Omar Khayyam
Omar Khayyam
His name is Abu al-Fath Omar ibn Ibrahim al-Khayyam al-Nisaburi, born in 1048 AD.
He was named after his profession as a young man. He worked in manufacturing and weaving tents.
Khayyam loved to travel to seek knowledge, and remained there until he settled in Baghdad, where it was at the height of its scientific prosperity.
Omar Khayyam (/kaɪˈjɑːm/; Persian: عمر خیّام [ˈoːmɒːɾ xæjˈjɒːm]; 18 May 1048 – 4 December 1131) was a Persian mathematician, astronomer, and poet. He was born in Nishapur, in northeastern Iran, and spent most of his life near the court of the Karakhanid and Seljuq rulers in the period which witnessed the First Crusade.
As a mathematician, he is most notable for his work on the classification and solution of cubic equations, where he provided geometric solutions by the intersection of conics. Khayyam also contributed to the understanding of the parallel axiom. As an astronomer, he designed the Jalali calendar, a solar calendar with a very precise 33-year intercalation cycle.
There is a tradition of attributing poetry to Omar Khayyam, written in the form of quatrains (rubāʿiyāt رباعیات). This poetry became widely known to the English-reading world in a translation by Edward FitzGerald(Rubaiyat of Omar Khayyam, 1859), which enjoyed great success in the Orientalism of the fin de siècle.
Life
Omar Khayyam was born in Nishapur, a leading metropolis in Khorasan during medieval times that reached its climax of prosperity in the eleventh century under the Seljuq dynasty. Nishapur was then religiously a major center of Zoroastrians. It is likely that Khayyam's father was a Zoroastrian who had converted to Islam. He was born into a family of tent-makers (Khayyam). His full name, as it appears in the Arabic sources, was Abu’l Fath Omar ibn Ibrāhīm al-Khayyām. In medieval Persian texts he is usually simply called Omar Khayyām. The historian Bayhaqi, who was personally acquainted with Omar, provides the full details of his horoscope: "he was Gemini, the sun and Mercury being in the ascendant[...]".This was used by modern scholars to establish his date of birth as 18 May 1048.
His boyhood was passed in Nishapur. His gifts were recognized by his early tutors who sent him to study under Imam Muwaffaq Nīshābūrī, the greatest teacher of the Khorasan region who tutored the children of the highest nobility. In 1073, at the age of twenty-six, he entered the service of SultanMalik-Shah I as an adviser. In 1076 Khayyam was invited to Isfahan by the vizier and political figure Nizam al-Mulk to take advantage of the libraries and centers in learning there. His years in Isfahan were productive. It was at this time that he began to study the work of Greek mathematicians Euclid and Apollonius much more closely. But after the death of Malik-Shah and his vizier (presumably by the Assassins' sect), Omar had fallen from favour at court, and as a result, he soon set out on his pilgrimage to Mecca. A possible ulterior motive for his pilgrimage reported by Al-Qifti, is that he was attacked by the clergy for his apparent skepticism. So he decided to perform his pilgrimage as a way of demonstrating his faith and freeing himself from all suspicion of unorthodoxy. He was then invited by the new Sultan Sanjar to Marv, possibly to work as a court astrologer. He was later allowed to return to Nishapur owing to his declining health. Upon his return, he seemed to have lived the life of a recluse. Khayyam died in 1131, and is buried in the Khayyam Garden.
Mathematics
"Cubic equation and intersection of conic sections" the first page of two-chaptered manuscript kept in Tehran University.Khayyam was famous during his life as a mathematician. His surviving mathematical works include: A commentary on the difficulties concerning the postulates of Euclid's Elements (Risāla fī šarḥ mā aškala min muṣādarāt kitāb Uqlīdis, completed in December 1077), On the division of a quadrant of a circle (Risālah fī qismah rub‘ al-dā’irah, undated but completed prior to the treatise on algebra), and On proofs for problems concerning Algebra (Maqāla fi l-jabr wa l-muqābala, most likely completed in 1079). He furthermore wrote a treatise on extracting the nth root of natural numbers, which has been lost.
