**The Method of Archimedes**

## 2. The "Method"

The "Method" takes the form of a letter from Archimedes to
his colleague Eratosthenes, in which he sets out to explain
how he discovered his theorems on volumes, and in particular
the volume of the sphere. He explains that he is presenting a
heuristic and physically based argument; his "published"
proof is different, completely geometric, in the style of Euclid.
This
document is precious because it gives us a look "behind the
scenes" at the working of the mind of a great mathematician.
It also has proleptic contemporary pre-echoes: we live
in a time where many mathematical insights come from
physical intuition; finding the completely mathematical
proofs is often not easy.
The Method, in short, involves equating area with mass, and
solving volume problems by balancing various areas at various distances
from a central fulcrum. As Archimedes was well aware, this does not make
sense, because if a volume is made up of stacked areas the areas
cannot have non-zero mass. Nevertheless this way of thinking led
him to the correct result.

© copyright 1999, American Mathematical Society.