Definition:Duplicate Ratio

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Let $a, b, c$ be magnitudes such that:

$a : b = b : c$

Then $a$ has the duplicate ratio to $c$ of the ratio it has to $b$.

That is:

$a : c$ is the duplicate ratio of $a : b$.

From definition of ratio:

\(\ds a : b\) \(=\) \(\ds b : c\)
\(\ds \leadsto \ \ \) \(\ds \dfrac a b\) \(=\) \(\ds \dfrac b c\)
\(\ds \leadsto \ \ \) \(\ds \dfrac a c\) \(=\) \(\ds \dfrac a b \dfrac b c\)
\(\ds \leadsto \ \ \) \(\ds \dfrac a c\) \(=\) \(\ds \paren {\dfrac a b}^2\)

That is:

$a : c = \paren {a : b}^2$

In the words of Euclid:

When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second.

(The Elements: Book $\text{V}$: Definition $9$)