Pythagoras was specifically known for accumulating the wisdom of diverse cultures—supposedly he met Thales, was initiated as Egyptian priest in Hermopolis, spent time in Babylon after being captured, and was initiated into every mystery cult he could. And as a boy on his home island of Samos, he would have been exposed to the building of the largest stone temple in Ancient Greece (to Hera) and the incredible engineering feat of the tunnel of Eupalinos.
Iamblichus’s “life of Pythagoras” [1] is worth a read as he had access to all the old sources now lost. The relationship between math and spirituality was very strong back then!
There are lots of fun stories that may be true but no one will ever know. In Diogenes Laertius’ “Lives of the Philosophers,” it is claimed that when Pythagoras made his discovery of what we call the Pythagorean theorem, he sacrificed 100 oxen (a hecatomb) [1]. As noted by Charles Dodgson (Lewis Carrol), “that would produce an inconvenient supply of meat” [3], especially for a vegetarian. Iamblichus, on the other hand, claims it was a single ox — and made of flour!
[1] Guthrie, K. S., & Fideler, D. R. (Eds.). (1987). The Pythagorean sourcebook and library: an anthology of ancient writings which relate to Pythagoras and Pythagorean philosophy. Red Wheel/Weiser.
[2] “he sacrificed a hecatomb, when he had discovered that the square of the hypotenuse of a right-angled triangle was equal to the squares of the sides containing the right angle.” DL, found in [1]
[3] Maor, E. (2019). The Pythagorean theorem: a 4,000-year history. Princeton University Press.
Oh how can you skip the most fun tale of them all. [1] Its authenticity is dubious, but there's probably at least parts of truth in it. In a nutshell, Pythagoras started a cult based around numbers, and the pseudo-divine purity of rational numbers - of which everything can be represented.
Hippasus, a member of said cult, however managed to compellingly demonstrate that the square root of 2 could not be a rational number. Pythagorus tried to swear him to silence. When that did no twork, he had him killed. Over the square root of 2.
I don’t know of any ancient sources claiming that Pythagoras killed Hippasus. Seems unlikely to me! In any case, Hippasus was a fascinating figure:
“Aristoxenus (Fr. 90 Wehrli = DK I 109. 31 ff.) reports that Hippasus prepared four bronze disks of equal diameters, whose thicknesses were in the given ratios, and it is true that, if free hanging disks of equal diameter are struck, the sound produced by, e.g., a disk half as thick as another will be an octave apart from the sound produced by the other disk (Burkert 1972a, 377). Hippasus, thus, may be the first person to devise an experiment to show that a physical law can be expressed mathematically (Zhmud 2012a, 310).” [1]
Also in [2] this experiment is claimed to be the first documented scientific experiment in history. After all, Hippasus took a mathematical model for a physical phenomenon (how consonance relates to the mathematical ratios of a musical string) and tests the generalization of that model in a another physical medium (viz. bronze chimes with the same ratios 1:2 and 2:3 make the octave and fifth).
I wish they’d put this stuff in elementary math books when kids learn about Pythagoras.
Well, they continue to introduce Pythagorean theorem through many stages of elementary math, so 8 is pretty early. But.. here could be a section of a textbook:
Pythagoras & The Pythagoreans: Mathematics, Music, and Mystery
Pythagoras was an ancient Greek mathematician and philosopher who lived around 500 BC. He traveled widely and gathered knowledge from diverse cultures, like Egypt. He also founded his own school of men and women in Italy. There, he taught that numbers held the key to understanding the whole universe. Pythagoras is best known for the Pythagorean theorem.
Fun Fact! There are stories that when Pythagoras discovered his famous theorem, he celebrated in a big way. Some say he sacrificed 100 oxen, while others claim it was just an ox made of flour. The Pythagoreans were famously vegetarian, so what do you think?
Math & Music: Pythagoreans explored the relationship between math and music.
They discovered that musical notes have mathematical relationships. Hippasus, a member of the Pythagorean community, used bronze disks to show that musical notes are connected to mathematical ratios. This is considered one of the first scientific experiments!
Activity: Using a stringed instrument, like a guitar, try plucking the strings when pressing at 1/2 the string or 2/3s the string. Try different fractions. Can you hear the mathematical relationships in the sounds?
I have thoroughly enjoyed all of your comments! You’re great at showing how learning can be fun. Do you have a blog where you compile these stories and anecdotes?
When I was 8 I would sometimes watch a Disney VHS tape that had Donald Duck exploring stuff like this. The golden ratio, octaves of sound/music, pythagorean theorem etc. I loved it.
Which is why hands-on learning combined with source material is best practice for children, imo. Taking several disks and demonstrating that the thicknesses double but diameter is the same, letting the kids try hitting them and noting the differences in tone.
