Construction of Incommensurable Lines

Proof
Let $A$ be the assigned straight line.

Let $B$ and $C$ be two numbers which do not have to one another the ratio which a square number has to a square number.

That is, from the lemma, that $B$ and $C$ are not similar plane numbers.

From Magnitudes with Rational Ratio are Commensurable: Porism, let $D$ be constructed such that:
 * $\dfrac {A^2} {D^2} = \dfrac B C$

Therefore from Magnitudes with Rational Ratio are Commensurable, the square on $A$ is commensurable with the square on $D$.

Since:
 * $B$ does not have to one $C$ the ratio which a square number has to a square number

then:
 * $A^2$ does not have to one $D^2$ the ratio which a square number has to a square number.

Therefore from Commensurability of Squares $A$ is incommensurable in length with $D$.

Let a mean proportional $E$ be taken between $A$ and $D$.

From :
 * $\dfrac A D = \dfrac {A^2} {E^2}$

But $A$ is incommensurable in length with $D$.

Therefore $A^2$ is incommensurable with $E^2$.

Therefore $A$ is incommensurable in square with $E$.

Therefore two straight lines $D$ and $E$ have been found, such that:
 * $D$ is incommensurable in length with $A$
 * $E$ is incommensurable in square with $A$.