GetWiki

{{Short description|9th-century Persian polymath}}{{pp-move-indef}}{{other uses}}

factoids

}}780}}850}}TOOMER >FIRST1=GERALD J. EDITOR1-LAST=GILLISPIE TITLE=DICTIONARY OF SCIENTIFIC BIOGRAPHY ISBN=978-0-684-16966-8PAGES=358–365 EDITOR1-LAST=GIBB EDITOR2-LAST=KRAMERS EDITOR3-LAST=LéVI-PROVENçAL EDITOR4-LAST=SCHACHT TITLE=THE ENCYCLOPAEDIA OF ISLAM PUBLISHER=BRILL VOLUME=IV EDITION=2ND OCLC=399624, (aged ~70)| era = Islamic Golden Age| alma_mater = | school_tradition = Mathematics in the medieval Islamic world>Astronomy in the medieval Islamic world>geography}}| notable_ideas = Treatises on algebra and the Hindu–Arabic numeral systemAl-Jabr (820)Zij as-Sindhind (820)>Kitab Surat al-Ard (833)}}| influences = Abu Kamil of EgyptO'Connor, John J.; Robertson, Edmund F., "AbÅ« Kāmil Shujā' ibn Aslam" {{Webarchive>url=https://web.archive.org/web/20131211214159weblink |date=11 December 2013 }}, MacTutor History of Mathematics archive, University of St Andrews.| awards = | birth_place = Khwarazm, Abbasid Caliphate| death_place = Abbasid CaliphateHouse of Wisdom in Baghdad (appt. {{circa>820}})| nationality = Persian}}{{Use dmy dates|date=March 2022}}{{Use Oxford spelling|date=December 2023}}Muhammad ibn Musa al-Khwarizmi{{refn|group=note|There is some confusion in the literature on whether al-KhwārizmÄ«'s full name is {{transliteration|ar|ALA|AbÅ« Ê¿Abdallāh Muḥammad ibn MÅ«sā al-KhwārizmÄ«}} or {{transliteration|ar|ALA|AbÅ« Ja'far Muḥammad ibn MÅ«sā al-KhwārizmÄ«}}. 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."Ibn KhaldÅ«n, The Muqaddimah: An introduction to history {{Webarchive|url=https://web.archive.org/web/20160917023325weblink |date=17 September 2016 }}, Translated from the Arabic by Franz Rosenthal, New York: Princeton (1958), Chapter VI:19. In the introduction to his critical commentary on Robert of Chester's Latin translation of al-KhwārizmÄ«'s Algebra, L.C. Karpinski notes that AbÅ« Ja'far Muḥammad ibn MÅ«sā refers to the eldest of the BanÅ« MÅ«sā brothers. Karpinski notes in his review on (Ruska 1917) that in (Ruska 1918): "Ruska here inadvertently speaks of the author as AbÅ« Ga'far M. b. M., instead of AbÅ« Abdallah M. b. M." Donald Knuth writes it as {{transliteration|ar|ALA|AbÅ« 'Abd Allāh Muḥammad ibn MÅ«sā al-KhwārizmÄ«}} and quotes it as meaning "literally, 'Father of Abdullah, Mohammed, son of Moses, native of Khwārizm,'" citing previous work by Heinz Zemanek.BOOK, Donald, Knuth, Basic Concepts, The Art of Computer Programming, 1, 3rd, 1997, Addison-Wesley, 978-0-201-89683-1, 1, }} (; {{circa|lk=off|780|850}}), often referred to as simply al-Khwarizmi, was a Persian polymath who produced vastly influential Arabic-language works in mathematics, astronomy, and geography. Hailing from Khwarazm, he was appointed as the astronomer and head of the House of Wisdom in the city of Baghdad around 820 CE.His popularizing treatise on algebra, compiled between 813–33 as Al-Jabr (The Compendious Book on Calculation by Completion and Balancing),Oaks, J. (2009), "Polynomials and Equations in Arabic Algebra", Archive for History of Exact Sciences, 63(2), 169–203.{{rp|171}} presented the first systematic solution of linear and quadratic equations. One of his achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications.