Hang a chain, flip the curve upside down, and you have a perfect arch.
In 1675 Robert Hooke claimed he had found the ideal shape for an arch, and published it as a scrambled Latin anagram. Decoded after his death, it read: as hangs a flexible cable, so, inverted, stand the touching pieces of an arch. The curve is the catenary, and it isn't a parabola.
A catenary, from the Latin catena, 'chain', is the curve a flexible chain or cable makes when it hangs under its own weight from two points. It looks like a U and is easily mistaken for a parabola. It is often said Galileo confused the two, but in 1638 he actually wrote that a hanging cord is only approximately parabolic, a good approximation for shallow curves. That it is not a parabola was proved by Joachim Jungius, published in 1669.
In 1671 Robert Hooke told the Royal Society he had solved the problem of the best shape for an arch, apparently in connection with rebuilding St Paul's Cathedral. In 1675 he published the answer hidden in a Latin anagram, a common way to claim priority without giving the idea away. His executor revealed it in 1705: 'ut pendet continuum flexile, sic stabit contiguum rigidum inversum'. Meanwhile, in 1691, Leibniz, Huygens and Johann Bernoulli independently derived the curve's equation in answer to a challenge by Jakob Bernoulli.
Why does flipping work? A hanging chain is in pure tension: every link pulls exactly along the line of the chain, with no bending. Turn the shape upside down and the tension becomes pure compression, just what stone and brick handle well, so the arch stands without bending forces trying to crack it. Kiln builders still make arches by tracing a hanging chain onto a form.
Real structures bend the rule. St Louis's Gateway Arch is often called a catenary, but it tapers towards the top, so it is a 'weighted catenary', the shape of a chain with lighter links in the middle. And a suspension bridge's main cables, carrying a flat, heavy deck, hang closer to a parabola. Some ancient builders found the shape by eye: the great arch of Taq Kasra at Ctesiphon approximates a catenary.
Source: Wikipedia — Catenary · Text summarised from Wikipedia (CC BY-SA 4.0)
