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The History of the Binomial Coefficients in the Middle East

            While the interest in the binomial coefficients in India dealt with choosing and arranging things, mathematicians in the Middle East were interested almost entirely in expansions of polynomials; the work of the Indian Brahmagupta, which included the expansion of  (a+b)³, was available to the scholars of the Middle East, and their expansions on this problem may well have been inspired by his work.  

            In his book Pascal's Arithmetical Triangle, A.W.F. Edwards postulates that the work of Al-Karaji (circa 1000 A.D.) in expanding the Binomial Triangle might have borrowed Brahmegupta's work, given that it was available and Al-Karaji definitely had read other Hindu texts then available in Baghdad.  There is some disagreement whether he was born in Karaj in Persia, hence the name Al-Karaji, or in Karkh, a suburb of Baghdad, which would have made him an Arab.  The name Al-Karkhi is also used in some older texts, though the modern consensus is that della Vida's scholarship in 1933 is probably correct and he was from Karaj; there is no dispute that he worked in Baghdad, which was the great cultural center of the Middle East even before the time of Mohammed.

            Others in the Middle East who worked with the binomial coefficients include Al-Samawal, a Jew born in Baghdad who died in 1180, the Persian Nasir Al-Din Al-Tusi, who published in 1265, and the Persian Al-Kashi, who died in 1429, whose work Key of Arithmetic has the triangle up to the ninth level, the additive rule and the multiplicative rule.  Unlike many of the others listed here, Al-Kashi did not go to Baghdad, but instead worked in Samarkand, the major city in the area now known as Uzbekistan.

            There is also another famous Persian scientist who makes a claim to knowledge.  Omar Khayyam (circa 1100) wrote a letter claiming to have been able to expand binomials to sixth power and higher, but the actual work does not survive; in the letter he mentions that he is aware not only of work done in India, but of Euclid's Elements.

            Khayyam was one of the great scientists of his day; among other works, he gave a very good approximation for the exact length of a year, one of the most exact in the world when he was alive.  He is best known in the West for his collection of poems The Rubaiyat, which was translated into English in 1859 by Edward Fitzgerald.  The most famous two of these are:

The Moving Finger writes, and, having writ,	A Book of Verses underneath the Bough,
Moves on: nor all thy Piety nor Wit	A Jug of Wine, a Loaf of Bread--and Thou
Shall lure it back to cancel half a Line,	Beside me singing in the Wilderness--
Nor all thy Tears wash out a Word of it.	Oh, Wilderness were Paradise enow!