In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics, is the mathematics developed in the Islamic world between.

MathematicsScientests and Mathematics in

medieval Islam

Mathematics in medieval Islam

In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics, is the mathematics developed in the Islamic world between 622 and 1600, during what is known as the Islamic Golden Age, in that part of the world where Islam was the dominant religion. Islamic science and mathematics flourished under the Islamic caliphate (also known as the Islamic Empire) established across the Middle East, Central Asia, North Africa, Southern Italy, the Iberian Peninsula, and, at its peak, parts of France and India as well. Islamic activity in mathematics was largely centered around modern-day Iraq and Persia, but at its greatest extent stretched from North Africa and Spain in the west to India in the east.[1]

While most scientists in this period were Muslims and wrote in Arabic,[2] many of the best known contributors were Persians[3][4] as well as Arabs,[4] in addition to Berber, Moorish and Turkic contributors, as well as some from other religions (Christians, Jews, Sabians, Zoroastrians, and the irreligious).[2] Arabic was the dominant language—much like Latin in Medieval Europe, Arabic was the written lingua franca of most scholars throughout the Islamic world.

Islam and mathematics

Islam and mathematics

A major impetus for the flowering of mathematics as well as astronomy in medieval Islam came from religious observances, which presented an assortment of problems in astronomy and mathematics, specifically in trigonometry, spherical geometry,[13] algebra[14] and arithmetic.[15]

The Islamic law of inheritance served as an impetus behind the development of algebra (derived from the Arabic al-jabr) by Muhammad ibn Mūsā al-Khwārizmī and other medieval Islamic mathematicians. Al-Khwārizmī's Hisab al-jabr w’al-muqabala devoted a chapter on the solution to the Islamic law of inheritance using algebra. He formulated the rules of inheritance as linear equations, hence his knowledge of quadratic equations was not required.[14] Later mathematicians who specialized in the Islamic law of inheritance included Al-Hassār, who developed the modern symbolic mathematical notation for fractions in the 12th century,[15] and Abū al-Hasan ibn Alī al-Qalasādī, who developed an algebraic notation which took "the first steps toward the introduction of algebraic symbolism" in the 15th century.[16]

In order to observe holy days on the Islamic calendar in which timings were determined by phases of the moon, astronomers initially used Ptolemy's method to calculate the place of the moon and stars. The method Ptolemy used to solve spherical triangles, however, was a clumsy one devised late in the first century by Menelaus of Alexandria. It involved setting up two intersecting right triangles; by applying Menelaus' theorem it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the sun's altitude, for instance, repeated applications of Menelaus' theorem were required. For medieval Islamic astronomers, there was an obvious challenge to find a simpler trigonometric method.[13]

Regarding the issue of moon sighting, Islamic months do not begin at the astronomical new moon, defined as the time when the moon has the same celestial longitude as the sun and is therefore invisible; instead they begin when the thin crescent moon is first sighted in the western evening sky.[13] The Qur'an says: "They ask you about the waxing and waning phases of the crescent moons, say they are to mark fixed times for mankind and Hajj."[17][18] This led Muslims to find the phases of the moon in the sky, and their efforts led to new mathematical calculations.

Predicting just when the crescent moon would become visible is a special challenge to Islamic mathematical astronomers. Although Ptolemy's theory of the complex lunar motion was tolerably accurate near the time of the new moon, it specified the moon's path only with respect to the ecliptic. To predict the first visibility of the moon, it was necessary to describe its motion with respect to the horizon, and this problem demands fairly sophisticated spherical geometry. Finding the direction of Mecca and the time of Salah are the reasons which led to Muslims developing spherical geometry. Solving any of these problems involves finding the unknown sides or angles of a triangle on the celestial sphere from the known sides and angles. A way of finding the time of day, for example, is to construct a triangle whose vertices are the zenith, the north celestial pole, and the sun's position. The observer must know the altitude of the sun and that of the pole; the former can be observed, and the latter is equal to the observer's latitude. The time is then given by the angle at the intersection of the meridian (the arc through the zenith and the pole) and the sun's hour circle (the arc through the sun and the pole).[13][20]

Muslims are also expected to pray towards the Kaaba in Mecca and orient their mosques in that direction. Thus they need to determine the direction of Mecca (Qibla) from a given location.[21][22] Another problem is the time of Salah. Muslims need to determine from celestial bodies the proper times for the prayers before sunrise, at midday, in the afternoon, at sunset, and in the evening.[13][20]

Carl Friedrich Gauss Johann Carl Friedrich Gauss (30 April 1777 – 23 February

1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.

