Mathematics in the medieval Islamic world: Difference between revisions
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In the [[history of mathematics]], '''Islamic mathematics''' or '''Arabic mathematics''' refers to the [[mathematics]] developed by the [[Islam]]ic civilization between [[622]] and [[1600]]. While most |
In the [[history of mathematics]], '''Islamic mathematics''' or '''Arabic mathematics''' refers to the [[mathematics]] developed by the [[Islam]]ic civilization between [[622]] and [[1600]]. While most scientists in this period were [[Muslim]]s and [[Arabic language|Arabic]] was the dominant language, contributions were made by people of different ethnic groups ([[Arab]]s, [[Persian people|Persians]], [[Turkish people|Turks]], [[Moors]]) and religions ([[Muslim]]s, [[Christian|Christians]], [[Jew]]s, [[Zoroastrianism|Zoroastrians]]).<ref>Hogendijk 1999</ref> The center of Islamic mathematics was located in present-day [[Iraq]] and [[Iran]], but at its greatest extent stretched from [[Turkey]], [[North Africa]] and [[Spain]] in the west, to the border of [[China]] in the east.<ref>O'Connor 1999</ref> |
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Islamic science and mathematics flourished under the Islamic [[caliph]]ate (also known as the [[Arab Empire]] or [[Islamic Empire]]) established across the [[Middle East]], [[Central Asia]], [[North Africa]], [[Sicily]], the [[Iberian Peninsula]], and in parts of [[France]] and [[Pakistan]] (known as [[History of India|India]] at the time) in the 8th century. Although most Islamic texts on mathematics were written in Arabic, they were not all written by [[Arab]]s, since—much like [[Latin language|Latin]] in [[Medieval Europe]]—Arabic was used as the written language of scholars throughout the [[Islamic world]] at the time. |
[[Islamic science]] and mathematics flourished under the Islamic [[caliph]]ate (also known as the [[Arab Empire]] or [[Islamic Empire]]) established across the [[Middle East]], [[Central Asia]], [[North Africa]], [[Sicily]], the [[Iberian Peninsula]], and in parts of [[France]] and [[Pakistan]] (known as [[History of India|India]] at the time) in the 8th century. Although most Islamic texts on mathematics were written in Arabic, they were not all written by [[Arab]]s, since—much like [[Latin language|Latin]] in [[Medieval Europe]]—Arabic was used as the written language of scholars throughout the [[Islamic world]] at the time. Besides Arabs, a large number of Islamic scientists in many disciplines, including mathematics, were also Persians.<ref>[http://globalthink.net/index.cfm/2006/9/12/The-Persistence-of-Cultures-in-World-History--PersiaIran The Persistence of Cultures in World History: Persia/Iran by Dr. Laina Farhat-Holzman]</ref> |
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J. J. O'Conner and E. F. Robertson wrote in the ''[[MacTutor History of Mathematics archive]]'': |
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Recent research paints a new picture of the debt that we owe to Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to [[Europe]]an mathematicians of the 16th, 17th, and 18th centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects, the mathematics studied today is far closer in style to that of Islamic mathematics than to that of [[Greek mathematics]]. |
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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 [[Europe]]an mathematicians of the 16th, 17th, and 18th centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects, the mathematics studied today is far closer in style to that of Islamic mathematics than to that of [[Greek mathematics]]."}} |
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R. Rashed wrote in ''The development of Arabic mathematics: between arithmetic and algebra'': |
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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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==Influences == |
==Influences == |
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{{unreferenced|section|date=December 2006}} |
{{unreferenced|section|date=December 2006}} |
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[[Hellenistic mathematics]] and [[Indian mathematics]] had an important role in the development of early Islamic mathematics, especially works such as [[Euclid]]'s classic [[geometry]], [[Aryabhata]]'s [[trigonometry]] and [[Brahmagupta]]'s [[arithmetic]], and it is thought that they contributed to the era of Islamic scientific innovation that lasted until the [[14th century]]. Many ancient [[Greek language|Greek]] texts have survived only as [[Arabic]] translations by Islamic scholars. Perhaps the most important mathematical contribution from [[India]] was the decimal [[place-value]] [[Hindu-Arabic numeral system|Indo-Arabic numeral system]], also known as the [[Hindu numerals]]. The [[Persian people|Persian]] historian [[al-Biruni]] (c. 1050) in his book ''Tariq al-Hind'' states that the great [[Abbasid]] [[caliph]] [[al-Ma'mun]] had an embassy from India and with them brought a book which was translated to Arabic as ''Sindhind''. It is assumed that ''Sindhind'' is none other than Brahmagupta's ''[[Brahmasphutasiddhanta|Brahmasphuta-siddhanta]]''. |
[[Hellenistic mathematics]] and [[Indian mathematics]] had an important role in the development of early Islamic mathematics, especially works such as [[Euclid]]'s classic [[geometry]], [[Aryabhata]]'s [[trigonometry]], and [[Brahmagupta]]'s [[arithmetic]], and it is thought that they contributed to the era of Islamic scientific innovation that lasted until the [[14th century]]. Many ancient [[Greek language|Greek]] texts have survived only as [[Arabic]] translations by Islamic scholars. Perhaps the most important mathematical contribution from [[India]] was the decimal [[place-value]] [[Hindu-Arabic numeral system|Indo-Arabic numeral system]], also known as the [[Hindu numerals]]. The [[Persian people|Persian]] historian [[al-Biruni]] (c. 1050) in his book ''Tariq al-Hind'' states that the great [[Abbasid]] [[caliph]] [[al-Ma'mun]] had an embassy from India and with them brought a book which was translated to Arabic as ''Sindhind''. It is assumed that ''Sindhind'' is none other than Brahmagupta's ''[[Brahmasphutasiddhanta|Brahmasphuta-siddhanta]]''. |
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== Biographies == |
== Biographies == |
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==Fields== |
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===Algebra=== |
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{{see|History of algebra}} |
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[[Image:Al-Kitab al-mukhtasar fi hisab al-jabr wa-l-muqabala.jpg|thumb|right|A page from ''[[The Compendious Book on Calculation by Completion and Balancing]]''.]] |
