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In his book "On the lunisolar forces which put the oceans in motion" Euler gets the following relation:

$r=b+\beta \cos ^2 (\psi)$

and according to his claim (because obviously he doesn't provide the proof...) this is the equation of an ellipse with semi-major axis $b$ and with difference $\beta$ between the semi-major and semi-minor axes.

I tried to prove that equality and failed.

Thanks for your help

Vincent ISOZ
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    That equation is a bit strange. What is $\psi$? There is a very similar equation, $r^2 = b^2 + (a^2 - b^2)\cos^2(\phi)$, where $\phi$ is the eccentric anomaly, and $a$ & $b$ are the semi-major & semi-minor axes. – PM 2Ring Jun 28 '25 at 20:34
  • According to Eulers $\psi$ is the angle at the center of the ellipse to the perimeter – Vincent ISOZ Jun 28 '25 at 20:38
  • Ok. $\tan(\psi)=\frac{b}{a}\tan(\phi)$ – PM 2Ring Jun 28 '25 at 20:45
  • Related: https://math.stackexchange.com/q/315386/207316 – PM 2Ring Jun 28 '25 at 20:54
  • Your equation isn't an ellipse. See https://sagecell.sagemath.org/?z=eJwtykEOgjAQBdB9T%5FHDhhZHYzAu5w4s3Bk1LVatAUqmVeH2BuPyJe9tRZe5NMoSHBh7wk4JGA4raIs1nKnamHQ251pNYMifav4hhWFBA8ZoxfY%2DS2gvYxezPk6E%2DUTQmbAl1NUYDCngtjQ%2DyMsT7hKuXRh84qK3zyiFUc0mPeJHmy%2DevywS&lang=sage – PM 2Ring Jun 28 '25 at 21:01
  • That's also what i concluded: not an ellipse. I'm highly suprised that Euler said some non-sense. Maybe by claiming it's an "ellipse" he was making an approximate analogy because the real shape doesn't have an official name. – Vincent ISOZ Jun 28 '25 at 21:04
  • Maybe it's a misprint, or something... The shape of the Earth is well-approximated by an ellipsoid. https://en.wikipedia.org/wiki/Earth_ellipsoid – PM 2Ring Jun 28 '25 at 21:08
  • I will continue to investigate... – Vincent ISOZ Jun 28 '25 at 21:15
  • Can you send a link to the text of Euler? Seems the book you cite does not exist. – Kosh M. Woldfrid Jun 28 '25 at 21:57
  • If you search in english you won’t found it because the book is in german and was never translated. I will provide you the link tomorrow – Vincent ISOZ Jun 28 '25 at 23:24

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