Commentary on Proposition X.76

Back to Proposition X.76.

[Folio 0168 | p. 164]

$x, y$ are here of the form found in x. 33, viz.

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

By hypothesis $(x^2 + y^2)$ is a rational, $xy$ a medial, area. Therefore

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

whence

$$(x - y)^2 \smile (x^2 + y^2).$$

Therefore $(x - y)^2$, and consequently $(x - y)$, is irrational. The minor (irrational) straight line is thus of the form

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

Observe the use of convertendo ($\alpha\nu\alpha\sigma\tau\rho\acute{\epsilon}\psi\alpha\nu\tau\iota$) for the inference that, since $(x^2 + y^2) \smile 2xy$, $(x^2 + y^2) \smile (x - y)^2$. The use of the word corresponds exactly to its use in proportions.