On Floating Bodies

Overview

About fifty pages in two books, written by Archimedes of Syracuse around 250 BCE, and one of the founding texts of what would later be called hydrostatics, the study of fluids at rest and the forces bodies experience in them. It survives, like most of Archimedes’ work, through a complicated later manuscript history; the Greek text was lost for centuries and much of what we now have was recovered through a Latin translation and, famously, through the Archimedes Palimpsest, a medieval prayer book whose parchment had been scraped and reused, discovered to contain an erased copy of several Archimedes treatises underneath, including portions of this one, only fully deciphered with advanced imaging techniques in the early twenty-first century.

The treatise is not sold separately in most modern editions; Thomas Heath’s collected translation of Archimedes’ complete surviving works remains the standard English source.

What distinguishes this work from earlier and contemporary approaches to natural phenomena is its method. Archimedes does not proceed by cataloguing observations about floating and sinking objects and then generalizing from them. He proceeds the way Euclid proceeds in the Elements, from an explicit postulate, stated at the very opening, about the fundamental behavior of a fluid at rest, and derives everything that follows by strict geometric deduction. This makes On Floating Bodies one of the earliest surviving examples of what we would now call mathematical physics, a physical science built with the rigor of geometry rather than assembled from empirical generalization alone.

The argument

The opening postulate. Book one begins with a single postulate characterizing the behavior of a fluid at rest: that the parts of such a fluid, lying evenly and continuously, are arranged so that the part under less pressure is driven along by the part under more pressure, and that any part of the fluid is pressed by whatever lies vertically above it, if the fluid is confined and subject to compression. This is Archimedes’ foundational assumption about how a fluid transmits pressure through itself, and it plays exactly the role in this treatise that Euclid’s postulates play in the Elements: everything subsequent is derived from it by rigorous argument, not simply asserted from observation.

From this single postulate, Archimedes derives, as an early and striking result, that the surface of any fluid at rest and in equilibrium must be spherical, with its center at the center of the earth, a genuinely significant early argument for the sphericity implied by fluid equilibrium under a central force, reached from pure deduction rather than direct observation of the earth’s shape.

Floating and the principle of buoyancy. The treatise’s central results follow directly from the opening postulate, established through careful geometric argument about what must happen when a solid body is placed in a fluid.

Proposition five establishes that a solid body lighter than the fluid it is placed in will sink only until the weight of the fluid displaced by the submerged portion equals the weight of the solid itself, at which point it will float in equilibrium, submerged exactly that far and no further. This is the precise geometric statement of what it means for something to float: not merely that it stays on the surface, but that a definite, calculable relationship holds between the body’s own weight and the weight of the fluid its submerged portion displaces.

Proposition seven, the result now universally known as Archimedes’ Principle, extends this to bodies denser than the fluid, which sink to the bottom rather than floating, but which nonetheless experience a definite, calculable upward force while submerged, equal in magnitude to the weight of the fluid displaced by their volume. This is why an object weighed while submerged in water registers lighter than its true weight in air, by exactly the weight of the water it displaces, a result with direct and immediately practical application to determining the density, and therefore the composition, of an object of unknown material, which is the technical content behind the crown story later attached to Archimedes’ name, whether or not the particular bathtub anecdote itself is historically accurate.

Stability of floating bodies. The remainder of book one, and book two in its entirety, extend this foundation to increasingly sophisticated questions about the stability of floating bodies, meaning not simply whether a body floats but whether, if disturbed or tilted from its resting position, it tends to return to that position or instead capsizes further.

Book two takes up specifically the case of a floating segment of a paraboloid of revolution, a shape chosen not arbitrarily but because it provides a tractable and mathematically rich model relevant to the actual problem of hull design: how the shape of a ship’s cross-section affects whether it rights itself when rocked by waves or instead tips over. Archimedes works through, with extraordinary geometric care, how the stability of such a floating segment depends on the specific relationship between the paraboloid’s dimensions and the density of the material relative to the surrounding fluid, determining precisely under what conditions the segment floats stably in a given orientation and under what conditions it does not.

This second book represents an extension of pure mathematical physics into something close to applied naval engineering, and its level of geometric sophistication, working out stability conditions for a genuinely three-dimensional curved solid under the combined constraints of weight, buoyancy and equilibrium, was not matched by later mathematicians for many centuries after Archimedes.

How to read it

Read the opening postulate first and sit with it before moving on. Everything in the treatise is derived from this single assumption about fluid pressure, and understanding exactly what it claims, and why it is a reasonable starting assumption about how a fluid at rest actually behaves, is the key to following everything that comes after.

Work through the proof of proposition five, the basic floating condition, with a diagram in hand, even a rough one you draw yourself. Archimedes’ geometric arguments are compressed and assume considerable familiarity with the conventions of Greek mathematical proof, and a visual aid makes the logic far easier to follow than the bare prose alone.

Notice what Archimedes does not do: he does not describe any experiment, does not report any measurement, does not appeal to observed instances of objects floating or sinking as evidence. The entire treatise proceeds by deduction from the opening postulate, in exactly the spirit of Euclidean geometry applied to a physical rather than a purely abstract subject matter. This is worth pausing on as a specific and unusual methodological choice, distinct from both purely empirical natural history and from later experimental physics.

Read book two, on the stability of floating paraboloids, as evidence of how far Archimedes was willing and able to push pure mathematical analysis toward a genuinely practical engineering problem. The choice of the paraboloid specifically, rather than some more arbitrary shape, reflects real thought about what shapes actually matter for hull design, even though Archimedes frames the entire discussion in the language of pure geometry rather than practical shipbuilding advice.

Keep the manuscript history in mind as you read. This is a text that survived by the narrowest of margins, partially recovered from a medieval prayer book whose original parchment had been scraped for reuse, and the fact that we can read it in anything like completeness today is itself a remarkable story of textual transmission and, in the case of the Palimpsest, of twenty-first century imaging technology recovering what had seemed permanently lost.

Questions it raises

• Is deriving a physical science entirely from a single postulate, in the manner of Euclidean geometry, a legitimate and reliable method for physics, or does it depend on getting the initial postulate exactly right in a way that empirical testing alone can verify?
• What does it mean that Archimedes reaches a correct and still-standard physical principle, buoyancy, through pure deduction rather than experiment? Does this vindicate rationalist confidence in reasoning from first principles, or was Archimedes simply fortunate that his postulate happened to be accurate?
• How much of modern engineering intuition about hull stability was genuinely anticipated by Archimedes’ analysis of the floating paraboloid, and how much depends on factors his purely geometric treatment could not have addressed?
• Why did this level of mathematical rigor applied to physical phenomena not become the ordinary standard for natural philosophy for many centuries after Archimedes, rather than being extended and built upon relatively quickly?
• What does the treatise’s survival, through a scraped and reused medieval manuscript only fully recovered through modern imaging technology, suggest about how much of ancient mathematical and scientific achievement may still be permanently lost?
• Does the crown legend, however historically dubious in its specific details, capture something true about the kind of practical problem that mathematical physics of this sort could genuinely solve?