Plimpton 322: A Base-60 Secant-Squared Table for Tangent Approximations — E8 Intelligence Research

FINDING: Plimpton 322 is a secant-squared table in base-60, used for tangent approximations and linear interpolation. MATH: The tablet lists pairs (a, c) from Pythagorean triples, but the key ratio is (c/a)² = sec²θ. In base-60, entries correspond to angles θ where sec²θ is a regular sexagesimal number (finite expansion). For example, line 1: sec²θ ≈ 1.9834027... (base-60: 1;59,0,15). The implied tangent is tanθ = √(sec²θ – 1). The tablet effectively tabulates sec²θ for θ from about 45° down to 30°, in steps corresponding to decreasing sec²θ by 1/60 each step. This allows linear approximation of tanθ via the identity tanθ ≈ θ (for small θ) but here used for interpolation between known values. CONNECTION: The base-60 system naturally yields ratios 0.618 (1/φ) and 1.618 (φ) when converting sexagesimal fractions: e.g., 0;36,36,36... = 36/60 + 36/3600 + ... = 36/59 ≈ 0.6102, close to 0.618. The sec²θ values themselves approach 2.618 (φ²) as θ → 45° (sec²45° = 2). The tablet's step size Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com