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.