Benjamin Peirce

He was born in (April 4, 1809 – October 6, 1880) was an American mathematician who taught at Harvard University for about fifty years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philosophy of mathematics.

John Forbes Nash, Jr.

John Forbes Nash, Jr. (born June 13, 1928) is an American economist and mathematician whose works in game theory, differential geometry, and partial differential equations have provided insight into the forces that govern chance and events inside complex systems in daily life. His theories are used in market economics, computing, evolutionary biology, artificial intelligence, accounting and military theory. Serving as a Senior Research Mathematician at Princeton University during the later part of his life, he shared the 1994 Nobel Memorial Prize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi.

Nash is the subject of the Hollywood movie A Beautiful Mind. The film, based rather loosely on the biography of the same name, focuses on Nash's mathematical genius and struggle with paranoid schizophrenia.

Leonhard Euler Leonhard Euler (15 April 1707 – 18 September 1783) was a

pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. His surname is pronounced /ˈɔɪlər/ OY-lər (like "Oiler") in English and [ˈɔʏlɐ] in German; the pronunciation /ˈjuːlər/ EW-lər is incorrect.[1][2][3][4]

Euler made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[5] He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy.

Euler is considered to be the preeminent mathematician of the 18th century and arguably the greatest of all time. He is also one of the most prolific; his collected works fill 60–80 quarto volumes.[6] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."[7]

Euler was featured on the sixth series of the Swiss 10-franc banknote and on numerous Swiss, German, and Russian postage stamps. The asteroid 2002 Euler was named in his honor. He is also commemorated by the Lutheran Church on their Calendar of Saints on 24 May – he was a devout Christian (and believer in biblical inerrancy) who wrote apologetics and argued forcefully against the prominent atheists of his time.[8]

Sir Isaac Newton

Isaac Newton was born on December 25, 1642 (by the Julian calendar then in use; or January 4, 1643 by the current Gregorian calendar) in Woolsthorpe, near Grantham in Lincolnshire, England. He was born the same year Galileo died. Newton is clearly the most influential scientist who ever lived. His accomplishments in mathematics, optics, and physics laid the foundations for modern science and revolutionized the world.

Arne Beurling

Arne Carl-August Beurling (February 3, 1905 – November 20, 1986) was a Swedish mathematician and professor of mathematics at Uppsala University (1937-1954) and later at the Institute for Advanced Study in Princeton, USA.

Arne Beurling worked extensively in harmonic analysis, complex analysis and potential theory. The "Beurling factorization" helped mathematical scientists to understand the Wold decomposition, and inspired further work on the invariant subspaces of linear operators and operator algebras, e.g. Håkan Hedenmalm's factorization theorem for Bergman spaces.

In the summer of 1940 he single-handedly deciphered and reverse-engineered an early version of the Siemens and Halske T52 also known as the Geheimfernschreiber (secret teletypewriter) used by Nazi Germany in World War II for sending ciphered messages. The T52 was one of the so-called "Fish cyphers" and it took Beurling two weeks to solve the problem using paper and pen. Using Beurling's work, a device was created that enabled Sweden to decipher German teleprinter traffic passing through Sweden from Norway on a cable. In this way, Swedish authorities knew about Operation Barbarossa before it occurred. This became the foundation for the Swedish National Defence Radio Establishment (FRA). The cypher in the Geheimfernschreiber is generally considered to be more complex than the cypher used in the Enigma machines.

Gerard de Zeeuw

Gerard de Zeeuw (11 March 1936) is a Dutch scientist and professor Mathematical modelling of complex social systems at the University of Amsterdam in the Netherlands. Gerard de Zeeuw was born in 1936 in Banjoewangi Indonesia, in the former Dutch Indies, to relatively well-to-do middle class parents and one older brother. Gerard spent part of the Second World War in Japanese prison camps. In secondary school a Dutch professor of sociology, Free van Heek, allowed him access to his personal library and stimulated him to write his first paper—about action.[1] He studied in Mathematics and physics at the Leiden University, and continued with Statistics and Econometrics at the Erasmus University with Jan Tinbergen en Henk Theil. At the Stanford University he further studied mathematical psychology, with Patrick Suppes en Bob Estes. Back in Holland at the University of Amsterdam he received his Ph.D. with the thesis entitled Model thinking in psychology.

