New Era

The death of the sultan, a month after his vizier Nizam al-Mulk was murdered on the road from Esfahan to Baghdad by the terrorist movement called the Assassins, ended Khayyam’s period of peaceful existence.

Malik-Shah’s second wife took over as ruler for two years and she had argued with Nizam al-Mulk. So support was withdrawn from his clients and funding for the Observatory ceased. Accordingly, Khayyam’s calendar reform was put on hold.

Khayyam also came under attack from the orthodox Muslims who felt that Khayyam’s questioning mind didn’t conform to the faith.

Despite being out of favor on all sides, Khayyam remained at the Court. He wrote a work in which he described former rulers in Iran as men of great honor who had supported public works, science, and scholarship.

The Mathematician

Malik-Shah’s third son, Sanjar, became the overall ruler of the Seljuk Empire in 1118. Sometime after this Khayyam left Esfahan and traveled to Merv (now Mary, Turkmenistan). This is the city which Sanjar had made the capital of the Seljuk Empire.

Sanjar created a great center of Islamic learning in Merv where Khayyam wrote further works on mathematics.

Khayyam produced his Treatise on Demonstration of Problems of Algebra that contained a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections.

Khayyam was the first to conceive a general theory of cubic equations, writing:

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 text is Khayyam’s realization that a cubic equation can have more than one solution. He demonstrated the existence of equations having two solutions. Yet, he does not appear to have found that a cubic can have three solutions.

Science Legacy

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. He also posed the question of whether a ratio can be regarded as a number but leaves the question unanswered.

Khayyam’s legacy remains largely in science with his work in geometry so far ahead of its time. It was not used again until René Descartes built upon Khayyam’s theories in 17^th century France.

Outside the world of mathematics, Khayyam is best known for nearly 600 short four-line poems in the Rubaiyat.

Interestingly, Khayyam’s poetry was not published in the Muslim world until 200 years after his death. (It would be another 500 years until it appeared in Europe).

These delays in publication lead some to doubt whether Khayyam actually wrote Rubaiyat, or it was a later author.

After careful analysis, however, most scholars now agree that he is the author. They revealed a philosophical side to Khayyam that few of his contemporaries knew. 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.

This article is from our archive, originally published on an earlier date, and now republished for its importance.