10 ms·

Pythagorean Theorem proof in a 2100 year old Chinese book

346 points by balele 10y ago

lacampbell 10y ago
Wouldn't something from that period be written with seal script? That script looks like modern Han characters with an archaic font - it's completely readable.
- 13of40 10y ago
  Someone commented on the site that it's a 17th century reproduction.
  - yeukhon 10y ago
    Also while those are Han, the language is sophisticated to regular Chinese readers. I can't even make sense of the words (individually yes) as a native Chinese speaker. A far as paper writing, I found [1] which said roughly 100 A.D. some paper production began. [1]: http://www.chinaheritagequarterly.org/tien-hsia.php?searchterm=020_chinese_book.inc&issue=020 http://www.chinaheritagequarterly.org/tien-hsia.php?searchte...
    - gus_massa 10y ago
      Can you make an annotated version? Nothing fancy, if you add the translation with paintbrush and host it in imgur, I'll be completely happy.
      - y2kenny 10y ago
        It's traditional chinese but it's a challenge. This is my best guess: http://imgur.com/a/rnXPW http://imgur.com/a/rnXPW Looks like they were using colour as variable/labels.
        gus_massa 10y ago
        Thanks! I replied in my other comment, because I made a previous explanation attempt but after looking at your translation I think that it was wrong and I wanted to highlight the differences. https://news.ycombinator.com/item?id=13953495 https://news.ycombinator.com/item?id=13953495
- kobeya 10y ago
  Seal script is significantly older than that. 2100 years ago is squarely in the middle of the Han dynasty, after the square script we are familiar with was already standardized.
- empath75 10y ago
  Keep in mind that the earliest copies of Euclid, Plato, etc, are often from several hundred or 1000 years later.
lacampbell 10y ago
Any ideas of it's authentic? China is hyper nationalist so I'm initially skeptical. Might be as fictitous as their GDP figures or sovereignty over Taiwan.
- deleted 10y ago
  [deleted]
- yeukhon 10y ago
  > sovereignty over Taiwan This is the claim that China and Taiwan should reunite since the PRC drove the Nationalist Party out of mainland China. It isn't fictitious, it's merely a political wish to merge the two from the PRC side. Similarly, some Nationalists in Taiwan are happy to take back mainland China, as much as the country is divided on whether Taiwan should remain as it is, or should declare its full independence.
  - PhasmaFelis 10y ago
    Doesn't the PRC's claim on Taiwan boil down to a combination of right-of-conquest ("Taiwan belonged to the Republic of China, we beat up the Republic of China, therefore Taiwan is ours") and Just Because? ("The PRC is the only legitimate Chinese government because the PRC says so") I'm very skeptical of purely historical claims of sovereignty. If the people of Taiwan legitimately vote to join the PRC, that's their right, but the PRC has no legitimate right to demand it.
    - my_first_acct 10y ago
      Descriptions of all possible points of view are presented in the Wikipedia article "Political status of Taiwan" [1]. Among these is the (fringe) theory that based on various post-WWII treaties, the US currently holds sovereignty over Taiwan [2] (a view not endorsed by the US government, by the way). [1] https://en.wikipedia.org/wiki/Political_status_of_Taiwan https://en.wikipedia.org/wiki/Political_status_of_Taiwan [2] https://en.wikipedia.org/wiki/Political_status_of_Taiwan#Arguments_for_United_States_sovereignty_claims https://en.wikipedia.org/wiki/Political_status_of_Taiwan#Arg...
chrishowlin 10y ago
I worry that China is outcompeting us.
- spuz 10y ago
  Who is "us"?
  - PhasmaFelis 10y ago
    This is what we call a "joke," playing on some Westerners' fears of modern Chinese global economic power.
- HillaryBriss 10y ago
  According to this editorial, China will be the global tech leader soon (if it isn't already): http://www.scmp.com/business/global-economy/article/2081771/be-afraid-china-path-global-technology-dominance http://www.scmp.com/business/global-economy/article/2081771/...
- jernfrost 10y ago
  Who gives a shit? It is almost always Americans making such statements, obsessed as they are about being number one. If Chine gets richer and more technologically advance that would be good for them and good for us westerners. It is not a zero sum game. Although to be realistic I don't think China will ever outcompete the US. Not because I don't want them to, but simply because so many fundamentals count in Americas favor and against China. China is facing a demographic catastrophe. The US can easily grow its population with immigration and has plenty of land and natural resources to do so.
  - crimsonalucard 10y ago
    Nothing wrong with worrying and wanting your home country to be the best either. Worrying leads to motivation which leads to competition which benefits everyone.
  - contingencies 10y ago
elevensies 10y ago
So if I'm understanding correctly, this is evidence that about 300 years after the death of Pythagoras, his proof "travelled" over the silk road to China?
- dmoy 10y ago
  Or it was independently recreated.
  - platz 10y ago
    It could've been used without being proved also
    - idreyn 10y ago
      The diagrams on the pages shown basically serve as a proof.
      - theoh 10y ago
        It's familiar from hands-on science museums to see a particular case of the Pythagorean theorem illustrated concretely with diagrams or liquid on a rotating plexiglass panel... But surely those aren't proofs, just illustrations. A proof needs to abstract over all possible right-angle triangles.
