Muhammad ibn Musa al-Khwarizmi,[note 1] or simplyal-Khwarizmi (c. 780 – c. 850) was amathematician active during theIslamic Golden Age, who produced Arabic-language works inmathematics,astronomy, andgeography. Around 820, he worked at theHouse of Wisdom inBaghdad, the contemporary capital city of theAbbasid Caliphate. One of the most prominent scholars of the period, his works were widely influential on later authors, both in the Islamic world and Europe.
His popularizing treatise onalgebra, compiled between 813 and 833 asAl-Jabr (The Compendious Book on Calculation by Completion and Balancing),[7]: 171 presented the first systematic solution oflinear andquadratic equations. One of his achievements inalgebra was his demonstration of how to solve quadratic equations bycompleting the square, for which he provided geometric justifications.[8]: 14 Because al-Khwarizmi was the first person 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),[9] he has been described as the father[10][11][12] or founder[13][14] of algebra. The English termalgebra comes from the short-hand title of his aforementioned treatise (الجبرAl-Jabr,transl. "completion" or "rejoining").[15] His name gave rise to the English termsalgorism andalgorithm; the Spanish, Italian, and Portuguese termsalgoritmo; and the Spanish termguarismo[16] and Portuguese termalgarismo, all meaning 'digit'.[17]
Al-Khwarizmi revisedGeography, the 2nd-century Greek-language treatise byPtolemy, listing the longitudes and latitudes of cities and localities.[23]: 9 He further produced a set of astronomical tables and wrote about calendric works, as well as theastrolabe and thesundial.[24] Al-Khwarizmi made important contributions totrigonometry, producing accuratesine and cosine tables.
Al-Tabari gives his name as Muḥammad ibn Musá al-Khwārizmī al-Majūsī al-Quṭrubbullī (محمد بن موسى الخوارزميّ المجوسـيّ القطربّـليّ). Theepithetal-Qutrubbulli could indicate he might instead have come from Qutrubbul (Qatrabbul),[34] near Baghdad. However,Roshdi Rashed denies this:[35]
There is no need to be an expert on the period or a philologist to see that al-Tabari's second citation should read "Muhammad ibn Mūsa al-Khwārizmīand al-Majūsi al-Qutrubbulli," and that there are two people (al-Khwārizmī and al-Majūsi al-Qutrubbulli) between whom the letterwa [Arabic 'و' for the conjunction 'and'] has been omitted in an early copy. This would not be worth mentioning if a series of errors concerning the personality of al-Khwārizmī, occasionally even the origins of his knowledge, had not been made. Recently,G.J. Toomer ... with naive confidence constructed an entire fantasy on the error which cannot be denied the merit of amusing the reader.
On the other hand,David A. King affirms hisnisba to Qutrubul, noting that he was called al-Khwārizmī al-Qutrubbulli because he was born just outside of Baghdad.[36]
Another epithet given to him by al-Ṭabarī, "al-Majūsī," would seem to indicate that he was an adherent of the oldZoroastrian religion. This would still have been possible at that time for a man of Iranian origin, but the pious preface to al-Khwārizmī'sAlgebra shows that he was an orthodoxMuslim, so al-Ṭabarī's epithet could mean no more than that his forebears, and perhaps he in his youth, had been Zoroastrians.
