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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.

Allawonder
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    Yes, the questions are similar, but no answer there is satisfactory. The best summary I got is that measurement (of n-cubes) involves multiplication. – Allawonder Jan 16 '18 at 20:21

2 Answers2

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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”].

Mauro ALLEGRANZA
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  • That would boil down to explaining the origin of the term 'geometric mean' -- what is geometric about this mean that isn't about the arithmetic and harmonic means too? What distinguishes it to make it befitting to call it 'geometric'? -- this is exactly what I want to get at. – Allawonder Jan 16 '18 at 15:43
  • @Allawonder - see above: 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 $b$. It is called "geometric" because has a geometric natural interpretation. – Mauro ALLEGRANZA Jan 16 '18 at 15:49
  • The arithmetic and harmonic means also have respective geometric interpretations. We can construct them all with ruler and compasses. Therefore that's not why exponential sequences would be termed geometric progressions. There must be some early historical reason. – Allawonder Jan 16 '18 at 15:53
  • @Allawonder - it will be hard to find something "earlier" than Archytas of Tarentum, fourth century BC. – Mauro ALLEGRANZA Jan 16 '18 at 16:01
  • As I said in the OP, someone might have figured out a plausible explanation. What I need is not more than that, so that even if we can't find historical evidence for its origin, so long as the explanation given is plausible enough then I'm good with it. – Allawonder Jan 16 '18 at 16:21
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    Jeff Miller's Earliest Known Uses lists 1695 as the earliest use of geometrical mean and 1543 as that of geometric progression. Your links about the Pythagorean school show that they were interested in the geometric mean, but not necessarily that they called it the geometric mean (rather than some other term, like mean proportional). I was not able to access Fragment 2 as I got a "page not available" error. –  Jan 16 '18 at 16:42
  • @Rahul - correct. The earliest use of the English term: geometrical mean. – Mauro ALLEGRANZA Jan 16 '18 at 16:58
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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.

marty cohen
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    That is not plausible. There is a context in which every mathematical term arose, even such funny-sounding ones (to the non-mathematician at least) as irrational, imaginary or surreal. It doesn't seem as if those who gave the sequence (and the other two kinds) its name just upped and said, let's call this a geometric sequence. I mean, why is it called a geometric sequence? Why not exponential sequence, say, or multiplicative sequence, or even some odd word that has no relation to anything whatever? There must be a reason why it's called a geometric sequence. – Allawonder Jan 16 '18 at 16:26