Islamic mathematics
   HOME

TheInfoList



OR:

Mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
during the Golden Age of Islam, especially during the 9th and 10th centuries, was built upon syntheses of
Greek mathematics Ancient Greek mathematics refers to the history of mathematical ideas and texts in Ancient Greece during Classical antiquity, classical and late antiquity, mostly from the 5th century BC to the 6th century AD. Greek mathematicians lived in cities ...
(
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
,
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
,
Apollonius Apollonius () is a masculine given name which may refer to: People Ancient world Artists * Apollonius of Athens (sculptor) (fl. 1st century BC) * Apollonius of Tralles (fl. 2nd century BC), sculptor * Apollonius (satyr sculptor) * Apo ...
) and
Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, ...
(
Aryabhata Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the '' Āryabhaṭīya'' (which mentions that in 3600 ' ...
,
Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...
). Important developments of the period include extension of the place-value system to include
decimal fractions The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of the ...
, the systematised study of
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
and advances in
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and
trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...
. The medieval Islamic world underwent significant developments in mathematics. Muhammad ibn Musa al-Khwārizmī played a key role in this transformation, introducing algebra as a distinct field in the 9th century. Al-Khwārizmī's approach, departing from earlier arithmetical traditions, laid the groundwork for the arithmetization of
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, influencing mathematical thought for an extended period. Successors like
Al-Karaji (; c. 953 – c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works are mathematical: ''Al-Badi' fi'l-hisab'' (''Wonderful on ...
expanded on his work, contributing to advancements in various mathematical domains. The practicality and broad applicability of these mathematical methods facilitated the dissemination of Arabic mathematics to the West, contributing substantially to the evolution of Western mathematics. Arabic mathematical knowledge spread through various channels during the
medieval era In the history of Europe, the Middle Ages or medieval period lasted approximately from the 5th to the late 15th centuries, similarly to the post-classical period of global history. It began with the fall of the Western Roman Empire and t ...
, driven by the practical applications of Al-Khwārizmī's methods. This dissemination was influenced not only by economic and political factors but also by cultural exchanges, exemplified by events such as the
Crusades The Crusades were a series of religious wars initiated, supported, and at times directed by the Papacy during the Middle Ages. The most prominent of these were the campaigns to the Holy Land aimed at reclaiming Jerusalem and its surrounding t ...
and the translation movement. The
Islamic Golden Age The Islamic Golden Age was a period of scientific, economic, and cultural flourishing in the history of Islam, traditionally dated from the 8th century to the 13th century. This period is traditionally understood to have begun during the reign o ...
, spanning from the 8th to the 14th century, marked a period of considerable advancements in various scientific disciplines, attracting scholars from medieval Europe seeking access to this knowledge. Trade routes and cultural interactions played a crucial role in introducing Arabic mathematical ideas to the West. The translation of Arabic mathematical texts, along with Greek and Roman works, during the 14th to 17th century, played a pivotal role in shaping the intellectual landscape of the
Renaissance The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
.


