Let us start with the bathtub. You know the story. King Hiero II of Syracuse gives a goldsmith a lump of pure gold and commissions a new crown. The crown comes back looking beautiful, weighing exactly what it should, but the king suspects the goldsmith has secretly mixed in cheaper silver and pocketed the difference. He asks Archimedes to figure out whether the crown is pure gold, without damaging it.
Archimedes puzzles over this. Then, one day, he steps into a full bath. Water spills over the sides. And in that instant, he sees the solution. He leaps from the tub and runs through the streets of Syracuse, dripping wet, shouting “Eureka! Eureka!”: “I have found it! I have found it!”
It is probably the most famous scientific anecdote in history. It is also probably not true, at least not in its popular form. The story comes from the Roman architect Vitruvius, writing two centuries after Archimedes’ death, and the method Vitruvius describes (comparing the overflow of water displaced by the crown versus an equal weight of gold) would have been impractical for detecting a small amount of silver. The real method likely involved weighing the crown in air and then in water, comparing its buoyancy to that of pure gold.
But here is what matters: whether or not Archimedes actually ran naked through Syracuse, the science behind the story is real and profound. His treatise On Floating Bodies established the principles of hydrostatics (the physics of fluids at rest), and those principles remain as valid today as they were in the 3rd century BCE.
Archimedes in Syracuse
Archimedes lived from roughly 287 to 212 BCE in Syracuse, a Greek city-state on the island of Sicily. He was, by any measure, one of the greatest minds of the ancient world, and possibly of any era. His contributions ranged across mathematics, physics, engineering, and astronomy.
In mathematics, he calculated the most accurate approximation of pi achieved in antiquity. He determined the areas and volumes of spheres, cylinders, and parabolic segments using methods that anticipated integral calculus by nearly two millennia. In engineering, he designed war machines that terrorized the Roman fleet during the siege of Syracuse: catapults, cranes that could lift and flip enemy ships, and possibly (though this is debated) mirrors that focused sunlight to set ships on fire.
But his work on fluids may be his most enduring contribution to physics. On Floating Bodies, written in two books, was the first rigorous mathematical treatment of how objects behave in liquids. It was, in effect, the founding document of an entire branch of physics.
The Principle Itself
Archimedes’ principle, as stated in modern terms, is beautifully simple: any object submerged in a fluid experiences an upward force equal to the weight of the fluid it displaces.
That is it. One sentence. But the implications are vast.
Think about what this means in practice:
- A steel ship floats because it is shaped to displace a volume of water that weighs more than the ship itself
- A submarine dives by taking on water to increase its weight beyond the weight of water it displaces, and surfaces by expelling that water
- A hot air balloon rises because heated air is less dense than cool air, so the balloon displaces a weight of atmosphere greater than its own weight
- Ice floats on water because frozen water is slightly less dense than liquid water, a quirk that keeps lakes from freezing solid and makes life on Earth possible
- You feel lighter in a swimming pool because the water is pushing you up with a force equal to the weight of the water your body has displaced
The principle also explains the crown problem. Gold is denser than silver. A crown made of pure gold will displace less water than a crown of the same weight that contains silver (because the mixed crown, being less dense, will have a greater volume). By measuring the displacement (or better yet, by weighing the crown underwater), Archimedes could detect the fraud without cutting the crown open.
The Mathematics Behind the Intuition
What makes On Floating Bodies remarkable is not just the principle itself but the rigor with which Archimedes proved it. He did not simply assert that objects are buoyed up by the weight of fluid they displace. He demonstrated it mathematically, starting from basic axioms about how fluids behave.
His approach was geometric. In Book I, he established that the surface of a fluid at rest is spherical (centered on the center of the Earth) and proved the fundamental buoyancy principle. In Book II, he tackled a much harder problem: the stability of floating bodies, specifically paraboloids of revolution. He showed that the orientation in which a floating object settles depends on the relationship between its shape, its density, and the density of the fluid.
This second book is astonishingly sophisticated. Archimedes analyzed what happens when a floating paraboloid is tilted: does it right itself or tip over? The answer depends on the precise geometry and density ratios involved. This is essentially the problem that naval architects grapple with when designing ships that will not capsize, and Archimedes solved it with nothing more than geometry and logic, over two thousand years ago.
From Ancient Syracuse to Modern Engineering
The principles Archimedes established have never been superseded. They have been refined and extended, certainly. Fluid dynamics became vastly more complex once scientists began studying fluids in motion. The work of Daniel Bernoulli, Leonhard Euler, and the Navier-Stokes equations took hydrodynamics far beyond anything Archimedes imagined. But his fundamental insight about buoyancy remains the starting point.
