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Proposition: Prop. 12.18: Volumes of Spheres are in Triplicate Ratio of Diameters

Euclid's Formulation

Spheres are to one another in the cubed ratio of their respective diameters. * Let the spheres $ABC$ and $DEF$ have been conceived, and (let) their diameters (be) $BC$ and $EF$ (respectively). * I say that sphere $ABC$ has to sphere $DEF$ the cubed ratio that $BC$ (has) to $EF$.

Modern Formulation

(not yet contributed)

Table of Contents

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