Muhammad ibn Mūsā alKhwārizmī
this wiki
 "AlKhwārizmī" redirects here. For other uses, see AlKhwārizmī (disambiguation).
{{{title}}}  
250px A stamp issued September 6, 1983 in the Soviet Union, commemorating alKhwārizmī's (approximate) 1200th birthday.  
Born  c. 780 

Died  c. 850 
Ethnicity  Persian 
Known for  Contributions to mathematics 
Abū ʿAbdallāh Muḥammad ibn Mūsā alKhwārizmī^{[1]} (Persian/Arabic: أبو عبد الله محمد بن موسى الخوارزمي) (c. 780, Khwārizm^{[2]}^{[3]}^{[4]} – c. 850) was a Persian^{[5]}^{[2]}^{[6]} mathematician, astronomer and geographer, a scholar in the House of Wisdom in Baghdad.
His Kitab alJabr walMuqabala presented the first systematic solution of linear and quadratic equations. He is considered the founder of algebra.^{[7]} In the twelfth century, Latin translations of his work on the Indian numerals, introduced the decimal positional number system to the Western world.^{[4]} He revised Ptolemy's Geography and wrote on astronomy and astrology.
His contributions had a great impact on language. "Algebra" is derived from aljabr, one of the two operations he used to solve quadratic equations. Algorism and algorithm stem from Algoritmi, the Latin form of his name.^{[8]} His name is the origin of (Spanish) guarismo^{[9]} and of (Portuguese) algarismo, both meaning digit.
LifeEdit
Few details of alKhwārizmī's life are known with certainty, even his birthplace is unsure. His name may indicate that he came from Khwarezm (Khiva), then in Greater Khorasan, which occupied the eastern part of the Greater Iran, now Xorazm Province in Uzbekistan. Abu Rayhan Biruni calls the people of Khwarizm "a branch of the Persian tree".^{[10]}
AlTabari gave his name as Muhammad ibn Musa alKhwārizmī alMajousi alKatarbali (Arabic: محمد بن موسى الخوارزميّ المجوسـيّ القطربّـليّ). The epithet alQutrubbulli could indicate he might instead have come from Qutrubbul (Qatrabbul)^{[11]}, a viticulture district near Baghdad. However, Rashed^{[12]} points out that:
There is no need to be an expert on the period or a philologist to see that alTabari's second citation should read “Muhammad ibn Mūsa alKhwārizmī and alMajūsi alQutrubbulli,” and that there are two people (alKhwārizmī and alMajūsi alQutrubbulli) between whom the letter wa [Arabic ‘و’ for the article ‘and’] has been omitted in an early copy. This would not be worth mentioning if a series of errors concerning the personality of alKhwārizmī, 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.
Regarding alKhwārizmī's religion, Toomer writes:
Another epithet given to him by alṬabarī, "alMajūsī," 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 alKhwārizmī's Algebra shows that he was an orthodox Muslim, so alṬabarī's epithet could mean no more than that his forebears, and perhaps he in his youth, had been Zoroastrians.^{[5]}
In Ibn alNadīm's Kitāb alFihrist we find a short biography on alKhwārizmī, together with a list of the books he wrote. AlKhwārizmī accomplished most of his work in the period between 813 and 833. After the Islamic conquest of Persia, Baghdad became the centre of scientific studies and trade, and many merchants and scientists from as far as China and India traveled to this city, as did AlKhwārizmī. He worked in Baghdad as a scholar at the House of Wisdom established by Caliph alMaʾmūn, where he studied the sciences and mathematics, which included the translation of Greek and Sanskrit scientific manuscripts.
ContributionsEdit
AlKhwārizmī'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 830 book on the subject, "The Compendious Book on Calculation by Completion and Balancing" (alKitab almukhtasar fi hisab aljabr wa'lmuqabalaالكتاب المختصر في حساب الجبر والمقابلة).
On the Calculation with Hindu Numerals written about 825, was principally responsible for spreading the Indian system of numeration throughout the Middle East and Europe. It was translated into Latin as Algoritmi de numero Indorum. AlKhwārizmī, rendered as (Latin) Algoritmi, led to the term "algorithm".
Some of his work was based on Persian and Babylonian astronomy, Indian numbers, and Greek mathematics.
