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Omar Khayyam combined his profound lyrical insights with groundbreaking mathematics

By Joseph A. Kechichian Senior Writer

The English-speaking world came to know him as a poet after Edward Fitzgerald published a translation of his “Rubaiyat” (Quatrains) in 1839, but Persian scholar Omar Khayyam was a leading mathematician, with key discoveries to his credit. Although the widely read and studied “Rubaiyat” influenced European ideas about Persian poetry and literature, little of the poetry was made public during his lifetime. The poetry revealed a philosophical side to Khayyam that few of his contemporaries knew, with many assuming that he was an agnostic hedonist or a mystical Sufi poet influenced by platonic traditions. Though he probably was both, Khayyam was, above all else, a happy scientist who made significant contributions to geometry. These were finally appraised by René Descartes and Blaise Pascal in the 17th century.

Early life and times

Khayyam was born Ghiyath Al Deen Abul-Fatah ‘Umar Ibn Ebrahim Al Nishapuri Al Khayyami in 1044 in Nishapur, Persia. His family name meant “tent maker”, which may have been his father’s trade. His son, though, spent most of his life in Persian intellectual centres such as Samarkand and Bukhara, and enjoyed the favour of the Seljuq sultans who ruled the region.

When Seljuq Turkish tribes invaded southwestern Asia in the 11th Century, encompassing Mesopotamia, Syria, Palestine and most of Iran, few could withstand the onslaught. In 1038, the Seljuq ruler Toghrïl proclaimed himself sultan at Nishapur, and occupied Baghdad in 1055. Khayyam grew up in this military empire, though he also managed to study under renowned scholars such as Shaikh Mohammad Mansouri and Imam Muwaffaq Nishapuri. A workaholic Omar Khayyam was soon splitting his days into three: by day he taught algebra and geometry, attended the Seljuq court as an adviser of Malek-Shah in the evening, and returned to his astronomy books at night to complete his Jalali calendar.

Khayyam went on to devise a remarkably accurate solar calendar, named after Malek-Shah Jalal Al Deen. The latter commissioned the almanac as he developed a new schedule for revenue collection. The result: a calendar that recorded a single error, of one day, in 3,770 years. While the Gregorian calendar — which was developed in 1582 under Pope Gregory XIII to replace the then widely used Julian calendar that dated from Julius Caesar — included an error of one day in 3,330 years, Khayyam measured the length of one year as 365.24219858156 days, which was remarkably accurate. The project was cancelled in 1092, when Malek-Shah died, but the Jalali calendar survived and is still in use in parts of Iran and Afghanistan.

Khayyam moved to Isfahan after the Seljuq Sultan was murdered in 1092, when, regrettably, the ruler’s widow turned against him too. After a brief visit to Makkah and Madinah, the mathematician-poet returned to Nishapur, where he died in 1131 and where his remains were buried in the Khayyam Garden at the mausoleum of Imamzadeh Mahruq.

Scholarly contributions

Khayyam’s best-known scientific contributions were in algebra and geometry. He turned out to be an outstanding mathematician despite hardly being able to concentrate on his work because of obstacles created by opportunists who pretended to acquire knowledge but who, in fact, were little more than idle minds anxious to ingratiate themselves at the court. His “Problems of Arithmetic”, and the equally impressive “Treatise on Demonstration of Problems of Algebra”, which he wrote in Samarkand starting in 1070, sealed his place in history.

Simply stated, Khayyam solved the cubic equation x3 + 200x = 20x2 + 2000 dilemma as he found its positive root by considering the intersection of a rectangular hyperbola and a circle. This was a useful tool to address conic-section calculations that arose in construction as in so many other applications. Khayyam calculated an approximate numerical solution by interpolation in trigonometric tables, leading him to postulate that the solution of this cubic required the use of conic sections and it could not possibly be solved by ruler-and-compass methods.

His treatise contained a complete classification of cubic equations, with geometric solutions found by means of intersecting conic sections, which was based on an elegant right triangle where the hypotenuse equalled the sum of one leg plus the altitude on the hypotenuse. Earlier, leading scientists such as Al Mahani and Al Khazin had translated geometric problems into algebraic equations (something which only became possible after the immensely important work of Al Khwarizmi was published, although Khayyam was the first to conceive of a general theory of cubic equations).

To his credit, Khayyam also postulated that a cubic equation could have more than one solution, demonstrating that equations could have more than one solution, though he did not advance the notion that a cubic could have three solutions. He foresaw the day when “arithmetic solutions” could be developed to find out what existed beyond “the first three classes of known powers, namely the number, the thing and the square”. Nearly seven centuries later, the mathematicians Del Ferro, Tartaglia and Ferrari provided the necessary arithmetic.

Khayyam addressed the triangular array of binomial coefficients, which later became known as Pascal’s triangle and that stood at the base of probability. His “Explanations of the Difficulties in the Postulates of Euclid”, which tackled the problems of parallels opened the floodgates of non-Euclidean geometries. This work, on fractions and the multiplication of ratios, allowed the development of an entirely new aspect of mathematics.

