And when three magnitudes are proportional, the first is said to have to the third the squared^{1} ratio of that (it has) to the second.
The ratio $\frac \alpha\gamma$ of positive real numbers $\alpha,\gamma$ is called a squared ratio if there is a positive real number $\beta$ sucht that $\alpha,\beta,\gamma$ are in a proportion in three terms. \[\frac\alpha\beta=\frac\beta\gamma.\]
Note: Note that in this case we have
\[\frac\alpha\gamma=\frac{\alpha^2}{\beta^2},\]
explaining the name "squared ratio".
Corollaries: 1
Proofs: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Propositions: 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46
Sections: 47
Literally, "double" (translator's note) ↩