Omar Khayyam

• 1. Introduction

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  Omar Khayyam, a Persian scholar born in Nishapur in 1048, led a team that reported the length of the solar year as 365.24219858156 days. The report came in 1079. The modern figure is 365.242190 days, so his number lands remarkably close to the truth. Yet his name travels widely for a different reason. Millions know him as a poet, through quatrains that reached English readers in a Victorian translation. Scholars disagree sharply about how many of those poems he actually wrote. The same man also produced mathematics that pointed toward whole new branches of geometry. How did a court scientist of the Seljuk era become a famous literary name? What did he really believe about God? And which of the verses carrying his name can be trusted?

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• 2. Student of Khorasan

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  The historian Bayhaqi knew Khayyam personally and recorded the details of his horoscope, noting that the sun and Mercury stood in Gemini at his birth. Modern scholars used those details to fix his birthday on the 18th of May 1048. His full name was Ghiyath al-Din Abu al-Fath Umar ibn Ibrahim Nishapuri, which medieval Persian texts shorten to Omar Khayyam. Khayyam means tent-maker in Arabic, so many have assumed his forebears pitched tents for a living, though that idea is open to doubt. He grew up in a leading city of the Seljuk Empire, one that had once been a major center of the Zoroastrian religion.

  As a boy he memorized much of the Quran. He then studied religious sciences, Arabic grammar and literature under Mawlana Qadi Muhammad. Next he moved to Khawjah Abu'l-Hasan al-Anbari for mathematics, astronomy and cosmology, including Ptolemy's great work, the Almagest. His early tutors saw his gifts and sent him on to Imam Muwaffaq Nishaburi, called the greatest teacher in all Khorasan. Muwaffaq taught the children of the highest nobility, and the two men stayed close friends over the years. Khayyam may also have studied with Bahmanyar, a disciple of Avicenna.

  Around 1068 the young scholar traveled to the province of Bukhara, where he spent time in the renowned library of the Ark. About 1070 he moved to Samarkand and began his famous Treatise on Algebra. His patron there was Abu Tahir Abd al-Rahman ibn Alaq, the city's governor and chief judge. The Karakhanid ruler Shams al-Mulk Nasr welcomed him too. According to Bayhaqi, Nasr honored him so much that he seated Khayyam beside him on his own throne.

  In 1073-4 peace was concluded with Sultan Malik-Shah I, whose forces had raided Karakhanid lands. In 1074 the Grand Vizier, Nizam al-Mulk, invited Khayyam to the city of Marv to meet the sultan. Khayyam entered Malik-Shah's service that year. The sultan had a large project in mind, and it involved the night sky.

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• 3. The Sultan's Calendar

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  Eight scholars worked under Khayyam's direction at a new observatory in Isfahan. Malik-Shah commissioned it in 1074-5, and the team's task was to make large-scale observations and revise the astronomical tables. The undertaking probably lasted about five years. Its aim was to reform the Persian calendar.

  The recalibrated calendar fixed the first day of the year at the exact moment the center of the Sun crosses the vernal equinox. That moment marks the start of spring, or Nowruz, when the Sun enters the first degree of Aries before noon. The new calendar was named Jalali, in honor of the sultan, and it was inaugurated on the 15th of March 1079. The observatory fell out of use after Malik-Shah died in 1092.

  The Jalali calendar was a true solar calendar, in which each month lasted as long as the Sun takes to cross the matching sign of the zodiac. Its unique 33-year cycle of leap days, described in the works of Khazini, mixed leap years that come every fourth year and every fifth year. Over each cycle there were 25 ordinary years of 365 days and 8 leap years of 366 days. An error of one day builds up over 5,000 years. The Gregorian calendar of 1582 slips a day every 3,330 years. The scholar Moritz Cantor considered the Jalali calendar the most perfect ever devised.

  The calendar stayed in use across Greater Iran from the 11th to the 20th centuries. In 1911 it became the official national calendar of Qajar Iran. In 1925 it was simplified and its month names were modernised, producing the modern Iranian calendar.

  One of Khayyam's pupils, Nizami Aruzi, reported that he saw no great belief in astrological predictions in his teacher. The scholar George Saliba says the Arabic term ilm al-nujum has sometimes been wrongly translated as astrology. By the middle of the tenth century, according to Farabi's Enumeration of the Sciences, that field was already split in two. One half dealt with astrology and the other with theoretical, mathematical astronomy. A later patron, Sultan Sanjar, once asked Khayyam to predict the weather, and by report he did the job poorly.

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• 4. Cubics and Numbers

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  Three of Khayyam's mathematical works survive. They are the Commentary on the Difficulties Concerning the Postulates of Euclid's Elements, the Treatise On the Division of a Quadrant of a Circle, and the Treatise on Algebra, most likely finished in 1079. The Division treatise is undated but was completed before the Algebra. A fourth work, on the binomial theorem and the extraction of roots, has been lost.

