Editing Catenary (section)

==History==
[[File:GaudiCatenaryModel.jpg|thumb|250px|[[Antoni Gaudí]]'s catenary model at [[Casa Milà]]]]

The word "catenary" is derived from the Latin word ''catēna'', which means "[[chain]]". The English word "catenary" is usually attributed to [[Thomas Jefferson]],<ref>{{cite web|url=http://www.pballew.net/arithme8.html#catenary |archive-url=https://archive.today/20120906145306/http://www.pballew.net/arithme8.html#catenary |url-status=usurped |archive-date=September 6, 2012 |title="Catenary" at Math Words |publisher=Pballew.net |date=1995-11-21 |access-date=2010-11-17}}</ref><ref>{{cite book| last = Barrow| first = John D.| title = 100 Essential Things You Didn't Know You Didn't Know: Math Explains Your World| year = 2010| publisher = W. W. Norton & Company| isbn = 978-0-393-33867-6| page = [https://archive.org/details/100essentialthin0000barr/page/27 27]| url = https://archive.org/details/100essentialthin0000barr/page/27}}</ref>
who wrote in a letter to [[Thomas Paine]] on the construction of an arch for a bridge:

{{Blockquote|I have lately received from Italy a treatise on the [[Mechanical equilibrium|equilibrium]] of arches, by the Abbé Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect equilibrium.<ref>{{cite book| last = Jefferson| first = Thomas| title = Memoirs, Correspondence and Private Papers of Thomas Jefferson| url = https://archive.org/details/memoirscorrespon02jeffuoft| year = 1829| publisher = Henry Colbura and Richard Bertley| page = [https://archive.org/details/memoirscorrespon02jeffuoft/page/419 419] }}</ref>}}

It is often said<ref name="Lockwood124"/> that [[Galileo Galilei|Galileo]] thought the curve of a hanging chain was parabolic. However, in his ''[[Two New Sciences]]'' (1638), Galileo wrote that a hanging cord is only an approximate parabola, correctly observing that this approximation improves in accuracy as the curvature gets smaller and is almost exact when the elevation is less than 45°.<ref>{{cite book| last = Fahie| first = John Joseph| title = Galileo, His Life and Work| url = https://archive.org/details/galileohislifea01fahigoog| year = 1903| publisher = J. Murray| pages = [https://archive.org/details/galileohislifea01fahigoog/page/n411 359]–360 }}</ref> The fact that the curve followed by a chain is not a parabola was proven by [[Joachim Jungius]] (1587–1657); this result was published posthumously in 1669.<ref name="Lockwood124">[[#Lockwood|Lockwood]] p. 124</ref>

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.<ref>{{Cite journal|jstor=532102 |title=Monuments and Microscopes: Scientific Thinking on a Grand Scale in the Early Royal Society |journal=Notes and Records of the Royal Society of London |volume=55 |issue=2 |pages=289–308 |first=Lisa |last=Jardine|year=2001 |doi=10.1098/rsnr.2001.0145 |s2cid=144311552 }}</ref> Some much older arches approximate catenaries, an example of which is the Arch of [[Taq-i Kisra]] in [[Ctesiphon]].<ref>{{cite book| last = Denny| first = Mark| title = Super Structures: The Science of Bridges, Buildings, Dams, and Other Feats of Engineering| year = 2010| publisher = JHU Press| isbn = 978-0-8018-9437-4| pages = 112–113 }}</ref>

[[File:Analogy between an arch and a hanging chain and comparison to the dome of St Peter's Cathedral in Rome.png|thumb|Analogy between an arch and a hanging chain and comparison to the dome of [[Saint Peter's Basilica]] in Rome ([[Giovanni Poleni]], 1748)]]
In 1671, Hooke announced to the [[Royal Society]] that he had solved the problem of the optimal shape of an arch, and in 1675 published an encrypted solution as a Latin [[anagram]]<ref>[[cf.]] the anagram for [[Hooke's law]], which appeared in the next paragraph.</ref> in an appendix to his ''Description of Helioscopes,''<ref>{{cite web |url=http://www.lindahall.org/events_exhib/exhibit/exhibits/civil/design.shtml |title=Arch Design |publisher=Lindahall.org |date=2002-10-28 |access-date=2010-11-17 |url-status=dead |archive-url=https://web.archive.org/web/20101113210736/http://www.lindahall.org/events_exhib/exhibit/exhibits/civil/design.shtml |archive-date=2010-11-13 }}</ref> where he wrote that he had found "a true mathematical and mechanical form of all manner of Arches for Building." He did not publish the solution to this anagram<ref>The original anagram was ''abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux'': the letters of the Latin phrase, alphabetized.</ref> in his lifetime, but in 1705 his executor provided it as ''ut pendet continuum flexile, sic stabit contiguum rigidum inversum'', meaning "As hangs a flexible cable so, inverted, stand the touching pieces of an arch."

In 1691, [[Gottfried Leibniz]], [[Christiaan Huygens]], and [[Johann Bernoulli]] derived the [[equation]] in response to a challenge by [[Jakob Bernoulli]];<ref name="Lockwood124"/> their solutions were published in the ''[[Acta Eruditorum]]'' for June 1691.<ref>{{citation|first=C.|last=Truesdell|title=The Rotational Mechanics of Flexible Or Elastic Bodies 1638–1788: Introduction to Leonhardi Euleri Opera Omnia Vol. X et XI Seriei Secundae|location=Zürich|
url=https://books.google.com/books?id=gxrzm6y10EwC&pg=PA66|page=66|publisher=Orell Füssli|date=1960|isbn=9783764314415}}</ref><ref name="calladine" >{{citation|first=C. R.|last=Calladine|title=An amateur's contribution to the design of Telford's Menai Suspension Bridge: a commentary on Gilbert (1826) 'On the mathematical theory of suspension bridges'|journal=Philosophical Transactions of the Royal Society A|date=2015-04-13|volume=373|issue=2039|page=20140346|doi=10.1098/rsta.2014.0346|pmid=25750153|pmc=4360092|bibcode=2015RSPTA.37340346C}}</ref> [[David Gregory (mathematician)|David Gregory]] wrote a treatise on the catenary in 1697<ref name="Lockwood124"/><ref>{{citation|first=Davidis|last=Gregorii|title=Catenaria|journal=Philosophical Transactions|volume=19|issue=231|date=August 1697|pages=637–652|doi=10.1098/rstl.1695.0114|doi-access=free}}</ref> in which he provided an incorrect derivation of the correct differential equation.<ref name="calladine" />

[[Leonhard Euler]] proved in 1744 that the catenary is the curve which, when rotated about the {{mvar|x}}-axis, gives the surface of minimum [[surface area]] (the [[catenoid]]) for the given bounding circles.<ref name="MathWorld"/> [[Nicolas Fuss]] gave equations describing the equilibrium of a chain under any [[force]] in 1796.<ref>[[#Routh|Routh]] Art. 455, footnote</ref>