Stone tablet shows Babylonians able to approximate √2 with 99.9999% accuracy
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https://scholarship.rice.edu/handle/1911/61898
Part of that thesis discusses the Babylonian use of the rule of the double false position and it's application to solving quadratics, which can be used to get the square root. Anyway, I'm not an expert in this history, but I believe she includes references to the related texts for those interested.
They might have used Newtons algorithm (not using that name probably), but then they would have had to divide, which is even more horrible by hand.
Consider that their notation was probably far worse than what we are used to today, and it all becomes quite a hassle.
Probably make a whole set for different aspect ratios. Carpenters might like them.
“I used to buy 1.5 ft cuttings of wood at the market to make the diagonals on my cabinets, but thanks to geysersam’s number blocks, I buy 1.42 ft cuttings of wood. It’s like every 17’th cabinet is free!”
Heck, just 1+2/5 is likely sufficient.
I was specifically talking about the carpenter use case.
I don't know exactly how large of a times table I have in my head, but it's larger than 10x10, more like 15x15. A 60x60 table is 16 times larger than that, but I also learned my times table in elementary school and being a scribe or whatever the job title was for a guy who multiplied was very likely a full time job for a grown Babylonian. Learning your times tables up to 60x60 seems totally plausible.
x → x – f(x)/f'(x) with f = x² – N does in fact give you
x → x – [x² – N]/(2 x)
Which quickly simplifies to,
x → (x + N/x)/2.
With that said I am not sure that they knew this coverages quadratically or any of the machinery needed to make Newton's method work... It's a great way to approximate square roots!
We might in fact have been looking at Nasir's method this whole time.
IIRC, before e.g. continued fraction representations/approximations became available, high precision estimates of pi were based upon geometric subdivisions of a circle and computation of hundreds of terms.
I didn’t realize Babylonians had a system for “decimals” in sexagesimal.
Edit: another comment pointed out the post is an extrapolation of this paper: https://www.sciencedirect.com/science/article/pii/S031508609...
(For why the day has 24 hours though, see https://www.scientificamerican.com/article/experts-time-divi....)
https://en.wikipedia.org/wiki/Methods_of_computing_square_ro...
It would seem we don’t know exactly what method they used.
I would like to propose: really, really long and accurate ruler, just because the mental image is funny. The Greeks hadn’t come around yet to make everybody’s homework harder by insisting on compass and straight-edge (Note: this comment is not historically accurate).
This would be useful for figuring out land rights. And for that, you need very long ropes.
You'd need to be accurate to a half mm per km of rope, which is well beyond what you need for practical surveying. You also need to keep the rope essentially perfectly straight, which means doing the experiment on almost perfectly flat ground. You'd need to keep the rope at a controlled tension so it doesn't stretch. You'd be hosed by temperature and humidity changes while walking those kilometers. The rope would also need to be marked or measured to < 1mm accuracy when you're done walking. You'd need to make sure the right angle is square to < 1ppm as well, though maybe there's a clever procedure that corrects this.
It's not even easy to do this with laser surveying equipment.
Real-world ropes flex and bend, which alone causes experimental error orders of magnitude larger than one in a million. There does not exist a rope material that doesn't flex at least one millimeter per a km of rope, and certainly did not exist 6000 years ago. Hell, even if we assume a perfectly flat plain, the curvature of Earth would cause an error larger than 1e-6. And obviously if you try to hang the rope from both ends and pull it taut, it will flex and it's also impossible to remove all the slack.
You need a laser, or at least something like the Michelson–Morley experiment, exploiting the wave nature of light, to measure any real-world distance to the precision required. And obviously for that to work, you also need to know the speed of light to the same precision. And given that we're not in a vacuum, even light bends as it travels through air, especially air near the ground. You know heat haze? Yeah, that's going to mess up your experiment too.
And how on Earth would they ensure the ropes form a right angle to a precision of one in a million? Given a 1km long side, the sideways margin of error is again 1mm. The angular precision needed is sub-arcsecond, a resolution that you need a large-ish optical telescope to achieve.
And obviously then there's the question of measuring the length of the hypothenuse.
Now, I wish substack had support for math. Adding formulas in images isn't that bad (and practically unavoidable if you want to draw functions - unless you write js), but, still.
http://web.archive.org/web/20221229110741/https://thepalindr...
