Curves and Surfaces II: Algebraic Calculus and Archimedes’ Parabolic Area Formula for Cubics

Abstract:

Archimedes’ famous determination of the relation between a parabolic arc and the area of a maximally inscribed triangle, involving the factor 4/3, is the earliest computation in Calculus. It reveals that there is something special about the parabola, not shared by circles and hyperbolas – namely we can get at their areas exactly! Perhaps this result shows that Calculus was not discovered by Leibniz and Newton and their immediate predecessors, but rather by Archimedes.

 We will give a modern proof of Archimedes’ formula. And we will show how the result extends, in a remarkable and beautiful way, to cubic functions. The interesting new technology of Algebraic Calculus will appear, and a projective view of planar curves will be important. The talk will be suitable for undergraduates.