Definition 20

Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.

Guide

Definition V.20 for proportionality of numbers is not the same as the definition of proportionality for magnitudes in Book V given in V.Def.5. This definition for numbers was probably the earlier one, but as not all magnitudes are commensurable, it cannot adequately define proportionality for magnitudes.

This definition VII.20 is given by cases. The various cases correspond to definitions VII.Def.3 through VII.Def.6 for part, parts, and multiple.

When four numbers, a, b, c, and d, are proportional, that will be written in these guides symbolically as a : b = c : d.

In the first case of Definition 20, a is the same multiple of b as c is of d. An example of this is the proportion 12 : 6 = 22 : 11, where 12 is twice 6 and 22 is twice 11.

The second case is inverse to the first, a is the same part of b as c is of d. For an example take the proportion 6 : 12 = 11 : 22, where 6 is one half of 12, and 11 is one half of 22.

For an example of the third case, consider 12 : 16 = 21 : 28. Since the first is the same parts of the second, namely 3 parts of 4, as the the third is of the fourth, the proportion holds.

Ratios of numbers

Although the word “ratio” doesn’t appear in this definition, it appears frequently beginning in proposition VII.14. In book VII ratio is restricted to the use of saying when one ratio is the same as another, that is, there is a proportion as defined in this definition. In Book VIII, duplicate ratios, triplicate ratios, and other compounded ratios appear. Definitions for these concepts are not explicitly given, but once the concept of proportion has been defined, they have the same definition given in Book V for duplicate and triplicate ratio in V.Def.9-10. Compound ratios aren’t defined in Book V, but they can be understood by their use. See the Guide to V.Def.3. Various other definitions that go along with ratios and proportions were given in Book V, for instance, alternate ratios, inverse ratios, taken jointly, taken separately, and ex aequali. These definitions are also not repeated here in Book VII.

Very soon in these books on number theory Euclid begins to rely on properties of proportion proved in Book V using the other definition of proportion. That these are valid for proportions of numbers could be verified individually or by showing that the two definitions of proportion are equivalent for numbers.

Use of this definition

Proportions of numbers first appear in proposition VII.11, but all the propositions from VII.4 through VII.10 are in preparation for the study of numeric proportions.