Omar Khayyám ( ), (born 1048
AD, Neyshapur, Iran—1131 AD, Neyshapur, Iran), was a
Persian polymath, mathematician, philosopher, astronomer, physician, and poet. He also wrote treatises on
mechanics,
geography, and
music.
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. He also contributed to the
calendar reform and may have proposed a
heliocentric theory well before
Copernicus.
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
Khayyám 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.
Outside Iran and Persian speaking countries, Khayyám 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) who
made Khayyám the most famous poet of the East in the West through
his celebrated translation and
adaptations of
Khayyám's rather small number of
quatrains
(
rubaiyaas) in
Rubáiyát of Omar
Khayyám.
Early life
Khayyám's
full name was Ghiyath al-Din Abu'l-Fath Umar ibn Ibrahim
Al-Nishapuri al-Khayyami ( ) and was born in Nishapur, Iran, then a
Seljuk capital in Khorasan (present Northeast Iran), rivaling
Cairo or Baghdad.
He is thought to have been born into a family of tent makers
(literally,
al-khayyami in Arabic means "tent-maker");
later in life he would make this into a play on words:
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.
Mathematician
Omar Khayyám 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.
"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)
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 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.
Theory of parallels
Khayyám 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 Khayyám's own words" and quotes Khayyám, 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 Khayyám can be considered as the first treatment of
parallels axiom which is not based on
petitio principii but on more intuitive
postulate. Khayyám
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,
Geometric algebra
This philosophical view of mathematics (see below) has had a
significant impact on Khayyám'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 Khayyám'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 Khayyám discovered in
20th century, where the above quote appears, Khayyám 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 x^3+200x=20x^2+2000. 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 Khayyám died. In this paper Khayyám 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.
Binomial theorem and extraction of roots
This particular remark of Khayyám and certain propositions found in
his Algebra book has made some historians of mathematics believe
that Khayyám 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.
Khayyam-Saccheri quadrilateral
The
Khayyam–Saccheri
quadrilateral was first considered by Omar Khayyám in the late
11th century in Book I of
Explanations of the Difficulties in
the Postulates of Euclid. Unlike many commentators on Euclid
before and after him (including of course Saccheri), Khayyám 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.
Khayyám 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 Khayyám
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.
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 365.2425 days. Omar's calendar was
more accurate than 500 years later the Gregorian calendar. The
modern Iranian calendar is based on his calculations
Calendar reform
Omar Khayyám 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.
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 isfollowed 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 areform 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.
Heliocentric theory
It is said that Omar Khayyám 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. 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.
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 Khayyám 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.
Omar Khayyám's personal beliefs are not known with certainty, but
much is discernible from his poetic oeuvre.
Poetry
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.
Views on religion
In his own writings, Khayyám describes himself as an atheist,
rejecting religion , and the concept of the afterlife.
There have been widely divergent views on Khayyám. According to
Seyyed Hossein Nasr no other
Iranian writer/scholar is viewed in such extremely differing ways.
At one end of the spectrum there are night clubs named after
Khayyám and he is seen as a agnostic hedonist. On the other end of
the spectrum, he is seen as a mystical
Sufi
poet influenced by platonic traditions.
Robertson (1914) believes that Omar Khayyám himself was undevout
and had no sympathy with popular religion, but the verse: "Enjoy
wine and women and don't be afraid, God has compassion," suggests
that he wasn't an atheist. He further believes that 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..
Roberston (1914) further opines that Khayyám 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 Khayyám "performed
pilgrimages not from piety but from fear" of
his contemporaries who divined his unbelief.
The following two quatrains are representative of numerous others
that serve to reject many tenets of religious 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.
But some specialists, like
Seyyed
Hossein Nasr who looks at the available philosophical works of
Omar Khayyám, maintain that it is really reductive to just look at
the poems (which are sometimes doubtful) to establish his personal
views about God or religion; in fact, he even wrote a treatise
entitled "al-Khutbat al-gharrå˘" (The Splendid Sermon) on the
praise of God, where he holds orthodox views, agreeing with
Avicenna on
Divine
Unity. In fact, this treatise is not an exception, and
S.H. Nasr gives an example where he
identified himself as a
Sufi, after criticizing
different methods of knowing God, preferring the intuition over the
rational (opting for the so-called "
kashf", or
unveiling, method):
The same author goes on by giving other philosophical writings
which are totally compatible with the religion of Islam, as the
"al-Risålah fil-wujud" (Treatise on Being), written in
Arabic, which begin with Quranic verses and asserting
that all things come from God, and there is an order in these
things. In another work, "Risålah jawåban li-thalåth maså˘il"
(Treatise of Responseto Three Questions), he gives a response to
question on, for instance, the becoming of the soul post-mortem.
