Mathematics in the medieval Islamic world: Difference between revisions
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In the [[history of mathematics]], '''mathematics in medieval Islam''', often termed '''Islamic mathematics''' or '''Arabic mathematics''', is the [[mathematics]] developed by the [[Muslim world|Islamic civilization]] between 622 and 1600. [[Islamic science]] and mathematics flourished under the Islamic [[caliphate]] established across the Middle East, [[Central Asia]], [[North Africa]], [[Southern Italy]], the [[Iberian Peninsula]], and, at its peak, parts of France and [[Indian subcontinent|India]]. |
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In the [[history of mathematics]], '''mathematics in medieval Islam''', often termed '''Islamic mathematics''' or '''Arabic mathematics''', is the [[mathematics]] developed in the [[Muslim world|Islamic world]] between 622 and 1600, during what is known as the [[Islamic Golden Age]], in that part of the world where [[Islam]] was the dominant religion. [[Islamic science]] and mathematics flourished under the Islamic [[caliphate]] (also known as the Islamic Empire) established across the Middle East, [[Central Asia]], [[North Africa]], [[Southern Italy]], the [[Iberian Peninsula]], and, at its peak, parts of France and [[Indian subcontinent|India]] as well. Islamic activity in mathematics was largely centered around modern-day [[Iraq]] and [[Greater Iran|Persia]], but at its greatest extent stretched from North Africa and Spain in the west to India in the east.<ref>O'Connor 1999</ref> |
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While most scientists in this period were [[Muslim]]s and wrote in [[Arabic language|Arabic]],<ref name=Hogendijk/> many of the best known contributors were [[Persian people|Persians]]{{sfn|Schumpeter, Moss|1994|p=64}}{{#tag:ref|"A great portion (and most of the best) of medieval Muslim philosophers, physicians, ethicists, scientists, Islamic jurists, historians, and geographers were Persian-speaking Iranians"{{sfn|Schumpeter, Moss|1994|p=64}}|group="note"}}{{sfn|Rosenthal, Dawood, Khaldun|1994|p=430}}{{#tag:ref|Only the Persians engaged in the task of preserving knowledge and writing systematic scholarly works. Thus, the truth of the following statement by the Prophet becomes apparent: If scholarship hung suspended in the highest parts of heaven, the Persians would attain it. [...] This situation continued in the cities as long as the Persians and the Persian countries, the 'Iraq, Khurasan, and Transoxania, retained their sedentary culture. But when those cities fell into ruins, sedentary culture, which God has devised for the attainment of sciences and crafts, disappeared from them. Along with it, scholarship altogether disappeared from among the non-Arabs (Persians), who were (now) engulfed by the desert attitude. Scholarship was restricted to cities with an abundant sedentary culture. Today, no (city) has a more abundant sedentary culture than Cairo (Egypt). It is the mother of the world, the great center (Iwan) of Islam, and the mainspring of the sciences and the crafts. Some sedentary culture has also survived in Transoxania, because the dynasty there provides some sedentary culture. Therefore, they have there a certain number of the sciences and the crafts, which cannot be denied. Our attention was called to this fact by the contents of the writings of a (Transoxanian) scholar, which have reached us in this country. He is [[Taftazani|Sa'd-ad-din at-Taftazani]]. As far as the other non-Arabs (Persians) are concerned, we have not seen, since the imam [[Ibn al-Khatib]] and [[Nasir al-Din al-Tusi|Nasir-ad-din at-Tusi]], any discussions that could be referred to as indicating their ultimate excellence."{{sfn|Rosenthal, Dawood, Khaldun|1994|p=430}}|group="note"}} as well as [[Arab]]s,{{sfn|Rosenthal, Dawood, Khaldun|1994|p=430}} in addition to [[Berber people|Berber]], [[Moors|Moorish]] and [[Turkic peoples|Turkic]] contributors, as well as some from other religions ([[Christian]]s, [[Jew]]s, [[Sabians]], [[Zoroastrianism|Zoroastrians]], and the [[Irreligion|irreligious]]).<ref name=Hogendijk>Hogendijk 1999</ref> Arabic was the dominant language—much like [[Latin language|Latin]] in [[Medieval Europe]], Arabic was the written ''[[lingua franca]]'' of most scholars throughout the Islamic world. In this article, "Islam" and the adjective "Islamic" are used to describe the [[civilization]] rather than the religion. |
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==Origins and influences== |
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The first century of the [[Islam]]ic [[Caliphate|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 8th century Islam had a cultural awakening, and research in mathematics and the sciences increased.{{sfn|Boyer|1991|p=227}}{{#tag:ref|"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."{{sfn|Boyer|1991|p=227}}|group="note"}} 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 ''[[Euclid's Elements|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.{{sfn|Boyer|1991|p=227}} 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 of Ascalon|Eutocius]].{{sfn|Boyer|1991|p=234}}{{#tag:ref|"[...] 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]]."{{sfn|Boyer|1991|p=234}}|group="note"}} Historians are in debt to many Islamic translators, for it is through their work that many ancient Greek texts have survived only through [[Arabic]] translations.{{Citation needed|date=November 2010}} |
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[[Greek mathematics|Greek]], [[Indian mathematics|Indian]] and [[Babylonian mathematics|Babylonian]] all played an important role in the development of mathematics, including the early Islamic mathematics. The works of mathematicians such as Euclid, 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]].{{sfn|Berggren|2007|p=516}}{{#tag:ref|"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."{{sfn|Berggren|2007|p=516}}|group="note"}} The [[Persian people|Persian]] historian [[al-Biruni]] (c. 1050) in his book ''Tariq al-Hind'' states that [[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]]''.{{sfn|Boyer|1991|p=226}}{{#tag:ref|"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."{{sfn|Boyer|1991|p=226}}|group="note"}} 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"/>{{Nonspecific|date=February 2011}} [[Abu Rayhan Biruni|Biruni]] described the [[Aryabhatiya]], a treatise by the Indian Mathematician [[Aryabhata]], as a "mix of common pebbles and costly crystals".{{sfn|Boyer|1991|p=211}} |
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Indian influences were later 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">{{Citation |
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|first=Kim |
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|last=Plofker |
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|title=??? |
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|year=2007}}</ref>{{Nonspecific|date=February 2011}}{{#tag:ref|"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 name="Plofker 434" />{{Nonspecific|date=February 2011}}|group="note"}} Another likely reason for the declining Indian influence in later periods was due to [[Sindh]] achieving independence from the [[Caliphate]], thus limiting access to Indian works. Nevertheless, Indian methods continued to play an important role in algebra, arithmetic and trigonometry.{{sfn|Haq|1996|pp=52–70}} |
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Besides the Greek and Indian tradition, a third tradition which had a significant influence on mathematics in medieval Islam was the "mathematics of practitioners", which included the applied mathematics of "surveyors, [[Islamic architecture|builders]], [[Islamic art|artisans]], in geometric design, [[Islamic economics in the world|tax and treasury officials, and some merchants]]". This applied form of mathematics transcended ethnic divisions and was a common heritage of the lands incorporated into the Islamic world.{{sfn|Berggren|2007|p=516}} This tradition also includes the religious observances specific to Islam, which served as a major impetus for the development of mathematics as well as astronomy.{{Sfn|Gingerich|1986}} |
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===Islam and mathematics=== |
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A major impetus for the flowering of mathematics as well as [[astronomy in medieval Islam]] came from religious observances, which presented an assortment of problems in astronomy and mathematics, specifically in [[trigonometry]], [[spherical geometry]],{{Sfn|Gingerich|1986}} [[algebra]]{{sfn|Gandz|1938|pp=319–391}} and [[arithmetic]].{{Citation needed|date=May 2010}} |
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The [[Islamic inheritance jurisprudence|Islamic law of inheritance]] served as an impetus behind the development of algebra (derived from the [[Arabic language|Arabic]] ''al-jabr'') by [[Muhammad ibn Mūsā al-Khwārizmī]] and other medieval Islamic mathematicians. Al-Khwārizmī's ''[[The Compendious Book on Calculation by Completion and Balancing|Hisab al-jabr w’al-muqabala]]'' devoted a chapter on the solution to the Islamic law of inheritance using algebra. He formulated the rules of inheritance as [[linear equation]]s, hence his knowledge of [[quadratic equation]]s was not required.{{sfn|Gandz|1938|pp=319–391}} Later mathematicians who specialized in the Islamic law of inheritance included [[Al-Hassār]], who developed the modern symbolic [[mathematical notation]] for [[Fraction (mathematics)|fractions]] in the 12th century,{{Citation needed|date=May 2010}} and [[Abū al-Hasan ibn Alī al-Qalasādī]], who developed an algebraic notation which took "the first steps toward the introduction of algebraic symbolism" in the 15th century.<ref name=Qalasadi/> |
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In order to observe holy days on the [[Islamic calendar]] in which timings were determined by [[phases of the moon]], astronomers initially used [[Ptolemy]]'s method to calculate the place of the [[moon]] and [[star]]s. The method Ptolemy used to solve [[spherical triangle]]s, however, was a clumsy one devised late in the 1st century by [[Menelaus of Alexandria]]. It involved setting up two intersecting [[right triangle]]s; by applying [[Menelaus' theorem]] it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the [[sun]]'s [[altitude]], for instance, repeated applications of Menelaus' theorem were required. For medieval Islamic astronomers, there was an obvious challenge to find a simpler [[trigonometric]] method.{{Sfn|Gingerich|1986}} |
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Regarding the issue of moon sighting, Islamic months do not begin at the astronomical [[new moon]], defined as the time when the moon has the same [[celestial longitude]] as the sun and is therefore invisible; instead they begin when the thin [[crescent moon]] is first sighted in the western evening sky.{{Sfn|Gingerich|1986}} The Qur'an says: "They ask you about the waxing and waning phases of the crescent moons, say they are to mark fixed times for mankind and [[Hajj]]."<ref>{{cite quran|2|189|style=ref}}</ref><ref>{{citation |
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|url=http://www.almizan.org/Tafseer/Volume3/Baqarah47.asp |
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|chapter=Volume 3: Surah Baqarah, Verse 189 |
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|author=Syed Mohammad Hussain Tabatabai |
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|title=Tafsir al-Mizan |
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|accessdate=2008-01-24}}{{Dead link|date=October 2010|bot=H3llBot}}</ref> This led Muslims to find the phases of the moon in the sky, and their efforts led to new mathematical calculations.<ref>{{cite web |
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|url=http://www.chowk.com/Views/Science/The-Science-of-Moon-Sighting |
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|title=The Science of Moon Sighting |
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|author=Khalid Shaukat |
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|date=September 23, 1997 |
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|accessdate=2008-01-24}}</ref> |
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Predicting just when the crescent moon would become visible is a special challenge to Islamic mathematical astronomers. Although Ptolemy's theory of the complex lunar motion was tolerably accurate near the time of the new moon, it specified the moon's path only with respect to the [[ecliptic]]. 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]] and the time of [[Salah]] are the reasons which led to Muslims developing spherical geometry. Solving any of these problems involves finding the unknown sides or angles of a triangle on the [[celestial sphere]] from the known sides and angles. A way of finding the time of day, for example, is to construct a triangle whose [[Vertex (geometry)|vertices]] are the [[zenith]], the north [[celestial pole]], and the sun's position. The observer must know the altitude of the sun and that of the pole; the former can be observed, and the latter is equal to the observer's [[latitude]]. The time is then given by the angle at the intersection of the [[Meridian (astronomy)|meridian]] (the [[Arc (geometry)|arc]] through the zenith and the pole) and the sun's hour circle (the arc through the sun and the pole).{{Sfn|Gingerich|1986}}<ref name=Tabatabai/> |
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Muslims are also expected to pray towards the [[Kaaba]] in [[Mecca]] and orient their [[mosque]]s in that direction. Thus they need to determine the direction of Mecca ([[Qibla]]) from a given location.<ref>{{cite quran|2|144|style=ref}}</ref><ref>{{cite quran|2|150|style=ref}}</ref> Another problem is the time of [[Salah]]. Muslims need to determine from [[celestial bodies]] the proper times for the prayers before [[sunrise]], at [[Noon|midday]], in the [[afternoon]], at [[sunset]], and in the [[evening]].{{Sfn|Gingerich|1986}}<ref name=Tabatabai>{{citation |
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|url=http://www.almizan.org/Tafseer/Volume2/Baqarah32.asp|author=Syed Mohammad Hussain Tabatabai|work=Tafsir al-Mizan |
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|chapter=Volume 2: Surah Baqarah, Verses 142–151 |
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|accessdate=2008-01-24}}{{Dead link|date=October 2010|bot=H3llBot}}</ref> |
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== Importance == |
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J. J. O'Conner and E. F. Robertson wrote in the ''[[MacTutor History of Mathematics archive]]'':<ref name="JOC-EFR">J. J. O'Connor and E. F. Robertson (November 1999). ''[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_mathematics.html Arabic Mathematics: Forgotten Brilliance?]'' MacTutor History of Mathematics.</ref> |
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{{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 European 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]]."}} |
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R. Rashed wrote in ''The development of Arabic mathematics: between arithmetic and algebra'':<ref name="JOC-EFR"/><ref>R. Rashed (1994). ''The development of Arabic mathematics : between arithmetic and algebra''. (London).</ref> |
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{{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."}} |
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==Algebra== |
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[[File:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|right|A page from the ''[[The Compendious Book on Calculation by Completion and Balancing|Al-jabr wa'l muqabalah]]'' by [[Muhammad ibn Mūsā al-Khwārizmī|Al-Khwarizmi]].]] |
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{{See also|Algebra}} |
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The term [[algebra]] is derived from the Arabic term ''al-jabr'' in the title of [[Al-Khwarizmi]]'s ''[[The Compendious Book on Calculation by Completion and Balancing|Al-jabr wa'l muqabalah]]''. He originally used the term ''al-jabr'' to describe the method of "[[Reduction (mathematics)|reduction]]" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation.{{sfn|Boyer|1991|p=229}} |
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There are three theories about the origins of Islamic algebra. The first emphasizes [[Hindu]] influence, the second emphasizes [[Mesopotamia]]n or Persian-Syriac influence, and the third emphasizes [[Greek mathematics|Greek]] influence. Many scholars believe that it is the result of a combination of all three sources.{{sfn|Boyer|1991|p=227}}{{#tag:ref|"We have said enough so far as numbers are concerned, about the six types of equations. Now, however, it is necessary that we should demonstrate 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."{{sfn|Boyer|1991|p=230}}|group="note"}} |
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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 (e.g. twenty-two) with [[Arabic numerals]] (e.g. 22), but the Arabs never adopted or developed a syncopated or symbolic algebra,{{sfn|Boyer|1991|p=234}}{{#tag:ref|"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."{{sfn|Boyer|1991|p=234}}|group="note"}} until the work of [[Ibn al-Banna al-Marrakushi]] in the 13th century and [[Abū al-Hasan ibn Alī al-Qalasādī]] in the 15th century.<ref name=Qalasadi/> |
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There were four conceptual stages in the development of algebra, three of which either began in, or were significantly advanced in, the Islamic world. These four stages were as follows:<ref>{{Citation |
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|last=Victor J. Katz |
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|first=Bill Barton |
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|last2=Barton |
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|first2=Bill |
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|title=Stages in the History of Algebra with Implications for Teaching |
