## Euclid's Elements Book VII
Definitions 3 through 5

Def. 3. A number is a part of a number, the less of the greater, when it measures the greater;

Def. 4. But parts when it does not measure it.

Def. 5. The greater number is a multiple of the less when it is measured by the less.

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These definitions assume that "measures" is understood. It is clear what is meant by these definitions, namely, m measures n if m divides n, that is, there is some number k (greater than 1) so that mk = n. But division and multiplication haven't been defined yet, so this definition is inadequate, too. The problem, of course, is that Euclid has no foundations for number theory. The concepts of arithmetic were as obvious to him as as they were all others after him until the skeptic formalists of the late nineteenth century recognized a need for foundations of number theory.
At any rate, the intent of definitions 3 and 5 is evident. The words "part" and "parts" in definitions 3 and 4 are literal translations from the Greek, but they're not usually used this way in English. Perhaps "unit fraction" would be better for "part," and "non-unit proper fraction" would be better for "parts." Thus, BC is a part of BE, a 1/3 part, and BD is parts of BE, in particular, 2/3 parts.

Book VII Introduction - Definitions 1 and 2 - Definitions 6 through 10.

© 1997
D.E.Joyce
Clark University