Archimedes: A Life in Five Chapters

Chapter 1 · Syracuse and the Letters to Alexandria

A mathematician on the edge of the Greek world

c. 287 – c. 250 BC · Syracuse · Alexandria

Archimedes was born at Syracuse, a Greek city on the east coast of Sicily, around 287 BC. His father Phidias was an astronomer, and he was related in some way to Hiero II, the city's ruler — the relationship is reported but not clear.

He seems to have spent time at Alexandria as a young man, and he corresponded with mathematicians there for the rest of his life: Conon of Samos, whose death he mourned; Dositheus; Eratosthenes, who measured the Earth. Several of his surviving works are letters, opening with a greeting and a summary of what he has proved since last writing.

Those letters show a distinct working personality. He was competitive about difficulty, and he had a habit that says a good deal about him: he would sometimes send his correspondents statements of theorems *without* proofs — including, on at least one occasion by his own account, a couple of false ones, to catch out those who claimed to have found everything and produced no demonstration.

Syracuse was not the centre of the mathematical world; Alexandria was. Archimedes worked at a distance from the people who could follow him, which may be why the correspondence mattered so much, and why his results reach us as a series of finished, self-contained treatises rather than as a school's accumulating body of work.

Why this matters

Archimedes worked far from the mathematical centre of his world and kept in contact by letter, which is why his results survive as complete, self-contained treatises.

You have the letters going back and forth across the Mediterranean. What would you ask?

Ask Archimedes

• “Did you really send false theorems to catch out pretenders?”
• “What was it like working away from Alexandria?”
• “Which of your correspondents could actually follow you?”
• “What did your father the astronomer teach you?”
• “Why write your results as letters rather than as a treatise?”

Chapter 2 · Give Me a Place to Stand

The lever proved, and the centre of gravity invented

c. 250 BC · Syracuse

In *On the Equilibrium of Planes* Archimedes proved the law of the lever: two weights balance when their distances from the pivot are inversely proportional to them. Double the distance, halve the weight.

What matters is the word *proved*. Levers had been used since before recorded history, and the rule was known empirically. Archimedes derived it from stated assumptions — chiefly that equal weights at equal distances balance, and that a weight can be treated as concentrated at a single point — in the manner of a geometrical theorem.

That second assumption required a new idea: the *centre of gravity*, the single point at which a body's whole weight may be treated as acting. He located it for triangles, parallelograms and parabolic segments. It remains one of the most useful simplifications in physics; every free-body diagram since depends on it.

The famous line — give me a place to stand and I will move the Earth — is reported by later writers rather than found in his own text, but it captures the claim exactly. There is no limit in principle. Given a long enough lever and somewhere to put the fulcrum, any weight can be lifted by any force.

Hiero is said to have challenged him to demonstrate it, whereupon Archimedes used a compound pulley to draw a fully laden ship out of the harbour single-handed. The story is late. The physics is right.

“Give me a place to stand and I will move the Earth.”

— Attributed to Archimedes by Pappus of Alexandria, Synagoge, Book VIII

Why this matters

The centre of gravity — treating a whole body's weight as acting at one point — is one of the most powerful simplifications ever introduced into physics.

You have the lever proved and the ship on the beach. What is your question?

Ask Archimedes

• “Why prove the law of the lever when everybody already used levers?”
• “What is a centre of gravity, and why did you need one?”
• “Is there really no limit to what a lever can lift?”
• “Did you actually pull a loaded ship out of the water?”
• “What assumptions does your proof of the lever start from?”

Chapter 3 · On Floating Bodies

The crown, the bath, and what he actually proved

c. 250 BC · Syracuse

In *On Floating Bodies* Archimedes established that a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid it displaces. From that single statement follow the conditions for floating, sinking and stability, and he worked out the stable positions of floating paraboloids — a problem in ship design.

The famous story belongs here and needs handling carefully. Vitruvius, writing about two hundred years later, tells it: Hiero suspected a goldsmith of substituting silver for some of the gold in a votive crown, and asked Archimedes to determine this without damaging it. Getting into a bath, Archimedes noticed the water rise, understood that volume could be measured by displacement, and ran into the street shouting *eureka* — I have found it.

Archimedes never mentions any of this. And the simple displacement method described by Vitruvius would be extremely difficult in practice: the volume difference for a realistic crown is tiny, and measuring the overflow accurately enough is beyond ordinary equipment.

A far better method follows directly from his own buoyancy principle: weigh the crown in air, weigh it again submerged, and compare the loss of weight with that of an equal weight of pure gold. That requires only a balance. Whether he used it is unknown — but it is the method his own book makes available, and it is what he would have had to hand.

Why this matters

The eureka story as usually told does not work in practice, while the principle Archimedes actually proved gives a method that does — a good case for reading the source rather than the legend.

You have the bath, the crown and the principle. What would you put to him?

