Omar Khayyam - 1

(1048 AD, Neyshapur, Iran—1123 AD, Neyshapur, Iran),

Umar Khayyam was born in Nishapur, Iran. His full name is Ghiyath al-Din Abu'l-Fath Umar ibn Ibrahim Al-Nisaburi al-Khayyami’ in that time Nishapur was Seljuk capital in Khorasan. It is said that Khayyam’s father may have been a convert from Zoroastrian religion to Sunni slam, and so he was a first generation Muslim. But other biographers of Khayyam have not supported this claim; they describe him as a Shi’a Muslim. 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:

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!

He is famous as a Persian polymath, mathematiciam, philosopher, astronomer and poet. He also wrote treatises on mechanics, geography, and music and was a physicist.
A part of his childhood was spent in the town of Balkh (present northern Afghanistan), In Balkh he study under the well-known scholar Sheikh Muhammad Mansuri. Subsequently, he studied under Imam Mowaffaq Nishapuri. In those times Imam Mowaffaq was considered one of the greatest teachers of the Khorassan region.

According to a well-known legend called Three Schoolmates, other then Umar Khayyam there was two other exceptional students studied under the Imam Mowaffaq at about the same time: 2. Nizam-ul-Mulk (b. 1018), who later become the Vizier to the Seljukid Empire, and 3. Hassan Bin Sabah (b.1034), who became the leader of the Hashshashin (Nizar Ismaili) sect. These three students became friends, and when Nizam-ul-Mulk became Vizier, Hassan Sabah and Omar Khayyam each went to him, and asked to share in his good fortune. As per his demand Hassan Sabah was granted a place in the government, but was removed from power after he participated in an unsuccessful coup against his benefactor, the Vizier. Omar Khayyam 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 rejected by many scholars, due to the 30-year age difference between Khayyam and Nizam-ul-Mulk. According to them it makes unlikely for the two to have attended school together. Also to consider this very fact that these three men were grew up in different parts of the country. But 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 Sabah, and strengthening the power establishment and the rule of law and order, represented by Nizam-ul-Mulk.

Omar Khayyam as a 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. He also derived the general methods for solving cubic equations and even some higher orders. 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). (The unknown factors are usually represented by an x.)

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 He also had other notable work in geometry, specifically on the theory of proportions.

Theory of parallels

His book entitled Explanations of the difficulties in the postulates in Euclid's Elements, covers the 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." 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 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. In a sense he made the first attempt at formulating a non-Euclidean postulate as an alternative to the parallel postulate,

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

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.

In an untitled writing on cubic equation by Khayyam discovered in 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 x3 + 200x = 20x2 + 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 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.

Binomial theorem and extraction of roots

From the Indians one has methods for obtaining square and cube roots, methods which are 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

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.
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