S.H. Nasr even gives some poetry where he is
perfectly in favor of Islamic orthodoxy, but also expressing
mystical views (God's goodness, the ephemerical state of this life,
...):
- Thou hast said that Thou wilt torment me,
- But I shall fear not such a warning.
- For where Thou art, there can be no torment,
- And where Thou art not, how can such a place exist?
- The rotating wheel of heaven within which we wonder,
- Is an imaginal lamp of which we have knowledge by
similitude.
- The sun is the candle and the world the lamp,
- We are like forms revolving within it.
- A drop of water falls in an ocean wide,
- A grain of dust becomes with earth allied;
- What doth thy coming, going here denote?
- A fly appeared a while, then invisible he became.
Giving some reasons of the misunderstaning about Omar Khayyám in
the West, but also elsewhere,
S.H. Nasr concludes by saying that if a
correct study of the authentical rubaiyat is done, but along with
the philosophical works, or even the spiritual biography entitled
Sayr wa sulak (Spiritual Wayfaring), we can no longer view
the man as a simple hedonistic wine-lover, or even an early
skeptic, but, by looking at the entire man, a profound mystical
thinker and scientist whose works are more important than some
doubtful verses. C.H.A. Bjerregaard has earlier resumed the
situation as such:
Philosopher
Khayyám 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".
However it
is now established that Khayyám taught for decades the philosophy
of Aviccena, especially "the Book of Healing", in his home town
Nishapur, till his
death. 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. Khayyám 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?"
Khayyám 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. 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, Khayyám 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 Khayyám.
- 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."
- 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.
- Clear distinction made by Khayyám, 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.
Legacy
See also
Notes
- Omar Khayyam and Max Stirner
- S. H. Nasr Chapter
9.
- Jos
Biegstraaten
- The Quatrains of Omar Khayyam E.H. Whinfield Pg 14
- . Excerpt: In some sense, his treatment was better than ibn
al-Haytham's because he explicitly formulated a new postulate to
replace Euclid's rather than have the latter hidden in a new
definition.
- Mathematical Masterpieces: Further Chronicles by
the Explorers, p. 92
- E. S. Kennedy, Chapter 10 in Cambridge History of Iran , p. 665.
- A. R. Amir-Moez, Khayyam's Solution of Cubic
Equations, Mathematics Magazine, Vol. 35, No. 5 (Nov., 1962),
pp. 269-271. This paper contains an extension by the late
M.
Hashtroodi of Khayyám's method to degree four equations.
- J. L. Coolidge, The Story of the Binomial Theorem,
Amer. Math. Monthly, Vol. 56,
No. 3 (Mar., 1949), pp. 147-157
- Boris Abramovich Rozenfelʹd (1988), A History of
Non-Euclidean Geometry: Evolution of the Concept of a Geometric
Space, p. 65. Springer, ISBN 0387964584.
- Boris A Rosenfeld and Adolf P Youschkevitch (1996),
Geometry, p.467 in Roshdi Rashed, Régis Morelon (1996),
Encyclopedia of the history of Arabic science, Routledge,
ISBN 0415124115.
- Here Omar Khayyám is described as "poet and mathematician",
i.e. poet appearing first.
- Molavi, Afshin, The Soul of Iran, Norton, (2005),
p.110
- Sadegh Hedayat, the greatest Persian novelist and short-story
writer of the twentieth century was at pains to point out that
Khayyám from "his youth to his death remained a materialist,
pessimist, agnostic". "Khayyam looked at all religions questions
with a skeptical eye", continues Hedayat, "and hated the
fanaticism, narrow-mindedness, and the spirit of vengeance of the
mullas, the so-called religious scholars".
- Bausani, A., Chapter 3 in Cambridge History of Iran , p. 289.
- S. H. Nasr Chapter
9, p. 170-1
- Dictionary of Minor Planet Names - p.255
References
- E.G. Browne. Literary History of Persia. (Four
volumes, 2,256 pages, and 25 years in the writing). 1998. ISBN
0-700-70406-X
- Jan Rypka, History of Iranian Literature. Reidel
Publishing Company. 1968 . ISBN 90-277-0143-1
External links