Is it a regional restriction?
edit: Thanks to the author for removing the paywall!
Journalists were moaning about 'having to' subscribe to his SubStack. It was actually the first I heard of the place and I thought it was the norm to charge.
It's interesting in that the paywall can be trivially circumvented... by asking a paid subscriber to forward the email of the full post. Costs nothing, but in practice, almost never happens.
Any information people consider important enough to share with the whole world will not be paywalled. Consequently, any paywalled information must not be important.
The important thing isn't that you promote the best worker; the key is that it isn't possible for the ambitious-but-incompetent to game their way up.
Edit: Wait, that was promoting.
(Also, if people are interested in randomness in antique decision making, me & my wife made a simulator of an ancient lot oracle unearthed in modern Turkey: https://sortesalearum.com)
The sad thing is, I feel that this method may actually produce better results than whatever popularity voting schemes we're currently using.
(It's somewhat enshrined in modern-day RNGs used in simulation and forecasting models.)
Michael Schulson's 2014 Aeon essay, "How to Choose" remains at the top of my best reads of the past decade:
... It makes sense that it should have taken Dove some 15 years to realise that randomness could be an asset. As moderns, we take it for granted that the best decisions stem from a process of empirical analysis and informed choice, with a clear goal in mind. That kind of decision-making, at least in theory, undergirds the ways that we choose political leaders, play the stock market, and select candidates for schools and jobs. It also shapes the way in which we critique the rituals and superstitions of others. But, as the Kantu’ illustrate, there are plenty of situations when random chance really is your best option. And those situations might be far more prevalent in our modern lives than we generally admit. ...
<https://aeon.co/essays/if-you-can-t-choose-wisely-choose-ran...>
Examples include swidden farmers in New Guinea, China's I Ching, the Athenian Greek elections (more a lottery than today's FPTP precise-count balloting), Renaissance Italy's Doges, and contemporary college admissions, with references for further reading.
https://imperiumromanum.pl/en/curiosities/alectriomancy-fort...
A few years ago I'd thought that the US Supreme Court might benefit by both expansion and selecting by lot the judicial panel for any given case. This would throw far more uncertainty into the outcome of any given Supreme Court case, as well as the capacity to throw the court in the benefit of one party or the other for years at a time. That's been proposed by others.
Another possibility might be to apply a similar approach to legislation, with an expanded Congress closer to the 30,000:1 constituent to representative ratio initially in the US Constitution rather than the 616,000:1 ratio presently in place. That would result in a House of 11,000 members, however.
(I'm willing to accept that this might prove infeasible, though I'd still like to see approaches enabling greater direct representation.)
What if, as with the Supreme Court reform proposal, legislative votes were based on a subset of representatives, again allocated by lot from amongst members? Again, though outcomes would over time average to partisan representation, any given vote might, or might not, favour the majority party.
One possible consequence: given the impossibility of guaranteeing a known outcome in votes, legislation might be written to more broadly consider overall views. This could avoid exremes, though it's worth noting that extremes gave rise to both highly discriminatory and civil-rights-reform practices. Some extremes seem less bad than others.
There's also the probable outcome that the locus of control would shift elsewhere. I'd still like to see some discussion of such concepts.
these tablets are close to 4000 years old.
being possesed by gods was an in thing, a believe demons were a prechristian idea as well, so if witch, and demon could be interchanged somehow, i think that fits the era.
A citation is needed to disprove the information that you refer to as "religious fabrication". Lack of information in Egyptian records is not a citation.
I know it is not satisfying, but essentially all of modern biblical scholarship no longer consider these (among others) are factual. This is held by believers and non-believers in the field alike. There is not a single paper to cite, its the result of decades of research with the field slowly settling on this as the result of hundreds of papers, studies, and books. In particular, you won't find a paper entitled "Modern Biblical Criticism rejects X", as it is not original research and worthy of publication. A book like Thomas Romer's "The Invention of God" goes through a bunch of the pro/con evidence and various theories and proposals put forth. And here is a paper by Romer that just summarily states that modern research has abandoned the documentary hypothesis, but doesn't really offer support for that as it is a well known point within the field that doesn't require re-litigation: https://hal.archives-ouvertes.fr/hal-03820791/document
It is only controversial to say this because it is a religious document.
