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Muḥammad ibn Mūsā al-Khwārizmī (Arabic: Template:Ar) was a Persian mathematician, astronomer, astrologer and geographer. He was born around 780, in either Khwarizm or Baghdad, and died around 850.

He was the author of al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala, the first book on the systematic solution of linear and quadratic equations. Consequently he is considered to be the father of algebra, a title he shares with Diophantus. The word algebra is derived from al-jabr, one of the two operations used to solve quadratic equations, as described in his book. Algoritmi de numero Indorum, the Latin translation of his other major work on the Indian numerals, introduced the positional number system and the number zero to the Western world in the 12th century. The words algorism and algorithm stem from Algoritmi, the Latinization of his name. His name is also the origin of the Spanish word guarismo, meaning digit.

Biography

Few details about al-Khwārizmī's life are known, it is not even certain where he was born. His name indicates he has come from Khwarizm (Khiva) in the Khorasan province of the Sassanid Persian Empire (now Xorazm Province of Uzbekistan).

His name is often given as either Abū ʿAbd Allāh Muḥammad ibn Mūsā al-Ḵwārizmī (Arabic: Template:Ar “Father of Abdullah, Muhammad, son of Moses, native of Khwārizm”) or Abū Ǧaʿfar Muḥammad ibn Mūsā al-Ḵwārizmī (Arabic: أبو جعفر محمد بن موسى الخوارزمي).

The historian al-Tabari gave his name as Muhammad ibn Musa al-Khwārizmī al-Majousi al-Katarbali (Arabic: Template:Ar). The epithet al-Qutrubbulli indicates he might instead have came from Qutrubbull, a small town near Baghdad. Regarding al-Khwārizmī's religion, Toomer writes:

Another epithet given to him by al-Ṭabarī, "al-Majūsī," 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-Khwārizmī's Algebra shows that he was an orthodox Muslim, so al-Ṭabarī's epithet could mean no more than that his forebears, and perhaps he in his youth, had been Zoroastrians.

In Ibn al-Nadīm's Kitāb al-Fihrist we find a short biography on al-Khwārizmī, together with a list the books he wrote. Al-Khwārizmī 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 such apparently so did Al-Khwārizmī. He worked in Baghdad as a scholar at the House of Wisdom established by Caliph al-Maʾmūn, where he studied and translated Greek scientific manuscripts.

Contributions

His major contributions to mathematics, astronomy, astrology, geography and cartography provided foundations for later and even more widespread innovation in algebra, trigonometry, and his other areas of interest. His systematic and logical approach to solving linear and quadratic equations gave shape to the discipline of algebra, a word that is derived from the name of his 830 book on the subject, al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala (الكتاب المختصر في حساب الجبر والمقابلة) or: "The Compendious Book on Calculation by Completion and Balancing". The book was first translated into Latin in the twelfth century.

His book On the Calculation with Hindu Numerals written about 825, was principally responsible for the diffusion of the Indian system of numeration in the Middle-East and then Europe. This book also translated into Latin in the twelfth century, as Algoritmi de numero Indorum. From the name of the author, rendered in Latin as algoritmi, originated the term algorithm.

Some of his contributions were based on earlier Iranian and Babylonian Astronomy, Indian numbers, and Greek sources.

Al-Khwārizmī systematized and corrected Ptolemy's data in geography as regards to Africa and the Middle east. Another major book was his Kitab surat al-ard ("The Image of the Earth"; translated as Geography), which presented the coordinates of localities in the known world based, ultimately, on those in the Geography of Ptolemy but with improved values for the length of the Mediterranean Sea and the location of cities in Asia and Africa.

He also assisted in the construction of a world map for the caliph al-Ma'mun and participated in a project to determine the circumference of the Earth, supervising the work of 70 geographers to create the map of the then "known world".

When his work was copied and transferred to Europe through Latin translations, it had a profound impact on the advancement of basic mathematics in Europe. He also wrote on mechanical devices like the astrolabe and sundial.

Algebra

al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala (Arabic: الكتاب المختصر في حساب الجبر والمقابلة “The Compendious Book on Calculation by Completion and Balancing”) is a mathematical book written approximately 830 AD.

