Definition:Geometric Mean/Mean Proportional

Definition
In the language of 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.

Historical Note
This definition is never made specifically in Euclid's, but introduced without definition in the porism to Perpendicular in Right-Angled Triangle makes two Similar Triangle:

It is mentioned again, in the same context, in Construction of Mean Proportional:

Also see

 * Definition:Mean