Archimedes playing with a computer

Extract

It was known before Archimedes (287-212 BC) that the circumference of a circle was proportional to its diameter and that the area was proportional to the square of its radius. It was Archimedes who first supplied a rigorous proof that these two proportionality constants were the same, now called π [1]. He started with inscribed and circumscribed hexagons and increased the number of sides from 6 up to 96 by successively doubling it. His result was not a single value. In fact he generated five intervals each of which contained π. He calculated a lower bound from the inscribed polygon and an upper bound from the circumscribed polygon of 96 sides. This gave him the interval () or (3.140845, 3.142857), which is less accurate than the interval bounded by half-perimeters of the inscribed and circumscribed 96-gons, which is (3.141031, 3.142714).