"What has been objected by an anonymous author, in the Leipsic Acts of Feb. 1699, in his animadversions on my demonstrations concerning the catenary, is this: that I have undertaken to demonstrate, after my manner, a matter found out and published by others seven years ago. This is true, and I cannot find any thing in it that is blame worthy. Those great men Huygens, Leibnitz, and Bernouilli, have discovered and communicated many properties of the catenaria, but without demonstration. I have contrived demonstrations, which was the thing I undertook to do. But was this matter that is the nature and primary properties of the catenaria all found out and published by others? ...From all ages architects have made use of arches in public buildings, as well for strength as beauty. Yet what was the true geometrical figure of an arch was not known before my demonstrations came out."
In physics and geometry, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field.
"The most difficult properties of the Catenary were revealed before the close of the seventeenth century. This curve is entitled to particular attention, not only because it throws light on the theory of arches, but because it applies directly to the construction of suspended bridges, which are now deservedly coming into repute."
"One can roll noncircular wheels over appropriate road surfaces. The most striking example of this is the fact that a can roll on a road that consists of linked catenaries (the catenaries are defined by y = - cosh x) ...(the ride is not smooth in the sense that the center does not move forward at a constant rate of rotation, but... the actual ride... feels quite smooth). This animation was inspired by an exhibit at San Franciscos ..."
"In 1690... Jacob Bernoulli brought up the problem of the catenary in a memoir... in the ...Huygens solution represents the past... a complex, though skillful, geometrical method. Leibniz, using his new [infinitesimal calculus] reaches a correct analytical formula...y/a = (b^\frac{x}{a} + b^\frac{-x}{a})/2 where a is [a] segment... and b... corresponds to... e... ...supplied two correct constructions ...presents valid statistical arguments and... new and important... equations of equilibrium in differential form. ...In 1697-1698, Jacob Bernoulli was the first to derive the general equations that not only solved the problem, but also permitted the treatment of the more general theme of the equilibrium of a flexible rope, subject to any distribution of tangential (f_t) and normal (f_n) forces. Bernoullis equations are...\frac{dT}{ds} + f_t = 0, \qquad \frac{T}{r} + f_n= 0where T is the tension, s the curvilinear abscissa, and r the radius of curvature."
"It was comparatively late that the theory of arches attracted the notice of mathematicians. Dr. Hooke gave the hint, that the figure of a perfectly flexible cord or chain, suspended from two points, was the proper form for an arch. Galileo considered the catenary as a parabolic curve, and John Bernouilli appears to have been the first who discovered its nature. Dr. Gregory (Phil. Trans. 1697) published an investigation of its properties, and observes that the inverted catenary is the best form for an arch on account of its lightness. This is true so long as it is not pressed by an extraneous weight. It is not, however, capable of bearing a load on any part, much less of being filled up on the spandrels, which must be the case in practice. Other considerations must be involved before it can be fitted to receive a roadway or other weight, either upon its crown or haunches."
"A non-catenary curve might be perfectly doable, but it takes more material, it has bigger beam sections, and overall it is much more complicated to construct... Even if the cladding falls out, the interiors and everything else falls away and the whole thing turns to dust and rubble and sand, [the catenary] should still stand."
"Within a shed erected on the construction site of the church of the Sagrada family... Gaudí... made an upside-down model using lightweight cables to represent the structural lines of the future church—a model based on the structural notion of the inverted catenary. ...Analogically represented by little pouches filled with lead pellets the action of the stresses has been done ...The resulting chain configurations are used to determine the geometrical shapes and structural profiles of columns, pillars, arches, and vaults. ...Vicens Vilarrubias i Valls took photos of the model ...Gaudí used these photos upside-down to draw over them the external and internal elevations, studies of details and sections of the building."