Ibn al-Nadīm'sAl-Fihrist 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 theMuslim conquest of Persia, Baghdad had become the centre of scientific studies and trade. Around 820 CE, he was appointed as the astronomer and head of the library of theHouse of Wisdom.[8]: 14 The House of Wisdom was established by theAbbasidCaliph al-Ma'mūn. Al-Khwārizmī studied sciences and mathematics, including the translation ofGreek andSanskrit scientific manuscripts. He was also a historian who is cited by the likes ofal-Tabari andIbn Abi Tahir.[38]
During the reign ofal-Wathiq, he is said to have been involved in the first of two embassies to theKhazars.[39]Douglas Morton Dunlop suggests that Muḥammad ibn Mūsā al-Khwārizmī might have been the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the threeBanū Mūsā brothers.[40]
Contributions
A page from al-Khwārizmī'sAlgebra
Al-Khwārizmī's contributions to mathematics, geography, astronomy, andcartography established the basis for innovation in algebra andtrigonometry. His systematic approach to solving linear and quadratic equations led toalgebra, a word derived from the title of his book on the subject,Al-Jabr.[41]
On the Calculation with Hindu Numerals, written about 820, was principally responsible for spreading theHindu–Arabic numeral system throughout the Middle East and Europe. When the work was translated into Latin in the 12th century asAlgoritmi de numero Indorum (Al-Khwarizmi on the Hindu art of reckoning), the term "algorithm" was introduced to the Western world.[42][43][44]
Al-Khwārizmī systematized and correctedPtolemy's data for Africa and the Middle East. Another major book wasKitab surat al-ard ("The Image of the Earth"; translated as Geography), presenting the coordinates of places based on those in theGeography of Ptolemy, but with improved values for theMediterranean Sea, Asia, and Africa.[46]
He wrote on mechanical devices like theastrolabe[47] andsundial.[24] He assisted a project to determine the circumference of the Earth and in making a world map foral-Ma'mun, the caliph, overseeing 70 geographers.[48] 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.[49]
Left: The original Arabic print manuscript of theBook of Algebra by Al-Khwārizmī. Right: A page fromThe Algebra of Al-Khwarizmi by Fredrick Rosen, in English.
Al-Jabr (The Compendious Book on Calculation by Completion and Balancing,Arabic:الكتاب المختصر في حساب الجبر والمقابلةal-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala) is a mathematical book written approximately 820 CE. It was written with the encouragement ofCaliph al-Ma'mun as a popular work on calculation and is replete with examples and applications to a range of problems in trade, surveying and legal inheritance.[50] The term "algebra" is derived from the name of one of the basic operations with equations (al-jabr, 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 asLiber algebrae et almucabala byRobert of Chester (Segovia, 1145) hence "algebra", and byGerard of Cremona. A unique Arabic copy is kept at Oxford and was translated in 1831 byF. Rosen.[51] A Latin translation is kept in Cambridge.[52]
It provided an exhaustive account of solving polynomial equations up to the second degree,[53] 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.[54]
Al-Khwārizmī's method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms (whereb andc are positive integers)
squares equal roots (ax2 =bx)
squares equal number (ax2 =c)
roots equal number (bx =c)
squares and roots equal number (ax2 +bx =c)
squares and number equal roots (ax2 +c =bx)
roots and number equal squares (bx +c =ax2)
by dividing out the coefficient of the square and using the two operationsal-jabr (Arabic:الجبر "restoring" or "completion") andal-muqābala ("balancing").Al-jabr is the process of removing negative units, roots and squares from the equation by adding the same quantity to each side. For example,x2 = 40x − 4x2 is reduced to 5x2 = 40x.Al-muqābala is the process of bringing quantities of the same type to the same side of the equation. For example,x2 + 14 =x + 5 is reduced tox2 + 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 notationhad 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 the 1831 "Rosen" translation)[50]
If some one say: “You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times.” Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts.
