Straight Line Commensurable with Medial Straight Line is Medial/Porism

Porism to Straight Line Commensurable with Medial Straight Line is Medial

In the words of Euclid:

From this it is manifest that an area commensurable with a medial area is medial.

(The Elements: Book $\text{X}$: Proposition $23$ : Porism)

Proof

In the words of Euclid:

And in the same way as was explained in the case of the rationals it follows, as regards medials, that a straight line commensurable in length with a medial straight line is called medial and commensurable with it not only in length but in square also, since, in general, straight lines commensurable in length are always commensurable in square also.

But, if any straight line be commensurable in square with a medial straight line , then, if it is also commensurable in length with it, the straight lines are called, in this case too, medial and commensurable in length and in square, but, if in square only, they are called medial straight lines commensurable in square only.

Historical Note

This proof is Proposition $23$ 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