Commensurability of Squares/Porism

Porism to Commensurability of Squares

In the words of Euclid:

And it is manifest from what has been proven that straight lines commensurable in length are always commensurable in square also, but those commensurable in square are not always commensurable in length also.

(The Elements: Book $\text{X}$: Proposition $9$ : Porism)

Proof

Follows directly from Commensurability of Squares.

$\blacksquare$

Historical Note

This proof is Proposition $9$ of Book $\text{X}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions