Commentary on Proposition X.78

Back to Proposition X.78.

[Folio 0171 | p. 167]

In this case $x, y$ have respectively the forms [cf. x. 35]

$$\frac{\rho \lambda^{\frac{1}{4}}}{\sqrt 2} \sqrt{1 + \frac{k}{\sqrt{1 + k^2}}}, \quad \frac{\rho \lambda^{\frac{1}{4}}}{\sqrt 2} \sqrt{1 - \frac{k}{\sqrt{1 + k^2}}}.$$

Suppose that

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

By hypothesis, the areas $\sigma u, \sigma v$ are medial; therefore $u, v$ are both rational and $\smile \sigma$ …………………………………..(1).

Further

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

so that

$$u \smile v$$

………………………………………….(2).

Hence [(1), (2)] $u, v$ are rational and $\sim$, so that $(u - v)$ is the irrational straight line called apotome [x. 73]. Thus $\sigma (u - v)$ is an irrational area, so that $(x - y)^2$, and consequently $(x - y)$, is irrational.

The irrational straight line

$$\frac{\rho \lambda^{\frac{1}{4}}}{\sqrt 2} \sqrt{1 + \frac{k}{\sqrt{1 + k^2}}} - \frac{\rho \lambda^{\frac{1}{4}}}{\sqrt 2} \sqrt{1 - \frac{k}{\sqrt{1 + k^2}}}$$

is called that which produces [i.e. when a square is described on it] with a medial area a medial whole, more literally that which with a medial area makes the whole medial (ἡ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα). A clearer phrase (to us) would be the “side” of the difference between two medial areas, corresponding to the “side” of (the sum of) two medial areas [x. 41].