"Riemann gave a rigorous definition of the integral by enclosing it between... the "lower sum"... the sum of the areas of the rectangles below the curve, and the "upper sum"... the sum of rectangles of somewhat greater height, which cover the area. The treatise on conoids and spheroids shows that Archimedes was familiar with this method of inclusion and... used it for the determination of volumes. But... one cannot say that he was familiar with the concept of the integral. His integrals always remained tied to a definite geometric interpretation, as volumes or as areas of plane figures. We have no evidence that he understood that one single concept is the foundation of all these geometric interpretations... he bases his rigorous proofs on totally different methods... Nevertheless, his rigorous determination of areas and volumes make Archimedes the precursor of the modern integral calculus."