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Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian
polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a substantial number of subjects, known to draw on complex bodies of knowledge to solve specific pro ...
from Khwarazm, who produced vastly influential works in
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
, and
geography Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, an ...
. Around 820 CE, he was appointed as the astronomer and head of the library of the
House of Wisdom The House of Wisdom ( ar, بيت الحكمة, Bayt al-Ḥikmah), also known as the Grand Library of Baghdad, refers to either a major Abbasid public academy and intellectual center in Baghdad or to a large private library belonging to the Abba ...
in
Baghdad Baghdad (; ar, بَغْدَاد , ) is the capital of Iraq and the second-largest city in the Arab world after Cairo. It is located on the Tigris near the ruins of the ancient city of Babylon and the Sassanid Persian capital of Ctesiphon ...
.Maher, P. (1998), "From Al-Jabr to Algebra", ''Mathematics in School'', 27(4), 14–15. Al-Khwarizmi's popularizing treatise on algebra (''
The Compendious Book on Calculation by Completion and Balancing ''The Compendious Book on Calculation by Completion and Balancing'' ( ar, كتاب المختصر في حساب الجبر والمقابلة, ; la, Liber Algebræ et Almucabola), also known as ''Al-Jabr'' (), is an Arabic mathematical treati ...
'', c. 813–833 CEOaks, J. (2009), "Polynomials and Equations in Arabic Algebra", ''Archive for History of Exact Sciences'', 63(2), 169–203.) presented the first systematic solution of
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
s. One of his principal achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications. Because he was the first to treat algebra as an independent discipline and introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation), he has been described as the father or founder of
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
. The term ''algebra'' itself comes from the title of his book (the word ''al-jabr'' meaning "completion" or "rejoining"). His name gave rise to the terms '' algorism'' and ''
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
,'' as well as Spanish, Italian and Portuguese terms ''algoritmo,'' and Spanish '' guarismo'' and Portuguese '' algarismo'' meaning " digit". In the 12th century,
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
translations of his textbook on arithmetic (''Algorithmo de Numero Indorum'') which codified the various Indian numerals, 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 of the Hindu–Arabic numeral ...
positional number system to the Western world. ''The Compendious Book on Calculation by Completion and Balancing'', translated into Latin by Robert of Chester in 1145, was used until the sixteenth century as the principal mathematical text-book of
European universities This is a list of lists of universities and colleges by country, sorted by continent and region. The lists represent educational institutions throughout the world which provide higher education in tertiary, quaternary, and post-secondary educatio ...
. In addition to his best-known works, he revised
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
's ''
Geography Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, an ...
'', listing the longitudes and latitudes of various cities and localities. He further produced a set of astronomical tables and wrote about calendaric works, as well as the astrolabe and the sundial. He also made important contributions to
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
, producing accurate
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
and cosine tables, and the first table of
tangents In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
.


