Can This Ancient Babylonian Tablet Improve Modern Math?

By Carl Engelking | August 24, 2017 2:24 pm

Plimpton 322 at the Rare Book and Manuscript Library at Columbia University in New York. (Credit: UNSW/Andrew Kelly)

Researchers in Australia say an ancient Babylonian tablet that’s considered the world’s oldest trigonometric table was a far more powerful tool than it’s given credit for.

The square tablet, known as Plimpton 322, is roughly the size of a cell phone and features four columns and 15 rows of cuneiform numbers written on it. Artifacts dealer Edgar Banks—the man who Indiana Jones is based off of— sold the 3,700-year-old artifact to New York publisher George Arthur Plimpton in 1923, and it was discovered near the ancient city of Larsa in Iraq. In the early 1940s, Brown University mathematics historian Otto Neugebauer studied Plimpton 322 and concluded that the rows and columns of numbers formed parts of Pythagorean triples, or three numbers that provide a solution to the equation a2 + b2 = c2. The number series 3,4,5 is a popular “triple” example, and it’s also the equation used to describe the dimensions of right-angle triangles.

Neugebauer’s theory was notable because it meant the Babylonians were articulating these integer solutions 1,000 years before Pythagoras was born. Therefore, the Babylonians, not the Greeks, were the pioneers of trigonometry.

Mathematical firsts aside, it still wasn’t clear how the Babylonians used this chunk of clay. In 2001, Eleanor Robson, an expert on Mesopotamian mathematics, offered a more mundane explanation for the tablet’s purpose. She demonstrated that Neugebauer’s theory doesn’t explain how certain values on the chart were chosen, how and why the table entries were ordered, or what purpose numbers in the first column served.


Daniel Mansfield (Credit: UNSW/Andrew Kelly)

Robson argued that historical evidence indicated this tablet was “a list of regular reciprocal pairs” and was used as a teaching tool to run students through a set of exercises. To solve the exercises on Plimpton 322, Robson wrote, a student would have relied on a method written on another Babylonian tablet, YBC6967. She went further, writing it is “unlikely that the author of Plimpton 322 was either a professional or amateur mathematician.” Robson received the Lester R. Ford Award for her work, and her theory gained prominence thereafter.

In a new study published in Historia Mathematica, University of New South Wales, Sydney lecturer Daniel Mansfield says Plimpton 322 was far more than a teaching tool. Based on his analysis, he says the table is a series of 15 right-triangles that decrease in inclination. He believes ancient engineers would have used Plimpton 322 to construct plans for palaces, canals or perhaps the Hanging Gardens of Babylon. Mansfield says the tablet represents the world’s oldest and most accurate working trigonometric table—even today. That’s because the Babylonians used a base 60—rather than a base 10 like we do—which permitted more accurate fractions.

“This is not only the world’s oldest trigonometric table; it is also the only completely accurate trigonometric table on record…this system has enormous potential for applications in surveying, computers and education. It’s rare that the ancient world teaches us something new, after 3,000 years Babylonian mathematics might be coming back into fashion,” Mansfield said.

So what did Robson think of this new challenge to her decade-old theory? She was a step ahead of journalists who were all likely asking the same question. She cut to the chase on Twitter, and didn’t seem convinced by the new finding:

There are myriad other Babylonian tablets that are worthy of attention from mathematical historians, but Plimpton 322 has captured a fair chunk of the limelight. Mansfield and colleagues interpretation of Plimpton 322 is yet another theory in a debate that has been waged among math historians for nearly a century. No doubt, the world will continue to learn more as the experts make their way through more of these ancient artifacts.

CATEGORIZED UNDER: Living World, top posts
  • Uncle Al

    a² + b² = c²
    (3,472,073)^7 + (4,627,011)^7 = (4,710,868)^7

    sqrt(2) = (152,139,002,499)/(107,578,520,350)
    e = {1 + 9^[(-4)^(42)]}^[(9^85)]

    We thus perceive Number Line social intent in which no decade receives less length than it deserves. Anything less would be historic European patriarchal aggression. Destroy All Slide Rules!

    • OWilson

      Any series of numbers, when isolated, gives explicit directions to the the buried treasure on Oak Island.

