# 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.

### General Definition

In the language of Euclid, the terms of a (finite) geometric progression 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 $13$)