Omar Khayyam

Omar Khayyam's full name was Ghiyath al-Din Abu'l-Fath Omar ibn Ibrahim Al-Nisaburi al-Khayyami. A literal translation of the name al-Khayyami (or al-Khayyam) means 'tent maker' and this may have been the trade of Ibrahim his father. Khayyam played on the meaning of his own name when he wrote:- The political events of the 11th Century played a major role in the course of Khayyam's life. The Seljuq Turks were tribes that invaded southwestern Asia in the 11th Century and eventually founded an empire that included Mesopotamia, Syria, Palestine, and most of Iran. The Seljuq occupied the grazing grounds of Khorasan and then, between 1038 and 1040, they conquered all of north-eastern Iran. The Seljuq ruler Toghrïl Beg proclaimed himself sultan at Nishapur in 1038 and entered Baghdad in 1055. It was in this difficult unstable military empire, which also had religious problems as it attempted to establish an orthodox Muslim state, that Khayyam grew up. Khayyam studied philosophy at Naishapur and one of his fellow students wrote that he was:- ... endowed with sharpness of wit and the highest natural powers ... However, this was not an empire in which those of learning, even those as learned as Khayyam, found life easy unless they had the support of a ruler at one of the many courts. Even such patronage would not provide too much stability since local politics and the fortunes of the local military regime decided who at any one time held power. Khayyam himself described the difficulties for men of learning during this period in the introduction to his Treatise on Demonstration of Problems of Algebra (see for example [1]):- I was unable to devote myself to the learning of this algebra and the continued concentration upon it, because of obstacles in the vagaries of time which hindered me; for we have been deprived of all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when time is asleep, to devote themselves meanwhile to the investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes; and if they see a certain person seeking for the right and preferring the truth, doing his best to refute the false and untrue and leaving aside hypocrisy and deceit, they make a fool of him and mock him. However Khayyam was an outstanding mathematician and astronomer and, despite the difficulties which he described in this quote, he did write several works including Problems of Arithmetic, a book on music and one on algebra before he was 25 years old. In 1070 he moved to Samarkand in Uzbekistan which is one of the oldest cities of Central Asia. There Khayyam was supported by Abu Tahir, a prominent jurist of Samarkand, and this allowed him to write his most famous algebra work, Treatise on Demonstration of Problems of Algebra from which we gave the quote above. We shall describe the mathematical contents of this work later in this biography. Toghril Beg, the founder of the Seljuq dynasty, had made Esfahan the capital of his domains and his grandson Malik-Shah was the ruler of that city from 1073. An invitation was sent to Khayyam from Malik-Shah and from his vizier Nizam al-Mulk asking Khayyam to go to Esfahan to set up an Observatory there. Other leading astronomers were also brought to the Observatory in Esfahan and for 18 years Khayyam led the scientists and produced work of outstanding quality. It was a period of peace during which the political situation allowed Khayyam the opportunity to devote himself entirely to his scholarly work. During this time Khayyam led work on compiling astronomical tables and he also contributed to calendar reform in 1079. Cowell quotes The Calcutta Review No 59:- When the Malik Shah determined to reform the calendar, Omar was one of the eight learned men employed to do it, the result was the Jalali era (so called from Page 1 Khayyam-CV Jalal-ud-din, one of the king's names) - 'a computation of time,' says Gibbon, 'which surpasses the Julian, and approaches the accuracy of the Gregorian style.' Khayyam measured the length of the year as 365.24219858156 days. Two comments on this result. Firstly it shows an incredible confidence to attempt to give the result to this degree of accuracy. We know now that the length of the year is changing in the sixth decimal place over a person's lifetime. Secondly it is outstandingly accurate. For comparison the length of the year at the end of the 19th century was 365.242196 days, while today it is 365.242190 days. In 1092 political events ended Khayyam's period of peaceful existence. Malik-Shah died in November of that year, a month after his vizier Nizam al-Mulk had been murdered on the road from Esfahan to Baghdad by the terrorist movement called the Assassins. Malik-Shah's second wife took over as ruler for two years but she had argued with Nizam al-Mulk so now those whom he had supported found that support withdrawn. Funding to run the Observatory ceased and Khayyam's calendar reform was put on hold. Khayyam also came under attack from the orthodox Muslims who felt that Khayyam's questioning mind did not conform to the faith. He wrote in his poem the Rubaiyat :- Indeed, the Idols I have loved so long Have done my Credit in Men's Eye much Wrong: Have drowned my Honour in a shallow cup, And sold my reputation for a Song. Despite being out of favour on all sides, Khayyam remained at the Court and tried to regain favour. He wrote a work in which he described former rulers in Iran as men of great honour who had supported public works, science and scholarship. Malik-Shah's third son Sanjar, who was governor of Khorasan, became the overall ruler of the Seljuq empire in 1118. Sometime after this Khayyam left Esfahan