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|journal=Educational Studies in Mathematics |
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|publisher=[[Springer Science+Business Media |
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|Springer Netherlands]] |
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|volume=66 |
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|issue=2 |
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|date=October 2007 |
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|doi=10.1007/s10649-006-9023-7 |
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|pages=185–201}}</ref> |
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*'''Geometric stage''', where the concepts of algebra are largely [[Geometry|geometric]]. This dates back to the [[Babylonian mathematics|Babylonians]] and continued with the [[Greek mathematics|Greeks]], and was revived by [[Omar Khayyam]]. |
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*'''Static equation-solving stage''', where the objective is to find numbers satisfying certain relationships. The move away from geometric algebra dates back to [[Diophantus]] and [[Brahmagupta]], but algebra didn't decisively move to the static equation-solving stage until [[Muhammad ibn Mūsā al-Khwārizmī|Al-Khwarizmi]]'s ''[[The Compendious Book on Calculation by Completion and Balancing|Al-Jabr]]''. |
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*'''Dynamic function stage''', where motion is an underlying idea. The idea of a [[Function (mathematics)|function]] began emerging with [[Sharaf al-Dīn al-Tūsī]], but algebra didn't decisively move to the dynamic function stage until [[Gottfried Leibniz]]. |
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*'''Abstract stage''', where mathematical structure plays a central role. [[Abstract algebra]] is largely a product of the 19th and 20th centuries. |
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===Static equation-solving algebra=== |
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;Al-Khwarizmi and ''Al-jabr wa'l muqabalah'' |
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The Muslim{{sfn|Boyer|1991|pp=228–229}}{{#tag:ref|"the author's preface in Arabic gave fulsome praise to Mohammed, the prophet, and to al-Mamun, "the Commander of the Faithful"."{{sfn|Boyer|1991|pp=228–229}}|group="note"}} Persian mathematician {{Unicode|[[Muhammad ibn Mūsā al-Khwārizmī]]}} (c. 780–850) 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 AD, wrote more than half a dozen mathematical and astronomical works; some of which were based on the Indian ''Sindhind''.{{sfn|Boyer|1991|p=227}}{{#tag:ref|"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."{{sfn|Boyer|1991|p=227}}|group="note"}} 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.{{sfn|Boyer|1991|p=228}}{{#tag:ref|"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."{{sfn|Boyer|1991|p=228}}|group="note"}} The book also introduced the fundamental method of "[[Reduction (mathematics)|reduction]]" and "balancing", referring to the transposition of subtracted terms to the other side of an [[equation]], that is, the cancellation of like terms on opposite sides of the equation. This is the operation which Al-Khwarizmi originally described as ''al-jabr''.{{sfn|Boyer|1991|p=229}}{{#tag:ref|"It is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the translation above. The word ''al-jabr'' presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word ''muqabalah'' is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation."{{sfn|Boyer|1991|p=229}}|group="note"}} |
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''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).{{sfn|Boyer|1991|p=229}}{{#tag:ref|"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."{{sfn|Boyer|1991|p=229}}|group="note"}} |
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J. J. O'Conner and E. F. Robertson wrote in the ''[[MacTutor History of Mathematics archive]]'':<ref name="JOC-EFR"/> |
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{{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."}} |
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The [[Hellenistic civilization|Hellenistic]] mathematician [[Diophantus]] was traditionally known as "the father of algebra"{{sfn|Boyer|1991|p=228}}{{#tag:ref|"Diophantus sometimes is called "the father of algebra," but this title more appropriately belongs to al-Khwarizmi. It is true that in two respects the work of al-Khwarizmi represented a retrogression from that of Diophantus. First, it is on a far more elementary level than that found in the Diophantine problems and, second, the algebra of al-Khwarizmi is thoroughly rhetorical, with none of the syncopation found in the Greek ''Arithmetica'' or in Brahmagupta's work. Even numbers were written out in words rather than symbols! It is quite unlikely that al-Khwarizmi knew of the work of Diophantus, but he must have been familiar with at least the astronomical and computational portions of Brahmagupta; yet neither al-Khwarizmi nor other Arabic scholars made use of syncopation or of negative numbers."{{sfn|Boyer|1991|p=228}}|group="note"}}<ref name="John Derbyshire For Diophantus">{{Harv|Derbyshire|2006|loc="The Father of Algebra" p. 31}} "Diophantus, the father of algebra, in whose honor I have named this chapter, lived in Alexandria, in Roman Egypt, in either the 1st, the 2nd, or the 3rd century CE."</ref> but debate now exists as to whether or not [[Muhammad ibn Mūsā al-Khwārizmī|Al-Khwarizmi]] deserves this title instead.{{sfn|Boyer|1991|p=228}} Those who support Diophantus point to the fact that the algebra found in ''Al-Jabr'' is more elementary than the algebra found in ''[[Arithmetica]]'' and that ''Arithmetica'' is syncopated while ''Al-Jabr'' is fully rhetorical.{{sfn|Boyer|1991|p=228}} Those who support Al-Khwarizmi point to the fact that he gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,{{sfn|Boyer|1991|p=230}}{{#tag:ref|"The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwarizmi's exposition that his readers must have had little difficulty in mastering the solutions."{{sfn|Boyer|1991|p=230}}|group="note"}} introduced the fundamental methods of reduction and balancing,{{sfn|Boyer|1991|p=229}} and was the first to teach algebra in an [[Elementary algebra|elementary form]] and for its own sake, whereas Diophantus was primarily concerned with the [[number theory|theory of numbers]].{{sfn|Gandz|1936|pp=263–277}}{{#tag:ref|"In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers"{{sfn|Gandz|1936|pp=263–277}}|group="note"}} In addition, R. Rashed and Angela Armstrong write: |
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{{quote|"Al-Khwarizmi's text can be seen to be distinct not only from the [[Babylonian mathematics|Babylonian tablets]], but also from Diophantus' ''Arithmetica''. It no longer concerns a series of [[problem]]s to be resolved, but an [[Expository writing|exposition]] which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."<ref>{{Citation |
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|last1=Rashed |
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|first1=R. |
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|last2=Armstrong |
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|first2=Angela |
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|year=1994 |
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|title=The Development of Arabic Mathematics |
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|publisher=[[Springer Science+Business Media|Springer]] |
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|isbn=0792325656 |
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|oclc=29181926 |
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|pages=11–2}}</ref>}} |
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;Ibn Turk and ''Logical Necessities in Mixed Equations'' |
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[['Abd al-Hamīd ibn Turk]] (fl. 830) 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''.{{sfn|Boyer|1991|p=234}}{{#tag:ref|"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."{{sfn|Boyer|1991|p=234}}|group="note"}} |
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The manuscript gives exactly the same [[Geometry|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.{{sfn|Boyer|1991|p=234}} The similarity between these two works has led some historians to conclude that Islamic algebra may have been well developed by the time of al-Khwarizmi and 'Abd al-Hamid.{{sfn|Boyer|1991|p=234}} |
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;Abū Kāmil and al-Karkhi |
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Arabic mathematicians were also the first to treat [[irrational number]]s as [[algebra]]ic objects.<ref name="ReferenceA">{{MacTutor |
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|class=HistTopics |
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|id=Arabic_mathematics |
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|title=Arabic Mathematics: forgotten brilliance? |
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|year=1999}}</ref> The [[Egypt]]ian mathematician [[Abū Kāmil Shujā ibn Aslam]] (c. 850–930) was the first to accept irrational numbers (often in the form of a [[square root]], [[cube root]] or [[Nth root|fourth root]]) as solutions to [[quadratic equation]]s or as [[coefficient]]s in an [[equation]].<ref name=Sesiano>Jacques Sesiano, "Islamic mathematics", p. 148, in {{citation |
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|title=Mathematics Across Cultures: The History of Non-western Mathematics |
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|first1=Helaine |
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|last1=Selin |
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|first2=Ubiratan |
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|last2=D'Ambrosio |
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|year=2000 |
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|publisher=[[Springer Science+Business Media|Springer]] |
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|isbn=1402002602}}</ref> He was also the first to solve three non-linear [[simultaneous equations]] with three unknown [[Variable (mathematics)|variables]].{{sfn|Berggren|2007|p=518}} |
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[[Al-Karaji|Al-Karkhi]] (953–1029), also known as Al-Karaji, was the successor of [[Abū al-Wafā' al-Būzjānī]] (940–998) and he was the first to discover the solution to equations of the form ax<sup>2n</sup> + bx<sup>n</sup> = c.{{sfn|Boyer|1991|p=239}}{{#tag:ref|"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)"{{sfn|Boyer|1991|p=239}}|group="note"}} Al-Karkhi only considered positive roots.{{sfn|Boyer|1991|p=239}} Al-Karkhi is also regarded as the first person to free algebra from [[Geometry|geometrical]] operations and replace them with the type of [[arithmetic]] operations which are at the core of algebra today. His work on algebra and [[polynomial]]s, gave the rules for arithmetic operations to manipulate polynomials. The [[History of mathematics|historian of mathematics]] F. Woepcke, in ''Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi'' (Paris, 1853), praised Al-Karaji for being "the first who introduced the theory of algebraic [[calculus]]". Stemming from this, Al-Karaji investigated [[binomial coefficients]] and [[Pascal's triangle]].<ref>{{MacTutor |id=Al-Karaji |title=Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji}}</ref> |
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===Linear algebra=== |
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In [[linear algebra]] and [[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 ''[[Encyclopedia of the Brethren of Purity|Rasa'il Ihkwan al-Safa]]'' (''Encyclopedia of the Brethren 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]{{dead link|date=February 2011}}.</ref> |
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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"/> |
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===Geometric algebra=== |
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[[Omar Khayyám]] (c. 1050–1123) wrote a book on Algebra that went beyond ''Al-Jabr'' to include equations of the third degree.{{sfn|Boyer|1991|pp=241–242}}{{#tag:ref|"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."{{sfn|Boyer|1991|pp=241–242}}|group="note"}} 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.{{sfn|Boyer|1991|pp=241–242}} 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.{{sfn|Boyer|1991|pp=241–242}} He only considered positive roots and he did not go past the third degree.{{sfn|Boyer|1991|pp=241–242}} He also saw a strong relationship between Geometry and Algebra.{{sfn|Boyer|1991|pp=241–242}} |
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===Dynamic functional algebra=== |
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[[File:Irakischer Maler von 1287 001.jpg|thumb|right|[[Arabic]] manuscript from the 12th century depicting the [[Brethren of Purity]].]] |
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In the 12th century, [[Sharaf al-Dīn al-Tūsī]] found algebraic and [[Numerical analysis|numerical]] solutions to cubic equations and was the first to discover the [[derivative]] of [[Cubic function|cubic polynomials]].{{sfn|Berggren|2007|p=516}} His ''Treatise on Equations'' dealt with [[equation]]s up to the third degree. The treatise does not follow [[Al-Karaji]]'s school of algebra, but instead represents "an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of [[algebraic geometry]]." The treatise dealt with 25 types of equations, including twelve types of [[linear equation]]s and [[quadratic equation]]s, eight types of [[cubic equation]]s with positive solutions, and five types of cubic equations which may not have positive solutions.<ref name=Sharaf>{{MacTutor |id=Al-Tusi_Sharaf |title=Sharaf al-Din al-Muzaffar al-Tusi}}</ref> He understood the importance of the [[discriminant]] of the cubic equation and used an early version of [[Gerolamo Cardano|Cardano]]'s formula<ref>{{Citation |
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|last1=Rashed |
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|first1=Roshdi |
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|last2=Armstrong |
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|first2=Angela |
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|year=1994 |
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|title=The Development of Arabic Mathematics |
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|publisher=[[Springer Science+Business Media|Springer]] |
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|isbn=0792325656 |
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|pages=342–3}}</ref> to find algebraic solutions to certain types of cubic equations.{{sfn|Berggren|2007|p=516}} |
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Sharaf al-Din also developed the concept of a [[Function (mathematics)|function]]. In his analysis of |
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the equation <math>\ x^3 + d = bx^2</math> for example, he begins by changing the equation's form to <math>\ x^2 (b - x) = d</math>. He then states that the question of whether the equation has a solution depends on whether or not the “function” on the left side reaches the value <math>\ d</math>. To determine this, he finds a maximum value for the function. He proves that the maximum value occurs when <math>x = \frac{2b}{3}</math>, which gives the functional value <math>\frac{4b^3}{27}</math>. Sharaf al-Din then states that if this value is less than <math>\ d</math>, there are no positive solutions; if it is equal to <math>\ d</math>, then there is one solution at <math>x = \frac{2b}{3}</math>; and if it is greater than <math>\ d</math>, then there are two solutions, one between <math>\ 0</math> and <math>\frac{2b}{3}</math> and one between <math>\frac{2b}{3}</math> and <math>\ b</math>. This was the earliest form of dynamic [[functional algebra]].<ref>{{Citation |
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|last=Katz |
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|first=Victor J. |
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|last2=Barton |
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|first2=Bill |
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|title=Stages in the History of Algebra with Implications for Teaching |
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|journal=Educational Studies in Mathematics |
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|publisher=[[Springer Science+Business Media |
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|Springer Netherlands]] |
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|volume=66 |
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|issue=2 |
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|date=October 2007 |
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|doi=10.1007/s10649-006-9023-7 |
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|pages=185–201 [192]}}</ref> |
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===Numerical analysis=== |
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<!-- Image with unknown copyright status removed: [[File:Jamshid al-Kashi (stamp 1).jpg|thumb|A stamp issued 1979 in [[Iran]] commemorating [[Jamshīd al-Kāshī]].]] --> |
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In [[numerical analysis]], the essence of [[Viète's formulas|Viète's method]] was known to [[Sharaf al-Dīn al-Tūsī]] in the 12th century, and it is possible that the algebraic tradition of Sharaf al-Dīn, as well as his predecessor [[Omar Khayyám]] and successor [[Jamshīd al-Kāshī]], was known to 16th century European algebraists, of whom [[François Viète]] was the most important.<ref>{{citation |
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|title=Historical Development of the Newton-Raphson Method |
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|first=Tjalling J. |
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|last=Ypma |
