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⚫ In the ], '''mathematics in medieval Islam''', often called '''Islamic mathematics''' or '''Arabic mathematics''', covers the body of ] preserved and advanced under the ] between circa 622 and c.1600.{{sfn|Hogendijk|1999}} The areas covers the Islamic ] established across the Middle East, extending from the ] in the west to the ] in the east and to the ] and ] in the south.
]'' by ].]]


[[File:Omar Kayyám - Geometric solution to cubic equation.svg|thumb|To solve the third-degree equation ''x''<sup>3
⚫ In the ], '''mathematics in medieval Islam''', often called '''Islamic mathematics''' or '''Arabic mathematics''', covers the body of ] preserved and advanced under the ] between circa 622 and c.1600.{{sfn|Hogendijk|1999}} ] and mathematics flourished under the Islamic ] established across the Middle East, extending from the ] in the west to the ] in the east and to the ] and ] in the south.
⚫ == History ==
⚫ === Induction ===
⚫ {{See also|Mathematical induction#History}}


⚫ The earliest implicit traces of mathematical induction can be found in ]'s ] (c. 300 BCE). The first explicit formulation of the principle of induction was given by ] in his ''Traité du triangle arithmétique'' (1665).
In his ''A History of Mathematics'', Victor Katz says that:{{sfn|Katz|1993}}


⚫ In between, implicit ] by induction for ] was introduced by ] (c. 1000) and continued by ], who used it for special cases of the ] and properties of ].
<blockquote>A complete history of mathematics of medieval Islam cannot yet be written, since so many of these Arabic manuscripts lie unstudied... Still, the general outline... is known. In particular, Islamic mathematicians fully developed the decimal place-value number system to include decimal fractions, systematised the study of algebra and began to consider the relationship between algebra and geometry, studied and made advances on the major Greek geometrical treatises of ], ], and ], and made significant improvements in plane and spherical geometry.</blockquote>


== Arithmetic ==
An important role was played by the translation and study of ], which was the principal route of transmission of these texts to Western Europe. Smith notes that:{{sfn|Smith|1958|loc=Vol. 1, Chapter VII.4}}
⚫ ===Irrationnals numbers===
<blockquote>The world owes a great debt to Arab scholars for preserving and transmitting to posterity the classics of Greek mathematics... their work was chiefly that of transmission, although they developed considerable ingenuity in algebra and showed some genius in their work in trigonometry.</blockquote>
====Magnitudes and irrationals====
According to Kline,{{quote|If ] had really offered a theory of irrationals}}, there are twa possible interpretations : {{quote|the first is that magnitudes themselves could be taken to be the irrational numbers, and the second is that the ratios of two magnitudes could be the irrational numbers}}<ref>Mathematical Thought from Ancient to Modern Times, Volume 1, page 72, Oxford University Press, 1972</ref>


