Islamic mathematics 3304216 225544515 2008-07-14T06:32:06Z SmackBot 433328 Date the maintenance tags or general fixes In the [[history of mathematics]], '''Islamic mathematics''' refers to the [[mathematics]] developed in the [[Muslim world|Islamic world]] between [[622]] and [[1600]], in the part of the world where [[Islam]] was the dominant religious and cultural influence. [[Islamic science]] and mathematics flourished under the Islamic [[caliph]]ate (also known as the [[Islamic Empire]]) established across the [[Middle East]], [[Central Asia]], [[North Africa]], [[Sicily]], the [[Iberian Peninsula]], and in parts of [[France]] and [[India]] in the 8th century. The center of Islamic mathematics was located in present-day [[Iraq]] and [[Iran]], but at its greatest extent stretched from [[Turkey]], [[North Africa]] and [[Spain]] in the west, to [[India]] in the east.<ref>O'Connor 1999</ref> While most scientists in this period were [[Muslim]]s and [[Arabic language|Arabic]] was the dominant language&mdash;much like [[Latin language|Latin]] in [[Medieval Europe]], Arabic was used as the written language of scholars throughout the [[Islamic world]] at the time&mdash;contributions were made by people of different ethnic groups ([[Arab]]s, [[Moors]], [[Persian people|Persians]], [[Turkic peoples|Turks]]) and religions ([[Muslim]]s, [[Christian]]s, [[Sabian]]s, [[Jew]]s, [[Zoroastrianism|Zoroastrians]]).<ref>Hogendijk 1999</ref> In particular, a large number of Islamic scientists in many disciplines, including mathematics, were Persians.<ref>[http://globalthink.net/index.cfm/2006/9/12/The-Persistence-of-Cultures-in-World-History--PersiaIran The Persistence of Cultures in World History: Persia/Iran by Dr. Laina Farhat-Holzman]</ref>{{Failed verification|date=July 2008}} ==Origins and influences == The first century of the [[Islam]]ic [[Arab Empire]] saw almost no scientific or mathematical achievements since the Arabs, with their newly conquered empire, had not yet gained any intellectual drive and research in other parts of the world had faded. In the second half of the eighth century Islam had a cultural awakening, and research in mathematics and the sciences increased.<ref name="Boyer Intro Islamic Algebra">{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=The Arabic Hegemony|pages=227|quote=The first century of the Muslim empire had been devoid of scientific achievement. This period (from about 650 to 750) had been, in fact, perhaps the nadir in the development of mathematics, for the Arabs had not yet achieved intellectual drive, and concern for learning in other parts of the world had faded. Had it not been for the sudden cultural awakening in Islam during the second half of the eighth century, considerably more of ancient science and mathematics would have been lost. [...] It was during the caliphate of al-Mamun (809-833), however, that the Arabs fully indulged their passion for translation. The caliph is said to have had a dream in which Aristotle appeared, and as a consequence al-Mamun determined to have Arabic versions made of all the Greek works that he could lay his hands on, including Ptolemy's ''Almagest'' and a complete version of Euclid's ''Elements''. From the Byzantine Empire, with which the Arabs maintained an uneasy peace, Greek manuscripts were obtained through peace treaties. Al-Mamun established at Baghdad a "House of Wisdom" (Bait al-hikma) comparable to the ancient Museum at Alexandria. Among the faculty members was a mathematician and astronomer, Mohammed ibn-Musa al-Khwarizmi, whose name, like that of Euclid, later was to become a household word in Western Europe. The scholar, who died sometime before 850, wrote more than half a dozen astronomical and mathematical works, of which the earliest were probably based on the ''Sindhad'' derived from India.}}</ref> The Muslim [[Abbasid]] [[caliph]] [[al-Mamun]] (809-833) is said to have had a dream where Aristotle appeared to him, and as a consequence al-Mamun ordered that Arabic translation be made of as many Greek works as possible, including Ptolemy's ''Almagest'' and Euclid's ''Elements''. Greek works would be given to the Muslims by the [[Byzantine Empire]] in exchange for treaties, as the two empires held an uneasy peace.<ref name="Boyer Intro Islamic Algebra" /> Many of these Greek works were translated by [[Thabit ibn Qurra]] (826-901), who translated books written by [[Euclid]], [[Archimedes]], [[Apollonius of Perga|Apollonius]], [[Ptolemy]], and Eutocius.<ref name="Boyer Islamic Rhetoric Algebra Thabit">{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=The Arabic Hegemony|pages=234|quote=but al-Khwarizmi's work had a serious deficiency that had to be removed before it could serve its purpose effectively in the modern world: a symbolic notation had to be developed to replace the rhetorical form. This step the Arabs never took, except for the replacement of number words by number signs. [...] Thabit was the founder of a school of translators, especially from Greek and Syriac, and to him we owe an immense debt for translations into Arabic of works by [[Euclid]], [[Archimedes]], [[Apollonius of Perga|Apollonius]], [[Ptolemy]], and [[Eutocius]].}}</ref> Historians are in debt to many Islamic translators, for it is through their work that many ancient [[Greek language|Greek]] texts have survived only through [[Arabic]] translations. [[Greek mathematics|Greek]], [[Indian mathematics|Indian]], and [[Mesopotamian]] mathematics all played an important role in the development of early Islamic mathematics. The works of mathematicians such as [[Euclid]], [[Apollonius of Perga|Apollonius]], [[Archimedes]], [[Diophantus]], [[Aryabhata]] and [[Brahmagupta]] were all acquired by the Islamic world and incorporated into their mathematics. Perhaps the most influential mathematical contribution from [[India]] was the decimal [[place-value]] [[Hindu-Arabic numeral system|Indo-Arabic numeral system]], also known as the [[Hindu numerals]].<ref name="Berggren Islamic mathematics">{{cite book | first=J. Lennart | last=Berggren | authorlink=J. Lennart Berggren | title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook | chapter=Mathematics in Medieval Islam | publisher=Princeton University Press | year=2007 | pages=516 | isbn=9780691114859 | quote=The mathematics, to speak only of the subject of interest here, came principally from three traditions. The first was Greek mathematics, from the great geometrical classics of Euclid, Apollonius, and Archimedes, through the numerical solutions of indeterminate problems in Diophantus's ''Arithmatica'', to the practical manuals of Heron. But, as Bishop Severus Sebokht pointed out in the mid-seventh century, "there are also others who know something." Sebokht was referring to the Hindus, with their in genius arithmetic system based on only nine signs and a dot for an empty place. But they also contributed algebraic methods, a nascent trigonometry, and methods from solid geometry to solve problems in astronomy. The third tradition was what one may call the mathematics of practitioners. Their numbers included surveyors, builders, artisans, in geometric design, tax and treasury officials, and some merchants. Part of an oral tradition, this mathematics transcended ethnic divisions and was common heritage of many of the lands incorporated into the Islamic world. }}</ref> The [[Persian people|Persian]] historian [[al-Biruni]] (c. 1050) in his book ''Tariq al-Hind'' states that the [[Abbasid]] [[caliph]] [[al-Ma'mun]] had an embassy in India from which was brought a book to Baghdad that was translated into Arabic as ''Sindhind''. It is generally assumed that ''Sindhind'' is none other than Brahmagupta's ''[[Brahmasphutasiddhanta|Brahmasphuta-siddhanta]]''.<ref name="Boyer Siddhanta">{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=The Arabic Hegemony|pages=226|quote=By 766 we learn that an astronomical-mathematical work, known to the Arabs as the ''Sindhind'', was brought to Baghdad from India. It is generally thought that this was the ''Brahmasphuta Siddhanta'', although it may have been the ''Surya Siddhanata''. A few years later, perhaps about 775, this ''Siddhanata'' was translated into Arabic, and it was not long afterwards (ca. 780) that Ptolemy's astrological ''Tetrabiblos'' was translated into Arabic from the Greek.