Prove the two lemmas (see pages 68 and 69) that Archimedes used to derive his al

Question

Prove the two lemmas (see pages 68 and 69) that Archimedes used to derive his algorithms for calculating pi.

Explanation / Answer

In Measurement of a Circle Archimedes derived that p was between 3 1/7 and 3 10/71. None of Archimedes original works exist; however, references to his works by near contemporaries give confidence that the collections written centuries later, essentially as university textbooks, are accurate records of his thoughts. His method was geometric. To establish a lower bound, he inscribes a regular hexagon inside a unit circle. By subdividing each face he arrives at a regular 12-gon, then 24-gon, and so on. As the number of faces increases, so does the perimeter of the polygons, and it approaches the value of 2p the circumference of the circle. The upper bound is established similarly using circumscribed polygons. The process of subdivision halves at each step the angle f formed by a polygon vertex, the center of the circle and the midpoint of a polygon face incident on the vertex. Archimedes did not use the notation of trigonometric functions. Greek mathematics used the ratios which only much later were given the names familiar today. For the process of subdivision, Archimedes derives the following two formulas, (1) cot f/2 = cot f + csc f, (2) csc2 f = 1 + cot2 f. Archimedes shows explicitly only the first of these identities. The second is inferred from his calculations. The two trigonometric identities can be combined into a single formula, (3) cot f/2 = cot f + v(1 + cot f2) . Let Pn be the semi-perimeter of a regular n-gon circumscribed around a unit circle. Let pn be the semi-perimeter of a regular n-gon inscribed in a unit circle. Then, (4) Pn = n /(cot f), (5) pn = n /(csc f). For a regular n-gon, f=p/n. Archimedes starts with the rational approximations for the hexagon and uses the half-angle formula to get cot f for f=p/12, p/24, and so on. From the cotangent the semi-perimeters are calculated. With each square root, he was required to round to a rational. He carefully rounds in the direction which keeps the inequality correct. The first calculation concerns the sequence of circumscribed polygons. To overestimate their area, it is necessary to underestimate cot f, beginning with the underestimate of this value for the hexagon, cot p/6 = v3 > 265/153 = 1.7320261438 The second calculation concerns the sequence of inscribed polygons. Although the formula for pn shows only the dependence on the cosecant, the calculation proceeds based on the cotangent. Therefore, an overestimate is required, cot p/6 = v3 < 1351/780 = 1.7320512821 Both calculations can begin with the exact value for the cosecant, csc p/6 = 2, so the first iteration gives the bounds, 2 + 265/153

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