◀ ▲ ▶Branches / Geometry / Elements-euclid / Book-12-proportional-stereometry / Proof: By Euclid

Proof: By Euclid

(related to Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides)

• Let there be similar, and similarly laid out, pyramids whose bases are triangles $ABC$ and $DEF$, and apexes the points $G$ and $H$ (respectively).
• I say that pyramid $ABCG$ has to pyramid $DEFH$ the cubed ratio of that $BC$ (has) to $EF$.

• For let the parallelepiped solids $BGML$ and $EHQP$ have been completed.
• And since pyramid $ABCG$ is similar to pyramid $DEFH$, angle $ABC$ is thus equal to angle $DEF$, and $GBC$ to $HEF$, and $ABG$ to $DEH$.
• And as $AB$ is to $DE$, so $BC$ (is) to $EF$, and $BG$ to $EH$ [Def. 11.9] .
• And since as $AB$ is to $DE$, so $BC$ (is) to $EF$, and (so) the sides around equal angles are proportional, parallelogram $BM$ is thus similar to paralleleogram $EQ$.
• So, for the same (reasons), $BN$ is also similar to $ER$, and $BK$ to $EO$.
• Thus, the three (parallelograms) $MB$, $BK$, and $BN$ are similar to the three (parallelograms) $EQ$, $EO$, $ER$ (respectively).
• But, the three (parallelograms) $MB$, $BK$, and $BN$ are (both) equal and similar to the three opposite (parallelograms), and the three (parallelograms) $EQ$, $EO$, and $ER$ are (both) equal and similar to the three opposite (parallelograms) [Prop. 11.24].
• Thus, the solids $BGML$ and $EHQP$ are contained by equal numbers of similar (and similarly laid out) planes.
• Thus, solid $BGML$ is similar to solid $EHQP$ [Def. 11.9] .
• And similar parallelepiped solids are in the cubed ratio of corresponding sides [Prop. 11.33].
• Thus, solid $BGML$ has to solid $EHQP$ the cubed ratio that the corresponding side $BC$ (has) to the corresponding side $EF$.
• And as solid $BGML$ (is) to solid $EHQP$, so pyramid $ABCG$ (is) to pyramid $DEFH$, inasmuch as the pyramid is the sixth part of the solid, on account of the prism, being half of the parallelepipedal solid [Prop. 11.28], also being three times the pyramid [Prop. 12.7].
• Thus, pyramid $ABCG$ also has to pyramid $DEFH$ the cubed ratio that $BC$ (has) to $EF$.
• (Which is) the very thing it was required to show.
  ∎

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References

Adapted from (subject to copyright, with kind permission)

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