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

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.