What Shape Does a Hanging Chain Make? Not the One Everybody Says

A hanging chain and a hanging cable carrying a level deck look alike and are governed by different conditions. A suspension bridge cable carries a roadway of uniform weight per horizontal distance, and that produces a parabola. A chain hanging freely carries only itself, and its weight is distributed uniformly along the length of the curve rather than along the horizontal, which is a different problem because the curve is longer than its horizontal span near the ends where it steepens. The resulting shape is described by a different function entirely, one built from exponentials, and it hangs slightly differently, though the two are close enough that the difference is invisible in a shallow hang.

The problem took a surprisingly long time and drew in the best mathematicians available:

• •Galileo asserted in 1638 that the shape was a parabola
• •Joachim Jungius showed before 1669 that it was not
• •Jakob Bernoulli posed it as a public challenge in 1690
• •Leibniz, Huygens and Johann Bernoulli each solved it in 1691
• •The solution required the newly developed calculus
• •The name was coined later, from the Latin for a chain

A hanging chain is in pure tension everywhere, since a chain cannot push, and it settles into the shape in which the forces along it are purely along its own length with no sideways component at any point. Inverting that shape reverses every force, turning the tension into compression, so an arch of that form carries its own weight in pure compression with no bending anywhere. That matters enormously in masonry, which is strong in compression and very weak in bending, so an arch of this shape can be built thin while any other shape needs enough thickness to contain the line of thrust. Robert Hooke stated the principle in 1675, publishing it as an anagram so he could claim priority later.

Comparing the two shapes directly makes the distinction concrete rather than a matter of assertion. Both are symmetrical about a vertical axis, both open upward and both flatten at the bottom, so at a shallow hang they overlie each other almost exactly and no eye could separate them. The difference grows with the steepness, and a chain hung deeply sags below a parabola through the same three points, because the extra length near the steep ends adds weight there that the level-loaded case does not have. A practical consequence is that treating a deeply hung cable as a parabola introduces real error, which matters for transmission lines, cableways and moorings.

The principle has been applied by people who knew the mathematics and by people who did not. Antoni Gaudi built hanging models from strings and small weighted bags, photographed them and inverted the photographs to design the arches and columns of his churches, which is a physical computation rather than a calculation and produces the correct answer automatically. Traditional builders arrived at similar forms by trial. The Gateway Arch in St Louis, completed in 1965, is a weighted version of the curve, with its equation inscribed on the structure. Overhead wires for electric railways hang in the curve and gave the arrangement its industry name. And the shape appears in soap films and in the path of a free-hanging rope anywhere.

A chain carries its weight along its own length rather than along the horizontal, which makes its shape an exponential curve rather than the parabola a loaded suspension cable forms. Leibniz, Huygens and Johann Bernoulli solved it independently in 1691 using the new calculus. Inverting the shape turns pure tension into pure compression, giving an arch with no bending anywhere, which Gaudi computed with hanging string models.