Definitions 3–5

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

4. But parts when it does not measure it.

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


These definitions are in preparation for the definition of proportion of numbers given in VII.Def.20. In the current definitions, the possible relations between a pair of numbers, a and b, are classified. Later in Book VII, the term “ratio” will be used for this relation.

In all three of these definitions, the concept of “measures“ is assumed to be understood. There is more to these definitions than meets the eye, though, at least part of the intent is evident.

On Definitions 3 and 5

To illustrate VII.Def.3, take a = 2u, which is a part of b = 6u, namely, a is the 1/3 part of b.

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If u is the unit, then a = 2u is represented as AB while b = 6u is represented by CF. As AB measures CF, a certain number of times, namely n = 3 times, by CD, DE, and DF, therefore a is a part of b, namely, the one-third part since it measures b three times. Note that although a and b are formal numbers, the number of times n that a measures b is not a formal number. This n doesn't appear in the definition.

We can also use the same figure as an illustration of VII.Def.5 to see that b is a multiple of a, in particular, the nth multiple of a.


Where Euclid would say that one number is a part of another number, modern mathematicians would say that the first number is a proper divisor of the second number. A divisor of n is any whole number m (including 1) that divides n in the sense that there is another number k such that mk = n. A proper divisor of n is any divisor except n itself.

For example, the proper divisors of the number 12 are 1, 2, 3, 4, and 6.

On Definition 4

Definition VII.Def.4 is less clear, but its intent can be read from the use to which it’s put in VII.Def.20 for proportions of numbers. For an example, consider the numbers a = 4u and b = 6u. The number a does not measure the number b, but it is parts of b.

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Here, a = 4u is represented as AC while b = 6u is represented as DG. Clearly, AC does not measure DG. The way this definition is used in VII.Def.4, just the knowledge that “a is parts of b” is not enough, what is also needed is how many parts of b is a. This will be needed to define a proportion such as 4 : 6 = 6 : 9. That proportion is supposed to hold since 4 is the same parts of 6 as 6 is of 9, namely 2 third parts. Thus, one number being parts of another also carries along with it how many of what parts.

There is one more difficulty with this definition. It seems obvious that when one number a =mu is less than another b =nu, then in all cases a would be parts of b, namely a consists of m one-nth parts of b. Yet, the proposition VII.4 has a proof to show that a is either a part or parts of b. The reason is that the desired parts should be in lowest terms. For our example, where a = 4u and b = 6u, it isn’t enough to say that a is 4 one-sixth parts of b; what’s needed is that a is 2 one-third parts of b.