Definition:Geometric Mean/Mean Proportional/Historical Note

Historical Note on Mean Proportional

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 $13$)