{{rp|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),(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." he has been described as the fatherBOOK,weblink The Voyage and the Messenger: Iran and Philosophy, Corbin, Henry, 1998, North Atlantic Books, 978-1-55643-269-9, en, 44, 19 October 2020, 28 March 2023,weblink live, 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." or founderJOURNAL, Katz, Victor J., Stages in the History of Algebra with Implications for Teaching,weblink VICTOR J.KATZ, University of the District of Columbia Washington DC, USA, 190, 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., 7 October 2017,weblink 27 March 2019, dead, BOOK,weblink The Oxford History of Islam, Esposito, John L., John Esposito, 6 April 2000, Oxford University Press, 978-0-19-988041-6, en, 188, Al-Khwarizmi is often considered the founder of algebra, and his name gave rise to the term algorithm., 29 September 2020, 28 March 2023,weblink live, of algebra. The English term algebra comes from the short-hand title of his aforementioned treatise ( {{Transliteration|ar|Al-Jabr}}, {{Translation|"completion" or "rejoining"}}).JOURNAL, Brentjes, Sonja, Sonja Brentjes, 1 June 2007, Algebra,weblink Encyclopaedia of Islam, THREE, en, 5 June 2019, 22 December 2019,weblink live, His name gave rise to the English terms algorism and algorithm; the Spanish, Italian, and Portuguese terms {{Text|algoritmo}}; and the Spanish term BOOK, Knuth, Donald,weblink Algorithms in Modern Mathematics and Computer Science, Springer-Verlag, 1979, 978-0-387-11157-5, Donald Knuth, dead,weblink" title="web.archive.org/web/20061107213306weblink">weblink 7 November 2006, and Portuguese term , both meaning "digit".JOURNAL, Gandz, Solomon, Solomon Gandz, 1926, The Origin of the Term "Algebra",weblink The American Mathematical Monthly, 33, 9, 437–440, 10.2307/2299605, 2299605, 0002-9890, In the 12th century, Latin-language translations of al-Khwarizmi's textbook on Indian arithmetic (), which codified the various Indian numerals, introduced the decimal-based positional number system to the Western world.{{harvnb|Struik|1987| p= 93}} Likewise, Al-Jabr, translated into Latin by the English scholar Robert of Chester in 1145, was used until the 16th century as the principal mathematical textbook of European universities.BOOK,weblinkweblink dead, 20 December 2019, History of the Arabs, Philip Khuri Hitti, 2002, 978-1-137-03982-8, 379, Palgrave Macmillan, BOOK,weblink registration, A History of the Islamic World, Hippocrene Books, Fred James Hill, Nicholas Awde, 2003, 978-0-7818-1015-9, 55, "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, WEB,weblink Al-Khwarizmi, Shawn Overbay, Jimmy Schorer, Heather Conger, University of Kentucky,weblink" title="web.archive.org/web/20131212235239weblink">weblink 12 December 2013, live, WEB,weblink Islam Spain and the history of technology, www.sjsu.edu, 24 January 2018, 11 October 2018,weblink" title="web.archive.org/web/20181011150650weblink">weblink live, Al-Khwarizmi revised Geography, the 2nd-century Greek-language treatise by the Roman polymath Claudius Ptolemy, listing the longitudes and latitudes of cities and localities.van der Waerden, Bartel Leendert (1985). A History of Algebra: From al–Khwarizmi to Emmy Noether. Berlin: Springer-Verlag.{{rp|9}} He further produced a set of astronomical tables and wrote about calendric works, as well as the astrolabe and the sundial.{{harvnb|Arndt|1983|p=669}} Al-Khwarizmi made important contributions to trigonometry, producing accurate sine and cosine tables and the first table of tangents.