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There are three theories about the origins of Arabic Algebra. The first emphasizes Hindu influence, the second emphasizes Mesopotamian or Persian-Syriac influence and the third emphasizes Greek influence. Many scholars believe that it is the result of a combination of all three sources.<ref name="Boyer Three Influences on al Jabr">{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=230|isbn=0471543977}} |
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{{quote|"Al-Khwarizmi continued: "We have said enough so far as numbers are concerned, about the six types of equations. Now, however, it is necessary that we should demonsrate geometrically the truth of the same problems which we have explained in numbers." The ring of this passage is obviously Greek rather than Babylonian or Indian. There are, therefore, three main schools of thought on the origin of Arabic algebra: one emphasizes Hindu influence, another stresses the Mesopotamian, or Syriac-Persian, tradition, and the third points to Greek inspiration. The truth is probably approached if we combine the three theories."}}</ref> |
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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 (eg. twenty-two) with [[Arabic numerals]] (eg. 22), but the Arabs never adopted or developed a syncopated or symbolic algebra.<ref name="Boyer Islamic Rhetoric Algebra Thabit">{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=234|isbn=0471543977}} |
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{{quote|"but al-Khwarizmi's work had a serious deficiency that had to be removed before it could serve its purpose effectively in the modern world: a symbolic notation had to be developed to replace the rhetorical form. This step the Arabs never took, except for the replacement of number words by number signs. ... Thabit was the founder of a school of translators, especially from Greek and Syriac, and to him we owe an immense debt for translations into Arabic of works by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius."}}</ref> |
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The Muslim<ref>{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=228-229|isbn=0471543977}} |
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{{quote|"the author's preface in Arabic gave fulsome praise to Mohammed, the prophet, and to al-Mamun, "the Commander of the Faithful"."}}</ref> |
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Persian mathematician {{Unicode|[[Muhammad ibn Musa al-Khwarizmi|Muhammad ibn Mūsā al-khwārizmī]]}} was a faculty member of the "House of Wisdom" (Bait al-hikma) in Baghdad, which was established by Al-Mamun. Al-Khwarizmi, who died around 850 A.D., wrote more than half a dozen mathematical and astronomical works; some of which were based on the Indian ''Sindhind''.<ref name="Boyer Intro Islamic Algebra">{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=227|isbn=0471543977}} |
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{{quote|"The first century of the Muslim empire had been devoid of scientific achievement. This period (from about 650 to 750) had been, in fact, perhaps the nadir in the development of mathematics, for the Arabs had not yet achieved intellectual drive, and concern for learning in other parts of the world had faded. Had it not been for the sudden cultural awakening in Islam during the second half of the eighth century, considerably more of ancient science and mathematics would have been lost. ... It was during the caliphate of al-Mamun (809-833), however, that the Arabs fully indulged their passion for translation. The caliph is said to have had a dream in which Aristotle appeared, and as a consequence al-Mamun determined to have Arabic versions made of all the Greek works that he could lay his hands on, including Ptolemy's ''Almagest'' and a complete version of Euclid's ''Elements''. From the Byzantine Empire, with which the Arabs maintained an uneasy peace, Greek manuscripts were obtained through peace treaties. Al-Mamun established at Baghdad a "House of Wisdom" (Bait al-hikma) comparable to the ancient Museum at Alexandria. Among the faculty members was a mathematician and astronomer, Mohammed ibn-Musa al-Khwarizmi, whose name, like that of Euclid, later was to become a household word in Western Europe. The scholar, who died sometime before 850, wrote more than half a dozen astronomical and mathematical works, of which the earliest were probably based on the ''Sindhad'' derived from India."}}</ref> |
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One of al-Khwarizmi's most famous books is entitled ''Al-jabr wa'l muqabalah'' or ''[[The Compendious Book on Calculation by Completion and Balancing]]'', and it gives an exhaustive account of solving polynomials up to the second degree.<ref>{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=228|isbn=0471543977}} |
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{{quote|"The Arabs in general loved a good clear argument from premise to conclusion, as well as systematic organization - respects in which neither Diophantus nor the Hindus excelled."}}</ref> |
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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 (ax<sup>2</sup> = bx), the second chapter deals with squares equal to number (ax<sup>2</sup> = c), the third chapter deals with roots equal to a number (bx = c), the fourth chapter deals with squares and roots equal a number (ax<sup>2</sup> + bx = c), the fifth chapter deals with squares and number equal roots (ax<sup>2</sup> + c = bx), and the sixth and final chapter deals with roots and number equal to squares (bx + c = ax<sup>2</sup>).<ref name="Al Jabr and its chapters">{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=229|isbn=0471543977 |
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|quote=in six short chapters, of the six types of equations made up from the three kinds of quantities: roots, squares, and numbers (that is x, x<sup>2</sup>, and numbers). Chapter I, in three short paragraphs, covers the case of squares equal to roots, expressed in modern notation as x<sup>2</sup> = 5x, x<sup>2</sup>/3 = 4x, and 5x<sup>2</sup> = 10x, giving the answers x = 5, x = 12, and x = 2 respectively. (The root x = 0 was not recognized.) Chapter II covers the case of squares equal to numbers, and Chapter III solves the cases of roots equal to numbers, again with three illustrations per chapter to cover the cases in which the coefficient of the variable term is equal to, more than, or less than one. Chapters IV, V, and VI are mor interesting, for they cover in turn the three classical cases of three-term quadratic equations: (1) squares and roots equal to numbers, (2) squares and numbers equal to roots, and (3) roots and numbers equal to squares.}}</ref> |
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'Abd al-Hamid ibn-Turk authored a manuscript entitled ''Logical Necessities in Mixed Equations'', which is very similar to al-Khwarzimi's ''Al-Jabr'' and was published at around the same time as, or even possibly earlier than, ''Al-Jabr''.<ref name="Boyer Ibn Turk">{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=234|isbn=0471543977 |
