Omar Khayyam ? 


Bust by Abolhassan Sadighi (c. 1960) in Nishapur, Iran


Born  18 May^{[1]} 1048^{[2]} Nishapur, Khorasan, Iran 
Died  4 December^{[1]} 1131 (aged 83)^{[2]} Nishapur, Khorasan, Iran 
Nationality  Persian 
School  Mathematics, Persian poetry, Persian philosophy 
Main interests

Mathematics, Astronomy/Astrology, Avicennism, Poetry 
Influences


Influenced

Omar Khayyam (Persian pronunciation: [xæj'j?:m]; (Persian); 18 May 1048  4 December 1131) was a Persian mathematician, astronomer, and poet.^{[3]}^{[4]}^{:7} As a mathematician, he is most notable for his work on the classification and solution of cubic equations, where he provided geometric solutions by the intersection of conics.^{[5]}^{[6]} As an astronomer, he composed a calendar which proved to be a more accurate computation of time than that proposed five centuries later by Pope Gregory XIII.^{[7]}^{:659}^{[8]} Omar was born in Nishapur, in northeastern Iran. He spent most of his life near the court of the Karakhanid and Seljuq rulers in the period which witnessed the First Crusade. There is a tradition of attributing poetry to Omar Khayyam, written in the form of quatrains (rubiy?t ?). This poetry became widely known due to the English translation by Edward FitzGerald (Rubaiyat of Omar Khayyam, 1859), which enjoyed great success in the Orientalism of the fin de siècle.
Omar Khayyam was born in Nishapur, a metropolis in Khorasan that was later destroyed by the Mongols in 1220.^{[9]}^{[10]}^{[11]} Nishapur was then a Seljuq capital and it was religiously a major center of Zoroastrians. It is likely that Khayyam's father was a Zoroastrian who had converted to Islam.^{[4]}^{:68} He was born into a family of tentmakers (Khayyam). His full name, as it appears in the Arabic sources, was Abu'l Fath Omar ibn Ibr?h?m alKhayy?m.^{[4]}^{:18}^{[a]} In medieval Persian texts he is usually simply called Omar Khayy?m.^{[7]}^{:658}^{[b]} The historian Bayhaqi, who was personally acquainted with Omar, provides the full details of his horoscope: "he was Gemini, the sun and Mercury being in the ascendant[...]".^{[12]}^{:471} This was used by modern scholars to establish his date of birth as 18 May 1048.^{[7]}^{:658}
He spent part of his childhood in the town of Balkh (in presentday northern Afghanistan), studying under the scholar Sheikh Muhammad Mansuri. He later studied under Imam Mowaffaq Nishapuri, one of the greatest teachers of the Khorasan region. In 1073, at the age of twentysix, he entered the service of Sultan MalikShah I as an adviser. In 1076 Khayyam was invited to Isfahan by Nizam alMulk to take advantage of the libraries and centers in learning there. His years in Isfahan were productive. It was at this time that he began to study the work of Greek mathematicians Euclid and Apollonius much more closely. But after the death of the MalikShah I (presumably by the Assassins' sect), Omar had fallen from favour at court, and as a result, he soon set out on his Hajj or pilgrimage to Mecca. A possible ulterior motive for his pilgrimage reported by AlQifti, is that he was attacked by the clergy for his apparent skepticism. So he decided to perform his pilgrimage as a way of demonstrating his faith and freeing himself from all suspicion of unorthodoxy.^{[4]}^{:29} He was then invited by the new Sultan Sanjar to Marv, possibly to work as a court astrologer.^{[1]} He was later allowed to return to Nishapur owing to his declining health. Upon his return, he seemed to have lived the life of a recluse.^{[13]}^{:99} Khayyam died in 1131, and is buried in the Khayyam Garden in Nishapur.
Khayyam was famous during his life as a mathematician. Three of his treatises are extant: A commentry on Euclid's Elements (Ris?la f? ?ar? m? a?kala min mudar?t kit?b Uql?dis, completed in December 1077^{[3]}), an essay on dividing the quadrant of a circle, and a treatise on algebra (Maq?la fi ljabr wa lmuq?bala, undated^{[3]}).^{[14]} He furthermore wrote a treatise on extracting the n^{th} root of natural numbers, which has been lost.
