(1048–1131) was a Persian poet and polymath, known for his contributions to mathematics, astronomy, philosophy, and Persian literature. He was born in Nishapur, Iran and lived during the Seljuk era, around the time of the First Crusade.
As a mathematician, Omar Khayyam was the first to provide a general solution for all third-degree polynomials by using the intersection of two conic sections, a method often later attributed to Descartes. Unlike Descartes, Khayyam performed these geometric calculations by selecting a unit length while strictly adhering to the rule of homogeneity. Additionally, in his work On the Division of a Quarter of a Circle, he attempted to derive approximate numerical solutions for cubic equations using trigonometric tables. He also contributed to a deeper understanding of Euclid's parallel axiom. The Saccheri quadrilateral is sometimes called a Khayyam–Saccheri quadrilateral to credit Omar Khayyam who described it in his 11th century book Risāla fī šarḥ mā aškala min muṣādarāt kitāb Uqlīdis (Explanations of the difficulties in the postulates of Euclid). As an astronomer, he calculated the duration of the solar year with remarkable precision and accuracy, and designed the Jalali calendar, a solar calendar with a very precise 33-year intercalation cycle
Although open to doubt, it has often been assumed that his forebears followed the trade of tent-making, since Khayyam means 'tent-maker' in Arabic. The historian Bayhaqi, who was personally acquainted with Khayyam, provides the full details of his horoscope: "he was Gemini, the sun and Mercury being in the ascendant[...]". This was used by modern scholars to establish his date of birth as 18 May 1048. which had earlier been a major center of the Zoroastrian religion. His full name, as it appears in Arabic sources, was Abu’l Fath Omar ibn Ibrahim al-Khayyam.
In effect, Khayyam's work is an effort to unify algebra and geometry. This particular geometric solution of cubic equations was further investigated by M. Hachtroudi and extended to solving fourth-degree equations. Although similar methods had appeared sporadically since Menaechmus, and further developed by the 10th-century mathematician Abu al-Jud, Khayyam's work can be considered the first systematic study and the first exact method of solving cubic equations. The mathematician Woepcke (1851) who offered translations of Khayyam's algebra into French praised him for his "power of generalization and his rigorously systematic procedure."
Binomial theorem and extraction of roots
In his algebraic treatise, Khayyam alludes to a book he had written on the extraction of the <math>n</math>th root of natural numbers using a law he had discovered which did not depend on geometric figures. One of Khayyam's predecessors, al-Karaji, had already discovered the triangular arrangement of the coefficients of binomial expansions that Europeans later came to know as Pascal's triangle; Khayyam popularized this triangular array in Iran, so that it is now known as Omar Khayyam's triangle. The resultant calendar was named in Malik-Shah's honor as the Jalālī calendar, and was inaugurated on 15 March 1079. The observatory itself was disused after the death of Malik-Shah in 1092. The calendar remained in use across Greater Iran from the 11th to the 20th centuries. In 1911, the Jalali calendar became the official national calendar of Qajar Iran. In 1925, this calendar was simplified and the names of the months were modernised, resulting in the modern Iranian calendar. The Jalali calendar is more accurate than the Gregorian calendar of 1582,
Other works
Khayyam has a short treatise devoted to Archimedes' principle (in full title, On the Deception of Knowing the Two Quantities of Gold and Silver in a Compound Made of the Two). For a compound of gold adulterated with silver, he describes a method to measure more exactly the weight per capacity of each element. It involves weighing the compound both in air and in water, since weights are easier to measure exactly than volumes. By repeating the same with both gold and silver one finds exactly how much heavier than water gold, silver and the compound were. This treatise was extensively examined by Eilhard Wiedemann who believed that Khayyam's solution was more accurate and sophisticated than that of Khazini and Al-Nayrizi who also dealt with the subject elsewhere. One of the earliest specimens of Omar Khayyam's Rubiyat is from Fakhr al-Din al-Razi. In his work (), he quotes one of his poems (corresponding to quatrain LXII of FitzGerald's first edition). Daya in his writings (, c. 1230) quotes two quatrains, one of which is the same as the one already reported by Razi. An additional quatrain is quoted by the historian Juvayni (, c. 1226–1283). A comparatively late manuscript is the Bodleian MS. Ouseley 140, written in Shiraz in 1460, which contains 158 quatrains on 47 folia. The manuscript belonged to William Ouseley (1767–1842) and was purchased by the Bodleian Library in 1844.
thumb|Inscription of a poem written by Omar Khayyam at [[Morića Han in Sarajevo, Bosnia and Herzegovina]]
There are occasional quotes of verses attributed to Khayyam in texts attributed to authors of the 13th and 14th centuries, but these are of doubtful authenticity, so that skeptical scholars point out that the entire tradition may be pseudepigraphic.<sup>:307</sup>
Five of the quatrains later attributed to Omar Khayyam are found as early as 30 years after his death, quoted in Sindbad-Nameh. While this establishes that these specific verses were in circulation in Omar's time or shortly later, it does not imply that the verses must be his. De Blois concludes that at the least the process of attributing poetry to Omar Khayyam appears to have begun already in the 13th century. and many more have been published since.
Religious views
A literal reading of Khayyam's quatrains leads to the interpretation of his philosophic attitude toward life as a combination of pessimism, nihilism, Epicureanism, fatalism, and agnosticism. This view is taken by Iranologists such as Arthur Christensen, Hans Heinrich Schaeder, John Andrew Boyle, Edward Denison Ross, Edward Henry Whinfield In addition to his Persian quatrains, J. C. E. Bowen mentions that Khayyam's Arabic poems also "express a pessimistic viewpoint which is entirely consonant with the outlook of the deeply thoughtful rationalist philosopher that Khayyam is known historically to have been." Edward FitzGerald emphasized the religious skepticism he found in Khayyam. In his preface to the Rubáiyát he claimed that he "was hated and dreaded by the Sufis", and denied any pretense at divine allegory: "his Wine is the veritable Juice of the Grape: his Tavern, where it was to be had: his Saki, the Flesh and Blood that poured it out for him." Sadegh Hedayat is one of the most notable proponents of Khayyam's philosophy as agnostic skepticism, and according to Jan Rypka (1934), he even considered Khayyam an atheist. Hedayat (1923) states that "while Khayyam believes in the transmutation and transformation of the human body, he does not believe in a separate soul; if we are lucky, our bodily particles would be used in the making of a jug of wine." Omar Khayyam's poetry has been cited in the context of New Atheism, such as in The Portable Atheist by Christopher Hitchens.
Al-Qifti () appears to confirm this view of Khayyam's philosophy.