(b) math of a catenary:
(i) catenary
https://en.wikipedia.org/wiki/Catenary
(section 5 Mathematical description: hyperbolic cosine function or cosh)
section 1 History: "The English word "catenary" is usually attributed to Thomas Jefferson * * * Galileo [1564 – 1642] thought the curve of a hanging chain was parabolic [he was wrong, but difference between parabolic and hyperbolic cosine is small around vertex; hyperbolic functions were invented after Galeo's death] * * * The fact that the curve followed by a chain is not a parabola was proven by Joachim Jungius (1587–1657) [Jungius disproved the curve as hyperbola, but did not now what it was]; this result was published posthumously in 1669. * * * The application of the catenary to the construction of arches is attributed to Robert Hooke, whose 'true mathematical and mechanical form' in the context of the rebuilding of St Paul's Cathedral alluded to a catenary. * * * In 1691, Gottfried Leibniz, Christiaan Huygens, and Johann Bernoulli [1667 – 1748; the Bernoulli clan of mathematicians were from Antwerp, the Netherlands, which was Spanish Netherlands (occupied by Spain); the Bernoullis, being protestants, escaped to Switzerland, where Johann was born. Johann was father of Daniel, who created Bernoulli principle] derived the equation in response to a challenge by Jakob Bernoulli
(A)
• "Joachim Jungius was given the name Joachim Junge (or Jung [German adjective masculine for young; j in German pronunciation is same as y in English] when he was born; Jungius is the Latin version of Junge which he used in all his publications. His father, Nicolaus Junge, was a teacher at the Gymnasium St Katharinen in Lübeck and his mother was Brigitte Holdmann, the daughter of Joachim Holdmann who was a minister in the Lutheran Cathedral in Lübeck. However, Nicolaus Junge was murdered when Joachim was only two years old and, in 1589, Brigitte married Martin Nordmann who was a teacher at the Gymnasium St Katharinen in Lübeck. Joachim was brought up by his mother and step-father. He attended the Gymnasium St Katharinen in Lübeck until 1605."
Joachim Jungius. In Edmund Robertson and John O'Connor, MacTutor: History of Mathematics Archive. School of Mathematics and Statistics, University of St Andrews, undated
https://mathshistory.st-andrews.ac.uk/Biographies/Jungius/
• "Joachim Jungius, of Lübeck, represents the German counterpart to Galileo Galilei in Italy, René Descartes [1596 – 1650] in France, and Francis Bacon [1561 – 1626] in England as an innovator in science and philosophy. Unlike these men, Jungius did not achieve an international reputation; even among scholars, interest in him has been largely confined to Germans, whose curiosity has been whetted by Gottfried Wilhelm Leibniz's enthusiastic praise of his merits as a philosopher." Encyclopedia.com
• Lübeck
https://en.wikipedia.org/wiki/Lübeck
The German letter ü (also found in über) is pronounced exactly like 魚 or 雨 in Mandarin Chinese.
(B) Gottfried Wilhelm Leibniz
https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz
(1646 – 1716; German)
(C) Bernoulli
https://www.ahdictionary.com/word/search.html?q=Bernoulli
(pronunciation)
(D)
• Huygens
https://www.merriam-webster.com/dictionary/Huygens
(1629 – 1695; Dutch)
• Huygens
https://www.merriam-webster.com/dictionary/Huygens
(pronunciation)
(E) Janet Heine Barnett, Enter, Stage Center: The Early Drama of the Hyperbolic Functions. Mathematics Magazine, 77: 15 (2004)
https://www.maa.org/sites/defaul ... 29.pdf.bannered.pdf
pages 15-16: "17th-century mathematicians focused their attention on the problem of the catenary when Jakob Bernoulli posed it as a challenge in a 1690 Acta Eruditorum paper * * * Issued at a time when the rivalry between Jakob and Johann Bernoulli [Johann was Jakob's younger brother; Johann rivaled Jakob and son Daniel for prizes top solve mathematical problems] was still friendly, this was one of the earliest challenge problems of the period. In June 1691, three independent solutions appeared in Acta Eruditorum * * * the crux of [Johann] Bernoulli's proof was to show that the curve in question satisfies the differential equation * * * . 17th-century solutions of the problem differed from those of today's calculus students in a particularly notable way: There was absolutely no mention of hyperbolic functions, or any other explicit
function, in the solutions of 1691!" (emphasis original).
page 18: "The hyperbolic functions did not, and could not, come into being until the full power of formal analysis had taken hold in the age of Euler.
• Leonhard Euler
https://en.wikipedia.org/wiki/Leonhard_Euler
(1707 – 1783; Swiss; "Euler is credited for popularizing the Greek letter π (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as well as first using the notation f(x) for the value of a function, the letter i to express the imaginary unit √−1, the Greek letter Σ (capital sigma) to express summations, the Greek letter Δ (uppercase delta) for finite differences, * * * He gave the current definition of the constant e, the base of the natural logarithm, now known as Euler's number. * * * Euler created * * * hyperbolic trigonometric functions")
(ii) Robert E Thurman (a postdoc then), The Forces on a Cable. The Geometry Center, University of Minnesota, undated
www.geom.uiuc.edu/~thurman/wonders/force.html
("The forces in the rope at [point] P are tangent to the rope: there is a force [tension] T pulling up and to the right, which is balanced by a force U pulling down and to the left. We will concentrate on the force U. We can break U into its horizontal (left) and vertical (down) components, as pictured above. * * * If you think about it, you can convince yourself that it's only the portion of the cable from P to the lowest point that is pulling down at P. * * * So, if w equals the the weight of the rope per unit length, and we let s be the length of the portion of the rope between the lowest point and P, then P feels a force of ws pointing straight down [ie, gravity]. Now what about the force pulling to the left? The only horizontal force anywhere on the rope is the tension due to stretching the rope between the pylons. Since this is the only horizontal force, and since the rope is in static equilibrium, every point on the rope feels the same horizontal tension H. We could measure H at the lowest point, where there are no other forces acting. * * * What changes when we hang a heavy road from our suspension cable? Not much, really! The only thing that changes in the above discussion is the weight pulling down at the point P. We will assume that the weight W per unit length of road is much heavier than the weight w per unit length of cable. In that case, the weight supported at P is essentially Wx, where x is the horizontal distance from the lowest point, instead of the distance s along the cable") |
发表于 12-28-2022 15:20:01
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