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.
Deep Dive into 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.
Archimedes' treatise "On Floating Bodies" (its customary Greek title is Πǫρὶ 'oχoυµǫνoν, which literally means "about hovering things", see [1]) has been highly esteemed by mathematicians over centuries. Book 2 of this treatise can be considered as a sort of crown of Archimedes' work. In this book he applies a number of his principal results about volumes and centers of gravity to a problem which is extremely difficult to handle under Greek premises: namely, the determination of the possible equilibrium positions of a floating paraboloid segment (for details see Section 1 below).
It seems, however, that even Archimedes was not in a position to treat this problem in full generality, since he restricts himself to the cases when the basis circle of the segment either lies outside of the fluid (i.e., the fluid touches this circle in at most one point) or is completely submerged.
The case not considered by Archimedes occurs when the basis circle is partially submerged and partially not. It is, indeed, of a different nature than the “archimedean” case. Whereas Archimedes gives ruler and compass constructions for the “tilt angle” of the segment, results of this kind cannot be expected in the “non-archimedean” case. However, it is possible to establish a mathematical model for this case, based on two equations E = 0 (the equilibrium condition) and F = 0 (the floating condition). Unlike the corresponding equations in Archimedes’ situation (see Section 1), E and F are no more purely algebraic expressions but also involve the function arctan(x). Nevertheless, these expressions are rather simple if established with care (see Theorem 1), and they do not involve “page-long monstrosities” of which Rorres [8] is warning.
The said paper of Rorres contains a graphic completion of the “equilibrium surface” (based on numerical integration) together with interesting observations of physical phenomena and inspiring examples (one of which we repeat here in a treatment different of his, see Example 1). However, [8] does not contain any closed formulas that would describe the non-archimedean case (which is what we expect from a mathematical model). From this point of view [8] appears only as a first step towards a completion of Archimedes’ treatise. We hope that the present paper forms a second step. Such a step also requires simple tools by which one can decide whether an equilibrium position is stable or unstable. In this connection our above equations E = 0 and F = 0 are, again, quite helpful, since they imply that the Hesse matrix of the potential function looks fairly simple for equilibria (see Section 5).
Maybe the most interesting question in the non-archimedean case concerns the number of possible equilibria for a paraboloid segment of a given shape and a given (relative) density. We have no mathematically rigorous answer to this question (which would constitute a third step). But we solve a related problem, namely, we determine the number of possible equilibria if the shape of the segment and the size of the submerged part of the basis are given, see Theorem 2. Combined with other devices like Proposition 2, this theorem allows finding “all” possible equilibria for a segment of given shape and density -not in the strict sense of the word but in a convincing manner, as we think.
The standard English translation of Archimedes’ treatise seems to be that of Heath [2] from 1897, which is based on Heiberg’s first edition of the Greek text. The German version [3] (with helpful notes) is a translation of Heiberg’s second edition [1], which considers the important Constantinople palimpsest discovered in 1899. We highly recommend the monograph [5] about Archimedes and his works. For a survey of the treatise on floating bodies the reader may also consult [9]. A number of problems of floating homogenous bodies are studied in [4] and [6], works which also provide a wider theoretical background than we use here. For additional important references see [8].
Throughout this paper we denote subsets of R 3 in an abbreviated way; for instance {x ≤ ay + b, y ≥ c, z ≤ dx 2 } stands for {(x, y, z) ∈ R 3 : x ≤ ay + b, y ≥ c, z ≤ dx 2 }.
In this section we give a modern paraphrase of Archimedes’ results (see [1], Lib. II) and, thereby, introduce some basic notations. It suffices to consider the fixed paraboloid {z = x 2 + y 2 } that arises from the parabola {y = 0, z = x 2 } in the xz-plane by rotation around the z-axis. Our paraboloid segment P is defined by
where a is the length of the axis {x = y = 0, 0 ≤ z ≤ a} of the segment. Hence the parameter (in the usual sense) of the rotating parabola equals 1/2 and the geometric properties of P are completely determined by a. The surface of the fluid is a plane
given by the parameters b, c. We assume b ≤ 0 throughout this paper. The segment P is said to be in right hand position if P ∩ {z > bx + c} lies outside the fluid (so in Diagram
In the case considered by Archimedes the intersection of
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
