Numbers are

proportionalwhen the first is the same multiple, or the same part, or the same parts, of the second that the third (is) of the fourth.

See rational numbers.

- 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

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

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