Archimedes (mathematician)

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Greek mathematician and philosopher who made major discoveries in geometry, hydrostatics, and mechanics, and established the sciences of statics and hydrostatics. He formulated a law of fluid displacement (Archimedes' principle), and is credited with the invention of the Archimedes screw, a cylindrical device for raising water. His method of finding mathematical proof to substantiate experiment and observation became the method of modern science in the High Renaissance.

Hydrostatics and Archimedes' principle
The best-known result of Archimedes' work on hydrostatics is Archimedes' principle, which states that a body immersed in water will displace a volume of fluid that weighs as much as the body would weigh in air. It is alleged that Archimedes' principle was discovered when he stepped into the public bath and saw the water overflow. He was so delighted that he rushed home naked, crying ‘Eureka! Eureka!’ (‘I have found it! I have found it!’).

He used his discovery to prove that the goldsmith of Hieron II, King of Syracuse, had adulterated a gold crown with silver. Archimedes realized that if the gold had been mixed with silver (which is less dense than gold), the crown would have a greater volume and therefore displace more water than an equal weight of pure gold. The story goes that the crown was found to be impure, and that the unfortunate goldsmith was executed.

Statics and the lever
In the field of statics, he is credited with working out the rigorous mathematical proofs behind the law of the lever. The lever had been used by other scientists, but it was Archimedes who demonstrated mathematically that the ratio of the effort applied to the load raised is equal to the inverse ratio of the distances of the effort and load from the pivot or fulcrum of the lever. Archimedes is credited with having claimed that if he had a sufficiently distant place to stand, he could use a lever to move the world.

This claim is said to have given rise to a challenge from King Hieron to Archimedes to show how he could move a truly heavy object with ease, even if he could not move the world. In answer to this, Archimedes developed a system of compound pulleys. According to Plutarch's Life of Marcellus (who sacked Syracuse), Archimedes used this to move with ease a ship that had been lifted with great effort by many men out of the harbour on to dry land. The ship was laden with passengers, crew and freight, but Archimedes – sitting at a distance from the ship – was reportedly able to pull it over the land as though it were gliding through water.

Mathematics
Archimedes wrote many mathematical treatises, some of which still exist in altered forms in Arabic. Archimedes' approximation for the value for π was more accurate than any previous estimate – the value lying between 223/71 and 220/70. The average of these two numbers is less than 0.0003 different from the modern approximation for π. He also examined the expression of very large numbers, using a special notation to estimate the number of grains of sand in the Universe. Although the result, 10⁶³, was far from accurate, Archimedes demonstrated that large numbers could be considered and handled effectively.

Archimedes also evolved methods to solve cubic equations and to determine square roots by approximation. His formulae for the determination of the surface areas and volumes of curved surfaces and solids anticipated the development of integral calculus, which did not come for another 2,000 years. Archimedes had decreed that his gravestone be inscribed with a cylinder enclosing a sphere together with the formula for the ratio of their volumes – a discovery that he regarded as his greatest achievement.