Theory of parallels
A part of Khayyam's commentary on Euclid's Elements deals with the parallel axiom. The treatise of Khayyam can be considered the first treatment of the axiom not based on petitio principii, but on a more intuitive postulate. Khayyam refutes the previous attempts by other mathematicians to provethe proposition, mainly on grounds that each of them had postulated something that was by no means easier to admit than the Fifth Postulate itself. Drawing upon Aristotle's views, he rejects the usage of movement in geometry and therefore dismisses the different attempt by Al-Haytham. Unsatisfied with the failure of mathematicians to prove Euclid's statement from his other postulates, Omar tried to connect the axiom with the Fourth Postulate, which states that all right angles are equal to one another.
Khayyam was the first to consider the three cases of acute, obtuse, and right angle for the summit angles of a Khayyam-Saccheri quadrilateral, three cases which are exhaustive and pairwise mutually exclusive. After proving a number of theorems about them, he proved that the Postulate V is a consequence of the right angle hypothesis, and refuted the obtuse and acute cases as self-contradictory.Khayyam's elaborate attempt to prove the parallel postulate was significant for the further development of geometry, as it clearly shows the possibility of non-Euclidean geometries. The hypothesis of the acute, obtuse, and that of the right angle are now known to lead respectively to the non-Euclidean hyperbolic geometry of Gauss-Bolyai-Lobachevsky, to that of Riemannian geometry, and to Euclidean geometry.
Tusi's commentaries on Khayyam's treatment of parallels made its way to Europe. John Wallis, the professor of geometry at Oxford, translated Tusi's commentary into Latin. Jesuit geometrician Girolamo Saccheri, whose work (euclides ab omni naevo vindicatus, 1733) is generally considered as the first step in the eventual development of non-Euclidean geometry, was familiar with the work of Wallis. The American historian of mathematics, David Eugene Smith mentions that Saccheri "used the same lemma as the one of Tusi, even lettering the figure in precisely the same way and using the lemma for the same purpose". He further says that "Tusi distinctly states that it is due to Omar Khayyam, and from the text, it seems clear that the latter was his inspirer."
The real number concept
This treatise on Euclid contains another contribution dealing with the theory of proportions and with the compounding of ratios. Khayyam discusses the relationship between the concept of ratio and the concept of number and explicitly raises various theoretical difficulties. In particular, he contributes to the theoretical study of the concept of irrational number. Displeased with Euclid's definition of equal ratios, he redefined the concept of a number by the use of a continuous fraction as the means of expressing a ratio. Rosenfeld and Youschkevitch (1973) argue that "by placing irrational quantities and numbers on the same operational scale, [Khayyam] began a true revolution in the doctrine of number." Likewise, it was noted by D. J. Struik that Omar was "on the road to that extension of the number concept which leads to the notion of the real number."
Geometric algebra
Omar Khayyam's construction of a solution to the cubic x3 + 2x = 2x2 + 2. The intersection point produced by the circle and the hyperbola determine the desired segment.Rashed and Vahabzadeh (2000) have argued that because of his thoroughgoing geometrical approach to algebraic equations, Khayyam can be considered the precursor of Descartes in the invention of analytic geometry. In The Treatise on the Division of a Quadrant of a Circle Khayyam applied algebra to geometry. In this work, he devoted himself mainly to investigating whether it is possible to divide a circular quadrant into two parts such that the line segments projected from the dividing point to the perpendicular diameters of the circle form a specific ratio. His solution, in turn, employed several curve constructions that led to equations containing cubic and quadratic terms.
The solution of cubic equations
Khayyam seems to have been the first to conceive a general theory of cubic equations and the first to geometrically solve every type of cubic equation, so far as positive roots are concerned. The treatise on algebra contains his work on cubic equations. It is divided into three parts: (i) equations which can be solved with compass and straight edge, (ii) equations which can be solved by means of conic sections, and (iii) equations which involve the inverse of the unknown.