The treatise on algebra contains his work on cubic equations.^{[15]} It is divided into three parts, (a) equations which can be solved with ruler and compass, (b) equations which can be solved by means of conic sections, and (c) equations which involve the inverse of the unknown.^{[16]} He was the first to geometrically solve every type of cubic equation, so far as positive roots are concerned.^{[17]} He wrote on the triangular array of binomial coefficients known as Pascal's triangle. The second of Omar's mathematical works deals with the the parallel axiom.^{[5]}^{:282}
Khayyam wrote 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 the Western world from a reproduction in a manuscript written in 138788 by the Persian mathematician Tusi. Tusi mentions explicitly that he rewrites 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."^{[18]} This proposition^{[19]} 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.^{[20]} The proof of Euclid uses the socalled 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 nonEuclidean geometry.
The treatise of Khayyam can be considered the first treatment of the parallels axiom not based on petitio principii, but on a more intuitive postulate. Khayyam 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 AlHaytham, too.^{[21]} In a sense he made the first attempt at formulating a nonEuclidean postulate as an alternative to the parallel postulate,^{[22]}
Khayyam illustrated Euclid's parallel postulate in Book I of Explanations of the Difficulties in the Postulates of Euclid.^{[23]} Khayyam was not trying to prove the parallel postulate as such but to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle):
Khayyam then considered the three cases (right, obtuse and acute) that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he (correctly) refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid.
Tusi's commentaries on Khayyam's treatment of parallels made its way to Europe. John Wallis, the professor of geometry at Oxford, translated Tusi's commentary into Latin. Jesuit geometrician Girolamo Saccheri, whose work (euclides ab omni naevo vindicatus, 1733) is generally considered as the first step in the eventual development of nonEuclidean 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."^{[13]}^{:104}^{[25]}^{[4]}^{:195}
This philosophical view of mathematics (see below) has influenced Khayyam's celebrated approach and method in geometric algebra and in particular in solving cubic equations. His solution is not a direct path to a numerical solution, and his solutions are not numbers but line segments. Khayyam's work can be considered the first systematic study and the first exact method of solving cubic equations,^{[27]} although similar methods had appeared sporadically since Menaechmus.
In work on cubic equations by Khayyam discovered in the 20th century,^{[26]} where the above quote appears, Khayyam works on problems of geometric algebra. First is the problem of "finding a point on a quadrant of a circle such that when a normal is dropped from the point to one of the bounding radii, the ratio of the normal's length to that of the radius equals the ratio of the segments determined by the foot of the normal." Again in solving this problem, he reduces it to another geometric problem: "find a right triangle having the property that the hypotenuse equals the sum of one leg (i.e. side) plus the altitude on the hypotenuse ".^{[28]} To solve this geometric problem, he specializes a parameter and reaches the cubic equation x^{3} + 2x = 2x^{2} + 2.^{[26]} Indeed, he finds a positive root for this equation by intersecting a hyperbola with a circle.
This particular geometric solution of cubic equations has been further investigated and extended to degree four equations.^{[29]}
Regarding more general equations he states that the solution of cubic equations requires the use of conic sections and that it cannot be solved by ruler and compass methods.^{[26]} A proof of this impossibility was only plausible 750 years after Khayyam died.^{[]} In this text Khayyam mentions his will to prepare a paper giving full solution to cubic equations: "If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements which are greatly useful in this art, will be prepared."^{[26]}
This refers to the book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of algebra, part of the body of Persian mathematics that was eventually transmitted to Europe.^{[27]} In particular, he derived general methods for solving cubic equations and even some higher orders.
In his algebraic treatise, Khayyam alludes to a book he had written on the extraction of the th root of the numbers using a law which he had discovered which did not depend on geometric figures.^{[8]} The manuscript of this book is not extant. Based on the context, some historians of mathematics such as Struik, believe that Omar must have known the formula for the expansion of the binomial , where n is a positive integer.^{[5]} The case of power 2 is explicitly stated in Euclid's elements and the case of at most power 3 had been established by Indian mathematicians. Khayyam was the mathematician who noticed the importance of a general binomial theorem. The argument supporting the claim that Khayyam had a general binomial theorem is based on his ability to extract roots.^{[31]}
Like most Persian mathematicians of the period, Khayyam was also an astronomer and achieved fame in that role, especially for his expertise in horoscopic astrology^{[clarification needed]}. In 1073, the Seljuq Sultan Jalal alDin MalikShah Saljuqi (MalikShah I, 107292), invited Khayyam to build an observatory, along with various other distinguished scientists. A popular claim to the effect that Khayyam believed in heliocentrism is based on Edward FitzGerald's popular but anachronistic rendering of Khayyam's poetry, in which the first lines are mistranslated with a heliocentric image of the Sun flinging "the Stone that puts the Stars to Flight".^{[36]}
Khayyam was a member of a panel that reformed the Iranian calendar. The panel was convened by Seljuk Sultan Malik Shah I, and completed its reforms in 1079, resulting in the Jalali calendar.^{[37]}^{[38]}
The calendar is based on actual solar transit, similar to Hindu calendars, and requires an ephemeris for calculating dates. The lengths of the months can vary between 29 and 31 days depending on the moment when the sun crosses into a new zodiacal area (an attribute common to most Hindu calendars). This meant that seasonal errors were lower than in the Gregorian calendar.