Modern applications are everywhere:
- Ship design relies on Archimedes’ principle to calculate displacement and predict how a vessel will sit in the water
- Submarine engineering uses controlled buoyancy to navigate at different depths
- Hydrometers (instruments that measure the density of liquids) work on the Archimedean principle and are used in brewing, winemaking, and petroleum refining
- Geological surveys use gravity measurements (related to density and displacement) to detect underground mineral deposits
- Medical imaging techniques, including some density-based scans, echo the basic principle of relating volume to weight
Even the concept of specific gravity (the ratio of a substance’s density to the density of water) is a direct descendant of Archimedes’ crown experiment. When a jeweler tests whether a gemstone is genuine by comparing its density to a known standard, that jeweler is doing exactly what Archimedes did for King Hiero.
The Lost and Found Texts
The survival of Archimedes’ works is itself a remarkable story. Like most ancient Greek texts, the originals were lost. What we have are copies of copies, transmitted through a chain of scribes spanning centuries. On Floating Bodies survived in Latin translation, and portions were rediscovered in the famous Archimedes Palimpsest, a 10th-century Byzantine manuscript that was scraped clean and overwritten with a prayer book in the 13th century.
In 1998, the palimpsest was sold at auction to an anonymous buyer who made it available for scholarly study. Using multispectral imaging and X-ray fluorescence, researchers were able to read the original Archimedean text beneath the prayers. It was one of the great feats of modern textual recovery, revealing mathematical arguments that had been hidden for seven hundred years.
This story of transmission and recovery connects Archimedes to a much broader tradition. Ancient mathematical knowledge survived only because scholars in each generation recognized its value and took the trouble to copy and preserve it. The geometric methods Archimedes used drew on the tradition established by Euclid, whose Elements (compiled about a century before Archimedes) codified the axiomatic approach to geometry that Archimedes employed throughout his work. Kronecker Wallis’s edition of Euclid’s Elements, completing Oliver Byrne’s visionary color-coded presentation, captures the elegance of this foundational text.
Archimedes and Newton
When Isaac Newton wrote his Principia nearly two thousand years after Archimedes, he chose to present his revolutionary physics in the geometric style that Archimedes had mastered. This was not nostalgia. Newton believed that geometric proofs carried a certainty and clarity that algebraic methods could not match. In many ways, the Principia is the spiritual successor to Archimedes’ work: both used rigorous geometry to describe the physical world.
Newton also extended the study of fluids far beyond Archimedes’ static analysis. Book II of the Principia deals with the motion of bodies through resisting fluids, a problem Archimedes never tackled. Newton’s treatment was not entirely successful (fluid dynamics turned out to be fiendishly difficult), but it set the agenda for centuries of subsequent research.
For those interested in how Newton built on the Archimedean tradition, Kronecker Wallis’s handcrafted edition of Newton’s Principia is an extraordinary way to engage with one of the most important scientific works ever written.
The Death of Archimedes
Archimedes died in 212 BCE, when Roman forces under General Marcellus finally captured Syracuse after a two-year siege. According to the most common account, a Roman soldier came upon Archimedes while he was working on a mathematical diagram drawn in the sand. “Do not disturb my circles,” Archimedes supposedly said. The soldier killed him.
Marcellus, who had given orders that Archimedes be taken alive, was reportedly grief-stricken. He honored the mathematician with a proper burial and a tomb decorated, at Archimedes’ own request, with a sphere inscribed in a cylinder, representing what Archimedes considered his greatest mathematical achievement: proving that the volume of a sphere is exactly two-thirds the volume of its circumscribing cylinder.
It is a fitting epitaph for a man who saw mathematics not as abstraction but as truth about the physical world. A sphere in a cylinder. The ratio is exact, beautiful, and eternal, like the principle that a body immersed in fluid is buoyed up by the weight of the fluid it displaces.
Holding the Thread
There is something thrilling about the fact that the physics you learned in school (the Archimedes principle from your textbook) was worked out by a single person on a Mediterranean island more than 2,200 years ago. The mathematical language has changed. The applications have multiplied beyond anything Archimedes could have imagined. But the core insight has not been revised, corrected, or overturned. It was right the first time.
That kind of durability is rare in any field of human endeavor. It speaks to something profound about the nature of mathematical physics: when you get it right, you get it right forever. Archimedes understood this. Euclid understood it before him. Newton understood it after. The thread runs unbroken from Syracuse to Cambridge to the modern world.
The tradition of preserving and celebrating these foundational scientific works (of recognizing that the physical book, the diagram, the handwritten proof all carry meaning beyond their mere content) is exactly what drives projects like Kronecker Wallis’s Portraying Science. Because science is not just what we know. It is how we came to know it, who figured it out, and the beautiful, fragile documents they left behind.
So the next time you step into a bath and watch the water rise, think of Archimedes. He may not have run through the streets shouting “Eureka.” But he did something better. He wrote it down, proved it, and made sure the truth would outlast him by millennia.
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