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Math Education

Ancient Tablet Reveals Babylonians Discovered Trigonometry (sciencemag.org) 83

An anonymous reader quotes a report from Science Magazine: Trigonometry, the study of the lengths and angles of triangles, sends most modern high schoolers scurrying to their cellphones to look up angles, sines, and cosines. Now, a fresh look at a 3700-year-old clay tablet suggests that Babylonian mathematicians not only developed the first trig table, beating the Greeks to the punch by more than 1000 years, but that they also figured out an entirely new way to look at the subject. However, other experts on the clay tablet, known as Plimpton 322 (P322), say the new work is speculative at best. Consisting of four columns and 15 rows of numbers inscribed in cuneiform, the famous P322 tablet was discovered in the early 1900s in what is now southern Iraq by archaeologist, antiquities dealer, and diplomat Edgar Banks, the inspiration for the fictional character Indiana Jones.

Now stored at Columbia University, the tablet first garnered attention in the 1940s, when historians recognized that its cuneiform inscriptions contain a series of numbers echoing the Pythagorean theorem, which explains the relationship of the lengths of the sides of a right triangle. (The theorem: The square of the hypotenuse equals the sum of the square of the other two sides.) But why ancient scribes generated and sorted these numbers in the first place has been debated for decades. Mathematician Daniel Mansfield of the University of New South Wales (UNSW) realized that the information he needed was in missing pieces of P322 that had been reconstructed by other researchers. He and UNSW mathematician Norman Wildberger concluded that the Babylonians expressed trigonometry in terms of exact ratios of the lengths of the sides of right triangles, rather than by angles, using their base 60 form of mathematics, they report today in Historia Mathematica.

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Ancient Tablet Reveals Babylonians Discovered Trigonometry

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  • As the bible says they all started to speak in different tongues.

    • After a cup a coffee, I figured it this way. Closed Source has a new shiny example of what works; right?
    • by Orgasmatron ( 8103 ) on Friday August 25, 2017 @10:22PM (#55087893)

      There is a version of that story in the Sumerian literature too. Here is the Electronic Text Corpus of the Sumerian Language (ETCSL) link:

      http://etcsl.orinst.ox.ac.uk/cgi-bin/etcsl.cgi?text=t.1.8.2.3# [ox.ac.uk]

      • by CODiNE ( 27417 )

        Seriously, I tried to read that, it's extremely verbose and repetitious. Can you quote the relevant section? Is it the part where somebody acted like a donkey?

        • by Orgasmatron ( 8103 ) on Saturday August 26, 2017 @07:26AM (#55089001)

          134-155. "Chant to him the holy song, the incantation sung in its chambers -- the incantation of Nudimmud: "On that day when there is no snake, when there is no scorpion, when there is no hyena, when there is no lion, when there is neither dog nor wolf, when there is thus neither fear nor trembling, man has no rival! At such a time, may the lands of Shubur and Hamazi, the many-tongued, and Sumer, the great mountain of the me of magnificence, and Akkad, the land possessing all that is befitting, and the Martu land, resting in security -- the whole universe, the well-guarded people -- may they all address Enlil together in a single language! For at that time, for the ambitious lords, for the ambitious princes, for the ambitious kings, Enki, for the ambitious lords, for the ambitious princes, for the ambitious kings, for the ambitious lords, for the ambitious princes, for the ambitious kings -- Enki, the lord of abundance and of steadfast decisions, the wise and knowing lord of the Land, the expert of the gods, chosen for wisdom, the lord of Eridug, shall change the speech in their mouths, as many as he had placed there, and so the speech of mankind is truly one.""

          Enmerkar is building a tower/temple to the goddess Inana at Eridu. He asks her for permission to collect a tribute from Aratta. The messenger is told to threaten to destroy Aratta and disperse the people if they don't pay up, and to chant a song asking Enki to fix the languages - "change the speech in their mouths, as many as he had placed there".

