Royal road or labyrinth? Luca Valerio’s De centro gravitatis solidorum and the beginnings of modern mathematics

When Luca Valerio published his De centro gravitatis in 1603, he claimed he was opening up what he called a `royal road' to the investigation of centers of gravity of solid figures. Although the topic had been treated earlier by his immediate predecessors, Commandino and Maurolico, Valerio felt, and the authors of this study agree, that his work marked an innovation in its investigation of all solids known at the time. One part of this innovation was his conception of a general class of objects (in this case, solids satisfying certain symmetry conditions), as opposed to more specific objects, such as conic sections. And the other was the development of a considerable body of theorems applicable to that class, including the invention of the method of exhaustion, an invention usually ascribed to Eudoxus, in the sense that Valerio was the first to systematize this ancient device, in the first three theorems of De centro.In applying these general theorems to the solution of a wide class of problems, Valerio thus advanced beyond his ancient Renaissance predecessors