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Proposition: Prop. 10.021: Medial is Irrational

(Proposition 21 from Book 10 of Euclid's “Elements”)

The rectangle contained by rational straight lines (which are) commensurable in square only is irrational, and its square root is irrational - let it be called medial.

• For let the rectangle $AC$ be contained by the rational straight lines $AB$ and $BC$ (which are) commensurable in square only.
• I say that $AC$ is irrational, and its square root is irrational - let it be called medial.

Modern Formulation

Thus, a medial straight line has a length expressible as \(\delta^{1/4},\) for some positive rational number $\delta$.

Table of Contents

Proofs: 1

1. Lemma: Lem. 10.021: Medial is Irrational

Mentioned in:

Corollaries: 1 2
Proofs: 3 4 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 73
Propositions: 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125

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References

Adapted from (subject to copyright, with kind permission)

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

Adapted from CC BY-SA 3.0 Sources:

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