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|journal=SIAM Review |
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|volume=37 |
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|issue=4 |
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|date=December 1995 |
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|publisher=Society for Industrial and Applied Mathematics |
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|pages=531–551 [534] |
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|doi=10.1137/1037125}}</ref> |
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A method algebraically equivalent to [[Newton's method]] was also known to Sharaf al-Dīn. In the 15th century, his successor al-Kashi later used a form of Newton's method to numerically solve <math>\ x^P - N = 0</math> to find roots of <math>\ N</math>. In [[western Europe]], a similar method was later described by Henry Biggs in his ''Trigonometria Britannica'', published in 1633.<ref>{{citation |
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|title=Historical Development of the Newton-Raphson Method |
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|first=Tjalling J. |
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|last=Ypma |
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|journal=SIAM Review |
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|volume=37 |
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|issue=4 |
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|date=December 1995 |
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|publisher=Society for Industrial and Applied Mathematics |
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|pages=531–551 [539] |
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|doi=10.1137/1037125}}</ref> |
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===Symbolic algebra=== |
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[[Al-Hassar]], a mathematician from [[Morocco]] specializing in [[Islamic inheritance jurisprudence]] during the 12th century, developed the modern symbolic [[mathematical notation]] for [[Fraction (mathematics)|fractions]], where the [[Fraction (mathematics)|numerator]] and [[denominator]] are separated by a horizontal bar. This same fractional notation appeared soon after in the work of [[Fibonacci]] in the 13th century.{{Citation needed|date=May 2010}} |
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[[Abū al-Hasan ibn Alī al-Qalasādī]] (1412–1482) was the last major medieval [[Arab]] algebraist, who improved on the [[Mathematical notation|algebraic notation]] earlier used in the [[Maghreb]] by [[Ibn al-Banna]] in the 13th century<ref name=Qalasadi>{{MacTutor Biography|id=Al-Qalasadi|title= Abu'l Hasan ibn Ali al Qalasadi}}</ref> and by [[Ibn al-Yāsamīn]] in the 12th century. In contrast to the syncopated notations of their predecessors, [[Diophantus]] and [[Brahmagupta]], which lacked symbols for [[Operation (mathematics)|mathematical operations]],{{sfn|Boyer|1991|p=178}}{{#tag:ref|"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."{{sfn|Boyer|1991|p=178}}|group="note"}} al-Qalasadi's algebraic 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/> |
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The [[symbol]] <math>\mathit{x}</math> now commonly [[denote]]s an unknown [[Variable (mathematics)|variable]]. Even though any letter can be used, <math>\mathit{x}</math> is the most common choice. This usage can be traced back to the [[Arabic language|Arabic]] word ''šay<nowiki>'</nowiki>'' شيء = “thing,” used in Arabic algebra texts such as the ''[[The Compendious Book on Calculation by Completion and Balancing|Al-Jabr]]'', and was taken into [[Old Spanish language|Old Spanish]] with the pronunciation “šei,” which was written ''xei,'' and was soon habitually abbreviated to <math>\mathit{x}</math>. (The Spanish [[pronunciation]] of “x” has changed since). Some sources say that this <math>\mathit{x}</math> is an abbreviation of [[Latin]] ''causa'', which was a translation of {{lang-ar|شيء}}. This started the habit of using letters to represent quantities in [[algebra]]. In mathematics, an “[[italic type|italicized]] x” (<math>x\!</math>) is often used to avoid potential confusion with the multiplication symbol. |
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==Arithmetic== |
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{{See also|Arithmetic}} |
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===Arabic numerals=== |
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{{See also|Arabic numerals}} |
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The [[Indian numerals|Indian numeral]] system came to be known to both the [[Persian people|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.<ref>[http://www-gap.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Indian_numerals.html Indian numerals]</ref> 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. |
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In the [[Arab world]]—until early modern times—the Arabic numeral system was often only used by mathematicians. [[Astronomy in medieval Islam|Muslim astronomers]] mostly used the [[Babylonian numerals|Babylonian numeral system]], and [[Islamic economics in the world|merchants]] mostly used the [[Abjad numerals]]. A distinctive "Western 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, which is the direct ancestor to the modern Western Arabic numerals now used throughout the world.{{sfn|Gandz|1931|pp=393–424}} |
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The first mentions of the numerals in the West are found in the ''[[Codex Vigilanus]]'' of 976.<ref>[http://www.mathorigins.com/V.htm Mathorigins.Com_V]</ref> 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]]. |
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[[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]] ({{lang-lat|algorithmus}}) with a meaning "calculation method". |
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Al-Hassār, a mathematician from the [[Maghreb]] ([[North Africa]]) specializing in [[Islamic inheritance jurisprudence]] during the 12th century, developed the modern symbolic [[mathematical notation]] for fractions, where the [[Fraction (mathematics)|numerator]] and [[denominator]] are separated by a horizontal bar. The "dust [[cipher]]s he used are also nearly identical to the digits used in the current Western Arabic numerals. These same digits and fractional notation appear soon after in the work of [[Fibonacci]] in the 13th century.{{Citation needed|date=May 2010}} |
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===Decimal fractions=== |
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In discussing the origins of [[decimal fractions]], [[Dirk Jan Struik]] states that:<ref>D. J. Struik, ''A Source Book in Mathematics 1200–1800'' (Princeton University Press, New Jersey, 1986). p. 7. ISBN 0-691-02397-2</ref> |
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{{quote|"The introduction of decimal fractions as a common computational practice can be dated back to the [[Flemish Region|Flemish]] pamphelet ''De Thiende'', published at [[Leiden|Leyden]] in 1585, together with a French translation, ''La Disme'', by the Flemish mathematician [[Simon Stevin]] (1548–1620), then settled in the Northern Netherlands. It is true that decimal fractions were used by the [[Chinese mathematics|Chinese]] many centuries before Stevin and that the Persian astronomer [[Al-Kāshī]] used both decimal and [[sexagesimal]] fractions with great ease in his ''Key to arithmetic'' (Samarkand, early fifteenth century)."<ref>P. Luckey, ''Die Rechenkunst bei Ğamšīd b. Mas'ūd al-Kāšī'' (Steiner, Wiesbaden, 1951).</ref>}} |
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While the [[Persian people|Persian]] mathematician [[Jamshīd al-Kāshī]] claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggrenn notes that he was mistaken, as decimal fractions were first used five centuries before him by the [[Baghdad]]i mathematician [[Abu'l-Hasan al-Uqlidisi]] as early as the 10th century.{{sfn|Berggren|2007|p=518}} |
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===Real numbers=== |
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The [[Middle Ages]] saw the acceptance of [[zero]], [[negative number|negative]], [[Integer|integral]] and [[fraction (mathematics)|fractional]] numbers, first by [[Indian mathematics|Indian mathematicians]] and [[Chinese mathematics|Chinese mathematicians]], and then by Arabic mathematicians, who were also the first to treat [[irrational number]]s as algebraic objects,<ref name="ReferenceA"/> which was made possible by the development of algebra. Arabic mathematicians merged the concepts of "[[number]]" and "[[Magnitude (mathematics)|magnitude]]" into a more general idea of [[real number]]s, and they criticized Euclid's idea of [[ratio]]s, developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude.<ref>{{citation |
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|last=Matvievskaya |
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|first=Galina |
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|year=1987 |
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|title=The Theory of Quadratic Irrationals in Medieval Oriental Mathematics |
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|journal=[[New York Academy of Sciences|Annals of the New York Academy of Sciences]] |
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|volume=500 |
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|pages=253–277 [254] |
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|doi=10.1111/j.1749-6632.1987.tb37206.x}}</ref> In his commentary on Book 10 of the ''Elements'', the [[Persian people|Persian]] mathematician [[Al-Mahani]] (d. 874/884) examined and classified [[quadratic irrational]]s and cubic irrationals. He provided definitions for rational and irrational magnitudes, which he treated as irrational numbers. He dealt with them freely but explains them in geometric terms as follows:<ref name=Matvievskaya-259>{{citation |
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|last=Matvievskaya |
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|first=Galina |
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|year=1987 |
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|title=The Theory of Quadratic Irrationals in Medieval Oriental Mathematics |
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|journal=[[New York Academy of Sciences|Annals of the New York Academy of Sciences]] |
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|volume=500 |
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|pages=253–277 [259] |
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|doi=10.1111/j.1749-6632.1987.tb37206.x}}</ref> |
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{{quote|"It will be a rational (magnitude) when we, for instance, say 10, 12, 3%, 6%, etc., because its value is pronounced and expressed quantitatively. What is not rational is irrational and it is impossible to pronounce and represent its value quantitatively. For example: the roots of numbers such as 10, 15, 20 which are not squares, the sides of numbers which are not cubes ''etc.''"}} |
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In contrast to Euclid's concept of magnitudes as lines, Al-Mahani considered integers and fractions as rational magnitudes, and square roots and [[cube root]]s as irrational magnitudes. He also introduced an [[arithmetic]]al approach to the concept of irrationality, as he attributes the following to irrational magnitudes:<ref name=Matvievskaya-259/> |
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{{quote|"their sums or differences, or results of their addition to a rational magnitude, or results of subtracting a magnitude of this kind from an irrational one, or of a rational magnitude from it."}} |
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The [[Egypt]]ian mathematician [[Abū Kāmil Shujā ibn Aslam]] (c. 850–930) was the first to accept irrational numbers as solutions to [[quadratic equation]]s or as [[coefficient]]s in an [[equation]], often in the form of square roots, cube roots and [[Nth root|fourth roots]].<ref name=Sesiano/> In the 10th century, the [[Iraqi people|Iraqi]] mathematician Al-Hashimi provided general proofs for numbers (rather than geometric demonstrations) as he considered multiplication, division, etc. for ”lines”. Using this method, he provided the first proof for irrational numbers.<ref>{{citation |
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|last=Matvievskaya |
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|first=Galina |
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|year=1987 |
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|title=The Theory of Quadratic Irrationals in Medieval Oriental Mathematics |
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|journal=[[New York Academy of Sciences |
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|Annals of the New York Academy of Sciences]] |
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|volume=500 |
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|pages=253–277 [260] |
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|doi=10.1111/j.1749-6632.1987.tb37206.x}}</ref> [[Abū Ja'far al-Khāzin]] (900–971) provides a definition of rational and irrational magnitudes, stating that if a definite [[quantity]] is:<ref>{{citation |
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|last=Matvievskaya |
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|first=Galina |
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|year=1987 |
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|title=The Theory of Quadratic Irrationals in Medieval Oriental Mathematics |
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|journal=[[New York Academy of Sciences|Annals of the New York Academy of Sciences]] |
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|volume=500 |
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|pages=253–277 [261] |
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|doi=10.1111/j.1749-6632.1987.tb37206.x}}</ref> |
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{{quote|"contained in a certain given magnitude once or many times, then this (given) magnitude corresponds to a rational number. [...] Each time when this (latter) magnitude comprises a half, or a third, or a quarter of the given magnitude (of the unit), or, compared with (the unit), comprises three, five, or three fifths, it is a rational magnitude. And, in general, each magnitude that corresponds to this magnitude (''i.e.'' to the unit), as one number to another, is rational. If, however, a magnitude cannot be represented as a multiple, a part (l/''n''), or parts (''m''/''n'') of a given magnitude, it is irrational, ''i.e.'' it cannot be expressed other than by means of roots."}} |
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Many of these concepts were eventually accepted by European mathematicians some time after the [[Latin translations of the 12th century]]. Al-Hassār, an Arabic mathematician from the [[Maghreb]] ([[North Africa]]) specializing in [[Islamic inheritance jurisprudence]] during the 12th century, developed the modern symbolic [[mathematical notation]] for fractions, where the [[Fraction (mathematics)|numerator]] and [[denominator]] are separated by a horizontal bar. This same fractional notation appears soon after in the work of [[Fibonacci]] in the 13th century.{{Citation needed|date=May 2010}} |
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===Number theory=== |
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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''−1</sup>(2<sup>''n''</sup> − 1) where 2<sup>''n''</sup> − 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> |
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In the early 14th century, [[Kamāl al-Dīn al-Fārisī]] made a number of important contributions to number theory. His most impressive work in number theory is on [[amicable number]]s. In ''Tadhkira al-ahbab fi bayan al-tahabb'' ("''Memorandum for friends on the proof of amicability''") introduced a major new approach to a whole area of number theory, introducing ideas concerning [[factorization]] and [[Combinatorics|combinatorial]] methods. In fact, al-Farisi's approach is based on the unique factorization of an [[integer]] into powers of [[prime number]]s. |
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==Geometry== |
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[[File:Durer astronomer.jpg|thumb|225px|An engraving by [[Albrecht Dürer]] featuring [[Mashallah ibn Athari|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.]] |
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{{See also|Geometry}} |
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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 [[Symmetric algebra|polynomial algebra]], [[Combinatorics|combinatorial analysis]], [[numerical analysis]], the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose. |
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[[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. |
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===Early Islamic geometry=== |
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:''See also [[#Applied mathematics|Applied mathematics]]'' |
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[[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 number|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 name=Sayili>[[Aydin Sayili]] (1960), "Thabit ibn Qurra's Generalization of the Pythagorean Theorem", ''[[Isis (journal)|Isis]]'' '''51''' (1): pp. 35–37</ref> |
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In some respects, Thabit is critical of the ideas of [[Plato]] and [[Aristotle]], particularly regarding motion. It would seem that here his ideas are based on an acceptance of using arguments concerning motion in his geometrical arguments. |
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[[Ibrahim ibn Sinan]] ibn Thabit (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]] (Alhazen), studied [[optics]] and investigated the optical properties of mirrors made from [[conic section]]s (see [[#Mathematical physics|Mathematical physics]]). |