          Sumerian translations aren't perfect, so we aren't positive if Enki is to fix the languages that he had broken earlier, or break the single language now.

          Oh, and towards the end, writing gets invented. The messages back and forth get longer and longer until the poor messenger can't remember it all - "The messenger, whose mouth was heavy, was not able to repeat it." - so the king invents writing, which makes the messenger positively giddy. That is in lines 500-514.

          • Enmerkar is building a tower/temple to the goddess Inana at Eridu. He asks her for permission to collect a tribute from Aratta. The messenger is told to threaten to destroy Aratta and disperse the people if they don't pay up, and to chant a song asking Enki to fix the languages - "change the speech in their mouths, as many as he had placed there".

            Right but this isn't like the Tower of Babel story at all.

            There is a structure, which may or may not be a tower. Languages and a dispersion of a group of people ar

            • Heh. You are asking a lot from 4000 year old literature. How many ancient stories do you think are out there concerning 1) a tall building, 2) in Babylon, and 3) the cause of language divergence?

              Think of them as cousins, descendants from a common ancestor-story, passed down from mouth to ear for thousands of years before the invention of writing.

              • the Ur-version to rule them all before the continents split up completely and you could walk from the UK to belgium on foot ? Shame ancient history is like cutting edge physics ... mostly theoretical lol , who has my time machine ? i got some catching up to do
          • by CODiNE ( 27417 )

            The lord of Kulaba inscribed the message like a tablet. It was just like that. The messenger was like a bird, flapping its wings; he raged forth like a wolf following a kid.

            That's pretty neat... earliest literary mention of someone geeking out.

    • So not only did they have tablets, but they had different and incompatible operating systems as well?

  • but who beat the Babylonians?
    • by Anonymous Coward

      but who beat the Babylonians?

      the green bay packers

  • by DontBeAMoran ( 4843879 ) on Friday August 25, 2017 @09:24PM (#55087729)

    Ancient tablet reveals Babylonians discovered trigonometry.

    What kind of tablet was it? iPad or Android?

  • Oh please. (Score:5, Insightful)

    by Anonymous Coward on Friday August 25, 2017 @09:31PM (#55087749)

    Equating a table of pythagorean triples to Euclid is like equating the wheel that Oog invented in 50,000 BC to a Ferrari.

    Babylonians don't get credit for trigonometry for being the first to use the concept of similar triangles.
    Persians don't get credit for all of algebra because they were the first to write down the quadratic formula.
    Aristotle doesn't get credit for absolutely all of math and logic because he invented modus ponens.
    Eudoxus and/or Archimedes don't get credit (though arguably they should) for inventing calculus for discovering the method of exhaustion.

    On the other hand, Euclid *does* get credit for inventing synthetic geometry and (basic) number theory, because Elements actually contains fleshed-out theories that still form a major part of what would be taught in a beginning course today.

    • Re:Oh please. (Score:4, Insightful)

      by Puff_Of_Hot_Air ( 995689 ) on Friday August 25, 2017 @11:50PM (#55088141)
      Where did you get Euclid from? No one is talking about Euclid except you. No one is suggesting the Babylonians were the first to "invent" trigonometry (mathematical concepts aren't invented, hence why you can't patent them). They were the first to use them, but we already knew that. What's interesting about this if that they used them way before we knew, but also that they were able to perform pratical trig (with triangles), in a completely different and novel way, without angles. What's more, this approach has pratical advantages over our current methods (albeit somewhat redundant since the calculator).
    • by Anonymous Coward

      THANKS! Guess I'll take your authoritative tone as the final word on the matter! Kudos tool!

  • by Anonymous Coward

    He and UNSW mathematician Norman Wildberger concluded that the Babylonians expressed trigonometry in terms of exact ratios of the lengths of the sides of right triangles, rather than by angles, using their base 60 form of mathematics, they report today in Historia Mathematica.

    So what's new about that, beyond the base 60 thing?

