# Magnitudes with Rational Ratio are Commensurable/Porism

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## Porism to Magnitudes with Rational Ratio are Commensurable

In the words of Euclid:

*From this it is manifest that, if there be two numbers, as $D$, $E$, and a straight line, as $A$, it is possible to make a straight line [$F$] such that the given straight line is to it as the number $D$ to the number $E$.*

And, if a mean proportional be also taken between $A$, $F$, as $B$,

as $A$ is to $F$, so will the square on $A$ be to the square on $B$, that is, as the first is to the third, so is the figure on the first to that which is similar and similarly described on the second.

But, as $A$ is to $F$, so is the number $D$ to the number $E$; therefore it has been contrived that, as the number $D$ is to the number $E$, so also is the figure on the straight line $A$ to the figure on the straight line $B$.

(*The Elements*: Book $\text{X}$: Proposition $6$ : Porism)

## Proof

Apparent from the construction.

$\blacksquare$

## Historical Note

This proof is Proposition $6$ 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