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In Weierstrass notation, the principal elliptic function $\wp$ is a solution of the 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The term "equianharmonic" seems out of date: checking Google ngram shows a strange pattern: its usage experienced a peak around 1860 and was much more frequent than "lemniscatic" until 1980s, and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.


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.

  • $\begingroup$ The term could not be possibly introduced so late (see Google ngram viewer). $\endgroup$ Dec 18, 2022 at 22:41
  • $\begingroup$ If the answer can't possibly be right, then why have you accepted it? $\endgroup$ Dec 20, 2022 at 23:38
  • $\begingroup$ edit: "introduced by Wiener" --> "explained by Wiener" $\endgroup$ Dec 21, 2022 at 7:24

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