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|quote=The ''Algebra'' of al-Khwarizmi usually is regarded as the first work on the subject, but a recent publication in Turkey raises some questions about this. A manuscript of a work by 'Abd-al-Hamid ibn-Turk, entitled "Logical Necessities in Mixed Equations," was part of a book on ''Al-jabr wa'l muqabalah'' which was evidently very much the same as that by al-Khwarizmi and was published at about the same time - possibly even earlier. The surviving chapters on "Logical Necessities" give precisely the same type of geometric demonstration as al-Khwarizmi's ''Algebra'' and in one case the same illustrative example x<sup>2</sup> + 21 = 10x. In one respect 'Abd-al-Hamad's exposition is more thorough than that of al-Khwarizmi for he gives geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution. Similarities in the works of the two men and the systematic organization found in them seem to indicate that algebra in their day was not so recent a development as has usually been assumed. When textbooks with a conventional and well-ordered exposition appear simultaneously, a subject is likely to be considerably beyond the formative stage. ... Note the omission of Diophantus and Pappus, authors who evidently were not at first known in Arabia, although the Diophantine ''Arithmetica'' became familiar before the end of the tenth century.}}</ref> |
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The manuscript gives the exact same geometric demonstration as is found in ''Al-Jabr'', and in one case the same example as found in ''Al-Jabr'', and even goes beyond ''Al-Jabr'' by giving a geometric proof that if the determinant is negative then the quadratic equation has no solution.<ref name="Boyer Ibn Turk" /> The similarity between these two works has led some historians to conclude that Arabic algebra may have been well developed by the time of al-Khwarizmi and 'Abd al-Hamid.<ref name="Boyer Ibn Turk" /> |
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[[Al-Karkhi]] was the successor of Abu'l-Wefa and he was the first to discover the solution to equations of the form ax<sup>2n</sup> + bx<sup>n</sup> = c.<ref name="Boyer al-Karkhi ax2n">{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=239|isbn=0471543977 |
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|quote=Abu'l Wefa was a capable algebraist aws well as a trionometer. ... His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus - but without Diophantine analysis! ... In particular, to al-Karkhi is attributed the first numerical solution of equations of the form ax<sup>2n</sup> + bx<sup>n</sup> = c (only equations with positive roots were considered),}}</ref> |
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Al-Karkhi only considered positive roots.<ref name="Boyer al-Karkhi ax2n"/> |
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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.<ref name="Boyer Omar Khayyam positive roots">{{cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=241-242|isbn=0471543977}} |
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{{quote|Omar Khayyam (ca. 1050-1123), the "tent-maker," wrote an ''Algebra'' that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the sixteenth century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved."}}</ref> |
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Omar Khayyám provided both arithmetic and geometric solutions for quadratic equations, but he only gave geometric solutions for general cubic equations since he mistakenly believed that arithmetic solutions were impossible.<ref name="Boyer Omar Khayyam positive roots" /> His method of solving cubic equations by using intersecting conics had been used by Menaechmus, Archimedes, and Alhazen, but Omar Khayyám generalized the method to cover all cubic equations with positive roots.<ref name="Boyer Omar Khayyam positive roots" /> He only considered positive roots and he did not go past the third degree.<ref name="Boyer Omar Khayyam positive roots" /> He also saw a strong relationship between Geometry and Algebra.<ref name="Boyer Omar Khayyam positive roots" /> |
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In the 12th century, [[Sharaf al-Din al-Tusi]] found algebraic and [[Numerical analysis|numerical]] solutions to cubic equations and was the first to discover the [[derivative]] of [[Cubic function|cubic polynomials]].<ref name=Berggren/> |
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J. J. O'Conner and E. F. Robertson wrote in the ''[[MacTutor History of Mathematics archive]]'': |
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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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===Arithmetic=== |
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{{main|Arabic numerals}} |
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The [[Indian numerals|Indian numeral]] system came to be known to both the [[Persians|Persian]] mathematician [[Muhammad ibn Musa al-Khwarizmi|Al-Khwarizmi]], whose book ''On the Calculation with Hindu Numerals'' written ''circa'' [[825]], and the [[Arab]] mathematician [[Al-Kindi]], who wrote four volumes, ''On the Use of the Indian Numerals'' (Ketab fi Isti'mal al-'Adad al-Hindi) ''circa'' [[830]], are principally responsible for the diffusion of the Indian system of numeration in the [[Middle-East]] and the West [http://www-gap.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Indian_numerals.html]. In the [[10th century]], [[Middle-East]]ern mathematicians extended the decimal numeral system to include fractions using [[Decimal separator|decimal point]] notation, as recorded in a treatise by [[Syrian]] mathematician [[Abu'l-Hasan al-Uqlidisi]] in [[952]]-[[953]]. |
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In the [[Arab world]]—until modern times—the Arabic numeral system was used only by mathematicians. Muslim scientists used the [[Babylonian numerals|Babylonian numeral system]], and merchants used the [[Abjad numerals]]. A distinctive "West Arabic" variant of the symbols begins to emerge in ca. the [[10th century]] in the [[Maghreb]] and [[Al-Andalus]], called the ''ghubar'' ("sand-table" or "dust-table") numerals. |
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The first mentions of the numerals in the West are found in the ''[[Codex Vigilanus]]'' of [[976]] [http://www.mathorigins.com/V.htm]. From the [[980s]], [[Pope Silvester II|Gerbert of Aurillac]] (later, Pope [[Silvester II]]) began to spread knowledge of the numerals in Europe. Gerbert studied in [[Barcelona]] in his youth, and he is known to have requested mathematical treatises concerning the [[astrolabe]] from [[Lupitus of Barcelona]] after he had returned to France. |
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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]] (Latin ''algorithmus'') with a meaning "calculation method". |
