In the said treatise Archimedes determines the equilibrium positions of a floating paraboloid segment, but only in the case when the basis of the segment is either completely outside of the fluid or completely submerged. Here we give a mathematical model for the remaining case, i.e., two simple conditions which describe the equilibria in closed form. We provide tools for finding all equilibria in a reliable way and for the classification of these equilibria. This paper can be considered as a continuation of Rorres's article [C. Rorres, Completing book II of Archimedes's on floating bodies, Math. Intell. 26(3) (2004) 32-42]. © 2007 Elsevier GmbH. All rights reserved.
Girstmair, K., & Kirchner, G. (2008). Towards a completion of Archimedes’ treatise on floating bodies. Expositiones Mathematicae, 26(3), 219–236. https://doi.org/10.1016/j.exmath.2007.11.002