Well, yes, suppose so

Stonehenge builders used Pythagoras’ theorem 2,000 years before Greek philosopher was born, say experts

We were all using gravity long before Newton too. It’s reality that matters, not the codification of it.

38 comments on “Well, yes, suppose so

  1. One contributor, megalithic expert Robin Heath has even proposed that there exists a great Pythagorean triangle in the British landscape linking Stonehenge, the site from which the Preseli bluestones were cut in Wales, and Lundy Island, an important prehistoric site.

    So it is what is technically known as Bloody Deluded Bullsh!t. Given that Britain probably didn’t have the technology to accurately lay out a triangle like that until well after World War Two.

    The new book, published today to coincide with today’s summer solstice, shows how within one of Stonehenge’s earliest incarnations, dating from 2750BC, there lies a rectangle of four Sarsen stones which when split in half diagonally forms a perfect Pythagorean 5:12:13 triangle.

    Well yes. I mean as long as you are working in two dimensions, it is a little hard to lay out a rectangle which when split in half diagonally did not form a perfect Pythagorean triangle. Indeed if anyone can think of one they might be in the running for some sort of award. As, you know, the hint is in the title. Methinks some girl work experience girl failed her primary school mathematics class.

    The eight lines which radiate from the rectangle and triangles also perfectly align to important dates in the Neolithic calendar, such as the summer and winter solstices and spring and autumn equinoxes.

    Well if you align something north-south and east-west that is kind of a given isn’t it?

    So ultimately what we have is the ground breaking news that they laid out their stones in a roughly evenly spaced north-south, east-west alignment. Hold the presses! I suspect my patio tiles do pretty much the same.

  2. Except, there is a difference between theorems and reality. You can’t avoid using gravity. Once you understand it, you can use it more cleverly, of course.

    But, no, they didn’t. They used one example of an integer fit to a generic theorem. This does not demonstrate use of the theorem themselves.

    Now, if they had other examples using different integer fits, that would be better evidence. Integer fits because those are easier to measure.

  3. I mean as long as you are working in two dimensions, it is a little hard to lay out a rectangle which when split in half diagonally did not form a perfect Pythagorean triangle

    Indeed. My assumption (and, remember, here we have crackpots being summarised by journalists) was that they assumed “perfect” to indicate an integer example of.

    In which case posultating (one of) the builders knowing 5:12:13 gives a right angle and having the “Great Rod of Ugg” (or a handy rope, we do know they had ropes?) to measure it out is interesting if hardly revolutionary.

  4. “there lies a rectangle of four Sarsen stones which when split in half diagonally forms a perfect Pythagorean 5:12:13 triangle.”
    Well, the builders provided the 5:12 rectangle, from which someone else might notice the right angle with integer ratio sides. However, if the 5:12 ratio is very precise, it might by worth noting the exact length of “1”, to see if other multiples of that crop up.
    As for Solstice and equinox lines, well that’s not quite as simple as knowing north/south, but neither is it very complicated. It’s just sustained observations with counting pebbles and sticks in the ground.

  5. From Wikipedia: “Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system. It was discovered by Edgar James Banks shortly after 1900, and sold to George Arthur Plimpton in 1922, for $10.[2]”

    Pythagoras lived circa 570–495 BCE.

    So maybe his discovery is now known to be pre-dated by more than it was known last week to be pre-dated.

    Best regards

  6. Individual examples of Pythagorean triangles are known from (almost) the earliest written records. Pythagoras is credited with discovering the general formula.

    That the builders of Stonehenge may have known that 5:12:13 (or 3:4:5) gives you a right-angled triangle (or it may be a coincidence) tells us little we didn’t know.

  7. One contributor, megalithic expert Robin Heath has even proposed that there exists a great Pythagorean triangle in the British landscape linking Stonehenge, the site from which the Preseli bluestones were cut in Wales, and Lundy Island, an important prehistoric site.

    I’m guessing he’s been drawing lots of lines on maps between ancient sites and come up with one that looks like it fits his theories.

  8. I see much of Professor Murphy’s theories in Pythagoras’ work.

    Richard Murphy invented Pythagoras’ Theorem.

  9. If you’ve ever practised any of the crafts & trades that use Greek science, it becomes quickly obvious the Greeks weren’t so much skilled scientists & simply the first people who wrote the principles down & whose writings have come down to the present. Because the science & the understanding of it gets generated in the process of accomplishing the crafts & trades. So all of it was around long before the Greeks.