Life

Few details of al-KhwārizmÄ«'s life are known with certainty. Ibn al-Nadim gives his birthplace as Khwarazm, and he is generally thought to have come from this region.BOOK, Oaks, Jeffrey A., Kalin, Ibrahim, The Oxford Encyclopedia of Philosophy, Science, and Technology in Islam, 2014, Oxford University Press, Oxford, 978-0-19-981257-8, 1, 451–459, KhwārizmÄ«, registration,weblink 6 September 2021, 30 January 2022,weblink live, "Ibn al-NadÄ«m and Ibn al-Qifá¹­Ä« relate that al-KhwārizmÄ«'s family came from Khwārizm, the region south of the Aral sea." Also → al-NadÄ«m, Abu'l-Faraj (1871–1872). Kitāb al-Fihrist, ed. Gustav Flügel, Leipzig: Vogel, p. 274. al-Qifá¹­Ä«, Jamāl al-DÄ«n (1903). TaʾrÄ«kh al-Hukamā, eds. August Müller & Julius Lippert, Leipzig: Theodor Weicher, p. 286.{{citation| editor-last=Dodge | editor-first=Bayard| translator-last=Dodge |title=The Fihrist of al-NadÄ«m: A Tenth-Century Survey of Islamic Culture | publisher=Columbia University Press | place=New York | year=1970 |volume=2 }} Of Persian stock,BOOK, Clifford A. Pickover, The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics,weblink 2009, Sterling Publishing Company, Inc., 978-1-4027-5796-9, 84, 19 October 2020, 28 March 2023,weblink live, JOURNAL, Saliba, George, Science and medicine, Iranian Studies, September 1998, 31, 3–4, 681–690, 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., 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)."BOOK, Ben-Menahem, Ari, Ari Ben-Menahem, Historical Encyclopedia of Natural and Mathematical Sciences, 2009, Springer, Berlin, 978-3-540-68831-0, 942–943, 1st, Persian mathematician Al-Khowarizmi, BOOK, Wiesner-Hanks, Merry E., Ebrey, Patricia Buckley, Beck, Roger B., Davila, Jerry, Crowston, Clare Haru, McKay, John P., Merry Wiesner-Hanks, Patricia Buckley Ebrey, John P. McKay, A History of World Societies, 2017, Bedford/St. Martin's, 419, 11th, 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., his name means 'of Khwarazm', a region that was part of Greater Iran,Encycloaedia Iranica-online, s.v. "CHORASMIA, ii. In Islamic times {{Webarchive|url=https://web.archive.org/web/20210902091627weblink |date=2 September 2021 }}," by Clifford E. Bosworth. and is now part of Turkmenistan and Uzbekistan.BOOK, Bosworth, Clifford Edmund, Clifford Edmund Bosworth, Gibb, H. A. R., Kramers, J. H., Lévi-Provençal, E., Schacht, J., The Encyclopaedia of Islam, 1960–2005, Brill, Leiden, IV, 1060–1065, 2nd, Khwārazm, 399624, Al-Tabari gives his name as Muḥammad ibn Musá al-KhwārizmÄ« al-MajÅ«sÄ« al-Quá¹­rubbullÄ« (). The epithet al-Qutrubbulli could indicate he might instead have come from Qutrubbul (Qatrabbul),"Iraq After the Muslim Conquest", by Michael G. Morony, {{isbn|1-59333-315-3}} (a 2005 facsimile from the original 1984 book), p. 145 {{Webarchive|url=https://web.archive.org/web/20140627081909weblink |date=27 June 2014 }} near Baghdad. However, Roshdi Rashed denies this:BOOK, Rashed, Roshdi, Roshdi Rashed,weblink Arab Civilization: Challenges and Responses : Studies in Honor of Constantine K. Zurayk, 1988, SUNY Press, 978-0-88706-698-6, Zurayq, Qusá¹­aná¹­Ä«n, 108, al-KhwārizmÄ«'s Concept of Algebra, Atiyeh, George Nicholas, Oweiss, Ibrahim M.,weblink 19 October 2015, 28 March 2023,weblink live, {{blockquote|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 MÅ«sa al-KhwārizmÄ« and al-MajÅ«si al-Qutrubbulli," and that there are two people (al-KhwārizmÄ« and al-MajÅ«si al-Qutrubbulli) between whom the letter wa [Arabic '' for the conjunction '(wikt:Ùˆ#Etymology 2|and)'] has been omitted in an early copy. This would not be worth mentioning if a series of errors concerning the personality of al-KhwārizmÄ«, 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 his nisba to Qutrubul, noting that he was called al-KhwārizmÄ« al-Qutrubbulli because he was born just outside of Baghdad.AV MEDIA, David A. King (historian), King, David A., Astronomy in the Service of Islam, 7 March 2018, 20:51, Al-Furqān Islamic Heritage Foundation – Centre for the Study of Islamic Manuscripts,weblink 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., 26 November 2021, 1 December 2021,weblink live, Regarding al-KhwārizmÄ«'s religion, Toomer writes:{{harvnb|Toomer|1990}}{{blockquote|Another epithet given to him by al-ṬabarÄ«, "al-MajÅ«sÄ«," would seem to indicate that he was an adherent of the old Zoroastrian religion. This would still have been possible at that time for a man of Iranian origin, but the pious preface to al-KhwārizmÄ«'s Algebra shows that he was an orthodox Muslim, so al-ṬabarÄ«'s epithet could mean no more than that his forebears, and perhaps he in his youth, had been Zoroastrians.|author=|title=|source=}}Ibn al-NadÄ«m's includes a short biography on al-KhwārizmÄ« together with a list of his books. Al-KhwārizmÄ« accomplished most of his work between 813 and 833. After the Muslim 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 the House of Wisdom.Maher, P. (1998), "From Al-Jabr to Algebra", Mathematics in School, 27(4), 14–15.{{rp|14}} The House of Wisdom was established by the Abbasid Caliph al-Ma'mÅ«n. Al-KhwārizmÄ« studied sciences and mathematics, including the translation of Greek and Sanskrit scientific manuscripts. He was also a historian who is cited by the likes of al-Tabari and Ibn Abi Tahir.{{The History of al-Tabari | volume = 32 | page = 158}}During the reign of al-Wathiq, he is said to have been involved in the first of two embassies to the Khazars.BOOK, BRILL, 978-90-474-2145-0, Golden, Peter, Ben-Shammai, Haggai, Roná-Tas, András, The World of the Khazars: New Perspectives. Selected Papers from the Jerusalem 1999 International Khazar Colloquium, 13 August 2007, 376, Douglas Morton Dunlop suggests that Muḥammad ibn MÅ«sā al-KhwārizmÄ« might have been the same person as Muḥammad ibn MÅ«sā ibn Shākir, the eldest of the three BanÅ« MÅ«sā brothers.{{harvnb|Dunlop|1943}}