⚫ The Greeks had discovered ]s, but were not happy with them and only able to cope by drawing a distinction between ''magnitude'' and ''number''. In the Greek view, magnitudes varied continuously and could be used for entities such as line segments, whereas numbers were discrete. Hence, irrationals could only be handled geometrically; and indeed Greek mathematics was mainly geometrical. The Roman mathematicians<ref>Also called Greek mathematician even if they belonged to the Roman Empire</reF> from the ]<ref>Amongst them Diophantus and Hippathia</ref> did not make any distinction between magnitude and number, allowing irrational quantities to appear as coefficients in equations and to be solutions of algebraic equations and works freely with irrationals as objects, without examining closely their nature.Islamic mathematicians including ] copied the greek works.<ref>http://www.math.tamu.edu/~dallen/history/infinity.pdf</ref>
] states regarding the role of Islamic mathematics:<ref>{{Citation|last=Sertima|first=Ivan Van|title=Golden age of the Moor, Volume 11|year=1992|publisher=Transaction Publishers|isbn=1-56000-581-5|authorlink=Ivan van Sertima|page=394}}</ref>
====The use of irrationnals====
<blockquote>The Islamic mathematicians exercised a prolific influence on the development of science in Europe, enriched as much by their own discoveries as those they had inherited by the Greeks, the Indians, the Syrians, the Babylonians, etc.</blockquote>
Babylonians used tables of cibic and square roots, providing accurates approximations like {{sqrt|2}} = 1,414213…<ref>According to {{Harvsp|Kline|1972}}, one cannot say if Babylonians were aware of the existence of an infinity of decimals or sexagesimals for the irrationnal numbers or if they believed they can convert them in a finite number of sexagesimal if they have more place in the board they used.</ref>{{,}}<ref>{{Harvsp|Kline|1972|p=8}}.</ref>. The discovery of irrationnal numbers was made by ] (V th century)<ref>{{Harvsp|Kline|1972|p=32}}.</ref>. The proof of the irrationality of {{sqrt|2}} was made by ] using an "ab absurdo" demonstration<ref>Reductio ad absurdum.</ref>{{,}}<ref name="mk33">{{Harvsp|Kline|1972|p=33}}.</ref>. ] has proved the irrationality of {{sqrt|3}}, {{sqrt|5}} and {{sqrt|7}}. ] realised a first classification of irrationnal numbers<ref name="mk33" /> . ] analysed all the possibilities for lines that could be mathematizied by <math>\sqrt{\sqrt{a}+\sqrt{b}}</math> (a et b are lines). Romans of Alexandria<ref>qualified often by Alexandrians, greek mathématicians belonging to the Roman Empire</ref>, at the opposite of the Greek who did not recognize the irrationals like numbers, used them like numbers for measuring the lenght, the area and the volume<ref>{{Harvsp|Kline|1972|p=104}}.</ref>. Hindus (200-1200) used correct operations with irrational numbers, like addition and substraction. The principle is : <math>\sqrt{a}+\sqrt{b}=\sqrt{(a+b)+2\sqrt{ab}}</math><ref name="mk185">{{Harvsp|Kline|1972|p=185}}.</ref>. The persian mathematician ] (1048-1122) stated that all ratio of measure, rational or irrational, could be named a number<ref>{{Harvsp|Kline|1972|p=191-192}}.</ref>. The Arabs copied the operations on irrationals introduced by Hindus<ref name="mk192"/>. One must wait until the XVIIIth century to have a good survey of irrational numbers. In 1727, Bernard le Bovier de Fontenelle stated that {{quote|the irrational numbers<ref>(French) les nombres irrationnes se trouvent en une quantité sans comparaison plus grande que les nombres rationnels</ref> are proportionnaly more and more important that rationnal}}<ref>Jean Mawhin, ''Analyse, fondements, technique, évolution'', De Boeck Université, Bruxelles, 1992, {{1re}} édition, p. 38.</ref>


Inside the work of synthesis made by Arabs, was ] who used the irrationnal numbers and ]. Another arab mathematician of the IX th century, ] who translated greek manuscripts but was also studying the ].
⚫ == History ==
] and ] based on the angle between the horizontal line and true horizon from the peak of a mountain with known height.<ref>{{MacTutor|id=Al-Biruni|title=Al-Biruni}}</ref><ref>{{Citation|last=Douglas|first=A. V.|title=R.A.S.C. Papers- Al-Biruni, Persian Scholar|journal=Journal of the Royal Astronomical Society of Canada|year=1973|volume=67|pages=973–1048|bibcode=1973JRASC..67..209D}}</ref>]]


===Positional numeration===
The most important contribution of the Islamic mathematicians was the development of algebra; combining Indian and Babylonian material with the Greek geometry to develop algebra.
⚫ In the twelfth century, ] translations of ]'s ] on the ] introduced the ] ] to the ].<ref name="Struik 93">{{harvnb|Struik|1987| p= 93}}</ref> His '']'' presented the first systematic solution of ] and ]s. In Renaissance Europe, he was considered the original inventor of algebra, although it is now known that his work is based on older Indian or Greek sources.<ref>{{harvnb|Rosen|1831|p=v&ndash;vi}}; {{harvnb|Toomer|1990}}</ref> He revised ]'s '']'' and wrote on astronomy and astrology.
In Algebra a mathematician substitutes symbols such as x, y or z for numbers in order to solve mathematical problems.


