
Pythagorean Theorem proof in a 2100 year old Chinese book - balele
http://fermatslibrary.com/s/pythagorean-theorem-proof-in-zhoubi-suanjing#email-newsletter
======
jernfrost
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.

~~~
sn41
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/)

~~~
marmaduke
Feyerabend's Against Method goes further: examining crucial discoveries in
science reveals no pattern which could be called a scientific method. Everyone
sort of freestyles it or does what works.

I think modern western science discounts other sources of knowledge, because
it is not just a platonic ideal but a sort of politics of the scientists,
which they need to justify the choices they make.

~~~
vixen99
This is waffle. 'Western Science' is an abstraction. The 'official position of
Western Science' is not an accredited publication as far as I know. More to
the point, if you have criticisms of individual scientists (the ones who
contribute to science [ i.e., people all nationalities and places on Earth) ]
who 'discount other sources of knowledge' please be specific. They may or may
not have a point. We can't know unless you specify.

~~~
marmaduke
Too much troll, didn't read

------
elevensies
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?

~~~
jacobolus
There’s evidence that the “Pythagorean Theorem” was known in Mesopotamia a
millennium before Pythagoras. It was likely widely known in the ancient world.

The Greeks wrote their mathematical ideas in a different way than other
cultures, so the answers to these questions aren’t completely clear-cut.

~~~
kobeya
But was the theorem proved?

~~~
jacobolus
It’s really hard to say. Cuneiform on clay tablets was mostly records of
finished computations, with the scratch work done with some kind of physical
manipulation of tokens or mentally, and most of the instruction and culture
transmitted orally.

The Sumerian/Babylonian style of written records did not match, say, Euclid’s
_Elements_. (They didn’t write books full of precise definitions and proofs.
They didn’t leave behind textbooks, only problem sets and worked examples.)
But note that neither did the style of the Pythagoreans necessarily, and the
notion that Pythagoras or his disciples made a proof comes from written
records from hundreds of years later.

------
mariodiana
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."

~~~
Houshalter
>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/Pythagor...](https://upload.wikimedia.org/wikipedia/commons/2/29/Pythagorean_Theorem_Proof.gif)
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
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.

~~~
pash
Search for the phrase "proofs without words" [0, 1]. There's long been a
column in _Mathematics Magazine_ by that name featuring proofs in this style,
and there is a book of the same title published by the Mathematical
Association of America (which I think collects proofs from the magazine
column). Lots of examples from other sources will also show up in a web
search.

0\.
[https://encrypted.google.com/search?hl=en&q=proofs%20without...](https://encrypted.google.com/search?hl=en&q=proofs%20without%20words)

1\. Image results:
[https://encrypted.google.com/search?q=proofs+without+words&n...](https://encrypted.google.com/search?q=proofs+without+words&num=30&hl=en&prmd=visn&source=lnms&tbm=isch&sa=X&ved=0ahUKEwiC6Kf95vDSAhVK4YMKHZpjCPIQ_AUICCgC&biw=1024&bih=704)

------
happy-go-lucky
[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
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.

------
lacampbell
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
Someone commented on the site that it's a 17th century reproduction.

~~~
yeukhon
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?searchte...](http://www.chinaheritagequarterly.org/tien-
hsia.php?searchterm=020_chinese_book.inc&issue=020)

~~~
gus_massa
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
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
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)

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

------
gus_massa
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
I made this for another comment. Here you go:
[http://imgur.com/a/rnXPW](http://imgur.com/a/rnXPW)

~~~
gus_massa
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.

~~~
y2kenny
I couldn't translate the middle left vertical sentence on the first diagram
because it's a bit too faded. (It said something about the corner...) There's
also references about 'this layer/level' all over the place that make me think
there may be some paper folding involve but it could simply mean 'this layer
is solid yellow' and so on.

------
moomin
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?

~~~
jeffwass
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
Yeah, the inner square is confusing.

------
partycoder
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
Ah, Stigler's Law of Eponymy

------
whatnotests
Why not 2500 years ago!?

------
good_vibes
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
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.

~~~
arjo129
Last I checked homeopathy was western. i think your bias towards eastern
medicine would be better founded if you were to say faith healing rather than
eastern. Within eastern medicine one also finds people who used early forms of
experimentation to derive working cures. Examples of working medical cures
include boswellia used to treat bursitis, artemisin used to treat malaria.
Also both Chinese and Indians had developed anaesthetic before Europeans.
Similarly the procedure of rhinoplasty was developed in India. When the
British came in western surgeons studied the technique and took it back to
Europe. In fact hygiene was a critical part of the ayurvedic surgeons life
well before Europeans realized the importance of hygiene in surgery. Of course
there are a lot bogus stuff in eastern cures as well but in the west you have
stuff like homeopathy. I think the key take away is that you cannot divide by
east and west. We are after all human and each culture has their own
contribution to the face of the earth.

