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Archimedes der Mathematiker Part 2 Measurement of a circle ( &Kappaύκλου μέτρησις): π is between 3 10/71 and 3 1/7 This approximation could be known before but Archimedes provided methods for better approximations for π that provide lower and upper limits. He obtained the approximation, circumscribing and inscribing the circle with regular polygons having 96 sides. He used for this method also the approximation 265/153 for the square root of 3. Ptolemy (85-165 AD) later obtained the approximation = 3 17/120 = 3.14167 using an inscribed 360-gon I have found the following information: What Archimedes did was to obtain nice recursion formulas for the perimeters in term of geometric and harmonic averages (Q_2n = H(P_n,Q_n) and P_2n = G(P_n,Q_2n), where P_n is the perimeter of the inscribed n-polygon and Q_n is the perimeter of the circumscribed n-polygon), which he had to use together with some fairly accurate numerical method for estimating square roots. Archimedes did not use any means of the inscribed and circumscribed lengths. He did the inscribed ones recursively and he did the outside ones similarly. By the way, he explicitly made the assumption that a convex curve inside another curve is shorter. Another mention of convexity in his work is his assumption (axiom) that the center of gravity of a convex set lies within the set. (or even that which are required to fill our larger universe as we know today ) is nothing compared to the population of the cattle. The cattle is far far larger than the number of all the sand grains that can fill the entire Universe! Homer describes the number of the cattle of Helios to 350 which is a more realistic number. If Archimedes indeed described this problem then in having this problem a integer solution and of such magnitude that seems not possible to solve without computer it would be really astonishing showing that Archimedes knew more that we know. It seems impossible that a solution was found by Archimedes considering the complex notation for Greek numbers as the smallest solution for this number in Greek notation would be described as: The 7th unit of 2 myriad 5819th numbers, and the 7602 myriad 7140th unit of 2 myriad 5815th numbers, and the 6486 myriad 8182nd unit of 2 myriad 5817th numbers,...,and the 9738 myriad 2340th unit of 3rd numbers, and the 6626 myriad 7194th unit of 2nd numbers, and 5508 myriad 1800. Many of the manuscripts of Archimedes were lost by the --A Wolfram Web Resource. |