}}</ref> The earliest translations from Sanskrit inspired several astronomical and astrological Arabic works, now mostly lost, some of which were even composed in verse.<ref name="Plofker 434"/> But Indian influences were soon overwhelmed by Greek mathematical and astronomical texts. It is not clear why this occurred but it may have been due to the greater availability of Greek texts in the region, the larger number of practitioners of Greek mathematics in the region, or because Islamic mathematicians favored the deductive exposition of the Greeks over the elliptic Sanskrit verse of the Indians. Regardless of the reason, Indian mathematics soon became mostly eclipsed by or merged with the "Graeco-Islamic" science founded on Hellenistic treatises.<ref name="Plofker 434">{{cite book | first = Kim | last = Plofker | year = 2007 | pages = 434 | title = | quote = The early translations from Sanskrit inspired several other astronomical/astrological works in Arabic; some even imitated the Sanskrit practice of composing technical treatises in verse. Unfortunately, the earliest texts in this genre have now mostly been lost, and are known only from scattered fragments and allusions in later works. They reveal that the emergent Arabic astronomy adopted many Indian parameters, cosmological models, and computational techniques, including the use of sines.<BR>These Indian influences were soon overwhelmed - although it is not completely clear why - by those of the Greek mathematical and astronomical texts that were translated into Arabic under the Abbasid caliphs. Perhaps the greater availability of Greek works in the region, and of practitioners who understood them, favored the adoption of the Greek tradition. Perhaps its prosaic and deductive expositions seemed easier for foreign readers to grasp than elliptic Sanskrit verse. Whatever the reasons, Sanskrit inspired astronomy was soon mostly eclipsed by or merged with the "Graeco-Islamic" science founded on Hellenistic treatises.}}</ref> == Importance == J. J. O'Conner and E. F. Robertson wrote in the ''[[MacTutor History of Mathematics archive]]'': {{quote|"Recent research paints a new picture of the debt that we owe to Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to [[Europe]]an mathematicians of the 16th, 17th, and 18th centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects, the mathematics studied today is far closer in style to that of Islamic mathematics than to that of [[Greek mathematics]]."}} R. Rashed wrote in ''The development of Arabic mathematics: between arithmetic and algebra'': {{quote|"[[Muhammad ibn Mūsā al-Khwārizmī|Al-Khwarizmi]]'s successors undertook a systematic application of [[arithmetic]] to [[algebra]], algebra to arithmetic, both to [[trigonometry]], algebra to the [[Euclid]]ean [[Number theory|theory of numbers]], algebra to [[geometry]], and geometry to algebra. This was how the creation of [[Symmetric algebra|polynomial algebra]], [[Combinatorics|combinatorial analysis]], [[numerical analysis]], the numerical solution of [[equation]]s, the new elementary theory of numbers, and the geometric construction of equations arose."}} == Biographies == ;{{transl|ar|ALA|[[Al-Ḥajjāj ibn Yūsuf ibn Maṭar]]}} (786 &ndash; 833) :Al-Ḥajjāj translated [[Euclid]]'s ''[[Euclid's Elements|Elements]]'' into Arabic. ;{{transl|ar|ALA|[[Muḥammad ibn Mūsā al-Khwārizmī]]}} (c. 780 [[Khwarezm]]/[[Baghdad]] – c. 850 Baghdad) <!-- JPH MT AMFB --> :Al-Khwārizmī was a Persian [[mathematician]], [[astronomy|astronomer]], [[astrology|astrologer]] and [[geographer]]. He worked most of his life as a [[scholar]] in the [[House of Wisdom]] in [[Baghdad]]. His ''Algebra'' was the first book on the systematic solution of [[linear equation|linear]] and [[quadratic equation]]s. [[Latin]] translations of his ''Arithmetic'', on the [[Indian numerals]], introduced the [[decimal]] [[Positional notation|positional number system]] to the [[Western world]] in the 12th century. He revised and updated [[Ptolemy]]'s ''Geography'' as well as writing several works on astronomy and astrology. ;{{transl|ar|ALA|[[Al-ʿAbbās ibn Saʿid al-Jawharī]]}} (c. 800 Baghdad? &ndash; c. 860 Baghdad?)<!-- MT --> :Al-Jawharī was a mathematician who worked at the House of Wisdom in Baghdad. His most important work was his ''Commentary on [[Euclid's Elements]]'' which contained nearly 50 additional [[proposition]]s and an attempted [[proof]] of the [[parallel postulate]]. ;{{transl|ar|ALA|[[ʿAbd al-Hamīd ibn Turk]]}} (fl. 830 Baghdad) <!-- JPH --> :Ibn Turk wrote a work on [[algebra]] of which only a chapter on the solution of [[quadratic equations]] has survived. ;{{transl|ar|ALA|[[Yaʿqūb ibn Isḥāq al-Kindī]]}} (c. 801 [[Kufah]] &ndash; 873 Baghdad) <!-- JPH MT AMFB --> :Al-Kindī (or Alkindus) was a [[philosopher]] and [[scientist]] who worked as the House of Wisdom in Baghdad where he wrote commentaries on many Greek works. His contributions to mathematics include many works on [[arithmetic]] and [[geometry]]. ;[[Hunayn ibn Ishaq]] (808 [[Al-Hirah]] &ndash; 873 Baghdad) <!-- MT --> : Hunayn (or Johannitus) was a translator who worked at the House of Wisdom in Baghdad. Translated many Greek works including those by [[Plato]], [[Aristotle]], [[Galen]], [[Hippocrates]], and the [[Neoplatonists]]. ;{{transl|ar|ALA|[[Banū Mūsā]]}} (c. 800 Baghdad &ndash; 873+ Baghdad) <!-- JPH AMFB --> :The Banū Mūsā where three brothers who worked at the House of Wisdom in Baghdad. Their most famous mathematical treatise is ''The Book of the Measurement of Plane and Spherical Figures'', which considered similar problems as [[Archimedes]] did in his ''[[On the Measurement of the Circle]]'' and ''On the sphere and the cylinder''. They contributed individually as well. The eldest, {{transl|ar|ALA|[[Ja'far Muhammad ibn Mūsā ibn Shākir|Jaʿfar Muḥammad]]}} (c. 800) specialised in geometry and astronomy. He wrote a critical revision on [[Apollonius of Perga|Apollonius]]' ''Conics'' called ''Premises of the book of conics''. {{transl|ar|ALA|[[Ahmad ibn Mūsā ibn Shākir|Aḥmad]]}} (c. 805) specialised in mechanics and wrote a work on [[pneumatic]] devices called ''On mechanics''. The youngest, {{transl|ar|ALA|[[Al-Hasan ibn Mūsā ibn Shākir|al-Ḥasan]]}} (c. 810) specialised in geometry and wrote a work on the [[ellipse]] called ''The elongated circular figure''. ;[[Al-Mahani]] <!-- MT --> ;[[Ahmed ibn Yusuf]] <!-- MT --> ;[[Thabit ibn Qurra]] (Syria-Iraq, 835-901) <!-- JPH MT --> ;[[Al-Hashimi]] (Iraq? ca. 850-900) <!-- JPH --> ;{{transl|ar|ALA|[[Muḥammad ibn Jābir al-Ḥarrānī al-Battānī]]}} (c. 853 [[Harran]] – 929 [[Qasr al-Jiss]] near [[Samarra]]) <!-- JPH (Syria, ca. 900) MT --> ;[[Abu Kamil]] (Egypt? ca. 900) <!-- JPH MT --> ;[[Sinan ibn Tabit]] (ca. 880 - 943) <!-- MT --> ;[[Al-Nayrizi]] <!-- MT --> ;[[Ibrahim ibn Sinan]] (Iraq, 909-946) <!-- JPH MT --> ;[[Al-Khazin]] (Iraq-Iran, ca. 920-980) <!-- JPH MT --> ;[[Al-Karabisi]] (Iraq? 10th century?) <!-- JPH --> ;[[Ikhwan al-Safa']] (Iraq, first half of 10th century) <!-- JPH --> :The Ikhwan al-Safa' ("brethren of purity") were a (mystical?) group in the city of Basra in Irak. The group authored a series of more than 50 letters on science, philosophy and theology. The first letter is on arithmetic and number theory, the second letter on geometry. ;[[Al-Uqlidisi]] (Iraq-Iran, 10th century) <!-- JPH MT --> ;[[Al-Saghani]] (Iraq-Iran, ca. 940-1000) <!-- JPH --> ;{{transl|ar|ALA|[[Abū Sahl al-Qūhī]]}} (Iraq-Iran, ca. 940-1000) <!-- JPH MT --> ;[[Al-Khujandi]] <!-- MT --> ;{{transl|ar|ALA|[[Abū al-Wafāʾ al-Būzjānī]]}} (Iraq-Iran, ca. 940-998) <!-- JPH MT --> ;[[Ibn Sahl]] (Iraq-Iran, ca. 940-1000) <!-- JPH --> ;[[Al-Sijzi]] (Iran, ca. 940-1000) <!-- JPH MT --> ;[[Labana of Cordoba]] (Spain, ca. 10th century) :One of the few Islamic female mathematicians known by name, and the secretary of the [[Caliphate of Cordoba|Umayyad Caliph]] al-Hakem II. She was well-versed in the exact sciences, and could solve the most complex geometrical and algebraic problems known in her time.<ref>{{citation|first=S.P.