The Antikythera Mechanism is the most sophisticated scientific instrument surviving from antiquity. It is an astronomical calculator with precision gearing, containing 32 bronze gears including a differential gear, and accurate to 1 part in 40,000. It was built in Rhodes in about 80 BC, and discovered in a sunken shipwreck. It has enabled historians of science to completely reassess the high technology of the ancient Greeks. The lecture will describe the machine, reveal the secrets of its intricate gearing, and prove a theorem to show how the Greeks were able to be so accurate.