Omar Khayyám

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For the protestor, see Omar Khayam (protestor). For other uses, see Khayyam (disambiguation)

Omar Khayyám

Portrait of Khayyam at his Mausoleum in Nishapur
Full name	Omar Khayyám
Birth	May 18, 1048
Death	December 4, 1122
School/tradition	Persian mathematics, Persian poetry, Persian philosophy
Main interests	Poetry, Mathematics, Philosophy, Astronomy
Influenced by Abū Rayhān al-Bīrūnī, Avicenna

Omar Khayyam (Persian: عمر خیام), (born 1048 AD, Neyshapur, Iran—1123 AD, Neyshapur, Iran), was a Persian polymath,^[1] mathematician, philosopher, astronomer and poet.

He has also become established as one of the major mathematicians and astronomers of the medieval period. Recognized as the author of the most important treatise on algebra before modern times as reflected in his Treatise on Demonstration of Problems of Algebra giving a geometric method for solving cubic equations by intersecting a hyperbola with a circle.^[2] He also contributed to calendar reform and may have proposed a heliocentric theory well before Copernicus.^{[citation needed]}

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. Zamakhshari referred to him as “the philosopher of the world”. Many sources have also testified that he taught for decades the philosophy of Ibn Sina in Nishapur where Khayyam lived most of his life, breathed his last, and was buried and where his mausoleum remains today a masterpiece of Iranian architecture visited by many people every year.^[3]

Outside Iran and Persian speaking countries, Khayyam has had impact on literature and societies through translation and works of scholars. The greatest such impact was in English-speaking countries; the English scholar Thomas Hyde (1636–1703) was the first non-Persian to study him. However the most influential of all was Edward FitzGerald (1809–83)^[4] 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 (rubaiyaas) in Rubaiyat of Omar Khayyam.

[edit] Early life

Khayyam's full name is Ghiyāth ad-Dīn Abul-Fat'h Umar ibn Ibrāhīm Khayyām Neyshābūri (Persian: غیاث الدین ابو الفتح عمر بن ابراهیم خیام نیشابوری) and was born in Nishapur, Iran, then a Seljuk capital in Khorasan (present Northeast Iran), rivaling Cairo or Baghdad. One dubious source mentions that Khayyam’s father may have been a convert, presumably from the Zoroastrian religion to Sunni Islam, and so Khayyam was a first generation Muslim. Other biographers of Khayyam^[5] have not supported this claim; most describe him as a Shi'a Muslim^[6][7]. He is thought to have been born into a family of tent makers (literally, al-khayyami means "tent maker"); later in life he would make this into a play on words:

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!^[2]

He spent part of his childhood in the town of Balkh (present northern Afghanistan), studying under the well-known scholar Sheik Muhammad Mansuri. Subsequently, he studied under Imam Mowaffaq Nishapuri, who was considered one of the greatest teachers of the Khorassan region.

According to a well-known legend called Three Schoolmates, two other exceptional students studied under the Imam Mowaffaq at about the same time: Nizam-ul-Mulk (b. 1018), who went on to become the Vizier to the Seljukid Empire, and Hassan-i-Sabah (b.1034), who became the leader of the Hashshashin (Nizar Ismaili) sect. It was said that these students became friends, and after Nizam-ul-Mulk became Vizier, Hassan-i-Sabah and Omar Khayyám each went to him, and asked to share in his good fortune. Hassan-i-Sabah demanded and was granted a place in the government, but he was ambitious, and was eventually removed from power after he participated in an unsuccessful coup against his benefactor, the Vizier. Omar Khayyám was more modest and asked merely for a place to live, to study science, and to pray. He was granted a yearly pension of 1,200 mithkals of gold from the treasury of Nishapur. He lived on this pension for the rest of his life.

This legend is dubious and is rejected by many scholars (e.g. Foroughi and Aghaeipour),^[8] in part due to the 30-year age difference between Khayyam and Nizam-ul-Mulk, which makes it unlikely for the two to have attended school together, also considering the fact that the three men grew up in different parts of the country. The popularity and spread of the legend, however, is notable and could perhaps be explained by the fact that the three men were the most prominent figures of their time and represented three dominant approaches to reform and betterment of the society, namely, scientific discovery, represented by Khayyam, armed rebellion, represented by Hassan-i-Sabah, and strengthening the power establishment and the rule of law and order, represented by Nizam-ul-Mulk.

