**Omar Khayyam** 18 May 1048 – 4 December 1131) was a Persian polymath, mathematician, astronomer, historian, philosopher, and poet.^{}^{}^{}^{} He was born in Nishapur, the initial capital of the Seljuk Empire. As a scholar, he was contemporary with the rule of the Seljuk dynasty around the time of the First Crusade.

His name Khayyam (“Tentmaker”) may have been derived from his father’s trade. He received a good education in the sciences and philosophy in his native Neyshābūr before traveling to Samarkand (now in Uzbekistan), where he completed the algebra treatise, *Risālah fiʾl-barāhīn ʿalā masāʾil al-jabr waʾl-muqābalah* (“Treatise on Demonstration of Problems of Algebra”), on which his mathematical reputation principally rests. In this treatise he gave a systematic discussion of the solution of cubic equations by means of intersecting conic sections. Perhaps it was in the context of this work that he discovered how to extend Abu al-Wafā’s results on the extraction of cube and fourth roots to the extraction of *n*th roots of numbers for arbitrary whole numbers *n*.

### Theory of parallels

A part of Khayyam’s commentary on Euclid’s Elements deals with the parallel axiom. ^{}The treatise of Khayyam can be considered the first treatment of the axiom not based on petitio principii, but on a more intuitive postulate. Khayyam refutes the previous attempts by other mathematicians to *prove* the proposition, mainly on grounds that each of them had postulated something that was by no means easier to admit than the Fifth Postulate itself. ^{}Drawing upon Aristotle’s views, he rejects the usage of movement in geometry and therefore dismisses the different attempt by Al-Haytham.^{}^{} Unsatisfied with the failure of mathematicians to prove Euclid’s statement from his other postulates, Omar tried to connect the axiom with the Fourth Postulate, which states that all right angles are equal to one another.^{}

Khayyam was the first to consider the three distinct cases of acute, obtuse, and right angle for the summit angles of a Khayyam-Saccheri quadrilateral After proving a number of theorems about them, he showed that Postulate V follows from the right angle hypothesis, and refuted the obtuse and acute cases as self-contradictory. ^{}His elaborate attempt to prove the parallel postulate was significant for the further development of geometry, as it clearly shows the possibility of non-Euclidean geometries. The hypotheses of acute, obtuse, and right angles are now known to lead respectively to the non-Euclidean hyperbolic geometry of Gauss-Bolyai-Lobachevsky, to that of Riemannian geometry, and to Euclidean geometry.

Tusi’s commentaries on Khayyam’s treatment of parallels made its way to Europe. John Wallis, professor of geometry at Oxford, translated Tusi’s commentary into Latin. Jesuit geometer Girolamo Saccheri, whose work (*euclides ab omni naevo vindicatus*, 1733) is generally considered the first step in the eventual development of non-Euclidean geometry, was familiar with the work of Wallis. The American historian of mathematics David Eugene Smith mentions that Saccheri “used the same lemma as the one of Tusi, even lettering the figure in precisely the same way and using the lemma for the same purpose”. He further says that “Tusi distinctly states that it is due to Omar Khayyam, and from the text, it seems clear that the latter was his inspirer.”^{}

## Astronomy

In 1074–5, Omar Khayyam was commissioned by Sultan Malik-Shah to build an observatory at Isfahan and reform the Persian calendar. There was a panel of eight scholars working under the direction of Khayyam to make large-scale astronomical observations and revise the astronomical tables.^{} Recalibrating the calendar fixed the first day of the year at the exact moment of the passing of the Sun’s center across vernal equinox. This marks the beginning of spring or Nowrūz, a day in which the Sun enters the first degree of Aries before noon.^{} 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 Jalālī calendar was a true solar calendar where the duration of each month is equal to the time of the passage of the Sun across the corresponding sign of the Zodiac. The calendar reform introduced a unique 33-year intercalation cycle. As indicated by the works of Khazini, Khayyam’s group implemented an intercalation system based on quadrennial and quinquennial leap years. Therefore, the calendar consisted of 25 ordinary years that included 365 days, and 8 leap years that included 366 days.^{} 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 modernized, resulting in the modern Iranian calendar. The Jalali calendar is more accurate than the Gregorian calendar of 1582, ^{}with an error of one day accumulating over 5,000 years, compared to one day every 3,330 years in the Gregorian calendar.^{}^{ } Moritz Cantor considered it the most perfect calendar ever devised.^{}

One of his pupils Nizami Aruzi of Samarcand relates that Khayyam apparently did not have a belief in astrology and divination: “I did not observe that he (*scil.* Omar Khayyam) had any great belief in astrological predictions, nor have I seen or heard of any of the great [scientists] who had such belief.”^{}^{ } While working for Sultan Sanjar as an astrologer he was asked to predict the weather – a job that he apparently did not do well.^{} George Saliba (2002) explains that the term *‘ilm al-nujūm*, used in various sources in which references to Omar’s life and work could be found, has sometimes been incorrectly translated to mean astrology. He adds: “from at least the middle of the tenth century, according to Farabi’s enumeration of the sciences, that this science, *‘ilm al-nujūm*, was already split into two parts, one dealing with astrology and the other with theoretical mathematical astronomy”