|last=Scott|title=History of the Moorish Empire in Europe|publisher=J.B. Lippincott Company|year=1904|pages=447-8 |url=http://www.1001inventions.com/index.cfm?fuseaction=main.viewBlogEntry&intMTEntryID=2580 |accessdate=2008-01-30}}</ref> ;[[Ibn Yunus]] (Egypt, ca. 950-1010) <!-- JPH MT --> ;[[Abu Nasr ibn `Iraq]] (Iraq-Iran, ca. 950-1030) <!-- JPH MT --> ;[[Kushyar ibn Labban]] (Iran, ca. 960-1010) <!-- JPH --> ;[[Al-Karaji]] (Iran, ca. 970-1030) <!-- JPH MT --> ;[[Ibn al-Haytham]] (Iraq-Egypt, ca. 965-1040) <!-- JPH MT --> ;{{transl|ar|ALA|[[Abū al-Rayḥān al-Bīrūnī]]}} ([[September 15]] [[973]] in Kath, [[Khwarezm]] &ndash; [[December 13]] [[1048]] in [[Gazna]]) <!-- JPH MT --> ;[[Ibn Sina]] <!-- MT --> ;[[Ibn Tahir al-Baghdadi|al-Baghdadi]] <!-- MT --> ;[[Al-Nasawi]] <!-- MT --> ;[[Al-Jayyani]] (Spain, ca. 1030-1090) <!-- JPH MT --> ;[[Ibn al-Zarqalluh]] (Azarquiel, al-Zarqali) (Spain, ca. 1030-1090) <!-- JPH --> ;[[Al-Mu'taman ibn Hud]] (Spain, ca. 1080) <!-- JPH --> ;[[al-Khayyam]] (Iran, ca. 1050-1130) <!-- JPH MT --> ;{{transl|ar|ALA|[[Ibn Yaḥyā al-Maghribī al-Samawʾal]]}} (c. [[1130]] [[Baghdad]] &ndash; c. [[1180]] [[Maragha]]) <!-- MT --> ;{{transl|ar|ALA|[[Sharaf al-Dīn al-Ṭūsī]]}} (Iran, ca. 1150-1215) <!-- JPH MT --> ;[[Ibn Mun`im]] (Maghreb, ca. 1210) <!-- JPH --> ;[[al-Marrakushi]] (Morocco, 13th century) <!-- JPH --> ;{{transl|ar|ALA|[[Naṣīr al-Dīn al-Ṭūsī]]}} ([[18 February]] [[1201]] in [[Tus]], [[Khorasan]] &ndash; [[26 June]] [[1274]] in [[Kadhimain]] near [[Baghdad]]) <!-- JPH MT --> ;{{transl|ar|ALA|[[Muḥyi al-Dīn al-Maghribī]]}} (c. 1220 Spain &ndash; c. 1283 [[Maragha]]) <!-- MT --> ;{{transl|ar|ALA|[[Shams al-Dīn al-Samarqandī]]}} (c. 1250 [[Samarqand]] &ndash; c. 1310) <!-- MT --> ;[[Ibn Baso]] (Spain, ca. 1250-1320) <!-- JPH --> ;[[Ibn al-Banna']] (Maghreb, ca. 1300) <!-- JPH MT --> ;[[Kamal al-Din Al-Farisi]] (Iran, ca. 1300) <!-- JPH MT --> ;[[Al-Khalili]] (Syria, ca. 1350-1400) <!-- JPH MT --> ;[[Ibn al-Shatir]] (1306-1375) <!-- JPH --> ;'''{{transl|ar|ALA|[[Qāḍī Zāda al-Rūmī]]}}''' (1364 [[Bursa]] &ndash; 1436 Samarkand) <!-- MT --> ;{{transl|ar|ALA|[[Jamshīd al-Kāshī]]}} (Iran, Uzbekistan, ca. 1420) <!-- JPH MT --> ;[[Ulugh Beg]] (Iran, Uzbekistan, 1394-1449) <!-- JPH MT --> ;[[Al-Umawi]] <!-- MT --> ;[[Abū al-Hasan ibn Alī al-Qalasādī]] (Maghreb, 1412-1482) <!-- JPH MT --> :Later major medieval [[Arab]] mathematician. Pioneer of [[Mathematical notation|symbolic algebra]] ==Fields== ===Algebra=== {{see also|History of algebra}} [[Image:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|right|A page from ''[[The Compendious Book on Calculation by Completion and Balancing]]''.]] There are three theories about the origins of Arabic Algebra. The first emphasizes Hindu influence, the second emphasizes Mesopotamian or Persian-Syriac influence and the third emphasizes Greek influence. Many scholars believe that it is the result of a combination of all three sources.<ref name="Boyer Three Influences on al Jabr">{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=230|isbn=0471543977}} {{quote|"Al-Khwarizmi continued: "We have said enough so far as numbers are concerned, about the six types of equations. Now, however, it is necessary that we should demonsrate geometrically the truth of the same problems which we have explained in numbers." The ring of this passage is obviously Greek rather than Babylonian or Indian. There are, therefore, three main schools of thought on the origin of Arabic algebra: one emphasizes Hindu influence, another stresses the Mesopotamian, or Syriac-Persian, tradition, and the third points to Greek inspiration. The truth is probably approached if we combine the three theories."}}</ref> Throughout their time in power, before the fall of Islamic civilization, the Arabs used a fully rhetorical algebra, where sometimes even the numbers were spelled out in words. The Arabs would eventually replace spelled out numbers (eg. twenty-two) with [[Arabic numerals]] (eg. 22), but the Arabs never adopted or developed a syncopated or symbolic algebra.<ref name="Boyer Islamic Rhetoric Algebra Thabit">{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=234|isbn=0471543977}} {{quote|"but al-Khwarizmi's work had a serious deficiency that had to be removed before it could serve its purpose effectively in the modern world: a symbolic notation had to be developed to replace the rhetorical form. This step the Arabs never took, except for the replacement of number words by number signs. ... Thabit was the founder of a school of translators, especially from Greek and Syriac, and to him we owe an immense debt for translations into Arabic of works by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius."}}</ref> The Muslim<ref>{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=228-229|isbn=0471543977}} {{quote|"the author's preface in Arabic gave fulsome praise to Mohammed, the prophet, and to al-Mamun, "the Commander of the Faithful"."}}</ref> Persian mathematician {{Unicode|[[Muhammad ibn Musa al-Khwarizmi|Muhammad ibn Mūsā al-khwārizmī]]}} was a faculty member of the "House of Wisdom" (Bait al-hikma) in Baghdad, which was established by Al-Mamun. Al-Khwarizmi, who died around 850 A.D., wrote more than half a dozen mathematical and astronomical works; some of which were based on the Indian ''Sindhind''.<ref name="Boyer Intro Islamic Algebra">{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=227|isbn=0471543977}} {{quote|"The first century of the Muslim empire had been devoid of scientific achievement. This period (from about 650 to 750) had been, in fact, perhaps the nadir in the development of mathematics, for the Arabs had not yet achieved intellectual drive, and concern for learning in other parts of the world had faded. Had it not been for the sudden cultural awakening in Islam during the second half of the eighth century, considerably more of ancient science and mathematics would have been lost. ... It was during the caliphate of al-Mamun (809-833), however, that the Arabs fully indulged their passion for translation. The caliph is said to have had a dream in which Aristotle appeared, and as a consequence al-Mamun determined to have Arabic versions made of all the Greek works that he could lay his hands on, including Ptolemy's ''Almagest'' and a complete version of Euclid's ''Elements''. From the Byzantine Empire, with which the Arabs maintained an uneasy peace, Greek manuscripts were obtained through peace treaties. Al-Mamun established at Baghdad a "House of Wisdom" (Bait al-hikma) comparable to the ancient Museum at Alexandria. Among the faculty members was a mathematician and astronomer, Mohammed ibn-Musa al-Khwarizmi, whose name, like that of Euclid, later was to become a household word in Western Europe. The scholar, who died sometime before 850, wrote more than half a dozen astronomical and mathematical works, of which the earliest were probably based on the ''Sindhad'' derived from India."}}</ref> One of al-Khwarizmi's most famous books is entitled ''Al-jabr wa'l muqabalah'' or ''[[The Compendious Book on Calculation by Completion and Balancing]]'', and it gives an exhaustive account of solving polynomials up to the second degree.<ref>{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=228|isbn=0471543977}} {{quote|"The Arabs in general loved a good clear argument from premise to conclusion, as well as systematic organization - respects in which neither Diophantus nor the Hindus excelled."}}</ref> ''Al-Jabr'' is divided into six chapters, each of which deals with a different type of formula. The first chapter of ''Al-Jabr'' deals with equations whose squares equal its roots (ax² = bx), the second chapter deals with squares equal to number (ax² = c), the third chapter deals with roots equal to a number (bx = c), the fourth chapter deals with squares and roots equal a number (ax² + bx = c), the fifth chapter deals with squares and number equal roots (ax² + c = bx), and the sixth and final chapter deals with roots and number equal to squares (bx + c = ax²).