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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. In particular, a large number of Islamic scientists in many disciplines, including mathematics, were Persians.[3]
J. J. O'Conner and E. F. Robertson wrote in the ''MacTutor History of Mathematics archive'':
R. Rashed wrote in ''The development of Arabic mathematics: between arithmetic and algebra'':
The first century of the Islamic Arab Empire saw almost no scientific or mathematical achievements since the Arabs, with their newly conquered empire, had not yet gained any intellectual drive and research in other parts of the world had faded. In the second half of the eighth century Islam had a cultural awakening, and research in mathematics and the sciences increased.[4] The Muslim Abbasid caliph al-Mamun (809-833) is said to have had a dream where Aristotle appeared to him, and as a consequence al-Mamun ordered that Arabic translation be made of as many Greek works as possible, including Ptolemy's ''Almagest'' and Euclid's ''Elements''. Greek works would be given to the Muslims by the Byzantine Empire in exchange for treaties, as the two empires held an uneasy peace.4 Many of these Greek works were translated by Thabit ibn Qurra (826-901), who translated books written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.[5] 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.
Greek, Indian and Mesopotamian mathematics all played an important role in the development of 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 Islamic mathematics. Perhaps the most influential mathematical contribution from India was the decimal place-value Indo-Arabic numeral system, also known as the Hindu numerals.[6] 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 to Baghdad which was translated into Arabic as ''Sindhind''. It is generally assumed that ''Sindhind'' is none other than Brahmagupta's ''Brahmasphuta-siddhanta''.[7] The earliest translations from Sanskrit inspired several astronomical and astrological Arabic works, now mostly lost, some of which were even composed in verse.
But Indian influences were soon overwhelmed by Greek mathematical and astronomical texts. It is not clear why this occurred but it may have been due to the greater availability of Greek texts in the region, the larger number of practitioners of Greek mathematics in the region, or because Islamic mathematicians favored the deductive exposition of the Greeks over the elliptic Sanskrit verse of the Indians. Regardless of the reason, Indian mathematics became eclipsed by or merged with the "Graeco-Islamic" science founded on Hellenistic treatises.[8]
; (786 – 833)
:Al-Ḥajjāj translated Euclid's ''Elements'' into Arabic.
; (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.
; (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.
; (fl. 830 Baghdad)
:Ibn Turk wrote a work on algebra of which only a chapter on the solution of quadratic equations has survied.
; (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.
; (c. 800 Baghdad – 873+ Baghdad)
:The Banū Mūsā where three brothers who worked at the House of Wisdom in Baghdad. Their most famous mathematical treatise is ''The Book of the Measurement of Plane and Spherical Figures'', which considered similar problems as Archimedes did in his ''On the measurement of the circle'' and ''On the sphere and the cylinder''. They contributed individually as well. The eldest, (c. 800) specialised in geometry and astronomy. He wrote a critical revision on Apollonius' ''Conics'' called ''Premises of the book of conics''. (c. 805) specialised in mechanics and wrote a work on pneumatic devices called ''On mechanics''. The youngest, (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)
; (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)
; (Iraq-Iran, ca. 940-1000)
;Al-Khujandi
; (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)
; (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)
; (c. 1130 Baghdad – c. 1180 Maragha)
; (Iran, ca. 1150-1215)
;Ibn Mun`im (Maghreb, ca. 1210)
;al-Marrakushi (Morocco, 13th century)
; (18 February 1201 in Tus, Khorasan – 26 June 1274 in Kadhimain near Baghdad)
; (c. 1220 Spain – c. 1283 Maragha)
; (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)
;'' (1364 Bursa – 1436 Samarkand)
; (Iran, Uzbekistan, ca. 1420)
;Ulugh Beg (Iran, Uzbekistan, 1394-1449)
;Al-Umawi
;Al-Qalasadi (Maghreb, 15th century)
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.[9]
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.[10]
The Muslim[11]
Persian mathematician 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''.[12]
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.[11]
''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).[14]
'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''.[15]
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.15 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.15
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.[16]
Al-Karkhi only considered positive roots.
