AnOldBabylonian coefficient, its origin and impact on our understanding of measures on circles, including the radian measure
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arXiv
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| Format: | Preprint |
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2026
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| author | Kleb, Jens |
| author_facet | Kleb, Jens |
| contents | This study reconstructs the origin of a constant, here called $Ξ$ (Xi), as a primary scaling factor in Old Babylonian mathematics and astronomy. $Ξ$ 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 $Ξ= 375/360$ 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 $Ξ= 377/360$. A further refinement of this coefficient led to our modern $π$, 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 $Ξ$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_08098 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | AnOldBabylonian coefficient, its origin and impact on our understanding of measures on circles, including the radian measure Kleb, Jens History and Overview 01A17(primary), 01A85(secondary) This study reconstructs the origin of a constant, here called $Ξ$ (Xi), as a primary scaling factor in Old Babylonian mathematics and astronomy. $Ξ$ 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 $Ξ= 375/360$ 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 $Ξ= 377/360$. A further refinement of this coefficient led to our modern $π$, 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 $Ξ$ 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. |
| title | AnOldBabylonian coefficient, its origin and impact on our understanding of measures on circles, including the radian measure |
| topic | History and Overview 01A17(primary), 01A85(secondary) |
| url | https://arxiv.org/abs/2604.08098 |