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This research aims to explore the advanced numerical systems of the Sumerians, focusing on their base-60 system's impact on trigonometry and square root calculations.

July 30, 2026Open Access

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Sumerian Base-60 System: Ancient Trigonometry and Square Root Algorithms Predating Modern Methods — E8 Intelligence Research

ACAndrew Stewart Caldin

Key Points

This research aims to explore the advanced numerical systems of the Sumerians, focusing on their base-60 system's impact on trigonometry and square root calculations.
Analysis of the base-60 place-value system and its applications in trigonometry.
Investigation of the Plimpton 322 tablet for Pythagorean triples and trigonometric ratios.
Examination of square root iteration methods, particularly Heron's method.
The base-60 system demonstrates advanced trigonometric applications, including a direct tie to Pythagorean triples.
Square root algorithms from the Sumerians, such as Heron's method, predate modern computation techniques.
The system facilitates geometric principles and fractions aligning with 360° circles and the golden ratio.

Abstract

FINDING: Sumerian base-60 (sexagesimal) system enabled advanced trigonometry and precise arithmetic, including square root algorithms, predating modern methods. MATH: Base-60 place-value system; Plimpton 322 tablet contains Pythagorean triples (e. g. , 3-4-5, 5-12-13) and ratios equivalent to sec² (θ) = 1 + tan² (θ) ; square root iteration: x₍+₁ = (xₙ + a/xₙ) /2 (Heron's method). CONNECTION: Base-60 yields fractions with divisors 2, 3, 5 — aligns with 360° circle, 60° angles in equilateral triangles, and golden ratio approximations (e. g. , 1. 618 ≈ 97/60). DEPTH: 8 — Foundational to geometric harmony, astronomical cycles, and modern computational algorithms. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com

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Sumerian Base-60 System: Ancient Trigonometry and Square Root Algorithms Predating Modern Methods — E8 Intelligence Research

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