Life

Monument to Muhammad ibn Musa al-Khwarizmi at Ciudad Universitaria of Madrid, 190px Few details of al-Khwārizmī's life are known with certainty.
Ibn al-Nadim Abū al-Faraj Muḥammad ibn Isḥāq al-Nadīm ( ar, ابو الفرج محمد بن إسحاق النديم), also ibn Abī Ya'qūb Isḥāq ibn Muḥammad ibn Isḥāq al-Warrāq, and commonly known by the ''nasab'' (patronymic) Ibn al-Nadīm ...
gives his birthplace as Khwarazm, and he is generally thought to have come from this region. His name means 'the native of Khwarazm', a region that was part of
Greater Iran Greater Iran ( fa, ایران بزرگ, translit=Irān-e Bozorg) refers to a region covering parts of Western Asia, Central Asia, South Asia, Xinjiang, and the Caucasus, where both Iranian culture and Iranian languages have had a s ...
, and is now part of
Turkmenistan Turkmenistan ( or ; tk, Türkmenistan / Түркменистан, ) is a country located in Central Asia, bordered by Kazakhstan to the northwest, Uzbekistan to the north, east and northeast, Afghanistan to the southeast, Iran to the s ...
, and
Uzbekistan Uzbekistan (, ; uz, Ozbekiston, italic=yes / , ; russian: Узбекистан), officially the Republic of Uzbekistan ( uz, Ozbekiston Respublikasi, italic=yes / ; russian: Республика Узбекистан), is a doubly landlocked co ...
.
Muhammad ibn Jarir al-Tabari ( ar, أبو جعفر محمد بن جرير بن يزيد الطبري), more commonly known as al-Ṭabarī (), was a Muslim historian and scholar from Amol, Tabaristan. Among the most prominent figures of the Islamic Golden Age, al-Tabari ...
gives his name as Muḥammad ibn Musá al-Khwārizmī al- Majūsī al-Quṭrubbullī (). The
epithet An epithet (, ), also byname, is a descriptive term (word or phrase) known for accompanying or occurring in place of a name and having entered common usage. It has various shades of meaning when applied to seemingly real or fictitious people, di ...
''al-Qutrubbulli'' could indicate he might instead have come from Qutrubbul (Qatrabbul), a viticulture district near Baghdad. However, Rashed denies this: On the other hand, David A. King affirms his nisba to Qutrubul, noting that he was called al-Khwārizmī al-Qutrubbulli because he was born just outside of Baghdad. Regarding al-Khwārizmī's religion, Toomer writes:
Ibn al-Nadīm Abū al-Faraj Muḥammad ibn Isḥāq al-Nadīm ( ar, ابو الفرج محمد بن إسحاق النديم), also ibn Abī Ya'qūb Isḥāq ibn Muḥammad ibn Isḥāq al-Warrāq, and commonly known by the ''nasab'' (patronymic) Ibn al-Nadīm ...
's ''
Kitāb al-Fihrist The ''Kitāb al-Fihrist'' ( ar, كتاب الفهرست) (''The Book Catalogue'') is a compendium of the knowledge and literature of tenth-century Islam compiled by Ibn Al-Nadim (c.998). It references approx. 10,000 books and 2,000 authors.''The ...
'' includes a short biography on al-Khwārizmī together with a list of his books. Al-Khwārizmī accomplished most of his work between 813 and 833. After the
Muslim conquest of Persia The Muslim conquest of Persia, also known as the Arab conquest of Iran, was carried out by the Rashidun Caliphate from 633 to 654 AD and led to the fall of the Sasanian Empire as well as the eventual decline of the Zoroastrian religion. The ...
, Baghdad had become the centre of scientific studies and trade, and many merchants and scientists from as far as China and India traveled there, as did al-Khwārizmī . He worked in the House of Wisdom established by the
Abbasid The Abbasid Caliphate ( or ; ar, الْخِلَافَةُ الْعَبَّاسِيَّة, ') was the third caliphate to succeed the Islamic prophet Muhammad. It was founded by a dynasty descended from Muhammad's uncle, Abbas ibn Abdul-Mutta ...
Caliph al-Ma'mūn, where he studied the sciences and mathematics, including the translation of Greek and
Sanskrit Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had Trans-cultural diffusion ...
scientific manuscripts. During the reign of al-Wathiq, he is said to have been involved in the first of two embassies to the
Khazars The Khazars ; he, כּוּזָרִים, Kūzārīm; la, Gazari, or ; zh, 突厥曷薩 ; 突厥可薩 ''Tūjué Kěsà'', () were a semi-nomadic Turkic people that in the late 6th-century CE established a major commercial empire coverin ...
.
Douglas Morton Dunlop Douglas Morton Dunlop (1909–1987) was a renowned British orientalist and scholar of Islamic and Eurasian history. Early life and education Born in England, Dunlop studied at Bonn and Oxford under the historian Paul Ernst Kahle (1875–1965). ...
suggests that Muḥammad ibn Mūsā al-Khwārizmī might have been the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the three
Banū Mūsā The Banū Mūsā brothers ("Sons of Moses"), namely Abū Jaʿfar, Muḥammad ibn Mūsā ibn Shākir (before 803 – February 873); Abū al‐Qāsim, Aḥmad ibn Mūsā ibn Shākir (d. 9th century); and Al-Ḥasan ibn Mūsā ibn Shākir (d. 9th ce ...
.


