Plimpton 322: Old Babylonian Proto-Trigonometry via Reciprocal Pairs — E8 Intelligence Research

Abstract

FINDING: Plimpton 322 is an Old Babylonian table of 15 Pythagorean triples, likely generated using a reciprocal-pair method in base-60, representing a proto-trigonometric table based on ratios (not angles). MATH: For a right triangle with sides \(a < b < c\), the tablet lists \( (b^2 - a^2)/c^2 \) (column 1), \(a\) (column 2), and \(c\) (column 3). All triples satisfy \(a^2 + b^2 = c^2\). The generating method: choose regular sexagesimal reciprocals \(p\) and \(q\) (with \(p > q\)), then set \(a = p^2 - q^2\), \(b = 2pq\), \(c = p^2 + q^2\). Column 1 equals \((b/a)^2\) or \((c/a)^2\) in simplified form. CONNECTION: The ratios in column 1 correspond to squared tangents or secants in base-60. The use of regular numbers (divisors of 60) links to base-60 harmonic fractions. The triple generation mirrors the parametrization of rational points on the unit circle, a precursor to trigonometric ratios. No explicit golden ratio or crystallographic symmetry is present, but the base-60 system Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com