Commentary on Proposition X.23¶
Back to Proposition X.23.
As explained in the bracketed passage following this proposition, a straight line commensurable with a medial straight line in square only, as well as a straight line commensurable with it in length, is medial.
Algebraical notation shows this easily.
If $k^{\frac{1}{4}} \rho$ be the given straight line, $\lambda k^{\frac{1}{4}} \rho$ is a straight line commensurable in length with it and $\sqrt{\lambda} \cdot k^{\frac{1}{4}} \rho$ a straight line commensurable with it in square only.
But $\lambda \rho$ and $\sqrt{\lambda} \cdot \rho$ are both rational [X. Def. 3] and therefore can be expressed by $\rho'$, and we thus arrive at $k^{\frac{1}{4}} \rho'$, which is clearly medial.
Euclid’s proof amounts to the following.
Apply both the areas $\sqrt{k} \cdot \rho^2$ and $\lambda^2 \sqrt{k} \cdot \rho^2$ (or $\lambda \sqrt{k} \cdot \rho^2$) to a rational straight line $\sigma$.
The breadths $\sqrt{k} \cdot \frac{\rho^2}{\sigma}$ and $\lambda^2 \sqrt{k} \cdot \frac{\rho^2}{\sigma}$ (or $\lambda \sqrt{k} \cdot \frac{\rho^2}{\sigma}$) are in the ratio of the areas $\sqrt{k} \cdot \rho^2$ and $\lambda^2 \sqrt{k} \cdot \rho^2$ (or $\lambda \sqrt{k} \cdot \rho^2$) themselves and are therefore commensurable.
Now [X. 22] $\sqrt{k} \cdot \frac{\rho^2}{\sigma}$ is rational but incommensurable with $\sigma$.
Therefore $\lambda^2 \sqrt{k} \cdot \frac{\rho^2}{\sigma}$ (or $\lambda \sqrt{k} \cdot \frac{\rho^2}{\sigma}$) is so also;
whence the area $\lambda^2 \sqrt{k} \cdot \rho^2$ (or $\lambda \sqrt{k} \cdot \rho^2$) is contained by two rational straight lines commensurable in square only, so that $\lambda k^{\frac{1}{4}} \rho$ (or $\sqrt{\lambda} \cdot k^{\frac{1}{4}} \rho$) is a medial straight line.
It is in the Porism that we have the first mention of a medial area. It is the area which is equal to the square on a medial straight line, an area, therefore, of the form $k^{\frac{1}{2}} \rho^2$, which is, as a matter of fact, arrived at, though not named, before the medial straight line itself (X. 21).
The Porism states that $\lambda k^{\frac{1}{2}} \rho^2$ is a medial area, which is indeed obvious.