# When did architects first become aware of the usefulness of the catenary arch?

The Wikipedia-entry for catenaries lists Robert Hooke as the first to study catenaries mathematically, in the 1670s. However, the dome of the Florence Cathedral, completed structurally in 1436, follows a catenary arch, which its builder seems to have been aware of, meaning that the principle of using a catenary as a good solution (although not at that time mathematically proven to be optimum) for self-supporting structures must have been known earlier than the time of Hooke. Many other examples exists, some of which even dates back several thousands of years.

When did knowledge of the above-mentioned principle first become widespread? Who discovered it first (if possible to answer)?

Note that I'm not only interested in Western architecture; for instance, did Persian architects building mosques pre-Renaissance know about it?

• Seems to me you gentlemen should be talking about the arc of the cycloid. As I recall (see Stillman Drake) Galileo remarked the arc of the cycloid was exactly correct for the arch of a bridge as the tangent to the arc of the cycloid guaranteed a uniform distribution of weight all along the arch. I suspect ignorance of the properties of the arc of the cycloid is the reason we have wobbly bridges in this the 'modern age'. Apr 5 '18 at 19:49
• According to Dati, Galileo did consider cycloidal arch for a bridge in Pisa, but it was "exactly correct" not because of weight distribution but because it provided "a bridge of most beautiful form", see Camerota's Geometric objects, p. 230. But how is this relevant to the asked question? Apr 5 '18 at 23:43
• Drake Stillman (1978) 'Galileo at Work His Scientific Biography' Dover Paperback page 406: "That arched line" (the cycloid) " occurred to me to describe more than fifty years ago, and I admired it as a very gracious curve to be adapted to the arches of a bridge." Apr 6 '18 at 4:19

What the builder Neri di Fioravanti was aware of is not catenary arch but quinto acuto (pointed fifth) or Gothic arch, as the linked article explicitly states. The arc sits on top of a diameter composed of two halves drawn with 4/5 of this diameter spliced at the cusped top (sometimes smoothed out). It is hard to say if this is a better approximation to the catenary than parabola, which Galileo described as approximating hung chain in Two New Sciences (1638). That the chain curve is not a parabola was shown by Joachim Jungius also before Hooke, but only published in 1669. It was Hooke who linked the catenary to architechture in connection with the rebuilding of St Paul's Cathedral , see Catenary: History.

Although Romans still used spherical domes, Taq Kasra, the Archway of Ctesiphon has the shape close to the "catenary" or pointed fifth, the method is conjectural. This seems to be the earliest known occurence, a c. 3-6th century AD Persian structure, presumably the main portico of the audience hall of the Sassanid shah's throne room, when Persia was still Zoroastrian. After the Islamic conquest it was converted into a mosque. Explicit uses of the pointed arch also predate the Florentine dome and Gothic architecture like Notre-Dame de Paris.

"The pointed arch is also a characteristic feature of Near Eastern pre-Islamic Sassanian architecture that was adopted in the 7th century by Islamic architecture and appears in structures like the Al-Ukhaidir Palace (775 AD), the Abbasid reconstruction of the Al-Aqsa mosque in 780 AD, Ramlah Cistern (789 AD), the Great Mosque of Samarra (851 AD), and the Mosque of Ibn Tulun (879 AD) in Cairo. It also appears in the Great Mosque of Kairouan, Mosque–Cathedral of Córdoba, and several structures of Norman Sicily...

The majority view of scholars however is the idea that the pointed arch was a simultaneous and natural evolution in Western Europe as a solution to the problem of vaulting spaces of irregular plan, or to bring transverse vaults to the same height as diagonal vaults, as evidenced by Durham Cathedral's nave aisles, built in 1093."