        greglindahl 10y ago
        Both diagrams are a proof over all right-angle triangles, if you know algebra. See the very first comment after the images.
        theoh 10y ago
        Hmmm. Maybe I'm just used to the convincing generality of the similarity argument (from Euclid, via Polya): https://math.berkeley.edu/~giventh/papers/eu.pdf https://math.berkeley.edu/~giventh/papers/eu.pdf But also, we can't trust geometrical diagrams: https://en.wikipedia.org/wiki/Missing_square_puzzle https://en.wikipedia.org/wiki/Missing_square_puzzle
partycoder 10y ago
It was true then... http://abstrusegoose.com/376 http://abstrusegoose.com/376 However paper itself was invented around the same time. How come bookbinding already existed then?
- cercatrova 10y ago
  Ah, Stigler's Law of Eponymy
whatnotests 10y ago
Why not 2500 years ago!?
moomin 10y ago
I know you can use the second diagram to prove the theorem, but it requires an algebraic argument (the other two squares aren't shown). Unless the text says something pretty interesting, this looks more like a proof just for a 3,4,5 triangle. The first diagram I can't get my head round at all, or is it proving something different?
- greglindahl 10y ago
  The top comment after the images of the two diagrams explains how the 1st diagram is another algebraic proof, similar to the 2nd diagram.
- jeffwass 10y ago
  The first and second diagrams are similar, just using the 'outside' vs 'inside' deconstruction. If we label the sides a=3, b=4, and c=5 : The first diagram shows that the big outer square of length (a+b) equals the area of the 4 triangles plus the rotated square of side c. (a+b)^2 = 2ab + c^2 [I'm not sure why they drew an inner 3x3 square on its own]. The second diagram puts the same four triangles inside the rotated square of side c, which leaves an extra smaller square of side (b-a) inside. c^2 = 2ab + (b-a)^2
  - moomin 10y ago
    Yeah, the inner square is confusing.
mariodiana 10y ago
My understanding is that the 3-4-5 relationship was known for a long, long time before Pythagorus. I think the ancient Egyptians knew of the relationship. But that's not the same thing as a proof: meaning, the fact has been integrated with the fundamentals of a wider body of knowledge. I see the illustration, but I don't understand how that's a "proof."
- dheera 10y ago
  Also, Pythagoras was ~2500 years ago, before this thing.
  - ekianjo 10y ago
    Exactly, by the time this book existed 2100 years ago, the idea should have been traveling all over the place anyway. It's not like ancient civilizations were not connected at all.
    - pjmlp 10y ago
      Yes, people travelled a lot, specially merchants, that is also how Alexandria got many of its manuscripts. The biggest difference was that nothing was assured about the safety of the travel.
- Houshalter 10y ago
  >I see the illustration, but I don't understand how that's a "proof." That's an illustration of the proof. Here is a more modern, animated example: https://upload.wikimedia.org/wikipedia/commons/2/29/Pythagorean_Theorem_Proof.gif https://upload.wikimedia.org/wikipedia/commons/2/29/Pythagor... The Chinese proof looks more like this one though: https://i.stack.imgur.com/Ylzhg.gif https://i.stack.imgur.com/Ylzhg.gif There is also explanation in the annotations.
  - gravypod 10y ago
    It's mind blowing when I see a new image like that and it completely changes how I understand the topic. I wish I could find more of those.
gus_massa 10y ago
I don't know Chinese, but I don't like the explanation in English at the side. I think that both diagrams are part of the same proof, not two independent proofs. I agree with the first comment by Martin Cohen http://disq.us/p/1h6y5qx http://disq.us/p/1h6y5qx > The way I read it is that the first diagram says that(a+b)^2 = 4(ab/2)+c^2 and the second says that c^2 = 4(ab/2)+(a-b)^2. Both of these simplify to c^2 = a^2+b^2. I only have a small proposed stylistic change that is to keep the are of the triangles as unknown values, let's call it T. So the first diagram shows that (a+b)^2 = 4T + c^2 and the second that c^2 = 4T + (a-b)^2. And simplifying you get simplify to c^2 = a^2 + b^2. Is there a transcript and translation of the Chinese text?
- y2kenny 10y ago
  I made this for another comment. Here you go: http://imgur.com/a/rnXPW http://imgur.com/a/rnXPW
  - gus_massa 10y ago
    Thanks! Copy of your previous comments in other thread for context: > Also while those are Han, the language is sophisticated to regular Chinese readers. I can't even make sense of the words (individually yes) as a native Chinese speaker. > It's traditional chinese but it's a challenge. This is my best guess: > Looks like they were using colour as variable/labels. -- Most of my previous guess is wrong :( . In the second diagrams, they use that the area of the triangles is 6, they are not ignoring the number. In both pages they ignore the outer triangles. They are not painted in neither graph. If they were using "my" solution I'd expect that the outer triangles in the first diagram were painted with the same color as the inner triangles in the second diagram. The first diagram makes no sense. It looks like a graphic to show a numerical equality, but the geometric properties give no insight of the numerical properties. Perhaps it was a usual writing style? There is no 49 in the first graphic (nor in the second) so my previous explanation doesn't fit with the text. -- My second unsupported attempt of translation is: [Second diagram] The area of each red triangle