I don't see why that was necessary. You can get pretty far with eyeballing it and custom cutting to fit.
But groups of people, "they", who all build a lot, tend to have a few people who make these observations, and then the observations proliferate, because they are both way easier to use than the alternative hack-work, and yield much more aesthetically pleasing results. Especially in the stone age when you nornally don't have easy access to anything at all with a right angle (unlike in modern construction) and you build stuff out of clay bricks, where minor inaccuracies inevitably add up and make your life much harder down the road. Tying together three pieces of string with prescribed ratios, pull it tight was a very easy way to get a right angle compared to anything that came before.
It's necessary in the sense that stone age construction is so much easier if you know about it, and so much harder if you don't. Those who didn't come up with it didn't do such construction, because doing difficult things is harder than doing easy things.
I was chatting with the person that was in charge of marking the fields for the local youth soccer league. I volunteered to help one weekend, and one of the first things he asked was if I knew what a 3/4/5 triangle was since it was the only way to know you'll be squared. I never did figure out to what level he was dead panning his joke or if he was even meaning for it to be a joke. Either way, I laughed.
1. lay out a straight line 2x in length.
2. find midpoint (easy by drawing an arc with a string from each end point. Basic compass & ruler technique.
3. draw another arc from the midpoint, of radius 1x. Try different spots on the arc until it is equidistant from each of the endpoints of the line.
4. voila! an accurate right angle. Laying out the rest of the square is now trivial.
The _article_ happens to be about written sources from the Bronze Age Babylonians, which lets us glimpse at their accumulated knowledge. But there's no reason to believe this knowledge was particularly new at their time, and this was a clay tablet equivalent of an arXiv preprint.
i'm not saying at all that I know the answer, but eyeballing and cutting is good for a patio, but on a ziggurat or pyramid scale it seems you don't do so much eyeballing or cutting, and more planning how much material and how many slaves you're going to need for how long, and where to put the doors so the passageways will meet up, that sort of thing.
The scale doesn't seem right to me. I still can't fathom how that much could be done without using slave labor. If I remember correctly, they layed a block like every 6 minutes for 20+ years.
Do you know how many tombs of laborers were found, or where I could find more aobut that information? I'm very novice when it comes to Egypt
Everything becomes a custom cut, often in multiple dimensions and all earlier errors cascade all the way to the end.
Unrelated, but that sounds exactly like many software projects I've been unfortunately part of.
A modern analogy: today, people are very happy to use LLMs and Transformers without anyone "proving" that they work. Right now, philosophically, they are at the level of empirical observations (perhaps not even that). Does that mean that today's AI researchers should get no credit when at a later date? I am not sure. Empirical discovery of a rule is also no trivial thing.
That's not in any sense a value judgment of the empirical work done at DeepMind. Nor does it dtop anybody from writing a better article which explains tha the Babylonians (and many others) used the empirical observations underlying the theorem, while explaining that this did not constitute mathematical proof.
If there's no proof, it's not a theorem, by definition. That doesn't make empirical observations worthless -- far from it!
> Pythagoras is immortally linked to the discovery and proof of a theorem that bears his name – even though there is no evidence of his discovering and/or proving the theorem.
Simply quoting. It seems like
> Greek definitely did
is not a universally held belief.
The Greeks definitely were able to prove the Pythagorean theorem, and the Greeks definitely though Pythagoras specifically knew how to prove it: e.g. Proclus II states this in writing explicitly, as does Euclid 300 years later, and numerous ancient sources inbetween. It would be hard for this not to be the consensus. We can't know if Pythagoras really did, simply because we have no surviving writings directly from Pythagoras - as I stated explicitly in my previous post.
That fact that the Greek could prove the theorem at some point is not unlikely even without evidence. It’s as likely as others doing it well before them.
according to the article, this Babylonian tablet does come closer to theorem than you are suggesting. They weren't using Pythagorean triples, rather they figured out that the diagonal of a unit square is the square root of 2, and knew how to calculate that:
1 + 24/60 + 51/602 + 10/603 = 1.414213
Wait, really? I thought the proofs back then were geometric, so only proved specific instances. And so it wasn't until algebra, maybe trig, was discovered it was properly proven?
But I don't see how it proves anything if $a$ were say doubled, and $b$ kept the same.
* For any real positive values $a$ and $b$
* let $T$ be the (unique) right-angled triangle with non-hypotenuse sides of length $a$ and $b$
* then $a$^2 = $b$^2 = $c$^2
* where $c$ is the (unique) length of the hypotenuse of $T$.
