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 Evenly-Odd Number
The evenly-odd number is that which is itself indeed allotted the nature and essence of even-ness, but is opposed in a contrary division to the nature of the evenly-even number. For it will be shown that it is divided in a very different way. For because it is even, it receives a section into equal parts, but the parts of it immediately become indivisible. And such are the numbers: 6, 10, 14, 18, 22 and others similar to these. For in dividing these numbers into equal parts, we are immediately stopped by the odd number which cannot be divided equally.
But it happens to these numbers that all their parts have denominations contrary to the quantities of the parts that are denominated; nor can it ever be possible that any part of an evenly-odd number should receive a denomination and quantity of the same kind. For always, if the denomination is even, the quantity of the part will be odd. Thus for instance, in the number 18 the half of it, which term half is the appellation of parity, is 9, which is odd in quantity. But its third part, which is an uneven denomination, is 6, which is even in quantity. Again, by conversion, the sixth part which is an even denomination is 3, but the ternary is odd. And the ninth part, which is an uneven appellation, is 2 which is an even number. And the same thing will be found to take place in all other evenly-odd numbers: It is not possible that the name and number of any part are of the same kind.
Producing Evenly-Odd Numbers
These numbers are produced by disposing from unity all the numbers that differ from each other by 2, viz. all the odd numbers in a natural series. For if each of these is multiplied by 2, all the evenly-odd numbers will be generated. Let there be given then the following odd numbers from unity in a natural order, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. These, if each of them is multiplied by 2, will form the series 2, 6, 10, 14, 18, 22, 26, 30, 34, 38. And each of these if divided will only receive one section, a second division being excluded by the intervention of iimparity.
Observations
1. These numbers differ from each other by 4: and this arises from the mode of their generation. For the odd numbers which are the foundation of them, exceed each other by 2; and because each of these is multiplied by 2, the progression receives a four-fold increase.
2. But these species of numbers, the evenly-even, and the evenly-odd, are said to be contrary, because in the evenly-odd number, the greater extremity alone receives division, and the less term is in this alone liberated from section; and because in the form of the evenly-even number, the product of the extremes is equal to the product of the means, as far as to the two media, if the number of terms is even; but if the number of terms is odd, the square of the medium is equal to the product of the extremes. In the evenly-odd number however, if the number of terms is odd and there is therefore only one middle term, this term will be the half of the sum of the terms placed about it. It will also be the half of the sum of the terms placed above these; and this will be the case as far as to the extremes of all the terms. Thus in the series of evenly-odd numbers 2, 6, 10, the middle term 6 is the half of 10 + 2. And if there are two media, the sum of these will be equal to the sum of the terms placed above them. Thus in the series 2, 6, 10, 14, the sum of the media 6 + 10 is equal to 2 + 14 the sum of the extremes.
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-odd number?
The evenly-off number is the even number that receives a section into equal parts, but the parts of it immediately become indivisible. - How are the evenly-odd numbers generated?
All numbers produced by multiplying odd numbers by 2 are evenly-odd numbers.
Assessment