is 6 [because 4 * 3 / 2 = 6]. The side of the inner yellow square is 1 [because 4 - 3 = 1], so the area is 1 [ 1 * 1 = 1]. The area of the [rotated] square is the sum of the four red triangles and the small yellow square, it is 25 [because 6 + 6 + 6 + 6 + 1 = 25]. So the side of the [rotated] square is 5. Then the sides of the triangle at the bottom are 3, 4, 5. [First diagram] If we subtract the area of a square with side 5 and a square of side 3 [painted in cyan], we get the area [painted in yellow] that is 16. [Notice that 16 is the area of a square of side 4, that we never draw here.] -- My current opinion: The second diagram is a proof that in a right triangle with legs 3 and 4, the hypotenuse is 5. I think that this can be extended to any other Pythagorean triple, but one at a time. This doesn't look like a general proof for all of them. This looks more like the verification of an example than a general proof. The first diagram is an auxiliary construction for the second diagram. It's not very enlightling. The explanation of the side added in Fermat's library is no a transcript or an explanation of the proof in the main text. It's another proof of the Pythagorean Theorem with a somewhat similar graphic, but it's unrelated.
jernfrost 10y ago
From what I understand, the Chinese had often recorded many facts about nature and mathematics very early and accurately. However they never really created much laws. Laws of nature was a western invention. E.g. they observed accurately planetary movements, but never attempted to formulate the laws governing those movements. That sort of thinking was crucial for scientific thinking.
- ue_ 10y ago
  This is an interesting idea, even if it does play into the "exotic Oriental thinking" meme. I've seen this line of thinking usually attached with rubbish like "The Westerner wants to control the world, the Eastener only wants to explain it." However I think that first making observations and then the crucial step of extrapolating and making predictions based on those is as good as any formally named 'law'.
  - jernfrost 10y ago
    Reading up on Copernicus, Kepler and astronomy gives a good idea I think about what I am getting at. Extrapolation isn't that easy to do with e.g. astronomy. Orbits don't go in straight predictable lines. They are ellipses where the speed will vary depending on where you are in the orbit. Kepler found that the are covered per time unit was constant, thus implying that when the orbits were closer to the sun they would speed up. I guess you could say extrapolation was what Kepler tried to do but it took him a lot of work to figure it out. The Chinese while very advance in many respect did not try doing that. What I've read is that they did not have the Judeo-Christian belief that the mind of God could be learned. The laws governing the motions of the heavens was assume inscrutable to the human mind.
- sn41 10y ago
  Maybe true. The emphasis was on calculations. For example, there was a heliocentric model in India [1], which was too complicated to come from a "simple" mechanical model, but was quite accurate. But I think we should not be too eager to jump on to "science is only from laws" bandwagon. Courses on the philosophy of science take this dogma to absurd extremes, ignoring how current science is often done. There is enough science done today where the laws are only vague, but we do predict useful stuff from simulations - climate models and earthquake modelling being the most prominent - despite not having "simple" laws behind them. As Freeman Dyson often observes, physicists still don't have a good explanation of why a bicycle doesn't topple when in motion - but we've been riding them for a century and half [2]. I think trial-and-error, and simulations also count as sophisticated science. It's more than a tad racist to insist that Egyptians, Babylonians, Chinese and Indians, as well as medieval Arab scientists were all child races incapable of truly knowing what they were doing. [1] https://en.wikipedia.org/wiki/Nilakantha_Somayaji https://en.wikipedia.org/wiki/Nilakantha_Somayaji [2] https://www.wired.com/1998/02/dyson/ https://www.wired.com/1998/02/dyson/
gnipgnip 10y ago
Much of modern history is a throwback of the colonial times when the "barbarians" from the West were trying to search for some great past of their people. Thus much of what is attributed to the Greeks and Romans today should belong to the civilizations of Asia (India, China, and Iran). Pythagorean theorem has nothing to do with Pythagoras or the Greeks even [3] ! There is nice interview of Manjul Bhargava (surrounding a dredged up controversy in India) that delves into the actual history (rather than mindlessly parroting the liberal propaganda). http://www.rediff.com/news/special/did-india-discover-pythogoras-theorem-a-top-mathematician-answers/20150109.htm http://www.rediff.com/news/special/did-india-discover-pythog... I'm glad to hear the Chinese school system does not brainwash people as does the Indian one. Then again, India is a country created by pedophiles [1] and drug-runners [2]. [1] Gandhi is well-known for giving women enemas and sleeping naked with under aged girls. This more than qualifies for the contemporary semantic definitions. There are many other rumors that one can find on the intertubes. [2] Tagores, Tatas, Birlas... all of them made their fortunes in the British opium trade... ironically to China (this is well-documented; even on Wiki). Interestingly, two of India's Nobels come from this coterie... and we're forced to sing a poorly written poem by one of