Definition:Duplicate Ratio

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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 the definition of ratio:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle a : b$	$=$	$\displaystyle b : c$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle \dfrac a b$	$=$	$\displaystyle \dfrac b c$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle \dfrac a c$	$=$	$\displaystyle \dfrac a b \dfrac b c$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle \dfrac a c$	$=$	$\displaystyle \left({\dfrac a b}\right)^2$	$\displaystyle $	$\displaystyle $	$\displaystyle $

That is:

$a : c = \left({a : b}\right)^2$

As Euclid defined it:

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.

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle a : b\)	\(=\)	\(\displaystyle b : c\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \dfrac a b\)	\(=\)	\(\displaystyle \dfrac b c\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \dfrac a c\)	\(=\)	\(\displaystyle \dfrac a b \dfrac b c\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \dfrac a c\)	\(=\)	\(\displaystyle \left({\dfrac a b}\right)^2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)