Asking the children "why do you think these sound different?" without telling them why first, letting them explore it themselves, gently correcting them along the way. Then pulling out another example and encouraging them to apply the same logic to the new example.
No, instead for the most part, children _still_ get what we all had: memorise these facts without explanation and fill out the blank space on the exam where the memorised words get you points. Most of our entire schooling is just an exercise in memorisation, not in thought and it's sad.
Pythagoras definitely did not have him killed since he had been dead for ~50 years. So even if the story is true it is about Pythagoreans, the followers of Pythagoras.
"How can it equal one? If one times one equals one that means that two is of no value because one times itself has no effect. One times one equals two because the square root of four is two, so what's the square root of two? Should be one, but we're told its two, and that cannot be."
But his success has not stopped the actor from claiming he spends up to 17 hours a day creating nameless plastic structures, which are made of cut up pieces of plastic and either stitched together with copper wire or soldered, that he believes prove his new form of mathematics.
Howard studied chemical engineering at the Pratt Institute in Brooklyn until he fell out with one of his professors over the answer to the 1x1=1 conundrum.
It says he drowned... assumed because of the reason OP gave. Besides, moot point giving you're referencing Wikipedia to begin with.
> Hippasus is sometimes credited with the discovery of the existence of irrational numbers, following which he was drowned at sea. Pythagoreans preached that all numbers could be expressed as the ratio of integers, and the discovery of irrational numbers is said to have shocked them. However, the evidence linking the discovery to Hippasus is unclear.
> Pappus merely says that the knowledge of irrational numbers originated in the Pythagorean school, and that the member who first divulged the secret perished by drowning.
It also says “one writer even has Pythagoras himself "to his eternal shame" sentencing Hippasus to death by drowning, for showing "that
2√2 is an irrational number”.”
> supposedly he met Thales, was initiated as Egyptian priest in Hermopolis, spent time in Babylon after being captured, and was initiated into every mystery cult he could
It should be noted that such claims were made about many ancient sages to boost their "wisdom pedigree". Plato is said by ancient sources to have traveled to Egypt and Italy, but my understanding is that most modern scholars doubt that those journeys really happened.
Pythagoras’s father was a gem trader from the Levantine coast so the likelihood of him traveling around the Mediterranean is pretty decent. His Egyptian connection is attested early by Herodotus and his devotion to diverse schools of wisdom is attested in a critical statement by his contemporary, Heraclitus.
Of course it’s hard to know. But Plato wrote several letters about his trip to Italy, because he was briefly enslaved there before being freed by his friend the Pythagorean Archytas (who is famous for creating a steam powered flying machine and wrote a work on mechanical engineering).
https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1...
> The relationship between math and spirituality was very strong back then!
On Seymour Cray:
Another favorite pastime was digging a tunnel under his home; he attributed the secret of his success to "visits by elves" while he worked in the tunnel: "While I'm digging in the tunnel, the elves will often come to me with solutions to my problem." [0]
That sounds like he was being figurative. He probably meant that he got ideas while working.
Every morning, I get up at 5, and take a 5K walk. During that time, I tend to "triage" the day ahead, and often solve problems that were vexing me, the night before.
Part of my walk is around a local high school track. There is a small flock of killdeer birds, that hang out there, and I guess they give me the ideas I have, as they often come to me, at that point in my walk.
I enjoyed this part:
> One story has it that when Cray was asked by management to provide detailed one-year and five-year plans for his next machine, he simply wrote, "Five-year goal: Build the biggest computer in the world. One year goal: One-fifth of the above." And another time, when expected to write a multi-page detailed status report for the company executives, Cray's two sentence report read: "Activity is progressing satisfactorily as outlined under the June plan. There have been no significant changes or deviations from the June plan."
I'm actually curious if "digging in the tunnel" is a euphemism for using psychedelics, which would have been severely frowned upon at the time. The elves are known for their wisdom in helping you see the world and solve problems in unique ways.
> FRANCIS CRICK, the Nobel Prize-winning father of modern genetics, was under the influence of LSD when he first deduced the double-helix structure of DNA nearly 50 years ago.
> The abrasive and unorthodox Crick and his brilliant American co- researcher James Watson famously celebrated their eureka moment in March 1953 by running from the now legendary Cavendish Laboratory in Cambridge to the nearby Eagle pub, where they announced over pints of bitter that they had discovered the secret of life.
> Crick, who died ten days ago, aged 88, later told a fellow scientist that he often used small doses of LSD then an experimental drug used in psychotherapy to boost his powers of thought. He said it was LSD, not the Eagle’s warm beer, that helped him to unravel the structure of DNA, the discovery that won him the Nobel Prize.
DMT is pretty famous for, oddly, inducing hallucinations of Machine Elves in people. Don't discard the building tunnels part though, it is a weirdly popular hobby among the very male-brained.
