Commentary on Proposition X.18

Back to Proposition X.18.

With the same notation as before, we have to prove in this proposition that

  1. if $(a - x)$, $x$ are incommensurable in length, so are $a$, $\sqrt{a^2 - b^2}$, and

  2. if $a$, $\sqrt{a^2 - b^2}$ are incommensurable in length, so are $(a - x)$, $x$. Or, with the equations

$$\left.\begin{aligned} xy &= \frac{b^2}{4} \\ x + y &= a \end{aligned}\right\},$$

  1. if $x \mathrel{⌣} y$, then $a \mathrel{⌣} \sqrt{a^2 - b^2}$, and

  2. if $a \mathrel{⌣} \sqrt{a^2 - b^2}$, then $x \mathrel{⌣} y$. The steps are exactly the same as shown under (1) and (2) of the last note, with $\mathrel{⌣}$ instead of $\mathrel{⌢}$, except only in the lines “$x \mathrel{⌢} 2x$” and “$2x \mathrel{⌢} x$” which are unaltered, while, in the references, X. 13, 16 take the place of X. 12, 15 respectively.

[Lemma. Since it has been proved that straight lines commensurable in length are always commensurable in square also, while those commensurable in square are not always commensurable in length also, but can of course be either commensurable or incommensurable in length, it is manifest that, if any straight line be commensurable in length with a given rational straight line, it is called rational and commensurable with the other not only in length but in square also, since straight lines commensurable in length are always commensurable in square also.

But, if any straight line be commensurable in square with a given rational straight line, then, if it is also commensurable in length with it, it is called in this case also rational and commensurable with it both in length and in square; but, if again any straight line, being commensurable in square with a given rational straight line, be incommensurable in length with it, it is called in this case also rational but commensurable in square only.]