And when four magnitudes are in (continued) proportion, the first is said to have to the fourth the cubed1 ratio of that (it has) to the second. And so on, similarly, in successive order, whatever the (continued) proportion might be.
The positive real numbers \(\alpha,\delta\) are said to be in cubed ratio, if there exist positive real numbers $\beta,\gamma$ such that (\alpha,\beta,\gamma,\delta) are in a continued proportion. \[\frac\alpha\beta=\frac\beta\gamma=\frac\gamma\delta.\]
Please note that in this case
\[\frac\alpha\delta=\frac{\alpha^3}{\beta^3},\] which explains the name "cubed ratio".
In general, if a proportion in \(n\) terms holds for some positive real numbers \(\alpha_0,\ldots,\alpha_n\)
\[\frac{\alpha_0}{\alpha_1}=\frac{\alpha_1}{\alpha_2}=\ldots=\frac{\alpha_{n-1}}{\alpha_{n}},\quad\quad ( * )\]
then the ratio \[\frac{\alpha_0}{\alpha_{n}}=\left(\frac{\alpha_0}{\alpha_1}\right)^n,\quad\quad ( * * )\] is said to be in \(n\)-th continued ratio.
Note that the formula in \( ( * ) \) is a special case of the continued proportion, as we can write for \( ( * * ) \) the equivalent formula
\[\alpha_{n}=\alpha_0\left(\frac{\alpha_1}{\alpha_0}\right)^n.\]
Corollaries: 1 2 3
Proofs: 4 5 6 7 8 9 10 11 12 13
Propositions: 14 15 16 17 18 19 20
Sections: 21
Literally, "tripple" (translator's note) ↩