Funny enough DMT is the only psychedelic that's ever given me real, true insight and legitimately changed my life after a single breakthrough dose, 99.5% of which I didn't remember even immediately after 'coming to' (and still don't). I was suicidally depressed and had been for a number of years, struggling with drugs and depression, had all but given up on life. I did it alone not meaning to break through.
Immediately after my breakthrough, I was so overjoyed and happy and couldn't help but bellow out "there is so much love in the universe!!" over and over -- right after a run to the washroom to purge a rainbow from my mouth. Thankfully I was home alone. I immediately called my loved ones to talk to them and tell them how much I loved them. My sister (psych major) thought I was going to end my life soon because I was suddenly so happy and unburdened. Suicidal ideation ceased for over 5 years, and still has only come out in the rare circumstance I was under extreme emotional distress. The majority of the lessons I learned -- many obvious in retrospect, I feel I was almost purposely blinding myself from seeing them because it was easier not to -- came in dreams and waking daydream 'visions' in the days after my trip, and my sudden ability to notice my problems weren't so serious, I just needed to look at them from a different viewpoint, of which there are many.
The only person who noticed (or mentioned) any difference right away in me, was the wife of a client -- a devout Hindu woman. As soon as she saw me a few days after my trip she said "MyName, are you alright?" with a concerned look on her face. I still wonder what she noticed or what concerned her.
I haven't felt the need to do it or any other psychedelics since, but for some reason I felt I'd share a quick tale of my story since the topic of machine elves and insights came up.
I was also, coincidentally (common as it is, apparently?) obsessed with digging a tunnel as far as I could as a 12-13 year old boy until it collapsed on me after about 5ft because I'd started on an unstable hill. That was the end of my tunneling, although I suddenly have a strange urge to grab a shovel.
I got another one - Kary Mullis, PCR inventor, revolutionized genetics and biochemistry:
„In it, he recounts his late-night encounter with a glowing raccoon that spoke to him, addressing him as "doctor" -- a raccoon that may or may not have been an alien. He tells of passing out after inhaling too much nitrous oxide and later learning that he'd been saved from a fatal overdose by a woman who traveled to him on an "astral plane."“ [0]
> "The relationship between math and spirituality was very strong back then!"
Seems to me this connection is having a resurgence now as people attempt to project the capabilities of computers beyond human capabilities, where the implications are necessarily bordering on spiritual.
Examples include transhumanism, Yudkowsky-style absurdly extrapolated rationalism (Basilisk etc), the simulation hypothesis, AGI salvation hopes...
George Boole (Who we get Boolean from), had a road to Damascus experience as a teenager and later wrote "An Investigation of the Laws of Thought" where we get our 0s and 1s from, and truth operations, largely to prove God is Good.
The concept of 0 and 1, absolute truth and false, was born from his spiritual views.
They are often ways to transmute spirituality into a form more palatable to contemporary sensibilities trained on technology.
Who runs the universal simulation? Another name for God. The eternal torment of Roko’s Basilisk? Another name for Hell. AGI will save us from ourselves with its incomprehensible intelligence? A cyber-Jesus. Moving your mind into a transhumanist body? Souls rising to Heaven. Etc.
"The relationship between math and spirituality was very strong back then!"
According to those whose communication ended up being indelible. I wonder how we'll be able to preserve digital info for millennia?
Now I’m very much interested in reading more about his life! Gotta say tho, sounds a bit like an ancient Marco Polo where maybe the myth is larger than the man at this point.
Still going to read more. Thanks for the citations.
Try to get a copy of the Pythagorean Sourcebook! It’s great reading original sources. So satisfying and often more interesting than modern scholarship about them.
Every group who ever managed to build a building with a rectangular foundation figured out the relation between the side lengths and the diagonal. The Egyptians used Pythagorean triples to measure right angles way before the birth of Pythagoras.
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).
> Pythagorean triples to measure right angles ... 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
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:
depends on the method they used to come up with the formula. It's not obvious that the sum of a "more complicated than you can do in your head" series of fractions would be either the square root of two or the the diagonal of a square, let alone both. This is a fragment of a single clay tablet. If they had a systematic method of coming up with this convergent series, there might have been a stack of tablets showing square root of three, five, etc. It seems pretty highly suggestive to me.
I agree.Obtaining this formula by observing is more difficult than obtaining it by proving it. Besides, it is an issue that is already needed in field work. If you have pen, paper and a good number system, it shouldn't be too difficult to find the formula with shapes.
That entirely depends on how close you are to high-clay-content muddy bank with a growth of reeds vs. how close you are to the nearest convenience store/office supply store.
Why would you need numbers for right angles? You need a straight line and a string.
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 project" 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've personally worked on a project where we used a 3-4-5 right triangle to lay it out on the ground. Straight lines alone do not get you right angles.
