This answer comes from perhaps a less traditional AskHistorians source, an engineering textbook: "Structures: Or Why Things Don't Fall Down" by J.E. Gordon.

Gordon has a number of chapters dedicated to classic structures such as arches and columns. Modern structural analysis for masonry is based on Hookian elasticity and were established by Thomas Young. Young was born a few centuries too late to have helped the Romans when they started building arches (1773). The book states that most masonry was performed by dead reckoning throughout history before that point. This was possible because stone is extremely strong under compression, so all the designer really had to do was make sure their building wasn't going to tip over.

I'd highly encourage everyone to read the book. It's quite accessible to non-engineers, and it's a wonderful read. It's also available for free: https://archive.org/details/StructuresOrWhyThingsDontFallDown