You could give me the instance
$a$ = 2, $b$ = 3
$a$ = 4, $b$ = 3
But I agree with the second paragraph: there's a huge difference between a procedure that is handed down as part of "this is how we estimate a building protect" and a theorem that is declared universal truth and base for all kinds of other theorems. Even if they are exactly the same thing.
I only see theorems as useful for complex societies. The son of a gem dealer would have the time to work out universal truths.
The reputation of everyone doing it without numbers would reveal the pattern of the universal truth.
Finally this looks like it was all done using cuneiform.
Which brings up questions of notation and the language to describe a square root.
Progress is a luxurious goal.
Observing that something always holds and even having a formula for it is not the same thing as having a proof, and the proof is what makes it mathematics and geometry and not "just" engineering. The babylonians had a rule of thumb -- the greeks discovered the theorem -- and more than that, they seem to have invented the mathematical/geometrical proof as a concept, along with formal logic.
Without that mental framework, it's hard to say that the babylonians proved anything or had any theorems at all, only collections of rules of thumb. It's quite likely that lots of babylonians sort of independently and intuitively understood _why_ it must be true, but they don't seem to have ever written it down.
Not that there's anything wrong with having rules of thumb -- it's a huge achievement to even notice and collect and teach those things, all the stuff around you is built relying on them.
I encourage everyone to watch this series of videos.
As I understand Greeks invented proofs because "universal thruths" they exported from Babylon and Egypt sometimes explicitly contradicted each other.
I believe such contradictions may be a great nuisance when you try to get shit done.
Egypt and Babylon were sufficiently "complex" societies for proofs, but their tradition treated mathematics as a bunch of useful facts about numbers and shapes. New generation just memorized them. We should think it worked for them in most cases, and when it didn't work it was not so often for them to start thinking a lot of reforming mathematics. Plus they were indoctrinated by the math they learned (authority of a teacher is above of anything else, i suppose) and to reform math was not a natural idea for them.
The nice thing about Pythagorean triples for drawing out foundations is that you don't need access to any ground outside the foundation of your building. Being integers, you also don't need any measuring device apart from some rope. You just pace out a bit under 1/3 of the shortest side (or a bit under 1/5 the longest side, whichever is shorter) (call this an "'bout-right") length of rope. You then use your 'bout-right to make a 3'bout-right, a 4'bout-right, and a 5'bout-right piece of rope. Pull the three ropes tight in your perimeter, and you've got your right-angle for your foundation.
And if you can determine that the diagonals are the same length, you have what you need to get close enough to a parallelogram in the first place.
> If you create a parallelogram
Parallelism requires affine geometry, which you can't get just with straight lines (and their meeting points). Here are couple of explanations:
- We can also get projective geometry by using great-circles on the surface of a sphere (e.g. "equators" at different angles around the Earth), instead of straight lines on a flat plane: both situations give rise to exactly the same theory. Parallelism doesn't exist on the surface of a sphere, since all great-circles will meet at two antipodal points, so projective geometry (which describes great-circles as well as straight lines) cannot be used to construct/ensure/check that two sides of a quadrilateral are parallel.
- Alternatively, consider that projective geometry is invariant to changes in perspective, whilst parallelism is not. For example, we can get two straight lines by tracing over a photo of train tracks. If the photo was taken top-down, then the lines we traced will be parallel; but if the photo was looking along the track then our traced lines will converge (in the photo, they "meet" at the horizon). Projective geometry (and hence straight lines) can't distinguish between these two scenarios, due to this invariance.
> And if you can determine that the diagonals are the same length
If we extend our straight-line setup with some way to determine parallelism, we still wouldn't be able to compare the lengths of the diagonals, since they go in different directions. Projective geometry + parallelism is affine geometry, which can only compare lengths in the same direction. Essentially, parallelism allows us to translate: we can use this to compare two line segments by translating one so they share a common starting point, then seeing whether the other end has landed closer or further than the first line's. The latter comparison only makes sense if all the points end up colinear (i.e. the original segments were parallel, unlike a pair of diagonals).
To compare the diagonals we also need some form of metric, e.g. like the distance between a pair of compasses.
But that's not a theorem, just an observation. It becomes a theorem when you prove (i.e. explain why) this relationship always holds, based on more evident things. The Babylonian tablet mentioned in the article doesn't seem to do anything like that, whereas the Greeks definitely did (we don't know whether Pythagoras himself did it, as no writing of his survives, but later Greeks knew how to do it, and attributed it to Pythagoras).