Commentary on Proposition X.30

Back to Proposition X.30.

[Folio 0073 | p. 69]

In this case we take $m^2$, $n^2$ such that $m^2 + n^2$ is not square.

Find $x$ such that

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

whence

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

or

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

Then $\rho$, $\frac{\rho}{\sqrt{1 + k^2}}$ satisfy the condition.

The proof is after the manner of the proof of the preceding proposition and need not be repeated.

According as $\rho$ is of the form $a$ or $\sqrt{A}$, the straight lines take the form (1) $a$, $\sqrt{a^2 - \frac{k^2 a^2}{1 + k^2}}$, that is, $a$, $\sqrt{a^2 - B}$, or (2) $\sqrt{A}$, $\sqrt{A - B}$ and $\sqrt{A}$, $\sqrt{A - b^2}$.