  • Time travel (Score:5, Informative)

    by jfdavis668 ( 1414919 ) on Friday August 25, 2017 @10:19PM (#55087885)
    The Babylonians didn't invent trigonometry, they invented time travel. They went into the future and stole trigonometry from us.
  • by Anonymous Coward on Friday August 25, 2017 @10:27PM (#55087911)

    There's a lot to be said, but Slashdot has been infested with such hilarious Redditors. I never thought I'd miss the days when Digg commenters infested slashdot. Says something when the likes of Digg is far preferred to the likes of Reddit.

    moving on...

    the famous P322 tablet was discovered in the early 1900s in what is now southern Iraq by archaeologist, antiquities dealer, and diplomat Edgar Banks

    From Plimpton 322 wiki, [wikipedia.org]

    According to Banks, the tablet came from Senkereh, a site in southern Iraq corresponding to the ancient city of Larsa.

    So my comment isn't earth shattering, but at least you're smarter than you were a moment ago, unlike after reading any of the comments above mine. Go home Redditors, you're drunk.

  • The tablet doesn't really contain trigonometry as we understand it today. There is no concept of angle, for instance. Some have convincingly suggested [maa.org] alternate interpretations. That paper, by the way, dates from 2002, so this isn't really news.
    • by Puff_Of_Hot_Air ( 995689 ) on Friday August 25, 2017 @11:27PM (#55088081)

      The tablet doesn't really contain trigonometry as we understand it today. There is no concept of angle, for instance.

      That's absolutely true and also why the discovery is so interesting. It is trigonometry. Trigonometry without angles. The authors have a YouTube video which is very informative Here [youtu.be]. There are so many interesting things about this. Angles are not needed to work with triangles. The sexagesimal numbering system had many advantages over our current decimal system from an application perspective. It's just a whole new way of thinking about trig. Anyway, it's well worth 20 minutes of your time.

    • by jdreyer ( 121294 )
      Agreed. In the video, they keep saying that this trigonometry is "different" from modern trigonometry because it's based on ratios, not angles. Hogwash! Modern trigonometry is all about ratios, but in relation to angles. It might be interesting to study triangle ratios without reference to angles, but it's the relationships between the angles and the ratios that makes it trigonometry. If you can't figure out how high up a wall a 10' ladder goes at a 70 angle, it's not trigonometry!
      • by Anne Thwacks ( 531696 ) on Saturday August 26, 2017 @04:53AM (#55088675)
        If you can't figure out how high up a wall a 10' ladder goes at a 70 angle, it's not trigonometry!

        That is what it does do. However, it requires that you have a ladder an integer number of units high, and place the bottom an integer number of units from the wall, and those integers must be in a predetermined ratio.

        Basically, it is a list of triangles like 3,4,5 and 13,12,5 but, because they used base 60, there are a lot more of them in a given range.

        However, the strategy for finding the answers in the table, and the way in which the table is laid out, are way more useful to builders and surveyors than the tables we used prior to calculators being invented, and the answers (for the values in the table) are more accurate than many tables generated before the use of computers, as the method relied entirely on manipulating integers, rather than 4 figure log tables generating decimals to a fixed number of figures by expounding a power series to a limited number of terms.

        I was skeptical at first, but I am inclined to agree that it actually IS a different trigonometry, and it gives useful and practical results - but it does so by not solving the general case. However, it covers most cases that would be encountered by people using the technology of the day. (eg pyramid builders, land surveyors, and very probably boat builders). It is likely woefully inadequate for celestial navigation, but I have not tried it ;-)

        • by jdreyer ( 121294 )
          I still don't see how this is trigonometry. When solving a right triangle and you have two of the sides, you can find the third side using the Pythagorean Theorem, without trigonometry. That's pretty much what this system does, normalized so the long leg is 1. The way they did it, taking advantage of terminating sexagesimal fractions coupled with an impressive collection of Pythagorean triples, is certainly amazing, given how long ago this was done, and it might have been extremely useful, but I wouldn't ca
    • It's stuff I didn't know, pertaining to history and mathematics. I'd say it counts more as news than some trial baloon or fud about how much the next iPhone will cost.

