Archimedes: The Greatest Mathematician of the Ancient World

Archimedes of Syracuse calculated pi more accurately than anyone before him, found the volume of a sphere, explained why ships float and designed machines that held off a Roman army. Working more than 2,200 years ago without algebra or calculus, he reached results that would not be matched until the 1600s, which is why he is often ranked alongside Newton and Gauss as one of the greatest mathematicians who ever lived.

Life in Syracuse

Archimedes was born around 287 BCE in Syracuse, a powerful Greek city on the island of Sicily. His father, Phidias, was an astronomer. As a young man he probably studied in Alexandria, the great centre of learning founded by Alexander the Great, and he kept up a correspondence with scholars there, including Eratosthenes, for the rest of his life.

He spent most of his career in Syracuse, close to its ruler, King Hieron II. During the Second Punic War, Rome besieged Syracuse from around 214 to 212 BCE. Ancient historians such as Plutarch and Polybius describe how Archimedes' catapults and giant cranes, the "claw of Archimedes", lifted Roman ships out of the water. When the city finally fell in 212 BCE, Archimedes was killed by a Roman soldier, reportedly while drawing geometric figures in the sand, despite orders from the Roman general Marcellus that he be spared.

Eureka! The Golden Crown

The most famous story about Archimedes comes from the Roman writer Vitruvius, around two centuries later, so it may be embellished. King Hieron gave a goldsmith pure gold to make a crown, then suspected the goldsmith had kept some gold and mixed in cheaper silver. Archimedes had to find out without damaging the crown.

As he stepped into a full bath and saw the water overflow, he realised that a body pushes aside its own volume of water. Since silver is less dense than gold, a crown of mixed metal would have a larger volume than pure gold of the same mass. He is said to have run through the streets shouting "Eureka!" ("I have found it!").

Worked Example 1: Testing the crown

Suppose the crown has mass 1000 g and displaces 60 cm³ of water. Use densities of 19.3 g/cm³ for gold and 10.5 g/cm³ for silver.

Step 1: Volume of 1000 g of pure gold:

$$ V = \frac{m}{\rho} = \frac{1000}{19.3} \approx 51.8 \text{ cm}^3 $$

The crown displaces 60 cm³, which is more than 51.8 cm³, so it is not pure gold.

Step 2: Find how much gold it contains. Let the mass of gold be \( g \) grams, so the silver is \( 1000 - g \) grams. Total volume:

$$ \frac{g}{19.3} + \frac{1000 - g}{10.5} = 60 $$

Step 3: Expand using \( \tfrac{1}{19.3} \approx 0.05181 \) and \( \tfrac{1000}{10.5} \approx 95.24 \), \( \tfrac{1}{10.5} \approx 0.09524 \):

$$ 0.05181g + 95.24 - 0.09524g = 60 $$

$$ -0.04343g = -35.24 $$

$$ g \approx 811 \text{ g} $$

So the crown would contain about 811 g of gold and 189 g of silver. Sanity check: \( \tfrac{811}{19.3} + \tfrac{189}{10.5} \approx 42.0 + 18.0 = 60.0 \) cm³. Correct.

Historians doubt Archimedes used exactly this overflow method, because the volume differences are tiny. He more likely used his principle of buoyancy, comparing the crown and an equal mass of gold on a balance under water, which is much more sensitive.

Archimedes' Principle and the Lever

In his book On Floating Bodies, Archimedes stated that a body immersed in a fluid experiences an upward force equal to the weight of the fluid it displaces. This explains why a steel ship floats and is the foundation of hydrostatics.

In On the Equilibrium of Planes, he proved the law of the lever: two weights balance when their distances from the pivot are inversely proportional to their weights. A 10 kg mass 1 m from the pivot balances a 2 kg mass 5 m away, because \( 10 \times 1 = 2 \times 5 \). This led to his famous boast, recorded by later writers: "Give me a place to stand, and I will move the Earth."

