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Proposition: Prop. 10.112: Square on Rational Straight Line applied to Binomial Straight Line

(Proposition 112 from Book 10 of Euclid's “Elements”)1

The (square) on a rational (straight line), applied to a binomial (straight line), produces as breadth an apotome whose terms are commensurable (in length) with the terms of the binomial, and, furthermore, in the same ratio. Moreover, the created apotome will have the same order as the binomial. * Let $A$ be a rational (straight line), and $BC$ a binomial (straight line), of which let $DC$ be the greater term. * And let the (rectangle contained) by $BC$ and $EF$ be equal to the (square) on $A$. * I say that $EF$ is an apotome whose terms are commensurable (in length) with $CD$ and $DB$, and in the same ratio, and, moreover, that $EF$ will have the same order as $BC$.

fig112e

Modern Formulation

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Table of Contents

Proofs: 1

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Proofs: 1

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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

Footnotes

  1. Heiberg considers this proposition, and the succeeding ones, to be relatively early interpolations into the original text (translator's note). ↩