Plimpton 322: Old Babylonian Trigonometry via Secant-Squared Ratios — E8 Intelligence Research

Abstract

FINDING: Plimpton 322 encodes Old Babylonian sexagesimal trigonometry using secant-squared ratios (1 + tan²θ) from Pythagorean triples, not modern sine/cosine functions. MATH: - Core relation: \( \sec^2 \theta = 1 + \tan^2 \theta \) (sexagesimal form). - Tablet rows list: (sexagesimal) \( \sec^2 \theta \) values (Column I), with corresponding \( \tan \theta \) (Column II) and \( \sec \theta \) (Column III) derived from reciprocal pairs \((p,q)\) generating triples: \( a = p^2 - q^2,\; b = 2pq,\; c = p^2 + q^2 \). - Ratios: \( \tan \theta = b/a \) or \( a/b \); \( \sec \theta = c/a \) or \( c/b \). - Key constants: 1;24,51,10 (≈1.4142, √2) and 1;59,0,15 (≈1.9834) appear as secant values. CONNECTION: - Geometric harmony: The triples correspond to rational approximations of \( \tan \theta \) for angles near 45° (0.618, 0.786 ratios emerge from reciprocal pairs). - Base-60 system: All ratios are expressed sexagesimally, linking to Sumerian metrology and circle division (360°) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com