The Ten Means of Ancient Greece

The ancient Greeks defined a list of ten distinct "means",
including most of the well-known means that we still use
today.  However, oddly enough, they never explicitly defined
what we call the root-mean-square (although it can be constructed
by composition of some of their means).  In Pythagoras's time 
there were just three means, which we call the arithmetic, 
the geometric, and the harmonic (originally known as the 
"subcontrary mean").  Later, three more "means" were added, 
possibly by Eudoxus.  These six are described in the article
Iterated Means.  The last four means were added by two 
later Pythagoreans, Myonides and Euphranor.

We actually have a listing of "The Ten Means" from two different 
ancient authors (Nicomachus and Pappus), but the lists are not 
quite identical.  They each give one "mean" that the other left 
out, so taking the two lists together, we have eleven distinct 
means in all.

Consider three quantites a,b,c such that a > b > c, where we 
wish to make b the "mean" of a and c.  Notice that we can form 
three positive differences with these quantites: (a-b), (b-c), 
and (a-c).  The Greeks worked on the idea of equating a ratio 
of two of these differences to a ratio of two of the original 
quantities (not necessarily distinct).  For example, if we set 
the ratio (a-b)/(b-c) equal to the ratio a/b, the result is 
b^2 = ac, which represents the geometric mean.

If you look at all the possible ways of doing this, several of 
them are automatically ruled out by the assumed inequalities 
on a,b,c.  The ones that are not (necessarily) ruled out are
the ten (actually eleven) means as summarized below:

(1)    (a-b)/(b-c) = a/a = b/b = c/c      b = (a+c)/2

(2)    (a-b)/(b-c) = b/c = a/b            b = sqrt(ac)

(3)    (a-b)/(b-c) = a/c               b = 2/(1/a + 1/c)

(4)    (a-b)/(b-c) = c/a            b = (a^2 + c^2)/(a + c)

(5)    (a-b)/(b-c) = c/b         b = ((a-c)+sqrt(a^2-2ac+5c^2))/2

(6)    (a-b)/(b-c) = c/b         b = ((c-a)+sqrt(5a^2-2ac+c^2))/2

(7n)   (b-c)/(a-c) = c/a               b = (2ac - c^2)/a

(8)    (b-c)/(a-c) = c/b            b = (c+sqrt(4ac-3c^2))/2

(9)    (a-b)/(a-c) = c/a            b = (a^2 - ac + c^2)/a

(10p)  (a-b)/(a-c) = b/a              b = (a^2)/(2a - c)

(11)   (a-b)/(a-c) = c/b                  b = a - c

Some of these are obviously not very robust definitions of "means".
For example, using the 11th mean we would have  m11(5,4) = 1.  This
mean presumably was included because it doesn't *necessarily* violate
the assumed inequalities, e.g., m11(5,1)=4, but it seems only marginally
acceptable.  On the other hand, it's interesting to note that m5(2,1) 
equals phi, the golden proportion (1.618..).  We might also note that 
the 2nd mean of the 3rd and 4th means on this list is equivalent to 
what we call the root-mean-square.

Anyway, it's not too surprising that only the arithmetic, geometric, 
and harmonic have survived in common usage.  This process of broad 
abstract definition followed by pragmatic selection reminds me of 
how Western music originally had seven distinctly identified "modes"
(Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian), 
and then over time we discarded all but two of them (the Ionian and 
Aeolian), which we call the "major" and natural "minor" scales.  
This is doubly fitting, considering that the original concept of 
numerical "means" among the Pythagoreans and others was closely 
involved with the study of musical tones and scales.