Contributions

Al-Khwārizmī's contributions to mathematics, geography, astronomy, and
cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an i ...
established the basis for innovation in algebra and
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
. His systematic approach to solving linear and quadratic equations led to ''algebra'', a word derived from the title of his book on the subject, "The Compendious Book on Calculation by Completion and Balancing". ''On the Calculation with Hindu Numerals,'' written about 820, was principally responsible for spreading the Hindu–Arabic numeral system throughout the Middle East and Europe. It was translated into Latin as ''Algoritmi de numero Indorum''. Al-Khwārizmī, rendered as (Latin) ''Algoritmi'', led to the term "algorithm". Some of his work was based on Persian and
Babylonia Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c ...
n astronomy, Indian numbers, and
Greek mathematics Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathe ...
. Al-Khwārizmī systematized and corrected
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
's data for Africa and the Middle East. Another major book was ''Kitab surat al-ard'' ("The Image of the Earth"; translated as Geography), presenting the coordinates of places based on those in the ''Geography'' of Ptolemy but with improved values for the
Mediterranean Sea The Mediterranean Sea is a sea connected to the Atlantic Ocean, surrounded by the Mediterranean Basin and almost completely enclosed by land: on the north by Western and Southern Europe and Anatolia, on the south by North Africa, and on ...
, Asia, and Africa. He also wrote on mechanical devices like the
astrolabe An astrolabe ( grc, ἀστρολάβος ; ar, ٱلأَسْطُرلاب ; persian, ستاره‌یاب ) is an ancient astronomical instrument that was a handheld model of the universe. Its various functions also make it an elaborate inclin ...
and
sundial A sundial is a horological device that tells the time of day (referred to as civil time in modern usage) when direct sunlight shines by the apparent position of the Sun in the sky. In the narrowest sense of the word, it consists of a f ...
. He assisted a project to determine the circumference of the Earth and in making a world map for
al-Ma'mun Abu al-Abbas Abdallah ibn Harun al-Rashid ( ar, أبو العباس عبد الله بن هارون الرشيد, Abū al-ʿAbbās ʿAbd Allāh ibn Hārūn ar-Rashīd; 14 September 786 – 9 August 833), better known by his regnal name Al-Ma'm ...
, the caliph, overseeing 70 geographers. When, in the 12th century, his works spread to Europe through Latin translations, it had a profound impact on the advance of mathematics in Europe.


