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Euclid, Elements, Book VII, Definitions©
translated by Henry Mendell

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Propositions

Definitions Brief comments

1 Unit is that according to which each of the things which are is one, This definition of 'unit' parallels Aristotle's definition of 'quality' at Categories 8

2. and the multitude composed from units is a number.

3. A number is a part of a number, the smaller of the larger, whenever it measures the larger,

4. and parts whenever it does not measure,

5. and the larger is a multiple of the smaller whenever it is measured by the smaller.

6. The number which is divided in two is an even number,

7. and the number which is not divided in two is odd, or the number which differs from an even number by a unit.

8. The number measured by an even number taken in groups of an even number is an even-times even number,

9. and the number measured by an even number taken in groups of an odd number is an even-times odd number,

10. [and the number measured by an odd number taken in groups of an even number is an odd-times even,]

11. and the number measured by an odd number taken in groups of an odd number is an odd-times odd number.

12. The number measured only by a unit is a prime number.

13. Prime numbers relative to one another are those measured only by a unit as a common measure.

14. Compound number is a number measured by a number,

15. and compound numbers relative to one another are those measured by a number as a common measure.

16. A number is said to multiply a number whenever as many units as there are in it, so many times the multiplied number is added and becomes some number.

17. Whenever two numbers multiply one another and make some number, the number which results is called plane, and it sides are the numbers multiplying one another,

18. and whenever three numbers multiply one another and make some number, the number which results is called solid, and its sides are the numbers multiplying one another.

19. A square number is the equal-times equal number or the number enclosed by two equal numbers,

20. and a cube is the equal-times equal equal-times or enclosed by three equal numbers.

21. Numbers are proportional whenever the first is an equal multiple or the same part or the same parts of the second as the third of the fourth.

22. Similar plane and solid numbers are those having proportional sides.

23. A perfect number is one which is equal to all its parts.

Propositions:

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	Definitions	Brief comments
1	Unit is that according to which each of the things which are is one,	This definition of 'unit' parallels Aristotle's definition of 'quality' at Categories 8
2.	and the multitude composed from units is a number.
3.	A number is a part of a number, the smaller of the larger, whenever it measures the larger,
4.	and parts whenever it does not measure,
5.	and the larger is a multiple of the smaller whenever it is measured by the smaller.
6.	The number which is divided in two is an even number,
7.	and the number which is not divided in two is odd, or the number which differs from an even number by a unit.
8.	The number measured by an even number taken in groups of an even number is an even-times even number,
9.	and the number measured by an even number taken in groups of an odd number is an even-times odd number,
10.	[and the number measured by an odd number taken in groups of an even number is an odd-times even,]
11.	and the number measured by an odd number taken in groups of an odd number is an odd-times odd number.
12.	The number measured only by a unit is a prime number.
13.	Prime numbers relative to one another are those measured only by a unit as a common measure.
14.	Compound number is a number measured by a number,
15.	and compound numbers relative to one another are those measured by a number as a common measure.
16.	A number is said to multiply a number whenever as many units as there are in it, so many times the multiplied number is added and becomes some number.
17.	Whenever two numbers multiply one another and make some number, the number which results is called plane, and it sides are the numbers multiplying one another,
18.	and whenever three numbers multiply one another and make some number, the number which results is called solid, and its sides are the numbers multiplying one another.
19.	A square number is the equal-times equal number or the number enclosed by two equal numbers,
20.	and a cube is the equal-times equal equal-times or enclosed by three equal numbers.
21.	Numbers are proportional whenever the first is an equal multiple or the same part or the same parts of the second as the third of the fourth.
22.	Similar plane and solid numbers are those having proportional sides.
23.	A perfect number is one which is equal to all its parts.