Pascal’s Triangle

Unlike many commentators on Euclid, Khayyam was not trying to prove the parallel postulate: “two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge”. What he did was quite astute. He considered three cases (right, obtuse, and acute) and the summit angles of a quadrilateral, as he (correctly) refuted the obtuse and acute cases to be based on his postulate. Six centuries later, Giordano Vitale used the quadrilateral to prove that if three points on the line AB are equidistant from the summit CD, then AB and CD are equidistant everywhere.

The philosopher-mathematician

Notwithstanding his theories in mathematics, Khayyam was a believer who saw mathematics as a form of creation since all beings enjoyed a “Divine Origin” through which they gained reality. Moreover, the significance of axioms in geometry led the mathematician to rely upon philosophy, since, to him, the world of geometry explained the intelligible and immaterial. Natural bodies, which relied on substance and stood by themselves, were different from mathematical bodies, he maintained. For this and other views, Khayyam came under attack from orthodox Muslim religious authorities who concluded that Khayyam’s questioning mind did not necessarily conform to the faith. The perceptive scholar addressed such concerns in his “Rubaiyat” with these eloquent words:

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.

The “Rubaiyat” contains approximately 600 short four-line poems, although 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.

Several contemporary scholars, including Seyyed Hussain Nasr, concluded that the poems clarified Khayyam’s personal views about God. In his “Khutbat Al Gharra” (The Splendid Sermon), the Nashipuri native praised God, which was an illustration of the classic view on divinity. Nasr confirmed Khayyam’s Sufi tendencies, opting the intuition over the rational:

“... Fourth, the Sufis, who do not seek knowledge by ratiocination or discursive thinking, but by purgation of their inner being and the purifying of their dispositions. They cleanse the rational soul of the impurities of nature and bodily form, until it becomes pure substance. When it then comes face to face with the spiritual world, the forms of that world become truly reflected in it, without any doubt or ambiguity. This is the best of all ways, because it is known to the servant of God that there is no reflection better than the Divine Presence and in that state there are no obstacles or veils in between. Whatever man lacks is due to the impurity of his nature. If the veil be lifted and the screen and obstacle removed, the truth of things as they are will become manifest and known. And ... [Prophet Mohammad, PBUH] ... indicated this when he said: ‘Truly, during the days of your existence, inspirations come from God. Do you not want to follow them?’ Tell unto reasoners that, for the lovers of God, intuition is guide, not discursive thought.”

Interestingly, Omar Khayyam addressed an eternal philosophical question that all serious thinkers pondered over time, namely, what becomes of the soul after death? In Khayyam’s own words:

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.

Legacy to Arabs and Muslims

Beyond sheer erudition, Khayyam was a true Sufi student — although poor translations of the “Rubaiyat” cheapened his prose — and not a leading teacher such as Rumi, and what made Khayyam’s work so relevant and accessible was its very humanity. The mathematician rejected the notion that he was a philosopher in the Aristotelian sense, even if he wished to know who he was. Still, he was comparable to Ibn Sina (Avicena), whom he admired. Khayyam the philosopher must therefore be understood from two distinct angles: through his “Rubaiyat” as well as through his work on mathematics. Above all else, he wanted to know where he was going, best described in Verse LXIV of his opus:

Strange, is it not? that of the myriads who

Before us pass’d the door of Darkness through,

Not one returns to tell us of the Road,

Which to discover we must travel too.

List of works

Two key books on mathematics that can be found in established libraries only are: “Treatise on Demonstration of Problems of Algebra”, 1070; and “Commentaries on the Difficult Postulates of Euclid’s Book”, 1075

“Rubaiyat”, translated by Edward Fitzgerald and edited by Christopher Decker, Charlottesville, Virginia: University of Virginia Press, 1997.

Selected readings

Ali Reza Amir-Moez, “A Paper of Omar Khayyám”, “Scripta Mathematica” 26 (1963), pp 323-337.

Ali Reza Amir-Moez, “Khayyam’s Solution of Cubic Equations”, “Mathematics Magazine”, 35: 5 (November 1962), pp 269-271.

Donald and Marilynn Olson, “Zodiac Light, False Dawn, and Omar Khayyam”, “The Observatory” 108 (1988), pp 181-182.

Seyyed Hossein Nasr, “Islamic Philosophy from Its Origin to the Present: Philosophy in the Land of Prophecy”, Albany, New York: State University of New York Press, 2006.

Carl Henry Andrew Bjerregaard, “Sufism: Omar Khayyam and E. Fitzgerald”, London: The Sufi Publishing Society, 1915.

Jos Biegstraaten, “Khayyam, Omar xi. Impact on Literature and Society in the West” in Encyclopaedia Iranica, volume 15, Encyclopaedia Iranica Foundation, 2008, at http://www.iranicaonline.org/articles/khayyam-xi
.

Dr Joseph A. Kéchichian is an author, most recently of, “Legal and Political Reforms in Sa‘udi Arabia”, London: Routledge, 2013.

This article is the sixteenth of a series on Muslim thinkers who greatly influenced Arab societies across the centuries. Omar Khayyam combined his profound lyrical insights with groundbreaking mathematics...

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