Algebra

''The Compendious Book on Calculation by Completion and Balancing'' ( ar, الكتاب المختصر في حساب الجبر والمقابلة ) is a mathematical book written approximately 820 CE. The book was written with the encouragement of
Caliph al-Ma'mun Abu al-Abbas Abdallah ibn Harun al-Rashid ( ar, أبو العباس عبد الله بن هارون الرشيد, Abū al-ʿAbbās ʿAbd Allāh ibn Hārūn ar-Rashīd; 14 September 786 – 9 August 833), better known by his regnal name Al-Ma'mu ...
as a popular work on calculation and is replete with examples and applications to a wide range of problems in trade, surveying and legal inheritance. The term "algebra" is derived from the name of one of the basic operations with equations (, meaning "restoration", referring to adding a number to both sides of the equation to consolidate or cancel terms) described in this book. The book was translated in Latin as ''Liber algebrae et almucabala'' by Robert of Chester ( Segovia, 1145) hence "algebra", and also by Gerard of Cremona. A unique Arabic copy is kept at Oxford and was translated in 1831 by F. Rosen. A Latin translation is kept in Cambridge. It provided an exhaustive account of solving polynomial equations up to the second degree, and discussed the fundamental method of "reduction" and "balancing", referring to the transposition of terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. "It is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the translation above. The word ''al-jabr'' presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word ''muqabalah'' is said to refer to "reduction" or "balancing" — that is, the cancellation of like terms on opposite sides of the equation." Al-Khwārizmī's method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms (where ''b'' and ''c'' are positive integers) * squares equal roots (''ax''2 = ''bx'') * squares equal number (''ax''2 = ''c'') * roots equal number (''bx'' = ''c'') * squares and roots equal number (''ax''2 + ''bx'' = ''c'') * squares and number equal roots (''ax''2 + ''c'' = ''bx'') * roots and number equal squares (''bx'' + ''c'' = ''ax''2) by dividing out the coefficient of the square and using the two operations ( ar, الجبر "restoring" or "completion") and ("balancing"). is the process of removing negative units, roots and squares from the equation by adding the same quantity to each side. For example, ''x''2 = 40''x'' − 4''x''2 is reduced to 5''x''2 = 40''x''. is the process of bringing quantities of the same type to the same side of the equation. For example, ''x''2 + 14 = ''x'' + 5 is reduced to ''x''2 + 9 = ''x''. The above discussion uses modern mathematical notation for the types of problems that the book discusses. However, in al-Khwārizmī's day, most of this notation had not yet been invented, so he had to use ordinary text to present problems and their solutions. For example, for one problem he writes, (from an 1831 translation) In modern notation this process, with ''x'' the "thing" ( ''shayʾ'') or "root", is given by the steps, :(10-x)^2=81 x :100 + x^2 - 20 x = 81 x :x^2+100=101 x Let the roots of the equation be ''x'' = ''p'' and ''x = q''. Then \tfrac=50\tfrac, pq =100 and :\frac = \sqrt=\sqrt=49\tfrac So a root is given by :x=50\tfrac-49\tfrac=1 Several authors have also published texts under the name of , including Abū Ḥanīfa Dīnawarī, Abū Kāmil Shujāʿ ibn Aslam, Abū Muḥammad al-'Adlī, Abū Yūsuf al-Miṣṣīṣī,
'Abd al-Hamīd ibn Turk ( fl. 830), known also as ( ar, ابومحمد عبدالحمید بن واسع بن ترک الجیلی) was a ninth-century Muslim mathematician. Not much is known about his life. The two records of him, one by Ibn Nadim and the other by al-Q ...
, Sind ibn 'Alī, Sahl ibn Bišr, and Sharaf al-Dīn al-Ṭūsī. S. Gandz has described Al-Khwarizmi as the father of Algebra : Victor J. Katz adds : J.J. O'Conner and E.F. Robertson wrote in the ''
MacTutor History of Mathematics archive The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biographies on many historical and contemporary mathem ...
'': R. Rashed and Angela Armstrong write: According to Swiss-American historian of mathematics,
Florian Cajori Florian Cajori (February 28, 1859 – August 14 or 15, 1930) was a Swiss-American historian of mathematics. Biography Florian Cajori was born in Zillis, Switzerland, as the son of Georg Cajori and Catherine Camenisch. He attended schools first ...
, Al-Khwarizmi's algebra was different from the work of
Indian mathematicians chronology of Indian mathematicians spans from the Indus Valley civilisation and the Vedas to Modern India. Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians ...
, for Indians had no rules like the ''restoration'' and ''reduction''. Regarding the dissimilarity and significance of Al-Khwarizmi's algebraic work from that of Indian Mathematician
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the '' Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tr ...
, Carl Benjamin Boyer wrote:
It is true that in two respects the work of al-Khowarizmi represented a retrogression from that of Diophantus. First, it is on a far more elementary level than that found in the Diophantine problems and, second, the algebra of al-Khowarizmi is thoroughly rhetorical, with none of the syncopation found in the Greek ''Arithmetica'' or in Brahmagupta's work. Even numbers were written out in words rather than symbols! It is quite unlikely that al-Khwarizmi knew of the work of Diophantus, but he must have been familiar with at least the astronomical and computational portions of Brahmagupta; yet neither al-Khwarizmi nor other Arabic scholars made use of syncopation or of negative numbers. Nevertheless, the ''Al-jabr'' comes closer to the elementary algebra of today than the works of either Diophantus or Brahmagupta, because the book is not concerned with difficult problems in indeterminant analysis but with a straight forward and elementary exposition of the solution of equations, especially that of second degree. The Arabs in general loved a good clear argument from premise to conclusion, as well as systematic organization – respects in which neither Diophantus nor the Hindus excelled.


