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Khayyam was a Persian polymath, mathematician, philosopher, astronomer, physician, and poet. He wrote treatises on mechanics, geography, and music.
Theory of parallels : (History of non-Euclidean geometry and Parallel postulate)
Khayyam wrote a book entitled Explanations of the difficulties in the postulates in Euclid's Elements. The book consists of several sections on the parallel postulate (Book I), on the Euclidean definition of ratios and the Anthyphairetic ratio (modern continued fractions) (Book II), and on the multiplication of ratios (Book III).
The first section is a treatise containing some propositions and lemmas concerning the parallel postulate. It has reached us from a reproduction in a manuscript written in 1387-88 AD by the Persian mathematician Tousi. Tousi mentions explicitly that he re-writes the treatise "in Khayyam's own words" and quotes Khayyam, saying that "they are worth adding to Euclid's Elements (first book) after Proposition 28." This proposition states a condition enough for having two lines in plane parallel to one another. After this proposition follows another, numbered 29, which is converse to the previous one. The proof of Euclid uses the so-called parallel postulate. Objection to the use of parallel postulate and alternative view of proposition 29 have been a major problem in foundation of what is now called non-Euclidean geometry.
The treatise of Khayyam can be considered as the first treatment of parallels axiom which is not based on petitio principii but on more intuitive postulate. Khayyαm refutes the previous attempts by other Greek and Persian mathematicians to prove the proposition. And he, as Aristotle, refuses the use of motion in geometry and therefore dismisses the different attempt by Ibn Haytham too. In a sense he made the first attempt at formulating a non-Euclidean postulate as an alternative to the parallel postulate.
( نيشابور - آرامگاه خيام )

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