back to article Forget trigonometry, 'cos Babylonians did it better 3,700 years ago – by counting in base 60!

Those of you who can remember trigonometry can feel free to forget it, because ancient Babylonian mathematicians had a better way of doing it – using base 60! That's the conclusion of a new paper, Plimpton 322 is Babylonian exact sexagesimal trigonometry, in the new issue of the journal Historia Mathematica. The “Plimpton 322 …

  1. HkraM

    Copyright

    The Babylonians knew a thing or two about copyright as well if the tablet is still protected after over 3000 years.

    1. eldakka Silver badge

      Re: Copyright

      While I did upvote....

      technically, I don't think it's the tablet that is protected by copyright, it's the images of the tablet that are copyrighted.

      If you had access to the physical tablet, you could take your own photos, and you would own the copyright in those photos (or release them public domain if you so choose).

      1. Pompous Git Silver badge

        Re: Copyright

        "If you had access to the physical tablet, you could take your own photos, and you would own the copyright in those photos (or release them public domain if you so choose)."

        There are images already in the public domain, so not much point. I suspect HkraM was only trying to get a laugh...

        The Babylonian tablet Plimpton 322

    2. Anonymous Coward
      Anonymous Coward

      Re: Copyright

      It's not well known but Walt Disney descended from the Babylonians.

      1. This post has been deleted by its author

        1. Ugotta B. Kiddingme

          Re: Copyright

          Symon: "Yep. http://cyberlaw.stanford.edu/blog/2007/03/fairy-use-tale"

          Wow, thank you. That is absolutely brilliant. And quite possibly the best example of passive-aggressive trolling I have ever seen.

          1. Destroy All Monsters Silver badge

            Re: Copyright

            That is absolutely brilliant.

            Andrew is probably already penning an article about the starving Babylonian artists and scribes forced to live in Akkad dumpsters.

      2. LDS Silver badge
        Joke

        Walt Disney descended from the Babylonians

        If so Babylonians would have used octal arithmetic....

        1. herman Silver badge

          Re: Walt Disney descended from the Babylonians

          The Babylonians must have been polydactyl.

          https://www.google.ae/search?q=cat+with+60+toes

    3. MyffyW Silver badge
      Paris Hilton

      Re: Copyright

      Paris COS she's bound to have a TAN

      1. Pompous Git Silver badge

        Re: Copyright

        "Paris COS she's bound to have a TAN"
        A bit hard to tell from the bedroom scenes she's made...

        1. Anonymous Coward
          Anonymous Coward

          Re: Copyright

          "Paris COS she's bound to have a TAN"

          "A bit hard to tell from the bedroom scenes she's made..."

          Well, that's the SIN right there.

          1. Antron Argaiv Silver badge
            Coat

            Re: Copyright

            "Paris COS she's bound to have a TAN"

            "A bit hard to tell from the bedroom scenes she's made..."

            "Well, that's the SIN right there."

            I'll watch the video again, and this time, I'll try to ignore the SECs and pay more ATAN-tion to the skin tones...

            1. handleoclast
              Coat

              Re: Copyright

              @Antron Argaiv

              With puns like that, don't be surprised if somebody hits you with a COSH. Then you'll end up in a hospital COT.

              BTW, what's a nortna? The second part of your name is obvious, but not the first (not to me, anyway, I expect about 200 replies pointing out what I've missed).

            2. DropBear Silver badge
              Paris Hilton

              Re: Copyright

              "I'll watch the video again, and this time, I'll try to ignore the SECs and pay more ATAN-tion to the skin tones..."

              There's a direct connection to ATAN insomuch every programmer unfortunate enough to have dabbled in geometry-handling maths knows it's an unreliable bitch - you really should use ATAN2 instead...

    4. Anonymous Coward
      Anonymous Coward

      Re: Copyright

      >The Babylonians knew a thing or two about copyright

      And like the Greeks their best days are far behind them and they have been coasting ever since.

  2. Nick Kew Silver badge

    So much for digital

    If the relative lack of divisors of 10 is a shortcoming, why has the modern world moved so far towards pure binary (and powers of 2 in specific contexts)?

    1. Dave 126 Silver badge

      Re: So much for digital

      I don't know... Something to do with Napoleon possibly?

      Whilst I work (measure and cut) in mm, I estimate in feet and inches. However I may be missing a trick because of the way that 12 can be easily divided by 3 and by 4 (and obviously 2 and 6). For centuries, carpenters have been able to make beautiful pieces without a unit of measurement by means of dividers - it is only important that they can express a length as a rational of another.

      1. Anonymous Coward
        Anonymous Coward

        Re: So much for digital

        Thought that was why there was 60 seconds / minutes in a hour. Because it was easy to work out divide by 2, 3, 4, 5, 6, 10, 12, 15, 20, 30

        1. alain williams Silver badge

          Re: So much for digital

          I always thought that they used 60 because 6*60 gives you about the number of days in a year and that a circle has 360 degrees because every day you move about one degree around the zodiac.

          1. Omgwtfbbqtime Silver badge
            Mushroom

            Re: So much for digital

            That's why I prefer mils for angles.

            6400 mils to a circle.

            1mil subtends 1m at 1km.

            or at least close enough for artillery.

        2. Doctor Syntax Silver badge

          Re: So much for digital

          "Thought that was why there was 60 seconds / minutes in a hour."

          I think that's also derived from the Babylonians as do the divisions of a circle. But, of course, it was they who had the wit to use a number base that was convenient for integer division rather than an inconvenient one based simply on counting their fingers.

        3. Anonymous Coward
          Anonymous Coward

          Re: So much for digital

          "Thought that was why there was 60 seconds / minutes in a hour. "

          And 360 degrees in a circle.

        4. Anonymous Coward
          Anonymous Coward

          Re: So much for digital

          > Thought that was why there was 60 seconds / minutes in a hour. Because it was easy to work out divide by 2, 3, 4, 5, 6, 10, 12, 15, 20, 30

          And by 37⅙, if you're not too bothered about integer results (or your time source shows serious short-term stability issues).

        5. TimNevins
          Thumb Up

          Re: So much for digital

          Correct. Numbers like 12 and 60 were used by street market traders due to their high level of divisibility.