Omar Khayyám (c. 1050-1123) wrote a book on Algebra that went beyond ''Al-Jabr'' to include equations of the third degree.[17]
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.17 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.17 He only considered positive roots and he did not go past the third degree.17 He also saw a strong relationship between Geometry and Algebra.17
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.
J. J. O'Conner and E. F. Robertson wrote in the ''MacTutor History of Mathematics archive'':
Main articles: Arabic numerals
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".
Around 1000 AD, Al-Karaji, using mathematical induction, found a proof for the sum of integral cubes.[18] The historian of mathematics, F. Woepcke,[19] 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.Victor J. Katz (1995). "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '68' (3), p. 163-174.
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.Glen M. Cooper (2003). "Omar Khayyam, the Mathmetician", ''The Journal of the American Oriental Society'' '123'. 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.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.
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 method whereby variations in the frequency of the occurrence of letters could be analyzed and exploited to break ciphers (i.e. crypanalysis by frequency analysis).[20] 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 statistical analysis of letters and letter combinations in Arabic.[21]
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 and 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.
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.[22]
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[23][24] and analytic geometry.
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.
In 1250, Nasir al-Din al-Tusi, 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 parallel postulate and on Omar Khayyám's attempted proof a century earlier. Nasir al-Din attempted to derive a contradiction of the parallel postulate. His son, Sadr al-Din wrote a book on the subject in 1298, based on Nasir al-Din's later thoughts, which presented an argument for a hypothesis equivalent to the parallel postulate. 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 discovery of non-Euclidean geometry.[25]
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.[26]
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.[27]
In number theory, Ibn al-Haytham solved problems involving 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 numbers. In his ''Analysis and synthesis'', Ibn al-Haytham was the first to realize that every even perfect number is of the form 2''n''−1(2''n'' − 1) where 2''n'' − 1 is prime, but he was not able to prove this result successfully (Euler later proved it in the 18th century).[24]
The early Indian works on trigonometry were translated and expanded in the Muslim world by Arab and Persian mathematicians. 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 2''x'' = 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.
★ List of Muslim mathematicians
★ Latin translations of the 12th century
★ Islamic science
★ Islamic Golden Age
1. Hogendijk 1999
2. O'Connor 1999
3. The Persistence of Cultures in World History: Persia/Iran by Dr. Laina Farhat-Holzman
4. , , , Boyer, , 1991,
5. , , , Boyer, , 1991,
6. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, , J. Lennart, Berggren, Princeton University Press, 2007,
7. , , , Boyer, , 1991,
8. , , Kim, Plofker, , 2007,
9. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
10. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
11. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
12. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
13. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
14. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
15. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
16. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
17. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
18. Victor J. Katz (1998). ''History of Mathematics: An Introduction'', p. 255-259. Addison-Wesley. ISBN 0321016181.
19. F. Woepcke (1853). ''Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi''. Paris.
20. Simon Singh. The Code Book. p. 14-20
21. Al-Kindi, Cryptgraphy, Codebreaking and Ciphers
22. Aydin Sayili (1960). "Thabit ibn Qurra's Generalization of the Pythagorean Theorem", ''Isis'' '51' (1), p. 35-37.
23. R. Rashed (1994). ''The development of Arabic mathematics: between arithmetic and algebra''. London.
24.
25. Victor J. Katz (1998). ''History of Mathematics: An Introduction'', p. 270-271. Addison-Wesley. ISBN 0321016181.
26. Victor J. Katz (1998). ''History of Mathematics: An Introduction'', p. 255-259. Addison-Wesley. ISBN 0321016181.
27. Victor J. Katz (1995). "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '68' (3), p. 163-174.
28.
