A nephroid is an epycloid that can be generated by rolling a circle on the outside of a circle with doubled radius. It was called by Richard Proctor (1878) because its shape looks like a kidney (see https://mathworld.wolfram.com/Nephroid.html).

A nephroid of Freeth (called from Thomas Jacob Freeth, 1879) is a strophoid of the circle and has nothing to do with Proctor's nephroid (from what I undertand) and doesn't look like a kidney, nor Proctor's nephroid (see https://mathworld.wolfram.com/FreethsNephroid.html)

So: why is Freeth's nephroid called a nephroid ?

Note: I could not find the original paper of Freeth where he may have explained the name.

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    $\begingroup$ The original "paper" is a summary in the Appendix to vol. 10 of the Proceedings of the London Mathematical Society (1878), p. 228ff. It gives no explanations for the naming. Perhaps, Freeth and Proctor had different ideas about the shape of the kidney :) $\endgroup$
    – Conifold
    Commented Feb 11, 2021 at 18:42
  • $\begingroup$ @Conifold And what is on p. 130? $\endgroup$ Commented Feb 11, 2021 at 21:05
  • $\begingroup$ @GeraldEdgar Just a mention with Freeth's name and title. It seems that Proctor had in mind the shape of a pair of kidneys, while Freeth the shape of a single kidney. $\endgroup$
    – Conifold
    Commented Feb 12, 2021 at 23:06

1 Answer 1


Doesn't look like a kidney?

Freeth's nephroid

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    $\begingroup$ Well, I guess it's all about imagination. And indeed with the two figures it can be better understood. Thanks :) $\endgroup$
    – irimias
    Commented Feb 12, 2021 at 12:04

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