In Seville, scholars led the science of astronomy, criticising earlier works on the basis of new observations and poetry was used to help people memorise the principles of algebra.
Summarised extracts from a full article:
Seville by Salah Zaimeche
It was no loss of one single science the loss of Muslim Seville caused, but all sciences, which thrived in that mighty city, including exact sciences, such as mathematics. One of the mathematicians of Seville is Ibn al-Yasamin al-Ishbilli, one of the so many neglected, and yet accomplished scholars, who had a great impact on the science, an impact brought to general knowledge by the excellent entry on him by A. Djebbar, and out of which the following is derived. Ibn al-Yasamin (fl second half of the 12th century; d. 1204) comes originally form Norh Africa, of Berber descent, and, of Black skin, just as his mother was. He was according to the historian Ibn Said educated in Seville, including on the hands of Ibn Qasim al-Shalubin, who taught him algebra and the science of calculation, an education which was not restricted to mathematics since we know he also became famous in literature and poetry, and also was a legal expert. According to Ibn al-Abar, Ibn Yasamin wrote his famous algebraic poems in Seville, poems which in 1190 he was using in his teaching. Like most scholars of the time, Ibn al-Yasamin was a welcome visitor of the Almohad court, especially under Abu Yusuf Waqub (Al-Mansur) (ruled 1184-1199).
The best known work of Ibn al-Yasamin is a poem of fifty three verses in rajaz meter entitled Al-Urjuza al-Yasminiya fi'l jabr wal Muqabala (Poem on Algebra and restoration). In it, Ibn al-Yasamin defines the algebra known in his time: number, root, and sequence, then the six canonical equations of al-Khwarzmi with the processes of solving them, and finally the operation of algebra-the restoration, comparison, multiplication, and division of monomials. This work has been widely read not just in Spain and the Maghrib, but much beyond.
The success of this work led Ibn al-Yasamin to write a second on irrational quadratic numbers and maybe a third on the method of false position, and a fourth work entitled Talqih al-afkar bi rushum huruf al-ghubar (Fertilisation of thoughts through the help of dust letters). This latter work is the most important of all for both its quantity as well as quality. It is a book of two hundred folios which contains classic chapters on the science of calculation and geometry, amongst the works of the Muslim West, which have come to us, the only which consolidates these two disciplines. Its importance is also due to the nature of its material and its mathematical tools, which make it an original book and also one which is totally representative of this period of transition in which three mathematical traditions were juxtaposed: of the east, Andalus, and the Maghrib, before they became blended in the same mold.
There are some modern studies on Ibn al-Yasamin, which require further explorations so as to understand not just the previous point made by Djebbar, but also understand the situation and role of Islamic mathematics in Spain and their wider impact.
Seville is by far the best successor to its predecessor, Toledo, and excelled that city in every single science it inherited from it. Thus, with regard to astronomy and instrument construction it even surpassed it, especially with regard to astronomy. It produced two of the greatest figures of the science of astronomy, who laid the foundation of astronomy as we know it today.
Abu Muhammad Jabir ibn Aflah. The astronomer Geber of Latin writers, who should not to be confused with the chemist Geber, Jabir ibn Hayyan (fl. second half of the eighth century). Jabir Ibn Aflah is a Hispano-Muslim astronomer and mathematician, born or lived in Seville; died probably about the middle of the twelfth century (this author is here correcting Sarton (Introduction. Vol2 2; p. 206), who by mistake places Jabir in the 13th century, whilst subsequent Muslim writers of the early 13th century were aware of his writing, which hence makes no sense. Sarton's mistake must be, however, due to lack of attention only, for he correctly lists Jabir in the volume he devotes to the twelfth century).
Jabir wrote a treatise on astronomy, Kitab al-haiaa, also called Correction of the Almagest, Islah al-Majisti, which was soon translated into Latin by Gherardo Cremonese under the title: Gebri filii Affla Hispalensis de astronomia libri IX in quibus Ptolemaeum, alioqui doctissimum, emendavit. This translation was published by Peter Apian in Nurnberg in 1534, together with the other treatise Instrumentum primi mobilis. Subsequent translations of Jabir Ibn Aflah were made into Hebrew by Moses ibn Tibbon in 1274, then again by Jacob ben Mahir (second half of the thirteenth century); this second translation was commenced by Samuel ben Judah and completed in 1335.
In this work Jabir severely criticised Ptolemy, and correctly asserted that the lower planets, Mercury and Venus have no visible parallaxes. Sarton makes the following point, that Jabir criticized vigorously the Ptolemaic theory of planets but did not propose a better one, that the lower planets (Mercury and Venus) at least must have a perceptible parallax; Venus may happen to be exactly on the line joining the sun and the earth. Sarton's criticism of Jabir for failing to propose something better than Ptolemy despite being severely critical of him misses the main point. The fact is, indeed, science and scholarship advance on the merits of those who criticize, and show the weakness of an established argument, and demonstrate it to be so. Those amongst scholars who highlight and demonstrate the weakness and deficiencies of a previous theory have the great merit of demolishing it, and preparing the ground for their followers to build upon their work. They do not need to do it themselves. Hence, Jabir, by destroying Ptolemaic astronomy and demonstrated its shortcomings, had set the foundations upon which his successors built and gave us modern astronomy. And in this respect, he deserves much more than the dismissive remark, which is, unfortunately, the case for most accounts of Muslim scholarship, even on the part of those who supposedly praise Islamic achievements, always dismissing them in the end as pale accomplishments in comparison to their Greek counterparts.
