Commentary on Proposition X.37

Back to Proposition X.37.

[Folio 0089 | p. 85]

Here $x$, $y$ have the forms $k^{\frac{1}{4}}\rho$, $k^{\frac{3}{4}}\rho$ respectively, as found in x. 27. Exactly as in the last case we prove that

$$x^2 + y^2 \mathbin{\smile} 2xy,$$

whence

$$(x + y)^2 \mathbin{\smile} 2xy.$$

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

The irrational straight line $k^{\frac{1}{4}}\rho + k^{\frac{3}{4}}\rho$ is called a first bimedial straight line.

This and the corresponding first apotome of a medial ($k^{\frac{1}{4}}\rho - k^{\frac{3}{4}}\rho$) found in x. 74 are the positive roots of the equation

$$x^4 - 2\sqrt{k}(1 + k)\rho^2 \cdot x^2 + k(1 - k)^2\rho^4 = 0.$$