★ Episodes in the Mathematics of Medieval Islam, , J. Lennart, Berggren, Springer-Verlag, 1986, ISBN 0-387-96318-9
★ The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, , J. Lennart, Berggren, Princeton University Press, 2007,
★ A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc, 1991,
★ The History of Mathematics: A Brief Course, , Roger, Cooke, Wiley-Interscience, 1997,
★ The Muslim contribution to mathematics, , Ali Abdullah al-, Daffa', Croom Helm, 1977, ISBN 0-85664-464-1
★ Studies in the exact sciences in medieval Islam, , Ali Abdullah al-, Daffa, Wiley, 1984, ISBN 0471903205
★ The Crest of the Peacock: Non-European Roots of Mathematics, , George Gheverghese, Joseph, Princeton University Press, 2000, ISBN 0691006598
★ Studies in the Islamic Exact Sciences, , E. S., Kennedy, Syracuse Univ Press, 1984, ISBN 0815660677
★ The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, , Kim, Plofker, Princeton University Press, 2007,
★
★ The Development of Arabic Mathematics: Between Arithmetic and Algebra, , Roshdi, Rashed, Springer, 2001, ISBN 0792325656
★ Biografías de Matemáticos Árabes que florecieron en España, , José A, Sánchez Pérez, Estanislao Maestre, 1921,
★ Geschichte Des Arabischen Schrifttums, , Fuat, Sezgin, Brill Academic Publishers, 1997, ISBN 9004020071
★ Die Mathematiker und Astronomen der Araber und ihre Werke, , Heinrich, Suter, , 1900,
★ Die Mathematik der Länder des Ostens im Mittelalter, , Adolf P., Youschkevitch, , 1960, Sowjetische Beiträge zur Geschichte der Naturwissenschaft pp. 62-160.
★ Les mathématiques arabes: VIIIe-XVe siècles, , Adolf P., Youschkevitch, Vrin, 1976, ISBN 978-2-7116-0734-1
★ Hogendijk, Jan P. (January 1999). ''Bibliography of Mathematics in Medieval Islamic Civilization''.
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. In particular, a large number of Islamic scientists in many disciplines, including mathematics, were Persians.[3]
J. J. O'Conner and E. F. Robertson wrote in the ''MacTutor History of Mathematics archive'':
R. Rashed wrote in ''The development of Arabic mathematics: between arithmetic and algebra'':
| Contents |
| Origins and influences |
| Biographies |
| Fields |
| Algebra |
| Arithmetic |
| Calculus |
| Cryptography |
| Geometry |
| Induction |
| Number theory |
| Trigonometry |
| See also |
| Notes |
| References and further reading |
| External links |
Origins and influences
The first century of the Islamic Arab Empire saw almost no scientific or mathematical achievements since the Arabs, with their newly conquered empire, had not yet gained any intellectual drive and research in other parts of the world had faded. In the second half of the eighth century Islam had a cultural awakening, and research in mathematics and the sciences increased.[4] The Muslim Abbasid caliph al-Mamun (809-833) is said to have had a dream where Aristotle appeared to him, and as a consequence al-Mamun ordered that Arabic translation be made of as many Greek works as possible, including Ptolemy's ''Almagest'' and Euclid's ''Elements''. Greek works would be given to the Muslims by the Byzantine Empire in exchange for treaties, as the two empires held an uneasy peace.4 Many of these Greek works were translated by Thabit ibn Qurra (826-901), who translated books written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.[5] 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.
Greek, Indian and Mesopotamian mathematics all played an important role in the development of 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 Islamic mathematics. Perhaps the most influential mathematical contribution from India was the decimal place-value Indo-Arabic numeral system, also known as the Hindu numerals.[6] 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 to Baghdad which was translated into Arabic as ''Sindhind''. It is generally assumed that ''Sindhind'' is none other than Brahmagupta's ''Brahmasphuta-siddhanta''.[7] The earliest translations from Sanskrit inspired several astronomical and astrological Arabic works, now mostly lost, some of which were even composed in verse.
But Indian influences were soon overwhelmed by Greek mathematical and astronomical texts. It is not clear why this occurred but it may have been due to the greater availability of Greek texts in the region, the larger number of practitioners of Greek mathematics in the region, or because Islamic mathematicians favored the deductive exposition of the Greeks over the elliptic Sanskrit verse of the Indians. Regardless of the reason, Indian mathematics became eclipsed by or merged with the "Graeco-Islamic" science founded on Hellenistic treatises.[8]
Biographies
; (786 – 833)
:Al-Ḥajjāj translated Euclid's ''Elements'' into Arabic.
; (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.
; (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.
