August 24th, 2017

The ancient history of math: Plimpton 322

Today is one of those days I look at the news and sigh.

I don’t want to see lists of the most recent statements by people who think Trump is unstable.

I don’t want to hear how the president is so very wrong to attack the MSM who are just doing their best at journalism and not out to get him at all.

I don’t want to read the latest spin on the actions of extreme leftists.

I’ve heard it all before. Sometimes I just want to take a break.

And so today I turn to math’s ancient history:

A 3,700-year-old clay tablet has proven that the Babylonians developed trigonometry 1,500 years before the Greeks and were using a sophisticated method of mathematics which could change how we calculate today.

The tablet, known as Plimpton 332 [sic], was discovered in the early 1900s in Southern Iraq by the American archaeologist and diplomat Edgar Banks, who was the inspiration for Indiana Jones.

The true meaning of the tablet has eluded experts until now but new research by the University of New South Wales, Australia, has shown it is the world’s oldest and most accurate trigonometric table, which was probably used by ancient architects to construct temples, palaces and canals.

However unlike today’s trigonometry, Babylonian mathematics used a base 60, or sexagesimal system, rather than the 10 which is used today. Because 60 is far easier to divide by three, experts studying the tablet, found that the calculations are far more accurate…

“This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3000 years, but it has possible practical applications in surveying, computer graphics and education.

“This is a rare example of the ancient world teaching us something new.”

That’s pretty stunning. However, I’m not sure it’s true that this is the final verdict on the subject. Apparently, the argument about Plimpton 322 and whether it’s a trig table or something else has been going on (who knew?) for decades. Perhaps the current researchers think they’ve solved it once and for all, however.

I can’t tell. What I remember most about Babylon was something I first encountered in Art History 101, the Ishtar Gate that marked the entrance to the ancient city. Here’s a reconstruction in a museum in Berlin:

Impressive in any era.

The following wasn’t Babylonian (it was Assyrian; same general area a while later), but it was the ancient work of art that impressed me the most when I first saw it in that same course so long ago. The Wounded Lioness bas-relief:

The acute observational powers and the carving skills of the artist are all on clear display, along with the pathos of a wounded animal.

28 Responses to “The ancient history of math: Plimpton 322”

  1. Mr. Frank Says:

    Is math racist?

  2. Oldflyer Says:

    Well, those ancients certainly had some rather sophisticated method for engineering calculations. Or maybe they just constructed their structures by trial and error, and tore down the ones that didn’t work out. After all, slave labor was cheap. Joking.

    What a striking work of art. I slept through what served as art appreciation, so I can discover works like that now with virgin wonder. The craft of the ancient artist is startling; and as you rightly point out, the pathos inherent in the wounded animal is wrenching.

  3. Geoffrey Britain Says:

    Just think. If the mentality of today’s environmental nazis had dominated the elite back then, lions would still be eating children in Iraq today.

  4. artemptydgr Says:

    (who knew?)



    The antikythera mechanism

    And French clockwork robitic trapeze doll from the 1600s

  5. Bill Says:

    It’s amazing to think what they were able to do without calculators and with only hand tools.

  6. Cornhead Says:

    Wounded Lioness? Hillary.

  7. AesopFan Says:

    I always knew we had kept the sexagesimal system for clocks and globes because of the divisibility factor; maybe some of our Western geniuses should have kept it for other things.

    PS Just because it happened in the Middle East does not mean the Muslims get to claim it, as Obama more-or-less suggested in one of his infamous speeches.

  8. Rachelle Says:

    I doubt that Babylonian trigonometry was as advanced as the article seems to suggest. Instead, it sounds like rather basic trigonometry which I would expect any building civilization to have developed empirically.

    In G.H. Hardy’s ‘A Mathematician’s Apology’ I was struck by his observation on the ancient mathematics of the Babylonians and Egyptians. He seemed to view them as artifacts of historical interest. By contrast, he said that when one gets to Greek mathematics one is in the presence of the real thing, of genuine mathematical thinking. Greek mathematics (geometry) is still a powerful tool and it is worth remembering that although Newton knew (and had invented) calculus he presented his masterpiece, Principia’ in the language of Euclid.

    Still, anything we can learn of the evolution of mathematical thought is interesting and I like that people are still examining these ancient texts.

