Between two Similar Solid Numbers exist two Mean Proportionals/Mistake

Source Work

Book $\text{VIII}$

Commentary on Proposition $19$

This mistake can be seen in the edition as published by Dover: ISBN 0-486-60089-0.

[Euclid] next forms the products of $c, f$ respectively into the mean $b d$.

Then $abc, cbd, fbd, def$ are in the ratio of $a$ to $d$ etc.

For

$\ds a b c$

$:$

$\ds c b d = a b : b d = a : d$

$\ds b d$

$:$

$\ds f b d = c : f$

$\ds f b d$

$:$

$\ds d e f = b d : d e = b : e$

The middle line of the above is incorrect -- it ought to read:

$c b d : f b d = c : f$