AlKhwārizmī systematized and corrected Ptolemy's data for Africa and the Middle east. Another major book was Kitab surat alard ("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.
He also wrote on mechanical devices like the astrolabe and sundial.
He assisted a project to determine the circumference of the Earth and in making a world map for alMa'mun, the caliph, overseeing 70 geographers.^{[13]}
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. He introduced Arabic numerals into the Latin West, based on a placevalue decimal system developed from Indian sources.^{[14]}
AlgebraEdit
Aljabr wa'l muqabalahEdit
AlKitāb almukhtaṣar fī ḥisāb aljabr walmuqābala (Arabic: الكتاب المختصر في حساب الجبر والمقابلة “The Compendious Book on Calculation by Completion and Balancing”) is a mathematical book written approximately 830 CE. The book was written with the encouragement of the Caliph AlMa'mun 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^{[15]}. The term algebra is derived from the name of one of the basic operations with equations (aljabr, meaning completion, or, subtracting a number from both sides of the equation) 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.^{[16]}
The aljabr is considered the foundational text of modern algebra. It provided an exhaustive account of solving polynomial equations up to the second degree,^{[17]} and introduced the fundamental methods of "reduction" and "balancing", referring to 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.^{[18]}
AlKhwārizmī'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 alǧabr (Arabic: الجبر “restoring” or “completion”) and almuqābala ("balancing"). Alǧabr 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} = 40x − 4x^{2} is reduced to 5x^{2} = 40x. Almuqābala 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 which the book discusses. However, in AlKhwārizmī'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)
"If some one say: "You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eightyone times." Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eightyone things. Separate the twenty things from a hundred and a square, and add them to eightyone. 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 fortynine 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."^{[15]}
In modern notation this process, with 'x' the "thing" (shay') or "root", is given by the steps,
Let the roots of the equation be 'p' and 'q'. Then , and
So a root is given by
Several authors have also published texts under the name of Kitāb alğabr walmuqābala, including Abū Ḥanīfa alDīnawarī, Abū Kāmil Shujā ibn Aslam, Abū Muḥammad alʿAdlī, Abū Yūsuf alMiṣṣīṣī, 'Abd alHamīd ibn Turk, Sind ibn ʿAlī, Sahl ibn Bišr, and Šarafaddīn alṬūsī.
J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive:
"Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of alKhwarizmi, 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."^{[19]}
Father of algebraEdit
AlKhwarizmi is widely considered "the father of algebra", though debate exists as to whether the Hellenized Babylonian mathematician Diophantus also deserves this title.^{[20]}^{[21]} Many agree that AlKhwarizmi deserves this title most.^{[20]}
Those who support Diophantus point to the fact that the algebra found in AlJabr is more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while AlJabr is fully rhetorical.^{[20]} However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to earlier Babylonian mathematics. For this, and other, reasons mathematical historian Kurt Vogel writes: “Diophantus was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.”^{[22]}
Those who support AlKhwarizmi point to the fact that he gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,^{[23]} introduced the fundamental methods of reduction and balancing,^{[18]} and was the first to teach algebra in an elementary form and for its own sake, whereas Diophantus was primarily concerned with the theory of numbers.^{[24]} The "novelty of AlKhwarizmi lies in his extremely systematic treatment, aiming at a general classification of linear and quadratic equations, and at general methods of solving them which are established with proofs." [1] In addition, R. Rashed and Angela Armstrong write:
"AlKhwarizmi'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 resolved, 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."^{[25]}
ArithmeticEdit
AlKhwārizmī's second major work was on the subject of arithmetic, which survived in a Latin translation but was lost in the original Arabic. The translation was most likely done in the twelfth century by Adelard of Bath, who had also translated the astronomical tables in 1126.
The Latin manuscripts are untitled, but are commonly referred to by the first two words with which they start: Dixit algorizmi ("So said alKhwārizmī"), or Algoritmi de numero Indorum ("alKhwārizmī on the Hindu Art of Reckoning"), a name given to the work by Baldassarre Boncompagni in 1857. The original Arabic title was possibly Kitāb alJamʿ waltafrīq biḥisāb alHind^{[26]} ("The Book of Addition and Subtraction According to the Hindu Calculation")^{[27]}
AlKhwarizmi's work on arithmetic was responsible for introducing the Arabic numerals, based on the HinduArabic numeral system developed in Indian mathematics, to the Western world. The term "algorithm" is derived from the algorism, the technique of performing arithmetic with HinduArabic numerals developed by alKhwarizmi. Both "algorithm" and "algorism" are derived from the Latinized forms of alKhwarizmi's name, Algoritmi and Algorismi, respectively.
