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where C is the circumference of a circle, d is the diameter, and r is the radius.More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width.
The ratio of the volume of a sphere to the volume of its circumscribed cylinder is 2:3, as was determined by Archimedes. The principal formulae derived in On the Sphere and Cylinder are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder.
But for =, it converges impractically slowly (that is, approaches the answer very gradually), taking about ten times as many terms to calculate each additional digit. [ 79 ] In 1699, English mathematician Abraham Sharp used the Gregory–Leibniz series for z = 1 3 {\textstyle z={\frac {1}{\sqrt {3}}}} to compute π to 71 digits, breaking the ...
Archimedes, in his Measurement of a Circle, created the first algorithm for the calculation of π based on the idea that the perimeter of any (convex) polygon inscribed in a circle is less than the circumference of the circle, which, in turn, is less than the perimeter of any circumscribed polygon. He started with inscribed and circumscribed ...
The calculations Archimedes used to approximate the area numerically were laborious, and he stopped with a polygon of 96 sides. A faster method uses ideas of Willebrord Snell ( Cyclometricus , 1621), further developed by Christiaan Huygens ( De Circuli Magnitudine Inventa , 1654), described in Gerretsen & Verdenduin (1983 , pp. 243–250).
Through proof by contradiction (reductio ad absurdum), 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 areas of figures and the value of π .
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De circulo & adscriptis liber (1619). Ludolph van Ceulen spent a major part of his life calculating the numerical value of the mathematical constant π, using essentially the same methods as those employed by Archimedes some seventeen hundred years earlier. [2]