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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 solut

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%AB%A0 https://zh.wikipedia.org/wiki/%E6%95%B0%E4%B9%A6%E4%B9%9D%E7...

leereeves 8y ago
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?
- fspeech 8y ago
  I am saying all the equations listed also had their solutions given.
  - leereeves 8y ago
    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 8y ago
      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 8y ago
        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 8y ago
        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_Unknowns https://en.wikipedia.org/wiki/Jade_Mirror_of_the_Four_Unknow...
fspeech 8y ago
The detail of the numerical algorithm is here: https://zh.wikipedia.org/wiki/%E7%A7%A6%E4%B9%9D%E9%9F%B6%E7%AE%97%E6%B3%95 https://zh.wikipedia.org/wiki/%E7%A7%A6%E4%B9%9D%E9%9F%B6%E7... As you can see Horner's method is a very important part of it because it makes the numerical steps practical to do with little memory.
wbhart 8y ago
There's a degree 10 polynomial here, but it has an integer root, which seems to be what they computed. This can be achieved just by factoring the leading and trailing coefficients, and by various other easy methods. It's not really anywhere near as difficult as computing an approximation to the root or solving in radicals (when that's possible). Edit: OK I see on one of the other pages they were finding rational approximations to a root. That is pretty clever for the time.

leereeves 8y ago

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?

fspeech 8y ago
I am saying all the equations listed also had their solutions given.
- leereeves 8y ago
  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 8y ago
    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 8y ago
      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 8y ago
        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_Unknowns https://en.wikipedia.org/wiki/Jade_Mirror_of_the_Four_Unknow...

fspeech 8y ago

The detail of the numerical algorithm is here: https://zh.wikipedia.org/wiki/%E7%A7%A6%E4%B9%9D%E9%9F%B6%E7%AE%97%E6%B3%95 https://zh.wikipedia.org/wiki/%E7%A7%A6%E4%B9%9D%E9%9F%B6%E7... As you can see Horner's method is a very important part of it because it makes the numerical steps practical to do with little memory.

wbhart 8y ago

There's a degree 10 polynomial here, but it has an integer root, which seems to be what they computed. This can be achieved just by factoring the leading and trailing coefficients, and by various other easy methods. It's not really anywhere near as difficult as computing an approximation to the root or solving in radicals (when that's possible). Edit: OK I see on one of the other pages they were finding rational approximations to a root. That is pretty clever for the time.