Jens Kleb | Searcharxiv

arXiv2026

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

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.

math.HO↗

arXiv2021

On the evidence of a trigonometric function value system in Babylon

Ratios and coefficients are used to simplify calculations. For geometric usage these values also called function values. Like in Egypt also in Babylon such a value system can be shown. The reconstructed calculation sequence, of the Plimpton 322 cuneiform tablet, presented and described here, shows in its completeness that, around 3800 years ago there already was a systematically applied exact measuring system, with the usage of trigonometric function values. With this approach one can plausibly explain that, as we still it practice today, a geometry of the circle has been used for this calculation. It is based on, but not only, of the usage of the regularities, which 1200 years later, were named after Pythagoras. During a second calculation step, for an intended scaled documentation, presentation or transfer to other locations, the dimensionless calculated function value, was extended, with a fractional part, or more exact spelled a common factor. This transformation creates a real measurable length from the ratios, always related to the smallest unit. The systematic usage by means of inclining triangles already in old Babylonian times, goes far beyond the previously known level of this age. The accuracy of the individual calculation is just as verifiable and on exactly the same level as the trigonometric functions used today. However, at least at this point in time, the Babylonians were content with dividing a quarter circle into at least 150 inclining triangles. This old Babylonian trigonometric function value system, together with the here described "Babylonian diagonal calculation", is thus the forerunner of the Greek chord calculation and the Indo-Arabic trigonometric functions of the early Middle Ages, which we still use today. In addition, the new approach to Plimpton 322 plausible explains why and how such values could have been used.

math.HO↗