
The Chinese were solving 14th degree polynomials in 1303. Why? - tombh
https://www.reddit.com/r/AskHistorians/comments/9bkfhj/by_1303_the_chinese_were_solving_equations_of_the/
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betterunix2
One of the things I find interesting about ancient Chinese math is how
computational it was (in modern terminology we would say they were only
interested in computable numbers); AFAIUI the ancient Chinese were vaguely
aware of numbers that could not be computed exactly (i.e. irrationals) but
mostly cared about being able to compute approximations up to whatever
precision was required for a given task. So while Western mathematicians were
struggling in vain to find a way to double a cube using a compass and
straightedge or to derive a quintic formula, the Chinese mathematicians were
satisfied with an algorithm that found useful approximations. There is a
profound difference in philosophy, as evidenced by the title of this entire
thread: the Chinese were "solving" 14th degree polynomials, but only in the
sense of having an algorithm for doing so (as opposed to having a formula like
the quadratic formula; in fact there is no such formula for degree 14
polynomials, or for any polynomials beyond the quartic).

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how_to_bake
Hate to be annoying, but I'm a native English speaker and I just skipped over
"AFAIUI" because it was too much work to guess what it meant.

"As far as I understand it" ? RIP if you're not a native speaker.

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betterunix2
I think it is a relatively common abbreviation used in informal writing
online, similar to "AFAIK" or "AFAICT" (maybe those are not so common?) but
admittedly less common than "QED" or "etc." Yes, it means "as far as I
understand it."

~~~
jerrycruncher
Anecdotally, I'm a native speaker, have been reading text on BBSes and then
the internet for well over 30 years, and have never seen that one. AFAIK has
been around for decades.

Like some other commenters, I skipped right over it rather than try to parse
it.

~~~
__sdegutis
I've been online for 20 years, and AFAIK was pretty common, although a few
years ago I came across AFAICT which was pretty easy to extrapolate as ending
with "can tell". From that I was able to recognize the pattern here and pretty
quickly got "understand it" for UI. I think it's pretty neat that we're able
to adapt language in this way, though obviously not foolproof or universal.

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leereeves
At first I thought the headline meant they were factoring or finding the roots
of 14th degree polynomials, which would have been quite impressive.

But no, the page just says they were calculating the answer for a particular
value of x, using "the method of fan fa, today called Horner's method". A
simple process, for example:

To compute 3x^2 + 12x + 12, start with 3, multiply by x, add 12, multiply the
result by x, and add 12 again.

~~~
fspeech
No the headline was not wrong. One of the answers explained Horner's method
and its relation to using abacus (which has a small register file, in modern
lingo). However that is not the extent of the book mentioned. You can find the
book here:
[https://zh.wikipedia.org/wiki/%E5%9B%9B%E5%85%83%E7%8E%89%E9...](https://zh.wikipedia.org/wiki/%E5%9B%9B%E5%85%83%E7%8E%89%E9%89%B4)
The equations dealt with have been transcribed into modern algebraic
expressions. They were definitely solved, not merely computed.

The more complete answer should have been "The higher order polynomial (as
high as 14) came from solving multivariate simultaneous equations (with
degrees as high as 4) by elimination of variables."

~~~
leereeves
I wasn't referring to the reddit page, but the Wikipedia page it's based on:

> Si-yüan yü-jian (《四元玉鑒》), or Jade Mirror of the Four Unknowns, was written
> by Zhu Shijie in 1303 AD and marks the peak in the development of Chinese
> algebra. The four elements, called heaven, earth, man and matter,
> represented the four unknown quantities in his algebraic equations. It deals
> with simultaneous equations and with equations of degrees as high as
> fourteen. The author uses the method of fan fa, today called Horner's
> method, to solve these equations.[48]

If there's more to the story you might want to correct Wikipedia.

~~~
fspeech
The page I linked to is the actual page for Si-yüan yü-jian (《四元玉鑒》). The
Wikipedia article you referred to may be slightly confusing but is not wrong.
If you tried to solve a polynomial numerically, Horner's method would be
handy. However we know there is no general solution per Abel and Galois.

~~~
leereeves
Which part are you're referring to when you say "The equations dealt with have
been transcribed into modern algebraic expressions. They were definitely
solved, not merely computed"?

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fspeech
The page I linked to listed the problems posed and solved. The Chinese version
is accompanied by modern notations using x,y,z...

By the way the numerical solution to high order one variable polynomial was
already known and given by another mathematician in an earlier (1247 AD) book:
[https://zh.wikipedia.org/wiki/%E6%95%B0%E4%B9%A6%E4%B9%9D%E7...](https://zh.wikipedia.org/wiki/%E6%95%B0%E4%B9%A6%E4%B9%9D%E7%AB%A0)

~~~
leereeves
I thought you were referring to a specific solution for a 14th degree
polynomial.

If you're just referring to the notation, that's a modern translation using
modern notation. What was in the original?

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fspeech
I am saying all the equations listed also had their solutions given.

~~~
leereeves
It's not clear to me what's the original and what's a modern addition. For
example:

Mixed question Straight paragraph source Eighteen questions.

Eighteenth question:

Today, there is a sum of money and multiplication, and the product is reduced
by the sum of the balances. It has a total of 170,162 steps. Only the cloud
and the benefits. The fourth is Yifang, the third is from the low, the second
is the benefit of the low, the first is the right, the three squares open,
such as a quarter of the flat. Question, long, flat geometry? Answer: Ping is
twelve steps and is thirty steps long.

Li Tianyuan is the opening number and has:

(equation)

The solution is x=3, multiplied by four, which is the flat number.

It seems to me that the text is the original Chinese (with only a numeric
solution for one case) and the general equation is a modern addition.

~~~
fspeech
I don't know what your confusion is. Every problem was given in the format of
"Question, Answer, Steps to arrive at the answer" in the Chinese original.
Numerically solving high order polynomials was already known so was not
explained. The explanations were focused on elimination of variables for
simultaneous equations (which can give you high order single variable
polynomials).

~~~
leereeves
It might just be lost in translation, since I'm using Google Translate, but I
don't see the "Question, Answer, Steps". I'll have to take your word for it.

~~~
fspeech
Ah I see. Sorry I missed that there is a English page (actually linked to from
the link referenced above). While having fewer problems than the original page
it may give you a better flavor than Google Translate (and shows how far
Google translate still has to go :-):
[https://en.wikipedia.org/wiki/Jade_Mirror_of_the_Four_Unknow...](https://en.wikipedia.org/wiki/Jade_Mirror_of_the_Four_Unknowns)

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samfriedman
AskHistorians is surely one of the great jewels of Reddit.

Fantastically moderated, and filled with an array of helpful experts happy to
research and write up answers to all sorts of interesting questions and
topics.

