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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 ...
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.
Archimedes wrote the first known proof that 22 / 7 is an overestimate in the 3rd century BCE, although he may not have been the first to use that approximation. His proof proceeds by showing that 22 / 7 is greater than the ratio of the perimeter of a regular polygon with 96 sides to the diameter of a circle it circumscribes.
The number π (/ p aɪ /; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter.It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.
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).
Archimedes did not admit the method of indivisibles as part of rigorous mathematics, and therefore did not publish his method in the formal treatises that contain the results. In these treatises, he proves the same theorems by exhaustion, finding rigorous upper and lower bounds which both converge to the answer required. Nevertheless, the ...
Calculation made in Philadelphia, Pennsylvania, giving the value of pi to 154 digits, 152 of which were correct. First discovered by F. X. von Zach in a library in Oxford, England in the 1780s, and reported to Jean-Étienne Montucla, who published an account of it. [20] 152: 1722: Toshikiyo Kamata: 24 1722: Katahiro Takebe: 41 1739: Yoshisuke ...
On the Sphere and Cylinder (Greek: Περὶ σφαίρας καὶ κυλίνδρου) is a treatise that was published by Archimedes in two volumes c. 225 BCE. [1] It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder, and was the first to do so. [2]
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