TrigonometryEdit
In trigonometry, alKhwārizmī (c. 780850) produced tables for the trigonometric functions of sines and cosine in the Zīj alSindhind,^{[28]} alongside the first tables for tangents. He was also an early pioneer in spherical trigonometry, and wrote a treatise on the subject.^{[19]}
AstronomyEdit
AlKhwārizmī's Zīj alSindhind^{[5]} (Arabic: زيج "astronomical tables of Sind and Hind") 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.^{[28]} 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. AlKhwarizmi's work marked the beginning of nontraditional methods of study and calculations.^{[29]}
The original Arabic version (written c. 820) is lost, but a version by the Spanish astronomer Maslamah Ibn Ahmad alMajriti (c. 1000) has survived in a Latin translation, presumably by Adelard of Bath (January 26, 1126).^{[30]} The four surviving manuscripts of the Latin translation are kept at the Bibliothèque publique (Chartres), the Bibliothèque Mazarine (Paris), the Bibliotheca Nacional (Madrid) and the Bodleian Library (Oxford).
AlKhwarizmi made several important improvements to the theory and construction of sundials, which he inherited from his Indian and Hellenistic predecessors. He made tables for these instruments which considerably shortened the time needed to make specific calculations. His sundial was universal and could be observed from anywhere on the Earth. From then on, sundials were frequently placed on mosques to determine the time of prayer.^{[31]} The shadow square, an instrument used to determine the linear height of an object, in conjunction with the alidade for angular observations, was also invented by alKhwārizmī in ninthcentury Baghdad.^{[32]}
The first quadrants and mural instruments were invented by alKhwarizmi in ninth century Baghdad.^{[33]} The sine quadrant, invented by alKhwārizmī, was used for astronomical calculations.^{[34]} The first horary quadrant for specific latitudes, was also invented by alKhwārizmī in Baghdad, then center of the development of quadrants.^{[34]} It was used to determine time (especially the times of prayer) by observations of the Sun or stars.^{[35]} The Quadrans Vetus was a universal horary quadrant, an ingenious mathematical device invented by alKhwarizmi in the ninth century and later known as the Quadrans Vetus (Old Quadrant) in medieval Europe from the thirteenth century. It could be used for any latitude on Earth and at any time of the year to determine the time in hours from the altitude of the Sun. This was the second most widely used astronomical instrument during the Middle Ages after the astrolabe. One of its main purposes in the Islamic world was to determine the times of Salah.^{[34]}
GeographyEdit
AlKhwārizmī's third major work is his Kitāb ṣūrat alArḍ (Arabic: كتاب صورة الأرض "Book on the appearance of the Earth" or "The image of the Earth" translated as Geography), which was finished in 833. It is a revised and completed version of Ptolemy's Geography, consisting of a list of 2402 coordinates of cities and other geographical features following a general introduction.^{[36]}
There is only one surviving copy of Kitāb ṣūrat alArḍ, which is kept at the Strasbourg University Library. A Latin translation is kept at the Biblioteca Nacional de España in Madrid. The complete title translates as Book of the appearance of the Earth, with its cities, mountains, seas, all the islands and rivers, written by Abu Ja'far Muhammad ibn Musa alKhwārizmī, according to the geographical treatise written by Ptolemy the Claudian.
