Commentary on Proposition X.24¶
Back to Proposition X.24.
There is the same difficulty in the text of this enunciation as in that of
The Greek says “medial straight lines commensurable in length in any of the aforesaid ways”; but straight lines can only be commensurable in length in one way, though they can be medial in two ways, as explained in the addition to the preceding proposition, i.e. they can be either commensurable in length or commensurable in square only with a given medial straight line. For the same reason as that explained in the note on X. 19 I have omitted “in any of the aforesaid ways” in the enunciation and bracketed the addition to X. 23 to which it refers.
$k^{\frac{1}{4}} \rho$ and $\lambda k^{\frac{1}{4}} \rho$ are medial straight lines commensurable in length. The rectangle contained by them is $\lambda k^{\frac{1}{2}} \rho^2$, which may be written $k'^{\frac{1}{2}} \rho'^2$ and is therefore clearly medial.
Euclid’s proof proceeds thus. Let $x$, $\lambda x$ be the two medial straight lines commensurable in length. Therefore
$$x^2 : x \cdot \lambda x = x : \lambda x.$$
But $x \frown \lambda x$, so that $x^2 \frown x \cdot \lambda x$. [X. 11] Now $x^2$ is medial [X. 21]; therefore $x \cdot \lambda x$ is also medial. [X. 23, Por.]
We may of course write two medial straight lines commensurable in length in the forms $m k^{\frac{1}{4}} \rho$, $n k^{\frac{1}{4}} \rho$; and these may either be $m \sqrt{a\sqrt{B}}$, $n \sqrt{a\sqrt{B}}$, or $m \sqrt[4]{AB}$, $n \sqrt[4]{AB}$.