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.
Binary search three 6 bit symbols, that is 18 multiplications. A bit less if you do not blindly pick the centre but estimate - or guess if you will - what the correct digit should approximately be. I would assume they could have done this to much larger precision if they had cared enough as I find it hard to imagine that they would have been limited by having to test every candidate. If I asked you to find the square root of two to eight fractional digits, you would probably come up with the idea of skipping ahead more than 0.00000001 each time pretty quickly. I guess my point is that it might be interesting that they knew about the idea of the square root of two but the precision does not really surprise me.
I would expect that, while it is a hassle, somebody would do it. And you only need to do the computation once. Then, stamp it on a square tablet and you can sell it!
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!”
I guess for them it would be more like a three - four with the leading one - digit number, would it not? Did they have a 60 x 60 multiplication table at hand? I am almost tempted to try it out, I would guess that it should be doable in two hours in modern notation.
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.
It's not that hard in base 10. It might take 5 minutes with the long multiplication technique. I'm assuming that there is an equivalent in the Babylonian base 60 system
Newton's method is in fact what is described in the Twitter thread.
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!
I was about to ask exactly this. I did a few iterations and it converges pretty fast, nice to know someone went the extra mile to see exactly how many iterations it would take. Annoying if you were doing it by hand, but not that bad really.
About one bit (in binary equivalent) per naive iteration? Better with some linear interpolation (pseudo-Newton)?
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.
Thanks for sharing. that did bit of information. I do not understand though. how is that related? because they are multiples of six? They are also multiples of four and two and three
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).
I would agree on the long and accurate ruler. The actual problem here is understanding what the square root is (and how it can be used), not finding it's value for some arbitrary number. And despite all even 4000 years ago our ancestors could make a very precise tools.
Entirely agree. To be honest, I think it rather marvellous that people were able to imagine, then reason about, so much precision so long before any physical representation of it was possible. The Babylonians could calculate to a ten thousandth of an inch, but no human could measure the physical equivalent until Henry Maudslay. In a way the whole Industrial Revolution rested on people realising that the five-thousand-year-old mathematical view of abstract truth of "shape" could be rendered as physical reality with the right techniques... but coming up with those abstractions in a world where the roundest thing came off a potter's wheel and the basic unit of length was some fellow's forearm was an even more impressive achievement. We should not take it for granted just because it was so long ago.
Why? Long rope, make a right triangle. The proportion of one of the equal sides to the remaining side is square root of 2. Seems more likely to be experimentally approximated before calculated.
This would be useful for figuring out land rights. And for that, you need very long ropes.
I would be shocked if anyone could measure sqrt(2) to half a part per million using 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.
Hi there, author here. I have removed the paywall for this and all future posts. I am new to Substack, and still experimenting with the proper format. This change has been in my mind for a while, so I went and removed the paywall from all of my posts, and left the paid subscription as a supporting option.
Hey, just wanted to chime in that your post is nice.
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.
You missed the whole Domonic Cummings UK thing then.
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.
Substack authors can choose which posts to paywall, and how far into the post to end the preview.
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.
> 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.
Essential for astronomical calculations, like forecasting the paths of Jupiter and Venus. A daily brief of "omens" was submitted to the Assyrian King at breakfast. Has anything changed in modern day? I watch the weather and traffic at 6am and likewise predict how "long and glorious my reign shall live" ;)
Romans relied on a few different omens for a lot of things which reduced to injecting randomness into a lot of decision making. I wonder if such things would be valuable to add back to modern life… roll dice to determine which tickets to add to the next sprint, legislatures have to flip a coin to trigger a floor vote for a piece of contentious legislation, that kind of thing.
I think that makes sense. If you cannot be very sure of actual evidence why rely on it anyway? But randomness might mean that every time you use the result you learn something, at least you would learn whether your rule-based default value for the control variable is systematically wrong and tilted to one direction or another.
I think companies could test their interview process this way. Fill some positions by choosing from the candidate pool at random (there could be a baseline filter to ensure a minimum level of competence) and compare the random hires with the regular hires to see how much of a difference the interview process really makes.
I've long advocated for building a candidate pool for both hires and promos from a low baseline (with the option to opt out for promo), then hire/promote randomly out of that pool. Then fire/demote based on demonstrated incompetence.
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.
That goes along with my thinking. Some things are obvious but when you move away from the top topic or two you’ll always have a long list of things where it’s not so clear which should be done next, and there’s plenty which is always important to do but never the most important thing. A bit of randomness injected into that process whether you’re dealing with software tickets, national legislation, or household chores might just lead to better outcomes than relying on the judgement of a few people without enough information to make nonarbitrary decisions and might have bad biases which lead to things getting overprioritized or neglected.
Athenian democracy worked this way--you had to opt-in to the candidate pool, but then the candidate was chosen randomly from that pool.
(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)
(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. ...
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.
Sortilege is a good thing. Right now we have an ostensibly democratic system where elected representatives take part in another election among themselves about which one of them gets to have pre-emptive veto power over all legislative proposals for 2-6 years (in the US).
As I noted upstream, sortition's a pretty fascinating concept.
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.
if you could predict when certain phenomenon will occur, then act as if you are somehow in control of these things, you can spin up public perceptions of god-hood
If I had to take a guess: the scene being depicted in the comment could apply to any powerful king/priest of the past - imagine halls full of incense and chanting creating a feeling of mass hypnosis/hysteria (possibly intensified with consumption of psychedelics) and the booming voice of the king/priest proclaims their power over celestial objects - and it comes true due to normal celestial events/eclipses!
im thinking in the context of pre-judeo christian society.