          This was especially important when buying selling items that could not be easily divided such as fruit/vegetables which would spoil once cut. A cart of 60 apples could be split into 10 different uniform units.

      2. Anonymous Coward
        Anonymous Coward

        Napoleon?

        Metric units came much later than base 10 arithmetic.. metric units match the base 10 positional system, while imperial units don't.

        1. PNGuinn
          Headmaster

          Re: Napoleon?

          Pedant.

          Why waste a good opportunity to blame the French?

          He's a pedant. It's Friday. He looks slightly French. >>

      3. Chris G Silver badge

        Re: So much for digital

        Plus carpenters and builders have used the 3.4.5. formula to get a right angle for centuries too.

        Dividers, a piece of string or a length of wood is enough to provide the measurements for building almost anything including round towers and staircases.

      4. Dazed and Confused

        Re: So much for digital

        > Whilst I work (measure and cut) in mm

        Is this for lengths or cross sections?

        I was amused to note when ordering a replacement thermostat for my posh German shower that the fittings are 3/4" ones.

    2. Pen-y-gors Silver badge

      Re: So much for digital

      Obvious. Using base 10 evolved because we have 10 digits (most of us). Using base two reflects the decline in education standards, which means that many people count hands rather than fingers.

      Of course, binary is also handy for smart-arse techies because we can count to 1023 on our fingers.

      1. DailyLlama

        Re: So much for digital

        1024 if you undo your fly

        1. Anonymous Coward
          Anonymous Coward

          Re: So much for digital

          Or 2047?

          Oh well, there are 10 types of people in the world.

        2. Doctor Syntax Silver badge

          Re: So much for digital

          "1024 if you undo your fly"

          No, that would be 2047.

          1. Anonymous Coward
            Anonymous Coward

            Re: So much for digital

            > No, that would be 2047.

            Depends on how much youthful vigour you have left.

        3. Pen-y-gors Silver badge

          Re: So much for digital

          1024 if you undo your fly

          No, 2047! But only if you're a bloke.

          1. herman Silver badge

            It is getting nippy

            Well, the ladies win. They can use their nips.

        4. Allan George Dyer Silver badge

          Re: So much for digital

          @DailyLlama - Downvoted for:

          1) sexism

          2) Failure to calculate 2^11-1

        5. This post has been deleted by its author

        6. Borg.King

          Re: So much for digital

          1024 if you undo your fly

          That must be your least significant digit.

        7. wayne 8

          Re: So much for digital

          That's sexist. Check your male patriarchal privilege in a safe space.

        8. Someone Else Silver badge
          Coat

          Re: So much for digital

          1024 if you undo your fly

          Uhhh...that would be 2047. Or maybe -1024? (Possibly depending on the stiffness of the digit...)

        9. herman Silver badge

          Re: So much for digital

          Hey don't be so misogynistic. How must the lady vultures feel now?

        10. Anonymous Coward
          Anonymous Coward

          Re: So much for digital

          Big or little Endian?

      2. Jos V

        Re: So much for digital

        Well, it might be because we have 10 fingers, but in the older days, trades folks used their thumbs to count the phalanx bones of the fingers. 12 on each hand, for a total of 12 x 12 =144.

        1. Justin Clift

          Re: So much for digital

          Interesting. With that method, using one hand to count units (up to 12), and the other hand to count the groups of 12... it's possible to count up to 60 using your hands.

          Hadn't realised that before, but it makes sense as to a practical way to count up to 60 in early historical times.

          1. Charles 9 Silver badge

            Re: So much for digital

            It's entirely possible to handle 0-9 with just one hand. Use the same system with the other hand and you can do 0-99 easily.

      3. Anonymous Coward
        Anonymous Coward

        Using base 10 evolved because we have 10 digits

        @Pen-y-gors

        I was waiting for someone to post that.

        Therefore if the Babylonians used base 60 they had 60 fingers.

        There we have it, proof Aliens built the pyramids and I didn't even need my trusty hat.

      4. Joe Harrison

        Re: So much for digital

        You can do better than 1023 if you count up to 2 on each finger, I mean finger down = 0, finger halfway up = 1, finger completely up = 2. You could probably have even more finger positions but then it gets confusing.

        1. handleoclast

          Re: So much for digital

          @Joe Harrison

          You are using ternary (or trinary) digits. Confusingly abbreviated as "tits."

    3. Doctor Syntax Silver badge

      Re: So much for digital

      "why has the modern world moved so far towards pure binary (and powers of 2 in specific contexts)?"

      Imperial measurement made considerable use of binary. Weights from pounds down to drachms were binary as were volumes from gallons down to gills. In general they seem to have been based on measures which were a convenient size for some purpose with a strong inclination to subdivide on a binary basis. It's a natural thing to do. If you have a standard of weight, for instance, you can weigh out that amount of sand, flour or whatever on scales and then, using the same scales, divide that into two equal portions and subdivide further.

      The problem arises when two different scales of measurement overlap and we end up with a stone of 14 pounds. Other stones were available - I've seen reference to a stone of 15lbs in the C18th - but I suppose a atone of 16lbs would have required too much adjustment to reconcile with the larger scales in use for other purposes.

      1. Pen-y-gors Silver badge

        Re: So much for digital

        volumes from gallons down to gills

        Reminds me of a wonderful old early C19 dictionary I have. Defines a pint as 'half a quart', a quart as 'a quarter of a gallon' and a gallon as 'eight pints' - no actual mention of them being units of volume!

      2. Glenturret Single Malt

        Re: So much for digital

        While living in SE Asia, we came across the kati (= 4/3 lbs) and the tahil (one sixteenth of a kati). Beer was sold in bottles containing 4/3 pints.

    4. Primus Secundus Tertius Silver badge

      Re: So much for digital

      Binary arithmetic is a myth.

      The commercial world, which calls itself the real world, runs on binary coded decimal. That means it can handle billions of (dollars/pounds/euros/yen) to the nearest (cent/penny/hundredth).

      1. Destroy All Monsters Silver badge

        Re: So much for digital

        The commercial world, which calls itself the real world, runs on binary coded decimal

        S'truth. Today on java.lang.BigDecimal until JSR 354 comes out. (actually you need a BigRational too)

        See also: Why not use Double or Float to represent currency?

    5. MartinT

      Re: So much for digital

      "why has the modern world moved so far towards pure binary (and powers of 2 in specific contexts)?"

