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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/m3. Ice (-4 C) has a density of 917 kg/m3. 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.

9. Though Archimedes discovered many curious matters which evince great intelligence, that which I am about to mention is the most extraordinary. Hiero, when he obtained the regal power in Syracuse, having, on the fortunate turn of his affairs, decreed a votive crown of gold to be placed in a certain temple to the immortal gods, commanded it to be made of great value, and assigned an appropriate weight of gold to the manufacturer. He, in due time, presented the work to the king, beautifully wrought, and the weight appeared to correspond with that of the gold which had been assigned for it.

10. But a report having been circulated, that some of the gold had been abstracted, and that the deficiency thus caused had been supplied with silver, Hiero was indignant at the fraud, and, unacquainted with the method by which the theft might be detected, requested Archimedes would undertake to give it his attention. Charged with this commission, he by chance went to a bath, and being in the vessel, perceived that, as his body became immersed, the water ran out of the vessel. Whence, catching at the method to be adopted for the solution of the proposition, he immediately followed it up, leapt out of the vessel in joy, and, returning home naked, cried out with a loud voice that he had found that of which he was in search, for he continued exclaiming, in Greek, eureka (I have found it out).

11. After this, he is said to have taken two masses, each of a weight equal to that of the crown, one of them of gold and the other of silver. Having prepared them, he filled a large vase with water up to the brim, wherein he placed the mass of silver, which caused as much water to run out as was equal to the bulk thereof. The mass being then taken out, he poured in by measure as much water as was required to fill the vase once more to the brim. By these means he found out what quantity of water was equal to a certain weight of silver.

12. He then placed the mass of gold in the vessel, and, on taking it out, found that the water which ran over was lessened, because, as the magnitude of the gold mass was smaller than that containing the same weight of silver. After again filling the vase by measure, he put the crown itself in, and discovered that more water ran over then than with the mass of gold that was equal to it in weight; and thus, from the superfluous quantity of water carried over the brim by the immersion of the crown, more than that displaced by the mass, he found, by calculation, the quantity of silver mixed with the gold, and made manifest the fraud of the manufacturer.

Vitruvius: De Architectura, Book IX, ch. 9-12 of the preface
(translation by Joseph Gwilt, London, 1826)

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