Feynman:"Tangent is the mean proportional between the two parts of the diameter cut by an equal chord"

Question

I cribbed this from Feynman http://www.feynmanlectures.caltech.edu/I_07.html Here's the situation: We have a Commie Rusky cosmonaut traveling along $x$, just fast enough to evade the Western imperialist law of gravity. Find how fast, in capitalist units, he is traveling.

In presenting a derivation, Feynman promulgates:

Now, if we use one of those wonderful theorems in geometry, which says that our tangent is the mean proportional between the two parts of the diameter cut by an equal chord, we see that the horizontal distance travelled is the mean proportional between the 16 feet fallen and the 8000-mile diameter of the earth.

I can look at the diagram (reproduced and attached) and figure out the value of $x$, but I do not understand Feynman's language in stating the theorem. Can someone please explain the meaning, in this context, of "mean proportional" and "cut by an equal chord"?

I am curious to know if this language is familiar to most (American educated) mathematicians.

Acceleration toward the center of a circular path. From plane geometry,

$$\frac{x}{S}=\frac{2R-S}{x}\approx\frac{2R}{x},$$

where $R$ is the radius of the earth, 4000 miles; $x$ is the distance “travelled horizontally” in one second; and $S$ is the distance “fallen” in one second (16 feet).

Answer 1

It would help to include the caption of the figure:

Fig. 7–4. Acceleration toward the center of a circular path. From plane geometry, $x/S=(2R−S)/x=2R/x$, where $R$ is the radius of the earth, $4000$ miles; $x$ is the distance “travelled horizontally” in one second; and $S$ is the distance “fallen” in one second ($16$ feet).

So the "wonderful theorem" is the geometric mean theorem mentioned by Arthur in comments beneath the OP. See also the proofwiki definition of the term.