Galileo's mechanical studies were deeply influenced by two historical precedents. The first was the investigative model of Archimedes, characterized by rigorously geometric analytical methods. The second was the detailed interpretative work of Archimedes's legacy performed by humanists with geometrical expertise, such as Federico Commandino and Guidobaldo del Monte. In his Mathematical discourses and demonstrations, Galileo inaugurated the science of the resistance of physical bodies. For the first time, strict methods were proposed for theoretically predicting the breaking point of bodies subjected to traction and of bodies to which weights had been applied. Galileo also made major contributions in the field of statics, through innovative analyses on the problems of equilibrium and the operation of simple machines. In hydrostatics, we should note his reconstruction of the method used by Archimedes to expose the craftsman who had sold the tyrant of Syracuse a crown in gold and silver alloy, passing it off as a crown of solid gold. And there are his ingenious reflections and demonstrations on the phenomena of floating.