Plimpton 322: A 3,700-Year-Old Babylonian Secant-Squared Trig Table — E8 Intelligence Research

Abstract

FINDING: Plimpton 322 encodes a sexagesimal trigonometric table based on secant-squared (sec²θ) and tangent (tanθ) ratios, predating Greek trigonometry by over 1,000 years and using a simpler, more accurate method. MATH: - Tablet rows correspond to Pythagorean triples (a, b, c) with a < b, c² = a² + b². - Key ratios: sec²θ = (c/b)², tanθ = a/b, tan²θ = (a/b)². - Sexagesimal (base-60) entries: e.g., row 1 gives sec²θ = 1;59,0,15 (≈ 1.9834) and tan²θ = 0;59,0,15 (≈ 0.9834). - Relationship: sec²θ = 1 + tan²θ (trigonometric identity). - Constants: 1, 60, 3600 (base-60 powers); ratios like 1.4142 (√2), 1.732 (√3) appear in related tablets. CONNECTION: - Geometric harmony: The ratios tanθ and sec²θ directly relate to the golden ratio φ (1.618) and its reciprocal 0.618 via the identity φ² = φ + 1, though not explicitly on Plimpton 322. - Base-60 fractions: 0;15 = 1/4, 0;30 = 1/2, 0;45 = 3/4, 0;50 = 5/6 — all appear in tablet computations, linking to regular numbers (2^a·3^b·5^ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com