◀ ▲ ▶Branches / Geometry / Elements-euclid / Book-10-incommensurable-magnitudes / Lemma: Lem. 10.13: Finding Pythagorean Magnitudes

Lemma: Lem. 10.13: Finding Pythagorean Magnitudes

(Lemma to Proposition 13 from Book 10 of Euclid's “Elements”)

For two given unequal straight lines, to find by (the square on) which (straight line) the square on the greater (straight line is) larger than (the square on) the lesser.

• Let $AB$ and $C$ be the two given unequal straight lines, and let $AB$ be the greater of them.
• So it is required to find by (the square on) which (straight line) the square on $AB$ (is) greater than (the square on) $C$.

Modern Formulation

That is, if $\alpha$ and $\beta$ are the positive real numbers denoting the lengths of two given straight lines, with $\alpha > \beta$, to find a straight line of length $\gamma$ such that $\alpha^2=\beta^2+\gamma^2$. Similarly, we can also find $\gamma$ such that $\gamma^2=\alpha^2+\beta^2$.

Table of Contents

Proofs: 1

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Proofs: 1 2 3 4 5 6 7 8

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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