Origin and spread of Arab-Islamic mathematics

Arabic mathematics, particularly algebra, developed significantly during the
medieval period In the history of Europe, the Middle Ages or medieval period lasted approximately from the 5th to the late 15th centuries, similarly to the post-classical period of World history (field), global history. It began with the fall of the West ...
. Muhammad ibn Musa al-Khwārizmī's (; ) work between AD 813 and 833 in Baghdad was a turning point. He introduced the term "algebra" in the title of his book, " Kitab al-jabr wa al-muqabala," marking it as a distinct discipline. He regarded his work as "a short work on Calculation by (the rules of) Completion and Reduction, confining it to what is easiest and most useful in arithmetic". Later, people commented his work was not just a theoretical treatise but also practical, aimed at solving problems in areas like commerce and land measurement. Al-Khwārizmī's approach was groundbreaking in that it did not arise from any previous "arithmetical" tradition, including that of
Diophantus Diophantus of Alexandria () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations. Although Jose ...
. He developed a new vocabulary for algebra, distinguishing between purely algebraic terms and those shared with arithmetic. Al-Khwārizmī noticed that the representation of numbers is crucial in daily life. Thus, he wanted to find or summarize a way to simplify the mathematical operation, so-called later, the algebra. His algebra was initially focused on linear and quadratic equations and the elementary arithmetic of binomials and trinomials. This approach, which involved solving equations using radicals and related algebraic calculations, influenced mathematical thinking long after his death. Al-Khwārizmī's proof of the rule for solving
quadratic equations In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...
of the form (ax^2 + bx = c), commonly referred to as "squares plus roots equal numbers," was a monumental achievement in the history of algebra. This breakthrough laid the groundwork for the systematic approach to solving quadratic equations, which became a fundamental aspect of algebra as it developed in the Western world. Al-Khwārizmī's method, which involved completing the square, not only provided a practical solution for equations of this type but also introduced an abstract and generalized approach to mathematical problems. His work, encapsulated in his seminal text "Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala" (The Compendious Book on Calculation by Completion and Balancing), was translated into
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
in the 12th century. This translation played a pivotal role in the transmission of algebraic knowledge to Europe, significantly influencing mathematicians during the Renaissance and shaping the evolution of modern mathematics. Al-Khwārizmī's contributions, especially his proof for quadratic equations, are a testament to the rich mathematical heritage of the Islamic world and its enduring impact on Western mathematics. The spread of Arabic mathematics to the West was facilitated by several factors. The practicality and general applicability of al-Khwārizmī's methods were significant. They were designed to convert numerical or geometrical problems into equations in normal form, leading to canonical solution formulae. His work and that of his successors like
al-Karaji (; c. 953 – c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works are mathematical: ''Al-Badi' fi'l-hisab'' (''Wonderful on ...
laid the foundation for advances in various mathematical fields, including
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
,
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
, and rational Diophantine analysis. Al-Khwārizmī's algebra was an autonomous discipline with its historical perspective, eventually leading to the "arithmetization of algebra". His successors expanded on his work, adapting it to new theoretical and technical challenges and reorienting it towards a more arithmetical direction for abstract algebraic calculation. Arabic mathematics, epitomized by al-Khwārizmī's work, was crucial in shaping the mathematical landscape. Its spread to the West was driven by its practical applications, the expansion of mathematical concepts by his successors, and the translation and adaptation of these ideas into the Western context. This spread was a complex process involving economics, politics, and cultural exchange, greatly influencing Western mathematics. The period known as the
Islamic Golden Age The Islamic Golden Age was a period of scientific, economic, and cultural flourishing in the history of Islam, traditionally dated from the 8th century to the 13th century. This period is traditionally understood to have begun during the reign o ...
(8th to 14th century) was characterized by significant advancements in various fields, including
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
. Scholars in the Islamic world made substantial contributions to
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
,
astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
,
medicine Medicine is the science and Praxis (process), practice of caring for patients, managing the Medical diagnosis, diagnosis, prognosis, Preventive medicine, prevention, therapy, treatment, Palliative care, palliation of their injury or disease, ...
, and other
sciences Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...
. As a result, the intellectual achievements of Islamic scholars attracted the attention of scholars in medieval Europe who sought to access this wealth of knowledge. Trade routes, such as the
Silk Road The Silk Road was a network of Asian trade routes active from the second century BCE until the mid-15th century. Spanning over , it played a central role in facilitating economic, cultural, political, and religious interactions between the ...
, facilitated the movement of goods, ideas, and knowledge between the East and West. Cities like
Baghdad Baghdad ( or ; , ) is the capital and List of largest cities of Iraq, largest city of Iraq, located along the Tigris in the central part of the country. With a population exceeding 7 million, it ranks among the List of largest cities in the A ...
,
Cairo Cairo ( ; , ) is the Capital city, capital and largest city of Egypt and the Cairo Governorate, being home to more than 10 million people. It is also part of the List of urban agglomerations in Africa, largest urban agglomeration in Africa, L ...
, and Cordoba became centers of learning and attracted scholars from different cultural backgrounds. Therefore, mathematical knowledge from the Islamic world found its way to Europe through various channels. Meanwhile, the
Crusades The Crusades were a series of religious wars initiated, supported, and at times directed by the Papacy during the Middle Ages. The most prominent of these were the campaigns to the Holy Land aimed at reclaiming Jerusalem and its surrounding t ...
connected Western Europeans with the Islamic world. While the primary purpose of the Crusades was military, there was also cultural exchange and exposure to Islamic knowledge, including mathematics. European scholars who traveled to the Holy Land and other parts of the Islamic world gained access to Arabic manuscripts and mathematical treatises. During the 14th to 17th century, the translation of Arabic mathematical texts, along with
Greek Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assumed last common ancestor of all kno ...
and Roman ones, played a crucial role in shaping the intellectual landscape of the Renaissance. Figures like
Fibonacci Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci ...
, who studied in North Africa and the Middle East, helped introduce and popularize Arabic numerals and mathematical concepts in Europe.