them, day in day out, by the "Indian nationalists". The banking overlords have also made sure Gandhi remains in our pockets. Eesh! what a tragic state of affairs. Also guess who called Gandhi a "Mahatma" ? Interestingly neither of them particularly cared for India becoming independent (if you look at their writing). Amusingly there is some controversy about India's anthem being written to commemorate George V's coronation. There are counterarguments for this, but the anthem is referring to a person like a King or a deva - the latter possibility would be very very weird to be honest [4]. Not surprising he got a Knighthood and a Nobel. All said, calling India a nation of idiots at this point would be a compliment. [3] From the horse's mouth, http://www.bbc.com/news/business-27923256 http://www.bbc.com/news/business-27923256 [4] "Bhagya Vidhata" is very very Abrahamic (whose view of God is that of a ill-tempered King). In canonical circles, a deva is mostly for support "उपदि", not someone who decides the outcomes or luck of people - this is completely antithetical to Indian thought. Even if we circulate in non-scholarly circles, the semantics implied by Jana Gana Mana would still not fit well to any one deva. P.S: What are you blue-pill muchers downvoting me for ? Can't you read? These things are referenced!
- quakeguy 10y ago
good_vibes 10y ago
Couldn't have said it better myself. I think about this stuff a lot, how much ancient and modern overlap, how much Eastern and western overlap, yet how people still want to divide everything. "The intuitive mind is a sacred gift and the rational mind is a faithful servant. We have created a society that honors the servant and has forgotten the gift." - Albert Einstein "The people in the Indian countryside don’t use their intellect like we do, they use their intuition instead, and the intuition is far more developed than in the rest of the world… Intuition is a very powerful thing, more powerful than intellect, in my opinion. That’s had a big impact on my work. Western rational thought is not an innate human characteristic, it is learned and it is the great achievement of Western civilization. In the villages of India, they never learned it. They learned something else, which is in some ways just as valuable but in other ways is not. That’s the power of intuition and experiential wisdom." - Steve Jobs Carl Sagan was really keen on seeing 'the bigger picture' too, can't find the particular quote I was looking for. Maybe later.
- bmh100 10y ago
  These divisions are very important. As long as mystical beliefs towards Eastern thought are used to cause real harm through herbal remedies and supplements, the public must be taught to give it less importance. Among critical thinkers, it is valuable to look for similarities in scientific histories.
  - good_vibes 10y ago
    Big Pharm and meat-based diet do a lot more harm to the entire biosphere and individuals than Ayurvedic medicine and yoga. Don't even get me started. I don't know much about Eastern medicine outside of India to be honest. One exception: Chinese people buy tiger bone Viagra, elephant tusk art, rhino horn something, and shark fin soup. I was born Hindu but became closer to Buddhist as I learned more about the scientific revolution and how it built the modern world.
    - E6300 10y ago
      > Big Pharm and meat-based diet do a lot more harm to the entire biosphere and individuals than Ayurvedic medicine and yoga. Cars kill more people per year than heroin, therefore heroin is healthier than cars. Let's wait until pseudo-scientific treatments become multi-billion dollar industries and then we'll talk about what does more harm to what.
happy-go-lucky 10y ago
https://en.wikipedia.org/wiki/Pythagorean_theorem https://en.wikipedia.org/wiki/Pythagorean_theorem > In India, the Baudhayana Sulba Sutra, the dates of which are given variously as between the 8th and 5th century BC, contains a list of Pythagorean triples discovered algebraically, a statement of the Pythagorean theorem, and a geometrical proof of the Pythagorean theorem for an isosceles right triangle. https://en.wikipedia.org/wiki/Baudhayana_sutras https://en.wikipedia.org/wiki/Baudhayana_sutras > A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together. > The lines are to be referring to a rectangle, although some interpretations consider this to refer to a square. In either case, it states that the square of the hypotenuse equals the sum of the squares of the sides. > If this refers to a rectangle, it is the earliest recorded statement of the Pythagorean theorem.
- ajarmst 10y ago
  It's generally understood that the theorem, and in particular the 1/1/2 and 3/4/5 integral triangles were known prehistorically and independently 'discovered' in several places. It's not possible to know what the earliest recorded statement was, because we only have tiny fragments (more commonly, fragments of copies of fragments of later documentation of oral traditions) of what actually was recorded that long ago. Finding such a statement only says that its the earliest we're aware of, and even then its always contingent on a variety of factors. These sorts of discussions always seem to devolve into "10,000 generations ago, someone who lived near where some of my ancestors later lived might have known more mathematics than people who lived near where your ancestors lived." [The 'That means I'm better than you.' is usually not explicit] Which is boring.
ajarmst 10y ago
It's probably of value to note that that diagrams that everyone is talking about are not part of the original text.