Contributions

(File:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|A page from al-Khwārizmī's Algebra)Al-Khwārizmī's contributions to mathematics, geography, astronomy, and cartography established the basis for innovation in algebra and trigonometry. His systematic approach to solving linear and quadratic equations led to algebra, a word derived from the title of his book on the subject, Al-Jabr.WEB,weblink Mathematics Education in Iran From Ancient to Modern, Yahya Tabesh, Shima Salehi, Sharif University of Technology, 16 April 2018, 16 April 2018,weblink" title="web.archive.org/web/20180416200423weblink">weblink live, 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 in Latin as Algoritmi, led to the term "algorithm".{{harvnb|Daffa|1977}}BOOK, Clegg, Brian,weblink Scientifica Historica: How the world's great science books chart the history of knowledge, 1 October 2019, Ivy Press, 978-1-78240-879-6, 61, en, 30 December 2021, 28 March 2023,weblink live, Some of his work was based on Persian and Babylonian astronomy, Indian numbers, and Greek mathematics.Al-Khwārizmī systematized and corrected Ptolemy'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, Asia, and Africa.WEB,weblink Al-Khwārizmī - Biography, Notable Achievements & Facts, World History, Edu, 28 September 2022, He wrote on mechanical devices like the astrolabeJoseph Frank, al-Khwarizmi über das Astrolab, 1922. and sundial. He assisted a project to determine the circumference of the Earth and in making a world map for al-Ma'mun, the caliph, overseeing 70 geographers.ENCYCLOPEDIA, 30 May 2008,weblink al-Khwarizmi, Encyclopædia Britannica, 5 January 2008,weblink" title="web.archive.org/web/20080105123350weblink">weblink live, 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.WEB,weblink Al-Khwarizmi | Biography & Facts | Britannica, 1 December 2023, www.britannica.com,

Algebra

{{Further|Latin translations of the 12th century|Mathematics in medieval Islam|Science in the medieval Islamic world}}{{multiple image| align = right| image1 = The Algebra of Mohammed ben Musa (Arabic).png| total_width = 250| alt1 =| caption1 =| image2 = The Algebra of Mohammed ben Musa (English).png| alt2 =| caption2 =| footer = Left: The original Arabic print manuscript of the Book of Algebra by Al-KhwārizmÄ«. Right: A page from The Algebra of Al-Khwarizmi by Fredrick Rosen, in English.}}Al-Jabr (The Compendious Book on Calculation by Completion and Balancing, {{transliteration|ar|ALA|al-Kitāb al-mukhtaá¹£ar fÄ« ḥisāb al-jabr wal-muqābala}}) is a mathematical book written approximately 820 CE. It was written with the encouragement of Caliph 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.WEB It provided an exhaustive account of solving polynomial equations up to the second degree,{{sfn|Boyer|1991|p=228|ps=: "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."}} 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.{{Harv|Boyer|1991|loc="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."Al-KhwārizmÄ«'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 (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 operations {{transliteration|ar|ALA|al-jabr}} ( "restoring" or "completion") and {{transliteration|ar|ALA|al-muqābala}} ("balancing"). {{transliteration|ar|ALA|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 âˆ’ 4x2 is reduced to 5x2 = 40x. {{transliteration|ar|ALA|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 to x2 + 9 = x.The above discussion uses modern mathematical notation for the types of problems that the book discusses. However, in al-KhwārizmÄ«'s day, most of this notation had not yet been invented, so he had to use ordinary text to present problems and their solutions. Forexample, for one problem he writes, (from an 1831 translation){{blockquote|If some one says: "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 a 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.}}In modern notation this process, with x the "thing" ( shayʾ) or "root", is given by the steps, Let the roots of the equation be x = p and x = q. Then tfrac{p+q}{2}=50tfrac{1}{2}, pq =100 and So a root is given by Several authors have published texts under the name of {{transliteration|ar|ALA|Kitāb al-jabr wal-muqābala}}, including AbÅ« ḤanÄ«fa DÄ«nawarÄ«, AbÅ« Kāmil, AbÅ« Muḥammad al-'AdlÄ«, AbÅ« YÅ«suf al-Miṣṣīṣī, 'Abd al-HamÄ«d ibn Turk, Sind ibn 'AlÄ«, Sahl ibn BiÅ¡r, and Sharaf al-DÄ«n al-ṬūsÄ«.Solomon Gandz has described Al-Khwarizmi as the father of Algebra:{{blockquote|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.Gandz, Solomon, The sources of al-Khwarizmi's algebra, Osiris, i (1936), 263–277}}Victor J. Katz adds :{{blockquote|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.JOURNAL, Katz, Victor J., Victor J. Katz, Stages in the History of Algebra with Implications for Teaching,weblink VICTOR J.KATZ, University of the District of Columbia Washington DC, USA, 190, University of the District of Columbia Washington DC, USA, 2017-10-07,weblink 2019-03-27, dead, }}John J. O'Connor and Edmund F. Robertson wrote in the MacTutor History of Mathematics Archive:{{blockquote|Perhaps one of the most significant advances made by Arabic 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 allowed rational 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.{{MacTutor|id=Al-Khwarizmi|title=Abu Ja'far Muhammad ibn Musa Al-Khwarizmi}}}}Roshdi Rashed and Angela Armstrong write:{{blockquote|Al-Khwarizmi's text can be seen to be distinct not only from the Babylonian tablets, but also from Diophantus' Arithmetica. It no longer concerns a series of problems to be solved, but an exposition 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.BOOK, Rashed, R., Roshdi Rashed, Armstrong, Angela, 1994, The Development of Arabic Mathematics, Springer Science+Business Media, Springer, 978-0-7923-2565-9, 29181926, 11–12, }}According to Swiss-American historian of mathematics, Florian Cajori, Al-Khwarizmi's algebra was different from the work of Indian mathematicians, for Indians had no rules like the restoration and reduction.BOOK,weblink A History of Mathematics, Florian Cajori, Macmillan, 1919, 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., Regarding the dissimilarity and significance of Al-Khwarizmi's algebraic work from that of Indian Mathematician Brahmagupta, Carl B. 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.BOOK,weblink A History of Mathematics, Boyer, Carl Benjamin, Carl Benjamin Boyer, 1968, 252,