      Because it is easier and more efficient to build computers that use binary.

  3. Pompous Git Silver badge

    "why has the modern world moved so far towards pure binary (and powers of 2 in specific contexts)?"
    Because of binary logic:

    True/False

    0/1

    Yes/No...

    Simples :-)

    1. bazza Silver badge

      For some applications, that's not entirely true anymore. Multi-Level Cells in FLASH are not 1 or 0, there's several inbetween.

      In asynchronous (i.e. clock-less) electronics there's been some different approaches too: 1, 0, or not sure yet...

      Binary logic dominates in CPUs because with volts/no volts representing 1 and 0, there's practically no calibration to incorporate into a manufacturing process. With multi-level logic, e.g. 1, 2/3, 1/3, 0 (2 bits) suddenly it all becomes a lot harder to build.

      However, in communications multi-level representation of data is pretty common. QAM (quadrature amplitude modulation, see Wikipedia) involves multiple signal amplitudes, a far remove from simple binary on/off keying. In communications such schemes are used to get more data through a given signal bandwidth. It's the equivalent of storing more than 1 bit in a single memory cell (which is what MLC Flash is doing).

      Of course, the reason such signalling isn't used on, say, the memory bus between RAM and CPU is because it takes a lot of electronics to generate and receive such a signal; not good for speed / power. RAM buses these days are complicated enough, what with their propagation de-skew delay lines, more than 1 bit on the PCB trace at a time, etc. But complex modulations are used on links like Thunderbolt, USB3, Ethernet. There's way more than one bit on the wire at any single point in time.

      Really these days most high speed buses inside and outside of computers are RF data links, not just a single voltage level on a PCB trace.

      1. Measurer

        Digital is just very fast analogue with switching thresholds

      2. Pompous Git Silver badge

        "For some applications, that's not entirely true anymore. Multi-Level Cells in FLASH are not 1 or 0, there's several inbetween..."
        You are of course correct; I was being a bit glib. In truth we have found quite a few uses for multivalent logics in recent time so we are moving away from binary logic rather than towards it. William of Ockham (ca. 1287–1347) did a bit of work on trivalent logic without finding a use for it. Probably too busy counting angels on pinheads and neglecting to write down the results.

        1. Hans 1 Silver badge
          Windows

          To get the Slurp handle ... Microsoft, in their infinite wisdom, have extended boolean to a tri-state, which has 5 states (Yeahhh, don't get me started), of which two, exactly true and false, are supported ... the other 3 ain't ... just useless ... seriously, no joke, here's the doc:

          https://msdn.microsoft.com/en-us/library/microsoft.office.core.msotristate.aspx

          When you do Office automation, Word accepts standard $true & $false and casts those to msotristate:msotrue/false, respectively, where as other Office applications, I am looking at YOU PowerPoint don't ... behavior changes depending on which language (VBA, VBS, PowerShell) you use... what a pile, hey ?

    2. herman Silver badge

      Ayup, that must be why Fuzzy Logic never went anywhere.

  4. jake Silver badge

    Strangely enough ...

    ... everybody's ::koff::favorite::koff:: wiki has an arguably more readable image.

    https://en.wikipedia.org/wiki/File:Plimpton_322.jpg

    I'm pretty sure the scribble at the top right corner says "Three shirts, no starch, Wednesday late".

    1. Simon Sharwood, Reg APAC Editor (Written by Reg staff)

      Re: Strangely enough ...

      FWIW Wikipedia isn't public domain - we can't just lift their stuff.

      1. Pompous Git Silver badge

        Re: Strangely enough ...

        "Wikipedia isn't public domain - we can't just lift their stuff."
        The photograph of the Plimpton tablet isn't "their stuff"; it's in the public domain. Wikipedia lifting stuff from the public domain doesn't mean it's copyright Wikimedia Foundation no matter what Jimmy Wale might believe.

        1. Pen-y-gors Silver badge

          Re: Strangely enough ...copyright complications

          The photograph of the Plimpton tablet isn't "their stuff"; it's in the public domain.

          Image copyright is horribly complicated. The maker of the image owns copyright in the image. If someone is permitted to take a photo of the image then they have copyright of the new image, but so (usually) does the original photographer!

          Artworks are even more fun. If I own a picture by a modern artist, and take a photo of it, unless the artist has assigned all rights to me as part of the sale, I can't publish that photo (on the web or anywhere) without the permission of the artist (or other copyright holder).

          So, does this tablet count as an original artwork? Although after 3700 years I suspect the carver's copyright has expired (depending on Babylonian copyright laws - are they in the Berne convention? After all, Disney characters are copyright for all time.)

          1. Pompous Git Silver badge

            Re: Strangely enough ...copyright complications

            "Image copyright is horribly complicated. The maker of the image owns copyright in the image. If someone is permitted to take a photo of the image then they have copyright of the new image, but so (usually) does the original photographer!"
            Ain't that the truth? I'm a member of the Australian Copyright Council and beneficiary of royalties therefrom.

            The photograph in question is old enough the photographer is more than likely dead. Nobody knows who the photographer was, or if they do they've stayed schtum for a very long time.

          2. LDS Silver badge
            Joke

            "depending on Babylonian copyright laws"

            What Hammurabi stele says about it? Also, beware punishment for copyright infringement could be quite harsh, compared to our times...

      2. Pen-y-gors Silver badge

        Re: Strangely enough ...

        FWIW Wikipedia isn't public domain - we can't just lift their stuff.

        True, but most of it you can lift, subject to their various licences, most of which permit use with attribution. Images all state what the copyright position is. Hell, there are people selling Print-on-demand books on Amazon that are just a printout of Wikipedia articles on a topic.

  5. Pompous Git Silver badge

    "I'm pretty sure the scribble at the top right corner says "Three shirts, no starch, Wednesday late"."
    There ya go! And I always thought it said: "Not tonight; it's my pyramid!"

    1. This post has been deleted by its author

  6. Pompous Git Silver badge

    Three-valued logic

    Meant to add this to my remark above. The Wiki-bloody-pedia's quite good on this. My last university class was taught by a bloke who did his PhD on 3-value logics.

    Three-valued logic

    1. pavel.petrman Bronze badge

      Re: Three-valued logic

      Yes, three valued logic for binary computers has already been described in high-impact journals!