      If reversed, it gives the exact number of Nebuchadnezzar’s concubines!

      If narrated slowly AND backwards, it says “Paul is dead”!

    • Debbie

      If 3,472,073)^7 + (4,627,011)^7 = (4,710,868)^7 is true, then Fermat’s last theorem is false.

      • Uncle Al

        You miss the point. Don’t calculate. Mod 7 the least significant digits. The expression is off by 5.3×10^(-26) relative.

        8 vs. 8,4,2,6,8,4,2

        Sqrt(2) cannot be a ratio. e cannot be an integer finite expression. The soft “sciences’ plus non-classical gravitation and the Standard Model: Not truth but ever more complex lies. Religion, advocacy, “rights.” This font ruins numbers.

        • Debbie

          I don’t have to calculate to know that it’s and false (because FLT has been proved) and that the limited precision of ordinary floating point calculations precludes demonstrating that it’s false without some very tedious hand calculations or special programming.

          You’re babbling, so I’ll leave you alone now, as I should have in the first place.

          • Uncle Al

            (because FLT has been proved)” Mathematical proof is not empirical proof. Euclid made no errors. Euclid cannot accurately navigate the seas. The are eight primary geometries of three-space. Euclid, like Newton, is often a facile approximation…until he isn’t. All the fun is in the footnotes. Cf: Heegner number.

        • TLongmire

          There is a slight decrimansee between the blueyes shift with euthogens🤘

        • TLongmire

          Obligatory gated thru extortion

  • Erik Bosma

    And I’ll bet the Babylonians received their instructions in trigonometry from the early inhabitants of India, namely Harappa and Mohenjo-daro. However, most of the earlier Indian written stuff was recorded on wood which don’t last too long in a wet jungle climate.

    • Just Saying

      No, had to be space aliens!!

  • TheShapeOfThings

    For the last three years I have been telling people that I can build the seven stars from the book of Revelation from scratch. No takers yet. I am agnostic. I do not follow any religion.

    The first star is in the tao, it creates the waters and land that appear without explanation in genesis. Heaven is the sixth star. It is an image of itself in every way.

    The alpha star has crosses walking on waters that look just like the one the carpenter was nailed to. I don’t think he ever got his message across. If he did, people would be much nicer to eachother.

    It is amazing that nobody ever figured this out before, it isn’t that hard.

    They are super structures and most of the parts are somewhat interchangeable meaning the stars are just the beginning. They have practical applications.

    I want to build a house out of them.

    • Uncle Al

      Solid print portable examples. Blueprints are not generally expressive.

  • Philip Olson

    I still have a book of trig tables I used when young. Never built any hanging gardens or palaces with them, but did put together a five sided house with a spiral staircase. Using math does help in planning the layouts.

  • polistra24

    Right triangles are always useful for builders and surveyors. The small size of the tablet would make it a sort of pocket reference card, which engineers and builders used until very recently when small calculators took over.

    I had to look up “regular reciprocal pairs”…. all references to the phrase lead exactly to this tablet and nothing else, which says the technique isn’t useful for any purpose except discussing this tablet.

    If you were accustomed to working in base 60, you wouldn’t need a written reference to the factors of 60, just as we don’t need a written reference to the factors of 10.

  • John C

    If they were that advanced 3,700 years ago, how old is the origin of their math I wonder?

  • Debbie

    “the world’s oldest and most accurate working trigonometric table—even today.”

    Nonsense. If anybody still used trig tables they could be calculated with arbitrarily high precision using Maclaurin series.

    • Toribio Matamoros

      I hope you’re not a math major because you completely misinterpreted the phrase. It’s a conjunction: It’s the world’s oldest trigonometric table AND the most accurate working trigonometric table.

      If we already had an accurate working trigonometric table, it certainly wasn’t the oldest. On the other hand, if before discovering Plimpton 322 we had discovered another trigonometric table, it’s possible it wasn’t as accurate as Plimpton 322.

      Suggested Mathematical Logic Course for you to take: “Logic and Set Theory” by Patrick Suppes.

  • jonathanpulliam

    “O Homem Que Calculava”
    “Morro da Babilonia”
    “O Meu Jardim Ja Tem Jardineiro”.


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