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IOS Minaret Vol-1, No.1 (March 2007)
Vol. 14    Issue 01   01 - 30 June 2019

Omar Khayyam

Professor A. R. MOMIN

The 971st birth anniversary of Omar Khayyam, the celebrated Persian mathematician, astronomer and poet, was remembered with a Google Doodle on 18 May 2019.

The period between the decline of the Greco-Roman civilization and the Renaissance, which spans nearly 1000 years, is generally described as the Dark Ages in European history, in which no note-worthy developments in science, medicine and technology took place. Interestingly, this period roughly coincides with the Golden Age of Islamic science.

During the Golden Age of Islamic science, between the 9th and 16th centuries, Muslim scientists made original, wide-ranging and pioneering contributions to science, medicine and technology, including botany, chemistry, medicine and surgery, optics, anatomy, astronomy, mathematics, technology and geography. There is now a substantial, and growing, literature on the subject in English, German, French, Spanish and other European languages as well as in Arabic, Turkish and Persian. Muslim scientists placed a great deal of emphasis on the careful observation of natural phenomenon, on an objective, dispassionate evaluation of every piece of scientific knowledge and, above all, on the confirmation of conclusions through the scientific method. Wiedemann categorically states that the credit for inventing the experimental method in science should go to Muslim scientists, such as Ibn al-Haytham, Al-Razi, Ibn Zuhr and Albiruni.

During the early phase of the intercultural encounter between Muslim scientists and Greek science, which was marked by the assimilation of Greek scientific knowledge, Muslim mathematicians derived much of their knowledge about the theory of numbers from Euclid and Nicomachus of Gerana. However, in the course of time, they adopted a more critical attitude towards the theories and conclusions of Greek mathematicians. The system of Indian reckoning reached Europe in the 12th century through the Latin translation of a mathematical treatise written by Al-Khwarizmi (d. 863). Muslim mathematicians invented the symbol x (or s, which stands for the Arabic shay), to express an unknown quantity. This symbol found its way into Europe via Islamic Spain. Ibn al-Haytham, Al-Tusi and Albiruni made highly original contributions to geometry and trigonometry, which surpassed the theories and methods of Euclid. For a long time, historians of mathematics have debated whether trigonometry was founded by Levi Ben Gerson or Regiomontanus. In 1900 the German mathematician V. Braunmuhl set this debate to rest by convincingly demonstrating that the credit for founding trigonometry goes to Al-Tusi and that both Gerson and Regiomontanus had drawn on his works.

Omar Khayyam

Ghiyath al-Din Abul Fath Omar ibn Ibrahim al-Khayyam, popularly known as Omar Khayyam, was born in a family of professional tent-makers (Khayyam) in Nishapur in northeastern Iran in 1048. His father Ibrahim Khayyami was a wealthy physician. He received his early instruction under the tutelage of Bahmanyar bin Marzban, a Zoroastrian mathematician. Thereafter he studied science, philosophy and medicine under some of the greatest teachers and scholars of his time, including Shaykh Muhammad Mansuri, Muwaffaq Nishapuri and Khwaja Al-Anbari. At the age of 20, Khayyam joined a caravan on a three-month journey to the famed city of Samarqand, which was then a flourishing centre of learning and scholarship. Khayyam spent most of his life at the court of the Karakhanid and Saljuq rulers in Isfahan. At the age of 23 Khayyam joined the service of Sultan Malik Shah I as an advisor. While in Isfahan, he studied the works of Greek mathematicians like Euclid and Apollonius. Later he was invited by Sultan Sanjar as a court astrologer.

In the history of Islamic science, Khayyam’s fame rests mainly on his outstanding and pioneering contributions to mathematics. He wrote several treatises on mathematics and astronomy. His works, which have survived the vicissitudes of time, include Risala fi sharh ma ashkala min musadarat Kitab Uqlidis (A commentary on the difficulties concerning the postulates of Euclid’s Elements), Risala fi qisma rub al-daira (On the division of a quadrant of a circle), and Maqala fil-jabr wal-muqabalah (On proofs for problems concerning Algebra). Khayyam also wrote a short treatise on the relationship between music and arithmetic. He provided a systematic classification of musical notes.

