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## Archimedes / ahr-kuh-MEE-deez /

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###### Mathematics Ranking 3rd of 46 Archimedes with the Archimedes principle in the background.                                        Greek stamp from 1983.

Archimedes is one of the top three mathematicians along with Isaac Newton and Carl Gauss. He was an early pioneer in mechanical engineering. Archimedes can be regarded as the father of mathematical physics.(1) David Rivault’s Latin translation (1615) of Archimedes’ complete works was enormously influential on the work of Rene Descartes and Pierre de Fermat. Without the background of the rediscovered ancient mathematicians, amongst whom Archimedes was paramount, the development of mathematics in Europe between 1550 and 1650 is inconceivable.(2)

Bio

Archimedes was born and lived most of his life in Syracuse, the principle city-state of Sicily. He was killed by a Roman soldier during the siege of Syracuse, a major event of one of the Punic Wars. This is essentially all we know of Archimedes’ life.

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Archimedes discovered the ratio of the radius of a circle to its circumference and formulas for the surface area and volume of a sphere and of a cylinder. The formula for the area of a sphere is 4 times pi times the radius squared, or A = 4pr². The formula for the volume of a sphere is four-thirds times pi times the radius cubed, or V = 4/3pr3 and the formula for the volume of a cylinder is pi times the radius squared times the height of the cylinder, or V = pr²h. One of the bounds he established for pi, 22/7, was adopted as the usual approximation to it in antiquity and the Middle Ages, i.e., for about 1,500 years. His determination of pi was the first scientific estimate of this very important constant that is used in many formulas.

The Greek Eudoxus was the first to use the method of exhaustion for determining areas and volumes of more sophisticated geometric figures. Eudoxus’ contribution was a significant one, and he is usually regarded as the finest mathematician of antiquity next to the unsurpassed Archimedes himself. The general strategy was to approach an irregular figure (such as a circle) by means of a succession of known elementary ones (such as a square, rectangle, or triangle), each proving a better approximation than its predecessor. For instance, think of a circle as being a totally curvilinear, and thus quite intractable, plane figure. But if we inscribe within it a square, and then double the number of sides of the square to get an octagon, and then again double the number of sides to get a 16-gon, and so on, we will find these relatively simple polygons ever more closely approximating the circle itself. In Eudoxean terms, the polygons are “exhausting” the circle from within. Archimedes adopted this process when he determined the area of a circle. In addition, Archimedes credited Eudoxus with using the method of exhaustion to prove that the volume of “any cone is one third part of the cylinder which has the same base with the cone and equal height, a theorem that is quite difficult.” The method of exhaustion is the geometric forerunner of the modern notion of “limit,” which in turn lies at the heart of the calculus.(3)

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During his time in the Nile Valley, Archimedes is said to have invented the “Archimedean screw,” a device for raising water from a low level to a higher one. The invention is still in use today. He is famous for his discovery of Archimedes’ principle, which states that a body totally or partially immersed in a fluid is subject to an upward force that is equal in magnitude to the weight of fluid it displaces. Archimedes’ On Floating Bodies is the first known work in hydrostatics, of which he is recognized as the founder. The work’s purpose is to determine the positions that various solids will assume when floating in a fluid according to their form and the variation in their specific gravities. Hydrostatics is the branch of mechanics concerned with the equilibrium of liquids and the pressure exerted by liquid at rest. He designed many weapons to fight against the Romans.

Footnotes:

(1) Carl B. Boyer, A History of Mathematics, 2nd Edition (New York, 1991), p. 122.

(2) Encyclopaedia Britannica, Macropaedia, Volume 13, 1993, 15th Edition p. 873.

(3) William Dunham, Journey Through Genius The Great Theorems of Mathematics (New York, 1990), p. 29.

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