Babylonian Trigonometry Links Secant-Squared Triples to Golden Ratio — E8 Intelligence Research

FINDING: Babylonian trigonometry (Plimpton 322) encodes secant-squared values of Pythagorean triples, with a novel theorem linking the tangent-secant theorem directly to the golden ratio. | MATH: Pythagorean triples: \(a^2 + b^2 = c^2\) (integers). Plimpton 322 lists 15 pairs \((b^2/c^2, a^2/c^2)\) — effectively \(^2θ = (c/b)^2\) and \(tan^2θ = (a/b)^2\) in base-60. The new theorem (arXiv:2201.08212) states: For a tangent of length \(t\) and secant segment \(s\) from an external point to a circle, if \(t^2 = s · (s + d)\) (where \(d\) is the chord inside), then the ratio \(s/d\) satisfies \(s/d = φ = (1+√5)/2\) when \(t^2 = s · (s+d)\) and \(d = s/φ\). Specifically, the golden ratio emerges when the tangent length squared equals the product of the external secant segment and the full secant, yielding \(φ = 1.618...\) and its reciprocal \(1/φ = 0.618...\). | CONNECTION: Direct: \(φ\) and \(1/φ\) (0.618) arise from tangent- Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com