From Robert Osserman’s paper:
It was Robert Hooke who in 1675 made the connection between the ideal shape of an arch and that of a hanging chain in an aphorism that says, in abbreviated form, “As hangs the chain, so stands the arch.” In other words, the geometry of a standing arch should mirror that of a hanging chain. The horizontal and vertical forces in a hanging chain must add to a force directed along the chain, since any component perpendicular to the chain would cause it to move in that direction to gain equilibrium. Similarly, one wants the combined forces at each point of an arch to add up to a vector tangent to the arch. In both cases, the horizontal component of the force is constant and simply transmitted along the arch or chain, while the vertical forces are mirror images.
Wikipedia provided a full quote from Hooke:
Ut pendet continuum flexile, sic stabit contiguum rigidum inversum,
“As hangs a flexible cable so, inverted, stand the touching pieces of an arch.”