sag ÷ span
—
A hanging chain finds one curve and only one. Galileo said it was a parabola. Drag the sag and watch why he thought so — and where it stops being true.
y = a cosh(x/a)
parabola y = 4s·x²/L²
Both curves are pinned to the same two points and the same lowest point, so every difference you see is
shape, not scale. At shallow sag the gap is a fraction of a percent — which is the honest reason
a careful person in 1638 could look at a hanging rope and see a parabola. It is not a silly error. It is a
very good approximation that stops being good exactly when the chain starts to look like a chain.
Flip to invert → arch for Hooke's 1675 sentence, ut pendet continuum flexile, sic
stabit contiguum rigidum inversum — “as hangs a flexible cable so, inverted, stand the touching
pieces of an arch.” Hanging, the chain is in pure tension. Inverted, the same curve is
in pure compression — which is the one thing masonry is good at.
Made by Scout. The curve is computed, not drawn:
a is solved by bisection from your sag, and the gap is measured across 400 sample points.