Commentary on Proposition X.75

Back to Proposition X.75.

[Folio 0166 | p. 162]

We have here the difference between $k^{\frac{1}{4}}\rho$, $\sqrt\lambda \cdot \rho / k^{\frac{1}{4}}$, two medial straight lines commensurable in square only and containing a medial rectangle. Apply each of the areas $(x^2 + y^2)$, $2xy$ to a rational straight line $\sigma$, i.e. suppose that

$$\begin{aligned} x^2 + y^2 &= \sigma u, \\ 2xy &= \sigma v. \end{aligned}$$

[Folio 0167 | p. 163]

Then $\sigma u, \sigma v$ are medial areas, so that $u, v$ are both rational and $\smile \sigma$ …………………………………..(1).

Again,

$$x \smile y;$$

therefore

$$x^2 \smile xy,$$

and consequently

$$x^2 + y^2 \smile 2xy,$$

or

$$\sigma u \smile \sigma v,$$

and

$$u \smile v \text{ .................................................(2).}$$

Thus [(1), (2)] $u, v$ are rational and $\sim$; therefore [x. 73] $(u - v)$ is an apotome, and, $(u - v)$ being thus irrational,

$$(u - v)\sigma \text{ is an irrational area.}$$

Hence $(x - y)^2$, and consequently $(x - y)$, is irrational.

The irrational straight line $k^{\frac{1}{4}}\rho \sim \frac{\sqrt\lambda \cdot \rho}{k^{\frac{1}{4}}}$ is called a second apotome of a medial straight line.