In 628, the Hindu astronomer and mathematician ] published ''Brahma Siddhanta'', treaty where he exposed a decimal numeration exactly the same that we use today<ref>{{Ouvrage|prénom=Fernando|nom1=Corbalan|titre=Le Nombre d'or|sous-titre=Le langage mathématique de la beauté|traducteur=Youssef Halaoua, Maguy Ly et Laurence Moinereau|mois=9|année=2011|éditeur={{Lien|lang=es|trad=RBA|Groupe RBA|texte=RBA}}-]|collection=Le Monde est mathématique|numéro dans collection=1||passage=18}}.</ref>. The ] was copied by arab mathematician.
⚫ === Irrational numbers ===
===The positional zero indicating a missing cipher===
⚫ The Greeks had discovered ]s, but were not happy with them and only able to cope by drawing a distinction between ''magnitude'' and ''number''. In the Greek view, magnitudes varied continuously and could be used for entities such as line segments, whereas numbers were discrete. Hence, irrationals could only be handled geometrically; and indeed Greek mathematics was mainly geometrical. Islamic mathematicians including ] slowly removed the distinction between magnitude and number, allowing irrational quantities to appear as coefficients in equations and to be solutions of algebraic equations. They worked freely with irrationals as objects, but they did not examine closely their nature.<ref>http://www.math.tamu.edu/~dallen/history/infinity.pdf</ref>
The positionnal zero was used for the first time by Babylonians during the Seleucid period the take over by the Greeks during Alexandrian period. They used different symbols for the representation of the positional zero<ref>{{Harvsp|Kline|1972|p=132}}.</ref> comme : <math>\overline{{0}}</math>. The arab mathematician copied the idea.
===The zero as number===
Hindus treated the zero as a number. The hindu mathematician announced that the multiplication of a number by zero is equal to zero and that substract zero do not diminish a number. According to him, to divide a number by zero maintain this number unchanged<ref>Today, mathematicians say that to divide a number by zero is equal to multiply it by infinity.</ref>{{,}}<ref name="mk185" />. The arab mathematiocian copied the hindu model.
===Calculus of Pi===
The persian astronomer and mathematician ] calculated an approached value of ]. ] has calculated 3 decimals.


