Proof
Suppose a smaller set in the same ratio existed.
By ex aequali (VII.14) the ratio of extremes equals , so by VII.21 (least in a ratio are coprime) the original
, would not be coprime — contradiction.
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Depends on (3)
- VII.14Proposition VII.14If there be as many numbers as we please, and others equal to them in multitude, which taken two and two are in the…
- VII.20Proposition VII.20The least numbers of those which have the same ratio with them measure those which have the same ratio the same number…
- VII.21Proposition VII.21Numbers prime to one another are the least of those which have the same ratio with them.
Required by (dependents) (5)
- VIII.2Proposition VIII.2To find numbers in continued proportion, as many as may be prescribed, and the least that are in a given ratio.
- VIII.3Proposition VIII.3If as many numbers as we please in continued proportion be the least of those which have the same ratio with them, the…
- VIII.6Proposition VIII.6If there be as many numbers as we please in continued proportion, and the first do not measure the second, neither will…
- VIII.9Proposition VIII.9If two numbers be prime to one another, and numbers fall between them in continued proportion, then, however many…
- IX.35Proposition IX.35If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last…
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