Commentary on Proposition X.74

Back to Proposition X.74.

[Folio 0164 | p. 160]

The first apotome of a medial straight line is the difference between straight lines of the form $k^{\frac{1}{4}}\rho, k^{\frac{3}{4}}\rho$, which are medial straight lines commensurable in square only and forming a rational rectangle.

By hypothesis, $x^2, y^2$ are medial areas.

And, since $xy$ is rational,

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

$$\smile 2xy,$$

whence

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

But $2xy$ is rational; therefore $(x - y)^2$, and consequently $(x - y)$, is irrational.

This irrational, which is of the form $(k^{\frac{1}{4}}\rho \sim k^{\frac{3}{4}}\rho)$, is the first apotome of a medial straight line; the term corresponding of course to first bimedial, which applies where the sign is positive.