<ref name="Al Jabr and its chapters">{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=229|isbn=0471543977 |quote=in six short chapters, of the six types of equations made up from the three kinds of quantities: roots, squares, and numbers (that is x, x<sup>2</sup>, and numbers). Chapter I, in three short paragraphs, covers the case of squares equal to roots, expressed in modern notation as x<sup>2</sup> = 5x, x<sup>2</sup>/3 = 4x, and 5x<sup>2</sup> = 10x, giving the answers x = 5, x = 12, and x = 2 respectively. (The root x = 0 was not recognized.) Chapter II covers the case of squares equal to numbers, and Chapter III solves the cases of roots equal to numbers, again with three illustrations per chapter to cover the cases in which the coefficient of the variable term is equal to, more than, or less than one. Chapters IV, V, and VI are mor interesting, for they cover in turn the three classical cases of three-term quadratic equations: (1) squares and roots equal to numbers, (2) squares and numbers equal to roots, and (3) roots and numbers equal to squares.}}</ref> 'Abd al-Hamid ibn-Turk authored a manuscript entitled ''Logical Necessities in Mixed Equations'', which is very similar to al-Khwarzimi's ''Al-Jabr'' and was published at around the same time as, or even possibly earlier than, ''Al-Jabr''.<ref name="Boyer Ibn Turk">{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=234|isbn=0471543977 |quote=The ''Algebra'' of al-Khwarizmi usually is regarded as the first work on the subject, but a recent publication in Turkey raises some questions about this. A manuscript of a work by 'Abd-al-Hamid ibn-Turk, entitled "Logical Necessities in Mixed Equations," was part of a book on ''Al-jabr wa'l muqabalah'' which was evidently very much the same as that by al-Khwarizmi and was published at about the same time - possibly even earlier. The surviving chapters on "Logical Necessities" give precisely the same type of geometric demonstration as al-Khwarizmi's ''Algebra'' and in one case the same illustrative example x<sup>2</sup> + 21 = 10x. In one respect 'Abd-al-Hamad's exposition is more thorough than that of al-Khwarizmi for he gives geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution. Similarities in the works of the two men and the systematic organization found in them seem to indicate that algebra in their day was not so recent a development as has usually been assumed. When textbooks with a conventional and well-ordered exposition appear simultaneously, a subject is likely to be considerably beyond the formative stage. ... Note the omission of Diophantus and Pappus, authors who evidently were not at first known in Arabia, although the Diophantine ''Arithmetica'' became familiar before the end of the tenth century.}}</ref> The manuscript gives the exact same geometric demonstration as is found in ''Al-Jabr'', and in one case the same example as found in ''Al-Jabr'', and even goes beyond ''Al-Jabr'' by giving a geometric proof that if the determinant is negative then the quadratic equation has no solution.<ref name="Boyer Ibn Turk" /> The similarity between these two works has led some historians to conclude that Arabic algebra may have been well developed by the time of al-Khwarizmi and 'Abd al-Hamid.<ref name="Boyer Ibn Turk" /> [[Al-Karkhi]] was the successor of Abu'l-Wefa and he was the first to discover the solution to equations of the form ax<sup>2n</sup> + bx<sup>n</sup> = c.<ref name="Boyer al-Karkhi ax2n">{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=239|isbn=0471543977 |quote=Abu'l Wefa was a capable algebraist as well as a trigonometer. ... His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus - but without Diophantine analysis! ... In particular, to al-Karkhi is attributed the first numerical solution of equations of the form ax<sup>2n</sup> + bx<sup>n</sup> = c (only equations with positive roots were considered),}}</ref> Al-Karkhi only considered positive roots.<ref name="Boyer al-Karkhi ax2n"/> Omar Khayyám (c. 1050-1123) wrote a book on Algebra that went beyond ''Al-Jabr'' to include equations of the third degree.<ref name="Boyer Omar Khayyam positive roots">{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=241-242|isbn=0471543977}} {{quote|Omar Khayyam (ca. 1050-1123), the "tent-maker," wrote an ''Algebra'' that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the sixteenth century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved."}}</ref> Omar Khayyám provided both arithmetic and geometric solutions for quadratic equations, but he only gave geometric solutions for general cubic equations since he mistakenly believed that arithmetic solutions were impossible.<ref name="Boyer Omar Khayyam positive roots" /> His method of solving cubic equations by using intersecting conics had been used by Menaechmus, Archimedes, and Alhazen, but Omar Khayyám generalized the method to cover all cubic equations with positive roots.<ref name="Boyer Omar Khayyam positive roots" /> He only considered positive roots and he did not go past the third degree.<ref name="Boyer Omar Khayyam positive roots" /> He also saw a strong relationship between Geometry and Algebra.<ref name="Boyer Omar Khayyam positive roots" /> In the 12th century, [[Sharaf al-Din al-Tusi]] found algebraic and [[Numerical analysis|numerical]] solutions to cubic equations and was the first to discover the [[derivative]] of [[Cubic function|cubic polynomials]].<ref name=Berggren/> J. J. O'Conner and E. F. Robertson wrote in the ''[[MacTutor History of Mathematics archive]]'': {{quote|"Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed [[rational numbers]], [[irrational number]]s, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before."}} [[Abū al-Hasan ibn Alī al-Qalasādī]] (1412-1482) was the last major medieval [[Arab]] algebraist, who made the first attempt at creating an [[Mathematical notation|algebraic notation]] since [[Ibn al-Banna]] two centuries earlier, who was himself the first to make such an attempt since [[Diophantus]] and [[Brahmagupta]] in ancient times.<ref name=Qalasadi>{{MacTutor Biography|id=Al-Qalasadi|title= Abu'l Hasan ibn Ali al Qalasadi}}</ref> The syncopated notations of his predecessors, however, lacked symbols for [[Operation (mathematics)|mathematical operations]].<ref>{{Harv|Boyer|1991|loc="Revival and Decline of Greek Mathematics" p. 178}} "The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for operations and relations, as well as of the exponential notation."</ref> Al-Qalasadi's [[algebra]]ic notation was the first to have symbols for these functions and was thus "the first steps toward the introduction of algebraic symbolism." He represented [[Table of mathematical symbols|mathematical symbols]] using characters from the [[Arabic alphabet]].<ref name=Qalasadi/> ===Arithmetic=== {{main|Arabic numerals}} The [[Indian numerals|Indian numeral]] system came to be known to both the [[Persians|Persian]] mathematician [[Muhammad ibn Musa al-Khwarizmi|Al-Khwarizmi]], whose book ''On the Calculation with Hindu Numerals'' written ''circa'' [[825]], and the [[Arab]] mathematician [[Al-Kindi]], who wrote four volumes, ''On the Use of the Indian Numerals'' (Ketab fi Isti'mal al-'Adad al-Hindi) ''circa'' [[830]], are principally responsible for the diffusion of the Indian system of numeration in the [[Middle-East]] and the West [http://www-gap.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Indian_numerals.html]. In the [[10th century]], [[Middle-East]]ern mathematicians extended the decimal numeral system to include [[Fraction (mathematics)|fractions]] using [[Decimal separator|decimal point]] notation, as recorded in a treatise by [[Demographics of Syria|Syrian]] mathematician [[Abu'l-Hasan al-Uqlidisi]] in [[952]]-[[953]]. In the [[Arab world]]&mdash;until modern times&mdash;the Arabic numeral system was used only by mathematicians. Muslim scientists used the [[Babylonian numerals|Babylonian numeral system]], and merchants used the [[Abjad numerals]]. A distinctive "West Arabic" variant of the symbols begins to emerge in ca. the [[10th century]] in the [[Maghreb]] and [[Al-Andalus]], called the ''ghubar'' ("sand-table" or "dust-table") numerals. The first mentions of the numerals in the West are found in the ''[[Codex Vigilanus]]'' of [[976]] [http://www.mathorigins.com/V.htm]. From the [[980s]], [[Pope Silvester II|Gerbert of Aurillac]] (later, Pope [[Silvester II]]) began to spread knowledge of the numerals in Europe. Gerbert studied in [[Barcelona]] in his youth, and he is known to have requested mathematical treatises concerning the [[astrolabe]] from [[Lupitus of Barcelona]] after he had returned to France. [[Al-Khwarizmi|Al-Khwārizmī]], the [[Persian people|Persian]] scientist, wrote in [[825]] a treatise ''On the Calculation with Hindu Numerals'', which was translated into [[Latin translations of the 12th century|Latin in the 12th century]], as ''Algoritmi de numero Indorum'', where "Algoritmi", the translator's rendition of the author's name gave rise to the word [[algorithm]] (Latin ''algorithmus'') with a meaning "calculation method". ===Calculus=== Around [[1000]] AD, [[Al-Karaji]], using [[mathematical induction]], found a [[Mathematical proof|proof]] for the sum of [[integral]] [[Cube (algebra)|cubes]].