Concepts


Algebra

The study of
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, the name of which is derived from the
Arabic Arabic (, , or , ) is a Central Semitic languages, Central Semitic language of the Afroasiatic languages, Afroasiatic language family spoken primarily in the Arab world. The International Organization for Standardization (ISO) assigns lang ...
word meaning completion or "reunion of broken parts", flourished during the
Islamic golden age The Islamic Golden Age was a period of scientific, economic, and cultural flourishing in the history of Islam, traditionally dated from the 8th century to the 13th century. This period is traditionally understood to have begun during the reign o ...
.
Muhammad ibn Musa al-Khwarizmi Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
, a Persian scholar in the
House of Wisdom The House of Wisdom ( ), also known as the Grand Library of Baghdad, was believed to be a major Abbasid Caliphate, Abbasid-era public academy and intellectual center in Baghdad. In popular reference, it acted as one of the world's largest publ ...
in
Baghdad Baghdad ( or ; , ) is the capital and List of largest cities of Iraq, largest city of Iraq, located along the Tigris in the central part of the country. With a population exceeding 7 million, it ranks among the List of largest cities in the A ...
was the founder of algebra, is along with the
Greek Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assumed last common ancestor of all kno ...
mathematician
Diophantus Diophantus of Alexandria () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations. Although Jose ...
, known as the father of algebra. In his book ''
The Compendious Book on Calculation by Completion and Balancing ''The Concise Book of Calculation by Restoration and Balancing'' (, ;} or ), commonly abbreviated ''Al-Jabr'' or ''Algebra'' (Arabic: ), is an Arabic mathematics, Arabic mathematical treatise on algebra written in Baghdad around 820 by the Pers ...
'', Al-Khwarizmi deals with ways to solve for the positive
root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...
s of first and second-degree (linear and quadratic) polynomial equations. He introduces the method of reduction, and unlike Diophantus, also gives general solutions for the equations he deals with. Al-Khwarizmi's algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the algebraic work of Diophantus, which was syncopated, meaning that some symbolism is used. The transition to symbolic algebra, where only symbols are used, can be seen in the work of
Ibn al-Banna' al-Marrakushi Ibn al‐Bannāʾ al‐Marrākushī (), full name: Abu'l-Abbas Ahmad ibn Muhammad ibn Uthman al-Azdi al-Marrakushi () (29 December 1256 – 31 July 1321), was an Arab Muslim polymath who was active as a mathematician, astronomer, Islamic schol ...
and Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī. On the work done by Al-Khwarizmi, J. J. O'Connor and Edmund F. Robertson said: Several other mathematicians during this time period expanded on the algebra of Al-Khwarizmi. Abu Kamil Shuja' wrote a book of algebra accompanied with geometrical illustrations and proofs. He also enumerated all the possible solutions to some of his problems. Abu al-Jud,
Omar Khayyam Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīshābūrī (18 May 1048 – 4 December 1131) (Persian language, Persian: غیاث الدین ابوالفتح عمر بن ابراهیم خیام نیشابورﻯ), commonly known as Omar ...
, along with Sharaf al-Dīn al-Tūsī, found several solutions of the
cubic equation In algebra, a cubic equation in one variable is an equation of the form ax^3+bx^2+cx+d=0 in which is not zero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
. Omar Khayyam found the general geometric solution of a cubic equation.


Cubic equations

Omar Khayyam Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīshābūrī (18 May 1048 – 4 December 1131) (Persian language, Persian: غیاث الدین ابوالفتح عمر بن ابراهیم خیام نیشابورﻯ), commonly known as Omar ...
(c. 1038/48 in
Iran Iran, officially the Islamic Republic of Iran (IRI) and also known as Persia, is a country in West Asia. It borders Iraq to the west, Turkey, Azerbaijan, and Armenia to the northwest, the Caspian Sea to the north, Turkmenistan to the nort ...
– 1123/24) wrote the ''Treatise on Demonstration of Problems of Algebra'' containing the systematic solution of cubic or third-order equations, going beyond the ''Algebra'' of al-Khwārizmī. Khayyám obtained the solutions of these equations by finding the intersection points of two
conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...
s. This method had been used by the Greeks, but they did not generalize the method to cover all equations with positive
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusin ...
. Sharaf al-Dīn al-Ṭūsī (? in
Tus, Iran Tus () was an ancient city in Khorasan near the modern city of Mashhad, Razavi Khorasan province, Iran. To the ancient Greeks, it was known as Susia (). It was also known as Tusa. The area now known as Tus was divided into four cities, Ta ...
– 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, with ''a'' and ''b'' positive, he would note that the maximum point of the curve \ y = b x - x^3 occurs at x = \textstyle\sqrt, 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.


