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Archimedes: Method
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Archimedes is the GNU package for the design and simulation of submicron semiconductor devices. It is a 2D Fast Monte Carlo simulator which can take into account all the relevant quantum effects, thank to the implementation of the Bohm effective potential method.
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One of the major contributions Archimedes made to mathematics was his method for approximating the value of pi. It had long been recognized that the ratio of the circumference of a circle to its diameter was constant, and a number of approximations had been given up to that point in time by the Babylonians, Egyptians, and even the Chinese. There are some authors who claim that a biblical passage1 ... implies an approximate value of 3 (and in fact there is an interesting story2 associated with that).
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Archimedes' proofs of formulas for areas and volumes set the standard for the rigorous treatment of limits until modern times. But the way he discovered these results remained a mystery until 1906, when a copy of his lost treatise The Method was discovered in Constantinople (now Istanbul, Turkey).
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Archimedes is pictured with the methods he used to find an approximation to the area of a circle and the value of pi. Archimedes was the first to give a scientific method for calculating pi. to arbitrary accuracy. The method used by Archimedes -- the measurement of inscribed and circumscribed polygons approaching a 'limit" (described as 'exhaustion') -- was one of the earliest approaches to "integration". It preceded by more than a millennia Newton, Leibniz, and modern calculus.
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The method of Archimedes involves approximating pi by the perimeters of polygons inscribed and circumscribed about a given circle. Rather than trying to measure the polygons one at a time, Archimedes uses a theorem of Euclid to develop a numerical procedure for calculating the perimeter of a circumscribing polygon of 2n sides, once the perimeter of the polygon of n sides is known. Then, beginning with a circumscribing hexagon, he uses his formula to calculate the perimeters of circumscribing polygons of 12, 24, 48, and finally 96 sides. He then repeats the process using inscribing polygons (after developing the corresponding formula). The truly unique aspect of Archimedes' procedure is that he has eliminated the geometry and reduced it to a completely arithmetical procedure, something that probably would have horrified Plato but was actually common practice in Eastern cultures, particularly among the Chinese scholars.
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