<ref>Victor J. Katz (1998). ''History of Mathematics: An Introduction'', p. 255-259. [[Addison-Wesley]]. ISBN 0321016181.</ref> The [[historian]] of mathematics, F. Woepcke,<ref>F. Woepcke (1853). ''Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi''. [[Paris]].</ref> praised Al-Karaji for being "the first who introduced the [[theory]] of [[algebra]]ic [[calculus]]." Shortly afterwards, [[Ibn al-Haytham]] (known as Alhazen in the West), an [[Iraq]]i mathematician working in [[History of Arab Egypt|Egypt]], was the first mathematician to derive the formula for the sum of the [[Quartic|fourth]] [[Exponentiation|powers]]. In turn, he developed a method for determining the general formula for the sum of any [[integral]] powers, which was fundamental to the development of [[Integral|integral calculus]].<ref name=Katz>Victor J. Katz (1995). "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '''68''' (3), p. 163-174.</ref> The [[eleventh century]] [[Persian Empire|Persian]] mathematician [[Omar Khayyám]] saw a strong relationship between geometry and algebra, and was moving in the right direction when he helped to close the gap between numerical and geometric algebra<ref name="Boyer Omar Khayyam positive roots"/> with his geometric solution of the general [[cubic equation]]s,<ref name=Cooper>Glen M. Cooper (2003). "Omar Khayyam, the Mathmetician", ''The Journal of the American Oriental Society'' '''123'''.</ref> but the decisive step in [[analytic geometry]] came later with Descartes.<ref name="Boyer Omar Khayyam positive roots">{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=The Arabic Hegemony|pages=241-242|quote=Omar Khayyam (ca. 1050-1123), the "tent-maker," wrote an ''Algebra'' that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the sixteenth century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved."}}</ref> In the 12th century, the [[Persian people|Persian]] mathematician [[Sharaf al-Din al-Tusi]] was the first to discover the [[derivative]] of [[Cubic function|cubic polynomials]], an important result in [[Differential (calculus)|differential calculus]].<ref name=Berggren>J. L. Berggren (1990). "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", ''Journal of the American Oriental Society'' '''110''' (2), p. 304-309.</ref> [[Image:Al-kindi cryptographic.gif|right|thumb|The first page of [[al-Kindi]]'s manuscript ''On Deciphering Cryptographic Messages'', containing the first descriptions of [[cryptanalysis]] and [[frequency analysis]].]] ===Cryptography=== In the 9th century, [[al-Kindi]] was a pioneer in [[cryptanalysis]] and [[cryptology]]. He gave the first known recorded explanation of [[cryptanalysis]] in ''A Manuscript on Deciphering Cryptographic Messages''. In particular, he is credited with developing the [[Frequency analysis (cryptanalysis)|frequency analysis]] method whereby variations in the frequency of the occurrence of letters could be analyzed and exploited to break [[Encryption|ciphers]] (i.e. crypanalysis by frequency analysis).<ref>Simon Singh. The Code Book. p. 14-20</ref> This was detailed in a text recently rediscovered in the Ottoman archives in Istanbul, ''A Manuscript on Deciphering Cryptographic Messages'', which also covers methods of cryptanalysis, encipherments, cryptanalysis of certain encipherments, and [[Statistics|statistical]] analysis of letters and letter combinations in Arabic.<ref>{{cite web |url=http://www.muslimheritage.com/topics/default.cfm?ArticleID=372 |title= Al-Kindi, Cryptgraphy, Codebreaking and Ciphers |accessdate=2007-01-12 |format= HTML |work= }}</ref> [[Ahmad al-Qalqashandi]] (1355-1418) wrote the ''Subh al-a 'sha'', a 14-volume encyclopedia which included a section on cryptology. This information was attributed to Taj ad-Din Ali ibn ad-Duraihim ben Muhammad ath-Tha 'alibi al-Mausili who lived from 1312 to 1361, but whose writings on cryptology have been lost. The list of ciphers in this work included both [[Substitution cipher|substitution]] and [[Transposition cipher|transposition]], and for the first time, a cipher with multiple substitutions for each [[plaintext]] letter. Also traced to Ibn al-Duraihim is an exposition on and worked example of cryptanalysis, including the use of tables of [[letter frequencies]] and sets of letters which can not occur together in one word. ===Geometry=== {{see also|History of geometry}} [[Image:Durer astronomer.jpg|thumb|225px|An engraving by [[Albrecht Dürer]] featuring [[Mashallah]], from the title page of the ''De scientia motus orbis'' (Latin version with engraving, 1504). As in many medieval illustrations, the [[Compass (drafting)|compass]] here is an icon of religion as well as science, in reference to God as the architect of creation]] The successors of [[Muhammad ibn Mūsā al-Khwārizmī]] (born [[780]]) undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the [[Euclid]]ean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose. [[Al-Mahani]] (born [[820]]) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. [[Al-Karaji]] (born [[953]]) completely freed algebra from geometrical operations and replaced them with the [[arithmetic]]al type of operations which are at the core of algebra today. Although [[Thābit ibn Qurra|Thabit ibn Qurra]] (known as Thebit in [[Latin]]) (born [[836]]) contributed to a number of areas in mathematics, where he played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to ([[positive]]) [[real number]]s, [[integral calculus]], theorems in [[spherical trigonometry]], [[analytic geometry]], and [[non-Euclidean geometry]]. An important geometrical aspect of Thabit's work was his book on the composition of ratios. In this book, Thabit deals with arithmetical operations applied to ratios of geometrical quantities. The Greeks had dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of arithmetic could be applied. By introducing arithmetical operations on quantities previously regarded as geometric and non-numerical, Thabit started a trend which led eventually to the generalization of the number concept. Another important contribution Thabit made to [[geometry]] was his generalization of the [[Pythagorean theorem]], which he extended from [[special right triangles]] to all [[right triangle]]s in general, along with a general [[mathematical proof|proof]].<ref>Aydin Sayili (1960). "Thabit ibn Qurra's Generalization of the Pythagorean Theorem", ''[[Isis (journal)|Isis]]'' '''51''' (1), p. 35-37.</ref> [[Ibrahim ibn Sinan]] (born [[908]]), who introduced a method of [[integral|integration]] more general than that of [[Archimedes]], and [[al-Quhi]] (born [[940]]) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. These mathematicians, and in particular [[Ibn al-Haytham]], studied [[optics]] and investigated the optical properties of mirrors made from [[conic section]]s. Astronomy, time-keeping and [[geography]] provided other motivations for geometrical and trigonometrical research. For example Ibrahim ibn Sinan and his grandfather [[Thabit ibn Qurra]] both studied curves required in the construction of sundials. [[Abu'l-Wafa]] and [[Abu Nasr Mansur]] pioneered [[spherical geometry]] in order to solve difficult problems in [[Islamic astronomy]]. For example, to predict the first visibility of the moon, it was necessary to describe its motion with respect to the [[horizon]], and this problem demands fairly sophisticated spherical geometry. Finding the direction of [[Mecca]] ([[Qibla]]) and the time for [[Salah]] prayers and [[Ramadan]] are what led to Muslims developing spherical geometry.