Arithmetic

(File:Gregor Reisch, Margarita Philosophica, 1508 (1230x1615).png|thumb|upright=.8|Algorists vs. abacists, depicted in a sketch from 1508 CE)(File:Dixit algorizmi.png|thumb|upright=.8|Page 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 text kitāb al-ḥisāb al-hindī ('Book of Indian computation'{{refn|group=note|Some scholars translate the title al-ḥisāb al-hindī as "computation with Hindu numerals", but Arabic Hindī 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.{{citation |title=The Earliest Extant Arabic Arithmetic: Kitab al-Fusul fi al Hisab al-Hindi of Abu al-Hasan, Ahmad ibn Ibrahim al-Uqlidisi |first=A. S. |last=Saidan |journal=Isis |volume=57 |pages=475–490 |number=4 |date=Winter 1966 |publisher=The University of Chicago Press |jstor=228518|doi=10.1086/350163 |s2cid=143979243 }}}}), and perhaps a more elementary text, kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī ('Addition and subtraction in Indian arithmetic').{{sfn|Burnett|2017|p=39}}{{citation |last=Avari |first=Burjor |author-link=Burjor Avari |title=Islamic Civilization in South Asia: A history of Muslim power and presence in the Indian subcontinent |publisher=Routledge |year=2013 |isbn=978-0-415-58061-8 |url=https://books.google.com/books?id=hGHpVtQ8eKoC |pages=31–32 |access-date=29 September 2020 |archive-date=28 March 2023 |archive-url=https://web.archive.org/web/20230328222543weblink |url-status=live }} 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.{{citation |first=Glen |last=Van Brummelen |author-link=Glen Van Brummelen |chapter=Arithmetic |editor=Thomas F. Glick |title=Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia |chapter-url=https://books.google.com/books?id=zTQrDwAAQBAJ&pg=PA46 |year=2017 |publisher=Taylor & Francis |isbn=978-1-351-67617-5 |page=46 |access-date=5 May 2019 |archive-date=28 March 2023 |archive-url=https://web.archive.org/web/20230328222557weblink |url-status=live }}As part of 12th century wave of Arabic science flowing into Europe via translations, these texts proved to be revolutionary in Europe.{{citation |chapter=Al-Khwarizmi |editor=Thomas F. Glick |title=Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia |chapter-url=https://books.google.com/books?id=zTQrDwAAQBAJ&pg=PA298 |year=2017 |publisher=Taylor & Francis |isbn=978-1-351-67617-5 |access-date=6 May 2019 |archive-date=28 March 2023 |archive-url=https://web.archive.org/web/20230328222552weblink |url-status=live }} Al-Khwarizmi's Latinized name, Algorismus, turned into the name of method used for computations, and survives in the term "algorithm". It gradually replaced the previous abacus-based methods used in Europe.{{citation |first=Glen |last=Van Brummelen |author-link=Glen Van Brummelen |chapter=Arithmetic |editor=Thomas F. Glick |title=Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia |chapter-url=https://books.google.com/books?id=zTQrDwAAQBAJ&pg=PA46 |year=2017 |publisher=Taylor & Francis |isbn=978-1-351-67617-5 |pages=46–47 |access-date=5 May 2019 |archive-date=28 March 2023 |archive-url=https://web.archive.org/web/20230328222557weblink |url-status=live }}Four Latin texts providing adaptions of Al-Khwarizmi's methods have survived, even though none of them is believed to be a literal translation:{{sfn|Burnett|2017|p=39}}

• Dixit Algorizmi (published in 1857 under the title Algoritmi de Numero Indorum{{citation |chapter=Algoritmi de numero Indorum |title=Trattati D'Aritmetica |year=1857 |publisher=Tipografia delle Scienze Fisiche e Matematiche |location=Rome |pages=1– |chapter-url=https://books.google.com/books?id=1J9GAAAAcAAJ&pg=PA1 |access-date=6 May 2019 |archive-date=28 March 2023 |archive-url=https://web.archive.org/web/20230328222535weblink |url-status=live }})
• 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, who had translated the astronomical tables in 1126. It is perhaps the closest to Al-Khwarizmi's own writings.{{citation |first1=John N. |last1=Crossley |first2=Alan S. |last2=Henry |journal=Historia Mathematica |volume=17 |issue=2 |pages=103–131 |year=1990 |title=Thus Spake al-Khwārizmī: A Translation of the Text of Cambridge University Library Ms. Ii.vi.5 |doi=10.1016/0315-0860(90)90048-I|doi-access=free }}Al-Khwarizmi's work on arithmetic was responsible for introducing the Arabic numerals, based on the Hindu–Arabic numeral system developed in Indian mathematics, 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.WEB,weblink How Algorithm Got Its Name, 8 January 2018, earthobservatory.nasa.gov,

