Plimpton 322 and Babylonian exact sexagesimal trigonometry

Last week on Aug 24 Daniel Mansfield and I published the paper “Plimpton 322 is Babylonian exact sexagesimal trigonometry” in Historia Mathematica online. The paper has had a huge media response, partly due to the excellent press release created for us by Deb Smith from the Faculty of Science, UNSW Sydney, and partly by the lovely video put together by Brad Hall at UNSWTV with Daniel presenting an overview of our discovery that Plimpton 322 (P322), the world’s most famous Old Babylonian (OB) clay tablet, is actually the world’s first trigonometric table, and also the world’s most exact trigonometric table!!


These are remarkable claims that are sure to raise eyebrows not just in the historical community, but also in the mathematical one. How can an unknown scribe, writing almost 4000 years ago with a cuneiform wedge on a small clay tablet, possibly have understood trigonometry not only before anyone else, but in a fashion quite different from anything since (at least, everything before my book on Rational Trigonometry published in 2005)?

Could it really be that this ancient form of ratio-based trigonometry, which completely avoids all mention of angles, actually contains a more profound understanding of this fundamental subject than all those hundreds of subsequent tables? Might it be that we are on the verge of a major shift in our understanding of how to teach trigonometry to high school students by incorporating this new/ very old understanding? And could it be that the powerful sexagesimal system that the ancient Sumerians first devised and that is essential to the understanding of P322 holds powerful advantages for modern computing?

And of course: do we need to seriously re-evaluate the role of OB mathematics in the history of the subject? How many other important mathematics that is currently credited to the Greeks actually is due to the much earlier cultures of the Sumerians/Akkadians/Babylonians and/or the Egyptians?

These are fascinating questions that we hope will be among those discussed as the result of our work. But we do hope that people debate these and other important issues after at least having looked at our paper in some detail. Unfortunately some serious historical academics, as well as at least one science journalist, have leapt to negative conclusions without giving our paper a serious reading.

Eleanor and Evelyn: here is the link to the paper again — please have a go at  digesting our arguments, which we have spent two years carefully crafting, and which we are confident will change your orientation to this tablet: Plimpton 322 is far more than a teaching aid for teachers to cook up quadratic problems for their students. It is a work of undisputed genius which required a deep understanding of the trigonometry of a right triangle, and took a huge amount of effort to compile.

Anyway, I anticipate quite a few more posts on this fascinating development.

5 thoughts on “Plimpton 322 and Babylonian exact sexagesimal trigonometry

  1. Tom W.

    Fascinated and eager to read more on this from you! I’m have expecting to see Rational Trig emerge from this, I know there’s a link in there somewhere!

  2. Anupama Singh

    I think the number system uses picture of nails which is an clue about how they measured.
    i have seen people measuring with strings and nails.
    so fractional number system may exist.
    like for any length of string as a whole then measuring the remaining by folding string and so on..

    1. Anupama Singh

      I think the number system uses picture of nails which is an clue about how they measured.
      i have seen people measuring with strings and nails.
      so fractional number system may exist.
      like for any length of string as a whole then measuring the remaining by folding string and so on..

      or more likely simplest way could be
      n n/2 n/4 n/6 …

  3. Michiel Karskens

    Fascinating indeed, however I believe the truth is sexagesimally more so.
    The Babylonians were so far ahead of the Greek that Euclid’s indirect proof of the infinity of prime numbers was at best pseudo science to them. Dr Feynman succinctly gave the reasons why science needs the Babylonian method and not the Greek method in his televised lectures.
    In this working paper I explain the simple order in the sequence of prime numbers and give a direct proof of the infinity of prime numbers.
    Einstein was right, God does not play dice with the Universe. For the same reason the Babylonians used base 60. For the same reason there were 6 generations of Gods in the Babylonian Pantheon. For the same reason in the Book of Genesis God is said to have created the Universe in 6 days. For the same reason there are six mountain ranges in the Riemann landscape. Prime Numbers wrap around modulo 6 as in Gauss’ modular arithmetic. The number line is not a straight line as a ruler. The number line spirals modulus 6, and the Babylonians, the Sumerians, and the Brahmans understood this.

  4. Jovan J.

    It is not an unknown scribe. It came from the Old European Vinca civilization:

    Click to access vincascript.pdf

    This script is estimated to be 5000 to 8000BC old and the oldest known sentence in human language written in this script states: “The Bear Goddess and the Bird Goddess are the Bear Goddess indeed.”

    It is quite likely that in some 4000 years after the Vinca script was invented in the pre-neolitic period (and Pelagic culture spread to Sumerian and Crete around 2000BC ) the mathematics could have developed to the level as seen in this Babylon artifacts. Babylon was founded by Nino Belov (Nimrod) “UR NINO SAR SERBULA” (Sumerian tablet, 1900BC), or “LORD NINO SERB TSAR”. The origins of the sixty-second minute and sixty-minute hour can be traced all the way back to ancient Mesopotamia. In the same way that modern mathematics is a decimal system based on the number ten, the Sumerians mainly used a sexigesimal structure that was based around groupings of 60. This easily divisible number system was later adopted by the ancient Babylonians, who used it to make astronomical calculations on the lengths of the months and the year. Base-60 eventually fell out of use, but its legacy still lives on in the measurements of the both hour and the minute. Other remnants of the Sumerian sexigesimal system have survived in the form of spatial measurements such as the 360 degrees in a circle and the 12 inches in a foot.


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