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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 [[Lunar phase|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.{{Sfn|Gingerich|1986}}<ref name=Tabatabai/> |
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===Algebraic and analytic geometry=== |
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[[File:Arthur Szyk02.jpg|thumb|Illustration by [[Arthur Szyk]] for the 1940 edition of the ''[[Rubaiyat of Omar Khayyam]]''.]] |
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In the early 11th century, [[Ibn al-Haytham]] (Alhazen) was able to solve by purely algebraic means certain cubic equations, and then to interpret the results geometrically.<ref>Kline, M. (1972), ''Mathematical Thought from Ancient to Modern Times'', Volume 1, p. 193, [[Oxford University Press]]</ref> Subsequently, [[Omar Khayyám]] discovered the general method of solving [[cubic equation]]s by intersecting a parabola with a circle.<ref>Kline, M. (1972), ''Mathematical Thought from Ancient to Modern Times'', Volume 1, pp. 193–195, [[Oxford University Press]]</ref> |
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[[Omar Khayyám]] (1048–1122) 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 name="ReferenceA"/><ref>R. Rashed (1994). ''The development of Arabic mathematics: between arithmetic and algebra''. London.</ref> and [[analytic geometry]].<ref name=Cooper/> |
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In a paper written by Khayyam before his famous algebra text ''Treatise on Demonstration of Problems of Algebra'', he considers the problem: ''Find a point on a quadrant of a circle in such manner that when a normal is dropped from the point to one of the bounding radii, the ratio of the normal's length to that of the radius equals the ratio of the segments determined by the foot of the normal.'' Khayyam shows that this problem is equivalent to solving a second problem: ''Find a right triangle having the property that the hypotenuse equals the sum of one leg plus the altitude on the hypotenuse.'' This problem in turn led Khayyam to solve the cubic equation x<sup>3</sup> + 200x = 20x<sup>2</sup> + 2000 and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables. Perhaps even more remarkable is the fact that Khayyam states that the solution of this cubic requires the use of conic sections and that it cannot be solved by compass and straightedge, a result which would not be proved for another 750 years. |
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His ''Treatise on Demonstration of Problems of Algebra'' contained a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. In fact Khayyam gives an interesting historical account in which he claims that the Greeks had left nothing on the theory of cubic equations. Indeed, as Khayyam writes, the contributions by earlier writers such as al-Mahani and [[al-Khazin]] were to translate geometric problems into algebraic equations (something which was essentially impossible before the work of [[Muḥammad ibn Mūsā al-Ḵwārizmī]]). However, Khayyam himself seems to have been the first to conceive a general theory of cubic equations. |
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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{{sfn|Boyer|1991|p=241–242}} 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 [[René Descartes]].{{sfn|Boyer|1991|p=241–242}}{{#tag:ref|"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."{{sfn|Boyer|1991|pp=241–242}}|group="note"}} |
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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, entitled ''Treatise on 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]].<ref name=Sharaf/> |
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===Non-Euclidean geometry=== |
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[[File:Nasir al-Din Tusi.jpg|thumb|[[Nasīr al-Dīn al-Tūsī]] commemorated on an Iranian stamp upon the 700th anniversary of his death.]] |
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In the early 11th century, [[Ibn al-Haytham]] (Alhazen) made the first attempt at proving the [[Euclidean geometry|Euclidean]] [[parallel postulate]], the fifth [[Axiom|postulate]] in [[Euclid's Elements|Euclid's ''Elements'']], using a [[proof by contradiction]],<ref>{{Harv|Eder|2000}}</ref> where he introduced the concept of [[Hyperbolic motion|motion]] and [[Transformation (geometry)|transformation]] into geometry.<ref>{{Harv|Katz|1998|p=269}}: {{quote|In effect, this method characterized parallel lines as lines always equidisant from one another and also introduced the concept of motion into geometry.}}</ref> He formulated the [[Lambert quadrilateral]], which Boris Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert quadrilateral",<ref name=Rozenfeld>{{Harv|Rozenfeld|1988|p=65}}</ref> and his attempted proof also shows similarities to [[Playfair's axiom]].<ref name=Smith>{{Harv|Smith|1992}}</ref> |
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In the late 11th century, [[Omar Khayyám]] 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, pp. 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> |
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In ''Commentaries on the difficult postulates of Euclid's book'' Khayyam made a contribution to non-Euclidean geometry, although this was not his intention. In trying to prove the parallel postulate he accidentally proved properties of figures in non-Euclidean geometries. Khayyam also gave important results on ratios in this book, extending Euclid's work to include the multiplication of ratios. The importance of Khayyam's contribution is that he examined both Euclid's definition of equality of ratios (which was that first proposed by [[Eudoxus of Cnidus|Eudoxus]]) and the definition of equality of ratios as proposed by earlier Islamic mathematicians such as al-Mahani which was based on [[continued fraction]]s. Khayyam proved that the two definitions are equivalent. He also posed the question of whether a ratio can be regarded as a number but leaves the question unanswered. |
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The [[Saccheri quadrilateral|Khayyam-Saccheri quadrilateral]] was first considered by Omar Khayyam in the late 11th century in Book I of ''Explanations of the Difficulties in the Postulates of Euclid''.<ref name=autogenerated2>Boris Abramovich Rozenfelʹd (1988), ''A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space'', p. 65. Springer, ISBN 0387964584.</ref> Unlike many commentators on Euclid before and after him (including of course Saccheri), Khayyam was not trying to prove the [[parallel postulate]] as such but to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" ([[Aristotle]]): |
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:Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge.<ref>Boris A Rosenfeld and Adolf P Youschkevitch (1996), ''Geometry'', p. 467 in Roshdi Rashed, Régis Morelon (1996), ''Encyclopedia of the history of Arabic science'', Routledge, ISBN 0415124115.</ref> |
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Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a [[Saccheri quadrilateral]] can take and after proving a number of theorems about them, he (correctly) refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid. It wasn't until 600 years later that [[Giordano Vitale]] made an advance on the understanding of this quadrilateral in his book ''Euclide restituo'' (1680, 1686), when he used it to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. [[Saccheri]] himself based the whole of his long, heroic and ultimately flawed proof of the parallel postulate around the quadrilateral and its three cases, proving many theorems about its properties along the way. |
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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/> |
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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, pp. 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'', pp. 270–271, [[Addison-Wesley]], ISBN 0321016181: {{quote|"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."}}</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, pp. 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> |
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The theorems of [[Ibn al-Haytham]] (Alhazen), [[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, pp. 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 this 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> |
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==Trigonometry== |
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{{See also|Trigonometry}} |
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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, who enunciated a large number of theorems which freed the subject of trigonometry from dependence upon the complete [[quadrilateral]], as was the case in [[Greek mathematics|Hellenistic mathematics]] due to the application of [[Menelaus' theorem]]. According to E. S. Kennedy, it was after this development in Islamic mathematics that "the first real trigonometry emerged, in the sense that only then did the object of study become the [[Spherical trigonometry|spherical]] or plane [[triangle]], its sides and [[angle]]s."<ref>{{citation |
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|first=E. S. |
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|last=Kennedy |
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|title=The History of Trigonometry |
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|journal=31st Yearbook |
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|publisher=National Council of Teachers of Mathematics, Washington DC |
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|year=1969}} ([[cf.]] {{sfn|Haq|1996|pp=52–70}})</ref> Another important development was the subject's separation from astronomy. All works on trigonometry up until the 12th century treated it mainly as an adjunct to astronomy; the first treatment of trigonometry as a subject in its own right was by [[Nasīr al-Dīn al-Tūsī]] in the 13th century.<ref name=trigonometry>{{cite web |
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|title=trigonometry |
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|url=http://www.britannica.com/EBchecked/topic/605281/trigonometry |
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|publisher=''[[Encyclopædia Britannica]]'' |
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|accessdate=2008-07-21}}</ref> |
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===Trigonometric functions=== |
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In the early 9th century, {{Unicode|[[Muhammad ibn Mūsā al-Khwārizmī]]}} (c. 780–850) produced tables for the [[trigonometric functions]] of sines and cosine,<ref name=Kennedy-1956>{{citation |
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|last=Kennedy |
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|first=E.S. |
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|title=A Survey of Islamic Astronomical Tables; Transactions of the American Philosophical Society |
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|year=1956 |
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|location=[[Philadelphia]] |
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|publisher=[[American Philosophical Society]] |
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|volume=46 |
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|issue=2 |
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|pages=26–29}}</ref> and the first tables for tangents.<ref name=MacTutor-Khwarizmi>{{MacTutor|id=Al-Khwarizmi|name=Abu Ja'far Muhammad ibn Musa Al-Khwarizmi}}</ref> In 830, [[Habash al-Hasib al-Marwazi]] produced the first tables of cotangents as well as tangents.<ref name=trigonometry/><ref name=Sesiano-157/> [[Muhammad ibn Jābir al-Harrānī al-Battānī]] (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants, which he referred to as a "table of shadows" (in reference to the shadow of a [[gnomon]]), for each degree from 1° to 90°.<ref name=trigonometry/> By the 10th century, in the work of [[Abū al-Wafā' al-Būzjānī]] (959–998), Muslim mathematicians were using all six trigonometric functions, and had sine tables in 0.25° increments, to 8 decimal places of accuracy, as well as accurate tables of [[Tangent (trigonometry)|tangent]] values. |
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[[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°.<ref name=Kashi>{{MacTutor|id=Al-Kashi|title=Ghiyath al-Din Jamshid Mas'ud al-Kashi}}</ref> In one of his [[numerical approximations of π]], he correctly computed 2π to 9 [[sexagesimal]] digits.<ref>''Al-Kashi'', author: Adolf P. Youschkevitch, chief editor: Boris A. Rosenfeld, p. 256</ref> In order to determine sin 1°, al-Kashi discovered the following [[triple-angle formula]] often attributed to [[François Viète]] in the 16th century:<ref>{{citation |
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|title=Sherlock Holmes in Babylon and Other Tales of Mathematical History |
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|last1=Anderson |
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|first1=Marlow |
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|last2=Katz |
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|first2=Victor |
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|last3=Wilson |
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|first3=Robin J. |
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|publisher=[[Mathematical Association of America]] |
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|year=2004 |
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|isbn=0883855461 |
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|page=139}}</ref> |
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:<math>\sin 3 \phi = 3 \sin \phi - 4 \sin^3 \phi\,.</math> |
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Al-Kashi, alongside his colleague [[Ulugh Beg]] (1394–1449), gave accurate tables of sines and tangents correct to 8 decimal places. [[Taqi al-Din Muhammad ibn Ma'ruf|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]] 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 1° to a precision of 8 sexagesimals (the equivalent of 14 decimals):{{Citation needed|date=May 2010}} |
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:<math> \sin 1^\circ = 1^P 2' 49'' 43''' 11'''' 14''''' 44''''''16''''''' \ (= 1/60 + 2/60^2 + 49/60^3 + \cdots)\,.</math> |
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===Laws and identities=== |
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[[Muhammad ibn Jābir al-Harrānī al-Battānī]] (853–929) formulated a number of important trigonometrical relationships such as: |
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:<math>\tan a = \frac{\sin a}{\cos a}</math> |
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:<math>\sec a = \sqrt{1 + \tan^2 a }</math> |
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Although there are several candidates for the title of discoverer,<ref>''Also the 'sine law' (of geometry and trigonometry, applicable to sperical trigonometry) is attributed, among others, to Alkhujandi. (The three others are Abul Wafa Bozjani, Nasiruddin Tusi and Abu Nasr Mansur).'' Razvi, Syed Abbas Hasan (1991) ''A history of science, technology, and culture in Central Asia, Volume 1'' University of Peshawar, Peshawar, Pakistan, p. 358, [http://www.worldcat.org/oclc/26317600 OCLC 26317600]</ref><ref>Bijli suggests that three mathematicians are in contention for the honor, Alkhujandi, Abdul-Wafa and Mansur, leaving out Nasiruddin Tusi. Bijli, Shah Muhammad and Delli, Idarah-i Adabiyāt-i (2004) ''Early Muslims and their contribution to science: ninth to fourteenth century'' Idarah-i Adabiyat-i Delli, Delhi, India, p. 44, [http://www.worldcat.org/oclc/66527483 OCLC 66527483]</ref> in the 10th century, [[Abū al-Wafā' al-Būzjānī]] significantly contributed to the discovery and use of the [[law of sines]] for [[spherical trigonometry]]:<ref name=Sesiano-157>Jacques Sesiano, "Islamic mathematics", p. 157, in {{citation |
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|title=Mathematics Across Cultures: The History of Non-western Mathematics |
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|first1=Helaine |
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|last1=Selin |
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|first2=Ubiratan |
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|last2=D'Ambrosio |
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|year=2000 |
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|publisher=[[Springer Science+Business Media|Springer]] |
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|isbn=1402002602}}</ref> |
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:<math>\frac{\ a}{\sin A} = \frac{\ b}{\sin B} = \frac{c}{\sin C}.</math> |
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Abū al-Wafā' also developed the following trigonometric formula: |
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:<math> \sin 2x = 2 \sin x \cos x \ </math> |
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Abū al-Wafā also established the angle addition identities, e.g. sin (''a'' ± ''b'').<ref name=Sesiano-157/> |
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:<math>\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b</math> |
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Also in the late 10th and early 11th centuries, the Egyptian astronomer [[Ibn Yunus]] performed many careful trigonometric calculations and demonstrated the following formula: |
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:<math>\cos a \cos b = \frac{\cos(a+b) + \cos(a-b)}{2}</math> |
