Approximating cube roots of integers, after Heron's Metrica III.20

TL;DR

This paper explores methods for approximating cube roots of integers inspired by Heron's ancient techniques, investigates a conjecture related to Heron's method, and contributes to historical mathematical understanding.

Contribution

It presents new findings on Heron's method for cube root approximation and examines a conjecture by Taisbak regarding this ancient technique.

Findings

Insights into Heron's cube root approximation method

Analysis of Taisbak's conjecture on Heron's technique

Historical mathematical implications of the findings

Abstract

Heron, in Metrica III.20-22, is concerned with the the division of solid figures - pyramids, cones and frustra of cones - to which end there is a need to extract cube roots. We report here on some of our findings on the conjecture by Taisbak in C.M.Taisbak, Cube roots of integers. A conjecture about Heron's method in Metrika III.20. Historia Mathematica, 41 (2014), 103-104.