Euclid's Elements Book VII : Fundamentals of number theory.

### Definitions

- Definition 1]
- A unit is that by virtue of which each of the things that exist is called one.
- Definition 2]
- A number is a multitude composed of units.
- Definition 3]
- A number is a part of a number, the less of the greater, when it measures the greater;
- Definition 4]
- But parts when it does not measure it.
- Definition 5]
- The greater number is a multiple of the less when it is measured by the less.
- Definition 6]
- An even number is that which is divisible into two equal parts.
- Definition 7]
- An odd number is that which is not divisible into two equal parts, or that which differs by a
unit from an even number.
- Definition 8]
- An even-times even number is that which is measured by an even number according
to an even number.
- Definition 9]
- An even-times odd number is that which is measured by an even number according
to an odd number.
- Definition 10]
- An odd-times odd number is that which is measured by an odd number according to an odd number.
- Definition 11]
- A prime number is that which is measured by a unit alone.
- Definition 12]
- Numbers relatively prime are those which are measured by a unit alone as a common measure.
- Definition 13]
- A composite number is that which is measured by some number.
- Definition 14]
- Numbers relatively composite are those which are measured by some number as a common measure.
- Definition 15]
- A number is said to multiply a number when that which is multiplied is added to itself
as many times as there are units in the other.
- Definition 16]
- 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.
- Definition 17]
- 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.
- Definition 18]
- A square number is equal multiplied by equal, or a number which is contained by two equal numbers.
- Definition 19]
- And a cube is equal multiplied by equal and again by equal, or a number which is
contained by three equal numbers.
- Definition 20]
- Numbers are proportional when the first is the same multiple, or the same part, or the
same parts, of the second that the third is of the fourth.
- Definition 21]
- Similar plane and solid numbers are those which have their sides proportional.
- Definition 22]
- A perfect number is that which is equal to the sum its own parts.

### Propositions

- Proposition 1
- 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.
- Proposition 2
- To find the greatest common measure of two given numbers not relatively prime.
Proposition 3

- To find the greatest common measure of three given numbers not relatively prime.
- Proposition 4
- Any number is either a part or parts of any number, the less of the greater.
- Proposition 5
- If a number is part of a number, and another is the same part of another, then the
sum is also the same part of the sum that the one is of the one.
- Proposition 6
- If a number is parts of a number, and another is the same parts of another, then the
sum is also the same parts of the sum that the one is of the one.
- Proposition 7
- 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.
- Proposition 8
- 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.
- Proposition 9
- If a number is a part of a number, and another is the same part 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.
- Proposition 10
- 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.
- Proposition 11
- If a whole is to a whole as a subtracted number is to a subtracted number, then the remainder is
to the remainder as the whole is to the whole.
- Proposition 12
- 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.
- Proposition 13
- If four numbers are proportional, then they are also proportional alternately.
- Proposition 14
- 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.
- Proposition 15
- 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.
- Proposition 16
- If two numbers multiplied by one another make certain numbers, then the numbers so produced
equal one another.
- Proposition 17
- If a number multiplied by two numbers makes certain numbers, then the numbers so produced
have the same ratio as the numbers multiplied.
- Proposition 18
- If two number multiplied by any number make certain numbers, then the numbers so produced
have the same ratio as the multipliers.
- Proposition 19
- 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.
- Proposition 20
- 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.
- Proposition 21
- Numbers relatively prime are the least of those which have the same ratio with them.
- Proposition 22
- The least numbers of those which have the same ratio with them are relatively prime.
- Proposition 23
- If two numbers are relatively prime, then any number which measures one of them is relatively
prime to the remaining number.
- Proposition 24
- If two numbers are relatively prime to any number, then their product is also relatively prime to the same.
- Proposition 25
- If two numbers are relatively prime, then the product of one of them with itself is relatively
prime to the remaining one.
- Proposition 26
- If two numbers are relatively prime to two numbers, both to each, then their products are also
relatively prime.
- Proposition 27
- 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.
- Proposition 28
- 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.
- Proposition 29
- Any prime number is relatively prime to any number which it does not measure.
- Proposition 30
- 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.
- Proposition 31
- Any composite number is measured by some prime number.
- Proposition 32
- Any number is either prime or is measured by some prime number.
- Proposition 33
- Given as many numbers as we please, to find the least of those which have the same ratio with them.
- Proposition 34
- To find the least number which two given numbers measure.
- Proposition 35
- If two numbers measure any number, then the least number measured by them also measures the same.
- Proposition 36
- To find the least number which three given numbers measure.
- Proposition 37
- If a number is measured by any number, then the number which is measured has a part
called by the same name as the measuring number.
- Proposition 38
- If a number has any part whatever, then it is measured by a number called by the same
name as the part.
- Proposition 39
- To find the number which is the least that has given parts.

Euclid's Elements: Book VI <--- Book VII ---> Euclid's Elements: Book VIII