[edit] Mathematician

Omar Khayyam 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. Also, he was the first Persian mathematician to call the unknown factor of an equation (i.e., the x) shiy (meaning thing or something in Arabic). This word was transliterated to Spanish during the Middle Ages as xay, and, from there, it became popular among European mathematicians to call the unknown factor either xay, or more usually by its abbreviated form, x, which is the reason that unknown factors are usually represented by an x.^[9]

"Cubic equation and intersection of conic sections" the first page of two-chaptered manuscript kept in Tehran University

In the Treatise he also wrote on the triangular array of binomial coefficients known as Pascal's triangle. In 1077, Omar wrote Sharh ma ashkala min musadarat kitab Uqlidis (Explanations of the Difficulties in the Postulates of Euclid). An important part of the book is concerned with Euclid's famous parallel postulate, which had also attracted the interest of Thabit ibn Qurra. Al-Haytham had previously attempted a demonstration of the postulate; Omar'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 Khayyám also had other notable work in geometry, specifically on the theory of proportions.

[edit] Theory of parallels

Omar 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 us 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."^[10] This proposition ^[11] 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.^[12] 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 Omar 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.^[13] In a sense he made the first attempt at formulating a non-Euclidean postulate as an alternative to the parallel postulate,^[14]

[edit] Geometric Algebra

Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved by propositions five and six of Book two of Elements.

—Omar Khayyam^[15]

This philosophical view of mathematics (see below) 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.^[16]

In an untitled writing on cubic equation by Khayyam discovered in 20th century^[15], 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.^[17] To solve this geometric problem, he specializes a parameter and reaches the cubic equation x³ + 200x = 20x² + 2000.^[15] 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.^[18]

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. ^[15] A proof of this impossibility was plausible only 750 years after Khayyam passed away. 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."^[15]

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.^[16] In particular, he derived general methods for solving cubic equations and even some higher orders.

[edit] Binomial theorem and extraction of roots

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. Omar was the mathematician who noticed the importance of a general binomial theorem. The argument supporting the claim that Omar had a general binomial theorem is based on his ability to extract roots. ^[20]

[edit] Khayyam-Saccheri quadrilateral

The Khayyam–Saccheri quadrilateral was first considered by Omar Khayyam in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid.^[21] Unlike many commentators on Euclid before and after him (including of course Saccheri), 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):

Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge.^[22]

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.

It wasn't until 600 years later that Giordano Vitale made an advance on Khayyam in his book Euclide restituo (1680, 1686), when he used the quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. Saccheri himself based the whole of his long, heroic, and ultimately flawed proof of the parallel postulate around the quadrilateral and its three cases, proving many theorems about its properties along the way.

[edit] Astronomer

Like most Persian mathematicians of the period, Omar Khayyám was also famous as an astronomer. In 1073, the Seljuk Sultan Sultan Jalal al-Din Malekshah Saljuqi (Malik-Shah I, 1072-92), invited Khayyám to build an observatory, along with various other distinguished scientists. One being Shamse Tabrizi, his mentor and the father of Kimia Khatoon,who he fell in love with. Eventually, Khayyám and his colleagues measured the length of the solar year as 365.24219858156 days (correct to six decimal places). This calendric measurement has only a one-hour error every 5,500 years, whereas the Gregorian Calendar, adopted in Europe four centuries later, has a 1-day error in every 3,330 years, but is easier to calculate.

[edit] Calendar reform

Statue of Omar Khayyam in Iran.

Omar Khayyam was part of a panel that introduced several reforms to the Persian calendar, largely based on ideas from the Hindu calendar. On March 15, 1079, Sultan Malik Shah I accepted this corrected calendar as the official Persian calendar.^[23]

This calendar was known as 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 32 days depending on the moment when the sun crossed 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 Khayyám also built a star map (now lost), which was famous in the Persian and Islamic world.

[edit] Heliocentric theory

It is said that Omar Khayyam also estimated and proved to an audience that included the then-prestigious and most respected scholar Imam Ghazali, that the universe is not moving around earth as was believed by all at that time.^{[citation needed]} By constructing a revolving platform and simple arrangement of the star charts lit by candles around the circular walls of the room, he demonstrated that earth revolves on its axis, bringing into view different constellations throughout the night and day (completing a one-day cycle). He also elaborated that stars are stationary objects in space which, if moving around earth, would have been burnt to cinders due to their large mass. Some of these ideas may have been transmitted to Western science after the Renaissance.

[edit] Poet

Omar Khayyám's poetic work has eclipsed his fame as a mathematician and scientist.

He is believed to have written about a thousand four-line verses or quatrains (rubaai's). In the English-speaking world, he was introduced through the Rubáiyát of Omar Khayyám which are rather free-wheeling English translations by Edward FitzGerald (1809-1883).

