The works translated here--the two books On the Sphere and Cylinder--were a source of great pride for Archimedes, the greatest scientist of antiquity. Accompanying this translation is the first scientific edition of the diagrams, which incorporates new information from the recently discovered Archimedes Palimpsest. The volume also includes the first English translation of Eutocius's commentary. Reviel Netz's commentary analyzes Archimedes's work from contemporary research perspectives such as scientific style and the cognitive history of mathematical texts.