and travelled to Merv (now Mary, Turkmenistan) which Sanjar had made the capital of the Seljuq empire. Sanjar created a great centre of Islamic learning in Merv where Khayyam wrote further works on mathematics. The paper [18] by Khayyam is an early work on algebra written before his famous algebra text. In it he considers the problem:- Find a point on a quadrant of a circle in such manner 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. Khayyam shows that this problem is equivalent to solving a second problem:- Find a right triangle having the property that the hypotenuse equals the sum of one leg plus the altitude on the hypotenuse. This problem in turn led Khayyam to solve the cubic equation x3 + 200x = 20x2 + 2000 and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables. Perhaps even more remarkable is the fact that Khayyam states that the solution of this cubic requires the use of conic sections and that it cannot be solved by ruler and compass methods, a result which would not be proved for another 750 years. Khayyam also wrote that he hoped to give a full description of the solution of cubic equations in a later work [18]:- 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. Indeed Khayyam did produce such a work, the Treatise on Demonstration of Problems of Algebra which contained a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. In fact Khayyam gives an interesting historical account in which he claims that the Greeks had left nothing on the theory of cubic equations. Indeed, as Khayyam writes, the contributions by earlier writers such as al-Mahani and al-Khazin were to translate geometric problems into algebraic equations (something which was essentially impossible before the work of al-Khwarizmi). However, Khayyam Page 2 Khayyam-CV himself seems to have been the first to conceive a general theory of cubic equations. Khayyam wrote (see for example [9] or [10]):- In the science of algebra one encounters problems dependent on certain types of extremely difficult preliminary theorems, whose solution was unsuccessful for most of those who attempted it. As for the Ancients, no work from them dealing with the subject has come down to us; perhaps after having looked for solutions and having examined them, they were unable to fathom their difficulties; or perhaps their investigations did not require such an examination; or finally, their works on this subject, if they existed, have not been translated into our language. Another achievement in the algebra text is Khayyam's realisation that a cubic equation can have more than one solution. He demonstrated the existence of equations having two solutions, but unfortunately he does not appear to have found that a cubic can have three solutions. He did hope that "arithmetic solutions" might be found one day when he wrote (see for example [1]):- Perhaps someone else who comes after us may find it out in the case, when there are not only the first three classes of known powers, namely the number, the thing and the square. The "someone else who comes after us" were in fact del Ferro, Tartaglia and Ferrari in the 16th century. Also in his algebra book, Khayyam refers to another work of his which is now lost. In the lost work Khayyam discusses the Pascal triangle but he was not the first to do so since al-Karaji discussed the Pascal triangle before this date. In fact we can be fairly sure that Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients. This follows from the following passage in his algebra book (see for example [1], [9] or [10]):- The Indians possess methods for finding the sides of squares and cubes based on such knowledge of the squares of nine figures, that is the square of 1, 2, 3, etc. and also the products formed by multiplying them by each other, i.e. the products of 2, 3 etc. I have composed a work to demonstrate the accuracy of these methods, and have proved that they do lead to the sought aim. I have moreover increased the species, that is I have shown how to find the sides of the square-square, quatro-cube, cubo-cube, etc. to any length, which has not been made before now. the proofs I gave on this occasion are only arithmetic proofs based on the arithmetical parts of Euclid's "Elements". In Commentaries on the difficult postulates of Euclid's book Khayyam made a contribution to non-euclidean geometry, although this was not his intention. In trying to prove the parallels postulate he accidentally proved properties of figures in non-euclidean geometries. Khayyam also gave important results on ratios in this book, extending Euclid's work to include the multiplication of ratios. The importance of Khayyam's contribution is that he examined both Euclid's definition of equality of ratios (which was that first proposed by Eudoxus) and the definition of equality of ratios as proposed by earlier Islamic mathematicians such as al-Mahani which was based on continued fractions. Khayyam proved that the two definitions are equivalent. He also posed the question of whether a ratio can be regarded as a number but leaves the question unanswered. Outside the world of mathematics, Khayyam is best known as a result of Edward Fitzgerald's popular translation in 1859 of nearly 600 short four line poems the Rubaiyat. Khayyam's fame as a poet has caused some to forget his scientific achievements which were much more substantial. Versions of the forms and verses used in the Rubaiyat existed in Persian literature before Khayyam, and only about 120 of the verses can be attributed to him with certainty. Of all the verses, the best known is the following:- 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. 