      1. Destroy All Monsters Silver badge
        Windows

        Re: Three-valued logic

        Three-valued (and even higher-valued) logics are nice but the thing is, you need to know what your logic operations are going to express, and there are quite a few of them (16 for binary, 19683 for ternary)

        As for formulas, undecidablility arrives fast, too fast.

      2. To Mars in Man Bras!

        Re: Three-valued logic

        The USSR did quite a lot of work on 'trinary' computing, back in the 1950s:

        https://dev.to/buntine/the-balanced-ternary-machines-of-soviet-russia

  7. Tim99 Silver badge
    Coat

    We use base 10 for a reason

    We can count off on 8 fingers and two thumbs (alright we can go to 20 in warmer climates if we can use our toes). Some people in the world still count in 60s using the same 8 fingers and two thumbs. If you are predominantly right handed, use your right thumb to count to 3 with the top, middle and lower phalanx of your right hand little finger, then three more with the ring finger joints, then the middle finger, then the index finger to give 12. Extend the little finger of your left hand to count off the first 12, the repeat for another 12 with the right hand and extend the ring finger for 24, then count another 12, and use the left hand middle finger for 36, then the left index finger for 48, and finally the left thumb for 60.

    It may be one reason why old farts like me were taught the duodecimal system. We bought things in dozens and paid for them in shillings and pence - Also ten is only divisible by the integers 1,2, and 5; twelve is divisible by 1,2,3,4, and 6; and sixty is divisible by 1,2,3,4,5,6,10,12,15, 20, and 30 - So very "handy" when selling items or dividing them up between people. There were 12 shillings (and 240 pennies in the pound), so we could divide a pound by 16, 24, 30, 40, and 60 as well.

    "Uphill both ways, in the snow, barefoot" might allow a higher count...

    1. jake Silver badge

      Re: We use base 10 for a reason

      When my daughter was learning to count (age 4ish), I taught her to count to 15 on four fingers. She added the thumb, and then the other hand, on her own. In highschool, she "invented" three extra digits on each extremity, for full 32-bit compatibility ... with her right eye as a carry-bit.

      Teach your kids alternates to decimal numbers early and often ...

    2. Jan 0
      Windows

      Re: We use base 10 for a reason

      > There were 12 shillings (and 240 pennies in the pound)

      Errm, we real oldies remember 12 pennies in a shilling and 20 shillings to the pound. Moreover, you could subdivide a pound into 960 farthings.

      1. Paul Cooper

        Re: We use base 10 for a reason

        I did O-levels (pre GCSE!) in the good old pounds, shillings and pence days. I had to learn how to add, subtract, multiply and divide sums in pounds, shillings and pence. However, the only sensible way to do compound interest calculations was to convert to pounds and decimals of a pound, do the calculation, then convert back! However, there were a lot of useful hacks for less demanding arithmetic, some of which I can still remember - the dozen rule, for example (things were often sold in multiples of 12 in those days, so knowing that the price of 12 items in shillings = price of one item in pennies was very useful), the score rule (same in shillings and pounds), and several others I've now forgotten!

        1. To Mars in Man Bras!

          Re: We use base 10 for a reason

          >I did O-levels (pre GCSE!) in the good old pounds, shillings and pence days. I had to learn how to add, subtract, multiply and divide sums in pounds, shillings and pence

          I was a kid in the 1970s and just started school the year before decimalisation. I remember our 'times tables' books went up to 12, but we only ever were taught multiplication up to 10 times.

          I guess learning to multiply by twelve was, all of a sudden, no longer a required skill, bit they hadn't updated the text books yet.

          BTW: being a child of the decimal era, I always just thought that pre-decimal coinage was just another example of the previous generation making things unnecessarily complicated. It wasn't til much later in adult life that it suddenly occurred to me one day that 12 is divisible by 2,3,4 and 6, whilst 10 is only divisible by 2 and 5 —and I realised there was a method to the old folks' madness, after all.

      2. Pompous Git Silver badge

        Re: We use base 10 for a reason

        "Moreover, you could subdivide a pound into 960 farthings."
        And they were still legal tender... Save up enough and you could buy a penny ha'penny's-worth of sweets on the way home with your dad's Sunday bottle of cider from the offie.

      3. Charles 9 Silver badge

        Re: We use base 10 for a reason

        "Errm, we real oldies remember 12 pennies in a shilling and 20 shillings to the pound. Moreover, you could subdivide a pound into 960 farthings."

        And isn't it funny that everyone wrote out half-crowns as 2s 6d (2 and 6) rather than as, well, half a crown.

        1. dajames Silver badge

          Re: We use base 10 for a reason

          And isn't it funny that everyone wrote out half-crowns as 2s 6d (2 and 6) rather than as, well, half a crown.

          Is it? Did they?

          I used to write 2/6, just as I'd have written 4/9 for four shillings and ninepence ... it's quicker than "half a crown" and more consistent ... and "1/2 crown" might have been confused with one and tuppence.

          I never wrote "one florin" when I meant 2/-, either.

  8. rmason Silver badge

    Interesting factoid not in the article

    Is that the person who originally found/stole this artifact is the guy on whom Indiana Jones was loosely based.

    Edgar J Banks. Quite an interesting chap.

    1. harmjschoonhoven
      Meh

      Re: Interesting factoid not in the article

      Indiana Jones was based on the "archeologist" Langdon Warner (1881-1955) who stole manuscripts and Buddha's from the Dunhuang caves in China for the Harvard University Fogg Art Museum in the early 20th C.

  9. wiggers

    Special cases

    Pythagoras noted a number of special triangles, such as 3:4:5, that have integer ratios in base 10.

    https://en.wikipedia.org/wiki/Pythagorean_triple

    The Babylonians had similar special cases that can be written in base 60 as integers as shown on the tablet. Very similar techniques.

    The breakthrough with sin/cos/tan is that any triangle can be described and calculated because they are continuous transcendental functions.

    1. Pompous Git Silver badge

      Re: Special cases

      "Pythagoras noted a number of special triangles, such as 3:4:5, that have integer ratios in base 10."
      The Egyptians beat Pythagoras to it; they used 3:4:5 for land surveying. Heck, I used it a lot when building my home 14 years ago.

