Omar Khayyam

Gıyaseddin Ebu'l-Feth Ömer ibni İbrahim Nişaburi (18 May 1048 - 4 December 1131), commonly known as Omar Khayyam (Persian: عمر خیّام), was a Persian polymath, mathematician, astronomer, historian, philosopher, and poet. He was born in Nishapur, the first capital of the Seljuk Empire. As a scholar, he was contemporary with the rule of the Seljuk dynasty during the First Crusade.

As a mathematician, he is best known for his work on the classification and solution of cubic equations, for which 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. This, about a thousand years later, formed the basis of the Persian calendar still in use. In Iran, in 1079, he announced that the length of a year was measured as 365.24219858156 days. Considering that the length of a year changes in the sixth decimal place over the course of a person's lifetime, this is extremely accurate. By comparison, while the length of a year was 365.242196 days at the end of the 19th century, today it is 365.242190 days.

As a poet, he is known for his quatrains (rubaiyat). It is observed that the number of the quatrains increases as one approaches the present, according to the copying dates of his Rubaiyat, and it is understood that many of them are poems of other poets later attributed to him. The number of the quatrains that reflect his own original style is around 100.

From the 19th century onward, with the growth of interest in Khayyam's quatrains in Turkey, many writings have been written on Khayyam's poems and many translations of the quatrains have been made. The oldest known Turkish translation of the quatrains belongs to Muallim Feyzi. His work titled Hayyam, which includes the prose translations of some quatrains as well as his articles about Khayyam, was first published in the 1885-1886 issues of Tercüman-ı Hakikat, and then made into a book (Istanbul 1303).

Life

Omar Khayyam was born in 1048 in Nishapur, one of the leading metropolises of Khorasan in the Middle Ages, which reached the height of its prosperity in the eleventh century under the Seljuk dynasty. Nishapur was also an important center of Zoroastrianism, and it is likely that Khayyam's father was a Zoroastrian who had chosen Islam. His full name, as it appears in Arabic sources, is Ebu'l Feth Ömer ibn İbrahim el-Hayyam. In medieval Persian texts he is usually called simply Omar Khayyam. Although open to doubt, since Khayyam means tent-maker in Arabic, it has been assumed that his ancestors followed the tent-making trade. The historian Beyhaki, who knew Khayyam personally, relates all the details of his horoscope as follows: "He was Gemini, the Sun and Mercury were in the ascendant[...]." This information would be used by modern scholars to determine his date of birth as 18 May 1048.

His childhood was spent in Nishapur. His abilities were noticed by his first teachers, and they sent him to study under the supervision of Imam Muvaffak Nishaburi, the famous teacher of the Khorasan region who gave lessons to the children of the nobility. Over the years, Khayyam formed a good friendship with him. Khayyam also took lessons from the Zoroastrian-convert mathematician Ebu Hasan Behmenyar. After studying the sciences, philosophy, mathematics, and astronomy in Nishapur, he went around 1068 to the province of Bukhara, where he frequently visited the famous library of the Ark. Around 1070 he moved to Samarkand, where he began to write his famous treatise on algebra under the patronage of the city's governor and qadi, Ebu Tahir Abdurrahman ibn 'Alak. Omar Khayyam was warmly received by the Karakhanid ruler Şemsü'l-Mülk Nasr. Beyhaki describes this reception thus: "He would show him the utmost honor, so much so that he would seat [Khayyam] beside himself on his throne."

His work in the field of mathematics

Khayyam was famous as a mathematician throughout his life. His surviving mathematical works are:

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 quarter of a circle (Risālah fī qismah rub' al-dā'irah, undated but completed before the treatise on algebra), and On the proofs of the problems concerning algebra (Maqāla fi l-jabr wa l-muqābala, most probably completed in 1079).

He also wrote a subsequently lost treatise on the derivation of the binomial theorem and on the roots of natural numbers.

Theory of parallels

Part of Khayyam's commentary on Euclid's Elements concerns the parallel postulate. Khayyam's treatise may be regarded as the first treatment of the axiom based not on a petitio principii but on a more intuitive postulate. Khayyam refutes the earlier attempts of other mathematicians to "prove" the postulate, essentially on the grounds that they assumed something no easier than the Fifth Postulate itself. Drawing on Aristotle's views, he rejects the use of motion in geometry, and therefore rejects the different attempt made by Al-Haytham.

Not satisfied that mathematicians had been unable to prove Euclid's statement from his other postulates, Omar attempted to combine 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 different cases of acute, obtuse, and right angle for the summit angles of the Khayyam-Saccheri quadrilateral. After proving a series of theorems about them, he proved that Postulate V follows from the right-angle hypothesis and refuted the obtuse- and acute-angle cases as being self-contradictory.

His detailed attempt to prove the parallel postulate was important for the further development of geometry, since it clearly demonstrated the possibility of a non-Euclidean geometry. It is known that the acute-, obtuse-, and right-angle hypotheses lead respectively to the non-Euclidean hyperbolic geometry of Gauss-Bolyai-Lobachevsky, to Riemannian geometry, and to Euclidean geometry.

Tusi's comments on Khayyam's approach to parallels reached Europe. John Wallis, professor of geometry at Oxford, translated Tusi's commentary into Latin.

Works

Risâle fî taḳsîmi rubʿi'd-dâʾire

Risâle fi'l-berâhîn ʿalâ mesâʾili'l-cebr ve'l-muḳābele

Risâle fî şerḥi mâ eşkele min müṣâderâti Kitâbi Öḳlîdes

Mîzânü'l-ḥikem fî İhtiyâli maʿrifeti miḳdârey eẕ-ẕeheb ve'l-fiḍḍa fî cismin mürekkebin minhümâ

Silsile-i Tertîb (Risâle fî Külliyyâti'l-vücûd)

el-Ḳavl ʿale'l-ecnâs elletî bi'l-erbaʿ

Cevâb ʿan s̱elâs̱i mesâʾil: Żarûretü't-teżâd fi'l-ʿâlem ve'l-cebr ve'l-beḳāʾ

eż-Żiyâʾ el-ʿaḳlî fî mevżûʿi'l-ʿilmi'l-küllî

Şerḥu'l-müşkil min Kitâbi'l-Mûsîḳā

Sources

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