Solomon Gandz has described Al-Khwarizmi as the father of Algebra:
Al-Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences. In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers.[55]
The first true algebra text which is still extant is the work on al-jabr and al-muqabala by Mohammad ibn Musa al-Khwarizmi, written in Baghdad around 825.[56]
Perhaps one of the most significant advances made byArabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowedrational numbers,irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before.[57]
Al-Khwarizmi's text can be seen to be distinct not only from theBabylonian tablets, but also fromDiophantus'Arithmetica. It no longer concerns a series ofproblems to be solved, but anexposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems.[58]
According to Swiss-American historian of mathematics,Florian Cajori, Al-Khwarizmi's algebra was different from the work ofIndian mathematicians, for Indians had no rules like therestoration andreduction.[59] Regarding the dissimilarity and significance of Al-Khwarizmi's algebraic work from that of Indian MathematicianBrahmagupta,Carl B. Boyer wrote:
It is true that in two respects the work of al-Khowarizmi represented a retrogression from that ofDiophantus. 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 GreekArithmetica 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, theAl-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.[60]
Arithmetic
Algorists vs. abacists, depicted in an etching from 1503 CEPage from a Latin translation, beginning with "Dixit algorizmi"
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 textkitāb al-ḥisāb al-hindī ('Book of Indian computation'[note 2]), and perhaps a more elementary text,kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī ('Addition and subtraction in Indian arithmetic').[62][63] These texts described algorithms on decimal numbers (Hindu–Arabic numerals) that could be carried out on a dust board. Calledtakht 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 byAl-Uqlidisi's algorithms that could be carried out with pen and paper.[64]
As part of 12th century wave of Arabic science flowing into Europe via translations, these texts proved to be revolutionary in Europe.[65] Al-Khwarizmi'sLatinized name,Algorismus, turned into thename of method used for computations, and survives in the term "algorithm". It gradually replaced the previous abacus-based methods used in Europe.[66]
Four Latin texts providing adaptions of Al-Khwarizmi's methods have survived, even though none of them is believed to be a literal translation:[62]
Dixit Algorizmi (published in 1857 under the titleAlgoritmi de Numero Indorum[67])[68]
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 titleAlgoritmi de Numero Indorum. It is attributed to theAdelard of Bath, who had translated the astronomical tables in 1126. It is perhaps the closest to Al-Khwarizmi's own writings.[68]
Al-Khwarizmi's work on arithmetic was responsible for introducing theArabic numerals, based on theHindu–Arabic numeral system developed inIndian mathematics, to the Western world. The term "algorithm" is derived from thealgorism, the technique of performing arithmetic with Hindu-Arabic numerals developed by al-Khwārizmī. Both "algorithm" and "algorism" are derived from theLatinized forms of al-Khwārizmī's name,Algoritmi andAlgorismi, respectively.[69]
Page fromCorpus Christi College MS 283, a Latin translation of al-Khwārizmī'sZīj
Al-Khwārizmī'sZīj as-Sindhind[37] (Arabic:زيج السند هند, "astronomical tables ofSiddhanta"[70]) 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 values. This is the first of many ArabicZijes based on theIndian astronomical methods known as thesindhind.[71] The word Sindhind is a corruption of theSanskritSiddhānta, 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) ofBrahmagupta.[72]
The work contains tables for the movements of theSun,Moon and the fiveplanets known at the time. This work marked the turning point inIslamic astronomy. Hitherto, Muslim astronomers had adopted a primarily research approach to the field, translating works of others and learning already discovered knowledge.
Al-Khwārizmī produced accurate sine and cosine tables.[74][75]
Geography
Gianluca Gorni's reconstruction of the section of al-Khwārizmī's world map concerning the Indian Ocean. The majority of the placenames used by al-Khwārizmī match those of Ptolemy,Martellus andBehaim. The general shape of the coastline is the same betweenTaprobane andCattigara. TheDragon's Tail, or the eastern opening of the Indian Ocean, which does not exist in Ptolemy's description, is traced in very little detail on al-Khwārizmī's map, although is clear and precise on the Martellus map and on the later Behaim version.A15th-century version ofPtolemy'sGeography for comparisonEarliest extant map of the Nile, in Al-Khwārazmī’s Kitāb ṣūrat al- arḍ.
