Ancient Babylonian Tablet Plimpton 322: A 3,800-Year-Old Trigonometric Table — E8 Intelligence Research

Abstract

FINDING: Plimpton 322 is a 3800-year-old Babylonian clay tablet listing 15 rows of Pythagorean triples, generated via a sexagesimal (base-60) algorithm that encodes exact ratios of a right triangle's sides — effectively a primitive trigonometric table. MATH: - The tablet lists pairs (a, c) with b² = c² − a², where a, b, c are integers. In modern notation, the rows correspond to: \( \frac{b^2}{c^2} = 1 - \frac{a^2}{c^2} \) — i.e., values of \( \sin^2\theta \) or \( \cos^2\theta \) in base-60. - The generating formula (attributed to Old Babylonian scribes, reconstructed by Neugebauer & Sachs, 1945): \( a = 2pq \), \( b = p^2 - q^2 \), \( c = p^2 + q^2 \) (with p > q, coprime, opposite parity). - The tablet's first column lists \( \frac{b^2}{c^2} \) (or \( \frac{a^2}{c^2} \)) as sexagesimal fractions — e.g., row 1 gives \( \frac{119}{169} \) and row 15 gives \( \frac{56}{106} \) (reduced). - Key constants: base-60 (sexagesimal) place-value system; ratios like \( \frac{1 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com