oddly-odd. And the species indeed which are contrary, and obtain the place of extremes are the evenly-even, and the evenly odd. But the species which is a certain medium, and participates of each of the extremes, is the number which is called oddly-odd.
Note: We will study the evenly-odd and oddly-odd numbers in coming lessons.
The Evenly-even Number
Again, the evenly-even number is that which may be divided into two equal parts, and each of these parts into two other equal parts, and each of these may be divided in a similar manner, and the division of the parts may be continued till it is naturally terminated by indivisible unity. Thus the number 64 has for its half 32, but the half of this is 16, the half of 16 is 8, the half of 8 is 4, of 4, two, and the half of 2 is 1, which naturally does not admit of division. To this number it happens that whatever may be its part is found to be evenly-even both in denomination and quantity. And it seems that this number was called evenly-even, because all its parts are found to be evenly-even both in name and quantity. We shall however hereafter show how this number has even parts both in quantity and appellation.
But the generation of these numbers is as follows: All numbers in a double ratio from unity, will always be found to be evenly-even; and it is not possible that they should be produced in any other way. Thus for instance, the numbers in a double ratio from unity are 1, 2, 4, 8, 16, 32, 64, 128, 256, 312, and so on ad infinitum; for they are all evenly-even, and the ratio of their progression is double.
Observations
1. It is remarkable in this series, that if the number of terms is even, the two middle terms correspond to each other, and this also will be the case with the terms outside these, and so on till each term meets with the extremities. Thus for instance, let there be given a series of evenly-even numbers from 1 to 128:
1, 2, 4, 8, 16, 32, 64, 128
In this series therefore, because the number of terms is even, one medium cannot be found. Hence there are two media 8 and 16, which mutually correspond to each other. For of the last term 128, 8 is the sixteenth part, and 16 is the eighth part. Again, the terms outside these will also be found to correspond to each other. For 32 is the fourth part of 128, and 4 is the thirty-second part of it. Of the terms likewise which are outside these, 64 is the second part of 128, and 2 is the sixty-fourth part of it: 128 also is once 128, and 1 is the one hundred and twenty-eighth part of 128.
But if the number of terms is odd, one medium only can be found, from the nature of the odd, and this corresponds to itself. For if this series be given 1, 2, 4, 8, 16, 32, 64, there will be one medium only which is 8, and this is the eighth part of 64, and thus is converted to itself both in denomination (value) and quantity (times multiplied to produce the last number in the series). After the same manner also as above, the terms, which are about it, confer on each other mutual appellations according to their proper quantities. For 4 is the sixteenth part of 64, and 16 is the fourth part of it. And again, above these terms, 32 is the second part of 64, and 2 is the thirty-second part of it. 1 also is the sixty-fourth part of 64, and 64 is once 64. Hence as we have said, all the parts of this series are found to be evenly-even both in appellation (value) and quantity (number of times taken).
2. This also is admirable that in any number of terms in this series, the sum of all the terms but the last, is equal to the last term less by one. Thus when the number of terms is 3, the sum of 1 and 2 is 3 which is less than 4 by 1 . In four terms likewise, the sum of 1, 2 and 4 is 7 which is less than 8 by 1. In five terms 1 + 2 + 4 + 8 = 15, which is less than 16 by 1; and so in any other finite number of terms. The first progeny of number also preserves and guards this property ; for unity is less than the following number 2 by unity alone. And hence, it is by no means wonderful that the sum of the other terms should accord with its proper principle. (We shall likewise find this property to be of the greatest advantage in ascertaining those numbers which are called superfluous, diminished, and perfect.
3. This also must not be passed over in silence, that in this series when the number of terms is even, the product of the extremes is equal to the rectangle under product of the two means; for when the series is even the media are two. Thus in that disposition of evenly-even numbers in which the last term is 128, the two means are 8 and 16, which multiplied by each other produce 128, equal to 1 x 128. The numbers likewise which are above these, if they are multiplied, will produce the same number. For 4 x 32 is equal to 1 x 128. But if the number of terms is odd, one middle term is found, and this multiplied into itself will be equal to the product of the two extremes. Thus in that series of terms, in which the extreme is 64, one middle term alone is found, and this is 8, which multiplied into itself is equal to 1 x 64. The terms also which are above this medium will, when multiplied into each other, give the same product; for 4 x 16 = 64. Thus also 32 multiplied by 2 and 1 multiplied by 64, produce the same number without any variation.
Memory Work
Recite the points below until you are able to give the answer exactly, from memory, when asked the question.
- What are the two divisions of number?
The two divisions of number are the even and the odd. - What are the three divisions of the even number?
The three divisions of the even number are the evenly-even, evenly-odd, and oddly-odd. - What is the evenly-even number?
The evenly-even number is the even number which may be divided into two equal parts, and each of these parts into two other equal parts, and each of these may be divided in a similar manner till it is naturally terminated by indivisible unity. - How are the evenly-even numbers generated?
All numbers in a double ratio from unity are evenly-even numbers: 1, 2, 4, 8, 16, 32, 64, 128, etc.
Assessment
