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 $NG$.
(not yet contributed)
Proofs: 1
This proposition assumes that if two ratios are equal then the cube of the former is also equal to the cube of the latter, and vice versa (translator's note). ↩