  9. Wry Mouth Says:

    Math teacher here. One of the things I like to point out to my students:

    The Babylonian mathematicians are still with us.

    Every time one is momentarily perplexed converting hours and minutes to decimals (whether in time or angles), one can acknowledge their invisible hand. 60 seconds in a minute and 60 minutes in an hour… thanks, Babylonians!


  10. Paul in Boston Says:

    If you are ever in Chicago go to the Oriental Istitute of the University of Chicago, It has a large collection of artifacts from Assyria, Persepolis, and elsewhere. It’s quite something to see.

  11. Wry Mouth Says:

    Also, an art P.S.

    I was struck, Neo, as you were by the wounded lion. For me, it would be that, and many other ancient artifacts, but I was most forcibly struck by the 10,000-year-old cave paintings in France.

    I realized I had more in common with those people than I might with many of my contemporaries.

    Thanks for the post!

  12. groundhog Says:

    AesopFan Says:
    PS Just because it happened in the Middle East does not mean the Muslims get to claim it, as Obama more-or-less suggested in one of his infamous speeches.,

    At least, it predates the pollution of Islam.

  13. Irv Says:

    Neo – Before I read your blog tonight I was telling my wife that I had gotten sick of all the news being so predictable and negative with little redeeming value. So I spent a couple of hours on YouTube reviewing and reliving election night coverage from the perspective of various networks.

    I stayed up and watched the election on election night but didn’t spend enough time on the stations other than Fox, including even the Canadian station, to get a real feel for the coverage. What an education and what an enjoyable afternoon. It was like watching a movie that you knew how it would end but watched it anyway just because you enjoyed the story. I highly recommend it.

    And, as a retired math teacher, I enjoyed your post for its content as well. Thanks

  14. Tim Turner Says:

    I’m not sure it’s “the most accurate” or that the benefits for computer graphics are all that great.

    Ultimately, computers work in a base-2 system. Any fantastic base-60 calculations have to be translated into base-2, and then used on a base-2 hardware to render base-2 graphics on a base-2 monitor.

    As far as “most accurate”, I’m sure there are a few university computer labs with tables calculated out to far greater number of decimal places than will fit on any clay tablet.

  15. Ben David Says:

    Babylonian astronomy was also very advanced. They made very accurate measurements of the solar/planetary movement patterns by tracking reference points in mountain ranges.

    There is also evidence that these measurements assume a spherical earth.

    Judaism still uses Babylonian names for the months of our corrective lunar calendar. Brought back with Ezra and Nehemia to Judea. And the corrective mathematics were worked out based on Babylonian calculations.

    When the time of the new moon is announced in synagogue we still use a Babylonian time measure called a “helek” (=”division”) that is 1/1080 of an hour.

    The lunar cycle is 29 days, 12 hours, 40 minutes, and 73 of these “divisions”.

  16. Tuvea Says:

    Wry Mouth:

    IMHO one of the base principles of conservativism is that human nature hasn’t changed. The philosophies of the Greek thinkers and men like Marcus Aurelius still hold true today.

    All Socialists, on the other hand, have to believe that men are putty. To be molded into the fashion of the day. ‘True Aryans’ or ‘New Soviet Man’.

    Homo Sapiens Sapiens haven’t evolved much in the last 10,000 years.

    But now we have a whole lot of really cool toys!

  17. Frog Says:

    I second Tuvea’s thoughts.

  18. Rachelle Says:

    Not true that we haven’t evolved in the last 10,000 years. Evolution takes place e when a population is exposed to selective pressure. The advent of civilization, writing, mathematics and commerce puts significant selective pressure on much of humanity. The idea that evolution has stopped is nonsense.

  19. DNW Says:

    ” G.H. Hardy’s ‘A Mathematician’s Apology’ I was struck by his observation on the ancient mathematics of the Babylonians and Egyptians. He seemed to view them as artifacts of historical interest. By contrast, he said that when one gets to Greek mathematics one is in the presence of the real thing, of genuine mathematical thinking. …”

    The concept of proofs comes to mind.

    There is a great deal you can do with a peg and a bob and a string that looks sophisticated once it is all done.

  20. DNW Says:

    My gosh … look at that circle of standing stones, originally set in a radius accurate to better than two inches (we estimate).

    Must have been aliens from the Dog Star.

