Archimedes is credited with quantifying the concept of the center of gravity of an object and in his works he determined the locations of the centers of gravity of many planar and solid bodies. His calculations, however, implicitly assumed that the body was immersed in a uniform gravitational field, so he was actually determining the location of the body´s center of mass (or centroid). He did not realize that the concept of a center of gravity is not applicable in a nonuniform gravitational field, a fact that many are not aware of even today. This led to his incorrectly proving an erroneous theorem at the end of Book 1 of his work "On Floating Bodies". His theorem states that a truncated sphere floating in a body of water on a spherical earth that attracts objects to its center will float stably with its base horizontally under very general conditions. I´ll discuss his error and suggest an alternate proof of a similar correct result.