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Binomial Coefficients and Nasir Al-Din Al-Tusi

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Binomial Coefficients and Nasir Al-Din Al-Tusi Scientific Research and Essay Vol. 1 (2), pp. 028 -032, November 2006 Available online at http://www.academicjournals.org/SRE ISSN 1992-2248 © 2006 Academic Journals Perspective Binomial coefficients and Nasir al-Din al-Tusi Othman Echi Faculty of Sciences of Tunis, Department of Mathematics, University Tunis-El Manar"Campus Universitaire", 2092,Tunis, TUNISIA. E-mail: [email protected],[email protected]. Accepted 3 November, 2006 A historical note is given about the scientist Nasir al-Din al-Tusi legitimating the introduction of a new concept related to binomial coefficients. Al-Tusi binomial coefficients and binomial formulas are introduced and studied. Key words: Binomial coefficients, binomial theorem, history of Mathematics. INTRODUCTION HISTORICAL NOTE AND NOTATIONS Al-Tusi has wrote important works on Astronomy, logic, Mathematics and Philosophy. The first of these works, Abu Jafar Muhammad Ibn Muhammad Ibn al-Hasan ``Akhlaq-i nasiri", was written in 1232; it was a work on Nasir al-Din al-Tusi was born in Tus, Khurasan (now, ethics which al-Tusi dedicated to the Isma'ili ruler Nasir Iran) in 17 February 1201 and died in Baghdad 25 June ad-Din Abd ar-Rahim. Al-Tusi was kidnaped by the 1274. Al-Tusi was one of the greatest scientists, mathe- Isma'ili Hasan Bin Sabah's agents and sent to Alamut maticians, astronomers, philosophers, theologians and where he remained until its capture by the Mongol physicians of his time. Al-Tusi was an Arabic scholar Halagu Khan. Impressed by Al-Tusi's exceptional abilities whose writings became the standard texts in several and astrological competency, Ilkhanid Halagu Khan disciplines for several centuries. They include editions of appointed him as one of his ministers. Later, has he Euclid's Elements and Ptolemy's Almagest, as well as been designated an administrator of Auqaf. other books on mathematics and astronomy, and books In 1262, al-Tusi built an observatory at Meragha (in the on logic, ethics, and religion. Al-Tusi was known by a Azerbaijan region of north-western Iran) and directed its number of different names during his lifetime such as activity. It was equipped with the best instruments from Muhaqqiq-i Tusi, Khwaja-yi Tusi and Khwaja Nasir. His Baghdad and other Islamic centers of learning. It proper name was Muhammad ibn Muhammad ibn al- contained a twelve-feet wall quadrant made from copper Hasan al-Tusi. and an azimuth quadrant and 'turquet' invented by al- In 1214, when al-Tusi was 13 years old, Genghis Tusi. Other instruments included Astrolabes, represent- Khan, who was the leader of the Mongols, turned away tations of constellation, epicycles, and shapes of sphe- from his conquests in China and began his rapid res. Al-Tusi designed several other instruments for the advance towards the west. Genghis Khan turned his Observatory. After putting his Observatory to good use, attention again towards the east leaving his generals and making very accurate tables of planetary movements, al- sons in the west to continue his conquests. The Mongol Tusi published his work “Zij-i ilkhani (the Ilkhanic Tables)" invasion caused much destruction in both west and east. (which was dedicated to Ilkhanid Halagu Khan), written Fortunately, al-Tusi was able to study more advanced first in Persian and later translated int Arabic. This work topics before seeing the effects of the Mongols con- contains tables for computing the positions of the pla- quests on his own regions. From Tus, al-Tusi went to nets, and it also contains a star catalogue. The tables Nishapur which is 75 km west of Tus; this was a good were developed from observations over a twelve-year choice for al-Tusi to complete his education since it was period and were primarily based on original observations. an important centre of learning. There, al-Tusi studied The major astronomical treatise of al-Tusi was “al- Philosophy, Medicine and Mathematics. Particularly, he Tadhkira fi'ilm al-hay'a". This was written to give serious was taught Mathematics by Kamal al-Din ibn Yunus, who students a detailed acquaintance with astronomical and himself had been a pupil of Sharaf al-Din al-Tusi. In cosmological theory. In this treatise Nasir, gave a new Nishapur, al-Tusi began to acquire a reputation as an model of lunar motion, essentially different from Pto- outstanding scholar and became well known throughout lemy's. In his model Nasir, for the first time in the history the area. of astronomy, employed a theorem invented by himself Echi 029 which, 250 years later, occurred again in Chapter IV of al-Karaji's school. In the manuscript, al-Tusi determined Book III of Copernicus' "De Revolutionibus" (On the the coefficients of the expansion of a binomial to any revolutions of the heavenly