The fact that ancient masonry structures are still standing testifies to the engineering skills of their builders. Gothic vaults represent such builders’ peak achievements, so this book approaches the principles underlying the longevity of masonry structures by applying structural analysis to Gothic architecture. Of the three main structural criteria of strength, stiffness and stability, only the last is usually critical for masonry structures. The semicircular arch shows the importance for stability of correct proportions: as realised by Hooke, and later exploited by Poleni and others, to be stable, an arch must contain its line of thrust, which follows a catenary (the shape of a hanging chain, inverted). Advances in the theory also came from, among others, Galileo (on strength) and Navier (on stability, with an emphasis on solving equations with boundary conditions). The examples of three- and four-legged tables show that small changes in boundary conditions of structures can lead to large changes in the positions of thrust lines. The theory abandons the quest to know the ‘actual’ state of a structure, instead examining (and avoiding) possible modes of collapse.