Domes built in the 18th century by the Western world, broadly defined, benefited from the newly developed mathematics of calculus and the study of statics. There were numerous studies on the catenary curve as the most stable form for arches and vaults and the ideal shape for a dome was debated in the late 18th century. Calculus allowed Guarini's experimental designs with intersections of spheres, barrel vaults, and oval domes to be proven and they would become the foundation of Rococo spatial arrangements.
In the areas of the Holy Roman Empire, rebuilding of the many palaces and churches destroyed in the Thirty Years' War had begun by the end of the seventeenth century. Ovals resolved the tension between longitudinal and centralized spaces in German Baroque architecture, but resting them on drums created problems and so the domes often rested directly on arches or pendentives. Bulbous domes were popular in central and southern Germany and in Austria in the 18th century, and influenced those in Poland and Eastern Europe in the Baroque period. However, many bulbous domes in the larger cities of Eastern Europe were replaced during the second half of the eighteenth century in favor of hemispherical or stilted cupolas in the French or Italian styles.
Domes in Mexico were single shell constructions, with few exceptions, and the domes and lanterns were commonly covered with glazed tiles. A wooden structure supporting a layer of earth fill to support the domes during construction, a technique used in Europe to build domes since the Romanesque era.
Although never very popular in domestic settings, domes were used in a number of 18th century homes built in the Neoclassical style. In the United States, small cupolas were used to distinguish public buildings from private residences. After a domed design was chosen for the national capitol, several states added prominent domes to their assembly buildings.
Developments
Over the course of the eighteenth century, developments in mathematics and the study of statics led to a more precise formalization of the ideas of the traditional constructive practices of arches and vaults, and there was a diffusion of studies on what was considered the most stable form for these structures: the catenary curve. Robert Hooke had published a Latin anagram in 1675 indicating that the perfect shape for an arch to resist a given load was the inversion of the shape a flexible cord assumes under such a load, but he was unable to define this catenary curve mathematically and the anagram was deciphered only after his death. In 1704, James Bernoulli wrote that an inverted catenary arch of any thickness will resist its own weight, which likely inspired Pierre Bouguer to conclude in his Mémoire sur le lignes courbes qui sont propres à former les voûtes en dômes (1734) that the optimal dome shape was a rotated inverted semi-catenary. Giovanni Poleni, tasked by the Pope in 1743 to study the stability of the dome of St. Peter's Basilica, built scale models to demonstrate that a catenary curve can be found to fit between the two shells of the dome and published drawings in 1748, "the very first example of conscious and documented use of catenary in Architecture".
The treatise by Guarino Guarini, published posthumously in 1737, included the way to draw various vaults, but not how to build them. Building on Fontana's work, Bernardo Antonio Vittone published Istruzioni elementari dell’architettura civile in 1760, in which he recommended using ogival or ellipsoidal profiles to increase the proportional height of domes and increasing the angle at which they intersected with their lanterns, for both structural and aesthetic reasons. To counter the negative impact this raised profile has on the interior appearance, domes with two or three layers of vaults were built, with openings in the lower levels to admit light.
Analytical approaches were also being developed and debated in the eighteenth century, particularly between French and Italian mathematicians and architects, but were considered too theoretical to be used in construction. The study of dome structures changed radically, with domes being considered as a composition of smaller elements, each subject to mathematical and mechanical laws and easier to analyze individually, rather than being considered as whole units unto themselves. In 1734, mathematician Pierre Bouguer (and later others) argued that the dome could be thought of as sliced into a series of independent wedged-shaped segments meeting as arches. Therefore, a dome as a whole was stable if each constituent arch was stable and analysis of a dome could be performed the same way as analysis of an arch. He published his Mémoire in 1736, "the first treatise on the theory of the dome", with possible stable forms for domes when ignoring friction. A. F. Frézier built on this work and used a method of analyzing the thrust of domes as made of a series of masonry arch elements that could then be compared to the known thrust behavior of barrel vaults of the same span. The arch-slices approach was applied in 1743 by the three mathematicians hired to advise on the stability of the cracking dome of St. Peter's Basilica: François Jacquier, , and Roger Joseph Boscovich. It would subsequently become the standard approach to dome analysis. Their use of a simplified mathematical model of the dome's structure to calculate the strength an additional chain would need to ensure the safety of the dome was an unprecedented step, although they did not provide enough detail of the calculations or the assumptions of their model for their result to be independently verified. A 1773 study by Charles-Augustin de Coulomb focused on the verification of static equilibrium in masonry structures and took friction into account.
In the late eighteenth century, the ideal shape for a dome was debated by Charles Bossut, Lorenzo Mascheroni, , and , among others. The approach of Bossut (1776), Salimbeni (1787), and Bérard (1810) combined the frictionless model of Bouguer with the arch-slicing technique of Frézier in purely mathematical research that was never applied to the design of actual domes.