      1. wiggers

        Re: Special cases

        The point is, the special cases were well known in the ancient world, pre-Greek. So this tablet simply confirms that and provides more evidence for how far back it was known.

        It is not 'better' than sin/cos/tan, it's simply a collection of useful special cases.

        1. Anonymous Coward
          Anonymous Coward

          "It is not 'better' than sin/cos/tan, it's simply a collection"

          You missed the whole point of the article, about a trigonometry not based on angles and sin/cos/etc functions....

          1. Anonymous Coward
            Anonymous Coward

            Re: "It is not 'better' than sin/cos/tan, it's simply a collection"

            From the original article, which many of course didn't read...

            "P322 is historically and mathematically significant because it is both the first trigonometric table and also the only trigonometric table that is precise. Irrational numbers and their approximations are seen as essential to classical metrical geometry, but here we have shown they are not actually necessary for trigonometry. If the dice of history had fallen a different way, and the deep mathematical understanding of the scribe who created P322 not been lost, then very possibly ratio-based trigonometry would have developed alongside our angle-based approach."

            "The discovery of trigonometry is attributed to the ancient Greeks, but this needs to be reconsidered in light of the much earlier, computationally simpler and more precise Babylonian style of exact sexagesimal trigonometry. In addition to being historically significant, P322 also brings the founding assumptions of our own mathematical culture into perspective. Perhaps this different and simpler way of thinking has the potential to unlock improvements in science, engineering, and mathematics education today."

            "The novel approach to trigonometry and geometrical problems encapsulated by P322 resonates with modern investigations centered around rational trigonometry both in the Euclidean and non-Euclidean settings"

            What is important is not just the tables in the tablet itself - it's the way the table may have been computed, and the underlying assumptions.

            It's also interesting the article refers to some Knuth's article - it looks he was also interested in Babylonian algorithms...

        2. John Brown (no body) Silver badge

          Re: Special cases

          Yes, but humanity "forgot" about it when the Great Library of Nabu was destroyed in the massive eruption of the great volcano, Sohcahtoa

      2. Doctor Syntax Silver badge

        Re: Special cases

        "The Egyptians beat Pythagoras to it; they used 3:4:5 for land surveying. Heck, I used it a lot when building my home 14 years ago."

        When we moved into our home some years ago after my parents had dies I wondered what became of the 3:4:5 wooden triangle my dad made to set out the walls when he built the house. A year or so ago I found it propped up against a boundary wall when I was cutting back a holly. The joints attaching the hypotenuse had rotted but I still have the right angle.

      3. Alistair Silver badge
        Pint

        Re: Special cases

        Home building and the 3:4:5

        Three cases of beer for each 5 builders.

        Works on a Friday.

    2. John Miles

      Re: that have integer ratios in base 10

      If it has an integer ratio - then it is irrelevant what base the number is in e.g. 3:4:5 is same whether it is decimal or hexadecimal and if binary is still an integer just has more digits 11:100:101, all a different base number does is change which numbers are easier to divide by

      1. wiggers

        Re: that have integer ratios in base 10

        Yes, sorry, I realised that after I posted. I was focussing on the bigger issue...

    3. nijam

      Re: Special cases

      > ... such as 3:4:5, that have integer ratios in base 10

      It's nothing to do with the base (either in your post or the main article).

      They're integer ratios in every base. (Excepting the deranged case of non-integer bases, but you don't use those, I hope.)

      1. Anonymous Coward
        Anonymous Coward

        Re: Special cases

        One of my first programs in Fortran was to create a table of logarithms in base π as it made geometric problems a snap. This was before the first calculators came out. Naturally my junior high school maths teacher loved it.

        1. Fruit and Nutcase Silver badge
          Happy

          Re: Special cases

          @Jack of Shadows

          Naturally my junior high school maths teacher loved it.

          First day of trigonometry, my maths teacher wrote on the board...

          "Some Of His Children Are Having Trouble Over Algebra"

          sin = O/H, cos = A/H, tan = O/A

          1. Pompous Git Silver badge
            Pint

            Re: Special cases

            ""Some Of His Children Are Having Trouble Over Algebra""
            Better:

            Smiles Of Happiness

            Come After Having

            Tankards Of Ale!!!

    4. Primus Secundus Tertius Silver badge

      Re: Special cases

      Well said, Wiggers.

      The original academic article makes clear that the tablet was effectively equivalent to the ready reckoners and log tables familiar to the oldies among readers of El Reg.

    5. Fruit and Nutcase Silver badge
      Paris Hilton

      Lissajous

      Off-topic with respect to this article:

      sin and cos -

      http://mathworld.wolfram.com/LissajousCurve.html

      Paris is welcome to come over and investigate osculation by fiddling with my oscilloscope

  10. sitta_europea Bronze badge

    It's not 60. It's 360/2PI. We call it a radian.

    1. Korev Silver badge
      Joke

      What are you arcing on about?

    2. hmv Bronze badge

      True, but 60 is a reasonable builder's approximation.

      1. allthecoolshortnamesweretaken

        Re: a reasonable builder's approximation

        SQR(50) = 7

        because 7 x 7 = 49

        and 49 is basically 50

        1. JulieM Silver badge

          Re: a reasonable builder's approximation

          Also, g is approximately equal to π ** 2. So a one-metre pendulum has a period of about 2 seconds, i.e. takes about 1" to swing from end to end.

        2. Adam 1 Silver badge

          Re: a reasonable builder's approximation

          7 * 7 = 50, but only for sufficiently large values of 7

      2. breakfast
        Coat

        Would it work for shipbuilding too?

        If so, it could constitute... NOAH'S ARC.

  11. I ain't Spartacus Gold badge
    Devil

    So this is like a fossilised Babylonian iPad right? So why hasn't Gilgamesh sued Apple for stealing his look and feel, it's got the rounded corners and everything!

    1. LDS Silver badge
      Joke

      Because Jobs was a descendant of Gilgamesh (remember when his father came from...) - which was in fact a quite nasty king oppressing his people - and also was looking for "ethernal life".

      The sexy, round corner design comes from Shamhat - which after all tamed the wild Enkidu and lead him to his death...

      Scribe Ziqquratberg made people record the story on its Faceclay application...

  12. Mage Silver badge
    Pint

    Gilgamesh sued Apple

    Well, Gilgamesh isn't Babylonian and he's been dead over 70 years.