~~~
jernfrost
You are trivializing the significant difference modern western science based
medicine and earlier medicine. Of course people could come up with all sorts
of practical solutions to various ailments and conditions through trial, error
and observations. That however does not make it scientific. There is no
scientific theory in you Chinese and Indian examples on which to make
predictions and explanations. One example would be e.g. germ theory. It allows
one to make predictions about things not yet observed. It helps explain why
washing your hands is good. Now you can learn this from practical
observations, but if you have no idea that the reason is due to germs being
removed, then it prevents you from experimenting with and finding alternative
cleaning agents or practices.

~~~
averagewall
Western medicine is often found by trial and error too. Scientists perform
tests of thousands of compounds just in case one of them happens to do
something that might turn out to be useful. Some drugs are used to treat
different diseases than what they were originally developed for because we
noticed that patients with those other diseases mysteriously showed
improvements. I can't think of which off the top of my head but perhaps a
heart medicine that stopped baldness.

The key difference is western medicine is tested objectively, while
traditional medicines are based on belief. A million people getting sick and
taking a treatment and recovering doesn't mean the treatment does any good,
but they'll believe it does because everyone else believes it too.

------
lacampbell
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.

~~~
yeukhon
> 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
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.

~~~
skinstudio
It must be clarified that mainland of China is also part of ROC according to
ROC government[1]. In fact, the mainland and taiwan are in war, just like the
Union and Confederacy in American civil war, even if there is not military
conflicts right now. More interestingly, ROC actually claims larger territory
than PRC (for example, part of Mongolia). It is ROC's constitution, not PRC's
government, demands taiwan is part of China. Though it's not the People's
Republic of China. And guess what, according to ROC's constitution, mainland
China is "enemy occupied area". Therefore people in mainland China does not
have right to vote, or legitimately declare independence.

[1]
[https://commons.wikimedia.org/wiki/File:ROC_Administrative_a...](https://commons.wikimedia.org/wiki/File:ROC_Administrative_and_Claims.svg)

~~~
PhasmaFelis
China lodged a formal complaint and warned of the "destr[uction of] Sino-US
ties" because the US president _admitted that the Taiwanese president spoke to
him._ If there's a reasonable party in all this, it is definitely not China.
[https://www.theguardian.com/world/2016/dec/03/china-
donald-t...](https://www.theguardian.com/world/2016/dec/03/china-donald-
trumps-taiwan-phone-call-complaint-us)

~~~
allenz
We are incredibly off topic, but I think it's a bit reductionist to call China
unreasonable. China is, above all, pragmatic. The Taiwan situation is a farce,
but the Communist Party has reasons to believe it necessary. Party sovereignty
and legitimacy are based on defending Chinese independence and restoring
national grandeur (强国梦) after colonial humiliation. Any change on the Taiwan
issue could cause both separatist and nationalist forces to escalate out of
control, potentially destabilizing the entire country.

China really doesn't want to find out what might happen if countries start
recognizing Taiwanese independence, so the US president poking at that bubble
is seen as a threat. China responds with assertive words because fighting with
words is better than fighting a war.

------
chrishowlin
I worry that China is outcompeting us.

~~~
jernfrost
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.

~~~
contingencies
I agree with the non zero-sum point, however...

 _China is facing a demographic catastrophe._

Strongly disagree. The view on the ground in China is great. Almost everyone
has their material needs exceeded, many sectors of infrastructure are second
to none, education and internationalization of tastes is increasing, and the
society is generally very safe. E-vehicles and e-commerce are everywhere. Rent
a bicycle for nothing. Visit the tropics, Himalayas, desert or snow with a
cheap domestic flight. Cuisine is arguably the most extensive in the world.
Mobile payments are ubiquitous. Internet is cheap to free and literally you
can get a signal _anywhere_. Yes, of course like anywhere it's not perfect,
but it's really a far cry from America's stagnant and elitist education
system, crumbling or monopolistically privatized infrastructure (eg.
transport, medicine), and widespread opiate, gun and race problems.

(FYI: minimum wage in most Chinese cities for an _unskilled_ foreigner with
any kind of get up and go is 200CNY/hr (~USD$28/hr) these days... a great many
western workers are on less than half of that, and would do very well to
migrate.)

 _so many fundamentals count in Americas favor_

I would like to hear what these are supposed to be. Having elite universities
doesn't count if you rent them out to all comers, default English doesn't
count if more people speak it overseas, VC doesn't count if it's easier to get
elsewhere, and land isn't needed if your population is already happy to
urbanize with extreme density. China owns the world's supply chain and
manufacturing, has a highly efficient centralized form of government that can
get huge or long term projects done without recourse to political terms, and
leads the world in output and deployment scale of renewable energy systems
(hydro, solar), modern transport infrastructure, robotics, internet, mobile
payment and e-vehicles. Those are some pretty big fundamentals, too. From my
perspective all the US really has are systemic global financial and
communications surveillance capabilities and a big fat army. Their diplomatic
trajectory is nothing to write home about, either.

~~~
IIAOPSW
>China is facing a demographic catastrophe.

you didn't refute that. you just said "life on the ground today is good."
That's why its called a cliff. Things are pretty good right up to the moment
you fall off.

>FYI: minimum wage in most Chinese cities for an unskilled foreigner with any
kind of get up and go is 200CNY/hr

But that's not the same thing as minimum wage. Many many Chinese people earn
less than that. When you're a Chinese person you can't just hit up your mate
on wechat for some bullshit English teaching job whenever you want beer money.

~~~
contingencies
There's really no need to refute a statement which goes directly against
sustained personal observation, especially when it is made without producing
any actual evidence.

Obviously not everyone earns the same, then again local people tend to have
more money than even wealthy foreigners at the high end, and at the low end at
least tend to benefit from free housing, familial support networks, and do not
have to fund international travel to maintain ties to home.

Furthermore, downvoting because you don't agree with someone is considered bad
form.