Astronomy

{{Further|Astronomy in the medieval Islamic world}}(File:Corpus Christ College MS 283 (1).png|thumb|upright=.7|Page from Corpus Christi College MS 283, a Latin translation of al-Khwārizmī's Zīj)Al-Khwārizmī's {{transliteration|ar|Zīj as-Sindhind}} (, "astronomical tables of Siddhanta"{{citation|last=Thurston|first=Hugh|title=Early Astronomy|url=https://books.google.com/books?id=rNpHjqxQQ9oC&pg=PP204|year=1996|publisher=Springer Science & Business Media|isbn=978-0-387-94822-5|pages=204–}}) 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 Arabic Zijes based on the Indian astronomical methods known as the sindhind.{{harvnb|Kennedy|1956|pp= 26–29}} The word Sindhind is a corruption of the Sanskrit 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.BOOK, van der Waerden, Bartel Leendert, Bartel Leendert van der Waerden,weblink A History of Algebra: From al-Khwārizmī to Emmy Noether, 1985, Springer-Verlag, 978-3-642-51601-6, Berlin Heidelberg, 10, en, 22 June 2021, 24 June 2021,weblink live, The work contains tables for the movements of the sun, the moon and the five planets known at the time. This work marked the turning point in Islamic astronomy. 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 {{Circa|820}}) is lost, but a version by the Spanish astronomer Maslama al-Majriti ({{Circa|1000}}) has survived in a Latin translation, presumably by Adelard of Bath (26 January 1126).{{harvnb|Kennedy|1956|p=128}} 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 as-Sindhind contained tables for the trigonometric functions of sines and cosine. A related treatise on spherical trigonometry is attributed to him.Al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents.Jacques Sesiano, "Islamic mathematics", p. 157, in BOOK, Mathematics Across Cultures: The History of Non-western Mathematics, Helaine, Selin, Helaine Selin, Ubiratan, D'Ambrosio, Ubiratan D'Ambrosio, 2000, Springer Science+Business Media, 978-1-4020-0260-1, ENCYCLOPEDIA, trigonometry,weblink Encyclopædia Britannica, 21 July 2008, 6 July 2008,weblink" title="web.archive.org/web/20080706200811weblink">weblink live,

Geography

File:World map by Al-Khwarizmi.svg|thumb|upright=1.3|Gianluca Gorni's reconstruction of the section of al-KhwārizmÄ«'s world map concerning the Indian Ocean. The majority of the placenames used by al-KhwārizmÄ« match those of Ptolemy, Martellus and Behaim. The general shape of the coastline is the same between Taprobane and Cattigara. The Dragon'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-KhwārizmÄ«'s map, although is clear and precise on the Martellus map and on the later Behaim version.]]File:PtolemyWorldMap.jpg|thumb|A 15th-century version of Ptolemy's Geography for comparison]] Al-KhwārizmÄ«'s third major work is his {{transliteration|ar|Kitāb Ṣūrat al-Arḍ}} (, "Book of the Description of the Earth"),{{refn|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-KhwārizmÄ«, according to the Geographical Treatise written by Ptolemy the Claudian", although due to ambiguity in the word surah 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".}} also known as his Geography, which was finished in 833. It is a major reworking of Ptolemy's second-century Geography, consisting of a list of 2402 coordinates of cities and other geographical features following a general introduction.WEB, 30 May 2008,weblink The history of cartography, GAP computer algebra system, dead,weblink" title="web.archive.org/web/20080524092016weblink">weblink 24 May 2008, There is one surviving copy of {{transliteration|ar|Kitāb Ṣūrat al-Arḍ}}, which is kept at the Strasbourg University Library. A Latin translation is at the Biblioteca Nacional de España in Madrid.BOOK Al-KhwārizmÄ« corrected Ptolemy's gross overestimate for the length of the Mediterranean SeaEdward S. Kennedy, Mathematical Geography, p. 188, in {{Harv|Rashed|Morelon|1996|pp=185–201}} from the Canary Islands to the eastern shores of the Mediterranean; Ptolemy overestimated it at 63 degrees of longitude, while al-KhwārizmÄ« almost correctly estimated it at nearly 50 degrees of longitude. He "depicted the Atlantic and Indian Oceans as open bodies of water, not land-locked seas as Ptolemy had done."JOURNAL, Richard, Covington, Saudi Aramco World, May–June 2007, 2007, 17–21,weblink The Third Dimension, 6 July 2008,weblink" title="web.archive.org/web/20080512022044weblink">weblink 12 May 2008, dead, Al-KhwārizmÄ«'s Prime Meridian at the Fortunate Isles was thus around 10° east of the line used by Marinus and Ptolemy. Most medieval Muslim gazetteers continued to use al-KhwārizmÄ«'s prime meridian.