    Have a beer.

    "If Gilgamesh existed, he probably was a king who reigned sometime between 2800 and 2500 BC. The Sumerian King List claims that Gilgamesh ruled the city of Uruk for 126 years."

    He probably did exist, but unlikely to have been as described in the Epic Sumerian/Akkadian poem developed over 100s of years (In that sense a bit like King Arthur, though evidence for him is more tenuous and most of what you read/see is stuff made up 100s of years after original Welsh legends.)

    1. Doctor Syntax Silver badge

      Re: Gilgamesh sued Apple

      (In that sense a bit like King Arthur, though evidence for him is more tenuous and most of what you read/see is stuff made up 100s of years after original Welsh legends.)

      And not actually a king either, assuming he existed.

      1. Pompous Git Silver badge

        Re: Gilgamesh sued Apple

        "And not actually a king either, assuming he existed."
        From the OED:

        " In OE. the title appears first as the name of the chiefs of the various Anglian and Saxon ‘kins’, tribes, or clans, who invaded Britain, and of the petty states founded by them, as well as of the native British chiefs or princes with whom they fought, and of the Danish chiefs who at a later time invaded and occupied parts of the country. Among the Angles and Saxons the kingship was not strictly hereditary, according to later notions; but the cyning was chosen or accepted in each case from a recognized kingly or royal cynn or family (usually tracing its genealogy up to Woden). With the gradual ascendancy and conquests of Wessex in the 9th and 10th c., the king of the West Saxons became the king of the Angelcynn, Angelþéode, or English (Angligenarum, gentis Angligenæ, Anglorum), and the tribal kings came to an end. But there still remained a King of Scotland, and several petty kings in Ireland. In European and other more or less civilized countries, king came to be the title of the ruler of an independent organized state called a kingdom; but in mediæval times, as subsequently in the German Empire, some kings were really or nominally subordinate to the Emperor (as ostensibly representing the Roman Cæsar or Imperator), and a King was held to rank below an Emperor. In reference to ancient times the name is applied, like L. rex, Gr. βασιλεύς, Heb. melek, to the more or less despotic rulers not only of great dominions like Assyria, Persia, Egypt, but of petty states or towns such as Jericho, Ai, Mycenæ, Ithaca, Syracuse, and Rome. It is still applied to the native rulers of petty African states, towns, or tribes, Polynesian islands, and the like. "

        1. Doctor Syntax Silver badge

          Re: Gilgamesh sued Apple

          "From the OED:" etc

          Yup, but AFAIK the historical thinking is that the title doesn't apply as no such ruler is known but there may have been a military commander of that name. Or maybe there wasn't. All the references are considerably later, and have a strong whiff of myth about them. The only battle attributed to him in these sources which can be matched in earlier sources is Mons Badonis and that earlier source, Gildas, doesn't attach any participant's names at all. In fact, although he says it was a siege it doesn't even say who besieged whom.

          1. Pompous Git Silver badge

            Re: Gilgamesh sued Apple

            "All the references are considerably later, and have a strong whiff of myth about them."
            More accurately, there are few references because not much writing has survived from the period. The heroic poem Y Gododdin is dated to anywhere between the 7th and 11th Century and that mentions Arthur only in passing. None of the sources are at all reliable of course. I sometimes wonder what future historians will make of today when they pore over fragments of The Grauniad, and The Daily Fail.

            1. TechnicalBen Silver badge

              Re: Gilgamesh sued Apple

              In all honesty, they will take one look at them, and not bother reading it!

    2. Anonymous Coward
      Anonymous Coward

      "Gilgamesh isn't Babylonian"

      Babylonia culture directly descended from Sumerian/Akkadian one - one of the sources of the epic is from Babylonia.

      Babylonians would rule upon Uruk in certain periods. So it's easy to make some confusion, and some old text may refer to Gilgamesh as Babylonian.

    3. Tom 7 Silver badge

      Re: Gilgamesh sued Apple

      I wonder if Glgamesh is a portmanteau of Akkadian words and means something - much like Archimedes means 'Top Thinker' and so the history too could be a portmanteau of several kings histories.

  13. Cuddles Silver badge

    Nothing to do with fingers

    People have come up with all kinds of clever ways of counting in various different bases using their fingers and/or other body parts, but there's no evidence that any base system was developed because of the ability to do so. Indeed, the fact that it's possible to count in so many different bases rather suggests no such preference even makes much sense. The best explanation for why different bases have been preferred at different times is exactly the same as why different languages, alphabets, and so on have also been used at different times - coincidence and habit. A language or numerical system or whatever evolves naturally, and then people keep using it because it's what they're used to, until it evolves into something else or gets pushed out by a new system for a variety of different reasons.

    As for the article itself, the tablet is interesting but the comparison to trigonometry makes no sense at all. The whole point of trigonometric functions is that they are the ratios of the sides of triangles (OK, it gets a bit more complicated, but that's how they were first developed). In fact, the Babylonian method is clearly more primitive since they only address special cases, much like the Egyptians and others who also knew a bit about triangles long before Pythagoras*. The big deal with trigonometry is that you're no longer stuck with a few special cases and laborious tables, but can instead use general functions to handle any case you like.

    * Pythagoras himself quite possibly knowing nothing about triangles at all, with no evidence linking him with mathematics at all until over five centuries after he died.

  14. zebm

    Base 60 making a comeback?

    At least that's what seems to be suggested. Better to think that base 10 rational numbers become more popular.

  15. Version 1.0 Silver badge
    Facepalm

    3700 year ago ... and before then?

    Everyone acts like this was brand new 3700 year ago - but stuff like this doesn't appear overnight like some mushroom. The chances are that it was well establish much earlier and we simply haven't found the evidence yet. Look at Göbekli Tepe ... 12,000 years ago - you think that was built by people counting on their fingers?

    1. Destroy All Monsters Silver badge

      Re: 3700 year ago ... and before then?

      Well, GR and QM were pretty much invented overnight.

      So no.

      1. TechnicalBen Silver badge

        Re: 3700 year ago ... and before then?

        This is a misconception. Based on long timescales it was "over night", but a LOT of the math and theory for GR and QM were already there. Einstine was around for the finalling of the theory, the observation of evidence, but his work was built on the backs of giants.

