Commentary on Proposition X.108

Back to Proposition X.108.

[Folio 0240 | p. 236]

A rational area being of the form $k\rho^2$, and a medial area of the form $\sqrt{\lambda}.\rho^2$, the problem is to classify

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

according to the different possible relations between $k$, $\lambda$.

Suppose that

$$\begin{aligned} \sigma u &= k\rho^2, \\ \sigma v &= \sqrt{\lambda}.\rho^2. \end{aligned}$$

Since $\sigma u$ is rational and $\sigma v$ medial, $u$ is rational and $\frown \sigma$, while $v$ is rational and $\smile \sigma$.

Therefore

$$u \smile v;$$

thus $u$, $v$ are rational and $\sim$, whence $(u - v)$ is an apotome.

The possibilities are now as follows.

1. $\sqrt{u^2 - v^2} \frown u$,

2. $\sqrt{u^2 - v^2} \smile u$.

In both cases $u \frown \sigma$, so that $(u - v)$ is either (1) a first apotome, or (2) a fourth apotome.

In case (1) $\sqrt{\sigma(u-v)}$ is an apotome [x. 91], but in case (2) $\sqrt{\sigma(u-v)}$ is a minor irrational straight line [x. 94].