Commentary on Proposition X.110

Back to Proposition X.110.

[Folio 0243 | p. 239]

We have to classify

$$\sqrt{\sqrt{k} \cdot \rho^2 - \sqrt{\lambda} \cdot \rho^2},$$

where $\sqrt{k} \cdot \rho^2$ is incommensurable with $\sqrt{\lambda} \cdot \rho^2$.

Put

$$\sigma u = \sqrt{k} \cdot \rho^2,$$

$$\sigma v = \sqrt{\lambda} \cdot \rho^2.$$

[Folio 0244 | p. 240]

Then $u$ is rational and $\smile \sigma$, $v$ is rational and $\smile \sigma$, and

$$u \smile v.$$

Therefore $u, v$ are rational and $\sim$, so that $(u - v)$ is an apotome.

Now either

1. $\sqrt{u^2 - v^2} \mathbin{\Cap} u$, or (2) $\sqrt{u^2 - v^2} \smile u$, while in both cases both $u$ and $v$ are $\smile \sigma$.

In case (1) $(u - v)$ is a third apotome, and in case (2) $(u - v)$ is a sixth apotome, so that $\sqrt{\sigma (u - v)}$ is either (1) a second apotome of a medial straight line [x. 93], or (2) a “side” of the difference between two medial areas [x. 96].