Archimedes Calculates Pi and Dies Defending Syracuse
A method of exhaustion narrows down pi to two fractions, then a Roman soldier ends the life that discovered it
Quick facts
- Dates
- c. 287 BCE to 212 BCE
- Pi bounds given
- Between 3 10/71 and 3 1/7
- Requested tomb inscription
- Sphere inscribed in a cylinder, 2:3 ratio
- Died during
- Roman capture of Syracuse, Second Punic War
What happened
Archimedes, born around 287 BCE, perfected a method of integration, now called the method of exhaustion, that let him calculate areas, volumes, and surface areas of curved shapes long before calculus existed as a formal discipline. In Measurement of the Circle he showed that pi lies strictly between 3 10/71 and 3 1/7, a bound tight enough to be useful for engineering for the next 1,800 years. He also proved that the volume of a sphere is two-thirds the volume of its circumscribing cylinder, and that the sphere's surface area holds the same two-thirds ratio to the cylinder's surface area, a result he considered his most significant accomplishment and asked to have inscribed, alongside a diagram of a sphere and cylinder, on his own tomb. Archimedes was killed in 212 BCE during the Roman capture of Syracuse in the Second Punic War.
Why it matters
Archimedes applied the same rigorous mathematics Euclid had used for pure geometry to concrete physical and engineering problems, treating mathematics as a tool for solving real questions about circles, spheres, and levers rather than only an abstract pursuit. His method of exhaustion is the closest ancient mathematics came to integral calculus, a gap that would not be closed again until Newton and Leibniz nearly nineteen centuries later.
How we know
Several of Archimedes's own treatises survive in Greek manuscript tradition, including Measurement of the Circle and On the Sphere and Cylinder, and the circumstances of his death during the Roman siege of Syracuse are recorded by multiple ancient historians writing within a few centuries of the event.
Sources
- MacTutor History of Mathematics, University of St Andrews. Archimedes of Syracuse · Reputable sourcemathshistory.st-andrews.ac.uk · The domain "mathshistory.st-andrews.ac.uk" is on our Reputable source registry. · Link is live and its text matches the event's key terms (Jul 2026)
- MacTutor History of Mathematics, University of St Andrews. Euclid of Alexandria · Reputable sourcemathshistory.st-andrews.ac.uk · The domain "mathshistory.st-andrews.ac.uk" is on our Reputable source registry. · Link is live and its text matches the event's key terms (Jul 2026)
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Related timelines
- Ancient Greece → · See the Ancient Greece timeline for Archimedes's engineering defense of Syracuse against Rome and the wider Hellenistic scientific tradition centered on Alexandria.