You can get right angles with a straightedge and compass, both of which you can make on a construction site with string and pegs in the ground. It's just an instance of bisecting the angle. Which is more convenient in practice is situational.
Right. The big problem with accurately bisecting a 180 degree angle is you need to have access to points rather far in both directions from both points. If you want an accurate right-angle corner near the edge of a property that's flanked on two sides by fences, rivers, busy roads, etc., then you might not have convenient access to one of the anchor points needed to perform your bisection. (Or semi trucks snagging your rope might be inconvenient.)
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.
IIRC, straight lines alone will give you projective geometry (which also has points, but they're the meet of two lines).
> 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.
> How useful are universal truths compared to getting shit done?
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.
What everyone is talking past here is that the Babylonians discovered what engineers call a "rule of thumb", and engineering is focused on getting things done using rules of thumb. The best possible rule of thumb is a mathematical proof or a scientific discovery, but it's by no means necessary. .
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.
Getting shit done is the antithesis of progress because it goes hand in hand with "if it was good enough for my father, it will be good enough for me, who are you to disrespect (the methods of) my father!"
I took a course at the university of Buffalo in the early 80s about pre-roman examples of Pythagorean geometry in henge monuments and early structures. Basically the class was watching slide shows of the professors vacations across Europe and his photos of old ruins. Then he'd show us a line drawing of the structure and pick out what at times seemed random points that created 3-4-5 triangles. He'd exclaim 'see? They knew about the Pythagorean theorem before Pythagoras!' Our only graded assignment was to write a paper where we "discovered" a similar example. I picked a picture of some random ancient church and "found" the right triangle in it's foundation. I got an A. Definitely just an "observation" no theorum involved.
A "you" can, and many individual "you"s presumably did that, to general satisfaction. But the implicit assumption you made is that eyeballing and cutting is easy, while triangles are hard, even though it's the other way around, especially with stone age tools.
But we're talking about groups of people, "they", who all build a lot. Such groups 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.
>A "you" can, and many individual "you"s presumably did that, to general satisfaction
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.
I don't think I do. I'm fairly confident that the people of the chalcolithic did enough large scale construction (and pottery and watching the night skies and so on) to have figured a whole lot of empirically accessible geometry out.
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 built a tree house for my kids. I didn't exactly eyeball it but my amateur technique and planning left me having to do a lot of compensation for things being out of square.
Everything becomes a custom cut, often in multiple dimensions and all earlier errors cascade all the way to the end.
> I don't see why that was necessary. You can get pretty far with eyeballing it and custom cutting to fit.
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.
I remember reading about the worker homes and tombs, and I know that skeletal records also showed the workers had much muscular strain. So, it's impossible to deny that there were skilled laborers working on the pyramids, but just because some of the laborers were proven to be skilled and not slaves, doesn't mean slaves weren't used.
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
I'm currently in a brick build house which has a good ten percent slope, because the earth sank, but it hasn't collapsed. I'm pretty sure its not older than a hundred years and I'm amazed it still stands because it sure wasn't designed like this.
Ancient Mesopotamian laws sometimes flayed people alive for incompetent or corrupt work. Those types of mild punishments are good incentive to use the right angles. Pun intended.
> It becomes a theorem when you prove (i.e. explain why) this relationship always holds, based on more evident things. [...] whereas the Greeks definitely did (we don't know whether Pythagoras himself did it
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?
Proofs are of course the gold standard. But even if others discovered the rule, their contribution should not be held to be trivial.
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.
Yes, when people in the 4000s write clickbait holonet articles about how the Master Theorem of Neural Network Scaling was not actually invented by the Muskovites of Mars, but was found written on an Ancient American tablet containing the works of Hoffmann et al., I hope somebody will offer substantial corrections.
That's not in any sense a value judgment of the empirical work done at DeepMind. Nor does it stop anybody from writing a better article which explains that the Babylonians (and many others) used the empirical observations underlying the theorem, while explaining that this did not constitute mathematical proof.
> 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.
Your quote from the article does not suggest what you seem to think it suggests.
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.
You are actually right. You didn’t claim that the Greek did it first and that is what I doubted in the first place. I misread your comment.
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.
For anyone wondering how they got the approximation sqrt(2)=1+24/60+51/60^2+10/60^3.
It's based on the simple idea that:
Z = (a + b)^2 = (a^2 + (2a+b)*b)
=> (2a+b)* b < Z-a^2
Given an initial estimate "a", we need to find the largest "b" such that the term on the left is less than the term on the right. Therefore our estimate will always be slightly less than the actual answer and we can repeat the process to get slightly closer.
For the first iteration, Z=2 and a=1. We choose b=x/60:
Base 60 was the number system for the Sumarians/Babylonians.
> Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form—for measuring time, angles, and geographic coordinates.