== Algebra ==
⚫ In the twelfth century, ] translations of ]'s ] on the ] introduced the ] ] to the ].<ref name="Struik 93">{{harvnb|Struik|1987| p= 93}}</ref> His '']'' presented the first systematic solution of ] and ]s. In Renaissance Europe, he was considered the original inventor of algebra, although it is now known that his work is based on older Indian or Greek sources.<ref>{{harvnb|Rosen|1831|p=v&ndash;vi}}; {{harvnb|Toomer|1990}}</ref> He revised ]'s '']'' and wrote on astronomy and astrology.
The attribution to ] of the origin of algebra is discuussed since the ]<ref>G. Cifoletti, La question de l'algèbre. Mathématiques et rhétorique des hommes de droit dans la France du {{XVIe|s}} siècle, Annales, 1995, vol. 50, p. 1385-1416, {{lire en ligne|url=http://www.persee.fr/web/revues/home/prescript/article/ahess_0395-2649_1995_num_50_6_279438}}.</ref>. According to ]<ref>« Algèbre d'al-ğabr » et « algèbre d'arpentage » au neuvième siècle islamique et la question de l'influence babylonienne, Fr. Mawet et Ph. Talon, D'Imhotep à Copernic. Astronomie et mathématiques des origines orientales au moyen âge. Actes du Colloque international, Université Libre de Bruxelles, 3‑4 novembre 1989. (Lettres Orientales, Leuven, Peeters, 1992 , p. 88.</ref>, {{quote|Al-Khwarismi informe us that he has not create this branch of mathematics}}<ref>(French) Al-Khwarismi nous informe qu'il n'a pas inventé la discipline </ref>{{,}}<ref>The algebra and the algorithms of calculus</ref>. He quote, giving strenght to his argumentation, the passage of '"Al-Gabr w'al-muqabala" where Al-Khwarismi refers to "arithmetics used by a notary sollicitor"<ref>(French) arithmétiques utilisées constamment dans les affaires d'héritage et de legs</ref>{{,}}<ref>Translation of Sayili, 1962.</ref>. According to him {{quote|Al-Kwarismi only poduced a synthesis work of disciplines and algorithm, including the genuine superstructure}}<ref>(French) al-Khwarismi n'a fait "que" produire une oeuvre de synthèse des disciplines et techniques des calculateurs pratiques, y compris la superstructure "pure"</ref>{{,}}<ref>By superstructure, one can understand problems encoutered during professional life and the skill to solve them. Genuine refers to genuine mathematics or scientific mathematics used by Greek. (Summary of a note of {{Harvsp|Høyrup|1992|p=89}}).</ref>{{,}}<ref name=Hoyrup88/>. This synthesis work will play a role during the ]<ref>{{Harvsp|Høyrup|1992|p=108}}.</ref>{{,}}<ref>The auteur say also that this role is probable concerning ], but may be also to ].</ref>.
===Resolution of a system of equations===
] solved equations with multiples unknows reducing the system to one unknown and solved it. ] copied the method<ref name="mk192">{{Harvsp|Kline|1972|p=192}}.</ref>. According to Kline, the rab algebra is only rhetorical and was a backstep if we copare it to indian algebra and to Diophantus's algebra.<ref name="mk192" />.
===Equation of second degree===
The resolution of second degree equation was well-known since Babylonians<ref>{{Harvsp|Kline|1972|p=193}}.</ref>. ]<ref>The name of this mathematician, latinised in ''Algoritmi'', gived the word ''algorithm''.</ref> only explained the algebraic process by geometry.


The second treaty, ''Kitab fi'l-jabr wa'l-muqabala'' <ref>(French) Livre sur la restauration et la confrontation</ref> expose only operations on equations. He propose the resolution of ] by a completion of squares.
⚫ === Induction ===

⚫ {{See also|Mathematical induction#History}}
===Using of latin letters to represent the unknowns===
Diophante used the first letter of a word state that it is a quantity, i.e. <math>s</math> for side and even <math>s^{2}</math> to represent what he called the "power" of this quantity. ] used the same words and the same sympbolic<ref name="mk192" />.
====Using of ''{{math|x}} '' to represent the unknown====
Many arab books use the ''{{math|x}} '' to represent the unknown. Given the great destruction of books of mathematics during "autodafés"<ref>Like in the book Farhenheit 351</ref> (this of general ] but also this of the "vanity butcher" of]), one cannot say precisely wich mathematician (he could be also greek, indian or arabic) who used for the first time the ''{{math|x}} '' to represent the unknown. According to the well-known group of mathematicians ], {{quote|Diophante uses, for the first time, a litteral symbol to represent the unknown}}<ref>Nicolas Bourbaki, '']'', Masson, Paris, 1994, {{3e}} tirage, p. 69.</ref>{{,}}<ref>The authors insinuate that Diophantus has created the ''{{math|x}} '' in mathematics because they don't quote any arab mathematician in the chapter devoluted to algebra.</ref>