  The Treatise on Algebra holds his most celebrated result. Khayyam seems to have been the first to conceive a general theory of cubic equations. He was also the first to solve every type of cubic geometrically, as far as positive roots go. The treatise has three parts. The first covers equations solvable with compass and straight edge, the second covers those needing conic sections, and the third covers equations involving the inverse of the unknown.

  Khayyam compiled an exhaustive list of equations built from lines, squares and cubes. It held three binomial equations, nine trinomial equations and seven tetranomial equations. He concluded that fourteen types of cubic cannot be reduced to a lower degree. For those, he could not construct the unknown segment with compass and straight edge. Instead he found the positive root as a point on the horizontal axis where two conic sections cross, such as two parabolas, or a parabola and a circle. Unlike Descartes, he did this by choosing a unit length and strictly respecting the rule of homogeneity.

  Khayyam admitted that the arithmetic side of these cubics was still unsolved. He wrote that "possibly someone else will come to know it after us". The task stayed open until the sixteenth century, when Cardano, Del Ferro and Tartaglia found a general algebraic solution in Renaissance Italy. Similar methods had appeared before, from Menaechmus and then the tenth-century mathematician Abu al-Jud. Even so, Khayyam's work is considered the first systematic study and the first exact method for cubics. Woepcke, who translated Khayyam's algebra into French in 1851, praised his "power of generalization and his rigorously systematic procedure."

  The Division treatise applied algebra to geometry. In it Khayyam asked whether a circular quadrant can be split in two so that lines dropped from the dividing point to the perpendicular diameters stand in a set ratio. His curve constructions led to equations with cubic and quadratic terms. Rashed and Vahabzadeh argued in 2000 that this geometric approach makes him a precursor of Descartes in inventing analytic geometry. Khayyam also tried to find approximate numerical solutions to cubics, using trigonometric tables.

  A third idea sits in his work on Euclid. Khayyam was displeased with Euclid's definition of equal ratios, so he redefined number itself, using a continued fraction to express a ratio. This let him place irrational quantities and ordinary numbers on the same scale. Youschkevitch and Rosenfeld argue that this "began a true revolution in the doctrine of number." D. J. Struik wrote that Omar was "on the road to that extension of the number concept which leads to the notion of the real number."

  In his algebra, Khayyam mentions a book on extracting the nth root of natural numbers by a law that needed no geometric figures. It was most likely titled Difficulties of Arithmetic, or Mushkilat al-Hisab, and it has not survived. Struik and others believe he must have known the expansion of (a+b) to the power n. Euclid had stated the case of power 2, and Indian mathematicians had established up to power 3. Al-Karaji had earlier found the triangular arrangement of binomial coefficients that Europeans later called Pascal's triangle. Khayyam popularized it in Iran, where it is known as Omar Khayyam's triangle.

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• 5. The Parallel Problem

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  In December 1077 Khayyam completed a commentary on the most troublesome rule in Euclid's geometry, the parallel axiom, also called the Fifth Postulate. He rejected earlier attempts to prove it, mainly because each one assumed something no easier to accept than the postulate itself. Following Aristotle, he refused to allow motion in geometry. That ruled out the different attempt made by Ibn al-Haytham. Frustrated that nobody had derived the axiom from Euclid's other postulates, Khayyam tried to link it to the Fourth Postulate, which says all right angles are equal. His treatment is considered the first not based on circular reasoning, resting instead on a more intuitive postulate.

  His central tool was a four-sided figure now called the Khayyam-Saccheri quadrilateral. Khayyam was the first to consider three separate cases for its top angles: acute, obtuse and right. After proving a series of theorems about them, he showed that the Fifth Postulate follows from the right-angle case. He dismissed the obtuse and acute cases as self-contradictory. We now know that the acute, obtuse and right-angle hypotheses lead respectively to hyperbolic geometry, elliptic geometry and Euclidean geometry. His elaborate attempt therefore showed clearly that non-Euclidean geometries were possible.

  The ideas traveled west through the commentaries of Tusi. John Wallis, professor of geometry at Oxford, translated Tusi's commentary into Latin. The Jesuit geometer Girolamo Saccheri, whose 1733 book Euclides ab omni naevo vindicatus is generally considered the first step toward non-Euclidean geometry, knew Wallis's work. The historian David Eugene Smith noted that Saccheri used the same lemma as Tusi, even lettering the figure in precisely the same way. Smith added that Tusi distinctly credits Omar Khayyam, and that it seems clear Khayyam was his inspirer.