spheres). This theorem runs power giving the binomial formula and the Pascal triangle as follows: If a point moves with uniform circular motion relations between binomial coefficients. clockwise around the epicycle while the center of the Edwards (2002) has postulated that the work of al- epicycle moves counterclock-wise with half this speed Karaji in expanding the Binomial Triangle might have along an equal deferent circle, the point will describe a borrowed Brahmegupta's work, given that it was avai- straight-line segment. lable and al-Karaji definitely had read other Hindu texts Many historians claim that the Tusi-couple result was available in Baghdad which was the great cultural and used by Copernicus after he discovered it in al-Tusi's scientific center of Muslims. The binomial coefficients work (see for example, Boyer (1947) and Dreyer (1953). have been studied in cultures around the world, both in However, Veselovsky (1973) shows that it is much more the context of binomial expansions and in the question of plausible to suppose that Copernicus took the argument how many ways to choose k items out of a collection of n he needed from Proclus' "Commentary on the first book things. Note also that, three names from China figure of Euclid", and not from al- Tusi. There are two other prominently in the story of the discovery of the binomial astronomical treatises of al-Tusi. The first treatise, called coefficients. These scientists are Chia Hsien, Yanghui the "Muiniya" and written in 1235, contains a standard (1261) and Chu Shih-chieh (1303). The idea of taking account of Ptolemaic lunar and planetary theory. In the “six tastes one at a time, two at a time, three at a time, second treatise, the "Hall" (written between 1235 and etc." was written down correctly in India 300 years before 1256), al-Tusi uses the plane version of his "Tusi-couple" the birth of Christ in a book called the ``Bhagabati Sutra". to explain the motion in longitude of the epicycle centre Thus the Indian civilization is the earliest one that has an of the moon and the planets. The "Hall" does not yet understanding of the binomial coefficients in their combi- include the spherical version of the "Tusi-couple", which natorial form ``n choose k". is used in the "Tadhkira" to describe the prosneusis of The interest in the binomial coefficients in India dealt the moon and the latitude theory of the planets. It is with choosing and arranging things. However, mathema- worth noting that the term "Tusi couple" is a modern one, ticians in the Middle East were interested almost entirely coined by Edward Kennedy (1966). in expansions of polynomials; the work of the Indian It is also worth noting that al-Tusi has written revised Brahmagupta, which included the expansion of (a + b)3 , Arabic versions of works by Autolycus, Aristarchus, Euc- was available to the scholars of the Middle East. Some lid, Apollonius, Archimedes, Hypsicles, Theodosius, Me- Historians consider that investigations on binomial coef- nelaus and Ptolemy. Ptolemy's Almagest was one of the ficients by mathematicians in the Middle East may well works which Arabic/Muslim scientists studied intently. In have been inspired Brahmagupta's work. There is anot- 1247 al-Tusi wrote ``Tahrir al-Majisti (Commentary on her scientist from the Middle East who worked with the the Almagest)" in which he introduced various trigono- binomial coefficients; namely, al-Samawal (a Jew born in metrical techniques to calculate tables of sines; al-Tusi Baghdad who died in 1180). Omar al-Khayyam is ano- gave tables of sine with entries calculated to three ther famous Persian scientist who makes a claim to sexagesimal places for each half degree of the argu- knowledge; he wrote a letter claiming to have been able ment. Ibadov (1968) has asserted that al-Tusi had found to expand binomials to sixth power and higher, but the the value of the sine of one degree (with the precision up actual work does not survive; in the letter he mentions to the fifth decimal place). Ibadov considers some trigo- that he is aware not only of work done in India, but of nometric propositions used for this purpose by Tusi and Euclid's Elements. Al-Khayyam is best known in the their relation with analogous results obtained by scien- West for his collection of poems “The Rubaiyat", which tists of Central Asia and Western Europe. was translated into English in 1859 by Edward Fitzge- An important mathematical contribution of al-Tusi was rald. the creation of trigonometry as a mathematical discipline In Europe, there are many authors who can fairly lay in its own right rather than as just a tool for astronomical claim to having made a serious study of the binomial applications. In “Treatise on the quadrilateral", al-Tusi coefficients, several of them long before Blaise Pascal gave the first extant exposition of the whole system of was even born.
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