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Also in the 11th century, [[Al-Jayyani]]'s ''The book of unknown arcs of a sphere'' introduced the general law of sines.<ref name="MacTutor Al-Jayyani"/> In the 13th century, [[Nasīr al-Dīn al-Tūsī]], in his ''On the Sector Figure'', stated the law of sines for plane and spherical triangles, discovered the [[law of tangents]] for spherical triangles, and provided proofs for these laws.{{sfn|Berggren|2007|p=518}} [[Jamshīd al-Kāshī]] (1393–1449) provided the first explicit statement of the [[law of cosines]] in a form suitable for [[triangulation]].<ref name=Kashi/> As such, the law of cosines is known the [[:fr:Théorème d'Al-Kashi|''théorème d'Al-Kashi'']] in France. |
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===Spherical trigonometry=== |
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Hellenistic methods dealing with spherical triangles were known, particularly the method of [[Menelaus of Alexandria]], who developed [[Menelaus' theorem]] to deal with spherical problems.<ref>{{MacTutor|id=Menelaus|title=Menelaus of Alexandria}} "Book 3 deals with spherical trigonometry and includes Menelaus's theorem."</ref>{{sfn|Boyer|1991|p=163}}{{#tag:ref|"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)."{{sfn|Boyer|1991|p=163}}|group="note"}} However, E. S. Kennedy points out that while it was possible in pre-lslamic mathematics to compute the magnitudes of a spherical figure, in principle, by use of the table of chords and Menelaus' theorem, the application of the theorem to spherical problems was very difficult in practice.<ref>{{citation |
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|first=E. S. |
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|last=Kennedy |
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|title=The History of Trigonometry |
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|journal=31st Yearbook |
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|publisher=National Council of Teachers of Mathematics, Washington DC |
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|year=1969 |
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|page=337}} ([[cf.]] {{sfn|Haq|1996|pp=52–70}})</ref> In order to observe holy days on the [[Islamic calendar]] in which timings were determined by [[phases of the moon]], astronomers initially used Menelaus' method to calculate the place of the [[moon]] and [[star]]s, though this method proved to be clumsy and difficult. It involved setting up two intersecting [[right triangle]]s; by applying Menelaus' theorem it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the [[sun]]'s [[altitude]], for instance, repeated applications of Menelaus' theorem were required. For medieval [[Astronomy in medieval Islam|Islamic astronomers]], there was an obvious challenge to find a simpler trigonometric method.{{Sfn|Gingerich|1986}} |
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In the early 9th century, [[Muhammad ibn Mūsā al-Khwārizmī]] was an early pioneer in [[spherical trigonometry]] and wrote a treatise on the subject.<ref name=MacTutor-Khwarizmi/> In the 10th century, [[Abū al-Wafā' al-Būzjānī]] discovered the [[law of sines]] for spherical trigonometry.<ref name=Sesiano-157/> In the 11th century, [[Al-Jayyani]] (989–1079) of [[Al-Andalus]] wrote ''The book of unknown arcs of a sphere'', which is considered "the first treatise on spherical trigonometry" in its modern form.<ref name="MacTutor Al-Jayyani">{{MacTutor|id=Al-Jayyani|title=Abu Abd Allah Muhammad ibn Muadh Al-Jayyani}}</ref> It "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 name="MacTutor Al-Jayyani"/> In the 13th century, [[Nasīr al-Dīn al-Tūsī]] developed spherical trigonometry into its present form,<ref name=trigonometry/> and listed the six distinct cases of a right-angled triangle in spherical trigonometry.{{sfn|Berggren|2007|p=518}} In his ''On the Sector Figure'', he also stated the law of sines for plane and spherical triangles, and discovered the [[law of tangents]] for spherical triangles.{{sfn|Berggren|2007|p=518}} |
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===Other advances=== |
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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, pp. 751–795 [769]</ref> and [[Islamic geography]], as described by [[Abu Rayhan Biruni]] in the early 11th century. Biruni employed triangulation techniques to measure the size of the Earth and the distances between places (see ''[[#Mathematical geography and geodesy|Mathematical geography and geodesy]]'' section).<ref>{{MacTutor|id=Al-Biruni|title=Abu Arrayhan Muhammad ibn Ahmad al-Biruni}}</ref> |
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In the late 11th century, [[Omar Khayyám]] (1048–1131) solved [[cubic equation]]s using approximate numerical solutions found by interpolation in trigonometric tables (see ''[[#Geometric algebra|Geometric algebra]]'' and ''[[#Algebraic and analytic geometry|Algebraic and analytic geometry]]'' sections). [[Jamshīd al-Kāshī]] (1393–1449) provided the first explicit statement of the [[law of cosines]] in a form suitable for triangulation.<ref name=Kashi/> |
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==Calculus== |
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[[File:Ibn al-Haytham.png|thumb|[[Ibn al-Haytham]] (Alhazen), author of the ''[[Book of Optics]]''.]] |
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===Integral calculus=== |
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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'', pp. 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 [[fourth power]]s, and using an early [[Mathematical proof|proof]] by [[mathematical induction]], he developed a method for determining the general formula for the sum of any integral powers. He used his result on sums of integral powers to perform an [[Integral|integration]], in order to find the volume of a [[paraboloid]]. He was thus able to find the [[integral]]s for [[polynomial]]s up to the [[Quadratic polynomial|fourth degree]], and came close to finding a general formula for the integrals of any polynomials. This was fundamental to the development of [[infinitesimal]] and integral calculus. His results were repeated by the [[Morocco|Moroccan]] mathematicians Abu-l-Hasan ibn Haydur (d. 1413) and Abu Abdallah ibn Ghazi (1437–1514), by [[Jamshīd al-Kāshī]] (c. 1380–1429) in ''The Calculator's Key'', and by the [[Indian mathematics|Indian mathematicians]] of the [[Kerala school of astronomy and mathematics]] in the 15th–16th centuries.<ref name=autogenerated4>Victor J. Katz (1995), "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '''68''' (3): 163–174 [165–169 & 173–174]</ref> |
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===Differential calculus=== |
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In the 12th century, the [[Persian people|Persian]] mathematician [[Sharaf al-Dīn al-Tūsī]] 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): pp. 304–309</ref> His ''Treatise on Equations'' developed concepts related to [[differential calculus]], such as the [[derivative]] function and the [[maxima and minima]] of curves, in order to solve cubic equations which may not have positive solutions. |
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For example, in order to solve the equation <math>\ x^3 + a = bx</math> with a and b positives, al-Tusi finds the maximum point of the curve <math>\ bx - x^3 = a</math>. He uses the derivative of the function to find that the maximum point occurs at <math>x = \sqrt{\frac{b}{3}}</math>, and then finds the maximum value for y at <math>2(\frac{b}{3})^\frac{3}{2}</math> by substituting <math>x = \sqrt{\frac{b}{3}}</math> back into <math>\ y = bx - x^3</math>. He finds that the equation <math>\ bx - x^3 = a</math> has a positive solution if <math>a \le 2(\frac{b}{3})^\frac{3}{2}</math>, and al-Tusi thus deduces that the equation has a positive root if <math>D = \frac{b^3}{27} - \frac{a^2}{4} \ge 0</math>, where <math>D</math> is the [[discriminant]] of the equation.<ref name=Sharaf/> |
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==Applied mathematics== |
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===Geometric art and architecture=== |
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{{Main|Arabesque|Girih tiles|Islamic art|Islamic architecture}} |
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[[Geometry|Geometric]] artwork in the form of the [[Arabesque]] was not widely used in the Middle East or [[Mediterranean Basin]] until the [[Islamic Golden Age|golden age of Islam]] came into full bloom, when Arabesque became a common feature of [[Islamic art]]. [[Euclidean geometry]] as expounded on by [[Al-Abbās ibn Said al-Jawharī]] (ca. 800–860) in his ''Commentary on Euclid's Elements'', the [[trigonometry]] of [[Aryabhata]] and [[Brahmagupta]] as elaborated on by [[Muhammad ibn Mūsā al-Khwārizmī]] (ca. 780–850), and the development of [[spherical geometry]]{{Sfn|Gingerich|1986}} by [[Abū al-Wafā' al-Būzjānī]] (940–998) and [[spherical trigonometry]] by [[Al-Jayyani]] (989–1079)<ref name="MacTutor Al-Jayyani"/> for determining the [[Qibla]] and times of [[Salah]] and [[Ramadan]],{{Sfn|Gingerich|1986}} all served as an impetus for the art form that was to become the Arabesque. |
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Recent 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>{{Citation |
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| author = Peter J. Lu and Paul J. Steinhardt |
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| year = 2007 |
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| title = Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture |
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| journal = [[Science (journal)|Science]] |
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| volume = 315 |
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| pages = 1106–1110 |
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| url = http://peterlu.org/content/decagonal-and-quasi-crystalline-tilings-medieval-islamic-architecture |
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|format=PDF| doi = 10.1126/science.1135491 |
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| pmid = 17322056 |
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| issue = 5815 |
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| postscript = . |
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}}</ref><ref>[http://www.physics.harvard.edu/~plu/publications/Science_315_1106_2007_SOM.pdf Supplemental figures]{{dead link|date=February 2011}}</ref> |
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===Mathematical astronomy=== |
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{{Main|Islamic astronomy|Zij}} |
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An impetus behind mathematical [[astronomy]] came from Islamic religious observances, which presented a host of problems in mathematical astronomy, particularly in [[spherical geometry]]. In solving these religious problems the Islamic scholars went far beyond the Greek mathematical methods.{{Sfn|Gingerich|1986}} For example, predicting just when the crescent moon would become visible is a special challenge to Islamic mathematical astronomers. Although [[Ptolemy]]'s theory of the complex lunar motion was tolerably accurate near the time of the new moon, it specified the moon's path only with respect to the [[ecliptic]]. 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]] and the time of [[Salah]] are the reasons which led to Muslims developing spherical geometry. Solving any of these problems involves finding the unknown sides or angles of a triangle on the [[celestial sphere]] from the known sides and angles. A way of finding the time of day, for example, is to construct a triangle whose [[Vertex (geometry)|vertices]] are the [[zenith]], the north [[celestial pole]], and the sun's position. The observer must know the altitude of the sun and that of the pole; the former can be observed, and the latter is equal to the observer's [[latitude]]. The time is then given by the angle at the intersection of the [[Meridian (astronomy)|meridian]] (the [[Arc (geometry)|arc]] through the zenith and the pole) and the sun's hour circle (the arc through the sun and the pole).{{Sfn|Gingerich|1986}}<ref name=Tabatabai/> |
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The ''[[Zij]]'' treatises were astronomical books that tabulated the parameters used for astronomical calculations of the positions of the Sun, Moon, stars, and planets. Their principal contributions to mathematical astronomy reflected improved trigonometrical, computational and observational techniques.<ref>Kennedy, ''Islamic Astronomical Tables'', p. 51</ref><ref>Benno van Dalen, PARAMS (Database of parameter values occurring in Islamic astronomical sources), [http://user.uni-frankfurt.de/%7Edalen/params.htm "General background of the parameter database"]</ref> The ''[[Zij]]'' books were extensive, and typically included materials on [[chronology]], geographical [[latitude]]s and [[longitude]]s, [[star]] tables, [[Trigonometry#Calculating trigonometric functions|trigonometrical functions]], functions in [[spherical astronomy]], the [[equation of time]], planetary motions, computation of [[eclipses]], tables for first visibility of the [[New moon|lunar crescent]], astronomical and/or [[Islamic astrology|astrological]] computations, and instructions for astronomical calculations using [[Epicycle|epicyclic]] [[geocentric]] models.<ref>Kennedy, ''Islamic Astronomical Tables'', pp. 17–23</ref> Some ''zījes'' go beyond this traditional content to explain or prove the theory or report the observations from which the tables were computed.<ref>Kennedy, ''Islamic Astronomical Tables'', p. 1</ref> |
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In [[observational astronomy]], [[Muhammad ibn Mūsā al-Khwārizmī]]'s ''Zij al-Sindh'' (830) contains trigonometric tables for the movements of the sun, the moon and the five planets known at the time.<ref>{{Harv|Dallal|1999|p=163}}</ref> [[Al-Farghani]]'s ''A compendium of the science of stars'' (850) corrected [[Ptolemy]]'s ''[[Almagest]]'' and gave revised values for the obliquity of the [[ecliptic]], the precessional movement of the [[apogee]]s of the sun and the moon, and the circumference of the earth.<ref>{{Harv|Dallal|1999|p=164}}</ref> [[Muhammad ibn Jābir al-Harrānī al-Battānī]] (853–929) discovered that the direction of the Sun's [[Orbital eccentricity|eccentric]] was changing,<ref>{{Harv|Singer|1959|p=151}} ([[cf.]] {{Harv|Zaimeche|2002}})</ref> and studied the times of the [[new moon]], lengths for the [[solar year]] and [[sidereal year]], prediction of [[eclipse]]s, and the phenomenon of [[parallax]].<ref>{{Harv|Wickens|1976|}} ([[cf.]] {{Harv|Zaimeche|2002}})</ref> Around the same time, Yahya Ibn Abi Mansour wrote the ''Al-Zij al-Mumtahan'', in which he completely revised the ''Almagest'' values.<ref>{{citation |
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|title=23rd Annual Conference on the History of Arabic Science |
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|date=October 2001 |
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|publisher=[[Aleppo]], [[Syria]]}} ([[cf.]] {{Harv|Zaimeche|2002}})</ref> In the 10th century, [[Abd al-Rahman al-Sufi]] (Azophi) carried out observations on the [[star]]s and described their [[position (vector)|position]]s, [[apparent magnitude|magnitude]]s, brightness, and [[colour]] and drawings for each constellation in his ''[[Book of Fixed Stars]]'' (964). [[Ibn Yunus]] observed more than 10,000 entries for the sun's position for many years using a large [[astrolabe]] with a diameter of nearly 1.4 meters. His observations on [[eclipse]]s were still used centuries later in [[Simon Newcomb]]'s investigations on the motion of the moon, while his other observations inspired [[Laplace]]'s ''Obliquity of the Ecliptic'' and ''Inequalities of Jupiter and Saturn's''.<ref name=Zaimeche>{{Harv|Zaimeche|2002}}</ref> |
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In the late 10th century, [[Abu-Mahmud al-Khujandi]] accurately computed the [[axial tilt]] to be 23°32'19" (23.53°),<ref>{{Citation |
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|first=Richard P. |
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|last=Aulie |
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|date=March 1994 |
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|title=Al-Ghazali Contra Aristotle: An Unforeseen Overture to Science In Eleventh-Century Baghdad |
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|journal=Perspectives on Science and Christian Faith |
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|volume=45 |
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|pages=26–46}} ([[cf.]] {{cite web |
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|url=http://www.1001inventions.com/index.cfm?fuseaction=main.viewSection&intSectionID=441 |
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|title=References |
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|publisher=1001 Inventions |
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|accessdate=2008-01-22}})</ref> which was a significant improvement over the Greek and Indian estimates of 23°51'20" (23.86°) and 24°,<ref>{{Harv|Saliba|2007}}</ref> and still very close to the modern measurement of 23°26' (23.44°). In 1006, the [[Egypt]]ian astronomer [[Ali ibn Ridwan]] observed [[SN 1006]], the brightest [[supernova]] in recorded history, and left a detailed description of the temporary star. He says that the object was two to three times as large as the disc of [[Venus]] and about one-quarter the brightness of the [[Moon]], and that the star was low on the southern horizon. In 1031, [[al-Biruni]]'s ''Canon Mas’udicus'' introduced the mathematical technique of analysing the [[acceleration]] of the planets, and first states that the motions of the [[Apsis|solar apogee]] and the [[precession]] are not identical. Al-Biruni also discovered that the distance between the Earth and the Sun is larger than [[Ptolemy]]'s estimate, on the basis that Ptolemy disregarded annular eclipses.<ref>{{Citation |
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|last=Saliba |
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|first=George |
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|authorlink=George Saliba |
