Commentary on Proposition X.39

Back to Proposition X.39.

[Folio 0092 | p. 88]

Here $x$, $y$ are 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, the rectangle $xy$ is medial; therefore $2xy$ is medial.

Also $(x^2+y^2)$ is a rational area.

Therefore

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

whence

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

so that $(x+y)^2$, and therefore $(x+y)$, is irrational.

The irrational straight line $\frac{\rho}{\sqrt{2}}\sqrt{1+\frac{k}{\sqrt{1+k^2}}} + \frac{\rho}{\sqrt{2}}\sqrt{1-\frac{k}{\sqrt{1+k^2}}}$ is called a major (irrational) straight line.

This and the corresponding minor irrational found in x. 76 are the positive roots of the equation

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