Arithmetic

Al-Khwārizmī's second most influential work was on the subject of arithmetic, which survived in Latin translations but is lost in the original Arabic. His writings include the text ''kitāb al-ḥisāb al-hindī'' ('Book of Indian computation'), and perhaps a more elementary text, ''kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī'' ('Addition and subtraction in Indian arithmetic'). These texts described algorithms on decimal numbers ( Hindu–Arabic numerals) that could be carried out on a dust board. Called ''takht'' in Arabic (Latin: ''tabula''), a board covered with a thin layer of dust or sand was employed for calculations, on which figures could be written with a stylus and easily erased and replaced when necessary. Al-Khwarizmi's algorithms were used for almost three centuries, until replaced by Al-Uqlidisi's algorithms that could be carried out with pen and paper. As part of 12th century wave of Arabic science flowing into Europe via translations, these texts proved to be revolutionary in Europe. Al-Khwarizmi's Latinized name, ''Algorismus'', turned into the name of method used for computations, and survives in the modern term "
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
". It gradually replaced the previous abacus-based methods used in Europe. Four Latin texts providing adaptions of Al-Khwarizmi's methods have survived, even though none of them is believed to be a literal translation: * ''Dixit Algorizmi'' (published in 1857 under the title ''Algoritmi de Numero Indorum'') * ''Liber Alchoarismi de Practica Arismetice'' * ''Liber Ysagogarum Alchorismi'' * ''Liber Pulveris'' ''Dixit Algorizmi'' ('Thus spake Al-Khwarizmi') is the starting phrase of a manuscript in the University of Cambridge library, which is generally referred to by its 1857 title ''Algoritmi de Numero Indorum''. It is attributed to the
Adelard of Bath Adelard of Bath ( la, Adelardus Bathensis; 1080? 1142–1152?) was a 12th-century English natural philosopher. He is known both for his original works and for translating many important Arabic and Greek scientific works of astrology, astronom ...
, who had also translated the astronomical tables in 1126. It is perhaps the closest to Al-Khwarizmi's own writings. Al-Khwarizmi's work on arithmetic was responsible for introducing the
Arabic numerals Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such a ...
, based on the Hindu–Arabic numeral system developed in
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 ...
, to the Western world. The term "algorithm" is derived from the algorism, the technique of performing arithmetic with Hindu-Arabic numerals developed by al-Khwārizmī. Both "algorithm" and "algorism" are derived from the Latinized forms of al-Khwārizmī's name, ''Algoritmi'' and ''Algorismi'', respectively.


Astronomy

Al-Khwārizmī's ( ar, زيج السند هند, " astronomical tables of '' Siddhanta''") is a work consisting of approximately 37 chapters on calendrical and astronomical calculations and 116 tables with calendrical, astronomical and astrological data, as well as a table of
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
values. This is the first of many Arabic '' Zijes'' based on the Indian astronomical methods known as the ''sindhind''. The word Sindhind is a corruption of the
Sanskrit Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had Trans-cultural diffusion ...
''Siddhānta'', which is the usual designation of an astronomical textbook. In fact, the mean motions in the tables of al-Khwarizmi are derived from those in the "corrected Brahmasiddhanta" ( Brahmasphutasiddhanta) of
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the '' Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tr ...
. The work contains tables for the movements of the sun, the
moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
and the five
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s known at the time. This work marked the turning point in
Islamic astronomy Islamic astronomy comprises the astronomical developments made in the Islamic world, particularly during the Islamic Golden Age (9th–13th centuries), and mostly written in the Arabic language. These developments mostly took place in the Middl ...
. Hitherto, Muslim astronomers had adopted a primarily research approach to the field, translating works of others and learning already discovered knowledge. The original Arabic version (written c. 820) is lost, but a version by the Spanish astronomer Maslamah Ibn Ahmad al-Majriti (c. 1000) has survived in a Latin translation, presumably by
Adelard of Bath Adelard of Bath ( la, Adelardus Bathensis; 1080? 1142–1152?) was a 12th-century English natural philosopher. He is known both for his original works and for translating many important Arabic and Greek scientific works of astrology, astronom ...
(26 January 1126). The four surviving manuscripts of the Latin translation are kept at the Bibliothèque publique (Chartres), the Bibliothèque Mazarine (Paris), the Biblioteca Nacional (Madrid) and the Bodleian Library (Oxford).


