# What exactly does this diagram of Omar Khayyam represent?

Evidently Omar Khayyam (1048-1131) was quite the mathematician. He did groundbreaking work on finding geometric solutions to the cubic equation, which is all the more notable since he did not have a good system of notation to work with.

As an example, suppose you want to solve $x^3 + 5x + 1 = 0$. Substituting $y = x^2$, one obtains $x(y + 5) + 1 = 0$, and so the solution lies at the intersection of a parabola and a hyperbola, which can be easily graphed.

More about his work can be found, e.g., here and here, and the ideas in his work also appear in modern papers such as this one of Wright and Yukie, and follow-up papers by Bhargava.

I was able to find this picture on Khayyam's Wikipedia page, which is possibly in Khayyam's own handwriting.

I would love to use it in a talk. But what precisely does this picture represent? Judging from the sources I quoted (among others), it seems possibly related to the solution to $x^3 + 200x = 20x^2 + 2000$, but it is not clear exactly how.

Do any MO users read Arabic or otherwise know what this picture is?

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I've asked someone from Syria who knows modern Arabic. He says it is a little challenging to read, that a ratio is involved, and that he might be able to say more tomorrow. My inference is that to compute the ratio of the two parts of the divided vertical radius in the top picture, a similar quantity involving the ratio of some areas in the bottom picture must be calculated. But I don't really know. – The Masked Avenger Sep 24 '13 at 2:18
Brin's answer below seems to be the best reference, judging from MathSciNet reviews. The translation of Al-Khayyam into French by Rashed and Vahabzadeh seems to be a scholarly critical edition... – Marty Sep 26 '13 at 7:32
The Stanford Encyclopedia of Philosophy discusses this document further: "Khayyam seems to have been attracted to cubic equations originally through his consideration of the following geometric problem: in a quadrant of a circle, drop a perpendicular from some point on the circumference to one of the radii so that the ratio of the perpendicular to the radius is equal to the ratio of the two parts of the radius on which the perpendicular falls. In a short, untitled treatise, Khayyam leads us from one case of this problem to the equation $x^3 + 200x = 20x^2 + 2000$... – Marty Sep 26 '13 at 7:33
...An approximation to the solution of this equation is not difficult to find, but Khayyam also generates a direct geometric solution: he uses the numbers in the equation to determine intersecting curves of two conic sections (a circle and a hyperbola), and demonstrates that the solution $x$ is equal to the length of a particular line segment in the diagram." (plato.stanford.edu/entries/umar-khayyam/#SolCubEqu) – Marty Sep 26 '13 at 7:33

A gentleman named "Hadi Jorati" should be able to help you. He got his mathematics Ph.D. from Princeton and is getting a new Ph.D. on ancient texts of this sort from Yale.