As Roman soldiers stormed Syracuse, a 75-year-old Archimedes reportedly told one of them not to disturb his circles drawn in the dust, and was killed on the spot. This episode builds a fuller portrait of the mathematician and engineer who lived from about 287 to 212 BC and bridged pure theory with practical invention.

We explore his correspondence with scholars in Alexandria, his work for King Hiero II, the golden wreath problem and the hydrostatic balance, launching the Syracusia with compound pulleys, and the war machines that held off Rome during the Second Punic War. We also discuss the method of exhaustion, The Sand Reckoner, and the rediscovered Archimedes Palimpsest.

• Why the naked Eureka bathtub story may be less plausible than the hydrostatic balance
• How the law of the lever made moving massive weights a matter of mathematics
• The catapults, cranes, and Claw of Archimedes that terrified the Roman fleet
• Why the burning mirror death ray likely never happened
• How The Method revealed he used imagined physics to discover mathematical truths