Al-Khwārizmī's third major work is hisKitāb Ṣūrat al-Arḍ (Arabic:كتاب صورة الأرض, "Book of the Description of the Earth"),[76] also known as hisGeography, which was finished in 833. It is a major reworking ofPtolemy's second-centuryGeography, consisting of a list of 2402 coordinates of cities and other geographical features following a general introduction.[77]
There is one surviving copy ofKitāb Ṣūrat al-Arḍ, which is kept at theStrasbourg University Library.[78][79] A Latin translation is at theBiblioteca Nacional de España in Madrid.[80] The book opens with the list oflatitudes andlongitudes, in order of "weather zones", that is to say in blocks of latitudes and, in each weather zone, by order of longitude. AsPaul Gallez notes, this 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; 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 deduced them from the context where they were not legible. He transferred the points ontograph paper and connected them with straight lines, obtaining an approximation of the coastline as it was on the original map. He did the same for the rivers and towns.[81]
Al-Khwārizmī corrected Ptolemy's gross overestimate for the length of theMediterranean Sea[82] from theCanary Islands to the eastern shores of the Mediterranean; Ptolemy overestimated it at 63 degrees oflongitude, while al-Khwārizmī almost correctly estimated it at nearly 50 degrees of longitude. He "depicted theAtlantic and Indian Oceans asopen bodies of water, not land-locked seas as Ptolemy had done."[83] Al-Khwārizmī'sPrime Meridian at theFortunate Isles 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.[82]
Jewish calendar
Al-Khwārizmī wrote several other works including a treatise on theHebrew calendar, titledRisāla fi istikhrāj ta'rīkh al-yahūd (Arabic:رسالة في إستخراج تأريخ اليهود, "Extraction of the Jewish Era"). It describes theMetonic cycle, a 19-year intercalation cycle; the rules for determining on what day of the week the first day of the monthTishrei shall fall; calculates the interval between theAnno Mundi or Jewish year and theSeleucid era; and gives rules for determining the mean longitude of the sun and the moon using theHebrew calendar. Similar material is found in the works ofAl-Bīrūnī andMaimonides.[37]
Other works
Ibn al-Nadim'sAl-Fihrist, an index of Arabic books, mentions al-Khwārizmī'sKitāb al-Taʾrīkh (Arabic:كتاب التأريخ), a book of annals. No direct manuscript survives; however, a copy had reachedNusaybin by the 11th century, where itsmetropolitan bishop, MarElias 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.[84]
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; theFihrist credits al-Khwārizmī withKitāb ar-Rukhāma(t) (Arabic:كتاب الرخامة). Other papers, such as one on the determination of the direction ofMecca, are on thespherical astronomy.
Two texts deserve special interest on themorning width (Ma'rifat sa'at al-mashriq fī kull balad) and the determination of theazimuth from a height (Ma'rifat al-samt min qibal al-irtifā'). He wrote two books on using and constructingastrolabes.
