Q. & A. 711 to 1707 with solved Papers css 1971 to date

MATHEMATICS
The Arabs were keen students of Mathematics. They made marked advance in this subject. The use of the cipher which was obtained by them from India, was passed on by them to Europe. Prof. Arnold says, ”They were undisputable the founders of plane and spherical trigonometry which properly speaking did not exist among the Greeks.” The Arabs gave the world the science of algebra in its perfect form. Algebra is one of the proudest achievements of the Arabs and it was cultivated with so much absorbing interest that within two centuries of its invention it had reached gigantic proportions Hajjaj bin Yusuf translated the first six books of Euclid into Arabic : he also wrote a treatise on geometrical problems.
Their contribution to the development of mathematics is great. Algebra, Statistics, conic sections and other branches of applied mathematics are the discoveries of the Muslims. They for the first time applied algebra to geometry. They discovered equations of the second degree and developed the theory of quadratic equations and the binomial theorem. The Arabs not only collected and translated the works of the Greek mathematicians but also illustrated them and made valuable commentaries of them. There were a good number of mathematicians among the Muslims who contributed much to mathematics during the middle ages.
The oldest extant Andalusi mathematical text is the unpublished treatise on land surveying (taksir) written by the physician Muhammad b. Abdun al-Jabali towards the middle of the

4th/10th century the book is of a practical nature, and this seems, indeed, to be one of the main characteristics of the first manifestations of Andalusi mathematics. The second half of the

4th/10th century witnessed the important mathematical and astronomical school founded by Abu ’I-Qasim Maslama b. Ahmad al-Majriti (d. 397/1007), three members of which-Maslama himself, Abu ’I-Qasim Ahmad b. Muhammad b. al-Samh (d.426/1035) and abu ’I Hasan Ali b. Sulayman ai-Zahrawi-wrote treatises on commercial arithmetic (mu’amalat). These texts do not seem to be extant, but we can gain an idea of their contents through the Liber Mahameleth, a Latin translation, ascribed to John of Seville, of an
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Andalusi treatise on the same subject’ The authorities quoted in it (Euclid, Archimedes, Nichomachos of Gerasa, Muhammad b. Musa al-Kharizmi and Abu Kamil Shuja b. Aslam al-Misri) are precisely those one would expect to be known in al-Andaius in the second half of the 4th/10th century. The book deals with elementary arithmetic (addition, subtraction, multiplication, division and extraction of the square root, together with adequate methods for obtaining good approximations to imperfect square roots) and algebra (equations of the first and second degrees), and ends with a long collection of practical problems which might be of interest to a merchant.10
Ibn al-Samh himself seems, apart from his kitab almu’amalat, to have written extensively on arithmetic and geometry, but his works on these subjects are apparently lost. Nothing, on the other hand, is known of the development of algebra in al-Andalus in this period, apart from what we can gather from the Liber Mahameleth and, possibly, from an analysis of the treatises on division of inheritance (’ilm al-fara’id)’ we have to wait till the last stage of Andalusi history the Granada of the Banu Nasr (631/1232-

897/1492) to find an abridgement of algebra (Ikhtisar al-jabr wa ’1muqabala), written by one Abu Abdullah Muhammad b. Umar b. Muhammad b. Badr of whom we know only that he wrote this book before 744/1343 and was (perhaps) an Andalusi author. The Ikhtisar is a treatise on elementary algebra dealing, among other things, with indeterminate equations in the Diophantine tradition, which are here documented for the first time in al-Andalus. Far more interesting is the work of the last important Andalusi mathematician, Abu IHassan Ali b. Muhammad al-Basti al-Qalasadi, who wrote extensively on arithmetic, algebra and fara’id. His mathematical works seem to have been strongly influenced by those of the Moroccan mathematician Abu ’l-’Abbas Ahmad b. Muhammad, known as Ibn al-Banna al-Marrakushi (654/1256, 725/1321). but his originality has been exaggerated by modern scholarship. Thus, although he did indeed make interesting improvements to the method of successive approximations of squares and cubes merely followed the lead of Abu Mansur al-Baghdadi and al-Umawi al-Anualusi. Again, although he has been regarded as the man who introduced algebraic symbolism and it is obvious that he did use it-he had numerous predecessors in this, both in the Mashriq and in the Maghrib.

