KHAYYÁM 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 Tusi. Tusi mentions explicitly that he re-writes the treatise “in Khayyám’s own words” and quotes Khayyám, 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 (numbered 5). 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 Khayyám can be considered as the first treatment of parallels axiom which is not based on petitio principii but on more intuitive postulate. Khayyám 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,
Omar Khayyam’s mathematical works is worth dwelling upon extensively. But our space here is limited. Let us defer this for another time in the future.

Philosopher
Khayyám himself rejects to be associated with the title falsafi- (lit. philosopher) in the sense of Aristotelian one and stressed he wishes “to know who I am”. In the context of philosophers he was labeled by some of his contemporaries as “detached from divine blessings”.
However it is now established that Khayyám taught for decades the philosophy of Aviccena, especially “the Book of Healing”, in his home town Nishapur, till his death
Khayyám the philosopher could be understood from two rather distinct sources. One is through his Rubaiyat and the other through his own works in light of the intellectual and social conditions of his time. The latter could be informed by the evaluations of Khayyam’s works by scholars and philosophers such as Bayhaqi, Nezami Aruzi, and Zamakhshari and also Sufi poets and writers Attar Nishapuri and Najmeddin Razi.
As a mathematician, Khayyám has made fundamental contributions to the Philosophy of mathematics especially in the context of Persian Mathematics and Persian philosophy with which most of the other Persian scientists and philosophers such as Avicenna, Biruni, and Tusi are associated. There are at least three basic mathematical ideas of strong philosophical dimensions that can be associated with Khayyám.
1.    Mathematical order: From where does this order issue, and why does it correspond to the world of nature? His answer is in one of his philosophical “treatises on being”. Khayyam’s answer is that “the Divine Origin of all existence not only emanates wojud or being, by virtue of which all things gain reality, but it is also the source of order that is inseparable from the very act of existence.”
2.    The significance of postulates (i.e. axiom) in geometry and the necessity for the mathematician to rely upon philosophy and hence the importance of the relation of any particular science to prime philosophy. This is the philosophical background to Khayyam’s total rejection of any attempt to “prove” the parallel postulate and in turn his refusal to bring motion into the attempt to prove this postulate as had Ibn al-Haytham because Khayyam associated motion with the world of matter and wanted to keep it away from the purely intelligible and immaterial world of geometry.
3.    Clear distinction made by Khayyám, on the basis of the work of earlier Persian philosophers such as Avicenna, between natural bodies and mathematical bodies. The first is defined as a body that is in the category of substance and that stands by itself, and hence a subject of natural sciences, while the second, also called “volume”, is of the category of accidents (attributes) that do not subsist by themselves in the external world and hence is the concern of mathematics. Khayyam was very careful to respect the boundaries of each discipline and criticized Ibn al-Haytham in his proof of the parallel postulate precisely because he had broken this rule and had brought a subject belonging to natural philosophy, that is, motion, which belongs to natural bodies, into the domain of geometry, which deals with mathematical bodies.
There is need for public understanding of the philosophical content of mathematical and other works of Omar Khayyam. Unfortunately, for a person like me who have stopped working on mathematical problems for over two decades now, it cannot be an easy self-imposed task. But one is abreast of the philosophy of science and one has been regularly consuming any new book available on the subject-matter in the last few years.

Khayyam, the Poet
Omar Khayyám’s poetic work has eclipsed his fame as a mathematician and scientist. He is believed to have written about a thousand four-line verses or quatrains (rubaai’s). In the English-speaking world, he was introduced through the Rubáiyát of Omar Khayyám which is rather free-wheeling English translations by Edward FitzGerald Other translations of parts of the rubáiyát (rubáiyát meaning “quatrains”) exist, but FitzGerald’s are the most well known. Translations also exist in languages other than English.
Ironically, FitzGerald’s translations reintroduced Khayyám to Iranians “who had long ignored the Neishapouri poet.” A 1934 book by one of Iran’s most prominent writers, Sadeq Hedayat, Songs of Khayyam, (Taranehha-ye Khayyam) is said have “shaped the way a generation of Iranians viewed” the poet.