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.
NOTE: In the construction we will examine, Archimedes relates the volume of the sphere to the volume of the cylinder and the volume of the cone. The volume of the cylinder is elementary once the area of the circle is known. The volume of the cone is established in Euclid by comparison with the volumes of inscribed and circumscribed pyramids, referring to the fact (elementary solid geometry) that the volume of a triangular (and thence arbitrary) pyramid is one third the volume of the prism on the same base to deduce that the same relation holds between the volume of the cone and the volume of the cylinder. Euclid's very elegant argument involves an arbitrarily large but always finite number of approximating steps. Archimedes' construction is fundamentally infinitesimal.