Editing Cycloid (section)

== History ==
{{quotebox|width=30%|
quote=It was in the left hand [[Try pot|try-pot]] of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along the cycloid, my soapstone for example, will descend from any point in precisely the same time.
|source=''[[Moby Dick]]'' by [[Herman Melville]], 1851}}

The cycloid has been called "The Helen of Geometers" as, like [[Helen of Troy]], it caused frequent quarrels among 17th-century mathematicians, while [[Sarah B. Hart|Sarah Hart]] sees it named as such  "because the properties of this curve are so beautiful".<ref>{{Cite book | last1=Cajori | first1=Florian | author1-link=Florian Cajori | title=A History of Mathematics | publisher=Chelsea | location=New York | isbn=978-0-8218-2102-2 | year=1999 | page=177 }}</ref><ref>{{Cite web |last=Hart |first=Sarah |date=7 April 2023 |title=The Wondrous Connections Between Mathematics and Literature |url=https://www.nytimes.com/2023/04/07/opinion/the-wondrous-connections-between-mathematics-and-literature.html |access-date=7 April 2023 |website=New York Times}}</ref>

Historians of mathematics have proposed several candidates for the discoverer of the cycloid.  Mathematical historian [[Paul Tannery]] speculated that such a simple curve must have been known to the [[Greek mathematics|ancients]], citing similar work by [[Carpus of Antioch]] described by [[Iamblichus]].<ref name=Tannery/>  English mathematician [[John Wallis]] writing in 1679 attributed the discovery to [[Nicholas of Cusa]],<ref name=Wallis/> but subsequent scholarship indicates that either Wallis was mistaken or the evidence he used is now lost.<ref name=Whitman/>  [[Galileo Galilei]]'s name was put forward at the end of the 19th century<ref name=Cajori/> and at least one author reports credit being given to [[Marin Mersenne]].<ref name=Roidt/> Beginning with the work of [[Moritz Cantor]]<ref name=Cantor/> and [[Siegmund Guenther|Siegmund Günther]],<ref name=Gunther/> scholars now assign priority to French mathematician [[Charles de Bovelles]]<ref name=Phillips/><ref name=Victor/><ref name=Martin/> based on his description of the cycloid in his ''Introductio in geometriam'', published in 1503.<ref name=Bovelles/>  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.<ref name=Whitman/>

Galileo originated the term ''cycloid'' and was the first to make a serious study of the curve.<ref name=Whitman />  According to his student [[Evangelista Torricelli]],<ref name=Torricelli/> in 1599 Galileo attempted the [[Quadrature (geometry)|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, which is the true value, but he incorrectly concluded the ratio was an irrational fraction, which would have made quadrature impossible.<ref name=Roidt/>  Around 1628, [[Gilles de Roberval|Gilles Persone de Roberval]] likely learned of the quadrature problem from [[Marin Mersenne|Père Marin Mersenne]] and effected the quadrature in 1634 by using [[Cavalieri's principle|Cavalieri's Theorem]].<ref name=Whitman />  However, this work was not published until 1693 (in his ''Traité des Indivisibles'').<ref name=Walker />

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 Viviani, who were able to produce a quadrature.  This result and others were published by Torricelli in 1644,<ref name=Torricelli/> 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.<ref name=Walker />

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 mass|center of gravity]], area and volume of the cycloid, with the winner or winners to receive prizes of 20 and 40 Spanish [[doubloon]]s.  Pascal, Roberval and Senator Carcavy were the judges, and neither of the two submissions (by [[John Wallis]] and [[Antoine de Lalouvère]]) was judged to be adequate.<ref name=Conner />{{rp|198}}  While the contest was ongoing, [[Christopher Wren]] sent Pascal a proposal for a proof of the [[arc length|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 ''Tractatus Duo'', giving Wren priority for the first published proof.<ref name=Walker />

Fifteen years later, [[Christiaan Huygens]] had deployed the cycloidal pendulum to improve chronometers and had discovered that a particle would traverse a segment of 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 curve|brachistochrone problem]], the solution of which is a cycloid.<ref name=Walker />