Trigonometry

Al-Khwārizmī's ''Zīj al-Sindhind'' also contained tables for the
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
of sines and cosine. A related treatise on spherical trigonometry is also attributed to him. Al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents.Jacques Sesiano, "Islamic mathematics", p. 157, in


Geography

Al-Khwārizmī's third major work is his ( ar, كتاب صورة الأرض, "Book of the Description of the Earth"), also known as his ''Geography'', which was finished in 833. It is a major reworking of
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
's second-century ''
Geography Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, an ...
'', consisting of a list of 2402 coordinates of cities and other geographical features following a general introduction. There is only one surviving copy of , which is kept at the Strasbourg University Library. A Latin translation is kept at the
Biblioteca Nacional de España The Biblioteca Nacional de España (''National Library of Spain'') is a major public library, the largest in Spain, and one of the largest in the world. It is located in Madrid, on the Paseo de Recoletos. History The library was founded by ...
in
Madrid Madrid ( , ) is the capital and most populous city of Spain. The city has almost 3.4 million inhabitants and a metropolitan area population of approximately 6.7 million. It is the second-largest city in the European Union (EU), and ...
. The book opens with the list of latitudes and longitudes, in order of "weather zones", that is to say in blocks of latitudes and, in each
weather Weather is the state of the atmosphere, describing for example the degree to which it is hot or cold, wet or dry, calm or stormy, clear or cloudy. On Earth, most weather phenomena occur in the lowest layer of the planet's atmosphere, the ...
zone, by order of longitude. As Paul Gallez points out, this excellent system allows the deduction of many latitudes and longitudes where the only extant document is in such a bad condition as to make it practically illegible. Neither the Arabic copy nor the Latin translation include the map of the world itself; however, Hubert Daunicht was able to reconstruct the missing map from the list of coordinates. Daunicht read the latitudes and longitudes of the coastal points in the manuscript, or deduces them from the context where they were not legible. He transferred the points onto
graph paper Graph paper, coordinate paper, grid paper, or squared paper is writing paper that is printed with fine lines making up a regular grid. The lines are often used as guides for plotting graphs of functions or experimental data and drawing curves ...
and connected them with straight lines, obtaining an approximation of the coastline as it was on the original map. He then does the same for the rivers and towns. Al-Khwārizmī corrected Ptolemy's gross overestimate for the length of the
Mediterranean Sea The Mediterranean Sea is a sea connected to the Atlantic Ocean, surrounded by the Mediterranean Basin and almost completely enclosed by land: on the north by Western and Southern Europe and Anatolia, on the south by North Africa, and on ...
Edward S. Kennedy, ''Mathematical Geography'', p. 188, in from the
Canary Islands The Canary Islands (; es, :es:Canarias, Canarias, ), also known informally as the Canaries, are a Spanish Autonomous communities of Spain, autonomous community and archipelago in the Atlantic Ocean, in Macaronesia. At their closest point to ...
to the eastern shores of the Mediterranean; Ptolemy overestimated it at 63 degrees of
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
, while al-Khwārizmī almost correctly estimated it at nearly 50 degrees of longitude. He "also depicted the
Atlantic The Atlantic Ocean is the second-largest of the world's five oceans, with an area of about . It covers approximately 20% of Earth's surface and about 29% of its water surface area. It is known to separate the " Old World" of Africa, Europe ...
and
Indian Ocean The Indian Ocean is the third-largest of the world's five oceanic divisions, covering or ~19.8% of the water on Earth's surface. It is bounded by Asia to the north, Africa to the west and Australia to the east. To the south it is bounded by ...
s as open bodies of water, not land-locked
sea The sea, connected as the world ocean or simply the ocean, is the body of salty water that covers approximately 71% of the Earth's surface. The word sea is also used to denote second-order sections of the sea, such as the Mediterranean Sea, ...
s as Ptolemy had done." Al-Khwārizmī's
Prime Meridian A prime meridian is an arbitrary meridian (a line of longitude) in a geographic coordinate system at which longitude is defined to be 0°. Together, a prime meridian and its anti-meridian (the 180th meridian in a 360°-system) form a great ...
at the
Fortunate Isles The Fortunate Isles or Isles of the Blessed ( grc, μακάρων νῆσοι, ''makárōn nêsoi'') were semi-legendary islands in the Atlantic Ocean, variously treated as a simple geographical location and as a winterless earthly paradise inhabit ...
was thus around 10° east of the line used by Marinus and Ptolemy. Most medieval Muslim gazetteers continued to use al-Khwārizmī's prime meridian.