• Thank you for the answer. I'm sorry, I should have been more explicit: When I referred to "its builder", I meant Brunelleschi, the builder of the dome, not Neri. In the linked article, it says: "As we will demonstrate, Brunelleschi and his friends mastered the physical principles of the catenary." Apr 3 '18 at 7:04
• @BobsonDugnutt On that the author is fairly speculative and anachronistic, involving Leibniz and the least action principle to reinterpret the old slacking line technique for laying bricks (corda da murare). That would still only give a hanging catenary for reinforcement, not a standing catenary of later arches, by her own admission. And even for that purpose she clearly believes that Brunelleschi discovered the "physical principle of the catenary" by himself. Apr 3 '18 at 22:05
• I guess you're right, the article is rather speculative - it should probably be changed on the wiki-entry. One last question though: You mention the Taq Kasra - is there any evidence of the principle of using catenaries being known to dome-builders after The Muslim Conquest, but before Hooke? Apr 4 '18 at 7:42
• @BobsonDugnutt It seems to me that people use the term "catenary arch" very loosely, Wikipedia states "catenary arch is a type of architectural pointed arch... used in Gothic architecture" (!) In this sense, sure. Muslim Heritage mentions "catenary vaulting" as a "Muslim feature" used in the church of Cluny c. 1090, but the supposed point of origin, Ibn Tulun Mosque, clearly uses pointed arch. It would have been hard for early builders to erect a standing catenary shape, two circular arcs are much easier to follow. Apr 4 '18 at 19:22
• Well, to be fair, the wiki-entry follows that up with "... that follows an inverted catenary curve," which does make the definition more precise. Thank you for the effort you put into this answer. Apr 4 '18 at 19:43

The following is not an answer; it's a comment.
Also, I'm not providing historical information.
However, I feel that the information that I present is interesting and useful to know.

Thinking that a catenary arch has general usefulness in architecture is actually problematic, I will argue. There is a wrong assumption at play.

A catenary is the mathematical solution to the problem of a structure (either hanging chain or standing arch) that has a uniform weight per unit of length. As we know, the catenary is the solution with equilibrium everywhere along the length.

However, in the case of any large building you make the walls thick at the base, so they can carry the total load, and you make the walls thinner as you go higher. That means that near the base the structure will have more weight per unit of arch length than the top section. And that means the mathematical solution will be somewhat different from a catenary.

Domes

With domes using a catenary cross-section is even less applicable.

First, take the case of a theoretical dome, a dome with a shell that has the same thickness everywhere. What can we say about its equilibrium shape? If you would take a catenary arch, and you would rotate that around its symmetry axis then the resulting dome that is not an equilibrium shape.

Demonstration:
Imagine two catenary arches, at right angles to each other, crossing each other at the top. That means the top stone is shared between the two arches. It follows that that in order to be overall in equilibrum the cross-section of that structure must be somewhat flatter than a catenary. Imagine adding more and more crossing arches (adding up to a dome). At the base of that structure weight will be contributed by each arch, but at the top most of the weight will be shared weight.

So: even in the case of a dome with a uniform thickness of the shell the equilibrium shape will not be one that has a catenary-shape cross section.

Let's take one of the most famous domes: the Pantheon in Rome. Naturally the builders of that structure constructed it with a tapered thickness of the shell. Very thick at the base, gradually thinning out towards the top. (Also, the aggregate in the concrete varies, with pumice used at the very top.) The result: different weight per unit of surface from base to top.

So, in the case of a real world dome there are two independent factors that affect what the equilibrium shape will be, and both of them will shift the mathematical solution away from a catenary cross-section.

Of course, this doesn't necessarily mean that you cannot build a dome with a catenary-shape cross-section. If the structure is strong enough then you can. The structure has to be able to withstand wind loads anyway. That is, the structure must be able to withstand bending stresses anyway, so chances are that the bending stresses that arise from not being an equilibrium shape are well within the range of what the structure is able to handle.

• +1 for the correction. The equilibrium solution for a (uniform) dome can be found here: Hooke's Cubico-Parabolical Conoid, J. Heyman. Apr 4 '18 at 19:54
• It has been used, though, in at least one architecturally significant building. If you visit the Basílica de la Sagrada Familia in Barcelona there is (or was, at any rate) an exhibition about the design process which includes a model made with (multiple) hanging cords. Apr 5 '18 at 7:15
• @PeterTaylor Yes, Gaudi was pretty fond of catenaries it seems. Apr 5 '18 at 20:00
• @BobsonDugnutt I didn't know about Hooke's Cubico-parabolical Conoid. Anyway, presumably Christopher Wren didn't use that because the stone lantern on top of the dome is 850 tonnes. As far as I can tell from a drawing of a cross section the load bearing structure is a straight cone. Not a fancy design, but I think the best design: it ensures that at every point the stress is compressive stress. Apr 7 '18 at 10:15
• Gaudi used catenaries, but he also attached weights to numerous places to model the weight of the structure at those points. Apr 20 '18 at 17:25