Babylonian Tablet Plimpton 322: Ancient Sexagesimal Trigonometry via Secant and Tangent Ratios — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Plimpton 322 is a 3800-year-old Babylonian sexagesimal table of 15 Pythagorean triples, now shown to encode exact ratios of a right triangle's sides — effectively a primitive trigonometric table based on the secant/tangent function, not angles. | MATH: The tablet lists columns of (a, c) pairs with b² = c² − a², all in base-60. The key ratios are the squared secant: (c/b)² and squared tangent: (a/b)², for angles decreasing from ~45° to ~30° in regular steps. The generating formula is the standard: a = 2pq, b = p² − q², c = p² + q², with p, q regular sexagesimal numbers (e.g., p/q = 12/5, 64/27, 75/32, 125/54, 9/4, 20/9, 54/25, 32/15, 25/12, 81/40, 54/25, 48/25, 50/27, 27/15, 25/12). The tablet's first column gives (c/b)² = 1 + (a/b)², i.e., 1 + tan²θ = sec²θ — the Pythagorean identity in sexagesimal form. | CONNECTION: The ratios 0.618 and 1.618 do NOT appear — the tablet uses regular sexagesimal fractions (e.g., 1.9834027…, 1.9491585…) that are exact in base-60 but not golden- Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Theoretical Computer Science
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204109393
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