Honours
A Sovietpostage stamp issued 6 September 1983, commemorating al-Khwārizmī's (approximate) 1200th birthday
13498 Al Chwarizmi — Main-belt Asteroid, Discovered 1986 Aug 6 by E. W. Elst and V. G. Ivanova at Smolyan.[86]
11156 Al-Khwarismi — Main-belt Asteroid, Discovered 1997 Dec 31 by P. G. Comba at Prescott.[87]
Notes
^There is some confusion in the literature on whether al-Khwārizmī's full name isابو عبدالله محمد بن موسى خوارزمیAbū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī orابوجعفر محمد بن موسی خوارزمیAbū Ja'far Muḥammad ibn Mūsā al-Khwārizmī.Ibn Khaldun notes in his Prolegomena: "The first to write on this discipline [algebra] was Abu 'Abdallah al-Khuwarizmi. After him, there was Abu Kamil Shuja' b. Aslam. People followed in his steps."[5] In the introduction to his critical commentary on Robert of Chester's Latin translation of al-Khwārizmī'sAlgebra, L. C. Karpinski notes that Abū Ja'far Muḥammad ibn Mūsā refers to the eldest of theBanū Mūsā brothers. Karpinski notes in his review on (Ruska 1917) that in (Ruska 1918): "Ruska here inadvertently speaks of the author as Abū Ga'far M. b. M., instead of Abū Abdallah M. b. M." Donald Knuth writes it asAbū 'Abd Allāh Muḥammad ibn Mūsā al-Khwārizmī and quotes it as meaning "literally, 'Father of Abdullah, Mohammed, son of Moses, native of Khwārizm,'" citing previous work by Heinz Zemanek.[6]
^Some scholars translate the titleal-ḥisāb al-hindī as "computation with Hindu numerals", but ArabicHindī means 'Indian' rather than 'Hindu'. A. S. Saidan states that it should be understood as arithmetic done "in the Indian way", with Hindu-Arabic numerals, rather than as simply "Indian arithmetic". The Arab mathematicians incorporated their own innovations in their texts.[61]
^Toomer, Gerald J. (1970–1980). "al-Khuwārizmī, Abu Ja'far Muḥammad ibn Mūsā". In Gillispie, Charles Coulston (ed.).Dictionary of Scientific Biography. Vol. VII. Scribner. pp. 358–365.ISBN978-0-684-16966-8.
^Vernet, Juan (1960–2005). "Al-Khwārizmī". In Gibb, H. A. R.; Kramers, J. H.; Lévi-Provençal, E.; Schacht, J. (eds.).The Encyclopaedia of Islam. Vol. IV (2nd ed.). Leiden: Brill. pp. 1070–1071.OCLC399624.
^Oaks, J. (2009), "Polynomials and Equations in Arabic Algebra",Archive for History of Exact Sciences, 63(2), 169–203.
^abMaher, P. (1998), "From Al-Jabr to Algebra",Mathematics in School, 27(4), 14–15.
^(Boyer 1991, "The Arabic Hegemony" p. 229) "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."
^Boyer, Carl B., 1985.A History of Mathematics, p. 252. Princeton University Press. "Diophantus sometimes is called the father of algebra, but this title more appropriately belongs to al-Khowarizmi...", "...the Al-jabr comes closer to the elementary algebra of today than the works of either Diophantus or Brahmagupta..."
^Gandz, Solomon, The sources of al-Khwarizmi's algebra, Osiris, i (1936), 263–277, "Al-Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences. In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers."
^Katz, Victor J."Stages in the History of Algebra with Implications for Teaching"(PDF).VICTOR J.KATZ, University of the District of Columbia Washington DC, USA: 190. Archived fromthe original(PDF) on 27 March 2019. Retrieved7 October 2017 – via University of the District of Columbia Washington DC, USA.The first true algebra text which is still extant is the work on al-jabr and al-muqabala by Mohammad ibn Musa al-Khwarizmi, written in Baghdad around 825.
^Hill, Fred James; Awde, Nicholas (2003).A History of the Islamic World. Hippocrene. p. 55.ISBN978-0-7818-1015-9."The Compendious Book on Calculation by Completion and Balancing" (Hisab al-Jabr wa H-Muqabala) on the development of the subject cannot be underestimated. Translated into Latin during the twelfth century, it remained the principal mathematics textbook in European universities until the sixteenth century
^abSaliba, George (September 1998). "Science and medicine".Iranian Studies.31 (3–4):681–690.doi:10.1080/00210869808701940.Take, for example, someone like Muhammad b. Musa al-Khwarizmi (fl. 850) may present a problem for the EIr, for although he was obviously of Persian descent, he lived and worked in Baghdad and was not known to have produced a single scientific work in Persian.