Jewish calendar

Al-Khwārizmī wrote several other works including a treatise on the Hebrew calendar, titled {{transliteration|ar|Risāla fi istikhrāj ta'rīkh al-yahūd}} (, "Extraction of the Jewish Era"). It describes the Metonic cycle, a 19-year intercalation cycle; the rules for determining on what day of the week the first day of the month Tishrei shall fall; calculates the interval between the Anno Mundi or Jewish year and the Seleucid era; 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 Al-Bīrūnī and Maimonides.

Other works

Ibn al-Nadim's {{transliteration|ar|Al-Fihrist}}, an index of Arabic books, mentions al-Khwārizmī's {{transliteration|ar|Kitāb al-Taʾrīkh}} (), a book of annals. No direct manuscript survives; however, a copy had reached Nusaybin by the 11th century, where its metropolitan bishop, 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.BOOK, LJ Delaporte, Chronographie de Mar Elie bar Sinaya, 1910, xiii, 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 {{transliteration|ar|Kitāb ar-Rukhāma(t)}} (). Other papers, such as one on the determination of the direction of Mecca, are on the spherical astronomy.Two texts deserve special interest on the morning width ({{transliteration|ar|Ma'rifat sa'at al-mashriq fī kull balad}}) and the determination of the azimuth from a height ({{transliteration|ar|Ma'rifat al-samt min qibal al-irtifā'}}). He wrote two books on using and constructing astrolabes.

Honours

• Al-Khwarizmi (crater) â€” A crater on the far side of the Moon. JOURNAL, El-Baz, Farouk, Al-Khwarizmi: A New-Found Basin on the Lunar Far Side, Science, 1973, 180, 4091, 1173–1176,weblink 10.1126/science.180.4091.1173, 1736378, 17743602, 1973Sci...180.1173E, 10623582, 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.WEB,weblink Small-Body Database Lookup, ssd.jpl.nasa.gov,
• 11156 Al-Khwarismi — Main-belt Asteroid, Discovered 1997 Dec 31 by P. G. Comba at Prescott.WEB,weblink Small-Body Database Lookup, ssd.jpl.nasa.gov,

Notes

{{reflist|group=note}}

References

{{Reflist}}

Sources

• JOURNAL, Arndt, A. B., Al-Khwarizmi, The Mathematics Teacher, December 1983, 668–670, 76, 9, 10.5951/MT.76.9.0668, 27963784,
• BOOK, Carl B., Boyer, Carl Benjamin Boyer, A History of Mathematics, Second, John Wiley & Sons, Inc., 1991, The Arabic Hegemony, 978-0-471-54397-8,weblink
• {{citation |first=Charles |last=Burnett |chapter=Arabic Numerals |editor=Thomas F. Glick |title=Routledge Revivals: Medieval Science, Technology and Medicine (2006): An Encyclopedia |chapter-url=https://books.google.com/books?id=zTQrDwAAQBAJ&pg=PA39 |year=2017 |publisher=Taylor & Francis |isbn=978-1-351-67617-5 |access-date=5 May 2019 |archive-date=28 March 2023 |archive-url=https://web.archive.org/web/20230328222536weblink |url-status=live }}
• BOOK, Daffa, Ali Abdullah al-, Ali Abdullah Al-Daffa, The Muslim contribution to mathematics, 1977, Croom Helm, London, 978-0-85664-464-1,
• JOURNAL, Dunlop, Douglas Morton, Muḥammad b. MÅ«sā al-KhwārizmÄ«, The Journal of the Royal Asiatic Society of Great Britain and Ireland, 1943, 2, 3–4, 248–250, 10.1017/S0035869X00098464, 25221920, 161841351,weblink 24 June 2021, 25 June 2021,weblink live,
• JOURNAL, Kennedy, E. S., A Survey of Islamic Astronomical Tables, Transactions of the American Philosophical Society, 1956, 46, 2, 123–177, 10.2307/1005726, 1005726, 2027/mdp.39076006359272,weblink free, 24 June 2021, 4 June 2021,weblink live,
• {{Citation

• BOOK, Struik, Dirk Jan, Dirk Jan Struik, A Concise History of Mathematics, 1987, 978-0-486-60255-4, 4th, Dover Publications, registration,weblink
• ENCYCLOPEDIA

Further reading

Biographical

• Brentjes, Sonja (2007). "KhwārizmÄ«: Muḥammad ibn MÅ«sā al‐KhwārizmÄ« {{Webarchive|url=https://web.archive.org/web/20110706185327weblink |date=6 July 2011 }}" in Thomas Hockey et al.(eds.). The Biographical Encyclopedia of Astronomers, Springer Reference. New York: Springer, 2007, pp. 631–633. (PDF version {{Webarchive|url=https://web.archive.org/web/20120114103045weblink |date=14 January 2012 }})
• Hogendijk, Jan P., Muhammad ibn Musa (Al-)Khwarizmi (c. 780–850 CE) {{Webarchive|url=https://web.archive.org/web/20180203102909weblink |date=3 February 2018 }} – bibliography of his works, manuscripts, editions and translations.
• {{MacTutor Biography|id=Al-Khwarizmi|title=Abu Ja'far Muhammad ibn Musa Al-Khwarizmi}}
• Sezgin, F., ed., Islamic Mathematics and Astronomy, Frankfurt: Institut für Geschichte der arabisch-islamischen Wissenschaften, 1997–99.

