Commentary on Proposition X.89

Back to Proposition X.89.

[Folio 0191 | p. 187]

Let $\rho$, $k\rho$ and the numbers $m$, $n$ of the last proposition be taken. Take $x$ such that

$$n : (m+n) = k^2\rho^2 : x^2 \text{ ..............................(1).}$$

In this case $x > k\rho$, and

$$x = k\rho \sqrt{\frac{m+n}{n}}$$

$$= k\rho \sqrt{1+\lambda}\text{, say.}$$

Then shall $(x - k\rho)$, or $(k\rho\sqrt{1+\lambda} - k\rho)$, be a fifth apotome. For, by (1), $x$ is rational and $\sim k\rho$. And since, by (1), $(m+n) : m = x^2 : (x^2 - k^2\rho^2)$, $\sqrt{x^2 - k^2\rho^2}$ is incommensurable with $x$.

Also $k\rho \sim \rho$. As explained in the note on x. 52, the fifth apotome

$$k\rho\sqrt{1+\lambda} - k\rho$$

is the lesser root of the quadratic

$$x^2 - 2k\rho\sqrt{1+\lambda}\cdot x + \lambda k^2\rho^2 = 0.$$