  10. “The new book, published today to coincide with today’s summer solstice, shows how within one of Stonehenge’s earliest incarnations, dating from 2750BC, there lies a rectangle of four Sarsen stones which when split in half diagonally forms a perfect Pythagorean 5:12:13 triangle.”

    As othere’s have noted, this is an integer fit. Like 3,4,5. Which could be genius or just luck. The problem with the genius argument is that there’s no benefit to doing this. It doesn’t make a building better.

  11. Anon – “As othere’s have noted, this is an integer fit. Like 3,4,5. Which could be genius or just luck.”

    Or it could be based on the length of Ooogh’s foot. 5 what? Feet? Metres? Cables?

    It is a ratio. You have to get that ratio or one a lot like it if you have a right handed triangle.

  12. BiND,

    There’s loads of bullshit in archaeology. Because it’s hard to prove, noone really cares that much of you’re wrong, so routes to success include entertainment and ticking the right political boxes.

  13. Anon – “There’s loads of bullshit in archaeology. Because it’s hard to prove, noone really cares that much of you’re wrong, so routes to success include entertainment and ticking the right political boxes.”

    Let me recommend Larence Keeley’s War Before Civilization as well as the relevant bits of John Keegan’s history of warfare for when archaeologists refuse to accept the evidence before them. Human bones show clear signs of butchery? Must be ritual! Text mention brown people eating other brown people. Must be racism!

    If I recall correctly it took someone inventing a modern test that showed someone killed the inhabitants of a pueblo, ate their bodies, slept in their beds over night, before taking a sh!t in their fireplace and moving on – the scientists proved the sh!t contained human remains.

  14. It’s reality that matters, not the codification of it.

    Au contraire. Reality is civilization creates models for understanding and controlling it.
    You may feel you are essentially the same as a weevil or stone , I do not .

  15. What utter bullsh*t! Everyone knows aliens helped them lay it out.

    The writers tortured the evidence until it confessed.

  16. What Chris Miller said.
    Plus we’re pretty sure that the builders of Stonehenge had a practical, even if not theoretical, means of getting an (almost) perfect right angle because they cut stones that would stand upright without leaning (OK, the vertical post-holes helped but getting those vertical…).
    What would be surprising would be the relevation that they engaged in abstract mathematical theory and used “12^2+5^2=13^2” in order to do this or that they had the concept of congruent triangles and drew squares in the grass to illustrate it.
    No way can you have the Pythagoras’ theorem before you learn a practical way of drawing right-angles because the proof involves drawing squares.
    The “Megalithic yard” is almost certainly some data-miner’s dream since estimates derived from different sites give different lengths for it.

  17. “The public is incredibly gullible about crazy theories around past events. Thank goodness.”

    Erich von Däniken

  18. SMFS, Sorry mate, not having a go at you, but surveyors could have laid that triangle out at pretty any point from the mid 19th Century without even breaking into a sweat. Britain was triangulated by the Board of Ordnance between 1791 and 1853 (see https://en.wikipedia.org/wiki/Principal_Triangulation_of_Great_Britain ) and after that you could refer to the OS Trig points. The problem is that you would have to take into account the curvature of the earth over that range of distance, and Pythagorus’ Theorem works in a plane. Incidentally, the sum of the angles in a triangle on a curved surface (like the earth) don’t sum to 180 degrees unless the triangle is very small. I very much doubt that such nuances occurred to dear old Pythag, or if they did, he couldn’t solve the problem.

  19. @john77
    “What would be surprising would be the relevation that they engaged in abstract mathematical theory and used “12^2+5^2=13^2” in order to do this or that they had the concept of congruent triangles and drew squares in the grass to illustrate it.
    No way can you have the Pythagoras’ theorem before you learn a practical way of drawing right-angles because the proof involves drawing squares.”

    You are, of course approaching the subject from entirely wrong & academic direction. Remembering that, until fairly recently, those with an academic bent were best whipped soundly & harnessed to a plough. Early peoples were much more interested in practical capabilities than intellectual masturbation.
    People had been tiling, using regular square tiles, for millennia before kebab cooks in dresses made their appearance. Which requires the setting out of geometric shapes & from which, if you ever do any, you’ll rapidly get your head around the concept of squares of numbers.

