\begin{definition}[Definition:Geometric Mean/Mean Proportional]
In the language of {{AuthorRef|Euclid}}, the geometric mean of two magnitudes is called the '''mean proportional'''.
Thus the '''mean proportional''' of $a$ and $b$ is defined as that magnitude $c$ such that:
:$a : c = c : b$
where $a : c$ denotes the ratio between $a$ and $c$.
From the definition of ratio it is seen that $\dfrac a c = \dfrac c b$ from which it follows that $c = \sqrt {a b}$ demonstrating that the definitions are logically equivalent.
\end{definition}