Definition:Geometric Mean/Mean Proportional

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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.

General Definition

In the language of Euclid, the terms of a (finite) geometric sequence of positive integers between (and not including) the first and last terms are called mean proportionals.

Historical Note

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

In the words of Euclid:

From this it is clear that, if in a right-angled triangle a perpendicular be drawn from the right angle to the base, the straight line so drawn is a mean proportional between the segments of the base.

(The Elements: Book $\text{VI}$: Proposition $8$ : Porism)

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

In the words of Euclid:

To two given straight lines to find a mean proportional.

(The Elements: Book $\text{VI}$: Proposition $13$)

Also see