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 zone, by order of longitude. As Paul Gallez points out, this excellent system allows us to deduce many latitudes and longitudes where the only document in our possession 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 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.^{[37]}
AlKhwārizmī corrected Ptolemy's gross overestimate for the length of the Mediterranean Sea^{[38]} (from the Canary Islands to the eastern shores of the Mediterranean); Ptolemy overestimated it at 63 degrees of longitude, while alKhwarizmi almost correctly estimated it at nearly 50 degrees of longitude. He "also depicted the Atlantic and Indian Oceans as open bodies of water, not landlocked seas as Ptolemy had done."^{[39]} AlKhwarizmi thus set the Prime Meridian of the Old World at the eastern shore of the Mediterranean, 10–13 degrees to the east of Alexandria (the prime meridian previously set by Ptolemy) and 70 degrees to the west of Baghdad. Most medieval Muslim geographers continued to use alKhwarizmi's prime meridian.^{[38]}
Jewish calendarEdit
AlKhwārizmī wrote several other works including a treatise on the Hebrew calendar (Risāla fi istikhrāj taʾrīkh alyahūd "Extraction of the Jewish Era"). It describes the 19year intercalation cycle, the rules for determining on what day of the week the first day of the month Tishrī shall fall; calculates the interval between the Jewish era (creation of Adam) and the Seleucid era; and gives rules for determining the mean longitude of the sun and the moon using the Jewish calendar. Similar material is found in the works of alBīrūnī and Maimonides.^{[5]}
Other worksEdit
Several Arabic manuscripts in Berlin, Istanbul, Tashkent, Cairo and Paris contain further material that surely or with some probability comes from alKhwārizmī. The Istanbul manuscript contains a paper on sundials, which is mentioned in the Fihirst. 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 (Maʿrifat saʿat almashriq fī kull balad) and the determination of the azimuth from a height (Maʿrifat alsamt min qibal alirtifāʿ).
He also wrote two books on using and constructing astrolabes. Ibn alNadim in his Kitab alFihrist (an index of Arabic books) also mentions Kitāb arRuḵāma(t) (the book on sundials) and Kitab alTarikh (the book of history) but the two have been lost.
The shaping of our mathematics can be attributed to AlKhwarizmi, the chief librarian of the observatory, research center and library called the House of Wisdom in Baghdad. His treatise, "Hisab aljabr w'almuqabala" (Calculation by Restoration and Reduction), which covers linear and quadratic equations, solved trade imbalances, inheritance questions and problems arising from land surveyance and allocation. In passing, he also introduced into common usage our present numerical system, which replaced the old, cumbersome Roman one.
See alsoEdit
Wikiquote has a collection of quotations related to: alKhwārizmī 
Wikimedia Commons has media related to: Muhammad ibn Musa alKhwarizmi 
 AlKhwarizmi (crater) — A crater on the far side of the moon named after alKhwārizmī.
 Khwarizmi International Award — An Iranian award named after alKhwārizmī.
 Mathematics in medieval Islam
 Astronomy in medieval Islam
NotesEdit
 ↑ There is some confusion in the literature on whether alKhwārizmī's full name is Abū ʿAbdallāh Muḥammad ibn Mūsā alKhwārizmī or Abū Jaʿfar Muḥammad ibn Mūsā alKhwārizmī. Ibn Khaldun notes in his encyclopedic work: "The first who wrote upon this branch (algebra) was Abu ʿAbdallah alKhowarizmi, after whom came Abu Kamil Shojaʿ ibn Aslam." (MacGuckin de Slane). (Rosen 1831, pp. xi–xiii) mentions that "[Abu Abdallah Mohammed ben Musa] lived and wrote under the caliphat of Al Mamun, and must therefore be distinguished from Abu Jafar Mohammed ben Musa, likewise a mathematician and astronomer, who flourished under the Caliph Al Motaded (who reigned A.H. 279289, A.D. 892902)." Karpinski notes in his review on (Ruska 1917) that in (Ruska 1918): "Ruska here inadvertently speaks of the author as Abū Gaʿfar M. b. M., instead of Abū Abdallah M. b. M."
 ↑ ^{2.0} ^{2.1} Hogendijk, Jan P. (1998). "alKhwarzimi" (^{[dead link]}). Pythagoras 38 (2): 4–5. ISSN 0033–4766.
 ↑ Berggren 1986
 ↑ ^{4.0} ^{4.1} Struik 1987, p. 93
 ↑ ^{5.0} ^{5.1} ^{5.2} ^{5.3} Toomer 1990
 ↑ Oaks, Jeffrey A.. "Was alKhwarizmi an applied algebraist?". University of Indianapolis. Retrieved on 20080530.