  • by Anonymous Coward

    (1) A table of Pythagorean triples does not Trigonometry make;

    (2) A table of Pythagorean triples, however ancient, is not going to change trigonometry today;

    (3) Credit for discoveries requires that you make your discoveries known. It's terribly fascinating that those people apparently figured this stuff out; but history will remember others as the founders of Trigonometry, and rightly so.

    • by PPH ( 736903 )

      History should also remember all those Dark Ages monks who scrapped the text off of ancient scientific texts to make prayer books. [wikipedia.org] You might want to give credit to mathematicians for (re)inventing this stuff a few hundred years ago. I'd like to know where we would be if science and math hadn't been set back by a few millennia.

      • I'm not sure I'd call math 'invented.' I'd prefer to call it discovered. I didn't invent the maths to model traffic. I discovered it. Well, technically I built off prior work. It existed before I made it known, however.

  • The tablet is missing the entire left section. It would be like finding part of a tablet that says "mc2" and decide that the person must have stumbled upon mass-energy equivalence long before Einstein. It's missing half the equation, and doesn't even define what "mc" is.
    • Clearly you have spent approximately 0 seconds reviewing this study. It is a table (much like a sine or cos table back in the pre-calculator era), and the missing bit was really just an explanation as to the why of the table. These guys have "reverse engineered" that with a combination of math (how the numbers are related) and an understanding of how the sexagesimal system and babloynian use of it influenced their practical use of math.
  • They were first or the invented an "they also figured out an entirely new way to look at the subject."
  • In addition discovering trigonometry, the tablet shows Babylonia showed up hung over to their trigonometry mid-term exam and barely got a passing grade. Lucky for Babylonia, grading on a curve [wikipedia.org] was millennia away from being invented.

  • > Ancient Tablet Reveals Babylonians Discovered Trigonometry

    And also invented the Tablet.

  • Imagine what humans in 3700 years will discover when they found our tablets!

  • I took an undergraduate degree in History solely for the opportunity to study the Code of Hammruabi [wikipedia.org], which, along with the Pentateuch of the Bible, is one of the earliest systems of law ever recorded. Only later as a much older man was I able to afford going to Paris to see one of the actual steles on which the Code was inscribed, which was an unforgettable experience (it's a solid piece of obsidian over 7 feet high, with deeply-cut symbols that look like they were made about a year ago. These were people who didn't believe in Agile, not even for a moment!)

    I am not terribly strong in mathematics, but I was able to follow the gist of the paper, and I do think their interpretations about the Babylonians preference for exactitude and integer divisions aligns quite well with the precision of their legal work.

    From a purely academic perspective, it was an enormously sophisticated and skilled culture for its time.

  • Trigonometry, the study of the lengths and angles of triangles, sends most modern high schoolers scurrying to their cellphones to look up angles, sines, and cosines.

    Not like in the old days, when we memorized those trig tables.

    Looking something up on your phone isn't any different than looking it up in a printed table. So looking up the sin of some particular angle (other than pi/2, pi/6 etc.) is something everyone has always done. The real challenge is memorizing trigonometric identities. But to be frank, if a student can find the trig identity he needs and use it successfully, who the hell cares? That's a very good result in itself.

  • It seems to me the dark ages started earlier than we think and have lasted longer.. So "we" are really more like 2000 year BEHIND times. Imagine if the questions of Lucretious about the property of atoms and the use of this trig had been continuous... That old saw about the ones unable to learn from history destined to always repeat it.
  • This is interesting stuff, but my question is, did they use this alternate trigonometry for anything. One alternative is some old crackpot in a Babylonian prison came up with this in his cell. Stamped it out on some old mud. The quest would be if they can find examples of these integer triangles used in construction. That would add a lot to the story. (Hmm, where do I find dimension drawings of some of their buildings accurate enough to test this?)

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