According to some accounts,^{[by whom?]} the version of the medieval Iranian calendar in which 2,820 solar years together contain 1,029,983 days (or 683 leap years, for an average year length of 365.24219858156 days) was based on the measurements of Khayyam and his colleagues.^{[39]} Another proposal is that Khayyam's calendar simply contained eight leap days every thirtythree years (for a year length of 365.2424 days).^{[40]}
The Jalali calendar is more accurate than the Gregorian calendar of 1582,^{[7]}^{:659} with an error of one day accumulating over 5,000 years, compared to one day every 3,330 years in the Gregorian calendar.^{[4]}^{:200}Moritz Cantor considered it the most perfect calendar ever devised.^{[13]}^{:101}
The Jalali 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 earliest allusion to Omar Khayyam's poetry is from the historian Imad adDin alIsfahani, a younger contemporary of Khayyam, who explicitly identifies him as both a poet and a scientist (Kharidat alqasr, 1174).^{[4]}^{:49}^{[41]}^{:35} One of the earliest specimens of Omar Khayyam's Rubiyat is from Fakhr alDin Razi. In his work Altanbih 'ala ba'd asrar almaw'dat fi'lQur'an (ca. 1160), he quotes one of his poems (corresponding to quatrain LXII of FitzGerald's first edition). Daya in his writings (Mirsad al'Ibad, ca. 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 (Tarikhi Jahangushay, ca. 12261283).^{[41]}^{:3637}^{[4]}^{:92} In 1340 Jajarmi includes thirteen quatrains of Khayyam in his work containing an anthology of the works of famous Persian poets (Munis alahr?r), two of which have hitherto been known from the older sources.^{[42]} 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 (17671842) and was purchased by the Bodleian Library in 1844.
There are occasional quotes of verses attributed to Omar in texts attributed to authors of the 13th and 14th centuries, but these are also of doubtful authenticity, so that skeptic scholars point out that the entire tradition may be pseudepigraphic.^{[41]}^{:11}Hans Heinrich Schaeder in 1934 commented that the name of Omar Khayyam "is to be struck out from the history of Persian literature" due to the lack of any material that could confidently be attributed to him. De Blois (2004) presents a bibliography of the manuscript tradition, concluding pessimistically that the situation has not changed significantly since Schaeder's time.^{[43]} Five of the quatrains later attributed to Omar are found as early as 30 years after his death, quoted in SindbadNameh. While this establishes that these specific verses were in circulation in Omar's time or shortly later, it doesn't 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.^{[44]}Edward Granville Browne (1906) notes the difficulty to disentangle the authentic poems from the inauthentic ones: "while it is certain that Khayyam wrote many quatrains, it is hardly possible, save in a few exceptional cases, to assert positively that he wrote any of those ascribed to him".^{[7]}^{:663}
In addition to the Persian quatrains, there are twentyfive Arabic poems attributed to Khayyam which are attested by historians such as alIsfahani, Shahrazuri (Nuzhat alArwah, ca. 12011211), Qifti (T?rikh alhukam?, 1255), and Hamdallah Mustawfi (Tarikhi guzida, 1339).^{[4]}^{:39}
Richard N. Frye (1975) emphasizes that there are a number of other Persian scholars who occasionally wrote quatrains, including Avicenna, Ghazzali, and Tusi. He concludes that it is also possible that poetry with Khayyam was the amusement of his leisure hours: "these brief poems seem often to have been the work of scholars and scientists who composed them, perhaps, in moments of relaxation to edify or amuse the inner circle of their disciples".^{[7]}^{:662}
The poetry attributed to Omar Khayyam has contributed greatly to his popular fame in the modern period as a direct result of the extreme popularity of the translation of such verses into English by Edward FitzGerald (1859). FitzGerald's Rubaiyat of Omar Khayyam contains loose translations of quatrains from The Bodleian manuscript. It enjoyed such success in the fin de siécle period that a bibliography compiled in 1929 listed more than 300 separate editions,^{[45]} and many more have been published since.^{[46]}
Khayyam himself rejected any association with the title falsaf? "philosopher" in the sense of Aristotelianism 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".^{[47]}