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===Calculus=== |
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{{see also|History of 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'', p. 255-259. [[Addison-Wesley]]. ISBN 0321016181.</ref> The [[historian]] of mathematics, F. Woepcke,<ref>F. Woepcke (1853). ''Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi''. [[Paris]].</ref> praised Al-Karaji for being "the first who introduced the [[theory]] of [[algebra]]ic [[calculus]]." Shortly afterwards, [[Ibn al-Haytham]] (known as Alhazen in the West), an [[Iraq]]i mathematician working in [[History of Arab Egypt|Egypt]], was the first mathematician to derive the formula for the sum of the [[Quartic|fourth]] [[Exponentiation|powers]]. In turn, he developed a method for determining the general formula for the sum of any [[integral]] powers, which was fundamental to the development of [[Integral|integral calculus]].<ref name=Katz>Victor J. Katz (1995). "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '''68''' (3), p. 163-174.</ref> |
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[[Analytic geometry]], an important part of calculus, began with [[Omar Khayyám]], a [[Persian literature|poet]]-mathematician in 11th century [[History of Iran|Persia]], who applied it to his general geometric solution of [[cubic equation]]s.<ref name=Cooper>Glen M. Cooper (2003). "Omar Khayyam, the Mathmetician", ''The Journal of the American Oriental Society'' '''123'''.</ref> In the 12th century, the [[Persian people|Persian]] mathematician [[Sharaf al-Din al-Tusi]] was the first to discover the [[derivative]] of [[Cubic function|cubic polynomials]], an important result in [[Differential (calculus)|differential calculus]].<ref name=Berggren>J. L. Berggren (1990). "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", ''Journal of the American Oriental Society'' '''110''' (2), p. 304-309.</ref> |
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===Geometry=== |
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{{see|History of geometry}} |
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[[Image:Durer astronomer.jpg|thumb|225px|An engraving by [[Albrecht Dürer]] featuring [[Mashallah]], from the title page of the ''De scientia motus orbis'' (Latin version with engraving, 1504). As in many medieval illustrations, the [[Compass (drafting)|compass]] here is an icon of religion as well as science, in reference to God as the architect of creation]] |
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The successors of [[Muḥammad ibn Mūsā al-Ḵwārizmī]] (born [[780]]) undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose. |
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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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Although [[Thābit ibn Qurra|Thabit ibn Qurra]] (known as Thebit in [[Latin]]) (born [[836]]) contributed to a number of areas in mathematics, where he played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to ([[positive]]) [[real number]]s, [[integral calculus]], theorems in [[spherical trigonometry]], [[analytic geometry]], and [[non-Euclidean geometry]]. An important geometrical aspect of Thabit's work was his book on the composition of ratios. In this book, Thabit deals with arithmetical operations applied to ratios of geometrical quantities. The Greeks had dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of arithmetic could be applied. By introducing arithmetical operations on quantities previously regarded as geometric and non-numerical, Thabit started a trend which led eventually to the generalisation 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 [[triangle]]s in general, along with a general [[mathematical proof|proof]].<ref>Aydin Sayili (1960). "Thabit ibn Qurra's Generalization of the Pythagorean Theorem", ''[[Isis (journal)|Isis]]'' '''51''' (1), p. 35-37.</ref> |
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[[Omar Khayyám]] (born [[1048]]) was a [[Persian people|Persian]] mathematician, as well as a poet. Along with his fame as a poet, he was also famous during his lifetime as a mathematician, well known for inventing the general method of solving [[cubic equation]]s by intersecting a parabola with a circle. In addition he discovered the [[binomial expansion]], and authored criticisms of Euclid's theories of parallels which made their way to England, where they contributed to the eventual development of [[non-Euclidean geometry]]. Omar Khayyam also combined the use of trigonometry and [[approximation theory]] to provide methods of solving algebraic equations by geometrical means. His work marked the beginnings of [[algebraic geometry]]<ref>R. Rashed (1994). ''The development of Arabic mathematics: between arithmetic and algebra''. [[London]].</ref><ref>{{MacTutor|class=HistTopics|id=Arabic_mathematics|title=Arabic mathematics: forgotten brilliance?|year=1999}}</ref> and [[analytic geometry]].<ref name=Cooper/> |
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[[Ibrahim ibn Sinan]] (born [[908]]), who introduced a method of [[integral|integration]] more general than that of [[Archimedes]], and [[al-Quhi]] (born [[940]]) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. These mathematicians, and in particular [[Ibn al-Haytham]], studied [[optics]] and investigated the optical properties of mirrors made from [[conic section]]s. |
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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]] both applied [[spherical geometry]] to astronomy. |
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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, which represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the study of algebraic geometry. |
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===Induction=== |
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The first known [[proof]] by [[mathematical induction]] appears in a book written by [[Al-Karaji]] around [[1000]] AD, who used it to prove the [[binomial theorem]], [[Pascal's triangle]], and the sum of [[integral]] [[Cube (algebra)|cubes]].<ref>Victor J. Katz (1998). ''History of Mathematics: An Introduction'', p. 255-259. [[Addison-Wesley]]. ISBN 0321016181. |
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{{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."}}</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]].<ref>Victor J. Katz (1995). "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '''68''' (3), p. 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> |
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===Trigonometry=== |
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{{see also|History of trigonometric functions}} |
