The relevant role of calculus in renaissance domes' design, before differential geometry was born

We take under consideration two masonry domes built during Italian Renaissance: the Santa Maria del Fiore dome in Florence by F. Brunelleschi and St. Peter's dome in Rome by Michelangelo. No original computation is known currently in reference to neither one nor the other. However present mathematical computations show that, even though probably unconsciously, the designs of both the domes were inspired by the use of catenary, a trascendent mathematical curve with excellent stability properties. It was defined and studied in details by Leibniz and Bernoulli brothers at the end of the 17th century (within the context of rising differential geometry) a long time later than both the domes were completely built. Therefore the two considered domes suggest that some practical applications of differential geometry in architecture design can be considered foregoing its theoretical formalization. Here we show some numerical computation which support this hypothesis. The aim of this work is to show that knowledge quite often is created in processes and not invented in a moment; in particular here we focus on the development of differential calculus related to its applications to equilibrium of masonry domes.