Commentary on Proposition X.115¶
Back to Proposition X.115.
Heiberg is clearly right in holding that this proposition, at all events, is alien to the general scope of Book X, and is therefore probably an interpolation, made however before Theon’s time. It is of the same character as a scholium at the end of the Book, which is (along with the interpolated proposition proving, in two ways, the incommensurability of the diagonal of a square with its side) relegated by August as well as Heiberg to an Appendix.
The proposition amounts to this.
The straight line $k^{\frac{1}{4}}\rho$ being medial, if $\sigma$ be a rational straight line, $\sqrt{k^{\frac{1}{4}}\rho\sigma}$ is a new irrational straight line. So is the mean proportional between this and another rational straight line $\sigma'$, and so on indefinitely.