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The Indian works on [[trigonometry]] were translated and expanded in the [[Muslim world]] by [[List of Arab scientists and scholars|Arab]] and [[List of Iranian scientists and scholars|Persian]] mathematicians. {{Unicode|[[Muhammad ibn Mūsā al-Khwārizmī]]}} produced tables of [[Trigonometric function|sines]] and [[tangent]]s, and also developed [[spherical trigonometry]]. By the 10th century, in the work of [[Abū al-Wafā' al-Būzjānī]], Muslim mathematicians were using all six [[trigonometric function]]s, and had sine tables in 0.25° increments, to 8 decimal places of accuracy, as well as tables of [[tangent (trigonometry)|tangent]] values. Abū al-Wafā' also developed the trigonometric formula sin 2''x'' = 2 sin ''x'' cos ''x''. |
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[[Omar Khayyam]] solved [[cubic equation]]s using approximate numerical solutions found by interpolation in trigonometric tables. [[Al-Jayyani]], an Arabic mathematician in [[Al-Andalus|Islamic Spain]], wrote the first treatise on [[spherical trigonometry]] in 1060. |
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All of these earlier works on trigonometry treated it mainly as an adjunct to astronomy; perhaps the first treatment as a subject in its own right was by [[Bhaskara II]] and [[Nasir al-Din al-Tusi]] (13th century). Nasir al-Din al-Tusi stated the [[law of sines]] and provided a proof for it, and also listed the six distinct cases of a right angled triangle in spherical trigonometry. |
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[[Ghiyath al-Kashi]] (14th century) gives trigonometric tables of values of the sine function to four [[sexagesimal]] digits (equivalent to 8 decimal places) for each 1° of argument with differences to be added for each 1/60 of 1°. [[Ulugh Beg]] (14th century) also gives accurate tables of sines and tangents correct to 8 decimal places. |
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==See also== |
==See also== |
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*[[List of Muslim mathematicians]] |
*[[List of Muslim mathematicians]] |
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*[[Latin translations of the 12th century]] |
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*[[Islamic science]] |
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*[[Islamic Golden Age]] |
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== Notes == |
== Notes == |
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* {{cite book|last=Joseph|first=George Gheverghese|title=The Crest of the Peacock: Non-European Roots of Mathematics|edition=2nd Edition|publisher=Princeton University Press|year=2000|id=ISBN 0691006598}}<!-- Reviewed: {{Cite review|first=Victor J.|last=Katz|journal=The College Mathematics Journal|volume=23|issue=1|year=1992|pages=82-84}}.--> |
* {{cite book|last=Joseph|first=George Gheverghese|title=The Crest of the Peacock: Non-European Roots of Mathematics|edition=2nd Edition|publisher=Princeton University Press|year=2000|id=ISBN 0691006598}}<!-- Reviewed: {{Cite review|first=Victor J.|last=Katz|journal=The College Mathematics Journal|volume=23|issue=1|year=1992|pages=82-84}}.--> |
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* {{cite book|first=E. S.|last=Kennedy|title=Studies in the Islamic Exact Sciences|year=1984|publisher=Syracuse Univ Press|id=ISBN 0815660677}} |
* {{cite book|first=E. S.|last=Kennedy|title=Studies in the Islamic Exact Sciences|year=1984|publisher=Syracuse Univ Press|id=ISBN 0815660677}} |
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* {{MacTutor|class=HistTopics|id=Arabic_mathematics|title=Arabic mathematics |
* {{MacTutor|class=HistTopics|id=Arabic_mathematics|title=Arabic mathematics: forgotten brilliance?|year=1999}} |
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* {{cite book|last=Rashed|first=Roshdi|others=Transl. by A. F. W. Armstrong|title=The Development of Arabic Mathematics: Between Arithmetic and Algebra|publisher=Springer|year=2001|id=ISBN 0792325656}} |
* {{cite book|last=Rashed|first=Roshdi|others=Transl. by A. F. W. Armstrong|title=The Development of Arabic Mathematics: Between Arithmetic and Algebra|publisher=Springer|year=2001|id=ISBN 0792325656}} |
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* {{cite book|last=Sánchez Pérez|first=José A|title=Biografías de Matemáticos Árabes que florecieron en España|location=Madrid|publisher=Estanislao Maestre|year=1921}} |
* {{cite book|last=Sánchez Pérez|first=José A|title=Biografías de Matemáticos Árabes que florecieron en España|location=Madrid|publisher=Estanislao Maestre|year=1921}} |
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Revision as of 22:58, 4 June 2007
In the history of mathematics, Islamic mathematics or Arabic mathematics refers to the mathematics developed by the Islamic civilization between 622 and 1600. While most scientists in this period were Muslims and Arabic was the dominant language, contributions were made by people of different ethnic groups (Arabs, Persians, Turks, Moors) and religions (Muslims, Christians, Jews, Zoroastrians).[1] The center of Islamic mathematics was located in present-day Iraq and Iran, but at its greatest extent stretched from Turkey, North Africa and Spain in the west, to the border of China in the east.[2]
Islamic science and mathematics flourished under the Islamic caliphate (also known as the Arab Empire or Islamic Empire) established across the Middle East, Central Asia, North Africa, Sicily, the Iberian Peninsula, and in parts of France and Pakistan (known as India at the time) in the 8th century. Although most Islamic texts on mathematics were written in Arabic, they were not all written by Arabs, since—much like Latin in Medieval Europe—Arabic was used as the written language of scholars throughout the Islamic world at the time. Besides Arabs, a large number of Islamic scientists in many disciplines, including mathematics, were also Persians.[3]
J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive:
"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."
R. Rashed wrote in The development of Arabic mathematics: between arithmetic and algebra:
"Al-Khwarizmi's successors undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose."
Influences
 |
Hellenistic mathematics and Indian mathematics had an important role in the development of early Islamic mathematics, especially works such as Euclid's classic geometry, Aryabhata's trigonometry, and Brahmagupta's arithmetic, and it is thought that they contributed to the era of Islamic scientific innovation that lasted until the 14th century. Many ancient Greek texts have survived only as Arabic translations by Islamic scholars. Perhaps the most important mathematical contribution from India was the decimal place-value Indo-Arabic numeral system, also known as the Hindu numerals. The Persian historian al-Biruni (c. 1050) in his book Tariq al-Hind states that the great Abbasid caliph al-Ma'mun had an embassy from India and with them brought a book which was translated to Arabic as Sindhind. It is assumed that Sindhind is none other than Brahmagupta's Brahmasphuta-siddhanta.
Biographies
- Al-Ḥajjāj ibn Yūsuf ibn Maṭar (786 – 833)
- Al-Ḥajjāj translated Euclid's Elements into Arabic.