Jewish calendar

Al-Khwārizmī wrote several other works including a treatise on the
Hebrew calendar The Hebrew calendar ( he, הַלּוּחַ הָעִבְרִי, translit=HaLuah HaIvri), also called the Jewish calendar, is a lunisolar calendar used today for Jewish religious observance, and as an official calendar of the state of Israel ...
, titled ( ar, رسالة في إستخراج تأريخ اليهود, "Extraction of the Jewish Era"). It describes the
Metonic cycle The Metonic cycle or enneadecaeteris (from grc, ἐννεακαιδεκαετηρίς, from ἐννεακαίδεκα, "nineteen") is a period of almost exactly 19 years after which the lunar phases recur at the same time of the year. The rec ...
, a 19-year intercalation cycle; the rules for determining on what day of the week the first day of the month
Tishrei Tishrei () or Tishri (; he, ''tīšrē'' or ''tīšrī''; from Akkadian ''tašrītu'' "beginning", from ''šurrû'' "to begin") is the first month of the civil year (which starts on 1 Tishrei) and the seventh month of the ecclesiastical year ...
shall fall; calculates the interval between the
Anno Mundi (from Latin "in the year of the world"; he, לבריאת העולם, Livryat haOlam, lit=to the creation of the world), abbreviated as AM or A.M., or Year After Creation, is a calendar era based on the biblical accounts of the creation of ...
or Jewish year and the
Seleucid era The Seleucid era ("SE") or (literally "year of the Greeks" or "Greek year"), sometimes denoted "AG," was a system of numbering years in use by the Seleucid Empire and other countries among the ancient Hellenistic civilizations. It is sometimes r ...
; and gives rules for determining the mean longitude of the sun and the moon using the Hebrew calendar. Similar material is found in the works of
Abū Rayḥān al-Bīrūnī Abu Rayhan Muhammad ibn Ahmad al-Biruni (973 – after 1050) commonly known as al-Biruni, was a Khwarazmian Iranian in scholar and polymath during the Islamic Golden Age. He has been called variously the "founder of Indology", "Father of ...
and
Maimonides Musa ibn Maimon (1138–1204), commonly known as Maimonides (); la, Moses Maimonides and also referred to by the acronym Rambam ( he, רמב״ם), was a Sephardic Jewish philosopher who became one of the most prolific and influential Torah ...
.


Other works

Ibn al-Nadim Abū al-Faraj Muḥammad ibn Isḥāq al-Nadīm ( ar, ابو الفرج محمد بن إسحاق النديم), also ibn Abī Ya'qūb Isḥāq ibn Muḥammad ibn Isḥāq al-Warrāq, and commonly known by the ''nasab'' (patronymic) Ibn al-Nadīm ...
's , an index of Arabic books, mentions al-Khwārizmī's ( ar, كتاب التأريخ), a book of annals. No direct manuscript survives; however, a copy had reached Nusaybin by the 11th century, where its
metropolitan bishop In Christian churches with episcopal polity, the rank of metropolitan bishop, or simply metropolitan (alternative obsolete form: metropolite), pertains to the diocesan bishop or archbishop of a metropolis. Originally, the term referred to the ...
, Mar Elias bar Shinaya, found it. Elias's chronicle quotes it from "the death of the Prophet" through to 169 AH, at which point Elias's text itself hits a lacuna. Several Arabic manuscripts in Berlin, Istanbul, Tashkent, Cairo and Paris contain further material that surely or with some probability comes from al-Khwārizmī. The Istanbul manuscript contains a paper on sundials; the ''Fihrist'' credits al-Khwārizmī with ( ar, كتاب الرخامة). Other papers, such as one on the determination of the direction of
Mecca Mecca (; officially Makkah al-Mukarramah, commonly shortened to Makkah ()) is a city and administrative center of the Mecca Province of Saudi Arabia, and the holiest city in Islam. It is inland from Jeddah on the Red Sea, in a narrow v ...
, are on the
spherical astronomy Spherical astronomy, or positional astronomy, is a branch of observational astronomy used to locate astronomical objects on the celestial sphere, as seen at a particular date, time, and location on Earth. It relies on the mathematical methods of ...
. Two texts deserve special interest on the morning width () and the determination of the
azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematical ...
from a height (). He also wrote two books on using and constructing
astrolabe An astrolabe ( grc, ἀστρολάβος ; ar, ٱلأَسْطُرلاب ; persian, ستاره‌یاب ) is an ancient astronomical instrument that was a handheld model of the universe. Its various functions also make it an elaborate inclin ...
s.