^Oaks, Jeffrey A. (2014)."Khwārizmī". In Kalin, Ibrahim (ed.).The Oxford Encyclopedia of Philosophy, Science, and Technology in Islam. Vol. 1.Oxford:Oxford University Press. pp. 451–459.ISBN978-0-19-981257-8.Archived from the original on 30 January 2022. Retrieved6 September 2021. "Ibn al-Nadīm and Ibn al-Qifṭī relate that al-Khwārizmī's family came from Khwārizm, the region south of the Aral sea." Also → al-Nadīm, Abu'l-Faraj (1871–1872).Kitāb al-Fihrist, ed. Gustav Flügel, Leipzig: Vogel, p.274. al-Qifṭī, Jamāl al-Dīn (1903).Taʾrīkh al-Hukamā, eds. August Müller & Julius Lippert, Leipzig: Theodor Weicher, p.286.
^Dodge, Bayard, ed. (1970),The Fihrist of al-Nadīm: A Tenth-Century Survey of Islamic Culture, vol. 2, translated by Dodge, New York:Columbia University Press
^A History of Science in World Cultures: Voices of Knowledge. Routledge. Page 228. "Mohammed ibn Musa al-Khwarizmi (780–850) was a Persian astronomer and mathematician from the district of Khwarism (Uzbekistan area of Central Asia)."
^Ben-Menahem, Ari (2009).Historical Encyclopedia of Natural and Mathematical Sciences (1st ed.). Berlin: Springer. pp. 942–943.ISBN978-3-540-68831-0.Persian mathematician Al-Khowarizmi
^Wiesner-Hanks, Merry E.;Ebrey, Patricia Buckley; Beck, Roger B.; Davila, Jerry; Crowston, Clare Haru;McKay, John P. (2017).A History of World Societies (11th ed.). Bedford/St. Martin's. p. 419.Near the beginning of this period the Persian scholar al-Khwarizmi (d. ca. 850) harmonized Greek and Indian findings to produce astronomical tables that formed the basis for later Eastern and Western research.
^Bosworth, Clifford Edmund (1960–2005). "Khwārazm". In Gibb, H. A. R.; Kramers, J. H.; Lévi-Provençal, E.; Schacht, J. (eds.).The Encyclopaedia of Islam. Vol. IV (2nd ed.). Leiden: Brill. pp. 1060–1065.OCLC399624.
^King, David A. (7 March 2018).Astronomy in the Service of Islam. Al-Furqān Islamic Heritage Foundation – Centre for the Study of Islamic Manuscripts. Event occurs at 20:51.Archived from the original on 1 December 2021. Retrieved26 November 2021.I mention another name of Khwarizmi to show that he didn't come from Central Asia. He came from Qutrubul, just outside Baghdad. He was born there, otherwise he wouldn't be called al-Qutrubulli. Many people say he came from Khwarazm, tsk-tsk.
^Golden, Peter; Ben-Shammai, Haggai; Roná-Tas, András (13 August 2007).The World of the Khazars: New Perspectives. Selected Papers from the Jerusalem 1999 International Khazar Colloquium. BRILL. p. 376.ISBN978-90-474-2145-0.
^Bradley, Michael John (2006).The Birth of Mathematics: Ancient Times To 1300. Infobase Publishing. p. 99.ISBN978-0-7910-9723-6.Al-Khwarizmi's most significant astronomical achievement was the creation of an extensive set of tables known as Zij al-sindhind (Astronomical tables of India). He based much of this work on similar tables that Indian astronomer Brahmagupta had included in his seventh-century work Brahmasphutasiddhānta (Improved system of Brahma), a work that Arabic translators had named Zij al-sindhind, combining the words sind, a region of Pakistan, and bind, the Arabic word for India. He also borrowed material from the massive second-century work Almagest (The greatest compilation), created by the Greek astronomer Claudius Ptolemy, and the more recent sixth-century work Zij al-shāh.
^Boyer 1991, p. 228: "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."
^(Boyer 1991, "The Arabic Hegemony" p. 229) "It is not certain just what the termsal-jabr andmuqabalah mean, but the usual interpretation is similar to that implied in the translation above. The wordal-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 wordmuqabalah is said to refer to "reduction" or "balancing" — that is, the cancellation of like terms on opposite sides of the equation."