 ↑ Gandz 1936
 ↑ Daffa 1977
 ↑ Knuth, Donald (1979). Algorithms in Modern Mathematics and Computer Science. SpringerVerlag. ISBN 0387111573.
 ↑ Abu Rahyan Biruni, "Athar alBaqqiya 'an alQurun alXaliyyah"(Vestiges of the past: the chronology of ancient nations), Tehran, MiraseMaktub, 2001. Original Arabic of the quote: "و أما أهل خوارزم، و إن کانوا غصنا ً من دوحة الفُرس" (pg. 56)
 ↑ "Iraq After the Muslim Conquest", by Michael G. Morony, ISBN 1593333153 (a 2005 facsimile from the original 1984 book), p. 145
 ↑ Rashed, Roshdi (1988), "alKhwārizmī's Concept of Algebra", in Zurayq, Qusṭanṭīn; Atiyeh, George Nicholas; Oweiss, Ibrahim M., Arab Civilization: Challenges and Responses : Studies in Honor of Constantine K. Zurayk, SUNY Press, p. 108, ISBN 0887066984, http://books.google.com/books?id=JXbXRKRY_uAC&pg=PA108&dq=Qutrubbulli#PPA108,M1
 ↑ "alKhwarizmi". Encyclopædia Britannica. Retrieved on 20080530.
 ↑ "Khwarizmi, Abu Jafar Muhammad ibn Musa al" in Oxford Islamic Studies Online
 ↑ ^{15.0} ^{15.1} Rosen, Frederic. "The Compendious Book on Calculation by Completion and Balancing The Compendious Book on Calculation by Completion and Balancing, AlKhwārizmī". 1831 English Translation. Retrieved on 20090914.
 ↑ Karpinski, L. C. (1912). "History of Mathematics in the Recent Edition of the Encyclopædia Britannica". American Association for the Advancement of Science.
 ↑ Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second, John Wiley & Sons, Inc., 228. ISBN 0471543977.
"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."
 ↑ ^{18.0} ^{18.1} (Boyer 1991, "The Arabic Hegemony" p. 229) "It is not certain just what the terms aljabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. The word aljabr 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."
 ↑ ^{19.0} ^{19.1} O'Connor, John J.; Robertson, Edmund F., "Muhammad ibn Mūsā alKhwārizmī", MacTutor History of Mathematics archive, University of St Andrews, http://wwwhistory.mcs.standrews.ac.uk/Biographies/AlKhwarizmi.html.
 ↑ ^{20.0} ^{20.1} ^{20.2} (Boyer 1991, "The Arabic Hegemony" p. 228) "Diophantus sometimes is called "the father of algebra," but this title more appropriately belongs to alKhwarizmi. It is true that in two respects the work of alKhwarizmi 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 alKhwarizmi 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 alKhwarizmi knew of the work of Diophantus, but he must have been familiar with at least the astronomical and computational portions of Brahmagupta; yet neither alKhwarizmi nor other Arabic scholars made use of syncopation or of negative numbers."
 ↑ (Derbyshire 2006, "The Father of Algebra" p. 31) "Diophantus, the father of algebra, in whose honor I have named this chapter, lived in Alexandria, in Roman Egypt, in either the 1st, the 2nd, or the 3rd century CE."
 ↑ Harald Kittel, Übersetzung: ein internationales Handbuch zur Übersetzungsforschung, Volume 2 p. 1123, 1124
 ↑ (Boyer 1991, "The Arabic Hegemony" p. 230) "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was alKhwarizmi's exposition that his readers must have had little difficulty in mastering the solutions."
 ↑ Gandz and Saloman (1936), The sources of alKhwarizmi's algebra, Osiris i, p. 263–277: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because 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".
 ↑ Rashed, R.; Armstrong, Angela (1994), The Development of Arabic Mathematics, Springer, pp. 11–2, ISBN 0792325656, OCLC 29181926
 ↑ Ruska
 ↑ Berggren 1986, p. 7
 ↑ ^{28.0} ^{28.1} Kennedy 1956, pp. 26–9
 ↑ (Dallal 1999, p. 163)
 ↑ Neugebauer
 ↑ (King 1999a, pp. 168–9)
 ↑ David A. King (2002), "A Vetustissimus Arabic Text on the Quadrans Vetus", Journal for the History of Astronomy 33: 237255 [2389]
 ↑ David A. King, "Islamic Astronomy", in Christopher Walker (1999), ed., Astronomy before the telescope, p. 167168. British Museum Press. ISBN 0714127337.
