Archimedes was the first to systematically find the centers of gravity of various solid bodies and to apply this concept in determining stable configurations of floating bodies. In this paper, we discuss an error in a proof developed by Archimedes that involves determining whether a uniform, spherical cap will float stably with its base horizontal in a liquid on a spherical Earth. We present a simpler, corrected proof and discuss aspects of his proof regarding a spherical cap that is not uniform.

## References

A Greek manuscript dating from about the ninth century and containing both books of *On Floating Bodies* was translated into Latin by the Flemish Dominican William of Moerbeke in 1269. Traces of the Greek manuscript were lost in the 14th century, but Moerbeke's holograph remains intact in the Vatican library (*Codex Ottobonianus Latinus* 1850) (Ref. 2). Moerbeke's Latin translation was the source of all versions of *On Floating Bodies* from his time until the 20th century. Moerbeke's translation of Book I of *On Floating Bodies* was first printed in 1543 by Niccolò Fontana Tartaglia in Venice. Books I and II were first printed in 1565, independently by Curtius Troianus in Venice and by Federico Commandino in Bologna (Ref. 3). A palimpsest (Refs. 14–16) from the tenth century, discovered and edited by Heiberg (Ref. 13) in 1906, contains the only extant Greek text. The texts by Dijksterhuis (Ref. 4) and Heath (Ref. 5) contain the only translations/paraphrases of *On Floating Bodies* presently available in English, although both have very incomplete treatments of Propositions 8 and 9 of Book I.

*The Works of Archimedes with The Method of Archimedes*(Dover Publications, New York, 1953).

*The Archimedes Palimpsest*(10th century) Private collection. Contents available on the website “The Digital Archimedes Palimpsest,” last accessed September

*Oeuvres complètes de Christian Huyghens publiées par la Sociéte hollandaise des Scien*ces, Vol. XI, Travaux mathématiques, 1645–1651. La Haye: Martinus Nijhoff, 1908 (

Archimedes was not able to determine all equilibrium configurations of a floating paraboloid on a flat Earth, because he was restricted to compass-and-straightedge constructions. However, it is possible to determine all of the equilibrium positions with the use of modern analytic geometry, numerical analysis, and computers (Ref. 25).

This torque on a body is an example of a *gravity*-gradient *torque*, a phenomenon that has been known since Newton's time. For a rigid body with rotational symmetry in Archimedes' universe (*n* = 0), an approximate formula for the gravity-gradient torque *Τ* is $T\u2245(W/2R0)(rmax2\u2212rmin2)\u2009sin\u20092\theta $, where *r*_{min} is the body's minimum radius of gyration, assumed to be about its axis of rotational symmetry, and *r*_{max} is the body's maximum radius of gyration, which is about any axis perpendicular to the axis of rotational symmetry through the body's center of mass. Additionally, *W* is the weight of the body if its entire mass were concentrated at its center of mass, *R*_{0} is the distance from the geocenter to the body's center of mass, and *θ* is the clockwise angle from the vertical to the body's axis of rotational symmetry. This formula is an excellent approximation if *R*_{0} is much larger than the dimensions of the solid. In his *Principia Mathematica* (Book III, Proposition 39, Ref. 18). Newton showed that the gravity-gradient torque exerted by the sun and the moon on Earth causes the precession of the equinoxes (Refs. 27–32). Later, the gravity-gradient torque exerted by Earth on the moon was shown to explain why the moon always points the same face toward Earth, a phenomenon called *tidal locking* (Refs. 28–32). Gravity-gradient torques have been exploited since the late 1960s to keep the axes of minimum moment of inertia of artificial near-Earth satellites always pointing towards Earth (Ref. 33).

Because of Earth's equatorial bulge, its rotational axis is its axis of maximum moment of inertia and any line through the geocenter perpendicular to its rotational axis is an axis of minimum moment of inertia. The central gravitational fields of the sun and moon exert separate gravity-gradient torques tending to orient Earth so that its equatorial plane passes through the sun or moon. Then, because the moon and sun revolve about Earth (relative to an observer on Earth), the effect of the two torques is to cause Earth's rotational axis to slowly precess about a line perpendicular to the ecliptic plane with a period of about 26,000 years.

For a hemisphere of uniform density and radius *r*, the center of mass is along its axis of symmetry a distance 3*r*/8 from its base, a fact first demonstrated by Archimedes (*The Method*, Ref. 5, p. 27). The maximum moment of inertia through the hemisphere's center of mass is about its axis of symmetry and its minimum moment of inertia through its center of mass is about any line parallel to its base. The corresponding moments are *I*_{max} = 2*mr*^{2}/5 and *I*_{min} = 83*mr*^{2}/320, where *m* is the mass of the hemisphere. For the lever in Appendix A, its moment of inertia about the line through its two point masses is zero, and so this line is its axis of minimum moment of inertia. Thus, as Fig. 9(c) illustrates, the lever will be in stable equilibrium when this line passes through the geocenter.

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