The formula wasn't why people cared at the time; it had been empirically known for centuries. What caused such a stir was he was the first person to prove that for "most" right triangles there are no rational numbers P, Q such that PA = QC for leg A and hypotenuse C. That was what was earth-shaking.
Its only fitting that the now defunct "Journal for Targeting, Measurement and Analysis for Marketing" would have a clickbaity and misleading title on an article that seems completely off-topic and of very poor quality. A Springer sponsored precursor to the SEO driven abominations of today maybe?
Confusing calculation with proof is an inexcusable mistake for any serious journal.
There can be little doubt that proving theorems is a cognitive tool that developed on the basis of observed regularities. But both asking the question why and, importantly, answering it using logic are highly non-trivial developmemts.
The mainstream is and was _enamored_ with Ancient Greece.
Our Western culture made Ancient Greek into the vocabulary root of our sciences.
We have a lineage of philosophy from Ancient Greece to the 19th century.
Only in the 19th century with archeology (again a neo-word made from Ancient Greek roots – it suggests to the mainstream that the Ancient Greek had a concept of archaeology, which they obviously didn’t have) we saw the truth: History goes thousands of years deeper, the origin of everything is thousands of years older.
Only 30 years ago the capital city of Hattuša was discovered; and only in the 20th century we gained an understanding of the multiple levels of the historic city of Troy.
Only recently we understand that "it didn’t start with Ancient Greece", but the mainstream still follows the tradition of medieval grammar schools and doesn’t look beyond Ancient Greece.
Is it not a simpler explanation that Greece simply did a better job of disseminating its own literature than prior cultures?
My understanding is that the “through line” of modern scientific progress begins in Greece because that’s where “trivially-legible to contemporary scholars” written history began.
Like, we always knew that it didn’t start with Ancient Greece (the Greeks themselves mention this), but because abundant primary sources prior to Ancient Greece don’t exist, there isn’t much we can do other than light a candle for their sake and use its light to read their thoughts as filtered through Plato, etc.
> Is it not a simpler explanation that Greece simply did a better job of disseminating its own literature than prior cultures?
That's a simpler explanation but it's not necessarily right.
What we can say for sure is that Greek thought has been easier for Western scientists and pseudo-scientists to learn. Availability and language are parts of that for sure. Geography is, too - it's easier for Europeans to excavate Europe than Iran.
But what does it mean to say "the Greeks" disseminated their knowledge better when virtually all of what we have comes from Roman citizens living centuries later?
> But what does it mean to say "the Greeks" disseminated their knowledge better when virtually all of what we have comes from Roman citizens living centuries later?
That Greek knowledge and culture survived the collapse of their prominence? Similar to the Romans after them and Babylonians before them.
Greek had been a lingua franca in the major ports of future empires for centuries [1], and Greek remained a spoken and written language in the Byzantine empire (the same was not true of ancient Egyptian or Babylonian – which were supplanted by Greek or other Aramaic languages during the hellenistic period).
I think at least as much credit is owed to the inheritors of Greek culture (Romans, Byzantines, the various Arabic empires), for preserving source material and references.
But I think Greece was seen as the original "filter" for civilization because it was both "successful" and comparatively extroverted to the great civilizations that came before.
Basically, it's what you said in your middle paragraph – wide availability, accessibility of language and culture. In other words, "better dissemination". :)
> I think at least as much credit is owed to the inheritors of Greek culture (Romans, Byzantines, the various Arabic empires), for preserving source material and references.
That's what I'm saying, though. Preservation is the work of the preservers. Many of the great thinkers of Greece didn't preserve a single word. Someone else did. Often other Greeks, often Romans (who obviously spoke Greek as you say, because they believed it to be a superior language).
> wide availability, accessibility of language and culture. In other words, "better dissemination".
But B is a subset of A here. Not all of those facets that I mentioned are due to the Greeks themselves, not even indirectly.
Sometimes, Western civilization sees "The Greeks" as a progenitor civilization because... we believe they're a progenitor civilization. It's a tradition to believe so, and it may well have started by mistake or for reasons of xenophobia or other bad motivations.
The Greek still did something remarkable that to my knowledge we don’t have record of from before: they started a culture of free thinking that grew into philosophy, logic, and scientific inquiry. They also sentenced some of their free thinkers to death for not worshipping the gods but that’s a separate issue.
> they started a culture of free thinking that grew into philosophy, logic, and scientific inquiry.
Culture of free thinking? Socrates, the most iconic free thinker of all time was forced to commit suicide by the greeks. So much for a culture of free thinking.
> They also sentenced some of their free thinkers to death for not worshipping the gods but that’s a separate issue.
It isn't. It directly contradicts and refutes your assertion.
> Maybe Greek society was not a homogeneous mass, and some parts of it nurtured free thinking while others reacted against it?
