Plimpton 322: Old Babylonian Trigonometry via Pythagorean Triples and Base-60 — E8 Intelligence Research

FINDING: Plimpton 322 is an Old Babylonian clay tablet (c. 1800 BCE) listing 15 rows of Pythagorean triples, likely representing the world's earliest known trigonometric table, based on base-60 (sexagesimal) arithmetic. MATH: - Pythagorean triples: \(a^2 + b^2 = c^2\), with \(a < b\). - Tablet entries correspond to normalized ratios: \(b/a\) and \(c/a\), specifically \(\frac{b}{a} = \frac{1}{2}(x - \frac{1}{x})\) and \(\frac{c}{a} = \frac{1}{2}(x + \frac{1}{x})\) for rational \(x = p/q\) in base-60. - Key constants: sexagesimal fractions like 0;30 (1/2), 0;45 (3/4), 0;50 (5/6), etc. - The tablet yields values of \(\tan^2\theta = (b/a)^2\) and \(\sec^2\theta = (c/a)^2\), effectively a table of secants. CONNECTION: - Ratios on Plimpton 322 approximate \(\sqrt{2}\), \(\sqrt{3}\), and the golden ratio \(\phi = 1.618...\) (e.g., row 11 gives \(b/a \approx 0.618\), the reciprocal of \(\phi\)). - Base-60 arithmetic naturally generates regular numbers (2,3,5) linked to crystallog Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com