Honors

* Al-Khwarizmi (crater) — A crater on the far side of the moon → NASA Portal
Apollo 11, Photography Index

13498 Al Chwarizmi
— Main-belt Asteroid, Discovered 1986 Aug 6 by E. W. Elst and V. G. Ivanova at Smolyan.
11156 Al-Khwarismi
— Main-belt Asteroid, Discovered 1997 Dec 31 by P. G. Comba at Prescott.


Notes


References


Further reading


Specific references


Biographical

* * Brentjes, Sonja (2007).
Khwārizmī: Muḥammad ibn Mūsā al‐Khwārizmī
in Thomas Hockey et al.(eds.). '' The Biographical Encyclopedia of Astronomers'', Springer Reference. New York: Springer, 2007, pp. 631–633.
PDF version
* * Hogendijk, Jan P.
Muhammad ibn Musa (Al-)Khwarizmi (c. 780–850 CE)
– bibliography of his works, manuscripts, editions and translations. * * * * * Sezgin, F., ed., ''Islamic Mathematics and Astronomy'', Frankfurt: Institut für Geschichte der arabisch-islamischen Wissenschaften, 1997–99.


Algebra

* * * * * Barnabas Hughes. ''Robert of Chester's Latin translation of al-Khwarizmi's al-Jabr: A new critical edition''. In Latin. F. Steiner Verlag Wiesbaden (1989). . * * *


Arithmetic

* * (This is a new edition of the complete medieval Latin translation of the Arithmetic of al-Khwarizmi, previous editions are all incomplete. This work is lost in Arabic). *


Astronomy

* * (Hogendijk's homepage. Publication in English, no. 25). * (Description and analysis of seven recently discovered minor works related to al-Khwarizmi). * * * Suter, Heinrich. d. Die astronomischen Tafeln des Muhammed ibn Mûsâ al-Khwârizmî in der Bearbeitung des Maslama ibn Ahmed al-Madjrîtî und der latein. Übersetzung des Athelhard von Bath auf Grund der Vorarbeiten von A. Bjørnbo und R. Besthorn in Kopenhagen. Hrsg. und komm. Kopenhagen 1914. 288 pp. Repr. 1997 (Islamic Mathematics and Astronomy. 7). . * (Van Dalen's homepage. List of Publications, Articles – no. 5).


Spherical trigonometry

* B.A. Rozenfeld. "Al-Khwarizmi's spherical trigonometry" (Russian), ''Istor.-Mat. Issled.'' 32–33 (1990), 325–339.


Jewish calendar

*


Geography

* * * * * * *


General references

* * * * * * * * * * * * {{DEFAULTSORT:Khwarizmi, Muhammad ibn Musa 780s births 850 deaths 8th-century Arabic writers 8th-century astrologers 8th-century Iranian astronomers 8th-century people from the Abbasid Caliphate 9th-century Arabic writers 9th-century astrologers 9th-century geographers 9th-century inventors 9th-century Iranian astronomers 9th-century people from the Abbasid Caliphate 9th-century Iranian mathematicians Astronomers from the Abbasid Caliphate Geographers from the Abbasid Caliphate Inventors of the medieval Islamic world Mathematicians from the Abbasid Caliphate Mathematicians who worked on Islamic inheritance Medieval Iranian astrologers Medieval Iranian geographers People from Xorazm Region Transoxanian Islamic scholars Persian physicists Scientists who worked on qibla determination Writers about religion and science