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Political and Cultural History of Islam

If this brief survey of arithmetic and algebra in al-Andalus is somewhat un-encouraging a different picture emerges when we consider geometry and spherical trigonometry. Apart from the lost geometrical works of Ibn al-Samh, we should here consider three important figures of the 5th/l 1th century: King Abu Amir Yusuf b. Ahmad al-Mu’taman of Saragossa, Abu Zayd ’Abdur Rahman b. Sayyid’-who flourished in Valencia between 456/1063 and 490/1096 and was the master of the famous philosopher and physicist Muhammad b. Yahya b. al-Sa’igh, known as Ibn Bajja (463/1070-

533/1138)-and the Qazi of Jaen Abu Abdullah Muhammad b. Mu’adh al-Jayyani (d. 486/1093). About al-Mu’taman we only knew, until very recently, that he had written an important treatise called Al-Istikmal, but this situation has now changed as a result of important works by A. Djebbar and J. P. Hogendijk. The latter has discovered four incomplete manuscripts of the Istikmal, containing fragments of the work dealing with number theory, plane geometry, study of the concepts of ratio and proportion following books V and VI of Euclid’s Elements, and the geometry of the sphere and of other solid bodies and conic sections. The extant parts of the Istikmal prove that al-Mu’taman had an important royal library containing the best books available in the 5th/11th century for the study of higher mathematics: Euclid’s Elements and Data, Archimedes, ’On the sphere and Cylinder (and, also, Eutocius’ commentary on the second book of this work), the books on Spherics by T’neodosius and Menelaos, Appolonius’ Conies, Ptolemy’s Almagest, Thabit b. Qurra’s treatises on amicable numbers and on Menelaos’ theorem, the treatise of the Banu Musa on the measurement of plane and spherical figures, Ibrahim b. Sinan’s book on the quadrature of the parabola, Ibn al-Haytham’s Optics, etc. Nevertheless, al-Mu’taman’s treatment of geometrical problems is not limited to mere reproduction of his sources, but, quite often, offers original solutions which prove that he was an excellent geometer.”

The diffusion of the Arabic numerals in non-Muslim Europe was incredibly slow. Christian arithmeticians throughout the eleventh, twelfth and part of the thirteenth centuries persisted in the use of the antiquated Roman numerals and the abacus or made a compromise and used the new algorisms together with their old system. It was in Italy that the new symbols were first employed for practical purposes. In 1202 Leonardo Fibonacci of Pisa, who was
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taught by a Muslim master and had travelled in North Africa, published a work which was the mam landmark in the introduction of the Arabic numerals. More than that, it marks the beginning of European mathematics. With the old type of numerals, arithmetical progress along certain lines would have been impossible. The zero and Arabic numerals lie behind the science of calculation as we know it today.12 ASTRONOMY
In Spam astronomical studies were cultivated assiduously after the middle of the tenth century and were regarded with special favour by the rulers of Cordova, Seville and Toledo. Following abuMa’shar of Baghdad, most of the Andalusian astronomers believed ip astral influence as the cause underlying the chief occurrences between birth and death on this earth. The study of this astral influence, i.e. astrology, necessitated the determining of the location of places throughout the world together with their latitudes and longitudes. Thus did astrology become the mother of astronomy. Finally it was through Spanish channels that the Latin West found its Oriental inspiration in astronomy and astrology. The leading Muslim astronomical works were translated in Spain into Latin, and the Alfonsine tables compiled by Alfonso X in the thirteenth century were nothing but a development of Arab astronomy. Spanish Arab astronomers built upon the preceding astronomical and astrological works of their co-religionists in the East. They reproduced the Aristotelian system, as distinguished from the Ptolemaic, and in the name of Aristotle attacked the Ptolemaic representation of the celestial movements. Outstanding among early Hispano-Arabic astronomers were al-Majriti of Cordova, al-Zarqali of Toledo of Seville.
Abu-al-Qasim Maslamah al-Majriti, the earliest Spainsh Muslim scientist of any importance, edited and corrected the planetary tables of al-Khwarizmi, the first table composed by a Muslim. He converted the basis of these tables from the era of Yazdagird into that of Islam and to some extent replaced the meridian of arin by that of Cordova. In 1126 Adelard of Bath made a Latin version of the tables ascribed to al-Khwarizmi. About fourteen years later another important zij, that of al-R.ntam. composed about