Euclid (left), in a portrait by 17th century painter, Antonio Cifrondi; and, Archimedes (right). (Source images, left and right, via Via Wikimedia Commons) |

This places the value of pi between 3.14085 and 3.14286. If we average these, we get 3.141855, which is within a hundredth of a percent of the value of pi. Richeson writes that an important part of Archimedes' method is that it involves straight lines, only. We now treat curves like strings that can be straightened into lines, and teachers often demonstrate the approximate value of pi by wrapping a string around a disk, and then unwrapping it against a ruler. Ancient geometers, perhaps thinking more mechanically than analytically, thought that curves and lines were fundamentally different. Archimedes' method was successful, since it wasn't based on any processing of curves. Not really defining pi, but talking around it, Archimedes showed that the circle and sphere constants were related; viz.,(223/71) ≤ π ≤ (22/7)

in whichπ = (C/D) = (A/r^{2}) = (6V/D^{3}) = (1/4)(S/r^{2})

A sphere inscribed in a cylinder, the supposed inscription at Archimedes' Tomb.Archimedes showed that the volume and surface area of the inscribed sphere are each two-thirds those of the cylinder. This was quite a feat in the days before calculus. (Illustration by the author using Inkscape.) |

- David Richeson, "Circular reasoning: who first proved that C/d is a constant?" arXiv Preprint Server, March 14, 2013.