346 points by balele 10y ago

Wouldn't something from that period be written with seal script? That script looks like modern Han characters with an archaic font - it's completely readable.

13of40 10y ago
Someone commented on the site that it's a 17th century reproduction.
- yeukhon 10y ago
  Also while those are Han, the language is sophisticated to regular Chinese readers. I can't even make sense of the words (individually yes) as a native Chinese speaker. A far as paper writing, I found [1] which said roughly 100 A.D. some paper production began. [1]: http://www.chinaheritagequarterly.org/tien-hsia.php?searchterm=020_chinese_book.inc&issue=020 http://www.chinaheritagequarterly.org/tien-hsia.php?searchte...
  - gus_massa 10y ago
    Can you make an annotated version? Nothing fancy, if you add the translation with paintbrush and host it in imgur, I'll be completely happy.
    - y2kenny 10y ago
      It's traditional chinese but it's a challenge. This is my best guess: http://imgur.com/a/rnXPW http://imgur.com/a/rnXPW Looks like they were using colour as variable/labels.
      - gus_massa 10y ago
        Thanks! I replied in my other comment, because I made a previous explanation attempt but after looking at your translation I think that it was wrong and I wanted to highlight the differences. https://news.ycombinator.com/item?id=13953495 https://news.ycombinator.com/item?id=13953495
kobeya 10y ago
Seal script is significantly older than that. 2100 years ago is squarely in the middle of the Han dynasty, after the square script we are familiar with was already standardized.
empath75 10y ago
Keep in mind that the earliest copies of Euclid, Plato, etc, are often from several hundred or 1000 years later.

lacampbell 10y ago

Any ideas of it's authentic? China is hyper nationalist so I'm initially skeptical. Might be as fictitous as their GDP figures or sovereignty over Taiwan.

deleted 10y ago
[deleted]
yeukhon 10y ago
> sovereignty over Taiwan This is the claim that China and Taiwan should reunite since the PRC drove the Nationalist Party out of mainland China. It isn't fictitious, it's merely a political wish to merge the two from the PRC side. Similarly, some Nationalists in Taiwan are happy to take back mainland China, as much as the country is divided on whether Taiwan should remain as it is, or should declare its full independence.
- PhasmaFelis 10y ago
  Doesn't the PRC's claim on Taiwan boil down to a combination of right-of-conquest ("Taiwan belonged to the Republic of China, we beat up the Republic of China, therefore Taiwan is ours") and Just Because? ("The PRC is the only legitimate Chinese government because the PRC says so") I'm very skeptical of purely historical claims of sovereignty. If the people of Taiwan legitimately vote to join the PRC, that's their right, but the PRC has no legitimate right to demand it.
  - my_first_acct 10y ago
    Descriptions of all possible points of view are presented in the Wikipedia article "Political status of Taiwan" [1]. Among these is the (fringe) theory that based on various post-WWII treaties, the US currently holds sovereignty over Taiwan [2] (a view not endorsed by the US government, by the way). [1] https://en.wikipedia.org/wiki/Political_status_of_Taiwan https://en.wikipedia.org/wiki/Political_status_of_Taiwan [2] https://en.wikipedia.org/wiki/Political_status_of_Taiwan#Arguments_for_United_States_sovereignty_claims https://en.wikipedia.org/wiki/Political_status_of_Taiwan#Arg...

chrishowlin 10y ago

I worry that China is outcompeting us.

spuz 10y ago
Who is "us"?
- PhasmaFelis 10y ago
  This is what we call a "joke," playing on some Westerners' fears of modern Chinese global economic power.
HillaryBriss 10y ago
According to this editorial, China will be the global tech leader soon (if it isn't already): http://www.scmp.com/business/global-economy/article/2081771/be-afraid-china-path-global-technology-dominance http://www.scmp.com/business/global-economy/article/2081771/...
jernfrost 10y ago
Who gives a shit? It is almost always Americans making such statements, obsessed as they are about being number one. If Chine gets richer and more technologically advance that would be good for them and good for us westerners. It is not a zero sum game. Although to be realistic I don't think China will ever outcompete the US. Not because I don't want them to, but simply because so many fundamentals count in Americas favor and against China. China is facing a demographic catastrophe. The US can easily grow its population with immigration and has plenty of land and natural resources to do so.
- crimsonalucard 10y ago
  Nothing wrong with worrying and wanting your home country to be the best either. Worrying leads to motivation which leads to competition which benefits everyone.
- contingencies 10y ago

elevensies 10y ago

So if I'm understanding correctly, this is evidence that about 300 years after the death of Pythagoras, his proof "travelled" over the silk road to China?

