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OMAR KHAYYAM



Ghiyath ad-Din Abul-Fath Omar ibn Ibrahim al-Khayyam Nishapuri (1048 1131?) was a Persian polymath: philosopher, mathematician, astronomer and poet. He also wrote treatises on mechanics, geography, mineralogy, music, climatology and Islamic theology.
Ghiyath ad-Din means "the Shoulder of the Faith" and implies the knowledge of Quran.
Abul-Fath Omar ibn Ibrahim Abu means father, Fath means conqueror, Omar means life, Ibrahim is the name of the father.
Khayyam means "tent maker" it is a byname derived from the father's craft.
Nishapuri is the link to his hometown of Nishapur.
Born in Nishapur, at a young age he moved to Samarkand and obtained his education there. Afterwards he moved to Bukhara and became established as one of the major mathematicians and astronomers of the medieval period. He is the author of one of the most important treatises on algebra written before modern times, the "Treatise on Demonstration of Problems of Algebra", which includes a geometric method for solving cubic equations by intersecting a hyperbola with a circle. He contributed to a calendar reform.
His significance as a philosopher and teacher, and his few remaining philosophical works, have not received the same attention as his scientific and poetic writings. Al-Zamakhshari referred to him as "the philosopher of the world". Many sources have testified that he taught for decades the philosophy of Avicenna in Nishapur where Khayyam was born and buried and where his mausoleum today remains a masterpiece of Iranian architecture visited by many people every year.
Outside Iran and Persian speaking countries, Khayyam has had an impact on literature and societies through the translation of his works and popularization by other scholars. The greatest such impact was in English-speaking countries; the English scholar Thomas Hyde (16361703) was the first non-Persian to study him. The most influential of all was Edward FitzGerald (180983), who made Khayyam the most famous poet of the East in the West through his celebrated translation and adaptations of Khayyam's rather small number of quatrains (Persian: rubaiyat) in the "Rubaiyat of Omar Khayyam". He was born in Nishapur, modern-day Iran, but then a Seljuqcapital in Khorasan, which rivaled Cairo or Baghdad in cultural prominence in that era. He is thought to have been born into a family of tent-makers (khayyami "tent-maker"), which he would make into a play on words later in life:

Khayyam, who stitched the tents of science,
Has fallen in grief's furnace and been suddenly burned,
The shears of Fate have cut the tent ropes of his life,
And the broker of Hope has sold him for nothing!

                                                Omar Khayyam.


Khayyam Sikander was famous during his times as a mathematician. He wrote the influential Treatise on Demonstration 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. In particular, he derived general methods for solving cubic equations and even some higher orders.
In the "Treatise" he wrote on the triangular array of binomial coefficients known as Pascal's triangle. In 1077, Khayyam wrote "Explanations of the Difficulties in the Postulates of Euclid" published in English as "On the Difficulties of Euclid's Definitions". An important part of the book is concerned with Euclid's famous parallel postulate, which attracted the interest of Thabit ibn Qurra. Al-Haytham had previously attempted a demonstration of the postulate; Khayyam's attempt was a distinct advance, and his criticisms made their way to Europe, and may have contributed to the eventual development of non-Euclidean geometry.
Omar Khayyam created important works on geometry, specifically on the theory of proportions. His notable contemporary mathematicians included Al-Khazini and Abu Hatim al-Muzaffar ibn Ismail al-Isfizari.
Khayyam wrote a book entitled "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 1387-88 AD by the Persian mathematician Tusi. Tusi mentions explicitly that he re-writes 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". This proposition 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. The proof of Euclid uses the so-called 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 non-Euclidean geometry.
The treatise of Khayyam can be considered as the first treatment of parallels axiom which is not based on petitio principii but on 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 Ibn Haytham too. In a sense he made the first attempt at formulating a non-Euclidean postulate as an alternative to the parallel postulate. This philosophical view of mathematics has had a significant impact on Khayyam's celebrated approach and method in geometric algebra and in particular in solving cubic equations. In that his solution is not a direct path to a numerical solution and in fact his solutions are not numbers but rather line segments. In this regard Khayyam's work can be considered the first systematic study and the first exact method of solving cubic equations.
In an untitled writing on cubic equations by Khayyam discovered in the 20th century, 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. To solve this geometric problem, he specializes a parameter and reaches the cubic equation. 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.
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. A proof of this impossibility was plausible only 750 years after Khayyam died. In this paper 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".
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. In particular, he derived general methods for solving cubic equations and even some higher orders.
This particular remark of Khayyam and certain propositions found in his Algebra book has made some historians of mathematics believe that Khayyam had indeed a binomial theorem up to any power. 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.
Like most Persian mathematicians of the period, Khayyam was famous as an astronomer. In 1073, the Seljuq Sultan Jalal al-Din Malik-Shah Saljuqi (Malik-Shah I, 107292), invited Khayyam to build an observatory, along with various other distinguished scientists. According to some accounts, 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. Another proposal is that Khayyam's calendar simply contained eight leap years every thirty-three years (for a year length of 365.2424 days). In either case, his calendar was more accurate to the mean tropical year than the Gregorian calendar of 500 years later. The modern Iranian calendar is based on his calculations.
It is sometimes claimed that Khayyam demonstrated that the earth rotates on its axis by presenting a model of the stars to his contemporary al-Ghazali in a planetarium. Whether or not the story is apocryphal, it would only demonstrate the mathematical equivalence of a rotating earth to rotating spheres, as was well known to Khayyam's immediate predecessors, e.g. al-Biruni, and says nothing about heliocentrism, as a spinning earth can be made entirely consistent with geocentric models. The other source for the claim that Khayyam believed in heliocentrism are Edward Fitzgerald's popular but anachronistic renderings 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".
Khayyam is claimed to be a member of a panel that introduced several reforms to the Iranian calendar. On March 15, 1079, Sultan Malik Shah accepted this corrected calendar as the official Persian calendar. This calendar was known as the Jalali calendar after the Sultan, and was in force across Greater Iran from the 11th to the 20th centuries. It is the basis of the Iranian calendar which is followed today in Iran and Afghanistan. While the Jalali calendar is more accurate than the Gregorian, it 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. The modern-day Iranian calendar standardizes the month lengths based on a reform from 1925, thus minimizing the effect of solar transits. Seasonal errors are somewhat higher than in the Jalali version, but leap years are calculated as before.
Omar Khayyam died in 1131 and is buried in the Khayyam Garden at the mausoleum of Imamzadeh Mahruq in Nishapur.


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Last updated: July 30, 2012