## Philosophy

Khayyam considered himself intellectually to be a student of Avicenna.^{} According to Al-Bayhaqi, he was reading the metaphysics in Avicenna’s *the Book of Healing* before he died. ^{}There are six philosophical papers believed to have been written by Khayyam. One of them, *On existence* (*Fi’l-wujūd*), was written originally in Persian and deals with the subject of existence and its relationship to universals. Another paper, titled *The necessity of contradiction in the world, determinism and subsistence* (*Darurat al-tadād fi’l-‘ālam wa’l-jabr wa’l-baqā’*), is written in Arabic and deals with free will and determinism. ^{} The titles of his other works are *On being and necessity* (*Risālah fī’l-kawn wa’l-taklīf*), *The Treatise on Transcendence in Existence* (*Al-Risālah al-ulā fi’l-wujūd*), *On the knowledge of the universal principles of existence*

## Poetry

* Rubáiyát of Omar Khayyám* is the title that Edward FitzGerald gave to his 1859 translation from Persian to English of a selection of quatrains (

*) attributed to Omar Khayyam (1048–1131), dubbed “the Astronomer-Poet of Persia”.Although commercially unsuccessful at first, FitzGerald’s work was popularised from 1861 onward by Whitley Stokes, and the work came to be greatly admired by the Pre-Raphaelites in England. FitzGerald had a third edition printed in 1872, which increased interest in the work in the United States. By the 1880s, the book was extremely popular throughout the English-speaking world, to the extent that numerous “Omar Khayyam clubs” were formed and there was a “fin de siècle cult of the Rubaiyat”.*

*rubāʿiyāt*^{}FitzGerald’s work has been published in several hundred editions and has inspired similar translation efforts in English, Hindi and in many other languages.

Khayyam was famous during his lifetime not as a poet but as an astronomer and mathematician. The earliest reference to his having written poetry is found in his biography by al-Isfahani, written 43 years after his death. This view is reinforced by other medieval historians such as Shahrazuri (1201) and Al-Qifti (1255). Parts of the *Rubaiyat* appear as incidental quotations from Omar in early works of biography and in anthologies. These include works of Razi (ca. 1160–1210), Daya (1230), Juvayni (ca. 1226–1283), and Jajarmi (1340). ^{}Also, five quatrains assigned to Khayyam in somewhat later sources appear in Zahiri Samarqandi’s *Sindbad-Nameh* (before 1160) without attribution.^{}

The number of quatrains attributed to him in more recent collections varies from about 1,200 (according to Saeed Nafisi) to more than 2,000. Skeptical scholars point out that the entire tradition may be pseudepigraphic.^{}^{ } The extant manuscripts containing collections attributed to Omar are dated much too late to enable a reconstruction of a body of authentic verses.

In the 1930s, Iranian scholars, notably Mohammad-Ali Foroughi, attempted to reconstruct a core of authentic verses from scattered quotes by authors of the 13th and 14th centuries, ignoring the younger manuscript tradition. After World War II, reconstruction efforts were significantly delayed by two clever forgeries. De Blois (2004) is pessimistic, suggesting that contemporary scholarship has not advanced beyond the situation of the 1930s, when Hans Heinrich Schaeder commented that the name of Omar Khayyam “is to be struck out from the history of Persian literature”.^{}

A feature of the more recent collections is the lack of linguistic homogeneity and continuity of ideas. Sadegh Hedayat commented that “if a man had lived for a hundred years and had changed his religion, philosophy, and beliefs twice a day, he could scarcely have given expression to such a range of ideas”.^{}^{ } Hedayat’s final verdict was that 14 quatrains could be attributed to Khayyam with certainty.^{} Various tests have been employed to reduce the quatrains attributable to Omar to about 100.^{}^{ } Arthur Christensen states that “of more than 1,200 ruba’is known to be ascribed to Omar, only 121 could be regarded as reasonably authentic”.^{}^{ } Foroughi accepts 178 quatrains as authentic, while Ali Dashti accepts 36 of them.^{}

FitzGerald’s source was transcripts sent to him in 1856–57, by his friend and teacher Edward B. Cowell, of two manuscripts, a Bodleian manuscript with 158 quatrains^{} and a “Calcutta manuscript”.

FitzGerald completed his first draft in 1857 and sent it to *Fraser’s Magazine* in January 1858. He made a revised draft in January 1859, of which he privately printed 250 copies. This first edition became extremely sought after by the 1890s, when “more than two million copies ha[d] been sold in two hundred editions”.^{}