Kingdom of France
Adjacent to a hospital and retirement home for injured war veterans, the royal chapel of Les Invalides in Paris, France, was begun in 1679 and completed in 1708. The dome was one of many inspired by that of St. Peter's Basilica and it is an outstanding example of French Baroque architecture. In 1861 the body of Napoleon Bonaparte was moved from St. Helena to the most prominent location under the dome.
The dome of Val-de-Grâce in Paris (1645–1710) was designed by Jacques Lemercier after having worked in Rome for seven years. It includes an inscription around the dome relating to the Bourbon kings. Inspired by St. Peter's Basilica, its dome likewise has two shells, but the outer shell is much taller in order to compensate for the foreshortening effect from viewing the exterior dome from nearby on the ground. The inner shell is made of stone and the outer shell is made of wood.
Plans for the Church of St. Genevieve, the patron saint of Paris, were approved in 1757 with a dome tall over a Greek cross plan. The architect of the church, Jacques-Germain Soufflot, wanted to surpass the dome of London's St. Paul's Cathedral and, like St. Paul's, the dome consisted of three shells. Unlike St. Paul's, and due to advances in mathematics and engineering, all three shells were built of stone and made a part of a structural system that permitted support by thinner piers and walls. Finished in 1790 by Jean-Baptiste Rondelet, the outward thrust of the domes were restrained by the use of metal rings but issues with the supporting pillars required Napoleon to have Rondelet strengthen the pillars in 1806. Later named the Panthéon, the crossing dome of the building has a span of and 8 inches and a height of . Two "pre-tensioned high resistance cables" were added to the dome in the early 21st century.
The Halle aux blés, a corn exchange in Paris, was a rotunda was built from 1763 to 1766 by Nicolas Le Camus de Mézières with a central circular courtyard wide. The courtyard was covered with a laminated timber and glass dome built from 1782 to 1783 by André Jacob Roubo, to the designs of Jacques-Guillaume Legrand and Jacques Molinos. It used the structural system published by Philibert de l'Orme in his 1561 treatise, with short sections of curved wood sandwiched together with staggered joints and held together with wooden pegs to form longer curved vertical beams braced horizontally by wooden purlins, which avoided the need for centering. The building became a popular sight in Paris and was seen and admired by Thomas Jefferson, who later used the same structural system for the small dome he added to his Monticello home. The 1783 dome burned down in 1803 and was replaced from 1808 to 1813 with an iron and glass dome designed by François-Joseph Bélanger.
Kingdom of Great Britain
London's Great Fire of 1666, following a devastating outbreak of plague in the city that killed a fifth of its population, spurred the commission of Christopher Wren to rebuild St. Paul's Cathedral, which occurred over the course of 35 years. Robert Hooke, who first articulated that a thin arch was comparable to an inverted hanging chain, may have advised Wren on how to achieve the crossing dome. Wren may also have been informed of the structural problems of the dome of St. Peter's Basilica by John Evelyn, who had examined it, and did not finalize his design for a dome three-quarters its size until shortly before its construction started in 1705.
When finished, the dome had three layers: an inner dome with an oculus, a decorative outer wooden dome covered in lead roofing, and a structural brick cone in between. The brick cone ends in a small dome that supports the cupola and outer roof and the decorated underside of which can be seen through the inner dome's oculus. The structure rises to the cross at its summit, but is evocative of the much smaller Tempietto by Bramante. The use of the brick cone, in addition to other innovations, allowed the piers beneath the dome to be reduced in size. The thickness of the brick cone is 450 millimeters. The inside wall of the drum supporting the inner dome slopes inwards to the base of the dome for structural reasons, a design feature seen in the 1672 model and apparently inspired by Robert Hooke's ideas on arch shape, and three encircling chains were used to resist horizontal thrust. The dome is supported by eight piers with a veneer of Portland stone over a core of rubble infill, which were damaged by the added pressure from the dome's construction and needed repairs in 1709. The dome was completed in 1710. The inner side of the dome was painted by James Thornhill from 1715 to 1717 with eight scenes of St. Paul from the Acts of the Apostles in a brown stone-like color mandated by the presiding committee. Wren's structural system became the standard for large domes well into the 19th century. The iron chains used to encircle the 34-meter-wide dome have since been replaced by stainless steel girdles. Damage to the outer timber truss dome during World War II resulted in the timber being replaced by reinforced concrete.
Although never very popular in domestic settings, domes were used in a number of 18th century homes built in the Neo-Classical style, including the 1720s Chiswick House, in West London. The Palladian mansion Penicuik House, built by Sir James Clerk, included a stable block with a domed dovecote built as a faithful imitation of the destroyed ancient monument Arthur's O'on. A chapel at Gibside by James Paine has a trefoil plan with a central dome. The domed mausoleum by Nicholas Hawksmoor at Castle Howard was built in 1742 and is similar to Bramante's Tempietto. Robert Adam referenced the Roman Pantheon dome in his 1767 design of Luton Hoo in Bedfordshire.