Archimedes' calculations of square roots

TL;DR

This paper reevaluates Archimedes' methods for calculating square roots, suggesting he likely used interpolation, which impacts understanding of his bounds on pi.

Contribution

It critically examines previous methods, proposes that Archimedes used interpolation, and clarifies his approach to calculating bounds on pi.

Findings

Previous methods for Archimedes' square roots are implausible.

Archimedes probably used an interpolation technique.

His approach influenced the precision of pi bounds.

Abstract

We reconsider Archimedes' evaluations of several square roots in 'Measurement of a Circle'. We show that several methods proposed over the last century or so for his evaluations fail one or more criteria of plausibility. We also provide internal evidence that he probably used an interpolation technique. The conclusions are relevant to the precise calculations by which he obtained upper and lower bounds on pi.