Babylonian Scribes Solve Quadratic Equations on Clay Tablets
A base-60 number system built for trade produces the standard formula centuries before algebra has a name
Quick facts
- Number base
- 60 (sexagesimal), positional
- Earliest tablets with square/cube tables
- c. 2000 BCE, Senkerah
- Earliest known cubic equation attempt
- Tablet BM 85200+, 36 problems
- Surviving legacy
- 60 minutes/hour, 360 degrees/circle
What happened
Babylonian mathematics, which replaced Sumerian mathematics in Mesopotamia from around 2000 BCE, used a positional number system with base 60 rather than base 10, the same system that survives today in 60 minutes to an hour and 360 degrees in a circle. Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 BCE and list squares and cubes of integers up to 60. Babylonian scribes went further than arithmetic tables: to solve a quadratic equation they used essentially the standard formula still taught today, working through problems where, for example, a rectangle's area and the amount by which its length exceeds its breadth were both given, leaving the breadth to satisfy a quadratic. A separate tablet catalogued as BM 85200+, containing 36 problems of this type, is the earliest known attempt to set up and solve cubic equations.
Why it matters
The Babylonian achievement shows that algebraic problem-solving predates the Greek geometric tradition by well over a thousand years, even though the Babylonians expressed their methods as step-by-step recipes rather than general proofs. Their base-60 system has outlasted almost every other ancient convention: it still structures how the modern world measures time and angles.
How we know
Tens of thousands of Babylonian clay tablets survive in museum collections and have been translated and analyzed by historians of mathematics including Otto Neugebauer, whose systematic decipherment of the cuneiform numerals established how the quadratic and cubic problems were actually solved.
Sources
- MacTutor History of Mathematics, University of St Andrews. Babylonian mathematics · Reputable sourcemathshistory.st-andrews.ac.uk · The domain "mathshistory.st-andrews.ac.uk" is on our Reputable source registry. · Link is live and its text matches the event's key terms (Jul 2026)
- MacTutor History of Mathematics, University of St Andrews. Pythagoras's theorem in Babylonian mathematics · Reputable sourcemathshistory.st-andrews.ac.uk · The domain "mathshistory.st-andrews.ac.uk" is on our Reputable source registry. · Link is live and its text matches the event's key terms (Jul 2026)
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