◀ ▲ ▶Branches / Geometry / Elements-euclid / Book-11-elementary-stereometry / Proposition: Prop. 11.37: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional

Proposition: Prop. 11.37: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional

Euclid's Formulation

If four straight lines are proportional then the similar, and similarly described, parallelepiped solids on them will also be proportional. And if the similar, and similarly described, parallelepiped solids on them are proportional then the straight lines themselves will be proportional.

• Let $AB$, $CD$, $EF$, and $GH$, be four proportional straight lines, (such that) as $AB$ (is) to $CD$, so $EF$ (is) to $GH$.
• And let the similar, and similarly laid out, parallelepiped solids $KA$, $LC$, $ME$ and $NG$ have been described on $AB$, $CD$, $EF$, and $GH$ (respectively).
• I say that as $KA$ is to $LC$, so $ME$ (is) to $NG$.

Modern Formulation

(not yet contributed)

Table of Contents

Proofs: 1

Thank you to the contributors under CC BY-SA 4.0!

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non-Github:

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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