- Muḥammad ibn Mūsā al-Khwārizmī (c. 780 Khwarezm/Baghdad – c. 850 Baghdad)
- Al-Khwārizmī was a mathematician, astronomer, astrologer and geographer. He worked most of his life as a scholar in the House of Wisdom in Baghdad. His Algebra was the first book on the systematic solution of linear and quadratic equations. Latin translations of his Arithmetic, on the Indian numerals, introduced the decimal 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.
- Al-ʿAbbās ibn Saʿid al-Jawharī (c. 800 Baghdad? – c. 860 Baghdad?)
- 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 propositions and an attempted proof of the parallel postulate.
- ʿAbd al-Hamīd ibn Turk (fl. 830 Baghdad)
- Ibn Turk wrote a work on algebra of which only a chapter on the solution of quadratic equations has survied.
- Yaʿqūb ibn Isḥāq al-Kindī (c. 801 Kufah – 873 Baghdad)
- Al-Kindī (or Alkindus) was a philosopher and scientist who worked as the House of Wisdom in Baghdad where he wrote commentaries on many Greek works. His contributions to mathematics include many works on arithmetic and geometry.
- Hunayn ibn Ishaq (808 Al-Hirah – 873 Baghdad)
- 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.
- Banū Mūsā (c. 800 Baghdad – 873+ Baghdad)
- The Banū Mūsā where three brothers who worked at the House of Wisdom in Baghdad. Their most famous treatise is called 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 oldest, Jaʿfar Muḥammad (c. 800) specialised in geometry and astronomy. He wrote a critical revision on Apollonius' Conics called Premises of the book of conics. Aḥmad (c. 805) specialised in mechanics and wrote a work on pneumatic devices called On mechanics. The youngest, al-Ḥasan (c. 810) specialised in geometry and wrote a work on the ellipse called The elongated circular figure.
- Al-Mahani
- Ahmed ibn Yusuf
- Thabit ibn Qurra (Syria-Iraq, 835-901)
- Al-Hashimi (Iraq? ca. 850-900)
- Muḥammad ibn Jābir al-Ḥarrānī al-Battānī (c. 853 Harran – 929 Qasr al-Jiss near Samarra)
- Abu Kamil (Egypt? ca. 900)
- Sinan ibn Tabit (ca. 880 - 943)
- Al-Nayrizi
- Ibrahim ibn Sinan (Iraq, 909-946)
- Al-Khazin (Iraq-Iran, ca. 920-980)
- Al-Karabisi (Iraq? 10th century?)
- Ikhwan al-Safa' (Iraq, first half of 10th century)
- The Ikhwan al-Safa' ("brethren of purity") were a (mystical?) group in the city of Basra in Irak. The group authored a series of more than 50 letters on science, philosophy and theology. The first letter is on arithmetic and number theory, the second letter on geometry.
- Al-Uqlidisi (Iraq-Iran, 10th century)
- Al-Saghani (Iraq-Iran, ca. 940-1000)
- Abū Sahl al-Qūhī (Iraq-Iran, ca. 940-1000)
- Al-Khujandi
- Abū al-Wafāʾ al-Būzjānī (Iraq-Iran, ca. 940-998)
- Ibn Sahl (Iraq-Iran, ca. 940-1000)
- Al-Sijzi (Iran, ca. 940-1000)
- Ibn Yunus (Egypt, ca. 950-1010)
- Abu Nasr ibn `Iraq (Iraq-Iran, ca. 950-1030)
- Kushyar ibn Labban (Iran, ca. 960-1010)
- Al-Karaji (Iran, ca. 970-1030)
- Ibn al-Haytham (Iraq-Egypt, ca. 965-1040)
- Abū al-Rayḥān al-Bīrūnī (September 15 973 in Kath, Khwarezm – December 13 1048 in Gazna)
- Ibn Sina
- al-Baghdadi
- Al-Nasawi
- Al-Jayyani (Spain, ca. 1030-1090)
- Ibn al-Zarqalluh (Azarquiel, al-Zarqali) (Spain, ca. 1030-1090)
- Al-Mu'taman ibn Hud (Spain, ca. 1080)
- al-Khayyam (Iran, ca. 1050-1130)
- Ibn Yaḥyā al-Maghribī al-Samawʾal (c. 1130 Baghdad – c. 1180 Maragha)
- Sharaf al-Dīn al-Ṭūsī (Iran, ca. 1150-1215)
- Ibn Mun`im (Maghreb, ca. 1210)
- al-Marrakushi (Morocco, 13th century)
- Naṣīr al-Dīn al-Ṭūsī (18 February 1201 in Tus, Khorasan – 26 June 1274 in Kadhimain near Baghdad)
- Muḥyi al-Dīn al-Maghribī (c. 1220 Spain – c. 1283 Maragha)
- Shams al-Dīn al-Samarqandī (c. 1250 Samarqand – c. 1310)
- Ibn Baso (Spain, ca. 1250-1320)
- Ibn al-Banna' (Maghreb, ca. 1300)
- Kamal al-Din Al-Farisi (Iran, ca. 1300)
- Al-Khalili (Syria, ca. 1350-1400)
- Ibn al-Shatir (1306-1375)
- Qāḍī Zāda al-Rūmī (1364 Bursa – 1436 Samarkand)
- Jamshīd al-Kāshī (Iran, Uzbekistan, ca. 1420)
- Ulugh Beg (Iran, Uzbekistan, 1394-1449)
- Al-Umawi
- Al-Qalasadi (Maghreb, 15th century)
Fields
Algebra
There are three theories about the origins of Arabic Algebra. The first emphasizes Hindu influence, the second emphasizes Mesopotamian or Persian-Syriac influence and the third emphasizes Greek influence. Many scholars believe that it is the result of a combination of all three sources.[4]
Throughout their time in power, before the fall of Islamic civilization, the Arabs used a fully rhetorical algebra, where sometimes even the numbers were spelled out in words. The Arabs would eventually replace spelled out numbers (eg. twenty-two) with Arabic numerals (eg. 22), but the Arabs never adopted or developed a syncopated or symbolic algebra.[5]
The Muslim[6] Persian mathematician Muhammad ibn Mūsā al-khwārizmī was a faculty member of the "House of Wisdom" (Bait al-hikma) in Baghdad, which was established by Al-Mamun. Al-Khwarizmi, who died around 850 A.D., wrote more than half a dozen mathematical and astronomical works; some of which were based on the Indian Sindhind.[7] 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.[8]
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).[9]
'Abd al-Hamid ibn-Turk authored a manuscript entitled Logical Necessities in Mixed Equations, which is very similar to al-Khwarzimi's Al-Jabr and was published at around the same time as, or even possibly earlier than, Al-Jabr.[10] The manuscript gives the exact same geometric demonstration as is found in Al-Jabr, and in one case the same example as found in Al-Jabr, and even goes beyond Al-Jabr by giving a geometric proof that if the determinant is negative then the quadratic equation has no solution.[10] The similarity between these two works has led some historians to conclude that Arabic algebra may have been well developed by the time of al-Khwarizmi and 'Abd al-Hamid.[10]
Al-Karkhi was the successor of Abu'l-Wefa and he was the first to discover the solution to equations of the form ax2n + bxn = c.[11] Al-Karkhi only considered positive roots.[11]
Omar Khayyám (c. 1050-1123) wrote a book on Algebra that went beyond Al-Jabr to include equations of the third degree.[12] 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.[12] 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.[12] He only considered positive roots and he did not go past the third degree.[12] He also saw a strong relationship between Geometry and Algebra.[12]
In the 12th century, Sharaf al-Din al-Tusi found algebraic and numerical solutions to cubic equations and was the first to discover the derivative of cubic polynomials.[13]
J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive:
"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 numbers, 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."