^Gandz, Solomon, The sources of al-Khwarizmi's algebra, Osiris, i (1936), 263–277
^Florian Cajori (1919).A History of Mathematics. Macmillan. p. 103.That it came from Indian source is impossible, for Hindus had no rules like "restoration" and "reduction". They were never in the habit of making all terms in an equation positive, as is done in the process of "restoration.
^Saidan, A. S. (Winter 1966), "The Earliest Extant Arabic Arithmetic: Kitab al-Fusul fi al Hisab al-Hindi of Abu al-Hasan, Ahmad ibn Ibrahim al-Uqlidisi",Isis,57 (4), The University of Chicago Press:475–490,doi:10.1086/350163,JSTOR228518,S2CID143979243
^Van Brummelen, Glen (2017),"Arithmetic", in Thomas F. Glick (ed.),Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia, Taylor & Francis, p. 46,ISBN978-1-351-67617-5,archived from the original on 28 March 2023, retrieved5 May 2019
^Thomas F. Glick, ed. (2017),"Al-Khwarizmi",Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia, Taylor & Francis,ISBN978-1-351-67617-5,archived from the original on 28 March 2023, retrieved6 May 2019
^Van Brummelen, Glen (2017),"Arithmetic", in Thomas F. Glick (ed.),Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia, Taylor & Francis, pp. 46–47,ISBN978-1-351-67617-5,archived from the original on 28 March 2023, retrieved5 May 2019
^"Algoritmi de numero Indorum",Trattati D'Aritmetica, Rome: Tipografia delle Scienze Fisiche e Matematiche, 1857, pp. 1–,archived from the original on 28 March 2023, retrieved6 May 2019
^abCrossley, John N.; Henry, Alan S. (1990), "Thus Spake al-Khwārizmī: A Translation of the Text of Cambridge University Library Ms. Ii.vi.5",Historia Mathematica,17 (2):103–131,doi:10.1016/0315-0860(90)90048-I
^The full title is "The Book of the Description of the Earth, with its Cities, Mountains, Seas, All the Islands and the Rivers, written by Abu Ja'far Muhammad ibn Musa al-Khwārizmī, according to the Geographical Treatise written by Ptolemy the Claudian", although due to ambiguity in the wordsurah it could also be understood as meaning "The Book of the Image of the Earth" or even "The Book of the Map of the World".
^abEdward S. Kennedy,Mathematical Geography, p. 188, in (Rashed & Morelon 1996, pp. 185–201)
^Covington, Richard (2007)."The Third Dimension".Saudi Aramco World, May–June 2007:17–21. Archived fromthe original on 12 May 2008. Retrieved6 July 2008.
^LJ Delaporte (1910).Chronographie de Mar Elie bar Sinaya. p. xiii.
Burnett, Charles (2017),"Arabic Numerals", in Thomas F. Glick (ed.),Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia, Taylor & Francis,ISBN978-1-351-67617-5,archived from the original on 28 March 2023, retrieved5 May 2019
Hughes, Barnabas (1989).Robert of Chester's Latin Translation of al-Khwarizmi's al-Jabr: A New Critical Edition (in Latin). Wiesbaden: F. Steiner Verlag.ISBN3-515-04589-9.
Neugebauer, Otto (1962).The Astronomical Tables of al-Khwarizmi.
Rosenfeld, Boris A. (1993). "'Geometric trigonometry' in treatises of al-Khwārizmī, al-Māhānī and Ibn al-Haytham". In Folkerts, Menso;Hogendijk, Jan P. (eds.).Vestigia Mathematica: Studies in Medieval and Early Modern Mathematics in Honour of H.L.L. Busard. Leiden: Brill. pp. 305–308.ISBN978-90-5183-536-6.
.jpg)
.png)
.png)
.png)
.png)