(Lest you think I’m taking sides, the Jewish account of exodus doesn’t line up with archeological evidence either.)
On the Egyptian side, basically no excavations or inscriptions anywhere match up with the biblical story. There were semitic-speaking slaves in Egypt, yes, but not chattel slavery as commonly depicted, but rather something more akin to domestic servants and feudal peasants. And they were never expelled, as there are ongoing references long after into the Ptolemaic kingdom. The only exodus on record of semitic peoples is the Hyksos, who weren't slaves but rather conquerors and the fifteenth dynasty of Egypt. When the Hyksos were eventually defeated, they were expelled and driven out as far as Syria. Maybe that was the origin of The Exodus? But if the story changed from a slave rebellion ("let my people go!") to a dynastic struggle by the ruling class, are we still talking about the same story?
Finally there's the question of when the story was written down in the first place. It is interesting to note that the first few prophets of Israel make no mention of the Exodus whatsoever. It is not part of the shared zeitgeist of early Israel. It is only a few hundred years later that the references to an Egyptian origin are made by some of the northern prophets, and then the story spreads to the south. The Exodus itself isn't written down until Babylonian captivity much, much later, at which point a liberation story starts sounding quite a bit like Mary Sue fan-fiction.
Wikipedia has a pretty good summary of the current state of research as well: https://en.wikipedia.org/wiki/The_Exodus
Please check this documentary on the new evidences regarding the Exodus [1].
[1]The Real Mount Sinai - Shocking Exodus Evidence in Saudi Arabia:
Anyway I was directly replying to my parent poster not to the article mentioning that if anyone especially kings that can sort of reliably predict the future they can claim divinity.
[1]Muhammad ibn Musa al-Khwarizmi:
https://en.m.wikipedia.org/wiki/Muhammad_ibn_Musa_al-Khwariz...
A simple search on Amazon on 'Egypt' and 'Exodus' returned several results of books written by the scholars including the following books [1],[2]. There is also a link to the video documentary regarding the Real Mount Sinai in my other comments.
[1] Israel in Egypt: The Evidence for the Authenticity of the Exodus Tradition:
https://academic.oup.com/book/25374
[2] Origin of the Hebrews: New Evidences of Israelites in Egypt From Joseph to The Exodus:
https://www.swrc.com/the-douglas-petrovich-collection-dougla...
swrc is not a valid source of biblical scholarshp, Douglas Petrovich publishes fringe theories and is not someone you want to be citing for biblical research. To wit, he argues that the tower of babel is the only reasonable explanation for the number of languages we have in the world. He is self proclaimed inerrantist and creationist[1]. Why? Because God talked to him and told him it was true. This is not scholarship. That linked video (mine, not yours) is pretty interesting - he talks about being rejected in academia, colleagues won't speak to him, etc. He unfortunately interprets this as rejection of the Christian God, not because his 'research' is shoddy and unsupported, and takes it as proof that he is right. I don't say this to mock him, he is in tears at the end, but to point out he isn't doing scholarship and isn't considered a reliable source in academia.
Please check videos of the Real Mount Sinai as explained in the Bilble (Old Testament) as being shared in my other comments and other sites.
https://johncarlosbaez.wordpress.com/2011/12/02/babylon-and-...
PS The method described to find roots is quite similar to the method used in Quake by Carmack, Newton-Raphson. That's one of those pieces of code that's oft-copied by copilot.
The ancients certainly knew about phi, because it defines the ratios between segments of a pentagram. They loved pentagrams. They probably would have considered the match above exact, having no independent method to estimate pi. (Phi is (sqrt(5)+1)/2. Phi²=phi+1, 1/phi=phi-1.)
A measurement system used by builders in medieval times had units that were related by powers of phi. Nobody knows how far back the convention goes, passed from master to apprentice.
They've found some tables of similar estimations, which apparently match the results you get when you apply the author's algorithm, suggesting that's what the Babylonians used as well. I don't think the method itself is actually attested anywhere in known texts.
But what about the second row?
The numbers on the second row are 42, 25, 35. So I tried working out 42/60 + 25/60^2 + 35/60^3 and it works out to 0.70710648, which is very close to 1/sqrt(2) ~= 0.70710678. So cool!
