Archimedes discovered many math and science concepts, including the Archimedes Screw and siege engines. Much of his work contributed to the development of later ideas such as hydrostatics, levers and many math concepts.
One of the discoveries Archimedes is most known for has to do with the concept of volume. When a king asked the scientist to determine if his crown was real gold, he used water displacement to work out the answer. Legend says that the idea came to him when he stepped into the bathtub and noticed how the water moved as his body entered. This led him to run through the streets shouting "Eureka." Other discoveries credited to Archimedes include math concepts such as the relationship between the surface area of a cylinder and sphere and improvements on the simple lever.