So then your assertion was incorrect: 'they started a culture of free thinking that grew into philosophy, logic, and scientific inquiry.' They didn't start anything? Some of them did? What are you even saying then?
It's idiot's logic to claim that greeks nurtured free thinkers while admitting they killed free thinkers.
Instead of digging yourself a bigger hole, just admit you were wrong.
Hey, I don’t know what’s happening in your life but all of your recent comments seem bitter and abrasive. I hope that you get to take a break in some way.
The article claims that the Babylonians "discovered" the Pythagorean theorem, but all it shows is that they (probably) believed it to be true.
Until we have better evidence, it still seems to be the case that (at least in the "West", I'm unfamiliar with e.g. Chinese mathematics) the Greeks were the first to come up with the concept of a mathematical proof that is valid deductively, and not inductively.
People typically wrongly attribute findings not to the person who first discovered it, but to the person who was able to most widely communicate/publish about it.
This is a particular difficult challenge in ancient times.
Knowledge was often shared verbally, not in written form.
Or if it was in written form, the material used has long since deteriorated.
So most of what we know about ancient thinking is based on knowledge that was so widely known and written that there's multiple copies of it; or the knowledge was communicated on a hard material like stone (egyptian hieroglyphics) ... but that doesn't mean it was the first ancients knew of it, it just means that the particular knowledge in written form has lasted the test of time the longest.
I know the top one is generally considered to be Euler's Equation.
e^(i * pi) + 1 = 0
It's considered incredibly elegant because it manages to combine multiple fundamental mathematical concepts into a single equation.
1 is the multiplicative identity, 0 is the additive identity, pi is the circle constant, e is euler's number, i is the square root of -1, the basic building block of complex numbers.
The 0 and the + are important: e^(i*pi) + 1 = 0 contains the 5 most important constants and the three most important operations. Getting them back with "+ 0" is quite inelegant.
Is there an easy explanation of why these are true? I already struggle grasping e^i, and I completely don't understand what e to an irrational power even means, let alone why it would be -1. Why the circle circumference ratio has anything to do with this is completely beyond me.
1. e^x, sin(x) and cos(x) can each be expanded out into an infinite sequence of fractions (Maclaurin series), each fraction of the form x^n / factorial(n). This relies on differential calculus, Taylor series, theory of limits, convergence etc.
2. The e series fractions contain all the integers in the numerator (x^0, x^1, x^2, etc), while the sin series has only odd integers and the cosine series has only even integers. Also, e series terms are all additive while the trig functions series alternate adding and subtracting each successive fraction.
3. Introducing complex numbers (i = square root of negative one), we can generate the series for e^ix, which can be shown to be equal to sin(x) + i * cos(x). Note that introducing i into the e series means we generate a negative term for the even fractions in the e series (squaring i gives us -1), which is why i is so necessary here.
4. Solving e^ix for x = pi, using sin(x) + icos(x), we get -1.
As far as why an exponential function like e^x should have anything fundamental linking it to trigonometric functions like sin(x) and cos(x), it is rather strange.
So that's the part I explain in my note. So if you combine the two explanations together I think we have the whole picture. Yours fills in the gap in my explanation.
So I haven't done complex analysis yet which I think you need to get the whole thing but I can get you some of the way there with basic trig.
If you take a unit circle and construct a radius to some point (x,y), if you drop a perpendicular line down to the x-axis, it's easy to see that the length of that perpendicular line is y and the distance you've gone across the x axis is x. So you have a right angled triangle where the hypotenuse is 1 (it's a unit circle) and the other two sides are x and y. Now consider the angle at the origin and call that theta.[1] You can do basic trig to show that the coordinates of your (x,y) point are (cos theta, sin theta), because sin is opposite (y) over hypoteneuse (1) and cos is adjacent (x) over hypotenuse. Ok cool. So x = cos theta and y = sin theta. If you measure in radians, then the angle of a full revolution is 2 * pi radians and the angle of a half revolution (180 degrees in other words) is pi. Now consider the point when you have gone around the unit circle 180o, Its coordinates are x=-1 and y=0. Remember this point - we'll come back to it in a minute.
Now imagine instead of your unit circle being just any old circle it's in the complex plane. This means that the x axis is the real part of some complex number and the y axis is the imaginary part. We now know that the coordinates of points on this circle are (cos theta, sin theta), but if you have a complex number z= a+bi, these correspond to a and b. So z = cos theta + i sin theta. Here's the bit where my current mathematical ability runs out of gas and you're just going to have to trust Euler, who showed that cos theta + i sin theta = e^(i theta).
Now remember our point from before where theta = 180 degrees? What was the angle in radians? It was pi. So e^(i pi) = -1 (because the real part of the number is the x coordinate, -1 and the imaginary part, the y coordinate is zero).