<ref name=Gingerich>{{Harvard reference |last=Gingerich |first=Owen |year=1986 |date=April 1986 |url=http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm |title=Islamic astronomy |journal=[[Scientific American]] |volume=254 |issue=10 |page=74 |accessdate=2008-05-18}}</ref><ref name=Tabatabai>{{citation|url=http://www.almizan.org/Tafseer/Volume2/Baqarah32.asp|author=Syed Mohammad Hussain Tabatabai|work=Tafsir al-Mizan|chapter=Volume 2: Surah Baqarah, Verses 142-151|accessdate=2008-01-24}}</ref> [[Omar Khayyám]] (born [[1048]]) was a [[Persian people|Persian]] mathematician, as well as a poet. Along with his fame as a poet, he was also famous during his lifetime as a mathematician, well known for inventing the general method of solving [[cubic equation]]s by intersecting a parabola with a circle. In addition he discovered the [[binomial expansion]], and authored criticisms of [[Euclid]]'s theories of [[Parallel postulate|parallels]] which made their way to England, where they contributed to the eventual development of [[non-Euclidean geometry]]. Omar Khayyam also combined the use of trigonometry and [[approximation theory]] to provide methods of solving algebraic equations by geometrical means. His work marked the beginnings of [[algebraic geometry]]<ref>R. Rashed (1994). ''The development of Arabic mathematics: between arithmetic and algebra''. [[London]].</ref><ref>{{MacTutor|class=HistTopics|id=Arabic_mathematics|title=Arabic mathematics: forgotten brilliance?|year=1999}}</ref> and [[analytic geometry]].<ref name=Cooper/> Khayyam also made the first attempt at formulating a [[non-Euclidean geometry|non-Euclidean]] [[postulate]] as an alternative to the [[Euclidean geometry|Euclidean]] [[parallel postulate]],<ref>Victor J. Katz (1998), ''History of Mathematics: An Introduction'', p. 270, [[Addison-Wesley]], ISBN 0321016181: {{quote|"In some sense, his treatment was better than ibn al-Haytham's because he explicitly formulated a new postulate to replace Euclid's rather than have the latter hidden in a new definition."}}</ref> and he was the first to consider the cases of [[elliptical geometry]] and [[hyperbolic geometry]], though he excluded the latter.<ref name=Rosenfeld>Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., ''[[Encyclopedia of the History of Arabic Science]]'', Vol. 2, p. 447-494 [469], [[Routledge]], London and New York: {{quote|"Khayyam's postulate had excluded the case of the hyperbolic geometry whereas al-Tusi's postulate ruled out both the hyperbolic and elliptic geometries."}}</ref> Persian mathematician [[Sharafeddin Tusi]] (born [[1135]]) did not follow the general development that came through [[al-Karaji]]'s school of algebra but rather followed Khayyam's application of algebra to geometry. He wrote a treatise on cubic equations, which represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the study of algebraic geometry. In 1250, [[Nasīr al-Dīn al-Tūsī]], in his ''Al-risala al-shafiya'an al-shakk fi'l-khutut al-mutawaziya'' (''Discussion Which Removes Doubt about Parallel Lines''), wrote detailed critiques of the [[Euclidean geometry|Euclidean]] [[parallel postulate]] and on [[Omar Khayyám]]'s attempted proof a century earlier. Nasir al-Din attempted to derive a [[proof by contradiction]] of the parallel postulate.<ref name=Katz/> He was one of the first to consider the cases of [[elliptical geometry]] and [[hyperbolic geometry]], though he ruled out both of them.<ref name=Rosenfeld/> His son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), wrote a book on the subject in 1298, based on al-Tusi's later thoughts, which presented one of the earliest arguments for a [[Non-Euclidean geometry|non-Euclidean]] hypothesis equivalent to the parallel postulate.<ref name=Katz/><ref>Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., ''[[Encyclopedia of the History of Arabic Science]]'', Vol. 2, p. 447-494 [469], [[Routledge]], London and New York: {{quote|"In ''Pseudo-Tusi's Exposition of Euclid'', [...] another statement is used instead of a postulate. It was independent of the Euclidean postulate V and easy to prove. [...] He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the ''Elements''."}}</ref> Sadr al-Din's work was published in [[Rome]] in 1594 and was studied by European geometers. This work marked the starting point for [[Giovanni Girolamo Saccheri]]'s work on the subject, and eventually the development of modern [[non-Euclidean geometry]].<ref name=Katz>Victor J. Katz (1998), ''History of Mathematics: An Introduction'', p. 270-271, [[Addison-Wesley]], ISBN 0321016181: <blockquote>"But in a manuscript probably written by his son Sadr al-Din in 1298, based on Nasir al-Din's later thoughts on the subject, there is a new argument based on another hypothesis, also equivalent to Euclid's, [...] The importance of this latter work is that it was published in Rome in 1594 and was studied by European geometers. In particular, it became the starting point for the work of Saccheri and ultimately for the discovery of non-Euclidean geometry."<blockquote></ref> A proof from Sadr al-Din's work was quoted by [[John Wallis]] and Saccheri in the 17th and 18th centuries. They both derived their proofs of the parallel postulate from Sadr al-Din's work, while Saccheri also derived his [[Saccheri quadrilateral]] from Sadr al-Din, who himself based it on his father's work.<ref>Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., ''[[Encyclopedia of the History of Arabic Science]]'', Vol. 2, p. 447-494 [469], [[Routledge]], London and New York: {{quote|"His book published in Rome considerably influenced the subsequent development of the theory of parallel lines. Indeed, J. Wallis (1616-1703) included a Latin translation of the proof of postulate V from this book in his own writing ''On the Fifth Postulate and the Fifth Definition from Euclid's Book 6'' (''De Postulato Quinto et Definitione Quinta lib. 6 Euclidis'', 1663). Saccheri quited this proof in his ''Euclid Cleared of all Stains'' (''Euclides ab omni naevo vindicatus'', 1733). It seems possible that he borrowed the idea of considering the three hypotheses about the upper angles of the 'Saccheri quadrangle' from Pseudo-Tusi. The latter inserted the exposition of this subject into his work, taking it from the writings of al-Tusi and Khayyam."}}</ref> The theorems of [[Ibn al-Haytham]] (Alhacen), [[Omar Khayyam]] and [[Nasir al-Din al-Tusi]] on [[quadrilateral]]s, including the [[Lambert quadrilateral]] and [[Saccheri quadrilateral]], were the first theorems on [[elliptical geometry]] and [[hyperbolic geometry]], and along with their alternative postulates, such as [[Playfair's axiom]], these works marked the beginning of [[non-Euclidean geometry]] and had a considerable influence on its development among later European geometers, including [[Witelo]], [[Levi ben Gerson]], [[Alfonso]], [[John Wallis]], and [[Giovanni Girolamo Saccheri]].<ref>Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., ''[[Encyclopedia of the History of Arabic Science]]'', Vol. 2, p. 447-494 [470], [[Routledge]], London and New York: {{quote|"Three scientists, Ibn al-Haytham, Khayyam and al-Tusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the ninteenth century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between tthis postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investiagtions of their European couterparts. The first European attempt to prove the postulate on parallel lines - made by Witelo, the Polish scientists of the thirteenth century, while revising Ibn al-Haytham's ''[[Book of Optics]]'' (''Kitab al-Manazir'') - was undoubtedly prompted by Arabic sources. The proofs put forward in the fourteenth century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that ''Pseudo-Tusi's Exposition of Euclid'' had stimulated borth J. Wallis's and G. Saccheri's studies of the theory of parallel lines."}}</ref> Recently discoveries have shown that geometrical [[quasicrystal]] patterns were first employed in the [[girih tiles]] found in medieval [[Islamic architecture]] dating back over five centuries ago. In 2007, Professor [[Peter Lu]] of [[Harvard University]] and Professor [[Paul Steinhardt]] of [[Princeton University]] published a paper in the journal ''Science'' suggesting that girih tilings possessed properties consistent with [[self-similar]] [[fractal]] quasicrystalline tilings such as the [[Penrose tiling]]s, predating them by five centuries.