However, whilst Sarton is only guilty of the mild crime of being somehow dismissive, and whilst he, Sarton, is by far, the most favourable Western scholar to Islamic civilization, others, such as the leading figure of Western history of science, Duhem, dismisses Muslim science as pure plagiarism of the Greeks as here with Jabir Ibn Aflah. In his famed, and the textbook of history of science for generations of followers, Le Systeme du Monde, Duhem dismisses Jabir's astronomy as simply a translation of Greek astronomy, and pure plagiarism. Yet, as just established, and as was subsequently made evident by the rare scholars not keen to take Duhem's words as established fact, Jabir Ibn Aflah precisely wrote his work in refutation of Greek astronomy, thus, its very reverse. How can a theory which demolishes another be a plagiarism of it? This is one of many contradictions writers on Islam and Islamic history are guilty of.
Duhem incidentally is the same leading historian who set his hordes of followers on the following line of thought, that the Muslims burnt Greek science, and yet, few lines down, the same Muslims plagiarized Greek science. Duhem goes, indeed:
`The revelations of Greek thought on the nature of the exterior world ended with the `Almagest,' (by Ptolemy) which appeared about A.D. 145, and then began the decline of ancient learning. Those of its works that escaped the fires kindled by Mohammedan warriors were subjected to the barren interpretations of Mussulman commentors and, like parched seed, awaited the time when Latin Christianity would furnish a favourable soil in which they could once more flourish and bring forth fruit.'
If Duhem is to be followed, the Muslims are responsible for one thing, and for its total opposite, both at once. Indeed, according to him the Muslims were fanatic, rampaging hordes, burners of Greek science, and also pale imitators, copiers of the Greeks. They cannot be both, though. How can you copy a book that you have burnt; or convey a science that you have destroyed on first contact? Incidentally, both these conflicting opinions can be found not just with Duhem, but also with his crowd of followers, who pursue the same aberrations.
Back to Jabir, who is specially noted for his work on spherical trigonometry, a science `in which the Arabs in general made great advances.' He introduced the equivalent of the formula: cos B = cos a. sin B for a spherical triangle rectangular in C.
Jabir was also the first to design a portable celestial sphere to measure and explain the movements of celestial objects. The invention of the astronomical instrument called turquet has been ascribed to him.
The point previously made, how after Jabir Ibn Aflah demolished Greek astronomy, rose al-Bitruji, who built on his work and set up the foundations of modern astronomy. Al-Bitruji (known as Alpetragius) was born in Morocco, lived in Seville and died around 1204. His biography is well summed up in the entry by Julio Samso in the Dictionary of Scientific Biography, upon which reliance is made here. Al-Bitruji's 'Kitab-al-Hay'ah' was popular in thirteenth century Europe, and was translated by the Sicilian based Michael Scot (who was either Irish or Scot) under the title `On the Sphere,' and was also translated into Hebrew by Moses Ibn Tibbon in 1259, whilst Yahuda ben Solomon Kohen produced an abridged version.
According to al-Bitruji, Ibn Tufayl (of Grenada) expounded an astronomical system that differed from Ptolemy's and did not use eccentrics or epicycles, and al-Bitruji was also aware of Jabir Ibn Aflah's criticism of Ptolemy, and of the problems of the order of the spheres of the inferior planets (Jabir's treatise is also one of the ways through which the sine theorem was introduced to Spain.) An important aspect of Al-Bitruji's planetary theory is his discussion of the order of the inferior planets. After presenting the history of the subject, he gives the order as moon, Mercury, sun, Venus, Mars, and so on: mercury was slower than the sun, which was slower than Venus. He rejected the objections made to the traditional order (moon, Mercury, Venus, sun) based on the fact that the transits of Mercury and Venus across the sun are not visible. He put it that Mercury and Venus have their own light and do not receive it from the sun, as the moon does. Therefore their transits cannot be perceptible.
Al-Bitruji's astronomical system spread through much of Europe in the 13th century; William the Englishman cited it, and Robert Grosseteste referred to it in many works, even plagiarizing from it in his refutation of the Ptolemaic system. The impact of al-Bitruji continued down the centuries, at the end of the 15th century impacting upon Simon de Phares, whilst Copernicus, in his De revolutionibus, cited his system in connection with theories of the order of the inferior planets.