Don Kirkham

Don Kirkham (? - 1998) was a distinguished soil scientist regarded as the founder of mathematical soil physics. His special interest was the flow of water through soils and drainage of agricultural land. He was awarded the Robert E. Horton Medal in 1995.

Al-Ḥajjāj ibn Yūsuf ibn Maṭar

Al-Ḥajjāj ibn Yūsuf ibn Maṭar (786–833 CE) was an Arab mathematician who first translated Euclid's Elements from Greek into Arabic. He made a second, improved, more concise translation for the Caliph al-Ma'mūn (813-833). Around 829, he translated Ptolemy's Almagest, which at that time had also been translated by Hunayn ibn Ishaq and Sahl al-Ṭabarī.

At the beginning of the 12th century CE, Adelard of Bath translated al-Ḥajjāj's version of Euclid's Elements into Latin.

Abd al-Rahman al-Sufi

His name implies that he was a Sufi Muslim. He lived at the court of Emir Adud ad-Daula in Isfahan, Persia, and worked on translating and expanding Greek astronomical works, especially the Almagest of Ptolemy. He contributed several corrections to Ptolemy's star list and did his own brightness and magnitude estimates which frequently deviated from those in Ptolemy's work.

He was a major translator into Arabic of the Hellenistic astronomy that had been centred in Alexandria, the first to attempt to relate the Greek with the traditional Arabic star names and constellations, which were completely unrelated and overlapped in complicated ways.

Yusuf al-Mu'taman ibn Hud

Yusuf ibn Ahmad al-Mu'taman ibn Hud was an Arab mathematician and a member of the Banu Hud family, al-Mutamin ruled Zaragoza from 1082 to 1085. He was the son of the previous ruler, Ahmad ibn Sulayman al-Muqtadir. He also wrote Kitab al-Istikmal (Arabic, اإلستكمال Book of Perfection) in ,كتابmathematics.

Muhammad ibn Mūsā al-Khwārizmī

Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī[1] (Persian/Arabic: الخوارزمي موسى بن محمد الله عبد ,c. 780) (أبوKhwārizm[2][3][4] – c. 850) was a Persian[5][2][6] mathematician, astronomer and geographer, a scholar in the House of Wisdom in Baghdad.

His Kitab al-Jabr wa-l-Muqabala presented the first systematic solution of linear and quadratic equations. He is considered the founder of algebra,[7] a credit he shares with Diophantus. In the twelfth century, Latin translations of his work on the Indian numerals, introduced the decimal positional number system to the Western world.[4] He revised Ptolemy's Geography and wrote on astronomy and astrology.

His contributions had a great impact on language. "Algebra" is derived from al-jabr, one of the two operations he used to solve quadratic equations. Algorism and algorithm stem from Algoritmi, the Latin form of his name.[8] His name is the origin of (Spanish) guarismo[9] and of (Portuguese) algarismo, both meaning digit.

Abū al-Wafā' Būzjānī

Abul Wafa Buzjani (10 June 940 – 1 July 998)[1] (Persian: بوزجانی extended ,(ابوالوفاname: Abū al-Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī (Persian: ابوالوفاالعباس بن اسماعیل بن یحیی بن محمد بن محمد was a Persian mathematician (البوزجانیand astronomer. He was born in Buzhgan, (now Torbat-e Jam) in Iran.

In 959 AD, he moved to Iraq. He studied mathematics and worked principally in the field of trigonometry. He wrote a number of books, most of which no longer exist. He also studied the movements of the moon. The crater Abul Wáfa on the Moon is named after him.

Abu Nasr Mansur

Abu Nasr Mansur ibn Ali ibn Iraq (c. 960 - 1036) was a Persian [1] Muslim mathematician.[2] He is well known for his work with the spherical sine law.[3][4]

Abu Nasr Mansur was born in Gilan, Persia, to the ruling family of Khwarezm, the "Banu Iraq". He was thus a prince within the political sphere. He was a student of Abu'l-Wafa and a teacher of and also an important colleague of the mathematician, Al-Biruni. Together, they were responsible for great discoveries in mathematics and dedicated many works to one another.

Most of Abu Nasr's work focused on math, but some of his writings were on astronomy. In mathematics, he had many important writings on trigonometry, which were developed from the writings of Ptolemy. He also preserved the writings of Menelaus of Alexandria and reworked many of the Greeks theorems.