<ref name=Lu>{{cite journal | author = Peter J. Lu and Paul J. Steinhardt | year = 2007 | title = Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture | journal = [[Science (journal)|Science]] | volume = 315 | pages = 1106–1110 | url = http://www.physics.harvard.edu/~plu/publications/Science_315_1106_2007.pdf | doi = 10.1126/science.1135491 }}</ref><ref>Supplemental figures [http://www.physics.harvard.edu/~plu/publications/Science_315_1106_2007_SOM.pdf]</ref> ===Mathematical induction=== The first known [[proof]] by [[mathematical induction]] was introduced in the ''al-Fakhri'' written by [[Al-Karaji]] around [[1000]] AD, who used it to prove [[Arithmetic progression|arithmetic sequences]] such as the [[binomial theorem]], [[Pascal's triangle]], and the sum formula for [[integral]] [[Cube (algebra)|cubes]].<ref>Victor J. Katz (1998), ''History of Mathematics: An Introduction'', p. 255-259, [[Addison-Wesley]], ISBN 0321016181: <blockquote>"Another important idea introduced by [[al-Karaji]] and continued by [[Ibn Yahyā al-Maghribī al-Samaw'al|al-Samaw'al]] and others was that of an inductive argument for dealing with certain arithmetic sequences. Thus al-Karaji used such an argument to prove the result on the sums of integral cubes already known to [[Aryabhata]] [...] Al-Karaji did not, however, state a general result for arbitrary ''n''. He stated his theorem for the particular integer 10 [...] His proof, nevertheless, was clearly designed to be extendable to any other integer.</blockquote></ref><ref>{{MacTutor|id=Al-Karaji|title=Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji}} {{quote|"Al-Karaji also uses a form of mathematical induction in his arguments, although he certainly does not give a rigorous exposition of the principle."}}</ref> His proof was the first to make use of the two basic components of an inductive proof, "namely the [[truth]] of the statement for ''n'' = 1 (1 = 1<sup>3</sup>) and the deriving of the truth for ''n'' = ''k'' from that of ''n'' = ''k'' - 1."<ref>Katz (1998), p. 255: <blockquote>"Al-Karaji's argument includes in essence the two basic components of a modern argument by induction, namely the truth of the statement for ''n'' = 1 (1 = 1<sup>3</sup>) and the deriving of the truth for ''n'' = ''k'' from that of ''n'' = ''k'' - 1. Of course, this second component is not explicit since, in some sense, al-Karaji's argument is in reverse; this is, he starts from ''n'' = 10 and goes down to 1 rather than proceeding upward. Nevertheless, his argument in ''al-Fakhri'' is the earliest extant proof of the sum formula for integral cubes."</blockquote></ref> Shortly afterwards, [[Ibn al-Haytham]] (Alhazen) used the inductive method to prove the sum of [[fourth power]]s, and by extension, the sum of any integral [[Exponentiation|powers]], which was an important result in [[integral]] [[calculus]]. He only stated it for particular integers, but his proof for those integers was by induction and generalizable.<ref>Victor J. Katz (1995), "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '''68''' (3), p. 163-174: {{quote|"The central idea in [[ibn al-Haytham]]'s proof of the sum formulas was the derivation of the equation [...] Naturally, he did not state this result in general form. He only stated it for particular integers, [...] but his proof for each of those ''k'' is by induction on ''n'' and is immediately generalizable to any value of ''k''."}}</ref><ref>Katz (1998), p. 255-259.</ref> [[Ibn Yahyā al-Maghribī al-Samaw'al]] came closest to a modern proof by mathematical induction in pre-modern times, which he used to extend the proof of the binomial theorem and Pascal's triangle previously given by al-Karaji. Al-Samaw'al's inductive argument was only a short step from the full inductive proof of the general binomial theorem.<ref>Katz (1998), p. 259: <blockquote>"Like the proofs of al-Karaji and ibn al-Haytham, al-Samaw'al's argument contains the two basic components of an inductive proof. He begins with a value for which the result is known, here ''n'' = 2, and then uses the result for a given integer to derive the result for the next. Although al-Samaw'al did not have any way of stating, and therefore proving, the general binomial theorem, to modern readers there is only a short step from al-Samaw'al's argument to a full inductive proof of the binomial theorem."</blockquote></ref> ===Number theory=== In [[number theory]], [[Ibn al-Haytham]] solved problems involving [[congruence relation|congruences]] using what is now called [[Wilson's theorem]]. In his ''Opuscula'', Ibn al-Haytham considers the solution of a system of congruences, and gives two general methods of solution. His first method, the canonical method, involved Wilson's theorem, while his second method involved a version of the [[Chinese remainder theorem]]. Another contribution to number theory is his work on [[perfect number]]s. In his ''Analysis and synthesis'', Ibn al-Haytham was the first to discover that every even perfect number is of the form 2<sup>''n''&minus;1</sup>(2<sup>''n''</sup>&nbsp;&minus;&nbsp;1) where 2<sup>''n''</sup>&nbsp;&minus;&nbsp;1 is [[Prime number|prime]], but he was not able to prove this result successfully ([[Leonhard Euler|Euler]] later proved it in the 18th century).<ref>{{MacTutor Biography|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham}}</ref> ===Recreational mathematics=== In [[recreational mathematics]], [[magic square]]s were known to [[Arab]] mathematicians, possibly as early as the 7th century, when the Arabs got into contact with Indian or South Asian culture, and learned Indian mathematics and astronomy, including other aspects of [[combinatorial mathematics]]. It has also been suggested that the idea came via China. The first magic squares of order 5 and 6 appear in an encyclopedia from [[Baghdad]] ''circa'' 983 AD, the [[Rasa'il Ihkwan al-Safa]] (the Encyclopedia of the Brethern of Purity); simpler magic squares were known to several earlier Arab mathematicians.<ref name="Swaney">Swaney, Mark. [http://www.arthurmag.com/magpie/?p=449 History of Magic Squares].</ref> The Arab mathematician [[Ahmad al-Buni]], who worked on magic squares around 1200 AD, attributed mystical properties to them, although no details of these supposed properties are known. There are also references to the use of magic squares in astrological calculations, a practice that seems to have originated with the Arabs.<ref name="Swaney"/> ===Trigonometry=== The early [[Indian mathematics|Indian works]] on [[trigonometry]] were translated and expanded in the [[Muslim world]] by [[List of Arab scientists and scholars|Arab]] and [[List of Iranian scientists and scholars|Persian]] mathematicians. {{Unicode|[[Muhammad ibn Mūsā al-Khwārizmī]]}} produced tables of [[Trigonometric function|sines]] and [[tangent (trigonometric function)|tangent]]s, and also developed [[spherical trigonometry]]. By the 10th century, in the work of [[Abū al-Wafā' al-Būzjānī]] (959-998), Muslim mathematicians were using all six [[trigonometric function]]s, and had sine tables in 0.25° increments, to 8 decimal places of accuracy, as well as tables of [[tangent (trigonometry)|tangent]] values. Abū al-Wafā' also developed the following trigonometric formula: :<math> \sin 2x = 2 \sin x \cos x \ </math> Also in the 10th century, [[Muhammad ibn Jābir al-Harrānī al-Battānī]] (853-929) formulated a number of important trigonometrical relationships such as: :<math>\tan a = \frac{\sin a}{\cos a}</math> :<math>\sec a = \sqrt{1 + \tan^2 a }</math> [[Al-Jayyani]] (989&ndash;1079) of [[al-Andalus]] wrote ''The book of unknown arcs of a sphere'', which is considered by some, but not others,<ref name="Boyer Menelaus of Alexandria"/> to be "the first treatise on [[spherical trigonometry]]"<ref name="MacTutor Al-Jayyani">{{MacTutor|id=Al-Jayyani|title=Abu Abd Allah Muhammad ibn Muadh Al-Jayyani}}</ref> although earlier mathematicians, such as [[Menelaus of Alexandria]], did have books that dealt with spherical trigonometry.