 ↑ ^{34.0} ^{34.1} ^{34.2} (King 2002, pp. 237–238)
 ↑ (King 1999a, pp. 167–8)
 ↑ "The history of cartography". GAP computer algebra system. Retrieved on 20080530.
 ↑ Daunicht
 ↑ ^{38.0} ^{38.1} Edward S. Kennedy, Mathematical Geography, p. 188, in (Rashed & Morelon 1996, pp. 185–201)
 ↑ Covington, Richard (2007), Saudi Aramco World, May–June 2007: 17–21, http://www.saudiaramcoworld.com/issue/200703/the.third.dimension.htm, retrieved 20080706
Further readingEdit
 Biographical
 Toomer, Gerald (1990). "AlKhwārizmī, Abu Jaʿfar Muḥammad ibn Mūsā". Dictionary of Scientific Biography 7. Ed. Gillispie, Charles Coulston. New York: Charles Scribner's Sons. ISBN 0684169622.
 Dunlop, Douglas Morton (1943). "Muḥammad b. Mūsā alKhwārizmī". Journal of the Royal Asiatic Society of Great Britain & Ireland: 248–250. Cambridge University.
 O'Connor, John J.; Robertson, Edmund F., "Abu Ja'far Muhammad ibn Musa AlKhwarizmi", MacTutor History of Mathematics archive, University of St Andrews, http://wwwhistory.mcs.standrews.ac.uk/Biographies/AlKhwarizmi.html.
 Fuat Sezgin. Geschichte des arabischen Schrifttums. 1974, E. J. Brill, Leiden, the Netherlands.
 Sezgin, F., ed., Islamic Mathematics and Astronomy, Frankfurt: Institut für Geschichte der arabischislamischen Wissenschaften, 1997–9.
 Algebra
 Gandz, Solomon (November 1926). "The Origin of the Term "Algebra"". The American Mathematical Monthly 33 (9): 437–440. doi: . ISSN 0002–9890.
 Gandz, Solomon (1936). "The Sources of alKhowārizmī's Algebra". Osiris 1 (1): 263–277. doi: . ISSN 0369–7827.
 Gandz, Solomon (1938). "The Algebra of Inheritance: A Rehabilitation of AlKhuwārizmī". Osiris 5 (5): 319–391. doi: . ISSN 0369–7827.
 Hughes, Barnabas (1986). "Gerard of Cremona's Translation of alKhwārizmī's alJabr: A Critical Edition". Mediaeval Studies 48: 211–263.
 Barnabas Hughes. Robert of Chester's Latin translation of alKhwarizmi's alJabr: A new critical edition. In Latin. F. Steiner Verlag Wiesbaden (1989). ISBN 3515045899.
 Karpinski, L. C. (1915). Robert of Chester's Latin Translation of the Algebra of AlKhowarizmi: With an Introduction, Critical Notes and an English Version. The Macmillan Company.
 Rosen, Fredrick (1831). The Algebra of Mohammed Ben Musa. Kessinger Publishing. ISBN 1417949147.
 Ruska, Julius. "Zur ältesten arabischen Algebra und Rechenkunst". Isis.
 Arithmetic
 Folkerts, Menso (1997). Die älteste lateinische Schrift über das indische Rechnen nach alḪwārizmī (in German and Latin). München: Bayerische Akademie der Wissenschaften. ISBN 3769601084.
 Astronomy
 Goldstein, B. R. (1968). Commentary on the Astronomical Tables of AlKhwarizmi: By Ibn AlMuthanna. Yale University Press. ISBN 0300004982.
 Hogendijk, Jan P. (1991). "AlKhwārizmī's Table of the "Sine of the Hours" and the Underlying Sine Table". Historia Scientiarum 42: 1–12.
 King, David A. (1983). AlKhwārizmī and New Trends in Mathematical Astronomy in the Ninth Century. New York University: Hagop Kevorkian Center for Near Eastern Studies: Occasional Papers on the Near East 2. LCCN 85150177.
