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The Treatise on Demonstration of Problems of Algebra

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  • #1

    The Treatise on Demonstration of Problems of Algebra

    I found this immensely interesting. It is written by Omar Khayyam.

    If you can find a copy in your local library, take a look at it even if you don't understand all of it.

    When you do, let's discuss.
    Tags: algebra, demonstration, problems, treatise
  • #2
    Apologies, that is the original name of the text.

    You should look for The algebra of Omar Khayyam by Daoud S.Kasir at your library.

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    • #3
      There is a copy at my library.

      Algebra of Omar Khayyam by - Powell's Books

      There is a link to a website with some more information on it including ISBN.

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      • #4
        I might be picking it up tomorrow. What in particular do you like about it Behzad?
        Even Adobe Photoshop can't change me.

        A line[Durrand line] of hatred that raised a wall between the two brothers -Hamid Karzai

        For generations, the Hindus of India prayed for deliverance from "the venom of the cobra, the teeth of the tiger and the vengeance of the Afghan."


        The men of Kábul and Khilj also went home; and whenever they were questioned about the Musulmáns of the Kohistán (the mountains), and how matters stood there, they said, "Don't call it Kohistán, but Afghánistán; for there is nothing there but Afgháns and disturbances." Thus it is clear that for this reason the people of the country call their home in their own language Afghánistán, and themselves Afgháns. The people of India call them Patán; but the reason for this is not known. But it occurs to me, that when, under the rule of Muhammadan sovereigns, Musulmáns first came to the city of Patná, and dwelt there, the people of India (for that reason) called them Patáns—but God knows!

        -Ferishta, 1560–1620

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        • #5
          Khayyam's remarkable solution of the cubic!

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          • #6
            I will try to get a few pictures of the book within the next week, since I couldn't find a pdf copy.

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            • #7
              I apologize for the extreme delay.

              The book is translated, prefaced and commented on by Daud S. Kasir.



              I still have the book, so if anyone has any specific questions I can write it out.

              Basicly Khayyam divides all cubic equation to 25 classes. So that all quantities remain positive. (He solved the cubic equation without having the notion of a negative number!).

              So, he has classes such as:

              ax^3 = d
              ax^3 = cx
              ax^3 = bx^2
              ax^3 = cx + d
              ax^3 = bx^2 + d
              ax^3 = cx^2 + bx + d

              These are 6 of the possible 25 classes. You can imagine others such as:

              ax^3 + d = bx^2 + cx

              He then individually solves each case using conic sections, namely parabolas and circles.

              For example the equation x^3 + 4x + 8

              We find the intersection of the parabola y = x^2/2 and the circle (x + 1)^2 + y^2 =1, there are two such intersections.

              The one with the positive x value, is the one we are looking, which is indeed a positive root of x^3 + 4x + 8.

              What Khayyam did is give a general method for such constructive solutions.

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