<ref>{{MacTutor|id=Menelaus|title=Menelaus of Alexandria}} Book 3 deals with spherical trigonometry and includes Menelaus's theorem.</ref><ref name="Boyer Menelaus of Alexandria">{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=Greek Trigonometry and Mensuration|pages=163|quote=In Book I of this treatise Menelaus establishes a basis for spherical triangles analogous to that of Euclid I for plane triangles. Included is a theorem without Euclidean analogue - that two spherical triangles are congruent if corresponding angles are equal (Menelaus did not distinguish between congruent and symmetric spherical triangles); and the theorem A + B + C > 180° is established. The second book of the ''Sphaerica'' describes the application of spherical geometry to astronomical phenomena and is of little mathematical interest. Book III, the last, contains the well known "theorem of Menelaus" as part of what is essentially spherical trigonometry in the typical Greek form - a geometry or trigonometry of chords in a circle. In the circle in Fig. 10.4 we should write that chord AB is twice the sine of half the central angle AOB (multiplied by the radius of the circle). Menelaus and his Greek successors instead referred to AB simply as the chord corresponding to the arc AB. If BOB' is a diameter of the circle, then chord A' is twice the cosine of half the angle AOB (multiplied by the radius of the circle).}}</ref> Al-Jayyani's work "contains formulae for [[Special right triangles|right-handed triangles]], the general [[law of sines]], and the solution of a [[spherical triangle]] by means of the polar [[triangle]]." This treatise later had a "strong influence on European mathematics", and his "definition of [[ratio]]s as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced [[Regiomontanus]].<ref>{{MacTutor|id=Al-Jayyani|title=Abu Abd Allah Muhammad ibn Muadh Al-Jayyani}}</ref> [[Omar Khayyám]] (1048-1131) solved [[cubic equation]]s using approximate numerical solutions found by interpolation in trigonometric tables. All of these earlier works on trigonometry treated it mainly as an adjunct to astronomy; perhaps the first treatment as a subject in its own right was by [[Bhaskara II]] and [[Nasīr al-Dīn al-Tūsī]] in the 13th century. Nasir al-Din al-Tusi stated the [[law of sines]] and provided a proof for it, and also listed the six distinct cases of a right-angled triangle in spherical trigonometry. The method of [[triangulation]], which was unknown in the [[Greco-Roman]] world, was also first developed by Muslim mathematicians, who applied it to practical uses such as [[surveying]].<ref>[[Donald Routledge Hill]] (1996), "Engineering", in Roshdi Rashed, ''Encyclopedia of the History of Arabic Science'', Vol. 3, p. 751-795 [769].</ref> [[Jamshīd al-Kāshī]] (1393-1449) gives trigonometric tables of values of the sine function to four [[sexagesimal]] digits (equivalent to 8 decimal places) for each 1° of argument with differences to be added for each 1/60 of 1°. [[Ulugh Beg]] (1394-1449) also gives accurate tables of sines and tangents correct to 8 decimal places. [[Taqi al-Din]] (1526-1585) contributed to trigonometry in his ''Sidrat al-Muntaha'', in which he was the first [[mathematician]] to compute a highly accurate numeric value for [[Trigonometric function|sin]]&nbsp;1°. He discusses the values given by his predecessors, explaining how [[Ptolemy]] (ca. 150) used an approximate method to obtain his value of Sin 1° and how Abū al-Wafā, [[Ibn Yunus]] (ca. 1000), al-Kashi, [[Qāḍī Zāda al-Rūmī]] (1337-1412), Ulugh Beg and Mirim Chelebi improved on the value. Taqi al-Din then solves the problem to obtain the value of sin&nbsp;1° to a precision of 8 sexagesimals (the equivalent of 14 decimals):<ref>{{cite web|title=Taqi al Din Ibn Ma’ruf's Work on Extracting the Cord 2° and Sin 1°|publisher=FSTC Limited|url=http://muslimheritage.com/topics/default.cfm?ArticleID=941|date=30 May 2008|accessdate=2008-07-04}}</ref> :<math> \sin 1^\circ = 1^P 2' 49'' 43''' 11'''' 14''''' 44''''''16''''''' \ (= 1/60 + 2/60^2 + 49/60^3 + \cdots)\,.</math> ==See also== *[[List of Muslim mathematicians]] *[[Latin translations of the 12th century]] *[[Islamic science]] *[[Islamic Golden Age]] *[[Inventions in the Muslim world]] == Notes == {{reflist|2}} == Further reading== <div class="references-2column"> *{{cite book|last=Berggren|first=J. Lennart|title=Episodes in the Mathematics of Medieval Islam|year=1986|publisher=Springer-Verlag|location=New York|id=ISBN 0-387-96318-9}}<!-- Reviewed: {{cite review|last=Toomer|first=Gerald J.|title=Episodes in the Mathematics of Medieval Islam|journal=American Mathematical Monthly|volume=95|issue=6|year=1988|url=http://links.jstor.org/sici?sici=0002-9890%28198806%2F07%2995%3A6%3C567%3AEITMOM%3E2.0.CO%3B2-3}}; {{cite review|first=Jan P.|last=Hogendijk|journal=Journal of the American Oriental Society|volume=109|issue=4|year=1989|pages=697-698}}.--> *{{cite book | first=J. Lennart | last=Berggren | authorlink=Len Berggren | editor=Victor J. Katz | title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook | chapter=Mathematics in Medieval Islam | publisher=Princeton University Press | year=2007 | isbn=9780691114859 }} *{{cite book | first=Carl B. | last=Boyer | authorlink=Carl Benjamin Boyer | title=A History of Mathematics | chapter=The Arabic Hegemony | edition=Second Edition | publisher=John Wiley & Sons, Inc | year=1991 | isbn=0471543977 }} *{{cite book | first=Roger | last=Cooke | authorlink=Roger Cooke | title=The History of Mathematics: A Brief Course | chapter=Islamic Mathematics | publisher=Wiley-Interscience | year=1997 | isbn=0471180823 }} *{{cite book|last=Daffa'|first=Ali Abdullah al-|title=The Muslim contribution to mathematics|year=1977|publisher=Croom Helm|location=London|id=ISBN 0-85664-464-1}} * {{cite book|first=Ali Abdullah al-|last=Daffa|first2=J.J.|last2=Stroyls|title=Studies in the exact sciences in medieval Islam|publisher=Wiley|location=New York|year=1984|id=ISBN 0471903205}} * {{cite book|last=Joseph|first=George Gheverghese|title=The Crest of the Peacock: Non-European Roots of Mathematics|edition=2nd Edition|publisher=Princeton University Press|year=2000|id=ISBN 0691006598}}<!-- Reviewed: {{Cite review|first=Victor J.|last=Katz|journal=The College Mathematics Journal|volume=23|issue=1|year=1992|pages=82-84}}.--> * {{cite book|first=E. S.|last=Kennedy|title=Studies in the Islamic Exact Sciences|year=1984|publisher=Syracuse Univ Press|id=ISBN 0815660677}} * {{MacTutor|class=HistTopics|id=Arabic_mathematics|title=Arabic mathematics: forgotten brilliance?|year=1999}} * {{cite book|last=Rashed|first=Roshdi|others=Transl. by A. F. W. Armstrong|title=The Development of Arabic Mathematics: Between Arithmetic and Algebra|publisher=Springer|year=2001|id=ISBN 0792325656}} * {{cite book|last=Sánchez Pérez|first=José A|title=Biografías de Matemáticos Árabes que florecieron en España|location=Madrid|publisher=Estanislao Maestre|year=1921}} * {{cite book|last=Sezgin|first=Fuat|title=Geschichte Des Arabischen Schrifttums|publisher=Brill Academic Publishers|language=German|year=1997|id=ISBN 9004020071}} * {{cite book|last=Suter|first=Heinrich|title=Die Mathematiker und Astronomen der Araber und ihre Werke|series=Abhandlungen zur Geschichte der Mathematischen Wissenschaften Mit Einschluss Ihrer Anwendungen, X Heft|location=Leipzig|year=1900}} * {{cite book|first=Adolf P.|last=Youschkevitch|coauthors=Boris A. Rozenfeld|title=Die Mathematik der Länder des Ostens im Mittelalter|year=1960|location=Berlin}} Sowjetische Beiträge zur Geschichte der Naturwissenschaft pp. 62-160. * {{cite book|first=Adolf P.|last=Youschkevitch|title=Les mathématiques arabes: VIII<sup>e</sup>-XV<sup>e</sup> siècles|others=translated by M. Cazenave and K. Jaouiche|publisher=Vrin|location=Paris|year=1976|id=ISBN 978-2-7116-0734-1}} </div> == External links == * Hogendijk, Jan P. (January 1999). [http://www.math.uu.nl/people/hogend/Islamath.html ''Bibliography of Mathematics in Medieval Islamic Civilization'']. {{Islamic mathematics}} [[Category:Islamic mathematics| ]] [[es:Matemática en el Islam medieval]] [[fa:ریاضیات اسلامی]] [[fr:Mathématiques arabes]] [[ms:Matematik Islam]] [[ja:アラビア数学]] [[ru:Математика исламского средневековья]]