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And, when two numbers having multiplied one another make some number, the number so produced be called plane, and its sides are the numbers which have multiplied one another.
And, when three numbers having multiplied one another make some number, the number so produced be called solid, and its sides are the numbers which have multiplied one another.
When two unequal numbers are set out, and the less is continually subtracted in turn from the greater, if the number which is left never measures the one before it until a unit is left, then the original numbers are relatively prime.
If a number is that part of a number which a subtracted number is of a subtracted number, then the remainder is also the same part of the remainder that the whole is of the whole.
If a number is the same parts of a number that a subtracted number is of a subtracted number, then the remainder is also the same parts of the remainder that the whole is of the whole.
If a number is a part of a number, and another is the same part of another, then alternately, whatever part or parts the first is of the third, the same part, or the same parts, the second is of the fourth.
If a number is a parts of a number, and another is the same parts of another, then alternately, whatever part of parts the first is of the third, the same part, or the same parts, the second is of the fourth.
If any number of numbers are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents.
If there are any number of numbers, and others equal to them in multitude, which taken two and two together are in the same ratio, then they are also in the same ratio ex aequali.
If a unit number measures any number, and another number measures any other number the same number of times, then alternately, the unit measures the third number the same number of times that the second measures the fourth.
If four numbers are proportional, then the number produced from the first and fourth equals the number produced from the second and third; and, if the number produced from the first and fourth equals that produced from the second and third, then the four numbers are proportional.
The least numbers of those which have the same ratio with them measure those which have the same ratio with them the same number of times; the greater the greater; and the less the less.
If two numbers are relatively prime, and each multiplied by itself makes a certain number, then the products are relatively prime; and, if the original numbers multiplied by the products make certain numbers, then the latter are also relatively prime.
If two numbers are relatively prime, then their sum is also prime to each of them; and, if the sum of two numbers is relatively prime to either of them, then the original numbers are also relatively prime.
If two numbers, multiplied by one another make some number, and any prime number measures the product, then it also measures one of the original numbers.
Book VII is the first of the three books on number theory. It begins with the 22 definitions used
throughout these books. The important definitions are those for unit and number, part and
multiple, even and odd, prime and relatively prime, proportion, and perfect number. The topics
in Book VII are antenaresis and the greatest common divisor, proportions of numbers, relatively
prime numbers and prime numbers, and the least common multiple.
The basic construction for Book VII is antenaresis, also called the Euclidean algorithm, a kind of reciprocal subtraction. Beginning with two numbers, the smaller, whichever it is, is repeatedly subtracted from the larger until a single number is left. This algorithm, studied in propositions VII.1 through VII.3, results in the greatest common divisor of two or more numbers.
Propositions V.5 through V.10 develop properties of fractions, that is,
they study how many parts one number is of another in preparation for ratios and proportions.
The next group of propositions VII.11 through VII.19 develop the
theory of proportions for numbers.
VII.11. a : b = c : d implies
both equal the ratio a – c : b – d. VII.12. a : b = c : d implies
both equal the ratio a + c : b + d. VII.13. a : b = c : d implies
a : c = b : d. VII.14. a : b = d : e and
b : c = e : f
imply a : c = d : f. VII.17. a : b = ca : cb. VII.19. a : b = c : d if an only if
ad = bc.
Propositions VII.20 through VII.29 discuss representing ratios in lowest terms as relatively prime numbers and properties of relatively prime numbers. Properties of prime numbers are presented in propositions VII.30 through VII.32. Book VII finishes with least common multiples in propositions VII.33 through VII.39.
Postulates for numbers
Postulates are as necessary for numbers as they are for geometry. Euclid, however, supplies none. Missing postulates occurs as early as proposition VII.2. In its proof, Euclid constructs a decreasing sequence of whole positive numbers, and, apparently, uses a principle to conclude that the sequence must stop, that is, there cannot be an infinite decreasing sequence of numbers. If that is the principle he uses, then it ought to be stated as a postulate for numbers.
Numbers are so familiar that it hardly occurs to us that the theory of numbers needs axioms, too. In fact, that field was one of the last to receive a careful scrutiny, and axioms for numbers weren’t developed until the late 19th century by Dedekind and others. By that time foundations for the rest of mathematics were laid upon either geometry or number theory or both, and only geometry had axioms. About the same time that foundations for number theory were developed, a new subject, set theory, was created by Dedekind and Cantor, and mathematics was refounded in terms of set theory.
The foundations of number theory will be discussed in the Guides to the various definitions and propositions.