Question: How could engineers and architects in ancient civilisations do their calculations without a calculator?

Alexandre Borovik answered on 16 Jun 2020:

Many trades of the past – not only architects and builders, but also, say, carpenters and tailors, apparently developed some tricks for measuring and estimating stuff they were working with, various mental shortcuts, mathematical by their nature, but never properly put in writing (for example, for reasons of professional secrecy or sacral nature of work) and therefore lost in history. Masonic lodges started as secret societies of stonemasons, for exactly this reason — but try to ask a modern days mason how he (they are always “he”, by the way) estimates stress forces in a stone arch, or how they shape the proverbial cornerstone of the Bible.Builders of Gothic cathedrals came with many counter-intuitive engineering solutions – for example, tall load bearing walls thickening from bottom up. I was intrigued by that and in cathedrals that I had a chance to visit, spent some time looking into every accessible for a visitor nook and corner for things which were put there by stone-carvers not for public, but for their own purposes. Quite common were small ugly creatures. apparently very strong, in darker corners of cellars, who were holding the weight of the colossal building on their necks and shoulders. It is my conjecture that gargoyles, jumping out of he walls of cathedrals with not very kind expressions on their faces, were another kind of mythological creatures who lived in the walls — they, I guess, were impersonations of the worst possible structural flaws in the building — stones which are squeezed out, shot out, by stresses in the wall — they might fly with the speed and momentum of a cannon ball. It is natural to suggest that when every stone had to be lifted and put in its place by human manual labour, the builders were thinking about stress and torque forces in the walls as of muscular tension in their own bodies. They were unable to measure these forces, but they could perhaps somehow imagine them and *compare* — the same way as we compare weights of stones, small – by weighing in a hand, big ones – by looking at them and trying to imagine that we are weighing them in hands. And ancient builders, most likely, had language, terminology for talking between themselves about all that. Mathematically, in modern terms, it is something that is called tensor calculus. What ancient builders did was proto-mathematics. This stuff deserves a better study.

An example from our times: Dame Celia Hoyles made a remarkably interesting study of arithmetic as it is actually used by nurses in hospitals when they have to compare, quickly, doses of drugs — and they do not what you perhaps expect — but correctly

I would love to read more about, say, folklore of odd mining communities – for example the 17th or 18th century Cornwall. In other countries, in similar folklore, signs of great respect to stress and torsion forces are quite noticeable. On British Isles, this stuff could happen to be much better recorded for history. Alas, I do not have time for that.