### Archimedes and Cavalieri's Principle

 Archimedes is considered to be one of the greatest mathematicians and scientists. In 212 BC, Romans stormed the city of Syracuse. Seventy-five year old Archimedes was so focused on his mathematical work that he ignored and hence enraged a soldier. The soldier then killed him. According to Plutarch (AD 45-120), Parallel Lives: Marcellus, Archimedes had requested that a pictorial representation of a sphere and a cylinder appear on his tombstone. From this, we can infer that he must have considered his work on a sphere and a cylinder to be one of his greatest accomplishments. Cicero (106-43 BC), in Tusculan Disputations, Book V, Sections 64-66, states that he went to Syracuse and indeed found the grave which contained the pictorial representation along with text verses. The formulas for the volume and surface area of a cylinder were known before Archimedes' time, but those for a sphere were not known. Archimedes wanted to find exact expressions for the volume and surface area of a sphere, and he did indeed do just this by using ideas related to Cavalieri's Principle.
1. Fill up the sphere with sand and pour it into the cylinder. Approximately what fraction of the cylinder does the sphere take up?
2. How many cones of sand does it take to fill up the cylinder?
What fraction of the cylinder does the cone take up?
3. Use only your answers in 1 and 2 to make (and write down) a conjecture relating the cylinder to the cone plus the sphere.