Seville accounted for a great instrument maker, Muhammad ibn Fattuh al-Khamairi , who flourished in the early 13th century. He is known for eight works at least. In the year 1212-3, he constructed an astrolabe in Seville, which in 1873 could be found in the French collection of H. Sauvaire, who had acquired it in Cairo, but today's possessor of the object is unknown. In the year 1216-7 he made a safiha (following in the tradition of al-Zarqali) also in Seville, which was moved between collections, in the Gengia collection, then in the Da Schio collection at Valdagno, and now in the Observatorio Astronomico in Roma, (No 694). The following year he made another Safiha, also in Seville, which was formerly in the Schultz collection before it was transferred to the French national Library (Bibliotheque Nationale de Paris). The same year he built an astrolabe in Seville, which is kept in the Collection of Cattaoui Pasha in Cairo, followed two years later (1221-2) by another astrolabe now kept in the Lewis Evans collection in the Museum of the History of Science. In the year 1224, he made an astrolabe in Seville, also kept in the same Lewis collection already cited, then six years after constructed another astrolabe formerly in the Harari collection, and finally in 1236, he made another astrolabe in Seville in the Mensing collection, now in the Alder Planetarium Chicago. Just a few years later, Seville fell, and this latter instrument could have been the last the Muslims constructed in Spain.
Seville fell in 1248 to Alfonso of Castile, who made a good use of what he inherited. During the reign of Alfonso el-Sabio (Alfonso the Wise,) King of Castile (1252 to 1284) in Spain, he commissioned works of history and science deeply reliant on Muslim sources. And during his reign was produced a collection of treatises on astronomy, and the famed Alphonsine tables; and writings on instruments mostly based on known Muslim works. Alfonso el-Sabio in 1254 established the Latin and Arabic college of Seville. Thus, just as with Toledo, Muslim loss was Christian gain. And this took place in the 13th century when Seville was wrested from the Muslims.
(Image in article is a detail from http://primates.ximian.com/~federico/photo/guadec-2002/)
 A. Djebbar: Ibn al-Yasamin; in Encyclopaedia of the history of Science, technology, and Medicine in Non Western Cultures; edited by H. Selin; Kluwer Academic Publishers. Dordrecht/Boston/London, 1997; pp. 414-5.
 A. Djebbar: Ibn al-Yasamin; p. 415.
 See, for instance:
-S. Jalal: Manzumat Ibn al-yasamin fi amal al-Jabr wal hisab; Kuwait; Mu'assassat al-Kuyat li taquadhum al-ilmi; 1988.
-M Souissi: Al-luma al-maradiniya fi sharh al-Yasminiyya; Kuwait; 198.
T. Zemouli: Mu'allafat Ibn al-Yasamin ar-riyaddiya; master thesis; E.N.S. Algiers; 1993..
 H. Suter: Die Mathematiker und Astronomen der Araber; 1900; p.119; Nachtrage, 1902; p. 174.
 G. Sarton: Introduction; vol 2; p. 206.
 M. Steinschneider: Eebraische tibersetzungen; 1893; pp. 543, 849.
 P.K.Hitti: History of the Arabs, MacMillan, London, 1970 edt. p. 572.
 G. Sarton: Introduction:; op cit; vol 2; p. 206.
 A. Duhem: Le Systeme du monde; vol. 2, Paris; 1914; pp. 172-179, in G. Sarton: Introduction; vol 2; p. 296.
 P. Duhem: Medieval Physics, in R. Palter edition: Toward Modern Science; The Noonday Press; New York; 1961; Vol 1; pp 141-159; Quote at p. 141; This article is a reprint from `Physics, history of,' Catholic Encyclopedia, XII (1911), pp 47-52.
 W. Montgomery Watt: The Influence of Islam on Medieval Europe, Edinburgh University Press; 1972. p. 35.
 Von Braunmuhl: Geschichte der Trigomometrie; vol. 1, 1900; pp. 81-3.
 W.M. Watt: Influence, op cit, p. 35.
 R.P. Lorch: The Astronomical Instruments of Jabir Ibn Aflah and the Torquetom; Centaurus, 1976; vol 20; pp 11-34.
 J. Samso: Al-Bitruji; in Dictionary of Scientific Biography; volume 15; supplement 1; Editor Charles C. Gillispie; Charles Scribner's Sons, New York, 1973 fwd. Pp. 33-6; at p. 33.
 J. Samso: Al-Bitruji; p. 33.
 J. Samso: Al-Bitruji; p. 33.
 J. Samso: Al-Bitruji; p. 35.
 J. Samso: Al-Bitruji; p. 35.
 J. Samso: Al-Bitruji; p. 35.
 L.A. Mayer: Islamic Astrolabists and their works; Albert Kundig; Geneva; 1956; p. 64.
 L.A. Mayer: Islamic Astrolabists; p. 65.
 L.A. Mayer: Islamic Astrolabists; p. 65.
 L.A. Mayer: Islamic Astrolabists; p. 65-6.
 L.A. Mayer: Islamic Astrolabists; p. 66.
 P.F. Kennedy: The Muslim sources of Dante? in The Arab influence in Medieval Europe, ed D.A. Agius and R. Hitchcock, Ithaca press, 1994, pp. 63-82. p. 72.