 Neugebauer, Otto (1962). The Astronomical Tables of alKhwarizmi.
 Rosenfeld, Boris A. (1993). ""Geometric trigonometry" in treatises of alKhwārizmī, alMāhānī and Ibn alHaytham". Vestiga mathematica: Studies in Medieval and Early Modern Mathematics in Honour of H. L. L. Busard. Amsterdam: Rodopi. ISBN 9051835361.
 Suter, H. [Ed.]: Die astronomischen Tafeln des Muhammed ibn Mûsâ alKhwârizmî in der Bearbeitung des Maslama ibn Ahmed alMadjrîtî 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). ISBN 382984008X.
 Van Dalen, B. AlKhwarizmi's Astronomical Tables Revisited: Analysis of the Equation of Time.
 Jewish calendar
 Kennedy, E. S. (1964). "AlKhwārizmī on the Jewish Calendar". Scripta Mathematica 27: 55–59.
 Geography
 Daunicht, Hubert (1968–1970). Der Osten nach der Erdkarte alḪuwārizmīs : Beiträge zur historischen Geographie und Geschichte Asiens (in German). Bonner orientalistische Studien. N.S.; Bd. 19. LCCN 71468286.
 Mžik, Hanz von (1915). "Ptolemaeus und die Karten der arabischen Geographen". Mitteil. D. K. K. Geogr. Ges. In Wien 58.
 Mžik, Hanz von (1916). "Afrika nach der arabischen Bearbeitung der γεωγραφικὴ ὑφήγησις des Cl. Ptolomeaus von Muh. ibn Mūsa alHwarizmi". Denkschriften d. Akad. D. Wissen. In Wien, Phil.hist. Kl. 59.
 Mžik, Hanz von (1926). Das Kitāb Ṣūrat alArḍ des Abū Ǧa‘far Muḥammad ibn Mūsā alḪuwārizmī.
 Nallino, C. A. (1896), "AlḪuwārizmī e il suo rifacimento della Geografia di Tolemo", Atti della R. Accad. dei Lincei, Arno 291, Serie V, Memorie, Classe di Sc. Mor., Vol. II, Rome
 Ruska, Julius (1918). "Neue Bausteine zur Geschichte der arabischen Geographie". Geographische Zeitschrift 24: 77–81.
 Spitta, W. (1879). "Ḫuwārizmī's Auszug aus der Geographie des Ptolomaeus". Zeitschrift Deutschen Morgenl. Gesell. 33.
General referencesEdit
 For a more extensive bibliography see: History of mathematics, Mathematics in medieval Islam, and Astronomy in medieval Islam.
 Berggren, J. Lennart (1986), Episodes in the Mathematics of Medieval Islam, New York: Springer Science+Business Media, ISBN 0387963189
 Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second, John Wiley & Sons, Inc.. ISBN 0471543977.
 Daffa, Ali Abdullah al (1977), The Muslim contribution to mathematics, London: Croom Helm, ISBN 0856644641
 Dallal, Ahmad (1999), "Science, Medicine and Technology", in Esposito, John, The Oxford History of Islam, Oxford University Press, New York
 Kennedy, E.S. (1956), A Survey of Islamic Astronomical Tables; Transactions of the American Philosophical Society, 46, Philadelphia: American Philosophical Society
 King, David A. (1999a), "Islamic Astronomy", in Walker, Christopher, Astronomy before the telescope, British Museum Press, pp. 143–174, ISBN 0714127337
 King, David A. (2002), "A Vetustissimus Arabic Text on the Quadrans Vetus", Journal for the History of Astronomy 33: 237–255
 Struik, Dirk Jan (1987), A Concise History of Mathematics (4th ed.), Dover Publications, ISBN 0486602559
 O'Connor, John J.; Robertson, Edmund F., "Abraham bar Hiyya HaNasi", MacTutor History of Mathematics archive, University of St Andrews, http://wwwhistory.mcs.standrews.ac.uk/Biographies/Abraham.html.
 O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive, University of St Andrews, http://wwwhistory.mcs.standrews.ac.uk/HistTopics/Arabic_mathematics.html.
 Roshdi Rashed, The development of Arabic mathematics: between arithmetic and algebra, London, 1994.