Algebra

• JOURNAL, Gandz, Solomon, Solomon Gandz, The Origin of the Term "Algebra, The American Mathematical Monthly, November 1926, 33, 9, 437–440, 10.2307/2299605, 2299605,weblink 24 June 2021, 25 June 2021,weblink live,
• JOURNAL, Gandz, Solomon, Solomon Gandz, The Sources of al-KhowārizmÄ«'s Algebra, Osiris, 1936, 1, 1, 263–277, 10.1086/368426, 301610, 60770737,weblink 24 June 2021, 25 June 2021,weblink live,
• JOURNAL, Gandz, Solomon, Solomon Gandz, The Algebra of Inheritance: A Rehabilitation of Al-KhuwārizmÄ«, Osiris, 1938, 5, 5, 319–391, 10.1086/368492, 301569, 143683763,weblink 24 June 2021, 25 June 2021,weblink live,
• JOURNAL, Hughes, Barnabas, Gerard of Cremona's Translation of al-KhwārizmÄ«'s al-Jabr, A Critical Edition, Mediaeval Studies, 1986, 48, 211–263, 10.1484/J.MS.2.306339,weblink
• Hughes, Barnabas. Robert of Chester's Latin translation of al-Khwarizmi's al-Jabr: A new critical edition. In Latin. F. Steiner Verlag Wiesbaden (1989). {{isbn|3-515-04589-9}}.
• BOOK, L.C., Karpinski, L. C. Karpinski, Robert of Chester's Latin Translation of the Algebra of Al-Khowarizmi: With an Introduction, Critical Notes and an English Version, 1915, The Macmillan Company,weblink 21 May 2020, 24 September 2020,weblink live,
• BOOK, Rosen, Fredrick, The Algebra of Mohammed Ben Musa, 1831, London,weblink

Astronomy

• BOOK, Commentary on the Astronomical Tables of Al-Khwarizmi: By Ibn Al-Muthanna, B.R., Goldstein, B. R. Goldstein, Yale University Press, 1968, 978-0-300-00498-4,
• JOURNAL, Hogendijk, Jan P., Jan Hogendijk, Al-KhwārizmÄ«'s Table of the "Sine of the Hours" and the Underlying Sine Table, Historia Scientiarum, 1991, 42, 1–12,weblink 24 June 2021, 7 May 2021,weblink" title="web.archive.org/web/20210507023229weblink">weblink live, (Hogendijk's homepage. Publication in English, no. 25).
• BOOK, King, David A., David A. King (historian), Al-KhwārizmÄ« and New Trends in Mathematical Astronomy in the Ninth Century, 1983, Hagop Kevorkian Center for Near Eastern Studies: Occasional Papers on the Near East 2, New York University,weblink 24 June 2021, 25 June 2021,weblink live, (Description and analysis of seven recently discovered minor works related to al-Khwarizmi).
• BOOK, Neugebauer, Otto, Otto Neugebauer, The Astronomical Tables of al-Khwarizmi, 1962,
• BOOK, Rosenfeld, Boris A., Folkerts, Menso, Hogendijk, Jan P., Jan Hogendijk, Vestigia Mathematica: Studies in Medieval and Early Modern Mathematics in Honour of H.L.L. Busard, 1993, Brill, Leiden, 978-90-5183-536-6, 305–308, 'Geometric trigonometry' in treatises of al-KhwārizmÄ«, al-MāhānÄ« and Ibn al-Haytham,
• BOOK, Van Dalen, Benno, Casulleras, Josep, Samsó, Julio, From Baghdad to Barcelona, Studies on the Islamic Exact Sciences in Honour of Prof. Juan Vernet, 1996, Instituto Millás Vallicrosa de Historia de la Ciencia Arabe, Barcelona, 195–252,weblink al-Khwârizmî's Astronomical Tables Revisited: Analysis of the Equation of Time,weblink 24 June 2021, 24 June 2021,weblink" title="web.archive.org/web/20210624203439weblink">weblink live, (Van Dalen's homepage. List of Publications, Articles – no. 5).

Jewish calendar

• JOURNAL, Kennedy, E. S., Edward Stewart Kennedy, Al-KhwārizmÄ« on the Jewish Calendar, 1964, Scripta Mathematica, 27, 55–59,

External links

• {{Commons category-inline|Muhammad ibn Musa al-Khwarizmi}}

{{Navboxes|title=Articles and topics related to al-Khwarizmi|state=collapsed|list1={{Islamic mathematics}}{{Mathematics in Iran}}{{Islamic astronomy}}{{Islamic geography}}{{People of Khorasan}}}}{{Authority control}}