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• 6. Beyond the Equations

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  A short treatise by Khayyam, titled On the Deception of Knowing the Two Quantities of Gold and Silver in a Compound Made of the Two, applies Archimedes' principle to a practical puzzle. The puzzle concerns gold adulterated with silver. Khayyam weighs the compound in air and then in water, because weights are easier to measure exactly than volumes. Repeating the process with pure gold and pure silver shows exactly how much heavier than water each substance is. Eilhard Wiedemann examined the treatise closely and believed Khayyam's solution was more accurate and sophisticated than those of Khazini and al-Nayrizi, who also tackled the subject.

  Another brief treatise explores the link between music and arithmetic. Khayyam gave a systematic classification of musical scales and discussed the mathematical relationships among notes, minor and major intervals, and tetrachords. A thirteenth-century account says two of his prose works became standard authorities, one on precious stones and the other on climatology.

  Six philosophical papers are believed to be his. One, On Existence, or Fi'l-wujud, was written in Persian and deals with existence and its relation to universals. Another, in Arabic, tackles free will and determinism under the title Darurat al-tadad fi'l-alam wa'l-jabr wa'l-baqa. The titles of the other four include On Being and Necessity and Abridgement concerning Natural Phenomena.

  Khayyam once complained about the era in which he lived. He said: "We are the victims of an age when men of science are discredited, and only a few remain who are capable of engaging in scientific research." He added that philosophers of his day mixed true with false and cared only for outward show. When they met a man sincere in his search for truth, he said, they mocked and despised him.

  Khayyam regarded himself as a student of Avicenna. According to Bayhaqi, he was reading the metaphysics in Avicenna's Book of Healing shortly before he died.

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• 7. Quatrains in Doubt

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  The earliest hint of Khayyam as a poet comes from the historian Imad al-Din al-Isfahani, a younger contemporary. In 1174 his Kharidat al-qasr named Khayyam as both a poet and a scientist. Around 1160, Fakhr al-Din al-Razi quoted one of Khayyam's poems, matching quatrain LXII of FitzGerald's first edition. Daya quoted two quatrains around 1230, one of them the same as Razi's. In 1340 Jajarmi included thirteen quatrains in an anthology called Mu'nis al-ahrar, two of them already known from older sources. The Bodleian manuscript Ouseley 140, written in Shiraz in 1460, contains 158 quatrains on 47 folia. It once belonged to William Ouseley, and the Bodleian Library bought it in 1844.

  The trail is shakier than it looks. Quotations of verses attributed to Khayyam in texts of the 13th and 14th centuries are of doubtful authenticity. Some skeptical scholars suggest the entire tradition may be pseudepigraphic. In 1934 Hans Heinrich Schaeder said Khayyam's name "is to be struck out from the history of Persian literature" for lack of material that can be firmly attributed to him. Five of the later attributed quatrains appear in the Sindbad-Nameh as early as 30 years after his death. That shows the verses were circulating, but not that he wrote them. Edward Granville Browne wrote in 1906 that it is certain Khayyam wrote many quatrains, yet hardly possible to say he wrote any specific one.

  Twenty-five Arabic poems are also attributed to him, attested by historians such as al-Isfahani and Qifti. John Andrew Boyle pointed out that other scholars, including Avicenna, Ghazali and Tusi, also wrote occasional quatrains. Khayyam's poems may have been a pastime for his leisure hours, composed to amuse the inner circle of his disciples.

  Edward FitzGerald's 1859 Rubaiyat of Omar Khayyam made the poet famous. It was a loose translation of quatrains from the Bodleian manuscript. It had earlier Western precursors. Thomas Hyde translated a single quatrain into Latin in 1700, and Joseph von Hammer-Purgstall produced German versions in 1818. Gore Ouseley followed with English poems in 1846. Even so, Khayyam remained little known in the West until FitzGerald's book appeared. At first it flopped. Whitley Stokes began promoting it from 1861, and the Pre-Raphaelites came to admire it. A third edition in 1872 raised interest in America. By the 1880s it was famous across the English-speaking world, with numerous Omar Khayyam Clubs and a so-called fin de siecle cult of the Rubaiyat. A bibliography compiled in 1929 listed more than 300 separate editions.

  FitzGerald rendered Khayyam's name as Tentmaker, and Omar the Tentmaker became a popular figure. Nathan Haskell Dole published a novel with that name in 1898, and John Smith Clarke followed with another in 1910. Richard Walton Tully's 1914 play was adapted as a silent film in 1922. In World War II, US General Omar Bradley picked up the nickname Omar the Tent-Maker.

  One quatrain became especially popular in English: "The Moving Finger writes; and having writ, Moves on: nor all your Piety nor Wit Shall lure it back to cancel half a Line, Nor all your Tears wash out a Word of it." Agatha Christie took the title of her 1942 novel The Moving Finger from it. Martin Luther King quoted it in his speech "Beyond Vietnam: A Time to Break Silence", saying, "Omar Khayyam is right: 'The moving finger writes, and having writ moves on.'" Bill Clinton cited it in an apologetic speech about the Clinton-Lewinsky scandal. Alan Watts cited it in a 1959 television series.

