According to a Greek text that was overwritten with Christian prayers, Archimedes worked out some of the principles of calculus over 1900 years before Newton and Leibniz. He called it The Method.
In The Method, Archimedes was working out a way to compute the areas and volumes of objects with curved surfaces, which was also one of the problems that motivated Newton and Leibniz. Ancient mathematicians had long struggled to "square the circle" by calculating its exact area. That problem turned out to be impossible using only a straightedge and compass, the only tools the ancient Greeks allowed themselves. Nevertheless, Archimedes worked out ways of computing the areas of many other curved regions.
Much more is explained in the book The Archimedes Codex. The entire text is available for free on Google Books (you might need this). (via long now)