Commentary on Proposition X.87¶

Let $\rho$ be a rational straight line. Take numbers $p$, $qm^2$, $q(m^2 - n^2)$ which have not to one another the ratio of square to square.

Now let $x$, $y$ be such that

$$p : qm^2 = \rho^2 : x^2 \text{ .................................................(1)}$$

and

$$qm^2 : q(m^2 - n^2) = x^2 : y^2 \text{ ...................................(2).}$$

Then shall $(x - y)$ be a third apotome.

For ($\alpha$), from (1), $x$ is rational but $\smile \rho$ ……………………………………………………….(3).

And, from (2), $y$ is rational but $\smile x$. Therefore $x, y$ are rational and $\sim$, so that $(x - y)$ is an apotome.

($\beta$) By (1), (2), ex aequali,

$$p : q(m^2 - n^2) = \rho^2 : y^2,$$

whence $y \smile \rho$.

Thus, by this and (3), $x, y$ are both $\smile \rho$ ………………………………………….(4).

Lastly, let $z^2 = x^2 - y^2$, so that, from (2), convertendo,

$$qm^2 : qn^2 = x^2 : z^2;$$

therefore $z$, or $\sqrt{x^2 - y^2}, \frown x$ ………………………………………….(5).

Thus [(4) and (5)] $(x - y)$ is a third apotome.

To find its form, we have, from (1) and (2),

$$x = \rho \cdot \frac{m\sqrt{q}}{\sqrt{p}},$$

$$y = \rho \cdot \frac{\sqrt{m^2 - n^2} \cdot \sqrt{q}}{\sqrt{p}},$$

so that

$$x - y = \sqrt{\frac{q}{p}} \cdot \rho (m - \sqrt{m^2 - n^2}).$$

This may be written in the form

$$m\sqrt{k} \cdot \rho - m\sqrt{k} \cdot \rho \sqrt{1 - \lambda^2}.$$

As explained in the note on x. 50, this is the lesser root of the equation

$$x^2 - 2m\sqrt{k} \cdot \rho x + \lambda^2 m^2 k\rho^2 = 0.$$