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The Method of Mechanical Theorems (Greek: Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as The Method, is one of the major surviving works of the ancient Greek polymath Archimedes. The Method takes the form of a letter from Archimedes to Eratosthenes, [1] the chief librarian ...
The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes, implementing the method of exhaustion. [48] This polygonal algorithm dominated for over 1,000 years, and as a result π is sometimes referred to as Archimedes's constant ...
The method of exhaustion typically required a form of proof by contradiction, known as reductio ad absurdum. This amounts to finding an area of a region by first comparing it to the area of a second region, which can be "exhausted" so that its area becomes arbitrarily close to the true area.
I was reading Method of exhaustion, and I wanted to try out Archimedes' method for calculating pi with a limit. Finding the limit as the number of sides of a regular polygon approaches infinity of the ratio of its area to its apothem squared gave me: Obviously, this equals pi, but I have no clue how to show that.
Cicero Discovering the Tomb of Archimedes (1805) by Benjamin West. Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek scholar John Tzetzes that Archimedes lived for 75 years before his death in 212 BC. [9]
The Archimedes Palimpsest is a parchment codex palimpsest, originally a Byzantine Greek copy of a compilation of Archimedes and other authors. It contains two works of Archimedes that were thought to have been lost (the Ostomachion and the Method of Mechanical Theorems) and the only surviving original Greek edition of his work On Floating ...
Used a geometrical method equivalent to Richardson extrapolation: 10 1665: Isaac Newton [2] 16 1681: Takakazu Seki [19] 11 16 1699: Abraham Sharp [2] Calculated pi to 72 digits, but not all were correct 71: 1706: John Machin [2] 100: 1706: William Jones: Introduced the Greek letter ' π ' 1719: Thomas Fantet de Lagny [2] Calculated 127 decimal ...
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.