Commentary on Proposition X.29¶

Take a rational straight line $\rho$ and two numbers $m^2$, $n^2$ such that $(m^2 - n^2)$ is not a square.

Take a straight line $x$ such that

$$m^2 : m^2 - n^2 = \rho^2 : x^2 \tag{1},$$

whence

$$x^2 = \frac{m^2 - n^2}{m^2}\rho^2,$$

and

$$x = \rho\sqrt{1 - k^2}, \quad \text{where } k = \frac{n}{m}.$$

Then $\rho$, $\rho\sqrt{1 - k^2}$ are the straight lines required.

It follows from (1) that

$$x^2 \sim \rho^2,$$

and $x$ is rational, but

$$x \smile \rho.$$

$$m^2 : n^2 = \rho^2 : \rho^2 - x^2,$$

so that $\sqrt{\rho^2 - x^2} \sim \rho$, and in fact $= k\rho$.

According as $\rho$ is of the form $a$ or $\sqrt{A}$, the straight lines are (1) $a$, $\sqrt{a^2 - b^2}$ or (2) $\sqrt{A}$, $\sqrt{A - k^2 A}$.