; (fl. 830 Baghdad)
:Ibn Turk wrote a work on algebra of which only a chapter on the solution of quadratic equations has survied.
; (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.
; (c. 800 Baghdad – 873+ Baghdad)
:The Banū Mūsā where three brothers who worked at the House of Wisdom in Baghdad. Their most famous mathematical treatise is ''The Book of the Measurement of Plane and Spherical Figures'', which considered similar problems as Archimedes did in his ''On the measurement of the circle'' and ''On the sphere and the cylinder''. They contributed individually as well. The eldest, (c. 800) specialised in geometry and astronomy. He wrote a critical revision on Apollonius' ''Conics'' called ''Premises of the book of conics''. (c. 805) specialised in mechanics and wrote a work on pneumatic devices called ''On mechanics''. The youngest, (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)
; (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)
; (Iraq-Iran, ca. 940-1000)
;Al-Khujandi
; (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)
; (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)
; (c. 1130 Baghdad – c. 1180 Maragha)
; (Iran, ca. 1150-1215)
;Ibn Mun`im (Maghreb, ca. 1210)
;al-Marrakushi (Morocco, 13th century)
; (18 February 1201 in Tus, Khorasan – 26 June 1274 in Kadhimain near Baghdad)
; (c. 1220 Spain – c. 1283 Maragha)
; (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)
;'' (1364 Bursa – 1436 Samarkand)
; (Iran, Uzbekistan, ca. 1420)
;Ulugh Beg (Iran, Uzbekistan, 1394-1449)
;Al-Umawi
;Al-Qalasadi (Maghreb, 15th century)
Fields
Algebra
A page from ''The Compendious Book on Calculation by Completion and Balancing''.
There are three theories about the origins of Arabic Algebra. The first emphasizes Hindu influence, the second emphasizes Mesopotamian or Persian-Syriac influence and the third emphasizes Greek influence. Many scholars believe that it is the result of a combination of all three sources.[9]
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.[10]
The Muslim[11]
Persian mathematician 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''.[12]
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.[11]
''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).[14]
'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''.[15]
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.15 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.15
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.[16]
Al-Karkhi only considered positive roots.
Omar Khayyám (c. 1050-1123) wrote a book on Algebra that went beyond ''Al-Jabr'' to include equations of the third degree.[17]
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.17 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.17 He only considered positive roots and he did not go past the third degree.17 He also saw a strong relationship between Geometry and Algebra.17
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.
J. J. O'Conner and E. F. Robertson wrote in the ''MacTutor History of Mathematics archive'':
Arithmetic
Main articles: Arabic numerals
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.[18] The historian of mathematics, F. Woepcke,[19] 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.Victor J. Katz (1995). "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '68' (3), p. 163-174.
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.Glen M. Cooper (2003). "Omar Khayyam, the Mathmetician", ''The Journal of the American Oriental Society'' '123'. 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.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.
Cryptography
In the 9th century, al-Kindi was a pioneer in cryptanalysis and cryptology. He gave the first known recorded explanation of cryptanalysis in ''A Manuscript on Deciphering Cryptographic Messages''. In particular, he is credited with developing the frequency analysis method whereby variations in the frequency of the occurrence of letters could be analyzed and exploited to break ciphers (i.e. crypanalysis by frequency analysis).[20] 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 statistical analysis of letters and letter combinations in Arabic.[21]
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 and transposition, and for the first time, a cipher with multiple substitutions for each plaintext letter. Also traced to Ibn al-Duraihim is an exposition on and worked example of cryptanalysis, including the use of tables of letter frequencies and sets of letters which can not occur together in one word.
Geometry
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 here is an icon of religion as well as science, in reference to God as the architect of creation
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.[22]
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[23][24] and analytic geometry.
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.
