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A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage. It is an example of a roulette, a curve generated by a curve rolling on another curve. The inverted cycloid (a cycloid rotated through 180°) is the solution to the brachistochrone problem (i.e., it is the curve of fastest descent under gravity) and the related tautochrone problem (i.e., the period of an object in descent without friction inside this curve does not depend on the object's starting position). == History == The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th-century mathematicians. 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 likely known in antiquity.〔 English mathematician John Wallis writing in 1679 attributed the discovery to Nicholas of Cusa,〔 but subsequent scholarship indicates Wallis was either mistaken or the evidence used by Wallis is now lost.〔 Galileo Galilei's name was put forward at the end of the 19th century〔 and at least one author reports credit being given to Marin Mersenne.〔 Beginning with the work of Moritz Cantor〔 and Siegmund Günther,〔 scholars now assign priority to French mathematician Charles de Bovelles〔〔〔 based on his description of the cycloid in his ''Introductio in geometriam'', published in 1503.〔 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.〔 Galileo originated the term ''cycloid'' and was the first to make a serious study of the curve.〔 According to his student Evangelista Torricelli,〔 in 1599 Galileo attempted the quadrature of the cycloid (constructing a square with area equal to 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.〔 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.〔 However, this work was not published until 1693 (in his ''Traité des Indivisibles'').〔 Constructing the tangent of the cycloid dates to August 1638 when Mersenne received unique methods from Roberval, Pierre de Fermat and René Descartes. Mersenne passed these results along to Galileo, who gave them to his students Torricelli and Viviana, who were able to produce a quadrature. This result and others were published by Torricelli in 1644,〔 which is also the first printed work on the cycloid. This led to Roberval charging Torricelli with plagiarism, with the controversy cut short by Torricelli's early death in 1647.〔 In 1658, Blaise Pascal had given up mathematics for theology but, while suffering from a toothache, began considering several problems concerning the cycloid. His toothache disappeared, and he took this as a heavenly sign to proceed with his research. Eight days later he had completed his essay and, to publicize the results, proposed a contest. Pascal proposed three questions relating to the center of gravity, area and volume of the cycloid, with the winner or winners to receive prizes of 20 and 40 Spanish doubloons. Pascal, Roberval and Senator Carcavy were the judges, and neither of the two submissions (by John Wallis and Antoine Lalouvère) were judged to be adequate.〔 While the contest was ongoing, Christopher Wren sent Pascal a proposal for a proof of the rectification of the cycloid; Roberval claimed promptly that he had known of the proof for years. Wallis published Wren's proof (crediting Wren) in Wallis's ''Tractus Duo'', giving Wren priority for the first published proof.〔 Fifteen years later, Christiaan Huygens had deployed the cycloidal pendulum to improve chronometers and had discovered that a particle would traverse an inverted cycloidal arch in the same amount of time, regardless of its starting point. In 1686, Gottfried Wilhelm Leibniz used analytic geometry to describe the curve with a single equation. In 1696, Johann Bernoulli posed the brachistochrone problem, the solution of which is a cycloid.〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cycloid」の詳細全文を読む スポンサード リンク
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