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|year=1980 |
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|contribution=Al-Biruni |
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|editor-last=Strayer |
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|editor-first=Joseph |
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|title=Dictionary of the Middle Ages |
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|volume=2 |
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|publisher=[[Charles Scribner's Sons]], New York |
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|page=249}}</ref> |
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During the "[[Maragheh observatory|Maragha Revolution]]" of the 13th and 14th centuries, Muslim astronomers realized that astronomy should aim to describe the behavior of [[Physical body|physical bodies]] in mathematical language, and should not remain a mathematical [[hypothesis]], which would only save the [[phenomena]]. The Maragha astronomers also realized that the [[On the Heavens|Aristotelian]] view of [[motion (physics)|motion]] in the universe being only [[Circular motion|circular]] or [[Linear motion|linear]] was not true, as the [[Tusi-couple]] showed that linear motion could also be produced by applying circular motions only.<ref>{{Harv|Saliba|1994b|pp=245, 250, 256–257}}</ref> Unlike the ancient [[Greek astronomy|Greek and Hellenistic astronomers]] who were not concerned with the coherence between the mathematical and physical principles of a planetary theory, Islamic astronomers insisted on the need to match the mathematics with the real world surrounding them,<ref>{{citation |
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|first=George |
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|last=Saliba |
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|author-link=George Saliba |
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|date=Autumn 1999 |
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|title=Seeking the Origins of Modern Science? |
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|journal=BRIIFS |
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|volume=1 |
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|issue=2 |
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|url=http://www.riifs.org/review_articles/review_v1no2_sliba.htm |
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|accessdate=2008-01-25}}</ref> which gradually evolved from a reality based on [[Aristotelian physics]] to one based on an empirical and mathematical [[physics]] after the work of [[Ibn al-Shatir]]. The Maragha Revolution was thus characterized by a shift away from the philosophical foundations of [[On the Heavens|Aristotelian cosmology]] and [[Ptolemaic astronomy]] and towards a greater emphasis on the empirical observation and mathematization of astronomy and of [[nature]] in general, as exemplified in the works of Ibn al-Shatir, [[Ali Qushji]], [[al-Birjandi]] and al-Khafri.<ref>{{Harv|Saliba|1994b|pp=42 & 80}}</ref><ref>{{citation |
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|first=Ahmad |
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|last=Dallal |
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|year=2001–2002 |
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|title=The Interplay of Science and Theology in the Fourteenth-century Kalam |
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|publisher=From Medieval to Modern in the Islamic World, Sawyer Seminar at the [[University of Chicago]] |
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|url=http://humanities.uchicago.edu/orgs/institute/sawyer/archive/islam/dallal.html |
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|accessdate=2008-02-02}}</ref><ref>{{Harv|Huff|2003|pp=217–218}}</ref> In particular, Ibn al-Shatir's [[Geocentrism|geocentric model]] was mathematically identical to the later [[Copernican heliocentrism|heliocentric Copernical model]].<ref>{{Harv|Saliba|1994b|pp=254, 256–257}}</ref> |
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===Mathematical geography and geodesy=== |
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[[File:Al-Biruni Afghan stamp.jpg|thumb|right|[[Abū Rayhān al-Bīrūnī]] was a [[polymath]] who is considered a pioneer in [[Islamic geography|mathematical geography]] and [[geodesy]].]] |
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{{Main|Islamic geography}} |
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The Muslim scholars, who held to the [[spherical Earth]] theory, used it in an impeccably Islamic manner, to calculate the distance and direction from any given point on the earth to [[Mecca]]. This determined the [[Qibla]], or Muslim direction of prayer. Muslim mathematicians developed [[spherical trigonometry]] which was used in these calculations.<ref>David A. King, ''Astronomy in the Service of Islam'', (Aldershot (U.K.): Variorum), 1993.</ref> |
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Around 830, Caliph [[al-Ma'mun]] commissioned a group of astronomers to measure the distance from Tadmur ([[Palmyra]]) to [[Ar Raqqah|al-Raqqah]], in modern [[Syria]]. They found the cities to be separated by one degree of latitude and the distance between them to be 66 2/3 miles and thus calculated the Earth's circumference to be 24,000 miles.<ref>[http://cosmos.bodley.ox.ac.uk/hms/home.php ''Gharā'ib al-funūn wa-mulah al-`uyūn'' (The Book of Curiosities of the Sciences and Marvels for the Eyes)], 2.1 "On the mensuration of the Earth and its division into seven climes, as related by Ptolemy and others," (ff. 22b–23a)</ref> Another estimate given by [[Al-Farghānī]] was 56 2/3 Arabic miles per degree, which corresponds to 111.8 km per degree and a circumference of 40,248 km, very close to the currently modern values of 111.3 km per degree and 40,068 km circumference, respectively.<ref>Edward S. Kennedy, ''Mathematical Geography'', pp. 187–188, in {{Harv|Rashed|Morelon|1996|pp=185–201}}</ref> |
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In mathematical [[geography]], [[Abū Rayhān al-Bīrūnī]], around 1025, was the first to describe a polar equi-[[azimuthal equidistant projection]] of the [[celestial sphere]].<ref>David A. King (1996), "Astronomy and Islamic society: Qibla, gnomics and timekeeping", in Roshdi Rashed, ed., ''[[Encyclopedia of the History of Arabic Science]]'', Vol. 1, pp. 128–184 [153]. [[Routledge]], London and New York.</ref> He was also regarded as the most skilled when it came to mapping [[City|cities]] and measuring the distances between them, which he did for many cities in the Middle East and western [[Indian subcontinent]]. He often combined astronomical readings and mathematical equations, in order to develop methods of pin-pointing locations by recording degrees of [[latitude]] and [[longitude]]. He also developed similar techniques when it came to measuring the heights of [[mountain]]s, depths of [[valley]]s, and expanse of the [[horizon]], in ''The Chronology of the Ancient Nations''. He also discussed [[human geography]] and the [[planetary habitability]] of the [[Earth]]. He hypothesized that roughly a quarter of the Earth's surface is habitable by [[human]]s, and also argued that the shores of Asia and Europe were "separated by a vast sea, too dark and dense to navigate and too risky to try" in reference to the Atlantic Ocean and Pacific Ocean.{{Citation needed|date=June 2010}} |
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[[Abū Rayhān al-Bīrūnī]] is considered the father of [[geodesy]] for his important contributions to the field,<ref name=Ahmed>Akbar S. Ahmed (1984). "Al-Beruni: The First Anthropologist", ''RAIN'' '''60''', pp. 9–10.</ref><ref>H. Mowlana (2001). "Information in the Arab World", ''Cooperation South Journal'' '''1'''.</ref> along with his significant contributions to geography and geology. At the age of 17, al-Biruni calculated the [[latitude]] of Kath, [[Khwarazm]], using the maximum altitude of the Sun. Al-Biruni also solved a complex [[Geodesy|geodesic]] equation in order to accurately compute the [[Earth]]'s [[circumference]], which were close to modern values of the Earth's circumference.<ref>James S. Aber (2003). Alberuni calculated the Earth's circumference at a small town of Pind Dadan Khan, District Jhelum, Punjab, Pakistan. [http://academic.emporia.edu/aberjame/histgeol/biruni/biruni.htm Abu Rayhan al-Biruni], [[Emporia State University]].</ref> His estimate of 6,339.9 km for the [[Earth radius]] was only 16.8 km less than the modern value of 6,356.7 km. In contrast to his predecessors who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, al-Biruni developed a new method of using [[trigonometric]] calculations based on the angle between a [[plain]] and [[mountain]] top which yielded more accurate measurements of the Earth's circumference and made it possible for it to be measured by a single person from a single location.<ref>Lenn Evan Goodman (1992), ''Avicenna'', p. 31, [[Routledge]], ISBN 041501929X.</ref><ref>{{citation |
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|title=Applicable Problems in History of Mathematics: Practical Examples for the Classroom |
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|author=Behnaz Savizi |
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|journal=Teaching Mathematics and Its Applications |
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|volume=26 |
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|issue=1 |
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|year=2007 |
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|pages=45–50 |
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|publisher=[[Oxford University Press]] |
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|doi=10.1093/teamat/hrl009}} ([[cf.]] {{cite web |
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|title=Applicable Problems in History of Mathematics; Practical Examples for the Classroom |
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|author=Behnaz Savizi |
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|publisher=[[University of Exeter]] |
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|url=http://people.exeter.ac.uk/PErnest/pome19/Savizi%20-%20Applicable%20Problems.doc |
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|accessdate=2010-02-21}})</ref><ref>{{citation |
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|title=Geometry Activities from Many Cultures |
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|author=Beatrice Lumpkin |
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|publisher=Walch Publishing |
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|year=1997 |
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|isbn=0825132851 |
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|pages=60, 112–113}} [http://books.google.co.uk/books?id=Xpr_rBdY9PwC&pg=RA1-PA60&lpg=RA1-PA60&dq=biruni+mountain+earth&source=bl&ots=tLz9hhn-2h&sig=aWdOXExGAuxWabQYYLXyLZeUEc4&hl=en&ei=W6yAS7LLMNH__AawyrH8Bg&sa=X&oi=book_result&ct=result&resnum=7&ved=0CBoQ6AEwBg#v=onepage&q=biruni%20mountain%20earth&f=false books.google.co.uk]</ref> |
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===Mathematical physics=== |
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{{Main|Islamic physics|Book of Optics}} |
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[[Ibn al-Haytham]]'s work on geometric [[optics]], particularly [[catoptrics]], in "Book V" of the ''[[Book of Optics]]'' (1021) contains the important mathematical problem known as "Alhazen's problem" (''Alhazen'' is the [[Latin]]ized name of Ibn al-Haytham). It comprises drawing lines from two points in the plane of a circle meeting at a point on the [[circumference]] and making equal angles with the normal at that point. This leads to an [[Quartic equation|equation of the fourth degree]]. This eventually led Ibn al-Haytham to derive the earliest formula for the sum of the [[fourth power]]s, and using an early [[Mathematical proof|proof]] by [[mathematical induction]], he developed a method for determining the general formula for the sum of any [[integral]] [[Exponentiation|powers]], which was fundamental to the development of [[infinitesimal]] and [[integral]] [[calculus]].<ref name=autogenerated1>Victor J. Katz (1995). "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '''68''' (3), pp. 163–174.</ref> Ibn al-Haytham eventually solved "Alhazen's problem" using [[conic section]]s and a geometric proof, but Alhazen's problem remained influential in Europe, when later mathematicians such as [[Christiaan Huygens]], [[James Gregory (mathematician)|James Gregory]], [[Guillaume de l'Hôpital]], [[Isaac Barrow]], and many others, attempted to find an algebraic solution to the problem, using various methods, including [[analytic geometry|analytic methods of geometry]] and derivation by [[complex number]]s.<ref name=autogenerated3>John D. Smith (1992). "The Remarkable Ibn al-Haytham", ''The Mathematical Gazette'' '''76''' (475), pp. 189–198.</ref> Mathematicians were not able to find an algebraic solution to the problem until the end of the 20th century.<ref name=Steffens>Bradley Steffens (2006), ''Ibn al-Haytham: First Scientist'', [http://www.ibnalhaytham.net/custom.em?pid=673906 Chapter Five], Morgan Reynolds Publishing, ISBN 1599350246</ref> |
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Ibn al-Haytham also produced tables of corresponding [[Angle of incidence|angles of incidence]] and [[refraction]] of [[light]] passing from one medium to another show how closely he had approached discovering the [[Snell's law|law of constancy of ratio of sines]], later attributed to [[Willebrord Snellius|Snell]]. He also correctly accounted for [[twilight]] being due to [[atmospheric refraction]], estimating the Sun's depression to be 19 degrees below the [[horizon]] during the commencement of the phenomenon in the mornings or at its termination in the evenings.<ref name=Sarton>[[George Sarton]], ''Introduction to the History of Science'', "The Time of Al-Biruni"</ref> |
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Ibn al-Haytham systematically endeavoured to mathematize physics in the context of his experimental research and controlled testing, which was oriented by geometric models of the structural mathematical principles that governed physical phenomena, particularly in relation to the explication of the behaviour and nature of vision and light.<ref>{{citation |
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|title=A Philosophical Perspective on Alhazen’s ''Optics'' |
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|first=Nader |
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|last=El-Bizri |
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|author-link=Nader El-Bizri |
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|journal=[[Arabic Sciences and Philosophy: A Historical Journal]] |
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|volume=15 |
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|issue=2 |
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|year= 2005 |
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|pages=189–218}}</ref> Ibn al-Haytham also advanced in his ''Discourse on Place'' (''Qawl fi al-makan'') a geometrical understanding of place as ''mathematical space'' that is akin to the 17th century conceptions of ''extensio'' by Descartes and ''analysis situs'' by Leibniz. Ibn al-Haytham established his geometrical thesis about ''place as space'' in the context of his mathematical refutation of the Aristotelian physical definition of ''topos'' as a ''boundary surface of a containing body'' (as argued in Book delta [IV] of Aristotle's ''Physics'').<ref>{{citation |
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|title=In Defence of the Sovereignty of Philosophy: al-Baghdadi’s Critique of Ibn al-Haytham’s Geometrisation of Place |
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|first=Nader |
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|last=El-Bizri |
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|author-link=Nader El-Bizri |
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|journal=[[Arabic Sciences and Philosophy (journal) |
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|Arabic Sciences and Philosophy: A Historical Journal]] |
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|volume=17 |
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|issue=1 |
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|year= 2007 |
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|pages=57–80}}</ref> |
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[[Abū Rayhān al-Bīrūnī]] (973–1048), and later [[al-Khazini]] (fl. 1115–1130), were the first to apply [[experiment]]al [[scientific method]]s to the [[statics]] and [[Dynamics (physics)|dynamics]] fields of [[mechanics]], particularly for determining [[specific weight]]s, such as those based on the theory of [[Beam balance|balances]] and [[Weighing scale|weighing]]. Muslim physicists applied the mathematical theories of [[ratio]]s and [[infinitesimal]] techniques, and introduced [[algebra]]ic and fine [[calculation]] techniques into the field of statics.<ref name=Rozhanskaya-642>Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", p. 642, in {{Harv|Morelon|Rashed|1996|pp=614–642}}</ref> |
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Abu 'Abd Allah Muhammad ibn Ma'udh, who lived in [[Al-Andalus]] during the second half of the 11th century, wrote a work on optics later translated into Latin as ''Liber de crepisculis'', which was mistakenly attributed to Alhazen. This was a "short work containing an estimation of the angle of depression of the sun at the beginning of the morning [[twilight]] and at the end of the evening twilight, and an attempt to calculate on the basis of this and other data the height of the atmospheric moisture responsible for the refraction of the sun's rays." Through his experiments, he obtained the accurate value of 18°, which comes close to the modern value.<ref>{{citation |
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|title=The Authorship of the Liber de crepusculis, an Eleventh-Century Work on Atmospheric Refraction |
