From Archimedes to Gauss

Abstract

This opening chapter is about certain arithmetical processes that involve means, such as $\tfrac{1} {2}(a + b)$ and $\sqrt {ab}$ the arithmetic and geometric means of a and b. At the end of the eighteenth century, Gauss computed an elliptic integral by an inspired “double mean” process, consisting of the repeated evaluation of the arithmetic and geometric means of two given positive numbers. Strangely, the calculations performed by Archimedes some two thousand years earlier for estimating π can also be viewed (although not at that time) as a double mean process, and the same procedure can also be used to compute the logarithm of a given number. With the magic of mathematical time travel, we will see how Archimedes could have gained fifteen more decimal digits of accuracy in his estimation of π if he had known of techniques for speeding up convergence. We also give a brief summary of other methods used to estimate π since the time of Archimedes. These include several methods based on inverse tangent formulas, which were used over a period of about 300 years, and some relatively more recent methods based on more sophisticated ideas pioneered by Ramanujan in the early part of the twentieth century.

Archimedes will be remembered when Aeschylus is forgotten because languages die and mathematical ideas do not.