**Omar Khayyam** was an Islamic scholar who was a poet as well as a mathematician. He compiled astronomical tables and contributed to calendar reform and discovered a geometrical method of solving cubic equations by intersecting a parabola with a circle.

- It was in this difficult unstable military empire, which also had religious problems as it attempted to establish an orthodox Muslim state, that Khayyam grew up.
- However, this was not an empire in which those of learning, even those as learned as Khayyam, found life easy unless they had the support of a ruler at one of the many courts.
- However Khayyam was an outstanding mathematician and astronomer and, despite the difficulties which he described in this quote, he did write several works including Problems of Arithmetic, a book on music and one on algebra before he was 25 years old.
- There Khayyam was supported by Abu Tahir, a prominent jurist of Samarkand, and this allowed him to write his most famous algebra work, Treatise on Demonstration of Problems of Algebra from which we gave the quote above.
- An invitation was sent to Khayyam from Malik-Shah and from his vizier Nizam al-Mulk asking Khayyam to go to Esfahan to set up an Observatory there.
- Other leading astronomers were also brought to the Observatory in Esfahan and for 18 years Khayyam led the scientists and produced work of outstanding quality.
- During this time Khayyam led work on compiling astronomical tables and he also contributed to calendar reform in 1079.
- Funding to run the Observatory ceased and Khayyam's calendar reform was put on hold.
- Khayyam also came under attack from the orthodox Muslims who felt that Khayyam's questioning mind did not conform to the faith.
- Despite being out of favour on all sides, Khayyam remained at the Court and tried to regain favour.
- Sometime after this Khayyam left Esfahan and travelled to Merv (now Mary, Turkmenistan) which Sanjar had made the capital of the Seljuq empire.
- Sanjar created a great centre of Islamic learning in Merv where Khayyam wrote further works on mathematics.
- Perhaps even more remarkable is the fact that Khayyam states that the solution of this cubic requires the use of conic sections and that it cannot be solved by ruler and compass methods, a result which would not be proved for another 750 years.
- Indeed Khayyam did produce such a work, the Treatise on Demonstration of Problems of Algebra which contained a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections.
- Khayyam gives an interesting historical account in which he claims that the Greeks had left nothing on the theory of cubic equations.
- Indeed, as Khayyam writes, the contributions by earlier writers such as al-Mahani and al-Khazin were to translate geometric problems into algebraic equations (something which was essentially impossible before the work of al-Khwarizmi).
- However, Khayyam himself seems to have been the first to conceive a general theory of cubic equations.
- Another achievement in the algebra text is Khayyam's realisation that a cubic equation can have more than one solution.
- Also in his algebra book, Khayyam refers to another work of his which is now lost.
- In the lost work Khayyam discusses the Pascal triangle but he was not the first to do so since al-Karaji discussed the Pascal triangle before this date.
- We can be fairly sure that Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients.
- In Commentaries on the difficult postulates of Euclid's book Khayyam made a contribution to non-euclidean geometry, although this was not his intention.
- Khayyam also gave important results on ratios in this book, extending Euclid's work to include the multiplication of ratios.
- The importance of Khayyam's contribution is that he examined both Euclid's definition of equality of ratios (which was that first proposed by Eudoxus) and the definition of equality of ratios as proposed by earlier Islamic mathematicians such as al-Mahani which was based on continued fractions.
- Khayyam proved that the two definitions are equivalent.
- Outside the world of mathematics, Khayyam is best known as a result of Edward Fitzgerald's popular translation in 1859 of nearly 600 short four line poems the Rubaiyat.
- Khayyam's fame as a poet has caused some to forget his scientific achievements which were much more substantial.
- Versions of the forms and verses used in the Rubaiyat existed in Persian literature before Khayyam, and only about 120 of the verses can be attributed to him with certainty.

Born 18 May 1048, Nishapur, Persia (now Iran). Died 4 December 1131, Nishapur, Persia (now Iran).

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Ancient Arab, Astronomy, Origin Iran, Special Numbers And Numerals

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive