User:Lesser Cartographies/sandbox/Cycloid

Updated Cycloid article 10:08, 9 October 2013 (UTC)

-

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 C. 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 andRené Descartes. Mersenne passed these results along to Galileo, who gave them to his 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.

- "[Historian of mathematics] Paul Tannery has discussed a passage by Iamblichus referring to double movement and has remarked that it is difficult to see how the cycloid could have escaped the notice of the ancients."

"John Wallis in a letter of 1679, ascribed the discovery to Nicolas Cusanus in 1450 and also mentioned Bouelles as one who in 1500 advanced the study of this curve. In the case of Cusanus, however, historians are agreed that Wallis was mistaken unless, says Cantor, he had access to some manuscript now lost."

"The most original contribution of Bovelles was his study of the cycloid or transcendental curve. This curve, represented mathematically by the formula X=a arc ccs (a-y)/a sqrt(2ay - y^2) he investigaed in the Introductio in geometriam.  Prior to the works of Günther and Cantor the postulation of this curve had been attributed incorrectly to Nicholas of Cusa, but recent research as confirmed Bovelles as its discoverer.  Although Bovelles did little more than postulate its existence, later mathematicians drew out its full implications.  [Footnote 86:  This work was written in 1501 but not published until 1503.] (pg 42)

“This curve, sometimes incorrectly attributed to Nicholas Cusa (c. 1450), was first studied by Charles de Bouelles (1501). It then attracted the attention of Galileo (1599), Mersenne (1628), and Roberval (1634). Pascal (1659) completely solved the problem of its quadrature, and found the center of gravity of a segment cut off by a line parallel to the base. Jean and Jacques Bernoulli showed that it is the brachistochrone curve, and Huygens (1673) showed how its properties of tautochronism might be applied to the pendulum.”

"This brings us to Galileo, who, according to Cantor, both popularized the curve and gave it its name. One of his pupils wrote that he first attempted the quadrature of the cycloid in 1599. (The term quadrature refers to “squaring” a shape by constructing a square with equal area. The quadrature of the circle is one of the great mathematical problems of antiquity, and has long been proven impossible.) As a method of finding the area under the arch, Galileo cut the shape out of a material (some say sheet metal) and compared its weight with that of a generating circle cut from the same material. Several experiments resulted in approximately the same ratio, 3 to 1, before Galileo gave up the study thinking (mistakenly) that the ratio was “incommensurable,” what we now call irrational. The clever approach of using weight to determine area empirically was a hallmark of Galileo's approach to science."

"Mersenne, who is also sometimes called the discoverer of the cycloid, can only truly be credited with being the first to give a precise mathematical definition of the curve. However, it was Mersenne who proposed the problem of the quadrature of the cycloid (and the construction of a tangent to a point on the curve) to at least three other very significant mathematicians: Roberval, Descartes, and Fermat. While all three responded with unique constructions, only Roberval was able to conquer the area problem. His ability to do so was based on a new way of finding areas under curves discovered by a student of Galileo. This student was named Bonaventura Cavalieri, and he is the namesake of the well-known Cavalieri’s principle. In a later section, we will examine this approach in detail."

"There were also some unusual curves known already in ancient Greece: quadratrix of Hippias, spiral of Archimedes, conchoid of Nicomedes, cissoid of Diocles. They were obtained as means to solve the classical problems of Greek mathematics (squaring the circle, duplicating the cube and trisecting an angle). No other curve was introduced in the Renaissance period, with the eventual exception of the cycloid. According to John Wallis, the cycloid was invented by Nicholas of Cusa in the middle of the fifteenth century [Cantor, 1892, p.185]. However, the prominent role, which the cycloid played in mathematics, started as late as in the seventeenth century."

"It has been called it the “Helen of Geometry,” not just because of its many beautiful properties but also for the conﬂicts it engendered."

"After demonstrating mathematical talent at an early age, Blaise Pascal turned his attention to theology, denouncing the study of mathematics as a vainglorious pursuit. Then one night, unable to sleep as the result of a toothache, he began thinking about the cycloid and to his surprise, his tooth stopped aching. Taking this as a sign that he had God’s approval to continue, Pascal spent the next eight days studying the curve. During this time he discovered nearly all of the geometric properties of the cycloid. He issued some of his results in 1658 in the form of a contest, offering a prize of forty Spanish gold pieces and a second prize of twenty pieces."

"It was Galileo who gave it the name "cycloid" and investigated some of its properties around 1599. he was looking for curves of least time descent, though without much success.  At the same time, Mersenne, Roberval, and Torricelli became intersted in the curve.  Pascal made some real contributions to the subject, primarily in calculating the length of the curve and various volumes as it is rotated about axes (though he often used the French word for it, roulette).  Torricelli correctly found the area under one arch to be three times the area of the generating circular disk.  In a series of papers starting with "Problemata de Cycloide proposita mense Junii 1658"[5], along with correspondence between Huygens and A. Dettonville (a pseudonym used by Pascal&mdash;an anagram of Louis de Montalte, the name under which he published his Lettres provinciales), Pascal proved various results and published a challenge to others to replicate them. In 1658, Christopher Wren correctly calculated the arc length of one arch, 8a, where a is the radius of the generating circle."

"Certainly the curve is described in a book by Charles Bouvelles written in 1501, and he is usually considered the true inventor. [citation to Cajori, F. A History of Mathematics (New York:The Macmillan Co., 1919), p 162.]