*Any object, wholly or partly immersed in a
fluid, is buoyed up by a force equal to the weight of the
fluid displaced by the object*

Archimedes' Principle was stated by the Greek mathematician Archimedes
of Syracuse (287-212 B.C.). Tradition tells us that King Heiro of
Syracuse was a close friend of Archimedes. Oftentimes Heiro advice the
mathematician to solve technical problems. One day, Heiro summoned a goldsmith to make a golden crown, and he
sent an exact amount of gold to the craftsman. However, upon returning
the crown, Heiro became suspicious about the amount of gold that was
used: perhaps the goldsmith kept apart some of the gold, and mixed the
remainder with a cheaper metal to make the crown. King Heiro summoned
Archimedes to determine whether the goldsmith had cheated him. The problem
proved seemed to be impossible, because nothing was known about
chemical analysis.

The solution came one night, when Archimedes took a
bath. The tub was filled to the brim, and as he submerged himself into
the water, he realized that the amount of water spilled was equal to his
own volume. Archimedes could now measure the density of the metal by
weighing the crown, and submerging it in water to obtain its volume.
*"Eureka"!* Archimedes rushed out of the bathtub and rushed
(supposedly naked) into the streets to announce that he had solved the
problem. Indeed, the crown did not contain enough gold, and the
goldsmith was beheaded.

Archimedes' Principle is the reason why boats remain buoyant
(or sometimes sink), balloons rise and ice floats. A body
will float in a given fluid depending on their relative
densities: both the apparent density (mass per unit of
volume) of the body, and that of the fluid determine the buoyant
force. If the body is less dense than the fluid, it will rise (or
float). If the body is denser than the fluid, it will drop (or sink).

The ratio of the two densities also determines *how much* of a
floating body will be submerged. For instance, sea water has a density
of 1024 kg/m^{3}. Ice (-4 C) has a density of 917
kg/m^{3}. Thus, an iceberg will be submerged for
917/1024 = 90%: only 10% is visible
above the surface.

When calculating the buoyant force on an object, the shape and
position of the object are also important. For instance, consider a
steel ship. Steel has a larger density than water, so a solid block of
steel would sink. However, a boat also has a large volume of air. The
apparent density of the ship is equal to the mass of the steel and
contained air divided by the entire volume of the ship. The apparent
density is less than the density of water, and thus the ship will float.