== Geometry ==
The property of plane geometry were well-known since ] (Books XI and XIII from Eléments)<ref>{{Harvsp|Kline|1972|p=85-86}}.</ref>. These properties were used by Arabs to develop ]s. The deutch artist ] used mathematics in his work and was interested by ] after having visit the Alhambra in 1936<ref>{{Harvsp|Corbalan|2011|p=81}}.</ref>.
== Trigonometry ==
The founder of trigonometry is ]<ref name="mk119">{{Harvsp|Kline|1972|p=119}}.</ref>. The division of the circonference of a circle in 360 degrees<ref>The circonference is divided in 60 parts, each part is divided in 60 parts of parts. The diametra is divided in 120 parts.</ref> is dued to ] of Alexandria<ref name="mk119"/>. Hipparchus calculated the number of units for a ] corresponding to a given number of degrees, and that's the same of our sinus function<ref>La corde est le double du côté d'un triangle rectangle inscrit dans un quadrant de cercle et ayant comme rayon l'hypothénuse. Un côté de l'angle droit d'un triangle rectangle est égal au produit de l'hypothénuse par le sinus de l'angle opposé.</ref>{{,}}<ref name="mk120">{{harvsp|Kline|1972|p=120}}.</ref>. ] (98 A.D.) déveloped the greek trigonometry in his books "Sphaerica" and "Chords in a circle" where one can find the concept of spherical triangle. His book "Sphaerica" has been translated in arab<ref name="mk120" />. The trigonometry was continued by Hindus. The astronomer and mathematician ] copied the works realised before<ref>According to {{Harvsp|Kline|1972|p=195}}, The Arabs made very little progress in astronomy.</ref> and uses the sinus function and tangente function in astronomy and wrote tables to calculate them. The ] is dued to chinese mathematician Kuo Chou-tching (1231-1316) who created the chou-chih calendar<ref>Philip J.Davis et Reuben Hersh, ''L'univers mathématique'', traduit par L. Chambadal, Gauthier-Villars, 1985, p. 26.</ref>.


⚫ The earliest implicit traces of mathematical induction can be found in ]'s ] (c. 300 BCE). The first explicit formulation of the principle of induction was given by ] in his ''Traité du triangle arithmétique'' (1665).


⚫ In between, implicit ] by induction for ] was introduced by ] (c. 1000) and continued by ], who used it for special cases of the ] and properties of ].


== Major figures and developments == == Major figures and developments ==
=== Omar Khayyám === === Omar Khayyám ===
] ''x''<sup>2</sup>&nbsp;=&nbsp;''ay'', a ] with diameter ''b''/''a''<sup>2</sup>, and a vertical line through the intersection point. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the ''x''-axis.]] </sup>&nbsp;+&nbsp;''a''<sup>2</sup>''x''&nbsp;=&nbsp;''b'' Khayyám constructed the ] ''x''<sup>2</sup>&nbsp;=&nbsp;''ay'', a ] with diameter ''b''/''a''<sup>2</sup>, and a vertical line through the intersection point. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the ''x''-axis.]]
] (c. 1038/48 in ] – 1123/24){{sfn|Struik|1987|p=96}} wrote the ''Treatise on Demonstration of Problems of Algebra'' containing the systematic solution of ]s, going beyond the ''Algebra'' of ].{{sfn|Boyer|1991|pp=241–242}} Khayyám obtained the solutions of these equations by finding the intersection points of two ]s. This method had been used by the Greeks,{{sfn|Struik|1987|p=97}} but they did not generalize the method to cover all equations with positive ].{{sfn|Boyer|19991|pp=241–242}} ] (c. 1038/48 in ] – 1123/24){{sfn|Struik|1987|p=96}} wrote the ''Treatise on Demonstration of Problems of Algebra'' containing the systematic solution of ]s, going beyond the ''Algebra'' of ].{{sfn|Boyer|1991|pp=241–242}} Khayyám obtained the solutions of these equations by finding the intersection points of two ]s. This method had been used by the Greeks,{{sfn|Struik|1987|p=97}} but they did not generalize the method to cover all equations with positive ].{{sfn|Boyer|19991|pp=241–242}}
<!-- <!--
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* {{Citation|last=Struik|year=1987|first=Dirk J.|authorlink=Dirk Jan Struik|title=A Concise History of Mathematics|edition=4th rev.|publisher=Dover Publications|isbn=0-486-60255-9|ref=harv}} * {{Citation|last=Struik|year=1987|first=Dirk J.|authorlink=Dirk Jan Struik|title=A Concise History of Mathematics|edition=4th rev.|publisher=Dover Publications|isbn=0-486-60255-9|ref=harv}}
{{refend}} {{refend}}