Induction

The earliest implicit traces of mathematical induction can be found in
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
's proof that the number of primes is infinite (c. 300 BCE). The first explicit formulation of the principle of induction was given by Pascal in his (1665). In between, implicit
proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...
by induction for arithmetic sequences was introduced by
al-Karaji (; c. 953 – c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works are mathematical: ''Al-Badi' fi'l-hisab'' (''Wonderful on ...
(c. 1000) and continued by al-Samaw'al, who used it for special cases of the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
and properties of
Pascal's triangle In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...
.


Irrational numbers

The Greeks had discovered
irrational number In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
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 Abū Kāmil Shujāʿ ibn Aslam and Ibn Tahir al-Baghdadi 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 mathematical objects, but they did not examine closely their nature. In the twelfth century,
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
translations of
Al-Khwarizmi Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
's
Arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
on the
Indian numerals Indian or Indians may refer to: Associated with India * of or related to India ** Indian people ** Indian diaspora ** Languages of India ** Indian English, a dialect of the English language ** Indian cuisine Associated with indigenous peopl ...
introduced the
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...
positional number system to the
Western world The Western world, also known as the West, primarily refers to various nations and state (polity), states in Western Europe, Northern America, and Australasia; with some debate as to whether those in Eastern Europe and Latin America also const ...
. His ''Compendious Book on Calculation by Completion and Balancing'' presented the first systematic solution of
linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
and
quadratic equation In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...
s. In
Renaissance The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
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. He revised
Ptolemy Claudius Ptolemy (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...
's ''
Geography Geography (from Ancient Greek ; combining 'Earth' and 'write', literally 'Earth writing') is the study of the lands, features, inhabitants, and phenomena of Earth. Geography is an all-encompassing discipline that seeks an understanding o ...
'' and wrote on astronomy and astrology. However, C.A. Nallino suggests that al-Khwarizmi's original work was not based on Ptolemy but on a derivative world map, presumably in Syriac or
Arabic Arabic (, , or , ) is a Central Semitic languages, Central Semitic language of the Afroasiatic languages, Afroasiatic language family spoken primarily in the Arab world. The International Organization for Standardization (ISO) assigns lang ...
.


Spherical trigonometry

The spherical
law of sines In trigonometry, the law of sines (sometimes called the sine formula or sine rule) is a mathematical equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\ ...
was discovered in the 10th century: it has been attributed variously to Abu-Mahmud Khojandi,
Nasir al-Din al-Tusi Muḥammad ibn Muḥammad ibn al-Ḥasan al-Ṭūsī (1201 – 1274), also known as Naṣīr al-Dīn al-Ṭūsī (; ) or simply as (al-)Tusi, was a Persians, Persian polymath, architect, Early Islamic philosophy, philosopher, Islamic medicine, phy ...
and Abu Nasr Mansur, with Abu al-Wafa' Buzjani as a contributor. Ibn Muʿādh al-Jayyānī's ''The book of unknown arcs of a sphere'' in the 11th century introduced the general law of sines. The plane law of sines was described in the 13th century by
Nasīr al-Dīn al-Tūsī Muḥammad ibn Muḥammad ibn al-Ḥasan al-Ṭūsī (1201 – 1274), also known as Naṣīr al-Dīn al-Ṭūsī (; ) or simply as (al-)Tusi, was a Persian polymath, architect, philosopher, physician, scientist, and theologian. Nasir al-Din al-Tu ...
. In his ''On the Sector Figure'', he stated the law of sines for plane and spherical triangles and provided proofs for this law.