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|first=A. I. |
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|last=Sabra |
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|author-link=A. I. Sabra |
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|journal=[[Isis (journal)|Isis]] |
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|volume=58 |
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|issue=1 |
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|date=Spring 1967 |
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|pages=77–85 [77] |
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|doi=10.1086/350185}}</ref> |
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In 1574, [[Taqi al-Din Muhammad ibn Ma'ruf|Taqi al-Din]] estimated that the [[star]]s are millions of kilometres away from the [[Earth]] and that the [[speed of light]] is constant, that if light had come from the eye, it would take too long for light "to travel to the star and come back to the eye. But this is not the case, since we see the star as soon as we open our eyes. Therefore the light must emerge from the object not from the eyes."{{Citation needed|date=May 2010}} |
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==Other fields== |
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===Cryptography=== |
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[[File: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]].]] |
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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 [[cipher]]s (i.e. crypanalysis by frequency analysis).<ref>Singh, Simon. ''The Code Book''. pp. 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.{{Citation needed|date=May 2010}} Al-Kindi also had knowledge of [[polyalphabetic cipher]]s centuries before [[Leon Battista Alberti]]. Al-Kindi's book also introduced the classification of ciphers, developed Arabic phonetics and syntax, and described the use of several statistical techniques for cryptoanalysis. This book apparently antedates other cryptology references by several centuries, and it also predates writings on [[probability]] and [[statistics]] by [[Blaise Pascal|Pascal]] and [[Fermat]] by nearly eight centuries.<ref>[[Ibrahim A. Al-Kadi]] (April 1992), "The origins of cryptology: The Arab contributions”, ''[[Cryptologia]]'' '''16''' (2): pp. 97–126</ref> |
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[[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. |
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===Mathematical induction=== |
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The first known [[Mathematical proof|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'', pp. 255–259, [[Addison-Wesley]], ISBN 0321016181: {{quote|"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."}}</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: {{quote|"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."}}</ref> |
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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), pp. 163–174: |
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{{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), pp. 255–259.</ref> |
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[[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: {{quote|"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."}}</ref> |
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== Astrolabe == |
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The [[astrolabe]] is a mathematical tool that could be used to solve all the standard problems of spherical astronomy in five different ways. |
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== Biographies == |
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;{{transl|ar|ALA|[[Al-Ḥajjāj ibn Yūsuf ibn Maṭar]]}} (786–833) |
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:Al-Ḥajjāj translated [[Euclid]]'s ''[[Euclid's Elements|Elements]]'' into Arabic. |
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;{{transl|ar|ALA|[[Muḥammad ibn Mūsā al-Khwārizmī]]}} (c. 780 [[Khwarezm]]/[[Baghdad]] – c. 850 Baghdad) <!-- JPH MT AMFB --> |
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: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 ''[[The Compendious Book on Calculation by Completion and Balancing|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. |
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;{{transl|ar|ALA|[[Al-ʿAbbās ibn Saʿid al-Jawharī]]}} (c. 800 Baghdad? – c. 860 Baghdad?)<!-- MT --> |
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: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 (mathematics)|proposition]]s and an attempted [[Mathematical proof|proof]] of the [[parallel postulate]]. |
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;{{transl|ar|ALA|[[ʿAbd al-Hamīd ibn Turk]]}} ([[floruit|fl.]] 830 Baghdad) <!-- JPH --> |
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:Ibn Turk wrote a work on [[algebra]] of which only a chapter on the solution of [[quadratic equations]] has survived. |
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;{{transl|ar|ALA|[[Yaʿqūb ibn Isḥāq al-Kindī]]}} (c. 801 [[Kufa]] – 873 Baghdad) <!-- JPH MT AMFB --> |
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: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]]. |
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;[[Hunayn ibn Ishaq]] (808 [[Al-Hirah]] – 873 Baghdad) <!-- MT --> |
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: 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]]. |
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;{{transl|ar|ALA|[[Banū Mūsā]]}} (c. 800 Baghdad – 873+ Baghdad) <!-- JPH AMFB --> |
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:The Banū Mūsā were 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''. |
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;[[Al-Mahani]] <!-- MT --> |
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;[[Ahmed ibn Yusuf]] <!-- MT --> |
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;[[Thabit ibn Qurra]] (Syria-Iraq, 835–901) <!-- JPH MT --> |
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;[[Al-Hashimi]] (Iraq? ca. 850–900) <!-- JPH --> |
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;{{transl|ar|ALA|[[Muḥammad ibn Jābir al-Ḥarrānī al-Battānī]]}} (c. 853 [[Harran]] – 929 [[Qasr al-Jiss]] near [[Samarra]]) <!-- JPH (Syria, ca. 900) MT --> |
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;[[Abu Kamil]] (Egypt? ca. 900) <!-- JPH MT --> |
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;[[Sinan ibn Tabit]] (ca. 880 – 943) <!-- MT --> |
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;[[Al-Nayrizi]] <!-- MT --> |
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;[[Ibrahim ibn Sinan]] (Iraq, 909–946) <!-- JPH MT --> |
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;[[Al-Khazin]] (Iraq-Iran, ca. 920–980) <!-- JPH MT --> |
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;[[Al-Karabisi]] (Iraq? 10th century?) <!-- JPH --> |
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;[[Ikhwan al-Safa']] (Iraq, first half of 10th century) <!-- JPH --> |
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: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. |
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;[[Al-Uqlidisi]] (Iraq-Iran, 10th century) <!-- JPH MT --> |
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;[[Al-Saghani]] (Iraq-Iran, ca. 940–1000) <!-- JPH --> |
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;{{transl|ar|ALA|[[Abū Sahl al-Qūhī]]}} (Iraq-Iran, ca. 940–1000) <!-- JPH MT --> |
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;[[Al-Khujandi]] <!-- MT --> |
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;{{transl|ar|ALA|[[Abū al-Wafāʾ al-Būzjānī]]}} (Iraq-Iran, ca. 940–998) <!-- JPH MT --> |
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;[[Ibn Sahl]] (Iraq-Iran, ca. 940–1000) <!-- JPH --> |
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;[[Al-Sijzi]] (Iran, ca. 940–1000) <!-- JPH MT --> |
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;[[Labana of Cordoba]] ([[Al-Andalus]], ca. 10th century) |
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: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|author=[http://www.worldcat.org/oclc/252003647&referer=brief_results Samuel Parsons Scott]|title=History of the Moorish Empire in Europe|publisher=J.B. Lippincott Company|location=Philadelphia & London|year=1904|edition=1|volume=3|page=447|chapter=xxix: Moorish art in southern Europe|isbn=978-1402144851 (published in 2004)|oclc=3522061|url=http://www.archive.org/stream/historyofmoorish03scotuoft#page/447/mode/1up|accessdate=2010-01-15}}</ref> |
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;[[Ibn Yunus]] (Egypt, ca. 950–1010) <!-- JPH MT --> |
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;[[Abu Nasr ibn `Iraq]] (Iraq-Iran, ca. 950–1030) <!-- JPH MT --> |
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;[[Kushyar ibn Labban]] (Iran, ca. 960–1010) <!-- JPH --> |
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;[[Al-Karaji]] (Iran, ca. 970–1030) <!-- JPH MT --> |
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;[[Ibn al-Haytham]] (Iraq-Egypt, ca. 965–1040) <!-- JPH MT --> |
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;{{transl|ar|ALA|[[Abū al-Rayḥān al-Bīrūnī]]}} (September 15, 973 in Kath, [[Khwarezm]] – December 13, 1048 in [[Gazna]]) <!-- JPH MT --> |
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;[[Ibn Sina]] (also known as Avicenna) ([[Al-Andalus]])<!-- MT --> |
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;[[Ibn Tahir al-Baghdadi|al-Baghdadi]] <!-- MT --> |
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;[[Al-Nasawi]] <!-- MT --> |
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;[[Al-Jayyani]] ([[Al-Andalus]], ca. 1030–1090) <!-- JPH MT --> |
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;[[Ibn al-Zarqalluh]] (Azarquiel, al-Zarqali) ([[Al-Andalus]], ca. 1030–1090) <!-- JPH --> |
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;[[Al-Mu'taman ibn Hud]] ([[Al-Andalus]], ca. 1080) <!-- JPH --> |
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;[[al-Khayyam]] (Iran, ca. 1050–1130) <!-- JPH MT --> |
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;{{transl|ar|ALA|[[Ibn Yaḥyā al-Maghribī al-Samawʾal]]}} (ca. 1130, [[Baghdad]] – c. 1180, [[Maragha]]) <!-- MT --> |
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;[[Al-Hassar]] (ca. 12th century, [[Morocco]]) |
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:Developed the modern [[mathematical notation]] for [[Fraction (mathematics)|fractions]] and the digits he uses for the ''ghubar'' numerals also closely resembles modern Western [[Arabic numerals]]. |
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;[[Ibn al-Yasamin]] (ca. 12th century, [[Morocco]]) |
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:The son of a [[Berber people|Berber]] father and [[black African]] mother, he was the first to develop a mathematical notation for algebra since the time of [[Brahmagupta]]. |
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;{{transl|ar|ALA|[[Sharaf al-Dīn al-Ṭūsī]]}} (Iran, ca. 1150–1215) <!-- JPH MT --> |
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;[[Ibn Munim]] ([[Morocco]], ca. 1210) <!-- JPH --> |
|||
;[[Ibn al-Banna al-Marrakushi]] ([[Morocco]], 1250–1320) <!-- JPH --> |
|||
;{{transl|ar|ALA|[[Naṣīr al-Dīn al-Ṭūsī]]}} (18 February 1201 in [[Tous, Iran|Tus]], [[Greater Khorasan|Khorasan]] – 26 June 1274 in [[Kadhimain]] near [[Baghdad]]) <!-- JPH MT --> |
|||
;{{transl|ar|ALA|[[Muḥyi al-Dīn al-Maghribī]]}} (c. 1220 Spain – c. 1283 [[Maragha]]) <!-- MT --> |
|||
;{{transl|ar|ALA|[[Shams al-Dīn al-Samarqandī]]}} (c. 1250 [[Samarqand]] – c. 1310) <!-- MT --> |
|||
;[[Ibn Baso]] ([[Al-Andalus]], ca. 1250–1320) <!-- JPH --> |
|||
;[[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]] – 1436 Samarkand) <!-- MT --> |
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;{{transl|ar|ALA|[[Jamshīd al-Kāshī]]}} (Iran, Uzbekistan, ca. 1420) <!-- JPH MT --> |
|||
;[[Ulugh Beg]] (Iran, Uzbekistan, 1394–1449) <!-- JPH MT --> |
|||
;[[Al-Umawi]] ([[Al-Andalus]])<!-- MT --> |
|||
;[[Abū al-Hasan ibn Alī al-Qalasādī]] ([[Al-Andalus]], 1412–1482) <!-- JPH MT --> |
|||
:Last major medieval [[Arab]] mathematician. Pioneer of [[Mathematical notation|symbolic algebra]]. |
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==See also== |
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{{Multicol}} |
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*[[Dharmic contribution to science in the caliphate]] |
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*[[Inventions in the Muslim world]] |
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*[[Islamic Golden Age]] |
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{{Multicol-break}} |
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*[[Islamic science]] |
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*[[Latin translations of the 12th century]] |
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*[[List of Muslim mathematicians]] |
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{{Multicol-end}} |
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==References== |
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===Footnotes=== |
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{{Reflist|group="note"}} |
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=== Notes === |
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{{Reflist|2}} |
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=== Sources === |
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;Books |
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{{refbegin}} |
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<!--REFBEGIN: CITED Books in alphabetical order--> |
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* {{cite book |ref={{SfnRef|Berggren|2007}} |last=Berggren |first=J. Lennart |editor=Victor J. Katz |title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook | chapter=Mathematics in Medieval Islam | edition=Second |year=2007 |publisher=[[Princeton University Press|Princeton University]] |location=Princeton, New Jersey |isbn=9780691114859 }} |
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* {{cite book |ref={{SfnRef|Boyer|1991}} |last=Boyer |first=Carl Benjamin |author-link=Carl Benjamin Boyer |title=A History of Mathematics | chapter=Greek Trigonometry and Mensuration, and The Arabic Hegemony | edition=Second |year=1991 |publisher=[[John Wiley & Sons]] |location=New York City |isbn=0471543977 }} |
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* {{cite book |ref={{SfnRef|Haq|1996}} |last1=Nasr |first1=Seyyed Hossein |author1-link=Seyyed Hossein Nasr |last2=Leaman |first2=Oliver |last3=Haq |first3=Syed Nomanul |title=Routledge History of World Philosophies |volume=1: History of Islamic Philosophy |chapter=4: The Indian and Persian background |year=1996 |publisher=[[Routledge]] |location=London |isbn=0415131596 }} |
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* {{cite book |ref={{SfnRef|Rosenthal, Dawood, Khaldun|1967}} |last1=Rosenthal |first1=Franz |last2=Dawood |first2=N. J. |last3=Ibn |first3=Khaldun |author3-link=Ibn Khaldun |title=The Muqaddimah: An Introduction to History |year=1967 |publisher=[[Princeton University Press|Princeton University]] |location=Princeton, New Jersey |isbn=0691017549 }} |
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<!--REFEND--> |
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{{refend}} |
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;Journals and magazines |
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{{refbegin}} |
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<!--REFBEGIN: CITED Journals/magazines... in alphabetical order--> |
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* {{cite journal |ref={{SfnRef|Gandz|1931}} |last=Gandz |first=Solomon |title=The Origin of the Ghubār Numerals, or the Arabian Abacus and the Articuli |volume=16 |issue=2 |pages=393–424 |journal=Isis |publisher=[[University of Chicago Press|University of Chicago]] |location=Chicago |year=1931 |month=November |doi=10.1086/346615 }} |
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* {{cite journal |ref={{SfnRef|Gandz|1936}} |last=Gandz |first=Solomon |title=1936 |volume= |issue= |pages=263–277 |journal=[[Osiris (journal)|Osiris]] |publisher=[[University of Chicago Press|University of Chicago]] |location=Chicago |year=1936 |doi=10.1086/368492 }} |
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* {{cite journal |ref={{SfnRef|Gandz|1938}} |last=Gandz |first=Solomon |title=The Algebra of Inheritance: A Rehabilitation of Al-Khuwarizmi |volume=5 |issue= |pages=319–391 |journal=[[Osiris (journal)|Osiris]] |publisher=[[University of Chicago Press|University of Chicago]] |location=Chicago |year=1938 |doi=10.1086/368492 }} |
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* {{cite journal |ref={{SfnRef|Gingerich|1986}} |last=Gingerich |first=Owen |title=Islamic Astronomy |volume=254 |issue=10 |page=74 |publisher=[[Scientific American]] |year=1986 |mont=April |url=http://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm |accessdate=2011-02-14 }} |
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* {{cite journal |ref={{SfnRef|Schumpeter, Moss|1994}} |last1=Schumpeter |first1=Joseph A. |last2=Moss |first2=Laurence S. |title=Historian of Economics: Selected Papers from the History of Economics Society Conference |page=64 |publisher=[[Routledge]] |location=London |year=1994 |isbn=041513353X |accessdate=2011-02-14 }} |
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<!--REFEND--> |
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{{refend}} |
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;Web |
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{{refbegin}} |
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<!--REFBEGIN: CITED Web sources in alphabetical order--> |
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<!--REFEND--> |
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{{refend}} |
