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.
  4. Test your conjecture and explain your results.
  5. Next, test your conjecture by using formulas for the volume of these three objects, in terms of r, the radius of the sphere. First sketch the figures and label the dimensions, in terms of r, and then write down the volume formulas and use them to test your conjecture.

    Archimedes was trying to derive the formula for the volume of a sphere, so he could not assume this formula anywhere in his work. I've found some interactive web pages that will give you an idea of his methods, but they are slightly different than his original construction, which is more accurately set up above. Instead of a sphere of radius r, begin with half a sphere (or hemisphere) of radius r. Also take a cylinder and a cone that are half as tall as before. You can come up with a conjecture that relates these three objects and test out your conjecture by using the corresponding formulas. You can look at the interactive explorations on
    http://mathcentral.uregina.ca/QQ/database/QQ.09.99/partridge1.html   and on
    http://www.walter-fendt.de/m14e/volsphere.htm   to see Archimedes' argument.