Arithmetic
The Indian numeral system came to be known to both the Persian mathematician 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 [1]. In the 10th century, Middle-Eastern mathematicians extended the decimal numeral system to include fractions using decimal point notation, as recorded in a treatise by Syrian mathematician Abu'l-Hasan al-Uqlidisi in 952-953.
In the Arab world—until modern times—the Arabic numeral system was used only by mathematicians. Muslim scientists used the Babylonian numeral system, and merchants used the Abjad numerals. A distinctive "West Arabic" variant of the symbols begins to emerge in ca. the 10th century in the Maghreb and Al-Andalus, called the ghubar ("sand-table" or "dust-table") numerals.
The first mentions of the numerals in the West are found in the Codex Vigilanus of 976 [2]. From the 980s, Gerbert of Aurillac (later, Pope Silvester II) began to spread knowledge of the numerals in Europe. Gerbert studied in Barcelona in his youth, and he is known to have requested mathematical treatises concerning the astrolabe from Lupitus of Barcelona after he had returned to France.
Al-Khwārizmī, the Persian scientist, wrote in 825 a treatise On the Calculation with Hindu Numerals, which was translated into Latin in the 12th century, as Algoritmi de numero Indorum, where "Algoritmi", the translator's rendition of the author's name gave rise to the word algorithm (Latin algorithmus) with a meaning "calculation method".
Calculus
Around 1000 AD, Al-Karaji, using mathematical induction, found a proof for the sum of integral cubes.[14] The historian of mathematics, F. Woepcke,[15] praised Al-Karaji for being "the first who introduced the theory of algebraic calculus." Shortly afterwards, Ibn al-Haytham (known as Alhazen in the West), an Iraqi mathematician working in Egypt, was the first mathematician to derive the formula for the sum of the fourth powers. In turn, he developed a method for determining the general formula for the sum of any integral powers, which was fundamental to the development of integral calculus.[16]
Analytic geometry, an important part of calculus, began with Omar Khayyám, a poet-mathematician in 11th century Persia, who applied it to his general geometric solution of cubic equations.[17] In the 12th century, the Persian mathematician Sharaf al-Din al-Tusi was the first to discover the derivative of cubic polynomials, an important result in differential calculus.[13]
Geometry
The successors of Muḥammad ibn Mūsā al-Ḵwārizmī (born 780) undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose.
Al-Mahani (born 820) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Al-Karaji (born 953) completely freed algebra from geometrical operations and replaced them with the arithmetical type of operations which are at the core of algebra today.
Although Thabit ibn Qurra (known as Thebit in Latin) (born 836) contributed to a number of areas in mathematics, where he played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, 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 generalisation 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 triangles in general, along with a general proof.[18]
Omar Khayyám (born 1048) was a 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 equations by intersecting a parabola with a circle. In addition he discovered the binomial expansion, and authored criticisms of Euclid's theories of 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[19][20] and analytic geometry.[17]
Ibrahim ibn Sinan (born 908), who introduced a method of integration more general than that of Archimedes, and al-Quhi (born 940) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. These mathematicians, and in particular Ibn al-Haytham, studied optics and investigated the optical properties of mirrors made from conic sections.
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 both applied spherical geometry to astronomy.
Persian mathematician Sharafeddin Tusi (born 1135) did not follow the general development that came through al-Karaji's school of algebra but rather followed Khayyam's application of algebra to geometry. He wrote a treatise on cubic equations, which represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the study of algebraic geometry.
Induction
The first known proof by mathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes.[21]
Shortly afterwards, Ibn al-Haytham (Alhazen) used the inductive method to prove the sum of fourth powers, and by extension, the sum of any integral powers, which was an important result in integral calculus.[22]
Trigonometry
The Indian works on trigonometry were translated and expanded in the Muslim world by Arab and Persian mathematicians. Muhammad ibn Mūsā al-Khwārizmī produced tables of sines and tangents, and also developed spherical trigonometry. By the 10th century, in the work of Abū al-Wafā' al-Būzjānī, 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 tables of tangent values. Abū al-Wafā' also developed the trigonometric formula sin 2x = 2 sin x cos x.
Omar Khayyam solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables. Al-Jayyani, an Arabic mathematician in Islamic Spain, wrote the first treatise on spherical trigonometry in 1060.
