Archimedes’ floating bodies on a spherical Earth Chris Rorresa) Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, Pennsylvania 19104 (Received 29 January 2015; accepted 14 October 2015) 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. VC 2016 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4934660] I. BACKGROUND We shall call the center of the sphere from which the cap is formed the cap’s sphere-center. If the cutting plane passes Archimedes’ Law of the Lever and Law of Buoyancy are through the sphere-center, either cap cut off is called a two of the most fundamental laws of nature. In two of his hemisphere. The flat portion of the surface of a cap is called works (On the Equilibrium of Planes I & II and On Floating 1–5 its base and the line through the sphere-center and perpen- Bodies I & II), Archimedes formulated these laws, dicular to its base is called its axis. deduced them from more fundamental principles, and In his Propositions 8 and 9, Archimedes assumed that applied them to determining the centers of gravity of various Earth was a sphere and that a body of liquid on it would rigid bodies and determining the stability configuration of assume a spherical shape, in accordance with his Proposition assorted floating bodies. These works of Archimedes are 2, On Floating Bodies I (cf., Ref. 5, p. 254). For a particular regarded as the foundations of the fields of static mechanics 6–10 cap, the vertical direction was determined by a line through and hydrostatics. its sphere-center and the center of Earth (the geocenter). He In discussing Archimedes’ results on floating bodies, we regarded a cap to be floating horizontally when its base was will use certain modern language; for example, we refer to perpendicular to the vertical line. uniform and central gravitational fields and describe a body Archimedes assumed throughout Book I that a particle is as having a weight vector pointing toward the center of attracted toward the center of a spherical Earth and that the Earth, although Archimedes and his contemporaries did not use or think in these terms. The use of such modern terminol- ogy, however, is for the convenience of the modern reader and does not invalidate the points we are trying to make. Also, although Archimedes and his colleagues were well aware that Earth was spherical and that objects fall toward its center, in most of his works, he tacitly assumed that Earth was flat. That is, he assumed that the forces of gravity on separate pieces of a rigid body were parallel to each other and consequently that the magnitude of the resultant force was the sum of the individual magnitudes. He also assumed that the magnitude of the gravitational force on a body—its weight—did not change with the position or orientation of the body. As we would say today, Archimedes assumed that the rigid bodies he was examining were in a uniform gravita- tional field. The one instance in his extant works where Archimedes did not assume a uniform gravitational field is in Propositions 8 and 9 at the end of his On Floating Bodies I (Ref. 5, pp. 261–262). These two propositions state that a solid body in the shape of a spherical cap will float stably on the surface of a spherical Earth with its base horizontal, ei- ther when the base is completely above the liquid level (Proposition 8) or below the liquid level (Proposition 9). Figure 1 shows photographs of such a body floating in water in these two horizontal-base positions. To be precise, by a spherical cap, we mean either piece of a solid sphere (or ball) cut off by a plane. A spherical cap is also called a spherical dome, spherical segment, spherical section,ortruncated sphere, although these usages are not consistent among authors. For brevity, we shall call a spheri- Fig. 1. Two photographs (taken by the author) of a wooden cap floating sta- cal cap simply a cap. bly in water. 61 Am. J. Phys. 84 (1), January 2016 http://aapt.org/ajp VC 2016 American Association of Physics Teachers 61 magnitude of the force does not vary with the position of the Newton works brilliantly for objects travelling at speeds particle or its distance from the geocenter. We make the much less than the speed of light, the mechanics of same assumption, referring to this situation as Archimedes’ Archimedes (i.e., that the line of the gravitational force universe, as opposed to Newton’s universe in which the grav- always passes through the center of mass) works brilliantly itational force varies with the inverse square of the distance for objects in a non-uniform gravitational field if the varia- of the particle from the geocenter. However, all of the results tion in the field is small with respect to the size of the object. in this paper are qualitatively true for both universes. (See Archimedes’ experience with objects moving in Earth’s cen- Ref. 11 for a derivation of Archimedes’ principle in a non- tral gravitational field was only with objects small with uniform gravitational field.) respect to Earth and such objects can be treated as if they Archimedes made a fundamental mistake in his proof of were in a uniform gravitational field. It would have been just Propositions 8 and 9, a mistake that is quite often made even as difficult for Archimedes in his day to develop the mechan- today. He assumed that the line of action of the gravitational ics of Newton as it was for Newton in his day to develop the force on a body in a spherical (or central) gravitational field mechanics of Einstein. Nevertheless, it is important for the goes through the body’s center of mass. This has been known modern reader to understand the differences among the laws to not be true since the time of Newton and a simple example of mechanics developed by these three individuals who lived is given in Appendix A. In Sec. II, it is shown how this centuries apart. mistake invalidates Archimedes’ proof. In his works, Archimedes uses the Greek phrase jesqo II. ARCHIMEDES’ PROOF OF PROPOSITION 8 sov~ baqe o1 (kentron tou bareos), which is usually translated 12 as “center of gravity” and occasionally as “center of mass.” Part I of supplementary material contains an English However, in this paper, we shall translate it as “center of translation of Proposition 8 and its proof from the Greek text 13 14–16 weight.” The word baq o1 (baros) has the meaning of weight, as edited by Heiberg from the Archimedes palimpsest. bulk, mass, heaviness, burden, etc., and any of the transla- Figure 2 displays digital scans of three diagrams illustrating tions is valid depending on context. In modern physics, only Archimedes’ proof, from a 1565 edition of Federico “center of gravity” and “center of mass” are used. The trans- Commandino.17 Commandino illustrated three cases: when lation “center of gravity” should only be used in the context the cap is (a) more than a hemisphere; (b) equal to a hemi- of a uniform gravitational field. In a uniform gravitational sphere; and (c) less than a hemisphere. field, the center of gravity and the center of mass of a body Archimedes gave a detailed proof only for case (a) and in reduce to the same thing, namely, a fixed point in the body his last line states that cases (b) and (c) can be proved simi- through which the line of action of the gravitational force on larly. Heiberg did not include diagrams for the cases when the body passes, regardless of the position or orientation of the base is above the fluid level, but Fig. 3(a) is our diagram the body. In a non-uniform gravitational field, the center of of case (a) based on Heiberg’s Greek text13 and mass remains the same point (its definition does not depend Commandino’s diagram. on the existence of a gravitational field), but the concept of a Figure 3(a) is a cross-section through the geocenter K of center of gravity no longer exists, as is discussed in detail Earth ABCD and the sphere-center K that is perpendicular to later on. In a central gravitational field, the line of action of the base EHH of the cap EZHH. The line HZ is the axis of the gravitational force on a body depends on the position and the cap and the line KK defines the vertical direction. The orientation of the body and generally these lines of action do axis HZ is shown tilted from the vertical direction with the not all pass through a fixed point in the body. In fact, a gravi- base completely above the liquid level. In his proof, tational field is uniform if and only if for any rigid body there Archimedes attempts to prove that the forces acting on the is a fixed point within it through which all lines of action of cap will cause it to rotate counterclockwise so that its axis the gravitational force pass, regardless of the position or ori- becomes aligned with the vertical (so that its base ends up entation of the body.