dmoy 10y ago
Or it was independently recreated.
- platz 10y ago
  It could've been used without being proved also
  - idreyn 10y ago
    The diagrams on the pages shown basically serve as a proof.
    - theoh 10y ago
      It's familiar from hands-on science museums to see a particular case of the Pythagorean theorem illustrated concretely with diagrams or liquid on a rotating plexiglass panel... But surely those aren't proofs, just illustrations. A proof needs to abstract over all possible right-angle triangles.
      - greglindahl 10y ago
        Both diagrams are a proof over all right-angle triangles, if you know algebra. See the very first comment after the images.
        theoh 10y ago
        Hmmm. Maybe I'm just used to the convincing generality of the similarity argument (from Euclid, via Polya): https://math.berkeley.edu/~giventh/papers/eu.pdf https://math.berkeley.edu/~giventh/papers/eu.pdf But also, we can't trust geometrical diagrams: https://en.wikipedia.org/wiki/Missing_square_puzzle https://en.wikipedia.org/wiki/Missing_square_puzzle

partycoder 10y ago

It was true then... http://abstrusegoose.com/376 http://abstrusegoose.com/376 However paper itself was invented around the same time. How come bookbinding already existed then?

cercatrova 10y ago
Ah, Stigler's Law of Eponymy

whatnotests 10y ago

Why not 2500 years ago!?

moomin 10y ago

I know you can use the second diagram to prove the theorem, but it requires an algebraic argument (the other two squares aren't shown). Unless the text says something pretty interesting, this looks more like a proof just for a 3,4,5 triangle. The first diagram I can't get my head round at all, or is it proving something different?

greglindahl 10y ago
The top comment after the images of the two diagrams explains how the 1st diagram is another algebraic proof, similar to the 2nd diagram.
jeffwass 10y ago
The first and second diagrams are similar, just using the 'outside' vs 'inside' deconstruction. If we label the sides a=3, b=4, and c=5 : The first diagram shows that the big outer square of length (a+b) equals the area of the 4 triangles plus the rotated square of side c. (a+b)^2 = 2ab + c^2 [I'm not sure why they drew an inner 3x3 square on its own]. The second diagram puts the same four triangles inside the rotated square of side c, which leaves an extra smaller square of side (b-a) inside. c^2 = 2ab + (b-a)^2
- moomin 10y ago
  Yeah, the inner square is confusing.

mariodiana 10y ago

My understanding is that the 3-4-5 relationship was known for a long, long time before Pythagorus. I think the ancient Egyptians knew of the relationship. But that's not the same thing as a proof: meaning, the fact has been integrated with the fundamentals of a wider body of knowledge. I see the illustration, but I don't understand how that's a "proof."

dheera 10y ago
Also, Pythagoras was ~2500 years ago, before this thing.
- ekianjo 10y ago
  Exactly, by the time this book existed 2100 years ago, the idea should have been traveling all over the place anyway. It's not like ancient civilizations were not connected at all.
  - pjmlp 10y ago
    Yes, people travelled a lot, specially merchants, that is also how Alexandria got many of its manuscripts. The biggest difference was that nothing was assured about the safety of the travel.
Houshalter 10y ago
>I see the illustration, but I don't understand how that's a "proof." That's an illustration of the proof. Here is a more modern, animated example: https://upload.wikimedia.org/wikipedia/commons/2/29/Pythagorean_Theorem_Proof.gif https://upload.wikimedia.org/wikipedia/commons/2/29/Pythagor... The Chinese proof looks more like this one though: https://i.stack.imgur.com/Ylzhg.gif https://i.stack.imgur.com/Ylzhg.gif There is also explanation in the annotations.
- gravypod 10y ago
  It's mind blowing when I see a new image like that and it completely changes how I understand the topic. I wish I could find more of those.

gus_massa 10y ago

I don't know Chinese, but I don't like the explanation in English at the side. I think that both diagrams are part of the same proof, not two independent proofs. I agree with the first comment by Martin Cohen http://disq.us/p/1h6y5qx http://disq.us/p/1h6y5qx > The way I read it is that the first diagram says that(a+b)^2 = 4(ab/2)+c^2 and the second says that c^2 = 4(ab/2)+(a-b)^2. Both of these simplify to c^2 = a^2+b^2. I only have a small proposed stylistic change that is to keep the are of the triangles as unknown values, let's call it T. So the first diagram shows that (a+b)^2 = 4T + c^2 and the second that c^2 = 4T + (a-b)^2. And simplifying you get simplify to c^2 = a^2 + b^2. Is there a transcript and translation of the Chinese text?