=== Bibliography ===
* Mathematical Tought from Ancient to Modern Times, Vol.1, Morris Kline, Oxford University Press, 1972
* Actes du Colloque International tenu à l'Université Libre de Bruxelles les 3-4 novembre 1989, D'Imhotep à Copernic, Astronomie et mathématiques des origines orientales au Moyen Âge, Lettres Orientales 2, Cahiers d'Altaïr, Édités par Fr.Mawet et Ph.Talon, Peeters-Leuven, 1992
* Le nombre d'Or, Fernando Corbalan, traduit par Youssef Halaoua, Maguy Ly, Laurence Moinereau, 2011, RBA Cillectionables S.A.
* Eléments d'histoire des mathématiques, Nicolas Bourbaki, Masson, Paris, 1994, 3eme tirage.


==Further reading== ==Further reading==

Revision as of 13:19, 30 July 2013

In the history of mathematics, mathematics in medieval Islam, often called Islamic mathematics or Arabic mathematics, covers the body of mathematics preserved and advanced under the Islamic civilization between circa 622 and c.1600. The areas covers the Islamic caliphate established across the Middle East, extending from the Iberian Peninsula in the west to the Indus in the east and to the Almoravid Dynasty and Mali Empire in the south.

To solve the third-degree equation x + ax = b Khayyám constructed the parabola x = ay, a circle with diameter b/a, and a vertical line through the intersection point. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the x-axis.

Omar Khayyám (c. 1038/48 in Iran – 1123/24) wrote the Treatise on Demonstration of Problems of Algebra containing the systematic solution of third-degree equations, going beyond the Algebra of Khwārazmī. Khayyám obtained the solutions of these equations by finding the intersection points of two conic sections. This method had been used by the Greeks, but they did not generalize the method to cover all equations with positive roots.

Sharaf al-Dīn al-Ṭūsī

Sharaf al-Dīn al-Ṭūsī (? in Tus, Iran – 1213/4) developed a novel approach to the investigation of cubic equations—an approach which entailed finding the point at which a cubic polynomial obtains its maximum value. For example, to solve the equation   x 3 + a = b x {\displaystyle \ x^{3}+a=bx} , with a and b positive, he would note that the maximum point of the curve   y = b x − x 3 {\displaystyle \ y=bx-x^{3}} occurs at x = b 3 {\displaystyle x=\textstyle {\sqrt {\frac {b}{3}}}} , and that the equation would have no solutions, one solution or two solutions, depending on whether the height of the curve at that point was less than, equal to, or greater than a. His surviving works give no indication of how he discovered his formulae for the maxima of these curves. Various conjectures have been proposed to account for his discovery of them.