        But those are two examples of very fine detail observations and mechanics. Building and construction, astronomy and planetary path prediction, medicine and health are all things every day people can make observations with. Math including. Which means it can get independent discovery, or just plain "common knowledge" use (such as Aboriginals in Australia not knowing what bacteria is, but knowing not to kill and eat meat near the camp for risk of disease).

      2. Pompous Git Silver badge

        Re: 3700 year ago ... and before then?

        "GR and QM were pretty much invented overnight"
        From a geological perspective... I think we can date the beginning of humans thinking quantitatively from a time before 3,700 years BP.

  16. Joe Harrison

    That's nice

    I posted earlier in the thread about counting on fingers and thumbs, now someone has given me 1 thumb up for it

  17. Anonymous Coward
    Anonymous Coward

    Counting on Fingers

    As for people counting using their fingers, they can also add their nose for an additional digit. Unfortunately, people doing this, and counting rapidly, appear to be thumbing their nose, and are often punched.

  18. Stevie Silver badge

    Bah!

    So now, instead of remembering the name of around thirty-two numbers to do sums I have to learn the names of 60 before we shift (presumably) to the left and incur a new number name?

    Count me out. In base ten.

    1. Anonymous Coward
      Anonymous Coward

      Re: Bah!

      If you look at Babylonian numbers, they used a base 10 representation for numbers, within a positional system for base 60. You just needed to know the symbols for 1-9 and 10. They also had a rudimentary understanding of zero. I do not know how the numbers were 'read', though.

      1. Stevie Silver badge

        Re: Bah!

        Not talking about symbols, names.

        In base ten, in English, nine plus one gives us not "One-zero", but "ten". Double that and we don't get "Two-zero", we get "twenty".

        There are 21 individual names for numbers between 0 and 20. There are another 12 names if you want to count up beyond 1 000 000 000.

        Now, how many names do you need for the sexadecimal digits that make life so much "easier".

        1. Pompous Git Silver badge

          Re: Bah!

          "There are 21 individual names for numbers between 0 and 20."
          You forgot two of a dozen, score, or twelvty.

        2. LDS Silver badge

          Re: Bah!

          AFAIK, number names are still subject to debate. That's because you need to find a phonetic representation of the number besides its symbolic representation (i.e. "two" <-> "2"). In some instance numbers were also used to refer to gods...

          Also AFAIK, they had no distinct names for numbers 1-60, but like those counting in base 10, had names for numbers up to 20, and then names for tens and so on. Remember the mathematics and positional system was base 60, the number symbols were not.

  19. Anonymous Coward
    Anonymous Coward

    Trigonometry is not about triangles...

    ...it's about circles.

    The hypotenuse is the radius of a unit circle, and the sin and cosine are the height and width of that line (the vertical and horizontal components of that radial line). As the tip of the radius follows the circumference of the circle, those horizontal and vertical components generate the sine and cosine waves at 90 degree phase to each other. Hence trigonometry is really about how circular motion relates to oscillating motion (think pistons and crankshafts); the stuff about triangles is all fine, but misses the bigger picture.

    This is also why the Babylonian table of ratios in the tablet are not going to replace Greek trigonometry any time soon, contrary to the assertion of the historian in the video - ratios of triangle sides miss out the actual point of what trigonometry is about.

    1. Charles 9 Silver badge

      Re: Trigonometry is not about triangles...

      If trigonometry really were about circles, then why the "tri"?

      I will agree with you, though, that there's a surprisingly strong connection between circles and triangles. Thus if you were to plot a graph of the Pythagoream Theorem (x^2 + y^2 = z^2), you get a perfect circle of radius z.

      PS. Interestingly, if you generalize the equation to x^n + y^n = z^n and plot their graphs for greater values of n, you find the graph morphs from a circle to a square as n approaches infinity.

      1. Deckard_C

        Re: Trigonometry is not about triangles...

        sin, cos, tan and the reciprocals and derivative are used heavly in design and describing AC circuits which includes audio and radio circuits and basically any circuit which has a signal. After all sin describes a pure AC signal or audio wave hence sine wave.

        To be honest I found the maths far to difficult just have a look at wikipedia https://en.wikipedia.org/wiki/Trigonometric_functions and scroll down

        I found using matrices to solve simultaneous equations with more than two unknowns to be much easier to get my head round not that I can remember how now.

        For example GPS uses trigonometric functions for the radio waves aspect, trigonometry to calculate your position using simultaneous equations to solve the 4 unknowns of your position in x, y, z and time.

        1. TheElder

          Re: Trigonometry is not about triangles...

          Try it with Spherical trigonometry. Three 90 degree angles may make a triangle.

    2. You aint sin me, roit
      Headmaster

      Re: Trigonometry is not about triangles...

      The Coward is right - and the diagrams show it.

      The authors seem a little confused...

      "it does not use angles and it does not use approximation": " A squared index and simplified values of b and d to help the scribe make their own approximation to b/d or d/b" - so did they approximate or not?

      As has been pointed out, the examples are just special cases of right angle triangle ratios, only relevant when processing those triangles, or the "half a rectangle", whereas the sine/cosine/tan ratio mechanism is not restricted to right-angled triangles, just to angles. Even better if you further generalize to the circle view and bring in radians...

      Then they say "The Babylonian approach is also much simpler because it only uses exact ratios. There are no irrational numbers and no angles, and this means that there is also no sin, cos or tan or approximation."

      Well a 30 degree angle, which has a lovely sine value of 0.5, would have an inconvenient "ratio" expression that is irrational in any base. So much for exact calculation.

      The only reason those examples are exact and don't involve irrationals is because they cannot handle the cases where irrationals are needed and so restrict themselves to a few special cases.

      There's a reason why we don't do things their way, and haven't for a long time. And it isn't because the ancients had a deeper understanding of trigonometry than we do... two thousand years ago the Greeks knew you can't square the circle.

  20. earl grey Silver badge
    Pint

    beer for boffins

    Ancient and modern. Here's to maths!

