Archimedes and Cavalieri's Principle
Archimedes is considered to be one of the greatest mathematicians and
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,
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.
- Fill up the sphere with sand and pour it into the cylinder.
Approximately what fraction of the cylinder does the sphere take up?
- How many cones of sand does it take to fill up the cylinder?
What fraction of the cylinder does the cone take up?
Use only your answers in 1 and 2 to
make (and write down) a conjecture
relating the cylinder to
the cone plus the sphere.
- Test your conjecture and explain your results.
- 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
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
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
to see Archimedes' argument.