Other major figures

See also

Notes

  1. Hogendijk 1999.
  2. Mathematical Thought from Ancient to Modern Times, Volume 1, page 72, Oxford University Press, 1972
  3. Also called Greek mathematician even if they belonged to the Roman Empire
  4. Amongst them Diophantus and Hippathia
  5. http://www.math.tamu.edu/~dallen/history/infinity.pdf
  6. According to Kline 1972 harvnb error: no target: CITEREFKline1972 (help), one cannot say if Babylonians were aware of the existence of an infinity of decimals or sexagesimals for the irrationnal numbers or if they believed they can convert them in a finite number of sexagesimal if they have more place in the board they used.
  7. Kline 1972, p. 8 harvnb error: no target: CITEREFKline1972 (help).
  8. Kline 1972, p. 32 harvnb error: no target: CITEREFKline1972 (help).
  9. Reductio ad absurdum.
  10. ^ Kline 1972, p. 33 harvnb error: no target: CITEREFKline1972 (help).
  11. qualified often by Alexandrians, greek mathématicians belonging to the Roman Empire
  12. Kline 1972, p. 104 harvnb error: no target: CITEREFKline1972 (help).
  13. ^ Kline 1972, p. 185 harvnb error: no target: CITEREFKline1972 (help).
  14. Kline 1972, p. 191-192 harvnb error: no target: CITEREFKline1972 (help).
  15. ^ Kline 1972, p. 192 harvnb error: no target: CITEREFKline1972 (help).
  16. (French) les nombres irrationnes se trouvent en une quantité sans comparaison plus grande que les nombres rationnels
  17. Jean Mawhin, Analyse, fondements, technique, évolution, De Boeck Université, Bruxelles, 1992, 1st édition, p. 38.
  18. Struik 1987, p. 93
  19. Rosen 1831, p. v–vi harvnb error: no target: CITEREFRosen1831 (help); Toomer 1990 harvnb error: no target: CITEREFToomer1990 (help)
  20. Corbalan, Fernando (9 2011). Le Nombre d'or: Le langage mathématique de la beauté. 1. Translated by Youssef Halaoua, Maguy Ly et Laurence Moinereau. RBA-Le Monde. p. 18. {{cite book}}: Check date values in: |date= (help).
  21. Kline 1972, p. 132 harvnb error: no target: CITEREFKline1972 (help).
  22. Today, mathematicians say that to divide a number by zero is equal to multiply it by infinity.
  23. G. Cifoletti, La question de l'algèbre. Mathématiques et rhétorique des hommes de droit dans la France du Template:XVIe siècle, Annales, 1995, vol. 50, p. 1385-1416, Template:Lire en ligne.
  24. « Algèbre d'al-ğabr » et « algèbre d'arpentage » au neuvième siècle islamique et la question de l'influence babylonienne, Fr. Mawet et Ph. Talon, D'Imhotep à Copernic. Astronomie et mathématiques des origines orientales au moyen âge. Actes du Colloque international, Université Libre de Bruxelles, 3‑4 novembre 1989. (Lettres Orientales, Leuven, Peeters, 1992 , p. 88.
  25. (French) Al-Khwarismi nous informe qu'il n'a pas inventé la discipline
  26. The algebra and the algorithms of calculus
  27. (French) arithmétiques utilisées constamment dans les affaires d'héritage et de legs
  28. Translation of Sayili, 1962.
  29. (French) al-Khwarismi n'a fait "que" produire une oeuvre de synthèse des disciplines et techniques des calculateurs pratiques, y compris la superstructure "pure"
  30. By superstructure, one can understand problems encoutered during professional life and the skill to solve them. Genuine refers to genuine mathematics or scientific mathematics used by Greek. (Summary of a note of Høyrup 1992, p. 89 harvnb error: no target: CITEREFHøyrup1992 (help)).
  31. Cite error: The named reference Hoyrup88 was invoked but never defined (see the help page).
  32. Høyrup 1992, p. 108 harvnb error: no target: CITEREFHøyrup1992 (help).
  33. The auteur say also that this role is probable concerning Cardan, but may be also to Viète.
  34. Kline 1972, p. 193 harvnb error: no target: CITEREFKline1972 (help).
  35. The name of this mathematician, latinised in Algoritmi, gived the word algorithm.
  36. (French) Livre sur la restauration et la confrontation
  37. Like in the book Farhenheit 351
  38. Nicolas Bourbaki, Éléments d'histoire des mathématiques, Masson, Paris, 1994, 3rd tirage, p. 69.
  39. The authors insinuate that Diophantus has created the x in mathematics because they don't quote any arab mathematician in the chapter devoluted to algebra.
  40. Kline 1972, p. 85-86 harvnb error: no target: CITEREFKline1972 (help).
  41. Corbalan 2011, p. 81 harvnb error: no target: CITEREFCorbalan2011 (help).
  42. ^ Kline 1972, p. 119 harvnb error: no target: CITEREFKline1972 (help).
  43. The circonference is divided in 60 parts, each part is divided in 60 parts of parts. The diametra is divided in 120 parts.
  44. La corde est le double du côté d'un triangle rectangle inscrit dans un quadrant de cercle et ayant comme rayon l'hypothénuse. Un côté de l'angle droit d'un triangle rectangle est égal au produit de l'hypothénuse par le sinus de l'angle opposé.
  45. ^ Kline 1972, p. 120 harvnb error: no target: CITEREFKline1972 (help).
  46. According to Kline 1972, p. 195 harvnb error: no target: CITEREFKline1972 (help), The Arabs made very little progress in astronomy.
  47. Philip J.Davis et Reuben Hersh, L'univers mathématique, traduit par L. Chambadal, Gauthier-Villars, 1985, p. 26.
  48. Struik 1987, p. 96.
  49. Boyer 1991, pp. 241–242.
  50. Struik 1987, p. 97.
  51. Boyer & 19991, pp. 241–242. sfn error: no target: CITEREFBoyer19991 (help)
  52. Berggren, J. Lennart; Al-Tūsī, Sharaf Al-Dīn; Rashed, Roshdi; Al-Tusi, Sharaf Al-Din (1990), "Innovation and Tradition in Sharaf al-Dīn al-Ṭūsī's al-Muʿādalāt", Journal of the American Oriental Society, 110 (2): 304–309, doi:10.2307/604533, JSTOR 604533