All of these earlier works on trigonometry treated it mainly as an adjunct to astronomy; perhaps the first treatment as a subject in its own right was by Bhaskara II and Nasir al-Din al-Tusi (13th century). Nasir al-Din al-Tusi stated the law of sines and provided a proof for it, and also listed the six distinct cases of a right angled triangle in spherical trigonometry.
Ghiyath al-Kashi (14th century) gives trigonometric tables of values of the sine function to four sexagesimal digits (equivalent to 8 decimal places) for each 1° of argument with differences to be added for each 1/60 of 1°. Ulugh Beg (14th century) also gives accurate tables of sines and tangents correct to 8 decimal places.
See also
- List of Muslim mathematicians
- Latin translations of the 12th century
- Islamic science
- Islamic Golden Age
Notes
- ^ Hogendijk 1999
- ^ O'Connor 1999
- ^ The Persistence of Cultures in World History: Persia/Iran by Dr. Laina Farhat-Holzman
- ^ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. p. 230. ISBN 0471543977.
{{cite book}}:|edition=has extra text (help)"Al-Khwarizmi continued: "We have said enough so far as numbers are concerned, about the six types of equations. Now, however, it is necessary that we should demonsrate geometrically the truth of the same problems which we have explained in numbers." The ring of this passage is obviously Greek rather than Babylonian or Indian. There are, therefore, three main schools of thought on the origin of Arabic algebra: one emphasizes Hindu influence, another stresses the Mesopotamian, or Syriac-Persian, tradition, and the third points to Greek inspiration. The truth is probably approached if we combine the three theories."
- ^ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. p. 234. ISBN 0471543977.
{{cite book}}:|edition=has extra text (help)"but al-Khwarizmi's work had a serious deficiency that had to be removed before it could serve its purpose effectively in the modern world: a symbolic notation had to be developed to replace the rhetorical form. This step the Arabs never took, except for the replacement of number words by number signs. ... Thabit was the founder of a school of translators, especially from Greek and Syriac, and to him we owe an immense debt for translations into Arabic of works by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius."
- ^ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. pp. 228–229. ISBN 0471543977.
{{cite book}}:|edition=has extra text (help)"the author's preface in Arabic gave fulsome praise to Mohammed, the prophet, and to al-Mamun, "the Commander of the Faithful"."
- ^ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. p. 227. ISBN 0471543977.
{{cite book}}:|edition=has extra text (help)"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."
- ^ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. p. 228. ISBN 0471543977.
{{cite book}}:|edition=has extra text (help)"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."
- ^ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. p. 229. ISBN 0471543977.
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, x2, and numbers). Chapter I, in three short paragraphs, covers the case of squares equal to roots, expressed in modern notation as x2 = 5x, x2/3 = 4x, and 5x2 = 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.
{{cite book}}:|edition=has extra text (help) - ^ a b c Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. p. 234. ISBN 0471543977.
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 x2 + 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.
{{cite book}}:|edition=has extra text (help) - ^ a b Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. p. 239. ISBN 0471543977.
Abu'l Wefa was a capable algebraist aws well as a trionometer. ... 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 ax2n + bxn = c (only equations with positive roots were considered),
{{cite book}}:|edition=has extra text (help) - ^ a b c d e Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. pp. 241–242. ISBN 0471543977.
{{cite book}}:|edition=has extra text (help)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."
- ^ a b J. L. Berggren (1990). "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", Journal of the American Oriental Society 110 (2), p. 304-309.
- ^ Victor J. Katz (1998). History of Mathematics: An Introduction, p. 255-259. Addison-Wesley. ISBN 0321016181.
- ^ F. Woepcke (1853). Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi. Paris.
- ^ Victor J. Katz (1995). "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3), p. 163-174.
- ^ a b Glen M. Cooper (2003). "Omar Khayyam, the Mathmetician", The Journal of the American Oriental Society 123.
- ^ Aydin Sayili (1960). "Thabit ibn Qurra's Generalization of the Pythagorean Theorem", Isis 51 (1), p. 35-37.
- ^ R. Rashed (1994). The development of Arabic mathematics: between arithmetic and algebra. London.
- ^ O'Connor, John J.; Robertson, Edmund F. (1999), "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics Archive, University of St Andrews
- ^ Victor J. Katz (1998). History of Mathematics: An Introduction, p. 255-259. Addison-Wesley. ISBN 0321016181.
"Another important idea introduced by al-Karaji and continued by al-Samaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences."
- ^ Victor J. Katz (1995). "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3), p. 163-174.
"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."
Further reading
- Berggren, J. Lennart (1986). Episodes in the Mathematics of Medieval Islam. New York: Springer-Verlag. ISBN 0-387-96318-9.
- Daffa', Ali Abdullah al- (1977). The Muslim contribution to mathematics. London: Croom Helm. ISBN 0-85664-464-1.
- Daffa, Ali Abdullah al-; Stroyls, J.J. (1984). Studies in the exact sciences in medieval Islam. New York: Wiley. ISBN 0471903205.
- Joseph, George Gheverghese (2000). The Crest of the Peacock: Non-European Roots of Mathematics (2nd Edition ed.). Princeton University Press. ISBN 0691006598.
{{cite book}}:|edition=has extra text (help) - Kennedy, E. S. (1984). Studies in the Islamic Exact Sciences. Syracuse Univ Press. ISBN 0815660677.
- O'Connor, John J.; Robertson, Edmund F. (1999), "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics Archive, University of St Andrews
- Rashed, Roshdi (2001). The Development of Arabic Mathematics: Between Arithmetic and Algebra. Transl. by A. F. W. Armstrong. Springer. ISBN 0792325656.
- 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.
{{cite book}}: CS1 maint: location missing publisher (link) - Youschkevitch, Adolf P. (1960). Die Mathematik der Länder des Ostens im Mittelalter. Berlin.
{{cite book}}: 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.
External links
- Hogendijk, Jan P. (January 1999). Bibliography of Mathematics in Medieval Islamic Civilization.