AnOldBabylonian coefficient, its origin and impact on our understanding of measures on circles, including the radian measure

TL;DR

This paper explores the origin of the Old Babylonian coefficient Xi, its role in ancient measurements, and how it influenced the development of the radian measure and modern π, highlighting its historical and mathematical significance.

Contribution

It reconstructs the historical origin of Xi, showing its connection to ancient measures and its impact on the evolution of circle measurement and the radian.

Findings

Xi = 375/360 from Old Babylonian measures

Refined to 377/360 in Ptolemy's work

Modern π incorporates ancient two-stage calculation

Abstract

This study reconstructs the origin of a constant, here called $\Xi$ (Xi), as a primary scaling factor in Old Babylonian mathematics and astronomy. $\Xi$ arises from the practical necessity of precise measurements on the sky or a circle, through the harmonization of length-measure systems. The analysis of the Nippur measure (with its famous cubit) and the Gudea measure shows that $\Xi = 375/360$ represents the ratio of these established Old Babylonian measure systems. As a precision factor for circumference calculations, it remained in use until today. In Ptolemy's work, we find a slightly refined value of $\Xi = 377/360$. A further refinement of this coefficient led to our modern $\pi$, which still incorporates the two Old Babylonian components of a demonstrably two-stage calculation and refinement process. The accuracy increased by only 0.5\% compared to the first ratio. This factor, attested on several Old Babylonian cuneiform tablets including those from Susa, demonstrates the profound understanding of sexagesimal logic. The relative sexagesimal notation (60 = 1 = 1/60) enabled the universal application of $\Xi$ and its reciprocal for highly accurate calculations of arc-length on circular segments. This investigation leads ultimately to a surprising consequence: the modern radian measure is a direct descendant of this Old Babylonian coefficient.