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.
This is how the self claimed god-king of the old namely Namrod and Pharaoh misled their subjects and claimed false deity. In the Bible (Old Testament) and Quran there are stories of Pharaoh during Moses time cruelly and pre-emptively killed all the Jewish new born sons based under the heavenly knowledge that he will be eventually overthrown by a Jewish prophet. According to legends and history, he gathered this prediction information from his sorcerers with their Genie partners eavesdropping the angels conversation because in the old days these eavesdropping activities was feasible and now abruptly stopped after the Quran revelation.
There are no detailed architectural diagrams documenting how the pyramids were built either. There were some papyrus scrolls discovered in 2013 which provide log information but still to this day, no details are known to exist.
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
Yeah but a source which is itself derived from an earlier source (Torah) but the earlier source lacks the relevant details is a shut and dry case of fabrication.
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.)
There's a lot stated in the article. There is absolutely no evidence for a Hebrew peoples invasion of Canaan. The cities mentioned in the Bible were excavated, but there is no record of any sieges. The transition from Canaanite to Israelite settlers is gradual over hundreds of years, and likely represents a transformation of the same culture and not an actual displacement of people. None of the places mentioned in Sinai that have been identified show any evidence of any people whatsoever having visited until hundreds of years after the absolute latest possible dates. "The conclusion – that Exodus did not happen at the time and in the manner described in the Bible – seems irrefutable [...] repeated excavations and surveys throughout the entire area have not provided even the slightest evidence." (Finkelstein and Silberman, 2002).
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.
Probably the histrorians were looking at the wrong place and perhaps the original Mount Sinai is not the one that's widely accepted. Any knowledge be it science, history, etc need to be revised and aligned to the newly discovery with more robust evidences.
Please check this documentary on the new evidences regarding the Exodus [1].
[1]The Real Mount Sinai - Shocking Exodus Evidence in Saudi Arabia:
Do you realized that most of ancient mathematics revolved around religions and one of the very first books on algebra was written by Al-Khwarizmi to accurately calculate inheritance for Islamic jurisprudence? [1]
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.
Time will tell whether it has zero correlation with Egyption sources and whether it's almost certainly invented.
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:
A simple google search reveals youtube videos and books on how the world is flat. It ain't flat.
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 do not conflate the case for Jewish settlement in Egypt with the flat earth theory. The former does not have enough convincing evidences yet to be found while the latter is straight up denial of the truth despite the numerous clear evidences.
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.
We have plenty of records from other societies indicating how absolute rulers did and didn't schedule their days. A king has a potentially limitless amount of stuff to do and also the power to set his own schedule as well as servants to help him stick to it. Whether he slept a lot or a little and whether his schedule was biased early or late was almost certainly a matter of personal preference and convenience. An absolute ruler also has to consider the signal being sent. Being awake and dressed in time to attend morning colors with regularity sends a message that a ruler might deem worth the effort to send.
The comments on that page are wonderful - Baez, Greg Egan and others working out together the explanation/theory of everything mentioned in the article.
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.
The details on the method are paywalled in his Substack, and it's not clear whether he has any evidence that the method he is proposing are actually how the Babylonians actually calculated square root 2.
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.
The substack loaded for me but all it says is that they used (a special case of) Newton-Raphson. No second tablet is mentioned showing that the Babylonians calculated that way.
There is a link to https://www.jstor.org/stable/10.4169/j.ctt19b9k86 which just says "this is an excellent approximation"
I suppose by scholarly standards we don't know that any Babylonians knew this method (the first of whom we're sure would be a 1st century Greek mathematician), and the result on the stone is generally attributed to an unknown source (wikipedia). But I guess you could call the attribution folk knowledge, I have often seen this method called something like Babylonian root-finding.
He explains the first row of symbols: 1, 24, 51, 10. This represents 1 + 24/60 + 51/60^2 + 10/60^3 ~= 1.41421296, very close to sqrt(2) ~= 1.41421356.
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!
That's because base 12 (dozens) and base 60 (minutes) are commonly used across cultures. So is 20 (vigesimal).
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.
- 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.
I find it fascinating that we have names for base 60 and base 12.
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.
That precision is fascinating. Other measurements of that age I'd guess were a lot less precise, such as time and distance. It seems like anything you used sqrt(2) for calculating, would be swamped with less-accurate factors and you'd get a much less accurate answer.
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)
Accuracy is best thought as relative, though the usage of percentages here is not a good fit imho. You compare your result to the wanted value by dividing the error by the value. In this case |x - √2|/√2 < 0.000001 (or 0.0001% if you insist)
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
That's what rubs me the wrong way I think, there is something off about a post like that, it just feels like bad faith "content" rather than an actual helpful write up about something, and I feel like more of this is ruining the internet. It's the equivalent of a more sophisticated listicle.
Your criticism feels entirely non-substantive. Of course it's lacking depth as its target audience is not historians. Maybe your issue is with the overall trend of "pop journalism" that puts out short, easy to ingest but ultimately reductive content (I think of outlets like Vox on YouTube). I personally think as long as it's not outright lying about the subject, exposing it to more people is better than gatekeeping it for academia.
Why oh why write a study on twitter, oh why?!
There are much more fitting mediums for that. I refuse to read it this way.
Would you release the next Marvell movie on TikTok or a book in text messages? Grrr.
Please don't complain about tangential annoyances—e.g. article or website formats, name collisions, or back-button breakage. They're too common to be interesting.
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.
I hear ya, but think it’s the media medium targeted at that audience. A TikTok version would appeal to younger people, which I think would be a great thing!
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.