In 1250, Nasir al-Din al-Tusi, 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 parallel postulate and on Omar Khayyám's attempted proof a century earlier. Nasir al-Din attempted to derive a contradiction of the parallel postulate. His son, Sadr al-Din wrote a book on the subject in 1298, based on Nasir al-Din's later thoughts, which presented an argument for a hypothesis equivalent to the parallel postulate. 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 discovery of non-Euclidean geometry.[25]
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.[26]
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.[27]
Number theory
In number theory, Ibn al-Haytham solved problems involving 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 numbers. In his ''Analysis and synthesis'', Ibn al-Haytham was the first to realize that every even perfect number is of the form 2''n''−1(2''n'' − 1) where 2''n'' − 1 is prime, but he was not able to prove this result successfully (Euler later proved it in the 18th century).[24]
Trigonometry
The early Indian works on trigonometry were translated and expanded in the Muslim world by Arab and Persian mathematicians. 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 2''x'' = 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
1. Hogendijk 1999
2. O'Connor 1999
3. The Persistence of Cultures in World History: Persia/Iran by Dr. Laina Farhat-Holzman
4. , , , Boyer, , 1991,
5. , , , Boyer, , 1991,
6. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, , J. Lennart, Berggren, Princeton University Press, 2007,
7. , , , Boyer, , 1991,
8. , , Kim, Plofker, , 2007,
9. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
10. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
11. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
12. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
13. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
14. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
15. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
16. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
17. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
18. Victor J. Katz (1998). ''History of Mathematics: An Introduction'', p. 255-259. Addison-Wesley. ISBN 0321016181.
19. F. Woepcke (1853). ''Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi''. Paris.
20. Simon Singh. The Code Book. p. 14-20
21. Al-Kindi, Cryptgraphy, Codebreaking and Ciphers
22. Aydin Sayili (1960). "Thabit ibn Qurra's Generalization of the Pythagorean Theorem", ''Isis'' '51' (1), p. 35-37.
23. R. Rashed (1994). ''The development of Arabic mathematics: between arithmetic and algebra''. London.
24.
25. Victor J. Katz (1998). ''History of Mathematics: An Introduction'', p. 270-271. Addison-Wesley. ISBN 0321016181.
26. Victor J. Katz (1998). ''History of Mathematics: An Introduction'', p. 255-259. Addison-Wesley. ISBN 0321016181.
27. Victor J. Katz (1995). "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '68' (3), p. 163-174.
28.
References and further reading
★ Episodes in the Mathematics of Medieval Islam, , J. Lennart, Berggren, Springer-Verlag, 1986, ISBN 0-387-96318-9
★ The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, , J. Lennart, Berggren, Princeton University Press, 2007,
★ A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc, 1991,
★ The History of Mathematics: A Brief Course, , Roger, Cooke, Wiley-Interscience, 1997,
★ The Muslim contribution to mathematics, , Ali Abdullah al-, Daffa', Croom Helm, 1977, ISBN 0-85664-464-1
★ Studies in the exact sciences in medieval Islam, , Ali Abdullah al-, Daffa, Wiley, 1984, ISBN 0471903205
★ The Crest of the Peacock: Non-European Roots of Mathematics, , George Gheverghese, Joseph, Princeton University Press, 2000, ISBN 0691006598
★ Studies in the Islamic Exact Sciences, , E. S., Kennedy, Syracuse Univ Press, 1984, ISBN 0815660677
★ The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, , Kim, Plofker, Princeton University Press, 2007,
★
★ The Development of Arabic Mathematics: Between Arithmetic and Algebra, , Roshdi, Rashed, Springer, 2001, ISBN 0792325656
★ Biografías de Matemáticos Árabes que florecieron en España, , José A, Sánchez Pérez, Estanislao Maestre, 1921,
★ Geschichte Des Arabischen Schrifttums, , Fuat, Sezgin, Brill Academic Publishers, 1997, ISBN 9004020071
★ Die Mathematiker und Astronomen der Araber und ihre Werke, , Heinrich, Suter, , 1900,
★ Die Mathematik der Länder des Ostens im Mittelalter, , Adolf P., Youschkevitch, , 1960, Sowjetische Beiträge zur Geschichte der Naturwissenschaft pp. 62-160.
★ Les mathématiques arabes: VIIIe-XVe siècles, , Adolf P., Youschkevitch, Vrin, 1976, ISBN 978-2-7116-0734-1
External links
★ Hogendijk, Jan P. (January 1999). ''Bibliography of Mathematics in Medieval Islamic Civilization''.
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