Commentary on Proposition X.27

Back to Proposition X.27.

[Folio 0065 | p. 61]

Euclid takes two rational straight lines commensurable in square only, say $\rho$, $k^{\frac{1}{2}}\rho$. Find the mean proportional, i.e. $k^{\frac{1}{4}}\rho$. Take $x$ such that

$$\rho : k^{\frac{1}{2}}\rho = k^{\frac{1}{4}}\rho : x \ \text{.........................................(1).}$$

This gives $x = k^{\frac{3}{4}}\rho$, and the lines required are $k^{\frac{1}{4}}\rho$, $k^{\frac{3}{4}}\rho$. For ($\alpha$) $k^{\frac{1}{4}}\rho$ is medial. And ($\beta$), by (1), since $\rho \sim k^{\frac{1}{2}}\rho$,

$$k^{\frac{1}{4}}\rho \sim k^{\frac{3}{4}}\rho,$$

whence [addition to X. 23], since $k^{\frac{1}{4}}\rho$ is medial,

$$k^{\frac{3}{4}}\rho \text{ is also medial.}$$

The medial straight lines thus found may take either of the forms

$$(1) \quad \sqrt{a\sqrt{B}}, \quad \sqrt{\frac{B\sqrt{B}}{a}} \quad \text{or} \quad (2) \quad \sqrt[4]{AB}, \quad \sqrt{B\sqrt{\frac{B}{A}}}.$$