Negative numbers

In the 9th century, Islamic mathematicians were familiar with negative numbers from the works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid.
Al-Khwarizmi Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
did not use negative numbers or negative coefficients. But within fifty years, Abu Kamil illustrated the rules of signs for expanding the multiplication (a \pm b)(c \pm d).
Al-Karaji (; c. 953 – c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works are mathematical: ''Al-Badi' fi'l-hisab'' (''Wonderful on ...
wrote in his book ''al-Fakhrī'' that "negative quantities must be counted as terms". In the 10th century, Abū al-Wafā' al-Būzjānī considered debts as negative numbers in ''A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen''. By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve polynomial divisions. As al-Samaw'al writes:
the product of a negative number—''al-nāqiṣ''—by a positive number—''al-zāʾid''—is negative, and by a negative number is positive. If we subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if we subtract a negative number from a lower negative number. If we subtract a negative number from a positive number, the remainder is their positive sum. If we subtract a positive number from an empty power (''martaba khāliyya''), the remainder is the same negative, and if we subtract a negative number from an empty power, the remainder is the same positive number.


Double false position

Between the 9th and 10th centuries, the
Egyptian ''Egyptian'' describes something of, from, or related to Egypt. Egyptian or Egyptians may refer to: Nations and ethnic groups * Egyptians, a national group in North Africa ** Egyptian culture, a complex and stable culture with thousands of year ...
mathematician Abu Kamil wrote a now-lost treatise on the use of double false position, known as the ''Book of the Two Errors'' (''Kitāb al-khaṭāʾayn''). The oldest surviving writing on double false position from the
Middle East The Middle East (term originally coined in English language) is a geopolitical region encompassing the Arabian Peninsula, the Levant, Turkey, Egypt, Iran, and Iraq. The term came into widespread usage by the United Kingdom and western Eur ...
is that of
Qusta ibn Luqa Qusta ibn Luqa, also known as Costa ben Luca or Constabulus (820912) was a Melkite Christian physician, philosopher, astronomer, mathematician and translator. He was born in Baalbek. Travelling to parts of the Byzantine Empire, he brought back Gre ...
(10th century), an
Arab Arabs (,  , ; , , ) are an ethnic group mainly inhabiting the Arab world in West Asia and North Africa. A significant Arab diaspora is present in various parts of the world. Arabs have been in the Fertile Crescent for thousands of years ...
mathematician from
Baalbek Baalbek (; ; ) is a city located east of the Litani River in Lebanon's Beqaa Valley, about northeast of Beirut. It is the capital of Baalbek-Hermel Governorate. In 1998, the city had a population of 82,608. Most of the population consists of S ...
,
Lebanon Lebanon, officially the Republic of Lebanon, is a country in the Levant region of West Asia. Situated at the crossroads of the Mediterranean Basin and the Arabian Peninsula, it is bordered by Syria to the north and east, Israel to the south ...
. He justified the technique by a formal, Euclidean-style geometric proof. Within the tradition of Golden Age Muslim mathematics, double false position was known as ''hisāb al-khaṭāʾayn'' ("reckoning by two errors"). It was used for centuries to solve practical problems such as commercial and juridical questions (estate partitions according to rules of Quranic inheritance), as well as purely recreational problems. The algorithm was often memorized with the aid of
mnemonics A mnemonic device ( ), memory trick or memory device is any learning technique that aids information retention or retrieval in the human memory, often by associating the information with something that is easier to remember. It makes use of e ...
, such as a verse attributed to Ibn al-Yasamin and balance-scale diagrams explained by al-Hassar and Ibn al-Banna, who were each mathematicians of Moroccan origin.


