The earth, if it were made of constant density fluid ("water") would presumably be a "Maclaurin [axisymmetric oblate] ellipsoid." Its dimensions were worked out already to good approximation by Newton in the Principia, but it was Colin Maclaurin (1698-1746) who provided exact solution and realized it was ellipsoid. Actually, however, both were wrong, in the sense that their prediction is that the eccentricity of Earth's ellipse is 1/233 whereas in fact it is 1/294. The reason is that the Earth does not have constant density; the inner part is considerably higher density. I do not know whether anybody has worked out the correct shape predictions for an inhomogenous planet like Earth, e.g. with a more realistic equation of state for the "water," but certainly I doubt anybody ever did it in closed form like Maclaurin did. It's really a miracle that the closed form solutions exist. It would be interesting to determine possible equations of state under which more such miracles occur... probably very few. Here's another thing Maclaurin did: https://en.wikipedia.org/wiki/Trisectrix_of_Maclaurin -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)