According to legend, Archimedes ran naked through the streets of ancient Syracuse shouting "Eureka!" after discovering his famous Law of Buoyancy, the basic law that determines how things float. He illustrated this law in his work "On Floating Bodies" by computing various floating positions of a solid paraboloid. With the geometric tools of his day Archimedes could only consider those cases when the flat base of the paraboloid is not cut by the water. However, as I show using modern computing power, the most interesting things happen when the base is cut by the water. For example, an iceberg that is slowly melting can suddenly overturn, or an obelisk originally sitting on solid ground can come crashing down when the soil under it liquefies during an earthquake. Such drastic phenomena are now studied in Catastrophe Theory, a field that Archimedes could have begun if he had had the computational tools to investigate all the possible cases of his floating paraboloids.