He is known for his formulation of a hydrostatic principle and a device for raising water still uses for developing countries known as Archimedes screw. Archimedes was popular as he gained reputation for his writings in his own time. The reason for this was not a widespread interest in new mathematical ideas but rather that Archimedes had invented many machines which were used as engines of war. These were particularly effective in the defence of Syracuse when it was attacked by the Romans under the command of Marcellus. .
Archimedes principle indicates that the upward buoyant force that is exerted on a body immersed in a fluid whether fully or partially submerged is equal to the weight of the fluid that the body displaces. He also studied the stability of various floating bodies of different shapes and different specific gravities.
In his first book "On the Sphere and Cylinder" Archimedes shows that the surface of a sphere is four times that of a great circle, he finds the area of any segment of a sphere, he shows that the volume of a sphere is two-third the volume of the circumscribed cylinder, and that the surface of a sphere is two-third the surfaces of a circumscribed cylinder including its bases. In the second book of this work Archimedes' most important result is to show how to cut a given sphere by a plane so that the ratio of the volumes of the two segments has a prescribed ratio.
In On spirals Archimedes defines a spiral. He gives fundamental properties connecting the length of the radius vector with the angles through which it has revolved. He gives results on tangents to the spiral as well as finding the area of portions of the spiral. In the work on conoids and spheroids Archimedes examines paraboloids of revolution, hyperboloids of revolution, and spheroids obtained by rotating an ellipse either about its major axis or about its minor axis.