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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Archimedes is said to have discovered buoyancy while he was taking a bath. He did not, however, discover the city of Bath.
Buoyancy always acts in the direction opposite to the direction of the gravitational force. We normally consider this direction as 'upward'.
buoyancy
Legend says that Archimedes discovered the principle of displacement while stepping into a full bath. He realized that the water that ran over equaled in volume the submerged part of his body. Through further experiments, he deduced the above mentioned Archimedes' principle.
People knew that some things floated in water long before writing was invented. The first person to establish the rule that controlled buoyancy was Archimedes.