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: The Approximation of Pi : The method of Archimedes involves approximating pi by the perimeters of polygons inscribed and cirscribed 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 cirscribing polygon of 2n sides, once the perimeter of the polygon of n sides is known. Then, beginning with a cirscribing hexagon, he uses his formula to calculate the perimeters of cirscribing 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. : : The Key Theorem : The key result used by Archimedes is Proposition 3 of Book VI of Euclid's Elements. The full statement of the theorem is as follows: : : If an angle of a triangle be bisected and the straight line cutting the angle cut the base also, the segments of the base will have the same ratio as the remaining sides of the triangle; and, if the segments of the base have the same ratio as the remaining sides of the triangle, the straight line joined from the vertex to the point of section will bisect the angle of the triangle.

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