Naming Geometric and Arithmetic Progressions

Date: 04/04/2003 at 01:04:45
From: Manny
Subject: Geometric progression

Why is an exponential progression called "geometric"? Does the name 
originate from the mean proportional relations among segments of right 
triangles? And why is a linear progression called "arithmetic"?

Date: 04/06/2003 at 23:03:45
From: Doctor Peterson
Subject: Re: Geometric progression

Hi, Manny.

I've often wondered about the exact history of these terms, but have 
never found any definitive explanation. The arithmetic, geometric, 
and harmonic sequences and means were all known and studied by the 
Greeks, but I don't have a source to show me what they called them in 
Greek, to confirm how far back the terms go. I have the general 
impression that a sequence that grows by addition was considered to be 
essentially based on arithmetic (where addition is fundamental), 
whereas one that grows by multiplication was important in geometry (in 
connection with the proportionality of similar figures, most likely). 
It is certainly possible that the simplicity of the construction of 
the geometric mean using a right triangle (which is based on the 
similarities you mention) may be involved.

The most useful information I've found is this, from D. E. Smith's 
History of Mathematics:

  The early writers often used _proportio_ to designate a series,
  and this usage is found as late as the 18th century. The most
  common use of the word, however, limited it to four terms. Thus
  the early writers spoke of an arithmetic proportion, meaning
  b-a=d-c, as in 2,3,4,5; and of a geometric proportion, meaning
  a:b=c:d, as in 2,4,5,10. To these proportions the Greeks added
  the harmonic progression 1/b - 1/a = 1/d - 1/c, as where a=1/2,
  b=1/3, c=1/4, and d=1/5. These three names are now applied to
  series. To them the Greeks added seven others, all of which go
  back at least to Eudoxus (c. 370 B.C.). The Renaissance writers
  began to exclude several of these, and at the present time we
  have only the geometric proportion left, and so the adjective
  has been dropped and we speak of proportion alone.

That is, the original idea was apparently that of a "proportion," or 
relation, among four items, which was considered in a broader sense 
than our current use of the word "proportion" as an equality of 
ratios. They used it also of the equality of differences (arithmetic) 
and of reciprocals (harmonic), as well as our "geometric" proportion. 
This still doesn't really tell us just why these terms were used, but 
it at least fits with my general impression.

I looked in the Oxford English Dictionary to see if it would shed any 
light on this question, and found this note under "geometrical":

  _Arithmetical progression_, _proportion_, _ratio_, etc. ... relate
  to differences instead of quotients. The term _geometrical_ points
  to the fact that problems involving multiplication were originally
  dealt with by geometry and not by arithmetic.

I'd still like to find more definitive information, but at least this 
more or less agrees with my impression.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/