Khayyam for decades taught the philosophy of Avicena, especially the Book of Healing, in his home town Nishapur, till his death.^{[48]} In an incident he had been requested to comment on a disagreement between Avicena and a philosopher called Abu'lBarak?t alBaghd?d? who had criticized Avicena strongly. Khayyam is said to have answered "[he] does not even understand the sense of the words of Avicenna, how can he oppose what he does not know?"^{[47]}
Khayyam's popularity in the West is closely associated with the reception of his poetry as extolling wine and praising a life of mystic ecstasy and hedonism. This reception also became prevalent in modern Iran, where nightclubs are named after Khayyam^{[]}. Sadegh Hedayat portrayed Khayyam as "a materialist, hedonist , agnostic", who "looked at all religions' questions with a skeptical eye".^{[]}^{[year needed]} Conversely, Khayyam has also been described as a mystical Sufi poet.
As a mathematician, Khayyam 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, Ab? Rayn alB?r?n? and Tusi are associated. There are at least three basic mathematical ideas of strong philosophical dimensions that can be associated with Khayyam.
Sadegh Hedayat is one of the most notable proponents of Khayyam's philosophy as agnostic skepticism. According to Jan Rypka (1934), Hedayat even considered Khayyam an atheist.^{[50]} Hedayat states in his introductory essay to his second edition of the Quatrains of the Philosopher Omar Khayyam 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".^{[51]} He further maintains that Khayyam's usage of Sufic terminology such as "wine" is literal, and that "Khayyam took refuge in wine to ward off bitterness and to blunt the cutting edge of his thoughts."^{[52]}
Edward FitzGerald emphasized the religious skepticism he found in Omar Khayyam.^{[53]} In his preface to the Rubáiyát, he describes Omar's philosophy as Epicurean and claims that Omar was "hated and dreaded by the Sufis, whose practice he ridiculed and whose faith amounts to little more than his own, when stripped of the Mysticism and formal recognition of Islamism under which Omar would not hide."^{[54]}
AlQifti (ca. 11721248) appears to confirm this view of Omar's philsophy. In his work The History of Learned Men he reports that Omar's poems were only outwardly in the Sufi style, but were written with an antireligious agenda.^{[55]} Furthermore, Frye emphasizes that Khayyam was intensely disliked by a number of celebrated Sufi mystics who belonged to the same century. This includes Shams Tabrizi (spiritual guide of Rumi)^{[4]}^{:58}, Najm alDin Daya, and Attar, who regarded Khayyam not as a fellowmystic, but a freethinking scientist who awaited punishments hereafter.^{[7]}^{:663}Mohammad Ali Foroughi concluded that Khayyam's ideas may have been consistent with that of Sufis at times but there is no evidence that he was formally a Sufi. Aminrazavi (2007) states that "Sufi interpretation of Khayyam is possible only by reading into his Rubaiyat extensively and by stretching the content to fit the classical Sufi doctrine".^{[4]}^{:128} In this tradition, Omar's poetry has been cited in the context of New Atheism, e.g. in The Portable Atheist by Christopher Hitchens.^{[56]}
Other commentators do not accept that Omar's poetry has an antireligious agenda and interpret his references to wine and drunkenness in the conventional metaphorical sense common in Sufism. Seyyed Hossein Nasr argues that it is "reductive" to use a literal interpretation of his verses (many of which are of uncertain authenticity to begin with) to establish Omar's philosophy. Instead, he adduces Khayyam's translation of Avicenna's treatise Discourse on Unity (AlKhutbat alTawh?d), where he expresses orthodox views on Divine Unity in agreement with the author.^{[48]} Omar's most important single work on philosophy is a treatise on existence (Fi'lwuj?d), which is explicitly theistic, citing Quranic verses and asserting that all things come from God and that there is an order to all things.^{[48]} In his book On the elaboration of the problems concerning the book of Euclid, Omar also praises a compassionate God and refers to "the master of prophets, Muhammad, and his pure companions."^{[4]}^{:64}