References

Bibliography

  • Mathematical Tought from Ancient to Modern Times, Vol.1, Morris Kline, Oxford University Press, 1972
  • Actes du Colloque International tenu à l'Université Libre de Bruxelles les 3-4 novembre 1989, D'Imhotep à Copernic, Astronomie et mathématiques des origines orientales au Moyen Âge, Lettres Orientales 2, Cahiers d'Altaïr, Édités par Fr.Mawet et Ph.Talon, Peeters-Leuven, 1992
  • Le nombre d'Or, Fernando Corbalan, traduit par Youssef Halaoua, Maguy Ly, Laurence Moinereau, 2011, RBA Cillectionables S.A.
  • Eléments d'histoire des mathématiques, Nicolas Bourbaki, Masson, Paris, 1994, 3eme tirage.

Further reading

Books on Islamic mathematics
Book chapters on Islamic mathematics
  • Berggren, J. Lennart (2007), "Mathematics in Medieval Islam", in Victor J. Katz (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Second ed.), Princeton, New Jersey: Princeton University, ISBN 978-0-691-11485-9{{citation}}: CS1 maint: ref duplicates default (link)
  • Cooke, Roger (1997), "Islamic Mathematics", The History of Mathematics: A Brief Course, Wiley-Interscience, ISBN 0-471-18082-3
Books on Islamic science
Books on the history of mathematics
  • Joseph, George Gheverghese (2000), The Crest of the Peacock: Non-European Roots of Mathematics (2nd ed.), Princeton University Press, ISBN 0-691-00659-8 (Reviewed: Katz, Victor J.; Joseph, George Gheverghese (1992), "The Crest of the Peacock: Non-European Roots of Mathematics by George Gheverghese Joseph", The College Mathematics Journal, 23 (1), Mathematical Association of America: 82–84, doi:10.2307/2686206, JSTOR 2686206)
  • Youschkevitch, Adolf P. (1964), Gesichte der Mathematik im Mittelalter, Leipzig: BG Teubner Verlagsgesellschaft
Journal articles on Islamic mathematics
Bibliographies and biographies
  • Brockelmann, Carl. Geschichte der Arabischen Litteratur. 1.–2. Band, 1.–3. Supplementband. Berlin: Emil Fischer, 1898, 1902; Leiden: Brill, 1937, 1938, 1942.
  • Sánchez Pérez, José A. (1921), Biografías de Matemáticos Árabes que florecieron en España, Madrid: Estanislao Maestre
  • Sezgin, Fuat (1997), Geschichte Des Arabischen Schrifttums (in German), Brill Academic Publishers, ISBN 90-04-02007-1
  • Suter, Heinrich (1900), Die Mathematiker und Astronomen der Araber und ihre Werke, Abhandlungen zur Geschichte der Mathematischen Wissenschaften Mit Einschluss Ihrer Anwendungen, X Heft, Leipzig{{citation}}: CS1 maint: location missing publisher (link)
Television documentaries

External links

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