 Name Ghiasedin Abolfatth Omer ibn Ebrahim Khayyam Naishburi Page 3 Khayyam-CV Birth May 18, 1048Ad Nayshbur, Khorasan, Iran Death 1123 Naishbur, Khorasan, Iran School/tradition Analytic philosophy Main interests Ethics, epistemology, logic, mathematics, philosophy of language, philosophy of science, religion Notable ideas Logical atomism, theory of descriptions, knowledge by acquaintance and knowledge by description, Russell's paradox, Russell's teapot. Influenced by Plato, Gottfried Leibniz, David Hume, G.E. Moore, John Stuart Mill, Frege, Ludwig Wittgenstein, A.N. Whitehead, Thomas Paine, Giuseppe Peano Influenced Wittgenstein, A. J. Ayer, Rudolf Carnap, Kurt Gödel, Karl Popper, W. V. Quine, N. Chomsky, J. L. Austin, Saul Kripke, Moritz Schlick, Alfred Tarski, Friedrich Waismann, Donald Davidson, John Searle, A. S. Eddington, Alan Turing, Daniel Dennett, Hannah Arendt, Jules Vuillemin, John McDowell, Satish Kumar =

Meanwhile, in Persia, Muslims came in contact with India. It was from Sanskrit writings that they acquired, during the eighth century, their first knowledge of astronomy. About 830, Muhammad ibn Musa al-Khwarazmi, a translator of mathematical and astronomical books from the Sanskrit, published a book which was translated into Latin in the twelfth century, under the title Algoritmi de numero Indorum. It was from this book that the West first learnt of what we call "Arabic" numerals, which ought to be called "Indian." The same author wrote a book on algebra which was used in the West as a text. book until the sixteenth century. Persian civilization remained both intellectually and artistically admirable until the invasion of the Mongols in the thirteenth century, from which it never recovered. Omar Khayyám, the only man known to me who was both a poet and a mathematician, reformed the calendar in 1079. His best friend, oddly enough, was the founder of the sect of the Assassins, the "Old Man of the Mountain" of legendary fame. The Persians were great poets: Firdousi (ca. 941), author of the Shahnama, is said by those who have read him to be the equal of Homer. They were also remarkable as mystics, which other Mohammedans were not. The Sufi sect, which still exists, allowed itself great latitude in the mystical and allegorical interpretation of orthodox dogma; it was more or less Neoplatonic. Two Mohammedan philosophers, one of Persia, one of Spain, demand special notice; they are Avicenna and Averroes. Of these the former is the more famous among Mohammedans, the latter Page 4 Khayyam-CV among Christians. Avicenna (Ibn Sina) ( 980-1037) spent his life in the sort of places that one used to think only exist in poetry. He was born in the province of Bokhara; at the age of twenty-four he went to Khiva--"lone Khiva in the waste"--then to Khorassan--"the lone Chorasmian shore." For a while he taught medicine and philosophy at Ispahan; then he settled at Teheran. He was even more famous in medicine than in philosophy, though he added little to Galen. From the twelfth to the seventeenth century, he was used in Europe as a guide to medicine. He was not a saintly character, in fact he had a passion for wine and women. He was suspect to the orthodox, but was befriended by princes on account of his medical skill. At times he got into trouble owing to the hostility of Turkish mercenaries; sometimes he was in hiding, sometimes in prison. He was the author of an encyclopædia, almost unknown to the East because of the hostility of theologians, but influential in the West through Latin translations. His psychology has an empirical tendency. His philosophy is nearer to Aristotle, and less Neoplatonic, than that of his Muslim predecessors. Like the Christian scholastics later, he is occupied with the problem of universals. Plato said they were anterior to things. Aristotle has two views, one when he is thinking, - Arabic philosophy is not important as original thought. Men like Avicenna and Averroes are essentially commentators. Speaking generally, the views of the more scientific philosophers come from Aristotle and the Neoplatonists in logic and metaphysics, from Galen in medicine, from Greek and Indian sources in mathematics and astronomy, and among mystics religious philosophy has also an admixture of old Persian beliefs. Writers in Arabic showed some originality in mathematics and in chemistry--in the latter case, as an incidental result of alchemical researches. Mohammedan civilization in its great days was admirable in the arts and in many technical ways, but it showed no capacity for independent speculation in theoretical matters. Its importance, which must not be underrated, is as a transmitter. Between ancient and modern European civilization, the dark ages intervened. The Mohammedans and the Byzantines, while lacking the intellectual energy required for innovation, preserved the apparatus of civilization--education, books, and learned leisure. Both stimulated the West when it emerged from barbarism--the Mohammedans chiefly in the thirteenth century, the Byzantines chiefly in the fifteenth. In each case the stimulus produced new thought better than any produced by the transmitters--in the one case scholasticism, in the other the Renaissance (which however had other causes