Ask Archimedes

• “Did the bath and the shout actually happen?”
• “How would you have tested the crown with only a balance?”
• “Why does a ship made of heavy material float at all?”
• “What makes a floating body stable rather than rolling over?”
• “Does it annoy you to be remembered for a bath?”

Chapter 4 · Squeezing the Answer

Ninety-six sides, and the theorem he wanted on his tomb

c. 240 BC · Syracuse

To find the ratio of a circle's circumference to its diameter, Archimedes drew a regular polygon inside the circle and another outside it, and calculated both perimeters. The inner one is too small, the outer too large, and the true value is trapped between them. Then he doubled the number of sides — repeatedly — until he reached ninety-six sides each way, and concluded that the ratio lies between 3¹⁰⁄₇₁ and 3¹⁄₇.

This is the method of exhaustion, and it is the ancestor of integral calculus. He used it to find the area of a parabolic segment, the volume of a sphere, the area of a spiral, and the surface of a cylinder — results that would not be recovered until the seventeenth century.

His own favourite was that a sphere has two-thirds the volume of the cylinder that just encloses it, and two-thirds of its total surface area as well. He asked for a sphere inscribed in a cylinder to be carved on his tomb. Cicero, serving as quaestor in Sicily in 75 BC, reported finding that neglected tomb overgrown with brambles and identifying it by the figure.

A further work, *The Method*, explained how he actually found these results before proving them: by imagining figures balanced on a lever and sliced into indivisible pieces. He regarded it as heuristic, not proof. It was lost, and was found in 1906 beneath the writing of a medieval prayer book — showing he had come closer to the calculus than anyone had believed.

Why this matters

The method of exhaustion is the direct ancestor of integration, and The Method shows Archimedes discovering results by one route and proving them by another.

You have ninety-six sides and a sphere in a cylinder. What would you ask?

Ask Archimedes

• “How does trapping a value between two polygons count as knowing it?”
• “Why did you want the sphere and cylinder on your tomb?”
• “How did you find these results before you could prove them?”
• “Why did you not regard The Method as a proof?”
• “How far could you have gone with ninety-six more sides?”

Chapter 5 · The Siege

Two years of machines, and a soldier with a sword

214 – 212 BC · Syracuse

When Syracuse changed sides in the Second Punic War, Rome sent Marcellus to take it. The siege lasted two years, and ancient historians — Polybius, Livy and Plutarch — attribute the delay largely to one man.

Archimedes had designed the defences. Catapults ranged for different distances so that ships were under fire at every point of approach. Cranes on the walls — the *iron hand* — dropped, gripped the prows of ships and lifted them until they capsized or swamped. Polybius, our earliest source and a serious historian, describes Roman troops so unnerved that a rope or a beam appearing over the wall would send them running.

The burning mirrors, focusing sunlight to set ships alight, appear only in much later sources and have repeatedly failed practical tests. Treat them as legend.

Plutarch says Archimedes regarded all of this engineering as beneath the dignity of mathematics, and undertook it only at Hiero's request. That may be Plutarch imposing a Platonic attitude on him; *The Method* shows a man perfectly willing to think mechanically.

The city fell in 212 BC, betrayed from within during a festival. Marcellus had given orders that Archimedes be taken alive. He was killed by a soldier — in the best-known version, while working at a diagram in the dust, having told the man not to disturb his circles. He was about seventy-five.

Why this matters

The two-year defence of Syracuse is the earliest well-documented case of a single scientist materially changing the course of a war.

You have the iron hand, the fall of the city and the diagram in the dust. What is your question?

Ask Archimedes

• “How did the iron hand actually lift a ship?”
• “Did you really think the machines beneath you?”
• “Should a mathematician build weapons for his city?”
• “What were you working on when the soldier came?”
• “Which of your results would you save if only one survived?”

What Archimedes changed

The law of the lever and the principle of buoyancy are still taught essentially in Archimedes' form, and the centre of gravity remains one of the most useful ideas in mechanics. His method of exhaustion is the direct ancestor of integral calculus, and the recovery of The Method in 1906, found beneath the writing of a medieval prayer book, showed he had come closer to the calculus than anyone had believed.

A debate that continues

The burning mirrors have repeatedly failed practical tests and are almost certainly legend; the eureka story as Vitruvius tells it would not work in practice; and Plutarch's claim that Archimedes despised his own engineering may be a later imposition.

Keep exploring — ask Archimedes

• “What would you have done with algebraic notation?”
• “Is a result discovered but unproved worth publishing?”
• “Which problem defeated you completely?”

Related lives

• Euclid — Who Made Proof The Standard
• Eratosthenes — Who Measured the World From a Shadow
• Galileo Galilei — Father of Modern Science
• Isaac Newton — Lucasian Professor · President of the Royal Society

Related themes

Moments and levers · Density and floating · Areas and volumes