Other translations of parts of the rubáiyát (rubáiyát meaning "quatrains") exist, but FitzGerald's are the most well known. Translations also exist in languages other than English.

Ironically, FitzGerald's translations reintroduced Khayyam to Iranians "who had long ignored the Neishapouri poet." A 1934 book by one of Iran's most prominent writers, Sadeq Hedayat, Songs of Khayyam, (Taranehha-ye Khayyam) is said have "shaped the way a generation of Iranians viewed" the poet.^[24]

Omar Khayyam's personal beliefs are not known with certainty, but much is discernible from his poetic oeuvre.

[edit] Poetry

(These poems were translated by Edward FitzGerald and are potentially more revealing of the thoughts of Edward than Omar.)

Monument to Omar Khayyám in Bucharest.

And, as the Cock crew, those who stood before
  The Tavern shouted - "Open then the Door!
You know how little time we have to stay,
  And once departed, may return no more."

Alike for those who for TO-DAY prepare,
  And that after a TO-MORROW stare,
A Muezzin from the Tower of Darkness cries
  "Fools! your reward is neither Here nor There!"

Why, all the Saints and Sages who discuss'd
  Of the Two Worlds so learnedly, are thrust
Like foolish Prophets forth; their Words to Scorn
  Are scatter'd, and their mouths are stopt with Dust.

Oh, come with old Khayyam, and leave the Wise
  To talk; one thing is certain, that Life flies;
One thing is certain, and the Rest is Lies;
  The Flower that once has blown for ever dies.

Myself when young did eagerly frequent
  Doctor and Saint, and heard great Argument
About it and about: but evermore
  Came out of the same Door as in I went.

With them the Seed of Wisdom did I sow,
  And with my own hand labour'd it to grow:
And this was all the Harvest that I reap'd -
  "I came like Water, and like Wind I go."

Into this Universe, and why not knowing,
  Nor whence, like Water willy-nilly flowing:
And out of it, as Wind along the Waste,
  I know not whither, willy-nilly blowing.

The Moving Finger writes; and, having writ,
  Moves on: nor all thy Piety nor Wit
Shall lure it back to cancel half a Line,
  Nor all thy Tears wash out a Word of it.

And that inverted Bowl we call The Sky,
  Whereunder crawling coop't we live and die,
Lift not thy hands to It for help - for It
  Rolls impotently on as Thou or I.

[edit] Views on religion

"At the Tomb of Omar Khayyam", by Jay Hambidge.

Despite strong Islamic training, it is clear that Omar Khayyam himself was undevout and had no sympathy with popular religion,^[25] but the verse: "Enjoy wine and women and don't be afraid, God has compassion," suggests that he wasn't an atheist. Some religious Iranians have argued that Khayyam's references to intoxication in the Rubaiyat were actually the intoxication of the religious worshiper with his Divine Beloved - a Sufi conceit. This however, is reportedly a minority opinion dismissed as wishful pious thinking by most Iranians.^[24]

It is almost certain that Khayyám objected to the notion that every particular event and phenomenon was the result of divine intervention. Nor did he believe in an afterlife with a Judgment Day or rewards and punishments. Instead, he supported the view that laws of nature explained all phenomena of observed life. One hostile orthodox account of him shows him as "versed in all the wisdom of the Greeks" and as insistent that studying science on Greek lines is necessary.^[25] He came into conflict with religious officials several times, and had to explain his views on Islam on multiple occasions; there is even one story about a treacherous pupil who tried to bring him into public odium. The contemporary Ibn al Kifti wrote that Omar Khayyam "performed pilgrimages not from piety but from fear" of his contemporaries who divined his unbelief.^[25]

Khayyám's disdain of Islam in general and its various aspects such as eschatology, Islamic taboos and divine revelation are clearly visible in his writings, particularly the quatrains, which as a rule reflect his intrinsic conclusions describing those who claim to receive God's word as maggot-minded fanatics (via Le Gallienne's translation):^[26]

Allah, perchance, the secret word might spell;

If Allah be, He keeps His secret well;

 What He hath hidden, who shall hope to find?

Shall God His secret to a maggot tell?

...

The Koran! well, come put me to the test—

Lovely old book in hideous error drest—

 Believe me, I can quote the Koran too,

The unbeliever knows his Koran best.

