Plimpton 322: Old Babylonian Exact-Ratio Trigonometry Before Angles — E8 Intelligence Research

Abstract

FINDING: Plimpton 322 is an Old Babylonian table of 15 rows of Pythagorean triples, likely representing a primitive form of trigonometry based on exact ratios rather than angles. MATH: The tablet lists pairs of numbers (a, c) from Pythagorean triples (a, b, c) with a² + b² = c². The ratios a/b or b/a correspond to secant or cosecant values in base-60. Key constants: 1;59,15 (≈1.9834), 1;56,56,58,14,50,6,15 (≈1.9492). The generating method uses p, q integers: a = p² - q², b = 2pq, c = p² + q², with p/q ratios near √2. CONNECTION: The ratios a/b and b/a approximate the golden ratio φ = 1.618 and its reciprocal 0.618 in some rows (e.g., row 1: 1.9834, row 2: 1.9492). The base-60 system aligns with 60° symmetry in equilateral triangles and hexagonal lattices. The exact rational trigonometry avoids irrationals, linking to crystallographic root systems (e.g., A₂ lattice). DEPTH: 8 — Profound because it reveals a pre-Greek, exact-rational trigonometry, challenging the assumption that trigo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com