  21. DNW Says:

    “Homo Sapiens Sapiens haven’t evolved much in the last 10,000 years. “

    Oddly I agree with both this and with Rachelle.

    I think that it is probable that we could pretty well understand the minds of persons more or less like ourselves, meaning direct ancestors, who existed 10,000 years ago, if given time to get on the same metaphysical or worldview page so to speak.

    On the other hand lactase persistence, probably white skin, and fair hair, are, according to current researchers, about as new, or newer than that. The first, much newer.

    Toleration of gluten, I cannot recall a date for at the moment.

    Blue eyes may be older, but the news articles seem to claim they were paired with a relatively dark complexion when compared with recently historical European populations..

    You could also look up plague and Aids resistance as well.

  22. huxley Says:

    This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3000 years, but it has possible practical applications in surveying, computer graphics and education.

    This is pure hoohah. Base-60 was hot stuff several thousand years ago and we still use it when we tell time from a clock or read a protractor, but I can’t imagine what Dr. Mansfield has in mind here and he neglects to explain.

  23. Chester Draws Says:

    Does Mansfield suggest we have 60 digits and we have to learn all 60 before we can start Maths?

    If not, then we won’t be working in base 60 any time. We would be working in some pastiche version where we count to 60 in tens, and then have to convert in and out of a system that is in two different bases. Madness.

    Base 10 won because it is far and away the most logical system for a 10 digit being. Not too many digits to learn, but not so few that you get the abomination of large base 2 numbers that are impossible to hold in your head.

    There is also evidence that these measurements assume a spherical earth.

    Different hemispheres see different stars. The sun is overhead at different times of the year in different places. The horizon on a flat area is well inside the range of vision. Why water doesn’t all just pour off the edge. We have seasons.

    The earth is obviously spherical. Anyone who has done any serious thinking about the subject on a scientific rather than theological basis works that out. So any advanced astronomical society will work it out pretty quickly.

  24. Richard Aubrey Says:

    Odd questions arise from time to time.
    The Indian settlement (city) of Cahokia is available for a visit near St. Louis. In his “1491”, Mann describes teh marvelous engineering.
    Makes you wonder where are the little Cahokias, the half-Cahokias, the practice Cahokias.
    Ditto Gobekli Tepe in Turkey. Among other things, it has been dated to before the Neolithic, which leads to the question of how what must have been millions of man hours had been fed.
    The Antikythera Mechanism is pretty slick. Where’s the Walmart model? The proto-Antikytheras?
    Stonehenge is not a perfect circle, which would only work on the equator. Further north, in the Orkneys, is a similar construction but which is further and appropriately so from a perfect circle.
    At the very least, these things tell us of cultures which can observe, record, and conserve recordings and techniques over substantial periods of time. IOW, stability.
    In Maps of The Ancient Sea Kings, we read of a number of clues to ancient and unknown cultures. I’m not sure I buy much of it, especially the map adjustments, my spherical trig being nonexistent.
    But one of the items they mention, which seems to be worth thinking about, is that many of what we think of as really ancient are both beyond the capabilities of the contemporary societies in which we find them, and seem to be mature items meaning a good deal of progress–unknown to us and likely taking time–went before.

  25. Richard Aubrey Says:

    Also a masterpiece of the pathos of death. Nothing heroic or defiant here.

  26. Rufus Firefly Says:

    I’ve been to the Pergamon in Berlin. Twice! Ausgezeichnet!

  27. Ymar Sakar Says:

    The ancients had technology that was far ahead of the moderns.

    Holes are drilled through thousand ton blocks of stone which would require laser or plasma drills, since diamond drills would be destroyed. Slave labor and hand tools?

    The Great Pyramid of Giza has a tolerance that aligns it to true north, at a precision level beyond what NASA puts on the heat tiles of the space shuttles.

    The entire trigonometric base is straight and slightly curved. Imagine foundation workers having to put in a foundation for a house that is a square that doesn’t deviate at those lengths. The ports in the pyramid also used to point to Polaris and Orion, so it was an astronomical clock used to observe the heavens. The constellations have changed over time, so they no longer point to those constellations exactly.

    People mentioned base 60 in our time keeping. That is a result of there being 360 degrees in a circle. Geometrically and astronomically, we still used degrees, which are base 60. This was originally created to measure the distance stars moved. A single degree was just about the most accurate they could get to at that time, so they divided things up by how far they could see with a human eye.