 

And do you think that unto such as you,

A maggot-minded, starved, fanatic crew,

 God gave the secret, and denied it me?—

Well, well, what matters it! believe that too.^[27]

Although a great number of quatrains erroneously attributed to Khayyam manifest a more colorful irreligiousness and hedonism, nevertheless, the number of his original quatrains that advocate laws of nature and deny the idea of resurrection and eternal life readily outweigh others that express the slightest devotion or praise to God or Islamic beliefs. The following two quatrains are representative of numerous others that serve to reject many tenets of Islamic dogma:

O cleric, we are more active than you,
even so drunk, we are more attentive than you,
You drink the blood of men, we drink the blood of grapes [wine],
Be fair, which one of us is more bloodthirsty?

خيام اگر ز باده مستى خوش باش

با ماه رخى اگر نشستى خوش باش

چون عاقبت كار جهان نيستى است

انگار كه نيستى، چو هستى خوش باش

which translates in Fitzgerald's work as:

And if the Wine you drink, the Lip you press,

End in the Nothing all Things end in — Yes —

Then fancy while Thou art, Thou art but what

Thou shalt be — Nothing — Thou shalt not be less.

A more literal translation could read:

If with wine you are drunk be happy,

If seated with a moon-faced (beautiful), be happy,

Since the end purpose of the universe is nothing-ness;

Hence picture your nothing-ness, then while you are, be happy!

آنانكه ز پيش رفته‌اند اى ساقى

درخاك غرور خفته‌اند اى ساقى

رو باده خور و حقيقت از من بشنو

باد است هرآنچه گفته‌اند اى ساقى

which Fitzgerald has boldy interpreted as:

Why, all the Saints and Sages who discuss’d

Of the Two Worlds so learnedly — are thrust

Like foolish Prophets forth; their Words to Scorn

Are scatter’d, and their Mouths are stopt with Dust.

A literal translation, in an ironic echo of "all is vanity", could read:

Those who have gone forth, thou cup-bearer,

Have fallen upon the dust of pride, thou cup-bearer,

Drink wine and hear from me the truth:

(Hot) air is all that they have said, thou cup-bearer.

[edit] Philosopher

Tomb of Omar Khayyam in Neishapur, Iran.

Khayyam himself rejects to be associated with the title falsafi- (lit. philosopher) in the sense of Aristotelian one 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".^[28]

However it is now established that Khayyam taught for decades the philosophy of Aviccena in his home towm Nishapur.^[3] In an incident he had been requested to comment on a disagreement between Aviccena and a philosopher called Abu'l-Barakat (known also as Nathanel) who had criticized Aviccena 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?"^[28]

Khayyam the philosopher could be understood from two rather distinct sources. One is through his Rubaiyat and the other through his own works in light of the intellectual and social conditions of his time. ^[29] The latter could be informed by the evaluations of Khayyam’s works by scholars and philosophers such as Bayhaqi, Nezami Aruzi, and Zamakhshari and also Sufi poets and writers Attar Nishapuri and Najmeddin Razi.

As a mathematician, Khayam 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, Biruni, and Tusi are associated. There are at least three basic mathematical ideas of strong philosophical dimensions that can be associated with Khayyam.

1. Mathematical order: From where does this order issue, and why does it correspond to the world of nature? His answer is in one of his philosophical "treatises on being". Khayyam’s answer is that "the Divine Origin of all existence not only emanates wojud or being, by virtue of which all things gain reality, but It is also the source of order that is inseparable from the very act of existence."^[29]
2. The significance of postulates (i.e. axiom) in geometry and the necessity for the mathematician to rely upon philosophy and hence the importance of the relation of any particular science to prime philosophy. This is the philosophical background to Khayyam's total rejection of any attempt to "prove" the parallel postulate and in turn his refusal to bring motion into the attempt to prove this postulate as had Ibn al-Haytham because Khayyam associated motion with the world of matter and wanted to keep it away from the purely intelligible and immaterial world of geometry.^[29]
3. Clear distinction made by Khayyam, on the basis of the work of earlier Persian philosophers such as Avicenna, between natural bodies and mathematical bodies. The first is defined as a body that is in the category of substance and that stands by itself, and hence a subject of natural sciences, while the second, also called “volume”, is of the category of accidents (attributes) that do not subsist by themselves in the external world and hence is the concern of mathematics. Khayyam was very careful to respect the boundaries of each discipline and criticized Ibn al-Haytham in his proof of the parallel postulate precisely because he had broken this rule and had brought a subject belonging to natural philosophy, that is, motion, which belongs to natural bodies, into the domain of geometry, which deals with mathematical bodies.^[29]

[edit] Legacy

• A lunar crater Omar Khayyam was named after him in 1970.
• A minor planet 3095 Omarkhayyam discovered by Soviet astronomer Lyudmila Zhuravlyova in 1980 is named after him.^[30]

[edit] Notes

[edit] References