The Ancient Romans have played an immeasurable role in shaping the modern world. It was the first truly global civilization, absorbing and Romanizing influences from all around it, and passing them on to us. Those qualities, although built upon an undeniably natively Roman foundation (Romanitas—“Roman-ness”—which, valued, for instance, masculinity/virtus and pietas), grew into something much broader, much more flexible, and ultimately, something very enduring.

As they grew from a small collection of villages in central Italy in the 8th century BCE to the dominant power in the Mediterranean by 146 BCE, the Romans openly used trial and error means to explore, refine, and adapt the strategic, operational, and tactical tools they (or their neighbors or their adversaries) had at their disposal. A “Roman stamp” was always placed upon these, but this modus operandi notably extended to military weapons and tactics, engineering, and architecture. In this last area, the Romans undoubtedly absorbed Greek and Etruscan influences. But the end product (often innovative or “kitschy”, depending on your perspective) was Roman.

Indeed, architecture is the most direct and readily visible element of the Romans’ vast cultural legacy for no reason other than we can see and sometimes touch the remains themselves. Its influence is of course much broader. In perhaps the majority of world capitals, edifices of obvious Roman (as distinguished from Greek) form can be found. Sports stadiums and arenas all over the world would appear to be mere copies of the Roman amphitheatrum. Two architectural forms, the arch and the dome (and more generally, vaulted architecture), were monumentalized to an unprecedented scale by the Romans and have been mainstays of the architectural repertoire over the past two millennia.

In this paper, I briefly explore two concepts related to vaulted architecture. In the first, main part, I discuss the evolution of vaulted architecture in Rome, and its massive increase in scale and its proliferation through the development of concrete. In the second, supplementary part, I derive the equations of shape for arches and axially-symmetric domes, and solve them in closed form.

 

Of course, the Romans did not have physics or infinitesimal calculus at their disposal as tools by which they could design. They did what made sense to them and what, through trial and error, remained within the scope of their cultural technological memory. The physical outcome of this is that Roman arches and domes are not strictly “optimal” mathematically, and in fact, some of the Romans’ techniques make no sense to the modern engineer. For instance, in the Pantheon, the utility of “building in” arches into walls as a stress reduction technique is negligible. On the other hand, some techniques (such as impregnating walls with amphorae, as at the Mausoleum of Helena) are of definite engineering value. The gradation of the Pantheon’s dome cross-section reveals a deep intuition of engineering principles, one that first year undergraduate engineers today learn to express mathematically in their study of hydrostatics.

In the case of a pure Roman arch, the individual arch elements do not transmit all of the weight they bear in a direction aligned with their orientation, as they would if their orientation matched the mathematical curves derived in the paper: Roman arches are usually based on circles and circular segments rather than the hyperbolic cosines, error functions, or Airy functions I show in the paper are solutions to the physical equations.

Since this is not the case, the material in Roman arches and domes must accommodate additional bending moments, which lead to internal shearing stresses (which, if stone is used, must also be borne by any cement at the stone interfaces). If the resultant shear stresses are sufficiently small, the stones will retain the shape they attain under mechano-elastic equilibrium without fracture. Of course, given the vast number of surviving Roman vaults, this has clearly been the case.