Archimedes

Give Me a Place to Stand · Mathematician & Engineer · c. 287–212 BC

The geometer who proved the law of the lever from first principles, invented the idea of a centre of gravity, and explained why things float — and who considered his war machines the least interesting thing he did.

Who was Archimedes?

Archimedes lived and worked in Syracuse, a Greek city on the coast of Sicily, and corresponded with the mathematicians of Alexandria. In On the Equilibrium of Planes he proved the law of the lever — that two weights balance when their distances from the pivot are inversely proportional to them — and in doing so developed the concept of a centre of gravity, the single point at which a body's whole weight may be treated as acting. In On Floating Bodies he established that a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced. Using inscribed and circumscribed polygons of ninety-six sides he trapped the ratio of a circle's circumference to its diameter between 3¹⁰⁄₇₁ and 3¹⁄₇, and by similar methods of exhaustion he found areas and volumes that would not be recovered until the calculus. He regarded as his finest result the proof that a sphere has two-thirds the volume of the cylinder that encloses it, and asked for that figure to be carved on his tomb. King Hiero II set him practical problems and the city set him defences, and he designed machines that held off a Roman siege for two years. When Syracuse finally fell in 212 BC a Roman soldier killed him.

Major achievements

• Proved the law of the lever from stated assumptions
• Developed the concept of the centre of gravity
• Established the principle of buoyancy in On Floating Bodies
• Determined the ratio of a circle's circumference to its diameter between close bounds
• Proved the volume of a sphere to be two-thirds that of its enclosing cylinder

Life in brief

• c. 287 BC — Born in Syracuse: Son of the astronomer Phidias.
• c. 265 BC — Studies at Alexandria: Begins the correspondence that lasted his life.
• c. 250 BC — On the Equilibrium of Planes: The law of the lever, and centres of gravity.
• c. 250 BC — On Floating Bodies: Establishes the principle of buoyancy.
• c. 240 BC — Measurement of a Circle: Traps the circle ratio between two close fractions.
• 214 BC — The siege of Syracuse: His machines hold off the Roman fleet for two years.
• 212 BC — Syracuse falls: He is killed by a Roman soldier during the sack.
• 1906 — The Method recovered: Found beneath the writing of a medieval prayer book.

Explore the life of Archimedes — five chapters

1. Syracuse and the Letters to Alexandria — A mathematician on the edge of the Greek world
2. Give Me a Place to Stand — The lever proved, and the centre of gravity invented
3. On Floating Bodies — The crown, the bath, and what he actually proved
4. Squeezing the Answer — Ninety-six sides, and the theorem he wanted on his tomb
5. The Siege — Two years of machines, and a soldier with a sword

Read the five-chapter life of Archimedes →

Where Archimedes appears in your course

Archimedes has a genuine claim on 4 lessons of the Pearson Edexcel International GCSE science course built into Incandio:

• Moments and Centre of Gravity — Physics: Levers had been used for thousands of years before Archimedes; what he did was prove why they work. In On the Equilibrium of Planes he starts from a small number of assumptions — chiefly that equal weights at equal distances balance — and derives that weights balance when their distances from the pivot are inversely proportional to them, which is the principle on this page. He also developed the idea of a centre of gravity in order to do it, because a proof about extended shapes needs a single point at which each shape's weight can be treated as acting.
• Work Done — Physics: A lever lets a small force lift a large weight, which looks very much like getting something for nothing — and this page explains why it is not. The small force acts through a long distance and the large weight moves only a short one, so the product, the work, comes out the same on both sides. Archimedes proved the law of the lever from first principles and understood exactly what it does and does not give you, which makes him the ideal figure for a page whose whole content is that force and distance trade against each other.
• Density — Physics: The displacement method on this page is the oldest idea in the topic and it is attributed to him. The problem set by King Hiero was whether a crown was pure gold, which requires its density, which requires its volume — and a crown has no dimensions you can measure with a rule. Submerging it converts the impossible into the trivial, because any object pushes aside exactly its own volume of water. He is also the right person to ask about the famous story, because the version everyone knows comes from Vitruvius two centuries later and the method it describes would have been very hard to perform accurately.
• Pressure, and Pressure in Liquids — Physics: Upthrust is the consequence of everything on this page, and On Floating Bodies is where it was first established: a body immersed in a fluid is pushed up by a force equal to the weight of the fluid it displaces. That follows directly from pressure increasing with depth and acting in all directions — the pressure on the bottom of an object exceeds the pressure on its top, and the difference is the upthrust. He proved it from stated assumptions rather than noticing it, which is why the result has survived unchanged for over two thousand years.

Begin

• Start a conversation with Archimedes — a dramatised, historically grounded AI portrayal
• Read the Historical Brief
• Explore the life of Archimedes — five chapters
• Find your exam topics · Meet all 208 figures