Influences

The influence of medieval Arab-Islamic mathematics to the rest of the world is wide and profound, in both the realm of science and mathematics. The knowledge of the Arabs went into the western world through
Spain Spain, or the Kingdom of Spain, is a country in Southern Europe, Southern and Western Europe with territories in North Africa. Featuring the Punta de Tarifa, southernmost point of continental Europe, it is the largest country in Southern Eur ...
and
Sicily Sicily (Italian language, Italian and ), officially the Sicilian Region (), is an island in the central Mediterranean Sea, south of the Italian Peninsula in continental Europe and is one of the 20 regions of Italy, regions of Italy. With 4. ...
during the translation movement. "The Moors (western Mohammedans from that part of North Africa once known as Mauritania) crossed over into Spain early in the seventh century, bringing with them the cultural resources of the Arab world". In the 13th century, King
Alfonso X of Castile Alfonso X (also known as the Wise, ; 23 November 1221 – 4 April 1284) was King of Castile, Kingdom of León, León and Kingdom of Galicia, Galicia from 1 June 1252 until his death in 1284. During the April 1257 Imperial election, election of 1 ...
established the Toledo School of Translators, in the
Kingdom of Castile The Kingdom of Castile (; : ) was a polity in the Iberian Peninsula during the Middle Ages. It traces its origins to the 9th-century County of Castile (, ), as an eastern frontier lordship of the Kingdom of León. During the 10th century, the Ca ...
, where scholars translated numerous scientific and philosophical works from
Arabic Arabic (, , or , ) is a Central Semitic languages, Central Semitic language of the Afroasiatic languages, Afroasiatic language family spoken primarily in the Arab world. The International Organization for Standardization (ISO) assigns lang ...
into
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
. The translations included Islamic contributions to
trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...
, which helps European mathematicians and astronomers in their studies. European scholars such as
Gerard of Cremona Gerard of Cremona (Latin: ''Gerardus Cremonensis''; c. 1114 – 1187) was an Italians, Italian translator of scientific books from Arabic into Latin. He worked in Toledo, Spain, Toledo, Kingdom of Castile and obtained the Arabic books in the libr ...
(1114–1187) played a key role in translating and disseminating these works, thus making them accessible to a wider audience. Cremona is said to have translated into Latin "no fewer than 90 complete Arabic texts." European mathematicians, building on the foundations laid by Islamic scholars, further developed practical trigonometry for applications in navigation, cartography, and celestial navigation, thus pushing forward the age of discovery and scientific revolution. The practical applications of trigonometry for navigation and astronomy became increasingly important during the Age of Exploration. Al-Battānī is one of the islamic mathematicians who made great contributions to the development of trigonometry. He "innovated new trigonometric functions, created a table of cotangents, and made some formulas in spherical trigonometry." These discoveries, together with his astronomical works which are praised for their accuracy, greatly advanced astronomical calculations and instruments. Al-Khayyām (1048–1131) was a Persian mathematician, astronomer, and poet, known for his work on algebra and geometry, particularly his investigations into the solutions of cubic equations. He was "the first in history to elaborate a geometrical theory of equations with degrees ≤ 3", and has great influence on the work of Descartes, a French mathematician who is often regarded as the founder of analytical geometry. Indeed, "to read Descartes' Géométrie is to look upstream towards al-Khayyām and al-Ṭūsī; and downstream towards Newton, Leibniz, Cramer, Bézout and the Bernoulli brothers". Numerous problems that appear in "La Géométrie" (Geometry) have foundations that date back to al-Khayyām. Abū Kāmil (, also known as Al-ḥāsib al-miṣrī—lit. "The Egyptian Calculator") (c. 850 – c. 930), was studied algebra following the author of ''Algebra'', al-Khwārizmī. His ''Book of Algebra'' (Kitāb fī al-jabr wa al-muqābala) is "essentially a commentary on and elaboration of al-Khwārizmī's work; in part for that reason and in part for its own merit, the book enjoyed widespread popularity in the Muslim world". It contains 69 problems, which is more than al-Khwārizmī who had 40 in his book. Abū Kāmil's Algebra plays a significant role in shaping the trajectory of Western mathematics, particularly in its impact on the works of the Italian mathematician Leonardo of Pisa, widely recognized as Fibonacci. In his ''Liber Abaci'' (1202), Fibonacci extensively incorporated ideas from Arabic mathematicians, using approximately 29 problems from ''Book of Algebra'' with scarce modification.


