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.
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" has two statements. The first can be taken as a definition of odd number, a number which is not divisible into two equal parts, that is to say not an even number.
The other statement is not a definition for odd number, since one has already been given, but an unproved statement. It is easy to recognize that something has to be proved, since if we make the analogous definitions for another number, say 10, then analogous statement is false. Suppose we say a "decade number" is one divisible by 10, and and "undecade number' is one not divisible by 10. Then it is not the case that an undecade number differs by a unit from a decade number; the number 13, for instance, is not within 1 of a decade number.
The unproved statement that a number differing from an even number by 1 is an odd number ought to be proved. That statment is used in proposition IX.22 and several propositions that follow it. It could be proved using, for instance, a 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.
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