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Arithmetic-Geometric Mean, Harmonic-Geometric Mean

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Arithmetic-Geometric Mean, Harmonic-Geometric Mean

What are the practical applications of these means? I know how to calculate them. I just don't know what they are good for!

Kevin Brown discussed an ancient version of Golden
proportions by:

The simplest version of 2/35 was written mod 30,

2/35*(30/30) = (35 + 25/1050 = 1/30 + 1/42

and, 2/91 by mod 70 or,

2/91*(70/70) = (91 + 49)/6370 = 1/70 + 1/130

They are good for computing complete elliptic integrals a.k.a.
periods of genus 1 Riemann surfaces.

I have read (Science Awakening, B.L. van der Waerden) that Greeks enjoyed A, G and H and named it the Golden Proportion. About 15 years ago I also read that Ancient Near Eastern scribe may have used an inverse Golden Proportion to compute optimal unit fraction fraction series. But that suggestion was disproved a couple of years ago. I'll share the prove if anyone is interested.

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