Towards a Completion of Archimedes Treatise on Floating Bodies

In his treatise on floating bodies 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 a recent article of Rorres.

💡 Research Summary

Archimedes’ treatise on floating bodies famously derived the equilibrium positions of a paraboloid segment when the segment’s base is either completely above the fluid or entirely submerged. The intermediate situation—where the paraboloid is partially immersed—was left untouched, creating a long‑standing gap in the classical theory. The present paper fills this gap by constructing a rigorous mathematical model for the “partial immersion” case, deriving two simple closed‑form conditions that fully characterize equilibrium, and providing reliable computational tools for locating and classifying all possible equilibria.

The authors begin by parametrizing the paraboloid segment: its axis aligns with the vertical z‑direction, the full height is (a), and the fluid surface cuts the segment at a height (h) (0 ≤ h ≤ a). The intersection curve is a circle of radius (r=\sqrt{2ah-h^{2}}). Using cylindrical coordinates, the submerged volume is expressed as
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