Proposition: Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals

(Proposition 19 from Book 8 of Euclid's “Elements”)

Two numbers fall (between) two similar solid numbers in mean proportion. And a solid (number) has to a similar solid (number) a cubed1 ratio with respect to (that) a corresponding side (has) to a corresponding side. * Let $A$ and $B$ be two similar solid numbers, and let $C$, $D$, $E$ be the sides of $A$, and $F$, $G$, $H$ (the sides) of $B$. * And since similar solid (numbers) are those having proportional sides [Def. 7.21] , thus as $C$ is to $D$, so $F$ (is) to $G$, and as $D$ (is) to $E$, so $G$ (is) to $H$. * I say that two numbers fall (between) $A$ and $B$ in mean proportion, and (that) $A$ has to $B$ a cubed ratio with respect to (that) $C$ (has) to $F$, and $D$ to $G$, and, further, $E$ to $H$.


Modern Formulation

(not yet contributed)

Proofs: 1

Proofs: 1 2 3 4 5

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



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",, 2016


  1. Literally "triple" (translator's note).