y2kenny 10y ago
I made this for another comment. Here you go: http://imgur.com/a/rnXPW http://imgur.com/a/rnXPW
- gus_massa 10y ago
  Thanks! Copy of your previous comments in other thread for context: > Also while those are Han, the language is sophisticated to regular Chinese readers. I can't even make sense of the words (individually yes) as a native Chinese speaker. > It's traditional chinese but it's a challenge. This is my best guess: > Looks like they were using colour as variable/labels. -- Most of my previous guess is wrong :( . In the second diagrams, they use that the area of the triangles is 6, they are not ignoring the number. In both pages they ignore the outer triangles. They are not painted in neither graph. If they were using "my" solution I'd expect that the outer triangles in the first diagram were painted with the same color as the inner triangles in the second diagram. The first diagram makes no sense. It looks like a graphic to show a numerical equality, but the geometric properties give no insight of the numerical properties. Perhaps it was a usual writing style? There is no 49 in the first graphic (nor in the second) so my previous explanation doesn't fit with the text. -- My second unsupported attempt of translation is: [Second diagram] The area of each red triangle is 6 [because 4 * 3 / 2 = 6]. The side of the inner yellow square is 1 [because 4 - 3 = 1], so the area is 1 [ 1 * 1 = 1]. The area of the [rotated] square is the sum of the four red triangles and the small yellow square, it is 25 [because 6 + 6 + 6 + 6 + 1 = 25]. So the side of the [rotated] square is 5. Then the sides of the triangle at the bottom are 3, 4, 5. [First diagram] If we subtract the area of a square with side 5 and a square of side 3 [painted in cyan], we get the area [painted in yellow] that is 16. [Notice that 16 is the area of a square of side 4, that we never draw here.] -- My current opinion: The second diagram is a proof that in a right triangle with legs 3 and 4, the hypotenuse is 5. I think that this can be extended to any other Pythagorean triple, but one at a time. This doesn't look like a general proof for all of them. This looks more like the verification of an example than a general proof. The first diagram is an auxiliary construction for the second diagram. It's not very enlightling. The explanation of the side added in Fermat's library is no a transcript or an explanation of the proof in the main text. It's another proof of the Pythagorean Theorem with a somewhat similar graphic, but it's unrelated.

jernfrost 10y ago

From what I understand, the Chinese had often recorded many facts about nature and mathematics very early and accurately. However they never really created much laws. Laws of nature was a western invention. E.g. they observed accurately planetary movements, but never attempted to formulate the laws governing those movements. That sort of thinking was crucial for scientific thinking.

ue_ 10y ago
This is an interesting idea, even if it does play into the "exotic Oriental thinking" meme. I've seen this line of thinking usually attached with rubbish like "The Westerner wants to control the world, the Eastener only wants to explain it." However I think that first making observations and then the crucial step of extrapolating and making predictions based on those is as good as any formally named 'law'.
- jernfrost 10y ago
  Reading up on Copernicus, Kepler and astronomy gives a good idea I think about what I am getting at. Extrapolation isn't that easy to do with e.g. astronomy. Orbits don't go in straight predictable lines. They are ellipses where the speed will vary depending on where you are in the orbit. Kepler found that the are covered per time unit was constant, thus implying that when the orbits were closer to the sun they would speed up. I guess you could say extrapolation was what Kepler tried to do but it took him a lot of work to figure it out. The Chinese while very advance in many respect did not try doing that. What I've read is that they did not have the Judeo-Christian belief that the mind of God could be learned. The laws governing the motions of the heavens was assume inscrutable to the human mind.
sn41 10y ago
Maybe true. The emphasis was on calculations. For example, there was a heliocentric model in India [1], which was too complicated to come from a "simple" mechanical model, but was quite accurate. But I think we should not be too eager to jump on to "science is only from laws" bandwagon. Courses on the philosophy of science take this dogma to absurd extremes, ignoring how current science is often done. There is enough science done today where the laws are only vague, but we do predict useful stuff from simulations - climate models and earthquake modelling being the most prominent - despite not having "simple" laws behind them. As Freeman Dyson often observes, physicists still don't have a good explanation of why a bicycle doesn't topple when in motion - but we've been riding them for a century and half [2]. I think trial-and-error, and simulations also count as sophisticated science. It's more than a tad racist to insist that Egyptians, Babylonians, Chinese and Indians, as well as medieval Arab scientists were all child races incapable of truly knowing what they were doing. [1] https://en.wikipedia.org/wiki/Nilakantha_Somayaji https://en.wikipedia.org/wiki/Nilakantha_Somayaji [2] https://www.wired.com/1998/02/dyson/ https://www.wired.com/1998/02/dyson/

