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On this and other Stack Exchange website, there have been question about the so-called geometric series, and where its name comes from. My problem is that most answers follow one of two different ideas.

  1. Either they show an ingenious, but obviously not ancient, picture that illustrates concrete examples of geometric series summation.
  2. Or they make some reference to the middle of three consecutive terms in the sequence being the "geometric mean" of the other two.

Well, I want to raise the question again, because neither of these prototype answers really explains where the use of "geometric" started, nor who (if known) came up with it.

Any good explanation?

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    $\begingroup$ The "why" is not clear. The origin is with the Pythagorean School (see also: Archytas. The early extant souce seems to be Fragment 2 of the lost work of OnMusic of Archytas. $\endgroup$
    – Mauro ALLEGRANZA
    Jul 22, 2017 at 8:23
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    $\begingroup$ @Mauro Fragment 2 seems to be the closest we can be to an answer! Wow! If you migrate your comment to an answer, I will gladly accept it. $\endgroup$
    – Rodrigo A. Pérez
    Jul 23, 2017 at 2:56
  • $\begingroup$ Although the phrases "arithmetic mean", "geometric mean", and "harmonic mean" have a history that go back to the Greeks, is it really a stretch that 16th and 17th century mathematicians (who knew their classical mathematics) decided to refer to $\sum(a+kb)$, $\sum x^k$, $\sum 1/k$ as "arithmetic series", "geometric series", and "harmonic series" respectively since the value between any two terms is that particular mean? $\endgroup$
    – user6918
    Jan 16, 2018 at 15:33
  • $\begingroup$ I can even imagine a history where the originator of the terms thought it was a natural thing to call them and only had to take 2 seconds to explain the nomenclature to an acquaintance; everyone immediately understood and employed the terms without ever thinking of writing down the explanation so that future generations could have no doubt behind their motivations. $\endgroup$
    – user6918
    Jan 16, 2018 at 15:36

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The "why" is not clear.

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.

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In former times sequence and series have often been used for the same purpose.

GEOMETRIC SERIES is found in 1723 in A System of the Mathematics James Hodgson [Google print search, James A. Landau].

The term GEOMETRIC PROGRESSION was used by Michael Stifel in 1543: "Divisio in Arethmeticis progressionibus respondet extractionibus radicum in progressionibus Geometricis" [James A. Landau].

Geometrical progression appears in English in 1557 in the Whetstone of Witte by Robert Recorde: "You can haue no progression Geometricalle, but it must be made either of square nombers, or els of like flattes" [OED].

Geometric progression is found in English in 1696 in A new theory of the earth, from its original, to the consummation of all things by William Whiston: “Which to how immense a Sum it would arise, those who know any thing of the nature of Geometrick Progressions will easily pronounce.” [OED]

Source: http://jeff560.tripod.com/g.html

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Geometric(al) is here opposed to arithmetic(al). These adjectives refer to how solutions were found in the past before algebraic methods were developed. Geometry was believed to be a more powerful instrument for solving complex problems.

Basically problems linked to sums were solved using arithmetic, and problems linked to products/ratios/exponents were solved by geometry. E.g. it is easy to "see" the product of two numbers by looking at the rectangle area, and the square root extraction was therefore a geometry problem.

Using this opposition, a proportion, a sequence (progression) or a series could be qualified as arithmetical or geometrical.

Other answers already dated the use of the wording, I'm just adding extracts of a book using this conventional split, Arithmetical Institutions. Containing a Compleat System of Arithmetic Natural, Logarithmical, and Algebraical in all Their Branches, by John Kirby in 1735:

Arithmetical vs. geometrical ratios

enter image description here

(Note how ratio was used both for the common difference and the common ratio. This is now different in En. but it persisted in my native language (Fr.): Latin ratus gave ratio but also raison (reason). Raison is the word used to refer to the common difference and common ratio.)

Arithmetical proportion and progression

enter image description here

enter image description here

Geometrical ratio and progression

enter image description here

(Note geometrical ratios are a family including simple, multiplicate, harmonical and contraharmonical. Details are found in the same book.)

enter image description here

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