This seems to be the original source https://physicsworld.com/a/the-greatest-equations-ever/ but it doesn't actually rank the equations. The other source commenting on that does, but only the sample is available on Google Scholar. From that, first is Euler's identity, second is Maxwell's four electromagnetic field equations, and third isn't in the sample. The NYT article also commenting on it https://archive.ph/H7ujx suggests the theory of relativity, F=ma, or amusingly 1+1=2.
Contributions to the body of knowledge is not and never ever been monopolized by any particular nation or civilization but if you read most modern textbooks you're forgiven to think that it was all started with Greek civilization and then nothing of significance happened until Renaissance movement in the 14th CE (conveniently skipping the contributions of the Roman (namely Rome and Byzantine) and Muslim empires (namely Rashidun, Umayyad, Abassid, Andalusian Spanish, Ottoman)). A popular Monty Python sketch of "What have the Roman's done to us" perfectly summarized this ridiculous sentiment.[1]
The fact that many contributions from other older civilizations for examples Indus Valley (Indian) where the original cuneiform alphabet was started and Phoenician (Arabic) where most of the modern alphabets (Latin, Greek, Arabic, Sanskrit) originated. The former Indus Valley script has not even been successfully deciphered yet until today (Nobel price in waiting for the ones who will deciphered them), perhaps they are some older proofs that are just waiting to be discovered upon the understanding of the Indus Valley scripts and languages.
But the nonsensical narrative of whom has the monopoly of contributions to knowledge will carry on until the end of time but the reality is that we are just standing on the shoulders of giants [2].
I recall seeing that there was a 14 000 year old dear scapula found in China that had the "simple proof" diagram engraved on it. The article seems to say that Pythagoras' Theorem was discovered by the Mesopotamians, but neglects mentioning it may be _much_ older.
Someone above commented that just by building ancient peoples would have discovered this relationship.
Has anyone found the source for that tablet? All I have found is this:
>Note that quite a few descriptions on Babylonian tablets seem to cite a translation of a Pythagorean algorithm from a ca. 1900BC tablet by a Dennis Ramsey – I have not been able to find the original source of this anywhere.[0]
The linked arctile cites wikipedia and bible-history.com. This book[1] misquotes the supposed tablet as being ycb 7289, probably because these tablets are referenced next to each other on wikipedia.
>4 is the length and 5 the diagonal. What is the breadth ?
Its size is not known.
4 times 4 is 16.
5 times 5 is 25.
You take 16 from 25 and there remains 9.
What times what shall I take in order to get 9 ?
3 times 3 is 9.
3 is the breadth.
That doesn't appear to be in your link – am I wrong? I only see numbers there and it appears to be talking about something else entirely.
did they know about the Pythagorean theorem, or did they know about the square root of two? It seems that people did puzzle over this quantity for a while.
Diogenes said that Pythagoras hated beans: “One should abstain from fava beans, since they are full of wind and take part in the soul, and if one abstains from them one’s stomach will be less noisy and one’s dreams will be less oppressive and calmer.”
I guess they hadn't yet figured out this little kernel of wisdom: Beans, beans, they're good for your heart. The more you eat, the more you fart. The more you fart the better you feel, so eat your beans with every meal.
This discussion leaves unclear the the role of Pythagoras of Samos in the discovery of the Pythagorean Theorem. The statement of the theorem appears to have been known before the time of Pythagoras. He is credited with giving the first proof of the theorem, within the framework of Greek mathematics. On that claim his fame rests, although no record of his proof has survived.
I was surprised to learn that U.S. President Garfield devised a proof of Pythagoras' theorem. It got me wondering what other U.S. Presidents had an aptitude for math.
Jefferson (3rd President) was quite fluent in geometry and surveying, designing his home in Monticello.
Herbert Hoover (31st) was a mining engineer.
Jimmy Carter (39th) was a nuclear engineer by training, having worked in the U.S. Navy's nuclear sub program.
George Washington was also a surveyor before his later career(s). It was a well-trodden path to wealth and status for people of middling backgrounds and aptitudes for math and law in the pre-revolutionary U.S.
My math is a bit rusty, for a moment I was wondering about the proof of his theorem but then I realized that I messed up his theorem with the angles in a triangle that sum to 180°, which does not always hold up.
If the angles in a triangle don't add up to 180o then I think that means the triangle is not on a flat plane. An example would be if you took a point on the equator of the earth, moved East or West 1/4 of the earth's circumference and took another point and then projected those two points North until they met at the pole, you would have a triangle with 3 interior right angles.
Names are important as they indicate things; maybe Triangle Theorems would be better to cover the properties of triangles; perhaps there are other ones which we missed discovering... hubris has no limits
basic geometry was probably fundamental to the architecture necessary to make civilizations
i'd bet that as far back as we can find large structure there would probably have been strong understandings of geometry to make them
plus, humans have been around for 200-300k years, what we can find is from ~12k-25k years ago at the very fringe of our investigations. no doubt people have been mathematically capable for longer than they've been able to take full advantage of the concepts they understand
It's interesting that Pythagoras gets credit for a theorem he may not have discovered, especially when there's proof that Babylonians knew it 1000 years earlier.
