Catenary
History

The word "catenary" is derived from the Latin word catēna, which means " chain ". The English word "catenary" is usually attributed to Thomas Jefferson, who wrote in a letter to Thomas Paine on the construction of an arch for a bridge:

I have lately received from Italy a treatise on the 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.

It is often said that 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°. 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.

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. Some much older arches approximate catenaries, an example of which is the Arch of Taq-i Kisra in Ctesiphon.

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 in an appendix to his Description of Helioscopes, 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 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; their solutions were published in the Acta Eruditorum for June 1691. David Gregory wrote a treatise on the catenary in 1697 in which he provided an incorrect derivation of the correct differential equation.

Euler proved in 1744 that the catenary is the curve which, when rotated about the x -axis, gives the surface of minimum surface area (the catenoid) for the given bounding circles. Nicolas Fuss gave equations describing the equilibrium of a chain under any force in 1796.
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