Just what did archimedes say about buoyancy?

"A body immersed in a fluid is buoyed up with a force equal to the weight of the displaced fluid." So goes a venerable textbook¹ statement of the hydrostatic principle that bears Archimedes' name. Archimedes' principle is often proved for the special case of a right-circular cylinder or rectangular solid by considering the difference in hydrostatic forces between the (flat, horizontal) upper and lower surfaces, and then generalized by the even more venerable "it can be shown..." that the principle is in fact true for bodies of arbitrary shape.