## Euclid's Elements Book VII
Definitions 6 through 10

Def. 6. An even number is that which is divisible into two equal parts.

Def. 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.

Def. 8. An even-times even number is that which is measured by an even number according to an even number.

Def. 9. An even-times odd number is that which is measured by an even number according to an odd number.

Def. 10. An odd-times odd number is that which is measured by an odd number according to an odd number.

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Definition 6 for "even number" is clear: the number n is even if it is of the form m + m.

Definition 7 for "odd number" includes the unproved statement: if a number is not even, then it differs from an even number by 1. This statement could be proved using the Euclid's principle that any decreasing sequence of numbers is finite.

Definitions 8-10 are also clear. A product of two even numbers is an even-times even number; a product of an even and an odd number is an even-times odd number; and a product of of two odd numbers is an odd-times odd number. Note that a number like 12 is both even-times even and even-times odd being at the same time 2 times 6 and 4 times 3.

The numbers which are even-times even but not even-times odd are just the powers of 2: 4, 8, 16, 32, etc. These are the numbers which are even-times even only, and they occur in proposition IX.32.

Book VII Introduction - Definitions 3 through 5 - Definitions 11 through 14.

© 1997
D.E.Joyce
Clark University