Definition:Geometric Mean

Definition

Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive.

The geometric mean of $x_1, x_2, \ldots, x_n$ is defined as:

$\displaystyle G_n := \left({\prod_{k=1}^n x_k}\right)^{1/n}$

That is, to find out the geometric mean of a set of $n$ numbers, multiply them together and take the $n$th root.

Mean Proportional

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.

Note that 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 (The Elements: Book VI: Proposition $8$) .

It is mentioned again, in the same context, in Construction of Mean Proportional (The Elements: Book VI: Proposition $13$) .

See Also

• Mean
• Arithmetic Mean Never Less than Geometric Mean

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 3.10$