I programmed an interactive version of Robert's "means" diagram in Firefox javascript, which Robert put on his website here: http://www.symbo1ics.com/files/means.htm I'm not proud of the code, but this was a good excuse to learn how to program in Javascript with JSXGraph. --- BTW, you should be aware that one of the coolest features of Javascript is _full closures_ (as found in Common Lisp & Scheme), so you can use function-returning functions to your heart's content, and even play with "Curried" functions. At 07:55 PM 3/2/2011, quad wrote:

On Wed, Mar 2, 2011 at 6:36 PM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:

...

OED doesn't have much to say about "geometric[al] mean", regrettably. The entry for "mean" has a citation from about 1450 concerning the "meene proporcionalle", which is clearly the same thing as the GM (the OED entry for "mean proportional" quotes the same thing and seems to think it might mean something other than the GM, but I've no idea what else they think it could be). There's nothing for "arithmetic[al] mean" or "geometric[al] mean" until much later.

So it's inconclusive, but seems to me like weak evidence against the theory that "geometric mean" came *before* "geometric progression".

With regards to the geometric mean, I thought you all might be interested in a construction I just made of four means. See the PDF file here:

<http://www.symbo1ics.com/files/means.pdf>

Key ==================================================== Light Blue : Arbitrary Value : x Dark Blue : Arbitrary Value : y Dark Green : Arithmetic Mean : (x+y)/2 Light Green : Harmonic Mean : 2/(1/x + 1/y) Red : Geometric Mean : sqrt(x*y) Orange : Root Mean Square : sqrt[(x^2 + y^2)/2] ====================================================

The constructions should be easy to infer from the diagram. :)

This does make for some nice identities, somewhat interesting methods of computation, and easy derivation of inequalities.

-Robert