Commentary on Proposition X.20

Back to Proposition X.20.

[Folio 0053 | p. 49]

The converse of the last. If $\rho$ is a rational straight line, any rational area is of the form $k\rho^2$. If this be “applied” to $\rho$, the breadth is $k\rho$ commensurable in length with $\rho$ and therefore rational. We should reach the same result if we applied the area to another rational straight line $\sigma$. The breadth is then

$$\frac{k\rho^2}{\sigma} = \frac{k\rho^2}{\sigma^2}\cdot \sigma = \frac{m}{n}k\cdot \sigma \text{ or } k'\sigma\text{, say.}$$