The origin and use of the term "equianharmonic" (elliptic function)

Question

In Weierstrass notation, the principal elliptic function $\wp$ is a solution of the differential equation $$ (\wp')^2= 4\wp^3 -g_2\wp -g_3. $$ The case when $g_3=0$ is called lemniscatic (it corresponds to a square lattice), and the case $g_2=0$ is called equianharmonic (it corresponds to a hexagonal lattice). The origin of the first term is clear: it comes from the problem on the length of the Bernoulli's lemniscate.

What is the origin of the term "equianharmonic"?

The term "equianharmonic" seems out of date: checking Google ngram shows a strange pattern: its usage experienced a peak around 1860 and was much more frequent than "lemniscatic" until 1980s, and nowadays it is used about 10 times less frequently than "lemniscatic". Also the sizes and details of Wikipedia articles confirm this.

The comment above also posted as an answer by @CarloBeenakker at MO. Perhaps it should also be posted here? I'm not sure if that's the etiquette, but it seems reasonable to me. — LSpice, Dec 18 '22 at 19:34
not sure either about "etiquette", but it might serve a person to remove this question from the "unanswered" list. — Carlo Beenakker, Dec 18 '22 at 20:29
English Google ngram viewer shows that the term is very much older than your dates may indicate. — Alexandre Eremenko, Dec 18 '22 at 22:38
@njuffa: I relied only on Google ngram, which shows a large peak of usage (in English) around 1860. Most of the literature of that time is not available to me, unfortunately. — Alexandre Eremenko, Dec 18 '22 at 22:53
@njuffa: This post seems relevant to this discussion: https://hsm.stackexchange.com/questions/11571/books-on-elliptic-functions — Alexandre Eremenko, Dec 19 '22 at 05:56

Carlo Beenakker · Accepted Answer · 2022-12-21T07:26:54.067

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An answer to remove this question from the "unanswered list": The term "equianharmonic" refers to "equal anharmonic ratio", as explained by Wiener in 1901, see this earlier MO post.

edited Dec 21 '22 at 07:26

answered Dec 18 '22 at 20:28

Carlo Beenakker

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I agree to remove it from the unanswered list but the term could not be possibly introduced as late as 1901, see Google ngram viewer:-) – Alexandre Eremenko Dec 18 '22 at 22:36
Hi Carlo, would you by any chance have any input into this ? – Mikhail Katz Nov 20 '23 at 14:59

The origin and use of the term "equianharmonic" (elliptic function)

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