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I am trying to understand the uses of the word "Geometric" throughout mathematics. I suspect that there may be some historical reasons which would tie things together and help my understanding. Supposedly, "Geometric" means "of or relating to Geometry".

"Geometry" is the study of points, lines, surfaces, shapes, etc.

A "Geometric mean" of n numbers is the nth root of the product of the numbers -- how is this related to "Geometry" (shapes, lines, points, etc.)?

A "Geometric series" consists of a sequence of numbers, with each successive number the product of the previous number and some constant -- how does this relate to Geometry?

I know that there have been some attempts by mathematicians to combine Geometry and Algebra into a single framework. Could terms like "Geometric mean" and "Geometric series" have come from these attempts?

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  • $\begingroup$ "geo- metry" and a little bit of Greek. $\endgroup$ – NeerajKumar Nov 13 '15 at 13:39
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"Geometry" literally translates from Greek as "land measurement", and reflected a popular Greek belief that they learned it from Egyptian "rope stretchers", land meausurers, who used ropes to perform what came to be called straightedge and compass constructions.

Geometric mean for $n=2$, or mean proportional as Greeks called it, was likely so named because it answers a geometric question: what is the side of a square having the same area as a rectangle with given sides. Euclid gives a straightedge and compass construction for it in Elements VI.13, but it was already known to early Pythagoreans in 6th century BC.

Geometric series was likely so named because each of its terms is the geometric mean of its immediate neighbors $a_n=\sqrt{a_{n-1}a_{n+1}}$ (assuming all terms are positive). The same is true of arithmetic and harmonic series, their terms are also the corresponding means of their neighbors. The notion if not the name is also attributed to early Pythagoreans, Euclid expresses in words what we call partial sum of the geometric series in Elements IX.35.

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Geometric mean can be understood in terms of geometry. 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.

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"Geometry" means literally "Earth measuring", i.e., what we today would call "topography". The term "geometric mean" is ancient Greek, while the use of algebra to solve geometric problems dates from the times of Descartes (1596-1650). No relation.

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For first uses of mathematical terms, you can try HERE

some items found there:

GEOMETRIC MEAN. The geometric mean is one of the three Pythagorean means. ...

The term geometrical mean is found in E. Halley "A Most Compendious and Facile Method for Constructing the Logarithms, Exemplified and Demonstrated from the Nature of Numbers, without any Regard to the Hyperbola, with a Speedy Method for Finding the Number from the Logarithm Given," Philosophical Transactions of the Royal Society of London, Vol. 19. (1695 - 1697), pp. 58-67.

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The term GEOMETRIC PROGRESSION was used by Michael Stifel in 1543: "Divisio in Arethmeticis progressionibus respondet extractionibus radicum in progressionibus Geometricis"

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The term GEOMETRY was in use in the time of Plato and Aristotle, and “doubtless goes back at least to Thales,” according to Smith (vol. 2, page 273). The Greek word $\gamma\epsilon\omega\mu\epsilon\tau\rho\hat\iota\alpha$ was formed from $\gamma\hat\eta$ earth and $\mu\epsilon\tau\rho\hat\iota\alpha$ measuring; transliteration produced the Latin gemetria which became géométrie in French and thence geometry in English.

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    $\begingroup$ Your last pasted passage has two spelling errors in the Greek and one in the Latin. Not probably a very reliable source. $\endgroup$ – fdb Nov 13 '15 at 20:05
  • $\begingroup$ Greek letters and letters with accents were done there in such a way that copying pasting here, followed by MathJax, made them come out wrong. $\endgroup$ – Gerald Edgar Nov 14 '15 at 16:16
  • $\begingroup$ What aboit "gemetria"? $\endgroup$ – fdb Nov 14 '15 at 16:20
  • $\begingroup$ In the original, it says "geōmetria" $\endgroup$ – Gerald Edgar Nov 14 '15 at 16:21
  • $\begingroup$ I guess this stems from pre-Unicode (or at least non-Unicode) text. $\endgroup$ – Gerald Edgar Nov 14 '15 at 16:27
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The word 'geometry' originates from the Sanskrit word gyaamiti - "measuring the earth"

gya means "earth" miti means "to measure"

Here is the link: http://veda.wikidot.com/sanskrit-origin-names

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  • $\begingroup$ Hi, Amit, and welcome to HSM. This is interesting; I hadn't heard of it before. Can you add some references to back up your point, like other answers (not just to this question, but site-wide) do? Thanks. $\endgroup$ – HDE 226868 Nov 15 '15 at 15:15
  • $\begingroup$ This is interesting, indeed, but we must keep in mind that all settled cultures (i.e. not nomadic or huntergatheres) could not get by without measuring land in some way, and thus must have had an expression for it. "Geometry" is well-attested in Greek (Herodot. 2.109 already refers to γεωμετρία as a math pursuit, likely). Note how English chainman is similar to ἁρπεδονάπται, “rope-stretchers” mentioned by Democr., and German Landmesser and Russian земелемер are literrally “land-measurers.” So the similarity proves nothing per se. They measured their land, thus had to call that somehow! $\endgroup$ – kkm Mar 4 '18 at 5:42
  • $\begingroup$ Auch! I just checked the link, and it is a typical linguistic BS freak science. Just like the Sanskrit word matri [...] became "mater" in Greek and "mother" in English. -- sheer nonsense. Current linguistic theory, and an extremely strong one indeed, establishes that Indic, Germanic and Greek, among others, are all daughters of the same ancestral language, called PIE. It's a common trend among some people with an agenda to claim that their language was the oldest. Do not fall into this trap please! To start, en.wikipedia.org/wiki/Proto-Indo-European_language $\endgroup$ – kkm Mar 4 '18 at 5:59

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