    1. Pompous Git Silver badge
      1. Fink-Nottle
        Pint

        Re: beer for boffins

        Yay, and another one for the brief appearance of Cuisenaire rods in the video

  21. Ask Noah

    What this implies

    The ancient Babylonians used this trigonometry for their advanced building projects. In the Biblical timeline, 3700 years ago is around the youth of Isaac, when Abraham was over 100 years old. So the Babylonians certainly had knowledge of this when Abraham was a young man, which was at the time when the Babylonians were building the Tower of Babel, according to the traditional Biblical timeline in the classical book "Seder Olam." How else could such a difficult engineering project have been undertaken, without such an advanced trigonometric knowledge of how to design it?

    Also, base-60 trigonometry would be a natural system for accurate star-charting in ancient astronomy and astrology (60 minutes per hour in a twelve hour day, during which the earth makes a full 360 degree axial rotation). The Babylonians were advanced in this knowledge at that time, as we know that Abraham was an expert astrologer. (See Rashi on Genesis 13:5,)

    1. Anonymous Coward
      Anonymous Coward

      Re: What this implies

      Genesis 13:5

      And Lot also, who went with Abram, had flocks, and herds, and tents.?

      Rashi interpretation,

      What brought this about? The fact that he traveled with Avram.

      I was curious so thought I would look it all up however it makes no sense to me in your context, I would like to ask why it's took 3700 years for this to now be known, surely records must have been kept by the scholars over the years.

      1. bitten

        Re: What this implies

        The camels bringing him back 2700 years ago at best.

    2. Ask Noah

      Re: What this implies

      Sorry for the typo. The reference is Rashi on Genesis 15:5.

    3. Anonymous Coward
      Anonymous Coward

      Re: What this implies

      > when Abraham was a young man

      ... around the time that Boney M song was cool.

  22. Herbert Meyer
    Boffin

    everybody knows

    The proper way to measure angles is in radians, not degrees. Since any measure is an approximation, "exact" divisors are a fiction. So an irrational measure is appropriate. Besides, any use in mechanics is going to involve a moment, so the pi factor will appear quickly.

    1. JulieM Silver badge

      Re: everybody knows

      This is true. You can only measure real-life things to limited precision, so you only need to calculate just a bit more precisely than you can measure.

  23. JulieM Silver badge

    Copyright and Mathematical Tables

    Talking about copyright and mathematical tables, set my mind off on a train of thought.

    What if the publisher of a set of mathematical tables incorporated deliberate errors to allow easy spotting of plagiarism? Cartographers are said to have done similar things with maps ..... And then what if one day, the designer of a bridge stumbled upon one of these deliberate errors whilst doing an innocent engineeering calculation, and ended up underspecifying something and then as a consequence, one day, the bridge collapses and a train falls into a river?

    What if they checked their calculations using a different book of tables, which happened to be a pirate copy of the original and therefore included the same mistake?

    It's probably a good job there were no modern-style lawyers in those days. (Another thought: Perhaps it's precisely because we have so many labour-saving devices at our disposal today, it is entirely possible for some people literally to have nothing better to do with their time .....)

    1. Pompous Git Silver badge

      Re: Copyright and Mathematical Tables

      "What if the publisher of a set of mathematical tables incorporated deliberate errors to allow easy spotting of plagiarism?"
      The preface to my Chambers' mathematical tables states there are known errors that would trap any plagiarism. The errors are presumably in the last decimal place and consequently unlikely to prove problematic.

      The book's in storage but would be forty years old at least.

      1. TheElder

        Re: Copyright and Mathematical Tables

        A simple way to identify tables that are printed as graphics is to sprinkle single yellow pixel codes over the image area. This was invented by Xerox for the early colour printers. The government insisted that the serial number of each machine be printed on the copies to make it possible to track money counterfeiters.

        Single yellow pixels are almost impossible to see visually. Privacy violation started a long time ago.

      2. Tom 7 Silver badge

        Re: Copyright and Mathematical Tables

        The deliberate errors are fairly easy to spot - a simple difference between the last two figures of two values will show an out of trend value quite quickly - something any anally retentive 12 year old would know half way through the first lesson they finished 30 minutes before the rest.

    2. J P

      Re: Copyright and Mathematical Tables

      "Cartographers are said to have done similar things with maps"

      Cartographers have definitely done similar things with maps - I used a non-existent barn in Bedfordshire (OS landranger sheet 165) as a tie breaker question in a club mini-rally (What's special about the barn at grid reference XXXYYY?), and also been caught out instructing my driver to turn left at the barn while rallying somewhere in Scotland (that was in the late 1990s; the driver besmirched my map-reading skills, but much to my delight there was an item in the national press a few days after the rally confirming legal action between two road atlas manufacturers on similar grounds, vindicating my original instruction; IIRC, it was the AA who'd been copying OS maps, including the errors).

  24. bitten

    Pesky base 60 trigonometry is not Babylonian?

    Glad to know that the minutes and seconds stuff was invented by those decadent Greeks. Did Babylonian kids memorise the base sixty multiplication table?

    1. Herbert Meyer

      Re: Pesky base 60 trigonometry is not Babylonian?

      No, they remembered the divisor table. 60 was chosen because it has lotsa useful divisors - 2,3,4,5,6,10,12,15,20,30.

      1. TheElder

        useful divisors...

        I prefer binary -_-

        Ancient Hexagrams

  25. TheElder

    Base 60?

    I like the idea. It means I am only 1.1166666666666666666666666666667 years old.

  26. Anonymous Coward
    Meh

    For every five new things we learn today...

    We forget three wise old things our grandparents knew by heart.

  27. iLurker

    Really Ancient News, El Reg...

    The details of the tablet - and its mathematical significance - were published by Otto Neugebauer in 1945, Neugebauer being a professor of astronomy, a mathematician AND sufficiently well educated in classics as to translate it directly.

    And republished https://arxiv.org/pdf/1004.0025.pdf

    Tsk Tsk Reg, almost as bad as Pythagoras himself.

    1. Tom 7 Silver badge

      Re: Really Ancient News, El Reg...

      Its really quite a stunning piece of work - glad its survived and Neugebauer et all became interested n it.

  28. nickx89

    why oh why?

    My whole teenage mathematics ruined. Why did I study hard just to know today that use base 60.

  29. captain_solo
    Alien

    Of course they had a better method, they had direct contact with the transdimensional beings who originally taught us all these complicated maths.

  30. teebie

    Is it weird that there are copyright issues relating to a tablet that just says 'EFF' over and over again?

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