Commentary on Proposition X.21

Back to Proposition X.21.

[Folio 0054 | p. 50]

A medial straight line, now defined for the first time, is so called because it is a mean proportional between two rational straight lines commensurable in square only. Such straight lines can be denoted by $\rho$, $\rho \sqrt{k}$. A medial straight line is therefore of the form $\sqrt{\rho^2 \sqrt{k}}$ or $k^{\frac{1}{4}} \rho$. Euclid’s proof that this is irrational is equivalent to the following. Take $\rho$, $\rho \sqrt{k}$ commensurable in square only, so that they are incommensurable in length.

Now

$$\rho : \rho \sqrt{k} = \rho^2 : \rho^2 \sqrt{k},$$

whence [X. 11] $\rho^2 \sqrt{k}$ is incommensurable with $\rho^2$ and therefore irrational [X. Def. 4], so that $\sqrt{\rho^2 \sqrt{k}}$ is also irrational [ibid.].

A medial straight line may evidently take either of the forms $\sqrt{a\sqrt{B}}$ or $\sqrt[4]{AB}$, where of course $B$ is not of the form $k^2 A$.

---

Lemma.

If there be two straight lines, then, as the first is to the second, so is the square on the first to the rectangle contained by the two straight lines.

Let FE, EG be two straight lines.

I say that, as FE is to EG, so is the square on FE to the rectangle FE, EG.

[Folio 0055 | p. 51]

For on FE let the square DF be described, and let GD be completed.

Since then, as FE is to EG, so is FD to DG, [VI. 1] and FD is the square on FE, and DG the rectangle DE, EG, that is, the rectangle FE, EG,

therefore, as FE is to EG, so is the square on FE to the rectangle FE, EG.

Similarly also, as the rectangle GE, EF is to the square on EF, that is, as GD is to FD, so is GE to EF.

Q. E. D.

---

If $a$, $b$ be two straight lines,

$$a : b = a^2 : ab.$$