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== Further reading== |
== Further reading== |
||
{{Refbegin|2}} |
{{Refbegin|2}} |
||
;Books on Islamic mathematics |
|||
;Bibliographies and biographies |
|||
*{{Citation|last=Berggren|first=J. Lennart|authorlink=Len Beggren|title=Episodes in the Mathematics of Medieval Islam|year=1986|publisher=Springer-Verlag|location=New York|isbn=0-387-96318-9}} |
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* [[Carl Brockelmann|Brockelmann, Carl]]. ''Geschichte der Arabischen Litteratur''. 1.–2. Band, 1.–3. Supplementband. Berlin: Emil Fischer, 1898, 1902; Leiden: Brill, 1937, 1938, 1942. |
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** Review: {{citation|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|doi=10.2307/2322777|page=567|publisher=Mathematical Association of America|last2=Berggren|first2=J. L.}} |
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* {{Citation|last=Sánchez Pérez|first=José A.|authorlink=José Augusto Sánchez Pérez|title=Biografías de Matemáticos Árabes que florecieron en España|location=Madrid|publisher=Estanislao Maestre|year=1921}} |
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** Review: {{citation|first=Jan P.|last=Hogendijk|title=''Episodes in the Mathematics of Medieval Islam'' by J. Lennart Berggren|journal=Journal of the American Oriental Society|volume=109|issue=4|year=1989|pages=697–698|doi=10.2307/604119|url=http://jstor.org/stable/604119|publisher=American Oriental Society|last2=Berggren|first2=J. L.}}) |
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* {{Citation|last=Sezgin|first=Fuat|authorlink=Fuat Sezgin|title=Geschichte Des Arabischen Schrifttums|publisher=Brill Academic Publishers|language=German|year=1997|isbn=9004020071}} |
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* |
*{{Citation|last=Daffa'|first=Ali Abdullah al-|authorlink=Ali Abdullah Al-Daffa|title=The Muslim contribution to mathematics|year=1977|publisher=Croom Helm|location=London|isbn=0-85664-464-1}} |
||
* {{Citation|last=Rashed|first=Roshdi|authorlink=Roshdi Rashed|others=Transl. by A. F. W. Armstrong|title=The Development of Arabic Mathematics: Between Arithmetic and Algebra|publisher=Springer|year=2001|isbn=0792325656}} |
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* {{Citation|first=Adolf P.|last=Youschkevitch|authorlink=Adolph Pavlovich Yushkevich|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. |
|||
;Overview |
|||
* {{Citation|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|isbn=978-2-7116-0734-1}} |
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*{{Citation|last=Berggren|first=J. Lennart|authorlink=Len Beggren|title=Episodes in the Mathematics of Medieval Islam|year=1986|publisher=Springer-Verlag|location=New York|isbn=0-387-96318-9}} (Reviewed: {{citation|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|doi=10.2307/2322777|page=567|publisher=Mathematical Association of America|last2=Berggren|first2=J. L.}}; {{citation|first=Jan P.|last=Hogendijk|title=''Episodes in the Mathematics of Medieval Islam'' by J. Lennart Berggren|journal=Journal of the American Oriental Society|volume=109|issue=4|year=1989|pages=697–698|doi=10.2307/604119|url=http://jstor.org/stable/604119|publisher=American Oriental Society|last2=Berggren|first2=J. L.}}) |
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* [[Jens Høyrup|Høyrup, Jens]]. “The Formation of «Islamic Mathematics»: Sources and Conditions”. ''Filosofi og Videnskabsteori på Roskilde Universitetscenter''. 3. Række: ''Preprints og Reprints'' 1987 Nr. 1. |
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; Chapters on Islamic mathematics |
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;Other |
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* {{cite book |ref={{SfnRef|Berggren|2007}} |last=Berggren |first=J. Lennart |editor=Victor J. Katz |title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook | chapter=Mathematics in Medieval Islam | edition=Second |year=2007 |publisher=[[Princeton University Press|Princeton University]] |location=Princeton, New Jersey |isbn=9780691114859 }} |
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*{{Citation |
*{{Citation |
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| first=Roger |
| first=Roger |
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| Line 745: | Line 24: | ||
| isbn=0471180823 |
| isbn=0471180823 |
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}} |
}} |
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; Books on Islamic science |
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*{{Citation|last=Daffa'|first=Ali Abdullah al-|authorlink=Ali Abdullah Al-Daffa|title=The Muslim contribution to mathematics|year=1977|publisher=Croom Helm|location=London|isbn=0-85664-464-1}} |
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* {{Citation|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|isbn=0471903205}} |
* {{Citation|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|isbn=0471903205}} |
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* {{Citation|first=E. S.|last=Kennedy|authorlink=Edward Stewart Kennedy|title=Studies in the Islamic Exact Sciences|year=1984|publisher=Syracuse Univ Press|isbn=0815660677}} |
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*{{Citation |
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; Books on the history of mathematics |
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|last=Eder |
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|first=Michelle |
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|year=2000 |
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|title=Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam |
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|url=http://www.math.rutgers.edu/~cherlin/History/Papers2000/eder.html |
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|publisher=[[Rutgers University]] |
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|accessdate=2008-01-23 |
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}} |
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* [[Jens Høyrup|Høyrup, Jens]]. “The Formation of «Islamic Mathematics»: Sources and Conditions”. ''Filosofi og Videnskabsteori på Roskilde Universitetscenter''. 3. Række: ''Preprints og Reprints'' 1987 Nr. 1. |
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* {{Citation|last=Joseph|first=George Gheverghese|authorlink=George Gheverghese Joseph|title=The Crest of the Peacock: Non-European Roots of Mathematics|edition=2nd|publisher=Princeton University Press|year=2000|isbn=0691006598}} (Reviewed: {{citation|first=Victor J.|last=Katz|title=''The Crest of the Peacock: Non-European Roots of Mathematics'' by George Gheverghese Joseph|journal=The College Mathematics Journal|volume=23|issue=1|year=1992|pages=82–84|doi=10.2307/2686206|url=http://jstor.org/stable/2686206|publisher=Mathematical Association of America|last2=Joseph|first2=George Gheverghese}}) |
* {{Citation|last=Joseph|first=George Gheverghese|authorlink=George Gheverghese Joseph|title=The Crest of the Peacock: Non-European Roots of Mathematics|edition=2nd|publisher=Princeton University Press|year=2000|isbn=0691006598}} (Reviewed: {{citation|first=Victor J.|last=Katz|title=''The Crest of the Peacock: Non-European Roots of Mathematics'' by George Gheverghese Joseph|journal=The College Mathematics Journal|volume=23|issue=1|year=1992|pages=82–84|doi=10.2307/2686206|url=http://jstor.org/stable/2686206|publisher=Mathematical Association of America|last2=Joseph|first2=George Gheverghese}}) |
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;Bibliographies and biographies |
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*{{Citation |
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* [[Carl Brockelmann|Brockelmann, Carl]]. ''Geschichte der Arabischen Litteratur''. 1.–2. Band, 1.–3. Supplementband. Berlin: Emil Fischer, 1898, 1902; Leiden: Brill, 1937, 1938, 1942. |
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|last=Katz |
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* {{Citation|last=Sánchez Pérez|first=José A.|authorlink=José Augusto Sánchez Pérez|title=Biografías de Matemáticos Árabes que florecieron en España|location=Madrid|publisher=Estanislao Maestre|year=1921}} |
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|first=Victor J. |
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* {{Citation|last=Sezgin|first=Fuat|authorlink=Fuat Sezgin|title=Geschichte Des Arabischen Schrifttums|publisher=Brill Academic Publishers|language=German|year=1997|isbn=9004020071}} |
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| authorlink = Victor J. Katz |
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* {{Citation|last=Suter|first=Heinrich|authorlink=Heinrich Suter|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}} |
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|year=1998 |
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|title=History of Mathematics: An Introduction |
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|publisher=[[Addison-Wesley]] |
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|isbn=0321016181 |
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|oclc=38199387 |
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}} |
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* {{Citation|first=E. S.|last=Kennedy|authorlink=Edward Stewart Kennedy|title=Studies in the Islamic Exact Sciences|year=1984|publisher=Syracuse Univ Press|isbn=0815660677}} |
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* {{MacTutor|class=HistTopics|id=Arabic_mathematics|title=Arabic mathematics: forgotten brilliance?|year=1999}} |
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* {{Citation|last=Rashed|first=Roshdi|authorlink=Roshdi Rashed|others=Transl. by A. F. W. Armstrong|title=The Development of Arabic Mathematics: Between Arithmetic and Algebra|publisher=Springer|year=2001|isbn=0792325656}} |
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* {{Citation|last=Rashed|first=Roshdi|others=Transl. by Judith Field with revision of trans. by [[Nader El-Bizri]]|title=Al-Khwarizmi:The Beginnings of Algebra|publisher=[[Saqi Books]]|year=2009|isbn=0863564305}} |
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*{{Citation |
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|last=Rozenfeld |
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|first=Boris A. |
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| authorlink = Boris A. Rozenfeld |
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|year=1988 |
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|title=A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space |
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|publisher=[[Springer Science+Business Media]] |
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|isbn=0387964584 |
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|oclc=15550634 |
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}} |
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*{{Citation |
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|last=Smith |
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|first=John D. |
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|year=1992 |
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|title=The Remarkable Ibn al-Haytham |
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|journal=The Mathematical Gazette |
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|volume=76 |
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|issue=475 |
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|pages=189–198 |
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|publisher=[[Mathematical Association]] |
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|doi=10.2307/3620392 |
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|url=http://jstor.org/stable/3620392 |
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}} |
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* {{Citation|first=Adolf P.|last=Youschkevitch|authorlink=Adolph Pavlovich Yushkevich|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. |
|||
* {{Citation|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|isbn=978-2-7116-0734-1}} |
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{{Refend}} |
{{Refend}} |
||
== External links == |
== External links == |
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{{Refbegin}} |
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* [http://www.aina.org/books/hgsptta.htm "How Greek Science Passed to the Arabs"] by De Lacy O'Leary |
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* Hogendijk, Jan P. (January 1999). [http://www.jphogendijk.nl/publ/Islamath.html ''Bibliography of Mathematics in Medieval Islamic Civilization'']. |
* Hogendijk, Jan P. (January 1999). [http://www.jphogendijk.nl/publ/Islamath.html ''Bibliography of Mathematics in Medieval Islamic Civilization'']. |
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* {{MacTutor|class=HistTopics|id=Arabic_mathematics|title=Arabic mathematics: forgotten brilliance?|year=1999}} |
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{{Refend}} |
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{{Islamic mathematics}} |
{{Islamic mathematics}} |
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{{DEFAULTSORT:Mathematics In Medieval Islam}} |
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[[Category:Islamic mathematics| ]] |
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[[Category:Islamic Golden Age]] |
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[[ca:Matemàtiques en l'islam medieval]] |
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[[es:Matemática en el islam medieval]] |
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[[fa:ریاضیات اسلامی]] |
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[[fr:Mathématiques arabes]] |
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[[ms:Matematik Islam]] |
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[[ja:アラビア数学]] |
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[[ru:Математика исламского средневековья]] |
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Revision as of 14:49, 15 February 2011
In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics or Arabic mathematics, is the mathematics developed by the Islamic civilization between 622 and 1600. Islamic science and mathematics flourished under the Islamic caliphate established across the Middle East, Central Asia, North Africa, Southern Italy, the Iberian Peninsula, and, at its peak, parts of France and India.
Further reading
- Books on Islamic mathematics
- Berggren, J. Lennart (1986), Episodes in the Mathematics of Medieval Islam, New York: Springer-Verlag, ISBN 0-387-96318-9
- Review: Toomer, Gerald J.; Berggren, J. L. (1988), "Episodes in the Mathematics of Medieval Islam", American Mathematical Monthly, 95 (6), Mathematical Association of America: 567, doi:10.2307/2322777
- Review: Hogendijk, Jan P.; Berggren, J. L. (1989), "Episodes in the Mathematics of Medieval Islam by J. Lennart Berggren", Journal of the American Oriental Society, 109 (4), American Oriental Society: 697–698, doi:10.2307/604119)
- Daffa', Ali Abdullah al- (1977), The Muslim contribution to mathematics, London: Croom Helm, ISBN 0-85664-464-1
- Rashed, Roshdi (2001), The Development of Arabic Mathematics: Between Arithmetic and Algebra, Transl. by A. F. W. Armstrong, Springer, ISBN 0792325656
- Youschkevitch, Adolf P. (1960), Die Mathematik der Länder des Ostens im Mittelalter, Berlin
{{citation}}: Unknown parameter|coauthors=ignored (|author=suggested) (help)CS1 maint: location missing publisher (link) Sowjetische Beiträge zur Geschichte der Naturwissenschaft pp. 62–160. - Youschkevitch, Adolf P. (1976), Les mathématiques arabes: VIIIe-XVe siècles, translated by M. Cazenave and K. Jaouiche, Paris: Vrin, ISBN 978-2-7116-0734-1
- Høyrup, Jens. “The Formation of «Islamic Mathematics»: Sources and Conditions”. Filosofi og Videnskabsteori på Roskilde Universitetscenter. 3. Række: Preprints og Reprints 1987 Nr. 1.
- Chapters on Islamic mathematics
- Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". In Victor J. Katz (ed.). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Second ed.). Princeton, New Jersey: Princeton University. ISBN 9780691114859.
{{cite book}}: CS1 maint: ref duplicates default (link) - Cooke, Roger (1997), "Islamic Mathematics", The History of Mathematics: A Brief Course, Wiley-Interscience, ISBN 0471180823
- Books on Islamic science
- Daffa, Ali Abdullah al-; Stroyls, J.J. (1984), Studies in the exact sciences in medieval Islam, New York: Wiley, ISBN 0471903205
- Kennedy, E. S. (1984), Studies in the Islamic Exact Sciences, Syracuse Univ Press, ISBN 0815660677
- Books on the history of mathematics
- Joseph, George Gheverghese (2000), The Crest of the Peacock: Non-European Roots of Mathematics (2nd ed.), Princeton University Press, ISBN 0691006598 (Reviewed: Katz, Victor J.; Joseph, George Gheverghese (1992), "The Crest of the Peacock: Non-European Roots of Mathematics by George Gheverghese Joseph", The College Mathematics Journal, 23 (1), Mathematical Association of America: 82–84, doi:10.2307/2686206)
- Bibliographies and biographies
- Brockelmann, Carl. Geschichte der Arabischen Litteratur. 1.–2. Band, 1.–3. Supplementband. Berlin: Emil Fischer, 1898, 1902; Leiden: Brill, 1937, 1938, 1942.
- Sánchez Pérez, José A. (1921), Biografías de Matemáticos Árabes que florecieron en España, Madrid: Estanislao Maestre
- Sezgin, Fuat (1997), Geschichte Des Arabischen Schrifttums (in German), Brill Academic Publishers, ISBN 9004020071
- Suter, Heinrich (1900), Die Mathematiker und Astronomen der Araber und ihre Werke, Abhandlungen zur Geschichte der Mathematischen Wissenschaften Mit Einschluss Ihrer Anwendungen, X Heft, Leipzig
{{citation}}: CS1 maint: location missing publisher (link)
External links
- Hogendijk, Jan P. (January 1999). Bibliography of Mathematics in Medieval Islamic Civilization.
- O'Connor, John J.; Robertson, Edmund F. (1999), "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics Archive, University of St Andrews
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