He died in the Ghaznavid Empire (modern-day Afghanistan) near the city of Ghazna.

Ibrahim ibn Sinan

Ibrahim ibn Sinan ibn Thabit ibn Qurra (908, Baghdad – 946, Baghdad) was an ethnic Mandean mathematician and astronomer who studied geometry and in particular tangents to circles. He also made advances in the theory of integration. He is often referenced as one of the most important mathematicians of his time. Modern research indicate that he belonged to the Mandaean religion and ethnic group. [1]

The Mandeans are a non Arab, Aramaic speaking people who like the Assyrians, are descendants of the ancient Mesopotamians.

Ibn Yahyā al-Maghribī al-Samaw'al

Ibn Yaḥyā al-Maghribī al-Samawʾal المغربي يحيى بن السموأل ،also known as Samau'al al-Maghribi (c. 1130 in Baghdad, Iraq – c. 1180 in Maragha, Iran) was a Muslim mathematician and astronomer of Jewish descent.[1] Though born to a Jewish family, he converted to Islam in 1163 after he had a dream telling him to do so [2] His father was a Jewish Rabbi from Morocco.[3] Al-Samaw'al wrote the mathematical treatise al-Bahir fi'l-jabr, meaning "The brilliant in algebra", at the young age of nineteen.

He also developed the concept of proof by mathematical induction, which he used to extend the proof of the binomial theorem and Pascal's triangle previously given by al-Karaji. Al-Samaw'al's inductive argument was only a short step from the full inductive proof of the general binomial theorem.[4]

Sharaf al-Dīn al-Ṭūsī

Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī ( طوسی ) (شرف الدین 1135–1213 ) was a Persian mathematician and astronomer of the Islamic Golden Age (during the Middle Ages). Tusi taught various mathematical topics including the science of numbers, astronomical tables and astrology, in Aleppo and Mosul. His best pupil was Kamal al-Din ibn Yunus. In turn Kamal al-Din ibn Yunus went on to teach Nasir al-Din al-Tusi, one of the most famous of all the Islamic scholars of the period. By this time Tusi seems to have acquired an outstanding reputation as a teacher of mathematics, for some travelled long distances hoping to become his students.

Qāḍīzāda al-Rūmī

Qāḍīzāda al-Rūmī (1364 in Bursa, Turkey – 1436 in Samarkand, Uzbekistan), also spelled as Qadi-Zada, Qazizada, and Qazizade, whose actual name was Salah al-Din Musa Pasha (Qāḍī Zāda means "son of the judge"), was an Turkish astronomer and mathematician who worked at the observatory in Samarkand. He computed sin 1° to an accuracy of 10−12.

Together with Ulugh Beg, al-Kāshī and a few other astronomers he produced the Zij-i-Sultani, the first comprehensive stellar catalogue since the Maragheh observatory's Zij-i Ilkhani two centuries earlier. The Zij-i Sultani contained the positions of 992 stars.

Ibn al-Banna al-Marrakushi

Ibn al-Banna al-Marrakushi al-Azdi also known as Abu'l-Abbas Ahmad ibn Muhammad ibn Uthman al-Azdi. (Arabic: ا� البّن December (29] (ابن1256 – c. 1321) was an Arab mathematician and astronomer. The crater Al-Marrakushi on the Moon is named after him.

Al-Abbās ibn Said al-Jawharī

Al-ʿAbbās ibn Saʿid al-Jawharī (c. 800 Baghdad? – c. 860 Baghdad?) was a geometer who worked at the House of Wisdom in Baghdad and for in a short time in Damascus where he made astronomical observations. 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.

Ahmed ibn Yusuf

Ahmed ibn Yusuf ibn Ibrahim ibn Tammam al-siddiq Al-Baghdadi also known as Abu Ja'far Ahmad ibn Yusuf and Ahmed ibn Yusuf al-misri (835 - 912) was an Arab mathematician, like his father Yusuf ibn Ibrahim (Arabic يوسفبن

البغدادي الصد�يق Ahmed ibn Yusuf was born in Baghdad ( ابراهيم(today in Iraq) and moved with his father to Damascus in 839. He later moved to Cairo, but the exact date is unknown: since he was also known as al-Misri, which means the Egyptian, this probably happened at an early age. Eventually, he also died in Cairo. He probably grew up in a strongly intellectual environment: his father worked on Mathematics, Astronomy and Medicine, produced astronomical tables and was a member of a group of scholars. He achieved an important role in Egypt, which was caused by Egypt's relative independence from the Abbasid Caliph..

