Commentary on Proposition X.38¶
Back to Proposition X.38.
After proving (l. 21) that each of the squares on AB, BC is medial, Euclid states (ll. 24, 26) that EH, which is equal to the sum of the squares, is a medial area, but does not explain why. It is because, by hypothesis, the squares on AB, BC are commensurable, so that the sum of the squares is commensurable with either [x. 15] and is therefore a medial area [x. 23, Por.].
In this case [x. 28, note] $x$, $y$ are of the forms $k^{\frac{1}{4}}\rho$, $\lambda^{\frac{1}{2}}\rho/k^{\frac{1}{4}}$ respectively. Apply each of the areas $(x^2 + y^2)$ and $2xy$ to a rational straight line $\sigma$, i.e. suppose
$$x^2 + y^2 = \sigma u,$$
$$2xy = \sigma v.$$
Now it follows from the hypothesis, x. 15 and x. 23, Por. that $(x^2 + y^2)$ is a medial area; and so is $2xy$, by hypothesis; therefore $\sigma u$, $\sigma v$ are medial areas. Therefore each of the straight lines $u$, $v$ is rational and $\mathbin{\smile} \sigma$ …….. (1). Again
$$x \mathbin{\smile} y;$$
therefore
$$x^2 \mathbin{\smile} xy.$$
But
$$x^2 \mathbin{\sim} x^2 + y^2 \text{ and } xy \mathbin{\sim} 2xy;$$
therefore
$$x^2 + y^2 \mathbin{\smile} 2xy,$$
or
$$\sigma u \mathbin{\smile} \sigma v,$$
whence
$$u \mathbin{\smile} v \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots(2).$$
Therefore, by (1), (2), $u$, $v$ are rational and $\sim$. It follows, by x. 36, that $(u + v)$ is irrational. Therefore $(u + v)\,\sigma$ is an irrational area [this can be deduced from x. 20 by reductio ad absurdum], whence $(x + y)^2$, and consequently $(x + y)$, is irrational.
The irrational straight line $k^{\frac{1}{4}}\rho + \frac{\lambda^{\frac{1}{2}}\rho}{k^{\frac{1}{4}}}$ is called a second bimedial straight line.
This and the corresponding second apotome of a medial $\left(k^{\frac{1}{4}}\rho - \frac{\sqrt{\lambda}}{k^{\frac{1}{4}}}\rho\right)$ found in x. 75 are the positive roots of the equation
$$x^4 - 2\,\frac{k+\lambda}{\sqrt{k}}\,\rho^2 \cdot x^2 + \frac{(k-\lambda)^2}{k}\,\rho^4 = 0.$$