The most famous story about Archimedes does not appear in any of his books. It was told two centuries later by the Roman architect Vitruvius, in book IX of his treatise On Architecture.

Hiero’s crown

King Hiero had commissioned a votive crown to offer to the gods and had given the goldsmith an exact quantity of gold. The crown was finished and weighed the same as the gold supplied. But a rumour spread that the goldsmith had kept part of the gold and put the same mass of silver in its place. Hiero asked Archimedes to find out the truth without damaging the crown.

According to Vitruvius, Archimedes was thinking about the problem when he got into a full bath and saw the water overflow as his body sank. He understood at once what to do and ran home naked, shouting in Greek heúreka, heúreka: “I have found it, I have found it”.

Vitruvius’s method

The idea, in Vitruvius’s version, is to compare volumes. Silver is less dense than gold: the same mass of silver takes up almost twice the space. If the crown has silver mixed in, it will have a larger volume than a block of pure gold of the same mass. To measure the volume of an irregular object, one simply immerses it in a vessel filled to the brim and measures the water that overflows.

Archimedes is supposed to have immersed a block of pure gold with the mass of the crown, then one of silver, then the crown itself. The crown spilled more water than the pure gold: the goldsmith had indeed cheated.

Why it hardly works

In 1586 the young Galileo Galilei wrote a short text, The Little Balance, pointing out the problem: the difference in volume is tiny, and measuring the spilled water precisely enough would have been almost impossible with the vessels of the time. Do the calculation in the exercise below. Galileo proposed that Archimedes would have used a much more sensitive method, based on a result he really did prove.

Archimedes’s principle

In the treatise On Floating Bodies, Archimedes proves that a body immersed in a liquid is pushed upward by a force equal to the weight of the liquid displaced. A body denser than the liquid sinks, but seems lighter; a body less dense floats, with a submerged part such that the displaced liquid weighs as much as the whole body.

The method suggested by Galileo uses a balance. Hang the crown on one pan and a block of pure gold of the same mass on the other. In air, the balance is level. Now immerse both in water. If the crown has a larger volume, it displaces more water, receives a larger upthrust and rises: the balance tips towards the pure gold. The difference in upthrust acts directly on the balance, and an ordinary balance is far more sensitive than a water level.

In modern terms: a body of mass $m$ and volume $V$ immersed in water (density $\rho_a = 1$ g/cm³) has apparent weight

$$m - \rho_a V.$$
Exercise: the volume of the crown

The crown has a mass of 1000 g. Gold has a density of 19.3 g/cm³ and silver 10.5 g/cm³. What is the volume of a crown of pure gold? And of a crown with 700 g of gold and 300 g of silver? If the vessel has a circular mouth 20 cm in diameter, how much higher will the water rise?

Pure gold: $\frac{1000}{19.3} \approx 51.8$ cm³. Mixed crown: $\frac{700}{19.3} + \frac{300}{10.5} \approx 36.3 + 28.6 = 64.8$ cm³. The difference is about 13 cm³. The mouth of the vessel has area $\pi \cdot 10^2 \approx 314$ cm², so the water would rise $\frac{13}{314} \approx 0.04$ cm more: less than half a millimetre. That is why Galileo doubted Vitruvius’s version.

Exercise: the balance in water

With the same numbers, what is the apparent weight in water of the pure gold crown and of the mixed crown? What is the difference?

Apparent weight = mass − volume (in grams, with $\rho_a = 1$). Pure gold: $1000 - 51.8 = 948.2$ g. Mixed crown: $1000 - 64.8 = 935.2$ g. The difference is 13 g, which a pan balance shows easily. With Archimedes’s principle, the fraud stands out clearly.

Sources: Vitruvius, De architectura, IX, preface, 9–12 · Archimedes, On Floating Bodies, book I, propositions 5–7 · Galileo Galilei, La bilancetta (1586) · Chris Rorres, Archimedes (website of the Courant Institute, New York University) · Thomas L. Heath, The Works of Archimedes (1897).

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