What question did this study set out to answer?

To explore the systematic algorithm for completing the square found on Old Babylonian tablets.

July 21, 2026Open Access

Cuneiform Discovery: Cuneiform algebraic identities completing the square — E8 Intelligence Research

Key Points

To explore the systematic algorithm for completing the square found on Old Babylonian tablets.
Analyzed BM 13901 and YBC 6967 tablets from c. 1800–1600 BCE.
Identified procedures resembling modern geometric methods for solving quadratic equations.
Found an explicit calculation process for completing the square in BM 13901.
Demonstrated that the arithmetic was conducted using sexagesimal (base-60) numeral system.

Abstract

**Discovery: Geometric Algebra on BM 13901 and YBC 6967** The Old Babylonian tablets **BM 13901** (British Museum) and **YBC 6967** (Yale Babylonian Collection), dating to c. 1800–1600 BCE, preserve the earliest known systematic algorithm for "completing the square"—a geometric procedure identical to the modern derivation of the quadratic formula. **BM 13901** (Problem 1) solves: "The surface and my confrontation I have accumulated: 45′. " In modern notation: x² + x = 3/4 (where 45′ = 45/60). The scribe instructs: "1 the projection you posit. The moiety of 1 you break: 30′. 30′ and 30′ you make hold: 15′. 15′ to 45′ you append: 1. The equalside of 1 is 1. 30′… from the body of 1 you tear out: 30′ the confrontation. " The calculation is explicit sexagesimal arithmetic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com

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