  FitzGerald's success also rekindled interest in Khayyam as a poet in Iran. Sadegh Hedayat's Songs of Khayyam, from 1934, reintroduced the poetic legacy to modern Iran. Turkish writers including Abdullah Cevdet, Riza Tevfik and Yahya Kemal used Khayyam's themes to defend progressive ideas. Over 150 composers have drawn on the Rubaiyat, and the earliest was Liza Lehmann.

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• 8. Faith and Doubt

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  A literal reading of the quatrains makes Khayyam look like a pessimist, an Epicurean and an agnostic. Iranologists including Arthur Christensen, Hans Heinrich Schaeder, John Andrew Boyle, Edward Denison Ross, Edward Henry Whinfield and George Sarton took that view. FitzGerald emphasized the skepticism he found, claiming Khayyam was "hated and dreaded by the Sufis". He insisted the poet's wine was "the veritable Juice of the Grape". The Persian novelist Sadegh Hedayat held that Khayyam believed in no separate soul, so that lucky bodily particles might end up in a jug of wine. In FitzGerald's English, one quatrain declares "there is no heaven but here" and "there is no hell but here". Christopher Hitchens included Khayyam's poetry in The Portable Atheist.

  A medieval writer, al-Qifti, appears to confirm this reading. In The History of Learned Men he says the poems were Sufi only in style, and that Khayyam was once indicted for impiety. Qifti says he went on pilgrimage to prove his piety and then lived a strictly religious life, attending the place of worship morning and evening. The pilgrimage followed a political collapse. After the death of Malik-Shah, and of his vizier, who was probably murdered by the Ismaili order of Assassins, Khayyam lost favor at court. Qifti suggests the journey to Mecca was partly a public display of faith, meant to calm suspicions of skepticism and rumored sympathy with Zoroastrianism raised by a hostile clergy. The new sultan, Sanjar, then invited him to Marv, possibly as court astrologer. Poor health later let him return to his hometown, where he seems to have lived as a recluse.

  Others read the verses as mystical Sufi poetry. The French translator J. B. Nicolas argued that the constant calls to drink wine are a metaphor for the rapture of the enlightened state. Bjerregaard, Idries Shah and Dougan defended the Sufi view, and Dougan blamed FitzGerald's translation for the poet's reputation for hedonism. Iranian experts Mohammad Ali Foroughi and Mojtaba Minovi found no evidence that Khayyam was formally a Sufi. Aminrazavi said that reading him as a Sufi requires stretching his words. Boyle noted that several celebrated Sufi mystics of the same century disliked him intensely. Shams Tabrizi, the spiritual guide of Rumi, was among them. Najm al-Din Daya called him "an unhappy philosopher, atheist, and materialist". Attar saw him as a free-thinking scientist, not a fellow mystic.

  Seyyed Hossein Nasr calls it "reductive" to judge Khayyam by verses of uncertain authenticity. He points instead to Khayyam's interpretive translation of Avicenna's Discourse on Unity, which expresses orthodox views on Divine Unity. The prose works attributed to Khayyam are explicitly theistic and take up subjects such as the existence of God. Aminrazavi notes that his treatises open with salutations and prayers praising God and Muhammad. Biographers called him Imam, Ghiyath al-Din, and Hujjat al-Haqq, the Evidence of Truth. Those who praise his religiosity tend to skip his poetry, and those who mention his poetry often skip his piety. Bayhaqi, writing earlier than most, called Omar a very pious man who held orthodox views to his last hour. The question remains open.

  Khayyam died at 83 in Nishapur, on the 4th of December 1131. His disciple Nizami Aruzi tells of a day in Balkh in 1112-3, when Khayyam sat with Isfizari, one of his colleagues on the Jalali calendar. There he predicted, "my tomb shall be in a spot where the north wind may scatter roses over it". Four years after the death, Aruzi found the grave in a cemetery on the road to Marv, at the foot of a garden wall. Pear and apricot trees leaned over the wall and dropped their blossoms until the tombstone was hidden beneath them.

  Under the Pahlavi dynasty, the architect Houshang Seyhoun designed a new white marble monument for the tomb. In the 1960s the sculptor Abolhassan Sadighi made a statue for Laleh Park in Tehran and a bust for Nishapur. In 2016 three statues of Khayyam were unveiled, at the University of Oklahoma, in Nishapur and in Florence. A lunar crater was named for him in 1970, and the minor planet 3095 Omarkhayyam, discovered by Lyudmila Zhuravlyova, followed in 1980. Google marked his 971st birthday with a Doodle on the 18th of May 2019.

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