Western historians' perception of the contribution of Arab mathematicians

Despite the fundamental works Arabic mathematicians have done on the development of Algebra and algebraic geometry, Western historians in the 18th and early 19th century still regarded it as a fact that Classical science and math were unique phenomena of the West. Even though some math contributions from Arab mathematicians are occasionally acknowledged, they are considered to be "outside history or only integrated in so far as it contributed to science, which is essentially European", and just some technical innovations to the
Greek Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assumed last common ancestor of all kno ...
heritage rather than open up a completely new branch of mathematics. In the French philosopher
Ernest Renan Joseph Ernest Renan (; ; 27 February 18232 October 1892) was a French Orientalist and Semitic scholar, writing on Semitic languages and civilizations, historian of religion, philologist, philosopher, biblical scholar, and critic. He wrote wo ...
's work, Arabic math is merely "a reflection of
Greece Greece, officially the Hellenic Republic, is a country in Southeast Europe. Located on the southern tip of the Balkan peninsula, it shares land borders with Albania to the northwest, North Macedonia and Bulgaria to the north, and Turkey to th ...
, combined with Persian and Indian influences". And according to Duhem, "Arabic science only reproduced the teachings received from Greek science". Besides being considered as merely some insignificant additions or reflections to the great tradition of Greek classical science, math works from Arabic mathematicians are also blamed for lacking rigor and too focused on practical applications and calculations, and this is why Western historians argued they could never reach the level of Greek mathematicians. As Tannery wrote, Arabic math "in no way superseded the level attained by Diophantus". On the other hand, they perceived that Western mathematicians went into a very different way both in its method employed and ultimate purpose, "the hallmark of Western science in its Greek origins as well as in its modern renaissance, is its conformity to rigorous standards". Thus, the perceived non-rigorous proof in Arabic mathematicians' book authorizes Bourbaki to exclude the Arabic period when he retraced the evolution of algebra. And instead, the history of classical algebra is written as the work of the
Renaissance The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
and the origin of algebraic geometry is traced back to Descartes, while Arabic mathematicians' contributions are ignored deliberately. In Rashed's words: "To justify the exclusion of science written in Arabic from the history of science, one invokes its absence of rigor, its calculatory appearance and its practical aims. Furthermore, strictly dependent on Greek science and, lastly, incapable of introducing experimental norms, scientists of that time were relegated to the role of conscientious guardians of the Hellenistic museum." In 18th century
Germany Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total popu ...
and
France France, officially the French Republic, is a country located primarily in Western Europe. Overseas France, Its overseas regions and territories include French Guiana in South America, Saint Pierre and Miquelon in the Atlantic Ocean#North Atlan ...
, the prevailing Orientalist view was "East and West oppose each other not as geographical but as historical positivities", which labeled "
Rationalism In philosophy, rationalism is the Epistemology, epistemological view that "regards reason as the chief source and test of knowledge" or "the position that reason has precedence over other ways of acquiring knowledge", often in contrast to ot ...
" as the essence of the West, while the "Call of the
Orient The Orient is a term referring to the East in relation to Europe, traditionally comprising anything belonging to the Eastern world. It is the antonym of the term ''Occident'', which refers to the Western world. In English, it is largely a meto ...
" movement emerged in the 19th century was interpreted as "against Rationalism" and a return to a more "spiritual and harmonious" lifestyle. Thus, the prevailing
Orientalism In art history, literature, and cultural studies, Orientalism is the imitation or depiction of aspects of the Eastern world (or "Orient") by writers, designers, and artists from the Western world. Orientalist painting, particularly of the Middle ...
in that period was one of the main reasons why Arabic mathematicians were often ignored for their contributions, as people outside the West were considered to be lacking the necessary rationality and scientific spirit to made significant contributions to math and science.


Conclusion

The medieval Arab-Islamic world played a crucial role in shaping the trajectory of mathematics, with al-Khwārizmī's algebraic innovations serving as a cornerstone. The dissemination of Greek mathematics to the West during the
Islamic Golden Age The Islamic Golden Age was a period of scientific, economic, and cultural flourishing in the history of Islam, traditionally dated from the 8th century to the 13th century. This period is traditionally understood to have begun during the reign o ...
, facilitated by cultural exchanges and translations, left a lasting impact on Western mathematical thought. Mathematicians like Al-Battānī, Al-Khayyām, and Abū Kāmil, with their contributions to
trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...
,
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, and
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, extended their influence beyond their time. Despite the foundational contributions of Arab mathematicians, Western historians in the 18th and early 19th centuries, influenced by Orientalist views, sometimes marginalized these achievements. The East lacking rationality and scientific spirit perpetuated a biased perspective, hindering the recognition of the significant role played by Arabic mathematics in the development of algebra and other mathematical disciplines. Reevaluating the history of mathematics necessitates acknowledging the interconnectedness of diverse mathematical traditions and dispelling the notion of a uniquely European mathematical heritage. The contributions of Arab mathematicians, marked by practical applications and theoretical innovations, form an integral part of the rich tapestry of mathematical history, and deserves recognition.