Measuring the Circle: Bounds on Pi

In Measurement of a Circle, Archimedes trapped the value of pi between two polygons. A polygon drawn inside a circle has a perimeter smaller than the circumference; a polygon drawn outside has a larger one. As the number of sides increases, both get closer to the circle.

Worked Example 2: Pi from hexagons and 12-gons

Take a circle of diameter 1, so its circumference is \( \pi \).

Inscribed hexagon: A regular hexagon inside a circle of radius \( r \) has sides equal to \( r \). With \( r = \tfrac{1}{2} \), the perimeter is

$$ 6 \times \tfrac{1}{2} = 3 \quad\Rightarrow\quad \pi > 3 $$

Circumscribed hexagon: Each side equals \( \tfrac{2r}{\sqrt{3}} \). With \( r = \tfrac{1}{2} \):

$$ 6 \times \frac{1}{\sqrt{3}} = \frac{6}{1.7321} \approx 3.464 \quad\Rightarrow\quad \pi < 3.464 $$

Doubling to 12 sides: For a regular \( n \)-gon around a circle of diameter 1, the inscribed perimeter is \( n\sin\left(\tfrac{180^\circ}{n}\right) \) and the circumscribed perimeter is \( n\tan\left(\tfrac{180^\circ}{n}\right) \). For \( n = 12 \):

$$ 12 \sin 15^\circ = 12 \times 0.25882 \approx 3.1058 $$

$$ 12 \tan 15^\circ = 12 \times 0.26795 \approx 3.2154 $$

So \( 3.1058 < \pi < 3.2154 \). Archimedes had no trigonometric tables; he computed each step by hand using clever square-root approximations. He kept doubling: 24, 48 and finally 96 sides, where the perimeters are about 3.1410 and 3.1427. He stated the result as

$$ 3\tfrac{10}{71} < \pi < 3\tfrac{1}{7} $$

that is, \( 3.1408 < \pi < 3.1429 \). The upper bound, \( \tfrac{22}{7} \), is still used in school today.

The Sphere and the Cylinder

Archimedes considered his greatest achievement the discovery, in On the Sphere and Cylinder, that a sphere has exactly two-thirds of the volume and two-thirds of the surface area of the smallest cylinder that contains it. In modern terms, for a sphere of radius \( r \) inside a cylinder of radius \( r \) and height \( 2r \):

$$ V_{\text{sphere}} = \frac{2}{3} \times \pi r^2 (2r) = \frac{4}{3}\pi r^3 $$

He asked for a sphere inside a cylinder to be carved on his tomb. In 75 BCE, the Roman orator Cicero found the overgrown tomb in Syracuse by spotting this very carving.

The Lost Method

In 1906 the Danish scholar Johan Ludvig Heiberg examined a 13th-century prayer book in Constantinople and found that it was a palimpsest: the parchment had been scraped and reused, and underneath were faint copies of Archimedes' works, including The Method, previously thought lost. It showed that Archimedes discovered results by imagining shapes sliced into infinitely many thin strips and balanced on a lever, an approach strikingly close to integral calculus.

Did You Know?

• In The Sand Reckoner, Archimedes invented a system for naming enormous numbers and estimated that about \( 10^{63} \) grains of sand would fill the universe as he understood it.
• The face of Archimedes appears on the Fields Medal, the highest honour in mathematics.
• The "Archimedes screw", a rotating spiral for lifting water, is still used in irrigation and wastewater plants.

Key Takeaways

• Archimedes (around 287–212 BCE) worked in Syracuse and was killed when the Romans captured the city.
• His principle of buoyancy and law of the lever founded hydrostatics and statics.
• Using 96-sided polygons, he proved \( 3\tfrac{10}{71} < \pi < 3\tfrac{1}{7} \).
• He showed a sphere has two-thirds the volume of its enclosing cylinder.
• His Method, rediscovered in 1906, used slicing ideas that anticipated integral calculus.

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