    As for why computers might want to calculate in base 60 instead of base 2, it is for the same reason why they use hexadecimal. Meaning 10+6. Graphic processing units are already calculating advanced geometries in order to simulate real life looking practicals and textures. Eventually they’ll need to expand the base to a higher one that uses a higher language to program in the software, to keep up with the hardware. The less calculations the cpu has to make, the more efficient the program can work. Our problem is that the human mind is still stuck on base 2 architecture, because that’s how we made the hardware originally.

    The heavens are not created by base 2 architecture, but a super dimension processing method. Hyper Euclidean or non Euclidean geometries will demand a different mathematics. Since mathematicians have already calculated out 10 dimensions, we aren’t all that far from putting them into practice. Quantum Standard Model particle accelerator projects have already figured out that the lower dimensions don’t even exist as we know of them.

    Physicists used to think that there was a fundamental building block of matter. If they could just observe it in smaller scales, they could see it. It turned out, that beyond the Planck length, things lose their locality, their space time signature. It is everywhere and nowhere. It is in Quantum terms, the cat is neither dead nor alive.

    The universe is closer to a virtual simulation that somebody digitally coded into existence. All the “parameters” of the universal constants, it is as if they were inputted precisely in order to create human life. If it had been coded into base 2 or base 60, humans could have cracked and reverse engineered the code by now, but we have not. That is because the code is far more complex than the DNA helix, and most people can’t decipher much of even DNA so they call it junk data. In a computer program, no line is junk, if it is used. Destroy even one line of code in an OS and it can crash. The DNA is a self replicating, self correcting, strand of digital code, pure information in a material dimension.

    The Babylonians wrote in their histories, that their technology was the best because they received it from what they call the Annunaki, or the sons of Anu; the gods who descended to Earth to help Babylon become the greatest human empire.

    In Hebrew Genesis, the Annunaki were the Watchers who fell from Heaven and disobeyed divine commandments, and transferred technology illegally to humans. This resulted in the Tower of Babel incident in which the original centralized Enochian divine language was shattered, making it impossible for humans to understand the mathematics and principles written on stone, metal, or paper. This is because the pre Flood technology level was even higher than Babylon’s. Babylon had what was left of a fallen civilization that was killed off, for the most part.

    The Great Pyramid of Giza wasn’t built by the Egyptian pharoahs, there is a lot of indication that was it a Pre Flood structure. Thus as time goes on, you will see that there is a trend. The technological capabilities of these human civilizations goes downwards as time proceeds. The theory of secular evolution should be the reverse, that humans started with low tech and advanced/evolved to high tech.

    Yet the opposite thing happens, in which the 1000 BC to 500 AD artifacts of the ancients in N America is more advanced than the tepee living raiders of the red American Indians found in 1800s.

    Why is angel and god tech forbidden from being transferred to mortal humans? Ever hear of something called the Prime DIrective? Prometheus? Those are concepts that people can comprehend more easily.

    As for that lion… there’s an interesting story about that one. The Lion represents Jesus of Nazareth, the Messiah and Savior of Humanity, as he was once known as. For those that think that is far fetched, consider why CS Lewis put Arslan in what body and why.

    The Lion is also associated with the tribe of Judah, and it is Nimrod himself, the totalitarian tyrant that created the Tower of Babel to attack heaven and replace Jehovah, who was considered the mighty hunter of lions. The Babylonians follow the same gods as Nimrod, in fact Gilgamesh and Isis are mirror templates to the historical Nimrod and his wife.

    For christians, Nimrod is an interesting figure that has many connections to John’s Revelations.

    The “mound builders” of ancient America, are also called Jaredites or the Adena. The red indians called them white skinned giants with yellow or red hair, that were warlike and also extremely dangerous, as they ate these human tribes. They had six fingers and six toes, a genetic indicator that the tribes used to distinguish who was safe and not safe to meet up with at a distance. Goliath’s brothers also had the same dna mark.

    Recently, Americans are in an uproar just because they realized political factions and the MSM lied to them their entire lives. I wonder what the look on their faces will be when they realize that the entirety of human history and the creation of the verse, was merely another fabrication public education fed them…

  28. AesopFan Says:

    Tim Turner Says:
    August 25th, 2017 at 2:20 am

    “..It’s just like base 10, if you’re missing 2 fingers…”

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