Definition:Duplicate Ratio

Definition

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.