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Definition: 7.20: Proportional Numbers

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.

Modern Definition

See rational numbers.

Notes

Euclid introduced proportionality for positive real numbers already in Book 5 (Def. 5.06).
However, now he restricts this definition to positive integers: Euclid calls the positive integers $a,b,c$ and $d$ "proportional" if the rectangular numbers \(a\cdot d\) and \(b\cdot c\) are equal, formally

\[a\cdot d = b\cdot c.\]

Please note that this is exactly the case if they represent the same rational numbers.
In fact, being proportional in the sense of the above definition is an equivalence relation which can be used to define rational numbers: Two rational numbers \(a/b\) and \(c/d\) are equivalent, if and only if $ad=bc.$

\[\frac ab\sim\frac cd\Longleftrightarrow a\cdot d = b\cdot c.\]

Corollaries: 1
Definitions: 2 3 4
Proofs: 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
Propositions: 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101

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References

Adapted from (subject to copyright, with kind permission)

Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016