also). =================================================Khayyam's view on religion Khayyam gives the impression that organised religion to be the enemy of mankind, the force that divides the believer from the infidel and which thereby both excites and authorises murder. [ example Hassan SaBBah, Islam ]. Page 5 Khayyam-CV =================================================Nowruz .. From the era of Keykhosrow till the days of Yazdegard, last of the pre-Islamic kings of Persia, the royal custom was thus: on the first day of the New Year, Nau Ruz, the King's first visitor was the High Priest of the Zoroastrians, who brought with him as gifts a golden goblet full of wine, a ring, some gold coins, a fistful of green sprigs of wheat, a sword, and a bow. In the language of Persia he would then glorify God and praise the monarch.. This was the address of the High Priest to the king : "O Majesty, on this feast of the Equinox, first day of the first month of the year, seeing that thou hast freely chosen God and the Faith of the Ancient ones; may Surush, the Angel-messenger, grant thee wisdom and insight and sagacity in thy affairs. Live long in praise, be happy and fortunate upon thy golden throne, drink immortality from the Cup of Jamshid; and keep in solemn trust the customs of our ancestors, their noble aspirations, fair gestes and the exercise of justice and righteousness. May thy soul flourish; may thy youth be as the new-grown grain; may thy horse be puissant, victorious; thy sword bright and deadly against foes; thy hawk swift against its prey; thy every act straight as the arrow's shaft. Go forth from thy rich throne, conquer new lands. Honor the craftsman and the sage in equal degree; disdain the acquisition of wealth. May thy house prosper and thy life be long!" ===

Khayyam came to this conclusion that: "It is no peace in heaven that religion is after, but political power here on earth", and may be it was for this reason why Hassan Sabbah went after religion to prsue a political power. [ibid]. Khayyam also concluded science and reason are above myth and faith. In his mathematical mind he deduced the law of relations of matter to number, space, and time " the ultimate data will be number, matter, space, and time themselves.When their relations shall be known, all physical phenomena will be a branch of pure mathematics" . William Hicks 1895 ========================================= Khayyam's view of the Universe Khayyam's view was that the objects in universe were non-centripetal [ moving toward centre ]. And no matter where the centre was assumed, the arethmatics were the same. ========================================== Monism: Theory; that there is realy only one thing, or the theory that there are many things but that they are all of one fundamntal kind. SPINOZA's philosophy is monism in the first of these senses. The one thing is God, a substance with infinte Attributes, two of which , the mental and bodily are known to us. A monism in the sense is, for example, the Materialism of those who hold that sensations are kentical with brain processes. The opposite form of monism to materialism is the theory that things that appear to us as material are, in theselves, spiritual. A monism which is neutral between the two ("neutral monism") is the theory that MINDS, on the one hand, are differently collected assemblage of things of one kind, SENSATIONS. - This theory was advance by William James & Bertran Russell. ========================= Ali's View Memory is a co-operation between the senses, i.e. { vision, smell } -> [food] Page 6 Khayyam-CV Thinking is an [act, exercise] of the memory (ies) refershing. Q. ? Is memory capacity bounded, [ finte], limited to a specific number. Q. ? Is there any object which not physical Quark: is the building block of hadrons weight of hyperon <= {proton, neutron} ============================================== Q. What can be placed between user eyes and computer screen, inorder to make money? ============================================== Patten recognition for PIN number; silently repeat the pin number in your head, the cfmg ifrg with match the pattern with stored one. ============================================== Universe is 13.5 bln years old Solar system is 4 bln years old q. What is the grain of Univers.? Vesta series } largest Astoriod Astoriod -> Metioroid -> Commet brought most of the water to Earth. YESTERDAY MAGIC IS TODAY'S REALITY. ====================== ==================================================== In search for the aethestic Khayyam, Sadeq Hedayat could identify only 13 quatrains attested to be Khayyam's. He finally concluded that Khayyam could have written as many as 143. Others who have undertaken a similar inquiry have arrived at different numbers: Hussein Shajareh gives 121; M. A. Forughi has 178, while Ali Dashti has only 81. The largest attested number is Arberry's 250. But there are others who include as many as 800 quatrains in their version of the Ruba'iyyat. The major themes of Khayyam's Ruba'iyyat are: 1. The secret of creation 2. The agony of existence 3. Predestination (life planned by the maker) 4. Time and tide (life created by Time) 5. Rotating particles (life consisting of particles) 6. Acquiescence to the fortuitous (life happening as an accident) 7. Seizing the moment