Though Khayyam avidly read the works of Greek mathematicians, he was critical of some of their assumptions and postulates. In fact, his original researches blazed a trail for the development of a non-Euclidian geometry. Khayyam was the first mathematician to consider the three cases of acute, obtuse and right angle for the summit angles of a quadrilateral. He started from a non-Euclidean concept that expressed ratios in terms of continued fractions. Nasir al-Din al-Tusi wrote a detailed commentary on the works of Khayyam. John Wallis, a professor of geometry at the University of Oxford, translated Tusi’s commentary into Latin. Khayyam’s original researches reached Europe through this translation and many European mathematicians were influenced by them. Giovanni Girolamo Saccheri (1667-1733), an eminent Italian mathematician and scholastic philosopher who is generally considered the forerunner of non-Euclidian geometry, was familiar with this translation and was influenced by the ideas of Khayyam’s and Tusi. In fact, as David Eugene Smith (1860-1944), an American historian of mathematics, has remarked, “Saccheri used the same lemma as the one used by Tusi, even lettering the figure in precisely the same way and using the lemma for the same purpose.”

Khayyam discussed at length the relationship between the concepts of ratio and numbers and pointed to theoretical difficulties about Euclid’s formulations. He rejected Euclid’s definition of equal ratios, redefined the concept of number and made an original and pioneering contribution to the concept of irrational number. He sought to apply and unify algebra and geometry in an ingenious way, seven centuries before Pierre de Fermat and Rene Descartes attempted to integrate the two branches of the mathematical sciences. Some historians of mathematics have pointed out that Khayyam was a forerunner of the French mathematician and philosopher in the invention of analytic geometry. Khayyam was the first mathematician to conceive a general theory of cubic equations. He observed that the solution of cubic equations requires the use of conic sections and not the traditional method base on the ruler and the compass. Franz Woepcke (1826-1864) was a German mathematician who had studied Arabic at the University of Bonn and spent most of his life in Paris. He published a French translation of Khayyam’s work on algebra L'algèbre d'Omar Alkhayyâmî, publiée, traduite et accompagnée d'extraits des manuscrits inédits, in 1851. Woepcke praised Khayyam for his “power of generalisation and his rigorously systematic methodology.”


In 1974, Sultan Malik Shah commissioned Khayyam to build an observatory at Isfahan and to reform the Persian calendar. A panel of six astronomers made astronomical observations and revised astronomical tables under the guidance and supervision of Khayyam. Khayyam measured the length of the year with astonishing accuracy. His observations were based on the sun’s movement and quadrennial and quinquennial leap years. The average tropical year length quoted today is 365.24219858156 days. Khayyam arrived at this calculation with amazing precision in 1079. He developed a set of astronomical tables called Al-Zij al-Malikshahi, named after his royal patron. His observations and calculations led to the development of a new Jalali calendar. It was a solar calendar in which the duration of each month is equal to the time of the passage of the sun across the corresponding sign of the zodiac. It introduced a unique 33-year intercalation cycle. The calendar consisted of 25 ordinary years that included 365 days, and 8 leap years that included 366 days. The Jalali calendar has 365 days with a very precise 33-year intercalation cycle. The Jalali calendar, which was inaugurated on 15 March 1079, remained in use throughout Iran from the 11th to the 20th century. The Jalali calendar surpasses the 1582 Gregorian calendar in accuracy. In 1925 this calendar was simplified and the traditional names of the months were replaced by modern names, and has since then been Iran’s national calendar.


Khayyam’s fame in the West rests mainly on the translation of his poetic compositions or quatrains into European languages. Thomas Hyde translated some of Khayyam’s quatrains into Latin in 1700. Joseph von Hammer Purgstall (1774-1856) translated some of his poems into German in 1818. Gore Ouseley (1770-1844) translated the poems into English in 1849. Edward FitzGerald published an English translation of Khayyam’s quatrains, called The Rubaiyat of Omar Khayyam, in 1859. FitzGerald’s translation became extremely popular in Europe. More than 300 separate editions of the translation were published up to 1929. Many more have been published since then.

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