Base 11 is called undecimal[1] and Wikipedia has a comprehensive list[2]. It doesn't list 59 but it would be enneaquinquagesimal if you follow the rules.
[1] https://en.wikipedia.org/wiki/Undecimal
[2] https://en.wikipedia.org/wiki/List_of_numeral_systems#Standa...
- Use Latin for 1-10
- Use Latin 10n, 10n+1 and 10n+2
- Use Greek for the units + Latin for the tens for 10n+3..10n+9
The Greek-Latin mixup was probably caused by whoever invented the word hexadecimal (16). The older "sexadecimal" was maybe avoided due to the word "sex". "sedecimal" would be a bit more correct Latin but it doesn't seem to be attested.
it was simply very common to use Greek prefixes in Latin to indicate measures.
Micro, mega, nano, etto, etc. are all Greek words.
micro -> mikrós (small)
mega -> mégas (big)
nano -> νάνος (small in its own kind)
etto -> ἑκατόν (100)
there were absolutely no issues with the word sex, sexagesimal (from the Latin sexagesimus which meant the 60th) is still a thing.
They seem very similar to ordinals and partitive numerals in Spanish. Those words are being quickly forgotten except low values and powers of ten. But they were common knowledge years ago, I guess it was the same for other romance languages.
Sexagésimo/a is 60th or 1/60. So sexagesimal is very natural.
√2 is just a name of a relationship, not a measurement, which implies finding its place on the number line.
No one will ever know √2 to 100% accuracy, in the sense of answering every question of the the form "is x^2 < 2^n", for all exactly known x (rationals), computing each within a constant time limit.
Actually evaluating the digits is for... technicians.
(Besides, just choose an irrational number base and sqrt 2 is = 1)
The thread shows what they mean a few tweets in.
The math, in general, isn’t as good as it, IMO, should be. The claim
“Thus, the interval [x₀, 2/x₀] envelopes √2.
From this, it follows that the mid-point of the interval [x₀, 2/x₀] is a better approximation to √2”
isn’t correct. The midpoint is a better approximation, but that doesn’t follow from “the interval [x₀, 2/x₀] envelopes √2.”. [1,999] envelopes √2, but 500 isn’t a better approximation of √2 than 1.
(Aside: https://en.wikipedia.org/wiki/Accuracy_and_precision is useless for finding a definition of accuracy. The chapter “Common technical definition” doesn’t define anything, and the picture in that section seems to imply that, for both precision and accuracy, lower is better)
I'm sure there's a better way to see the conclusion stated in the tweet, but you can convince yourself with the brute-force way of graphing:
- `abs(x-sqrt(2))` (the distance between x and sqrt(x))
- `abs(2/x-sqrt(2))` (the distance between 2/x and sqrt(x))
- `abs((x/2+1/x)-sqrt(2))` (the distance between the midpoint and sqrt(2))
and seeing that the last one is always smaller than the first two between 1 and sqrt(2), the values for x stated in a previous tweet.
Read the thread he explains it very well. The percentage is 1 - (absolute error divided by the true value) * 100
But it's a common note that that's really the inaccuracy.
For comparison, the initial value of the iteration (1.2) would be 85% accurate, which without context might sounds like a lot but is pretty abysmal
https://thepalindrome.substack.com/p/how-did-the-babylonians...
If someone publishes original content for free and in the end puts a link to their paid blog, I really don't mind it. I've seen far worse advertising.
It explains and illustrates a fact beautifully. You're not meant to be able to do the calculation yourself if you found yourself back in time
Plus I found the read fairly interesting.
It was not about the font used or the colour.
It isn't on HN, there are zillions of moderator comments about it. It's repetitive and a settled thing to not do.
However I do not think this is wise shutting up about unconsumable content and broad audiance annoyances and leave it be. Equally could ban post of improper format as to make shut up people who do not ignore well known (#1!) problems .... How about adjusting posting guidelines instead when this is the peek (#1) bother triggering hundreds of complaints throughout the years? Hmm?!
May I please disagree about the necessity of the complaint? Or better go away instead?
Btw. is it that hard ignoring complaints about twitter annoyances when twitter annoyances itself are ignored? Pretty strong double standard and partiality here.
(/s)