Commentary on Proposition X.77

Back to Proposition X.77.

[Folio 0169 | p. 165]

Here $x, y$ are of the form [cf. x. 34]

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

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

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

and therefore

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

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

The irrational straight line

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

is called that which produces with a rational area a medial whole or more literally that which with a rational area makes the whole medial (ἡ μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσα). Here “produces” means “produces when a square is described on it.” A clearer way of expressing the meaning would be to call this straight line the “side” of a medial minus a rational area corresponding to the “side” of a rational plus a medial area [x. 40].