Plimpton 322: The Old Babylonian Ratio-Based Trigonometric Tablet — E8 Intelligence Research

Abstract

FINDING: Plimpton 322 is a 3800-year-old Old Babylonian cuneiform tablet containing 15 rows of Pythagorean triples, likely representing the earliest known trigonometric table based on ratios rather than angles. MATH: Each row corresponds to a normalized right triangle with short side (s) and long side (l) such that (l² - s²) / (l² + s²) yields a regular sexagesimal fraction. The tablet lists pairs (a, c) from triples (a, b, c) with a² + b² = c², where a = 2pq, b = p² - q², c = p² + q² for integers p > q. The key ratio is (b/a) = (p² - q²)/(2pq), which generates the slope (or "Plimpton ratio") used for trigonometric calculations. Base-60 (sexagesimal) arithmetic is used throughout. CONNECTION: The ratios on Plimpton 322 correspond to the secant or tangent of angles in a right triangle, but expressed as rational numbers in base-60. The underlying structure mirrors the golden ratio family: for p/q near φ (1.618), the triple approximates a golden triangle. Specifically, if p/q = φ, then Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com