Khayyam produced an exhaustive list of all possible equations involving lines, squares, and cubes. He considered three binomial equations, nine trinomial equations, and seven tetranomial equations. For the first and second degree polynomials, he provided numerical solutions by geometric construction. He concluded that there are fourteen different types of cubics that cannot be reduced to an equation of a lesser degree. For these he could not accomplish the construction of his unknown segment with compass and straight edge. He proceeded to present geometric solutions to all types of cubic equations using the properties of conic sections. The prerequisite lemmas for Khayyam’s geometrical proof include Euclid VI, Prop 13, and Apollonius II, Prop 12. The positive root of a cubic equation was determined as the abscissa of a point of intersection of two conics, for instance, the intersection of two parabolas, or the intersection of a parabola and a circle, etc. However, he acknowledged that the arithmetic problem of these cubics was still unsolved, adding that "possibly someone else will come to know it after us". This task remained open until the sixteenth century, where algebraic solution of the cubic equation was found in its generality by Cardano, Del Ferro, and Tartagliain Renaissance Italy.
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
In effect, Khayyam's work is an effort to unify algebra and geometry. This particular geometric solution of cubic equations has been further investigated by M. Hachtroudiand extended to solving fourth-degree equations. Although similar methods had appeared sporadically since Menaechmus, Khayyam's work can be considered the first systematic study and the first exact method of solving cubic equations. The mathematician Woepcke (1851) who offered translations of Khayyam's algebra into French praised him for his "power of generalization and his rigorously systematic procedure."
Binomial theorem and extraction of roots
From the Indians one has methods for obtaining square and cube roots, methods based on knowledge of individual cases—namely the knowledge of the squares of the nine digits 12, 22, 32 (etc.) and their respective products, i.e. 2 × 3 etc. We have written a treatise on the proof of the validity of those methods and that they satisfy the conditions. In addition we have increased their types, namely in the form of the determination of the fourth, fifth, sixth roots up to any desired degree. No one preceded us in this and those proofs are purely arithmetic, founded on the arithmetic of The Elements.
Omar Khayyam Treatise on Demonstration of Problems of Algebra
In his algebraic treatise, Khayyam alludes to a book he had written on the extraction of the 
th root of the numbers using a law which he had discovered which did not depend on geometric figures.This book was most likely titled The difficulties of arithmetic(Moškelāt al-hesāb), and is not extant. Based on the context, some historians of mathematics such as D. J. Struik, believe that Omar must have known the formula for the expansion of the binomial 
, where nis a positive integer.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.The arrangement of numbers known as Pascal's triangle enables one to write down the coefficients in a binomial expansion. This triangular arraysometimes is known as Omar Khayyam's triangle.
th root of the numbers using a law which he had discovered which did not depend on geometric figures.This book was most likely titled The difficulties of arithmetic(Moškelāt al-hesāb), and is not extant. Based on the context, some historians of mathematics such as D. J. Struik, believe that Omar must have known the formula for the expansion of the binomial , where nis a positive integer.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.The arrangement of numbers known as Pascal's triangle enables one to write down the coefficients in a binomial expansion. This triangular arraysometimes is known as Omar Khayyam's triangle.He invented tents in several fields and sciences:
language, astronomy, jurisprudence, and mathematics.
Khayyam distinguished himself from the rest of the scholars that he was able to collect two different qualities from each other.
He combined his creativity and his proficiency in mathematics, in addition to his literature and poetry, and if this signifies something, it indicates his genius.
As for his work in mathematics, he was a pioneer in the field of algebra. He went on to study equations of the third and fourth degrees. He also studied geometry and analytical geometry.
He was aware of the X-ray and the Sisi. Omar Khayyam is one of the most important geniuses who appeared after al-Khawarizmi in algebra.
Al-Khawarizmi was the ideal of the age of Khiam, and his example was in many things. As for the death of Omar Khayyim, may God have mercy on him, he passed away at the age of 83 years, full of giving and achievement.
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