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Lesson
The generation however and origin of these numbers is obtained by the following method, which Eratosthenes denominates a sieve; because all the odd numbers being placed in the middle by the art which we shall shortly unfold, those numbers which are of the first, or second, or third kind are distinguished. For let all the odd numbers in an orderly series be disposed from 3, to any extent whatever, namely:
3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49.
These, therefore, being thus disposed, it must be considered what is the first number of the series which 3 will measure. And it will be found, that two numbers being omitted, it will measure that number which immediately follows them, viz. it will measure 9. If also after 9 two others are omitted, it will measure the following number 15. Again, if beginning from 15 two numbers are omitted, it will measure the following number 21. And thus it will be found ad infinitum, that the first number 3, by omitting two numbers will measure all the following numbers posterior to itself, according to the quantity of the orderly series of odd numbers.
But in order to find the numbers of which 5 (the second odd number) is the measure, four terms must be omitted, and the number that immediately follows will be measured by 5. Thus by omitting the four odd numbers 7, 9, 11, 13, the next term will be 15, which 5 measures according to the quantity of the first odd number 3; for the fifth part of 15 is 3. But if after this, the four following numbers are omitted, viz. 17, 19, 21, 23, the number five will measure by its plurality the next number 25. And if after this four numbers are omitted, the same constancy of order being preserved, 5 will measure 35 which is the next following number. And this is the infinite procession.
If again, it is inquired what the third number is which may be measured, six terms must be omitted, and that which is the seventh term in order is to be measured by the quantity of the first number i.e., by three. And after this six other terms being omitted, the number which immediately occurs will be measured by the third number, and will have for its quotient the second number. Thus after 7, omitting six numbers, the number 21 which immediately occurs will be measured by 3; and after 21, six numbers being omitted, the next number 35 will be measured by 7 five times. But if again other six terms are omitted, the number which next occurs, viz. 49, will be measured by the same 7 seven times, which is the quantity of the third term. And this established order will proceed to the most extended number of terms.
Hence, they will receive a vicissitude of measuring; just as they are naturally constituted odd numbers in an orderly series. But they will be measured by the intermission of terms according to an even number, beginning from an intermission of two terms. Thus the first odd number will measure the odd numbers that follow after an omission of two terms; the second, those that follow after an omission of four terms; the third, when 6, the fourth, when 8, and the fifth when 10 terms are omitted, will measure the numbers that follow in an orderly series. And so of the rest ad infinitum.
This will also be effected if the terms double their places, and numbers are omitted conformably to the duplication. Thus 3 is the first term and one, for every first is one. If therefore this multiplies its own place twice, it will produce twice one. And since twice one is two, two terms must be omitted. Again, if the second term which is 5 doubles its place, it will produce 4, and four terms must be omitted. If 7 likewise, which is the third term, doubles its place, it will produce 6; and therefore six terms are to be omitted in an orderly series. The fourth term also, if it doubles its place, will produce 8, and 8 terms must be omitted. And the like will be found to take place in all the other terms. The series however will give the mode of measuring according to the order of collocation .
For the first term numbers according to the first, i.e. according to itself, the first term which it numerates; but it numbers the second, which it numerates, by the second , the third, by the third ; and the fourth by the fourth. When the second however begins to measure, it measures the first which it numerates according to the first; but it measures the second which it numerates by itself, i.e. by the second term; and the third by the third; and so of the rest. Thus 3 measures 9 by 3; 15 by 5; 21 by 7; 27 by 9, and so on. But 5 measures 15 by 3; 25 by 5; 35 by 7; 45 by 9, and so of the rest. If therefore we direct our attention to the other terms, either those that measure others, or that are themselves measured by others, we shall find that there cannot be at one and the same time a common measure of all of them, nor that all of them at the same time measure any other number; but it will appear that some of them may be measured by another number, so as only to be numbered by one term; others, so as to be numbered by many terms; and some, so as to have no other measure than unity. Hence, those that receive no measure besides unity, are said to be first and incomposite numbers; but those that receive a certain measure besides unity, or are allotted the appellation of a foreign parts these are said to be second and composite numbers.
The third species however, which is of itself second and composite, but when one number is compared to the other is first and incomposite, is obtained by the following method: The squares of the first and incomposite numbers, when compared to each other will be found to have no common measure. Thus the square of 3 is 9, and the square of 5 is 25. These there fore have no common measure. Again, the square of 5 is 25, and of 7 is 49: and these compared to each other will be found to be incommensurable. For there is no common measure of these except unity which is the generator and mother of all these.
Assessment
