Wikipedia: Cycloid

In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve.

The cycloid, with the cusps pointing upward, is the curve of fastest descent under constant gravity (the brachistochrone curve). It is also the form of a curve for which the period of an object in simple harmonic motion (rolling up and down repetitively) along the curve does not depend on the object's starting position (the tautochrone curve).

The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th-century mathematicians.[1]

Historians of mathematics have proposed several candidates for the discoverer of the cycloid. Mathematical historian Paul Tannery cited similar work by the Syrian philosopher Iamblichus as evidence that the curve was known in antiquity.[2] English mathematician John Wallis writing in 1679 attributed the discovery to Nicholas of Cusa,[3] but subsequent scholarship indicates that either Wallis was mistaken or the evidence he used is now lost.[4] Galileo Galilei's name was put forward at the end of the 19th century[5] and at least one author reports credit being given to Marin Mersenne.[6] Beginning with the work of Moritz Cantor[7] and Siegmund Günther,[8] scholars now assign priority to French mathematician Charles de Bovelles[9][10][11] based on his description of the cycloid in his Introductio in geometriam, published in 1503.[12] In this work, Bovelles mistakes the arch traced by a rolling wheel as part of a larger circle with a radius 120% larger than the smaller wheel.[4]

Galileo originated the term cycloid and was the first to make a serious study of the curve.[4] According to his student Evangelista Torricelli,[13] in 1599 Galileo attempted the quadrature of the cycloid (determining the area under the cycloid) with an unusually empirical approach that involved tracing both the generating circle and the resulting cycloid on sheet metal, cutting them out and weighing them. He discovered the ratio was roughly 3:1 but incorrectly concluded the ratio was an irrational fraction, which would have made quadrature impossible.[6] Around 1628, Gilles Persone de Roberval likely learned of the quadrature problem from Père Marin Mersenne and effected the quadrature in 1634 by using Cavalieri's Theorem.[4] However, this work was not published until 1693 (in his Traité des Indivisibles).[14]

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