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Archimedes: Problems
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Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. By assuming a proposition to be true and showing that this would lead to a contradiction, he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the method of exhaustion, and he employed it to approximate the value of π (Pi). He did this by drawing a larger polygon outside a circle and a smaller polygon inside the circle. As the number of sides of the polygon increases, it becomes a more accurate approximation of a circle. When the polygons had 96 sides each, he calculated the lengths of their sides and showed that the value of π lay between
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Archimedes was killed in the aftermath of the Battle of Syracuse -- a siege won by the Romans using war machines many of which had been invented by Archimedes himself. Archimedes was killed by a Roman soldier who likely had no idea who Archimedes was. At the time of his death Archimedes was reputedly sketching a geometry problem in the sand, his last words to the Roman soldier being "don't disturb my circles".
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In The Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He expressed the solution to the problem as a geometric series that summed to infinity with the ratio
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Euclid's Elements had catalogued practically all the results of Greek geometry up to Archimedes' time. Archimedes adopted Euclid's uniform and rigorously logical form: axioms followed by theorems and their proofs. But the problems Archimedes set himself and his solutions were on another level from any that preceded him.
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