  20. Incidentally:
    “Early peoples were much more interested in practical capabilities than intellectual masturbation.”

    A point of view has much to recommend it for the modern world.

  21. “I very much doubt that such nuances occurred to dear old Pythag”: they possibly did. Yer Greeks knew the world was round.

    Some centuries after Pyth’s time they had an excellent approximation to its circumference, courtesy of Eratosthenes.

  22. Was there ever a time ordinary people thought the world was anything but round? One only needs to sail a boat until the land disappears below the horizon…
    Apart from intellectuals of the period, of course.

  23. @ bis
    The Grauniad claims that the builders of Stonehenge used Pythagoras’ theorem, which an academic construct. If you take the trouble to read what I said, I claim that they didn’t need to know the academic theory used by Pythagoras as they obviously had a practical means of generating a right angle – and that – contrary to the view of the Grauniad, construction of squares, using right-angles, *must* have preceded Pythagoras’ theorem since he uses squares in his proof of the theorem.
    I am not the one going at it from an entirely wrong and academic direction – that’s the Grauniad.
    Incidentally, congruent triangles are an academic topic, not something that can be stumbled on while making clay tiles. There is no reason to expect the builders of Stonehenge to have discussed congruent triangles over their roast auroch

  24. @john77
    If you’re laying tiles, congruent triangles are about the second thing you discover when you get to a corner.
    Some mosaics are all congruent triangles. Common design feature in antiquity.

  25. The Egyptians knew 3:4:5. Indians and Chinese and Japanese were well aware of Pythagorean triples (without calling them that, natch). But the Greeks really did go further. They proved that a² + b² = c² for a right triangle. Euclid gave us the generating method m > n > 1 ∈ ℤ, gcd(m,n) = 1; a = m² – n², b = 2mn, c = m² + n² yields (a, b, c) as a triple e.g. m = 3089, n = 2351 ⇒ 4014720² + 14524478² = 15069122². You can systematically generate all Pythagorean triples with this. That’s a big leap over tying knots in a piece of string.

  26. But you can create a perfect square with strings. I did it when I was laying out the foundation for a garage I built.

    You verify it is square with a string. It must be the same distance from opposite corners. Simples.

    Say you have a square:

    AB
    CD

    A-D must equal B-C. Else it’s not square.

    Also, ABC gives you a 45° right triangle. You do not have to know the length of the sides or hypotenuse.

  27. Also, Mr Gamecock, your piece of string will give you a measured pi if you wrap it around a circular object. And a measured pi is perfectly acceptable for most of the practical uses of pi.

  28. @ bis
    No, because you can flip a congruent triangle over and it is still congruent but you cannot lay it on top of the mirror image.

  29. @john77
    Many tiles do not have a “face” side, so for the practical purposes of the tiler it makes no difference which side is up.
    But for strict congruence, many tiling patterns – which can involve triangular cuts – have mirror symmetry in both the north/south & also the east/west directions.
    All done without the input of a single academic.. In fact, it’s the mark of a competent tiler that he can visualise congruence without even thinking about it. Certainly without any irritating drawings on bits of paper.

  30. BiS : “Was there ever a time ordinary people thought the world was anything but round? One only needs to sail a boat until the land disappears below the horizon…
    Apart from intellectuals of the period, of course.”

    The intellectuals of the period *knew* the earth was round. The whole flat-earth idea in its earliest incarnation started late 17th/early 18th C, and *that* was based on the way medieval world maps are projected: flat, with Jerusalem as the center.
    Later on the Romanticists ran with the idea, and the Victorians managed to nail it into Gospel Truth because it fit with their ideas of superiority.

    It’s a common affliction of all things medieval, or earlier periods for that matter.

  31. The “flat earth” notion was publicised particularly by the American writer Washington Irving. He intended it as an attack on the Roman Catholic Church – a noble cause, to be sure, but not one that should be pursued by ignoble means. Put otherwise, he was a lying bastard.

  32. Correct, dearieme. Writing about Christopher Columbus, Irving made it up. And it stuck. Fake news from early America.

    Columbus, by the way, was an idiot. People had a general idea of how big the earth is. Columbus thought it smaller. His smaller earth made it easy to sail west to reach Asia. He got lucky. Very lucky. But for the quirks of geography, they would have all died.

    Must have caused quite a stir in Europe when he returned, claiming he had reached Asia, hence, his small earth theory was proven.

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