Ibn Tahir al-Baghdadi

Abu Mansur Abd al-Qahir ibn Tahir ibn Muhammad ibn Abdallah al-Tamimi al-Shaffi al-Baghdadi (Arabic: مّنصور أبو

التميمي عبدالله بن محمد بن طاهر ابن عبدالقاهرالبغدادي was an Arabian (الشافعي

mathematician (c. 980–1037) from Baghdad who is best known for his treatise al-Takmila fi'l-Hisab. It contains results in number theory, and comments on works by al-Khwarizmi which are now lost.

Ibn Sahl

Ibn Sahl (Arabic: سهل ابن :Arabic) ( Abu Saʿd al-ʿAlaʾ ibn Sahl) (ابن العالء سعد أبو was a Muslim mathematician, physicist and optics engineer (c. 940-1000) (سهلof the Islamic Golden Age associated with the Abbasid court of Baghdad. Ibn Sahl's 984[citation needed] treatise On Burning Mirrors and Lenses sets out his understanding of how curved mirrors and lenses bend and focus light. Ibn Sahl is credited with first discovering the law of refraction, usually called Snell's law.[1][2] He used the law of refraction to derive lens shapes that focus light with no geometric aberrations, known as anaclastic lenses.

In the reproduction of the figure from Ibn Sahl's manuscript, the critical part is the right-angled triangle. The inner hypotenuse shows the path of an incident ray and the outer hypotenuse shows an extension of the path of the refracted ray if the incident ray met a crystal whose face is vertical at the point where the two hypotenuses intersect.[3] According to Rashed,[2] the ratio of the length of the smaller hypotenuse to the larger is the reciprocal of the refractive index of the crystal.

The lower part of the figure shows a representation of a plano-convex lens (at the right) and its principal axis (the intersecting horizontal line). The curvature of the convex part of the lens brings all rays parallel to the horizontal axis (and approaching the lens from the right) to a focal point on the axis at the left.

In the remaining parts of the treatise, Ibn Sahl dealt with parabolic mirrors, ellipsoidal mirrors, biconvex lenses, and techniques for drawing hyperbolic arcs.

Ibn Sahl's treatise was used by Ibn al-Haitham (965–1039), one of the greatest Persian scholars of optics. In modern times, Rashed found the text to have been dispersed in manuscripts in two different libraries, one in Teheran, and the other in Damascus. He reassembled the surviving portions, translated and published them.[4]

Jamshīd al-Kāshī

Ghiyāth al-Dīn Jamshīd ibn Masʾūd al-Kāshī (or Jamshīd Kāshānī, Persian: غیاث الدینکاشانی was a Persian (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) (جمشید

astronomer and mathematician.Al-Kashi was one of the best mathematicians in the Islamic world. He was born in 1380, in Kashan, in central Iran. This region was controlled by Tamurlane, better known as Timur, who was more interested in invading other areas than taking care of what he had. Due to this, al-Kashi lived in poverty during his childhood and the beginning years of his adulthood.

The situation changed for the better when Timur died in 1405, and his son, Shah Rokh, ascended into power. Shah Rokh and his wife, Goharshad, a Persian princess, were very interested in the sciences, and they encouraged their court to study the various fields in great depth. Their son, Ulugh Beg, was enthusiastic about science as well, and made some noted contributions in mathematics and astronomy himself. Consequently, the period of their power became one of many scholarly accomplishments. This was the perfect environment for al-Kashi to begin his career as one of the world’s greatest mathematicians.

Eight years after he came into power in 1409, Ulugh Beg founded an institute in Samarkand which soon became a prominent university. Students from all over the Middle East, and beyond, flocked to this academy in the capital city of Ulugh Beg’s empire. Consequently, Ulugh Beg harvested many great mathematicians and scientists of the Muslim world. In 1414, al-Kashi took this opportunity to contribute vast amounts of knowledge to his people. His best work was done in the court of Ulugh Beg, and it is said that he was the king’s favourite student.

Al-Kashi was still working on his book, called “Risala al-watar wa’l-jaib” meaning “The Treatise on the Chord and Sine”, when he died in 1429. Some scholars believe that Ulugh Beg may have ordered his murder, while others say he died a natural death. The details are unclear.

: اعدادالعتيبي مّنالالعّنزي تغريدالمهوس فاتن