What about the 'geometry' in 'geometric progression'?

Question

Every time I have come upon a discussion of the geometric sequence, I have often wondered (in vain) about the qualifier 'geometric' since such sources never explained the origin of the term. Naturally, I have wondered about the related terms 'arithmetic' and 'harmonic' in the name of their respective sequences too; while I have been able to find some plausible explanation of the origin of these latter two terms, I have however wondered on end about the historical origin of the 'geometric' in the term geometric progression, without appreciable success.

You might have come across, or thought of, a plausible connection between exponential sequences and geometry that gave such sequences their collective name. Please share these below.

Thanks plenty.

Answer 1

In the geometric progression we have that every term is the geometric mean of its predecessor and successior:

$a_n^2=a_{n-1}a_{n+1}$

The geometric mean (or mean proportional) of two numbers, $a$ and $b$, is the length of the side of a square whose area is equal to the area of a rectangle with sides of lengths $a$ and $b$.

In other terms, is a number $c$ such that :

$a \times b = c \times c$

that comes from :

$$\frac a c = \frac c b.$$

See Euclid's Elements VI.13.

The origin is with the Pythagorean School (see also: Archytas).

The early extant souce seems to be Fragment 2 of the lost work of On Music of Archytas [cited by Porphyry, On Ptolemy’s Harmonics, 1.5] :

And Archytas speaking about the means writes these things:

“There are three means in music: one is the arithmetic [αριθμητικά], the second geometric [γεωμετρικά] and the third sub-contrary [, which they call “harmonic”].

Answer 2

It's just a convention. Arithmetic means you take the sum and then average. Geometric means you take the product and then the nth root. Harmonic means you take the sum of reciprocals, average and then take the reciprocal of that.

The main thing that has to happen is that the resulting average has to be from the min to the max. In particular, if all the values are the same, the average has to be that same value.