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Definitions 20 and 21

Def. 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.

Def. 21. Similar plane and solid numbers are those which have their sides proportional.

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The definition of proportionality of numbers given here is different than that given for proportionality of magnitudes in Book V. It is also simpler yet given by cases. For an example of the first case, 12:6 = 22:11 since the first is the same multiple of the second, namely twice, that the third is of the fourth. The second case is inverse to the first. For an example of the third case, 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. Actually, there should be a fourth case (inverse to the third case) when the second is the same parts of the first as the fourth is of the third, as 16:12 = 28:21. Of course, these cases could be merged into one by considering 1 to be a number and not distinguishing when the first is greater or less than the second.

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.

To illustrate definition 21, the two numbers 240 and 810 are similar solid numbers when represented as 4 times 6 times 10 and 6 times 9 times 15, respectively.

The various 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 not repeated here in Book VII, but they continue to apply.

Book VII Introduction - Definitions 15 through 19 - Definition 22.

© 1997
D.E.Joyce
Clark University