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

Abstract

FINDING: Plimpton 322 is a sexagesimal table of 15 Pythagorean triples, likely generated by reciprocal pairs (1/x, x) to produce exact ratios for right triangles — effectively a proto-trigonometric table. | MATH: Each row yields a triple (a, b, c) with a² + b² = c². In base-60, the tablet lists (s² − r², 2rs, s² + r²) for regular numbers r, s. The key ratios are the squared secant/tangent values: (c/a)² and (b/a)², with entries like 1;59,15, 1;56,56,58,14,50,6,15 — all exact in sexagesimal. The generating formula: for p > q, coprime, opposite parity, triple = (p² − q², 2pq, p² + q²). The tablet's column I is (c/a)², decreasing from ~1.983 to ~1.387, corresponding to angles from ~45° to ~30°. | CONNECTION: The ratios (c/a)² and (b/a)² are precisely the squares of cosecant and cotangent. In base-60, these yield exact finite sexagesimal expansions — the same harmonic structure as 1/φ² ≈ 0.382, 1/φ ≈ 0.618, φ ≈ 1.618, φ² ≈ 2.618. The tablet's angle range (30°–45°) maps to cotangent² values Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com