Other major figures

*
'Abd al-Hamīd ibn Turk (fl. 830), known also as () was a ninth-century mathematician. Not much is known about his life. The two records of him, one by Ibn Nadim and the other by al-Qifti are not identical. Al-Qifi mentions his name as ʿAbd al-Hamīd ibn Wase ibn Tu ...
(fl. 830) (quadratics) * Sind ibn Ali (d. after 864) * Thabit ibn Qurra (826–901) * Al-Battānī (before 858 – 929) * Abū Kāmil (c. 850 – c. 930) *
Abu'l-Hasan al-Uqlidisi Abū al-Ḥassan, Aḥmad Ibn Ibrāhīm, al-Uqlīdisī (, ) was a Muslim Arab mathematician of the Islamic Golden Age, possibly from Damascus, who wrote the earliest surviving book on the use of decimal fractions with Hindu–Arabic numerals, ''K ...
(fl. 952) (arithmetic) * 'Abd al-'Aziz al-Qabisi (d. 967) * Abū Sahl al-Qūhī (c. 940–1000) (centres of gravity) *
Ibn al-Haytham Ḥasan Ibn al-Haytham (Latinization of names, Latinized as Alhazen; ; full name ; ) was a medieval Mathematics in medieval Islam, mathematician, Astronomy in the medieval Islamic world, astronomer, and Physics in the medieval Islamic world, p ...
(c. 965–1040) * Abū al-Rayḥān al-Bīrūnī (973–1048) (trigonometry) * Al-Khayyām (1048–1131) * Ibn Maḍāʾ (c. 1116–1196) * Ismail al-Jazari (1136–1206) *
Jamshīd al-Kāshī Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) ( ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxiana) was a Persian astronomer and mathematician during the reign of Tamerlane. ...
(c. 1380–1429) (decimals and estimation of the circle constant)


Gallery

File:Gravure originale du compas parfait par Abū Sahl al-Qūhī.jpg, Engraving of Abū Sahl al-Qūhī's perfect compass to draw conic sections File:Theorem of al-Haitham.JPG, The theorem of Ibn Haytham


See also

*
Arabic numerals The ten Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) are the most commonly used symbols for writing numbers. The term often also implies a positional notation number with a decimal base, in particular when contrasted with Roman numera ...
* Indian influence on Islamic mathematics in medieval Islam *
History of calculus Calculus, originally called infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Many elements of calculus appeared in ancient Greece, then in China and the Middle East ...
*
History of geometry Geometry (from the ; '' geo-'' "earth", '' -metron'' "measurement") arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmeti ...
*
Science in the medieval Islamic world Science in the medieval Islamic world was the science developed and practised during the Islamic Golden Age under the Abbasid Caliphate of Baghdad, the Caliphate of Córdoba, Umayyads of Córdoba, Spain, Córdoba, the Abbadid dynasty, Abbadids ...
* Timeline of science and engineering in the Muslim world


References


Sources

* * * * * * *


Further reading

;Books on Islamic mathematics * ** Review: ** Review: * * * * Sowjetische Beiträge zur Geschichte der Naturwissenschaft pp. 62–160. * ; Book chapters on Islamic mathematics * Lindberg, D.C., and M. H. Shank, eds. ''The Cambridge History of Science. Volume 2: Medieval Science'' (Cambridge UP, 2013), chapters 2 and 3 mathematics in Islam. * ; Books on Islamic science * * ; Books on the history of mathematics * (Reviewed: ) * ;Journal articles on Islamic mathematics * Høyrup, Jens
“The Formation of «Islamic Mathematics»: Sources and Conditions”
''Filosofi og Videnskabsteori på Roskilde Universitetscenter''. 3. Række: ''Preprints og Reprints'' 1987 Nr. 1. ;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. * * * ; Television documentaries *
Marcus du Sautoy Marcus Peter Francis du Sautoy (; born 26 August 1965) is a British mathematician, Simonyi Professor for the Public Understanding of Science at the University of Oxford, Fellow of New College, Oxford and author of popular mathematics and popula ...
(presenter) (2008). "The Genius of the East". '' The Story of Maths''.
BBC The British Broadcasting Corporation (BBC) is a British public service broadcaster headquartered at Broadcasting House in London, England. Originally established in 1922 as the British Broadcasting Company, it evolved into its current sta ...
. * Jim Al-Khalili (presenter) (2010). '' Science and Islam''.
BBC The British Broadcasting Corporation (BBC) is a British public service broadcaster headquartered at Broadcasting House in London, England. Originally established in 1922 as the British Broadcasting Company, it evolved into its current sta ...
.


External links

* *
Richard Covington, ''Rediscovering Arabic Science'', 2007, Saudi Aramco World
{{DEFAULTSORT:Mathematics In Medieval Islam Islamic Golden Age