The view of Omar Khayyam as a Sufi was defended by Bjerregaard (1915).^{[57]} Dougan (1991) likewise attributes the reputation of hedonism to the failings of FitzGerald's translation, arguing that Omar's poetry is to be understood as "deeply esoteric".^{[58]}Idries Shah (1999) similarly accuses FitzGerald of misunderstanding Omar's poetry.^{[59]}Edward Henry Whinfield (2000) concludes that "we must not run away with the idea that he was himself a Sufi."^{[60]}
Omar Khayyam was held in very high esteem by his contemporaries. He was primarily venerated as a scholar and scientist, according to contemporary anecdotes primarily for his expertise in horoscopic astrology.^{[61]} It was not unusual for him to be given flowery epithets such as "Proof of Truth", "Philosopher of the World", "Lord of the Wise Men of East and West". Shahrazuri (d. 1300) calls him "Successor to Avicenna", and AlQifti (d. 1248) even though disagreeing with his views concedes he was "unrivalled in his knowledge of natural philosophy and astronomy". AlZamakhshari (d. 1143/4) calls him "Sage of the World" or "Philosopher of the Universe".^{[62]}
The first nonPersian scholar to study Omar Khayyam was the English orientalist Thomas Hyde, who in his Historia religionis veterum Persarum eorumque magorum (1700) translated some of his verses into Latin. Western interest in Persia increased with the Orientalism movement in the 19th century. Joseph von HammerPurgstall (17741856) translated some of Khayyam's poems into German in 1818, and Gore Ouseley (17701844) into English in 1846, but Khayyam remained relatively unknown in the West until after the publication of Edward FitzGerald's Rubaiyat of Omar Khayyam in 1859. FitzGerald's work at first was unsuccessful, but was popularised by Whitley Stokes from 1861 onward, and the work came to be greatly admired by the PreRaphaelites. In 1872 FitzGerald had a third edition printed which increased interest in the work in America. By the 1880s, the book was extremely well known throughout the Englishspeaking world, to the extent of the formation of numerous "Omar Khayyam Clubs" and a "fin de siècle cult of the Rubaiyat"^{[63]}
FitzGerald rendered Omar's name as "Tentmaker", and the anglicized name of "Omar the Tentmaker" resonated in Englishspeaking popular culture for a while. Thus, Nathan Haskell Dole published a novel called Omar, the Tentmaker: A Romance of Old Persia in 1898. Omar the Tentmaker of Naishapur is a historical novel by John Smith Clarke, published in 1910. "Omar the Tentmaker" is also the title of a 1914 play by Richard Walton Tully in an oriental setting, adapted as a silent film in 1922. US General Omar Bradley was given the nickname "Omar the TentMaker" in World War II.^{[64]} The name has also been recorded as a slang expression for "penis".^{[65]}
Omar Khayyam's poems have been translated into many languages, many of the more recent ones more literal than the 1850s translation by FitzGerald.^{[66]}
FitzGerald's translations also reintroduced Khayyam to Iranians, "who had long ignored the Neishapouri poet".^{[67]}Sadeq Hedayat in his Songs of Khayyam (Taranehhaye Khayyam, 1934) first introduced Omar's poetry to modern Iran, and under the Pahlavi dynasty, he was shaped into one of Iran's national icons along with Hafez, Saadi, Avicenna and others. The Mausoleum of Omar Khayyam was reconstructed under Reza Shah, by architect Hooshang Seyhoun, completed in 1963. A statue by Abolhassan Sadighi was erected in Laleh Park, Tehran in the 1960s. A bust by the same sculptor was erected near Khayyam's mausoleum in Nishapur.
A lunar crater Omar Khayyam was named after him in 1970 and a minor planet called 3095 Omarkhayyam, discovered by Soviet astronomer Lyudmila Zhuravlyova in 1980, is named after him.^{[68]}
In 2009, the state of Iran donated a pavilion to the United Nations Office in Vienna, erected at Vienna International Center.^{[69]} The pavilion includes four statues of medieval Persian scholars, one of Omar Khayyam alongside statues of Avicenna, Abu Rayhan Biruni, Zakariya Razi.^{[70]}^{[71]} In 2016, three statues of Khayyam was by Iranian sculptor Hossein Fakhimi were unveiled: one at the University of Oklahoma, one in Nishapur and one to in Florence, Italy.^{[72]}^{[73]}
If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another.
A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles.
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