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I believe Archimedes (a Greek mathematician) discovered buoyancy in 212-250 B.C.. He was hired by the king of Syracuse on the island of Sicily, to find out if his crown was made of true gold or just another metal with a similar physical state. He took a bath one day and realized that his mass displaced the water in the bath tub when he sat down. He then measured a bar of true gold with the same mass as the crown comparability in water. The goldsmith had made the crown with both gold and metal. For a full explanation of how the buoyancy principle works go to the related question "What is the buoyancy principle?" in the Related Questions section below.

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Q: Who discovered buoyancy?
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