On this page
Everyone knows the story: Archimedes steps into a bath, sees the water rise, and runs through the streets shouting "Eureka". Fewer people know that the story was written down by a Roman architect long after Archimedes died, while the actual law of buoyancy is in a book Archimedes wrote himself. This post separates the two and works through the physics you need from each.
What do we actually know about Archimedes?
Much less than we would like. The MacTutor biography of Archimedes from the University of St Andrews gives his dates as 287 BC to 212 BC, both in Syracuse in Sicily. A biography written by his friend Heracleides is lost. Most of what we know of his life comes from his own prefaces and from later writers such as Plutarch and Livy.
He was killed in 212 BC when the Romans captured Syracuse. Plutarch recorded three different versions of how it happened, which tells you how uncertain even that is.
His books are on firmer ground. MacTutor lists the works that survive, including *On Plane Equilibriums*, *On Floating Bodies*, *On the Sphere and Cylinder* and *The Sandreckoner*. A tenth-century manuscript containing *The Method*, which shows how he discovered results before proving them, was found by J. L. Heiberg in 1906.
Where does the Eureka story come from?
It comes from Vitruvius, a Roman architect whose *The Ten Books on Architecture* is generally dated, as its translator's preface notes, to the time of the emperor Augustus, close to two centuries after Archimedes' death. In Book IX of Morgan's translation, Vitruvius says King Hiero suspected that a goldsmith had replaced some gold in a votive crown with an equal weight of silver. Archimedes, getting into a bath, noticed that the deeper he sank the more water ran out, and ran home crying "Eureka, eureka".
Look at the method Vitruvius then describes. Archimedes made a lump of gold and a lump of silver, each the same weight as the crown, and measured how much water each pushed out of a full vessel. The crown pushed out more than the gold. That is a comparison of volumes, not the buoyancy principle. The story may be true, partly true or invented; there is no way to check it.
What does On Floating Bodies actually say?
Here we have Archimedes' own words, in T. L. Heath's translation of 1897. Proposition 5 says a solid lighter than a fluid will sink until the weight of the solid equals the weight of the fluid displaced. Proposition 7 says:
"A solid heavier than a fluid will, if placed in it, descend to the bottom of the fluid, and the solid will, when weighed in the fluid, be lighter than its true weight by the weight of the fluid displaced."
That is Archimedes' principle as you use it: buoyant force = weight of displaced fluid.
Worked example: could he catch the goldsmith?
Use densities from the Royal Society of Chemistry: gold 19.3 g/cm³ and silver 10.5 g/cm³. Water is 1.00 g/cm³. Assume the volumes of gold and silver simply add.
Question. A crown has a mass of 1000 g. Compare a pure gold crown with one made of 700 g gold and 300 g silver.
Pure gold volume = 1000 ÷ 19.3 ≈ 51.8 cm³.
Alloy volume = 700 ÷ 19.3 + 300 ÷ 10.5 ≈ 36.3 + 28.6 = 64.8 cm³.
Extra water displaced by the alloy ≈ 13 cm³, about 13 g.
Using Proposition 7, the apparent mass in water is 1000 − 51.8 = 948.2 g for gold and 1000 − 64.8 = 935.2 g for the alloy.
A 13 g difference is easy to see on a balance with the crown hanging in water. Measuring 13 cm³ of overflow from a vessel big enough for a crown would be much harder. The common exam mistake is subtracting the volume instead of the weight of displaced fluid: in SI units the buoyant force is ρ_fluid × V × g, in newtons.
Worked example: the law of the lever
In *On Plane Equilibriums*, Propositions 6 and 7 state that two magnitudes balance at distances reciprocally proportional to the magnitudes. In modern form, W₁d₁ = W₂d₂: the torques balance.
Question. A 600 N stone sits 0.2 m from the fulcrum of a rigid bar. How much force is needed 1.2 m from the fulcrum on the other side?
F × 1.2 = 600 × 0.2, so F = 120 ÷ 1.2 = 100 N.
The famous boast about moving the Earth reaches us through Plutarch, who wrote that Archimedes claimed any given weight could be moved by a given force, and that if there were another earth, he could go to it and move this one. MacTutor quotes the passage, but it is Plutarch's account, not a line from Archimedes' own books.
Where does buoyancy appear in your syllabus?
In NCERT's Class 11 chapter on fluids, buoyancy appears inside the terminal velocity of a falling drop, where viscous force plus buoyant force balance gravity. Expect it in fluid statics and viscosity problems. Our post on common mistakes with fluids and Bernoulli's equation covers more.
What can a student take from Archimedes' way of working?
Discover, then prove. In *The Method*, he describes finding results by a mechanical method and proving them by geometry afterwards. Guess with a quick sketch, then check with a full argument.
State the principle exactly. Proposition 7 says weight of fluid displaced, not volume. Precise wording prevents precise mistakes.
Ask where a story comes from. Knowing the source is part of knowing the science.
If the forces on a submerged object are hard to picture, the Fermi Simulation Lab has interactive simulations for ideas easier to see than read.
Try it now: find the apparent mass in water of the crown if it held 500 g of silver.
