Plimpton 322: Ancient Babylonian Exact Trigonometry via Sexagesimal Ratios — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Plimpton 322 is a 3800-year-old Old Babylonian cuneiform tablet containing 15 rows of sexagesimal Pythagorean triples, now understood as an exact trigonometric table based on ratios of sides, not angles. | MATH: The tablet lists triples (a, b, c) satisfying a² + b² = c² in base-60. Each row corresponds to a ratio (b/c)² or (a/c)² — specifically, the tablet's columns encode the squared secant and tangent functions: (c/b)² and (a/b)². The generating formula is the standard parameterization: a = 2pq, b = p² − q², c = p² + q², with p, q coprime, one even. The tablet's rows are ordered by decreasing (b/c)² — i.e., by decreasing angle from ~45° to ~30°. The sexagesimal entries include exact values like 1;59,15,0,0 (≈1.9834) and 1;49 (≈1.8167), which are (c/b)² for specific triples. | CONNECTION: The ratios on Plimpton 322 are directly related to the golden ratio family: the tablet's angle range (45° to 30°) brackets the angle whose tangent is 1/√φ ≈ 0.786 (≈38.17°), and the sexagesi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Theoretical Computer Science
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204451698
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