gnipgnip 10y ago

Much of modern history is a throwback of the colonial times when the "barbarians" from the West were trying to search for some great past of their people. Thus much of what is attributed to the Greeks and Romans today should belong to the civilizations of Asia (India, China, and Iran). Pythagorean theorem has nothing to do with Pythagoras or the Greeks even [3] ! There is nice interview of Manjul Bhargava (surrounding a dredged up controversy in India) that delves into the actual history (rather than mindlessly parroting the liberal propaganda). http://www.rediff.com/news/special/did-india-discover-pythogoras-theorem-a-top-mathematician-answers/20150109.htm http://www.rediff.com/news/special/did-india-discover-pythog... I'm glad to hear the Chinese school system does not brainwash people as does the Indian one. Then again, India is a country created by pedophiles [1] and drug-runners [2]. [1] Gandhi is well-known for giving women enemas and sleeping naked with under aged girls. This more than qualifies for the contemporary semantic definitions. There are many other rumors that one can find on the intertubes. [2] Tagores, Tatas, Birlas... all of them made their fortunes in the British opium trade... ironically to China (this is well-documented; even on Wiki). Interestingly, two of India's Nobels come from this coterie... and we're forced to sing a poorly written poem by one of them, day in day out, by the "Indian nationalists". The banking overlords have also made sure Gandhi remains in our pockets. Eesh! what a tragic state of affairs. Also guess who called Gandhi a "Mahatma" ? Interestingly neither of them particularly cared for India becoming independent (if you look at their writing). Amusingly there is some controversy about India's anthem being written to commemorate George V's coronation. There are counterarguments for this, but the anthem is referring to a person like a King or a deva - the latter possibility would be very very weird to be honest [4]. Not surprising he got a Knighthood and a Nobel. All said, calling India a nation of idiots at this point would be a compliment. [3] From the horse's mouth, http://www.bbc.com/news/business-27923256 http://www.bbc.com/news/business-27923256 [4] "Bhagya Vidhata" is very very Abrahamic (whose view of God is that of a ill-tempered King). In canonical circles, a deva is mostly for support "उपदि", not someone who decides the outcomes or luck of people - this is completely antithetical to Indian thought. Even if we circulate in non-scholarly circles, the semantics implied by Jana Gana Mana would still not fit well to any one deva. P.S: What are you blue-pill muchers downvoting me for ? Can't you read? These things are referenced!

quakeguy 10y ago

good_vibes 10y ago

Couldn't have said it better myself. I think about this stuff a lot, how much ancient and modern overlap, how much Eastern and western overlap, yet how people still want to divide everything. "The intuitive mind is a sacred gift and the rational mind is a faithful servant. We have created a society that honors the servant and has forgotten the gift." - Albert Einstein "The people in the Indian countryside don’t use their intellect like we do, they use their intuition instead, and the intuition is far more developed than in the rest of the world… Intuition is a very powerful thing, more powerful than intellect, in my opinion. That’s had a big impact on my work. Western rational thought is not an innate human characteristic, it is learned and it is the great achievement of Western civilization. In the villages of India, they never learned it. They learned something else, which is in some ways just as valuable but in other ways is not. That’s the power of intuition and experiential wisdom." - Steve Jobs Carl Sagan was really keen on seeing 'the bigger picture' too, can't find the particular quote I was looking for. Maybe later.

bmh100 10y ago
These divisions are very important. As long as mystical beliefs towards Eastern thought are used to cause real harm through herbal remedies and supplements, the public must be taught to give it less importance. Among critical thinkers, it is valuable to look for similarities in scientific histories.
- good_vibes 10y ago
  Big Pharm and meat-based diet do a lot more harm to the entire biosphere and individuals than Ayurvedic medicine and yoga. Don't even get me started. I don't know much about Eastern medicine outside of India to be honest. One exception: Chinese people buy tiger bone Viagra, elephant tusk art, rhino horn something, and shark fin soup. I was born Hindu but became closer to Buddhist as I learned more about the scientific revolution and how it built the modern world.
  - E6300 10y ago
    > Big Pharm and meat-based diet do a lot more harm to the entire biosphere and individuals than Ayurvedic medicine and yoga. Cars kill more people per year than heroin, therefore heroin is healthier than cars. Let's wait until pseudo-scientific treatments become multi-billion dollar industries and then we'll talk about what does more harm to what.

happy-go-lucky 10y ago

https://en.wikipedia.org/wiki/Pythagorean_theorem https://en.wikipedia.org/wiki/Pythagorean_theorem > In India, the Baudhayana Sulba Sutra, the dates of which are given variously as between the 8th and 5th century BC, contains a list of Pythagorean triples discovered algebraically, a statement of the Pythagorean theorem, and a geometrical proof of the Pythagorean theorem for an isosceles right triangle. https://en.wikipedia.org/wiki/Baudhayana_sutras https://en.wikipedia.org/wiki/Baudhayana_sutras > A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together. > The lines are to be referring to a rectangle, although some interpretations consider this to refer to a square. In either case, it states that the square of the hypotenuse equals the sum of the squares of the sides. > If this refers to a rectangle, it is the earliest recorded statement of the Pythagorean theorem.

ajarmst 10y ago
It's generally understood that the theorem, and in particular the 1/1/2 and 3/4/5 integral triangles were known prehistorically and independently 'discovered' in several places. It's not possible to know what the earliest recorded statement was, because we only have tiny fragments (more commonly, fragments of copies of fragments of later documentation of oral traditions) of what actually was recorded that long ago. Finding such a statement only says that its the earliest we're aware of, and even then its always contingent on a variety of factors. These sorts of discussions always seem to devolve into "10,000 generations ago, someone who lived near where some of my ancestors later lived might have known more mathematics than people who lived near where your ancestors lived." [The 'That means I'm better than you.' is usually not explicit] Which is boring.

ajarmst 10y ago

It's probably of value to note that that diagrams that everyone is talking about are not part of the original text.