This challenges the idea that ancient Greek mathematicians were always ahead of others.
It’s interesting that people in a CS forum don’t seem to know what a theorem is. The tablet shows that the relationship was known, a theorem needs a proof
It's in the tradition of maths that things are named after the second person to prove them/make them well known. There are numerous examples, but one of my favourites is Venn diagrams, which were called (by Venn) "Eulerean Circles" because he took them from Euler. Everyone else is like "Nope - Venn Diagrams."
It's probably just as well because otherwise just about everything would be named after Gauss which would make learning maths even more difficult than it already is.
The Pythagoreans did make a number of important discoveries to do with number theory, the ratios between string lengths for various musical notes (eg twice as long is an octave lower etc), cosmology and some other results in geometry to do with the properties of various 3-d shapes and stuff.
The journal is named „Journal of Targeting, Measurement and Analysis for Marketing“, which may be the scientific equivalent of a clickbait blogspam site (but I haven’t checked more deeply).
I've seen this tablet before, and an not convinced that it is actually the Pythagorean theorem. Its just a tablet with some tick marks inscribed onto it, along with a circular looking thing. It's very much a stretch to say the person who etched those markings intended to express the Pythagorean theorem.
There was a point in time when I was very interested in ancient civilizations from Mesopotamia, but in more recent years I an way less interested in it. The scholarship in that field is just terrible. In my opinion, a lot of the stuff is on par with alien "investigators" and stuff like that, yet for some reason the general public sees the field as totally legit.
How is this different from seeing a fuzzy video of some lights in the sky and then coming to the conclusion that it is definitely a UFO? If you're so convinced that this carving definitely proves that the carver was intending to express the Pythagorean formula, then what is the evidence?
Some people's definition of "evidence" is different from my own. If somebody really wants to believe something, then just about anything qualifies as evidence. This is why UFO people consider literally every single fuzzy video as undeniable proof that aliens exist.
… because those “squiggles” are just “words” in a “language” you can’t “read”?
And if you could read it, you would find it contains a lot of relevant things, concluding with:
> … 1.414213, which is nothing other than the decimal value of the square root of 2, accurate to the nearest one hundred thousandth.
You might then think to yourself:
> The conclusion is inescapable. The Babylonians knew the relation between the length of the diagonal of a square and its side
Which is all clearly explained in the article you’re commenting on. Do you have anything else meaningful to add, beyond “it’s nuffin’ but squiggles mate” and “aliens”?
It's just a bunch of numbers scribbled onto a tablet. For all we know it could just be some guy writing down the number of sheep he is willing to sell to his neighbor or something. To say this tablet proves the Mesopotamian knew about Pythagorean's theorem is quite a stretch.
To the people who want to believe, there is nothing that can be said. Believe what you want.
Also, this tablet has no provenance. According to the wikipedia page on this tablet, it says "It is unknown where in Mesopotamia YBC 7289 comes from" Basically it just magically appeared one day. For all we know it could be faked. In any other field, this artifact would be ruled inauthentic. But in this field, for some reason it just doesn't matter.
Imagine for a moment that the people who created the tablet used a different number system than us, and also imagine that we knew that number system and could convert it.
Then those “bunch of numbers” becomes something else entirely. Specifically, they become a bunch of numbers that highly relate to the Pythagorean theorem.
I don’t mind the poor scholarship it offers opportunities to have better ideas. What I love about the ancients is that they are just like us with fewer objects.
Iamblichus’s “life of Pythagoras” [1] is worth a read as he had access to all the old sources now lost. The relationship between math and spirituality was very strong back then!
There are lots of fun stories that may be true but no one will ever know. In Diogenes Laertius’ “Lives of the Philosophers,” it is claimed that when Pythagoras made his discovery of what we call the Pythagorean theorem, he sacrificed 100 oxen (a hecatomb) [1]. As noted by Charles Dodgson (Lewis Carrol), “that would produce an inconvenient supply of meat” [3], especially for a vegetarian. Iamblichus, on the other hand, claims it was a single ox — and made of flour!
[1] Guthrie, K. S., & Fideler, D. R. (Eds.). (1987). The Pythagorean sourcebook and library: an anthology of ancient writings which relate to Pythagoras and Pythagorean philosophy. Red Wheel/Weiser.
[2] “he sacrificed a hecatomb, when he had discovered that the square of the hypotenuse of a right-angled triangle was equal to the squares of the sides containing the right angle.” DL, found in [1]
[3] Maor, E. (2019). The Pythagorean theorem: a 4,000-year history. Princeton University Press.