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The term "equianharmonic" seems out of date: checking Google ngram shows a strange pattern: its usage experienced a peak around 1860 and was much more frequent than "lemniscatic" until 1980s, and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.

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    $\begingroup$ see mathoverflow.net/a/385121/11260 $\endgroup$ Dec 18, 2022 at 18:56
  • $\begingroup$ @Carlo Beenakker: Thanks This answers the question. $\endgroup$ Dec 18, 2022 at 18:59
  • $\begingroup$ 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. $\endgroup$
    – LSpice
    Dec 18, 2022 at 19:34
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    $\begingroup$ not sure either about "etiquette", but it might serve a person to remove this question from the "unanswered" list. $\endgroup$ Dec 18, 2022 at 20:29
  • $\begingroup$ English Google ngram viewer shows that the term is very much older than your dates may indicate. $\endgroup$ Dec 18, 2022 at 22:38

1 Answer 1


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.

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    $\begingroup$ 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:-) $\endgroup$ Dec 18, 2022 at 22:36
  • $\begingroup$ Hi Carlo, would you by any chance have any input into this ? $\endgroup$ Nov 20, 2023 at 14:59

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