The book is considered to have defined algebra. The word algebra is derived from the name of one of the basic operations with equations (al-jabr) 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.

Al-Khwārizmī's method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms (where b and c are positive integers)

• squares equal roots (ax = bx)
• squares equal number (ax = c)
• roots equal number (bx = c)
• squares and roots equal number (ax + bx = c)
• squares and number equal roots (ax + c = bx)
• roots and number equal squares (bx + c = ax)

by dividing out the coefficient of the square and using the two operations al-ǧabr (Arabic: الجبر “restoring” or “completion”) and al-muqābala ("balancing"). Al-ǧabr is the process of removing negative units, roots and squares from the equation by adding the same quantity to each side. For example, x = 40x - 4x is reduced to 5x = 40x. Al-muqābala is the process of bringing quantities of the same type to the same side of the equation. For example, x+14 = x+5 is reduced to x+9 = x.

Several authors have published texts under the name of Kitāb al-ǧabr wa-l-muqābala, including Abū Ḥanīfa al-Dīnawarī, Abū Kāmil (Rasāla fi al-ǧabr wa-al-muqābala), Abū Muḥammad al-ʿAdlī, Abū Yūsuf al-Miṣṣīṣī, Ibn Turk, Sind ibn ʿAlī, Sahl ibn Bišr (author uncertain), and Šarafaddīn al-Ṭūsī.

Arithmetic

Algoritmi de numero Indorum ("al-Khwārizmī on the Hindu Art of Reckoning") on Arithmetic, which survived in a Latin translation but was lost in the original Arabic. The translation was most likely done in the 12th century by Adelard of Bath, who had also translated the astronomical tables in 1126. The original Arabic title was possibly Kitāb al-Ǧamʿ wa-al-tafrīq bi-ḥisāb al-Hind.

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Geography

Al-Khwārizmī's third major work is his Kitāb ṣūrat al-Arḍ (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.

There is only one surviving copy of Kitāb ṣūrat al-Arḍ, 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 al-Khwārizmī, 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.

One of the corrections which al-Khwārizmī made in Ptolemy's work is the reduction of the latitude of the Mediterranean from 62° to 52° when, in actual fact, it should be only 42°. The Arab opts for the same zero meridian as Ptolemy, that of the Canaries. The amount of inhabited land extends over 180°.

The majority of the placenames used by al-Khwārizmī match those of Ptolemy, Martellus and Behaim. The general shape of the coastline is the same between Taprobane and Cattigara. The Atlantic coast of the Dragon's Tail, which does not exist in Ptolemy's map, is traced in very little detail on al-Khwārizmī's map, but is clear and precise on the Martellus map and on the later Behaim version.

Finally, al-Khwarizmi also compiled a set of astronomical tables (Zij), based on a variety of Hindu and Greek sources. This work included a table of sines, evidently for a circle of radius 150 units. Like his treatises on algebra and Hindu-Arabic numerals, this astronomical work (or an Andalusian revision thereof) was translated into Latin.

Astronomy

Al-Khwārizmī's Zīj al-sindhind (Arabic: زيج "astronomical tables") 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.

The original Arabic version (written c. 820) is lost, but a version by the Spanish astronomer Maslama al-Majrīṭī (c. 1000) has survived in a Latin translation, presumably by Adelard of Bath (January 26, 1126). 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).

Jewish calendar

Al-Khwārizmī has written several other works including a treatise on the Jewish calendar (Risāla fi istikhrāj taʾrīkh al-yahūd "Extraction of the Jewish Era"). It describes the 19-year 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 al-Bīrūnī and Maimonides.

Other works

Several Arabic manuscripts in Berlin, Istanbul, Taschkent, Cairo and Paris contain further material that surely or with some probability comes from al-Khwārizmī. 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 al-mashriq fī kull balad) and the determination of the azimuth from a height (Maʿrifat al-samt min qibal al-irtifāʿ).

He also wrote two books on using and constructing astrolabes. Ibn al-Nadim in his Kitab al-Fihrist (an index of Arabic books) also mentions Kitāb ar-Ruḵāma(t) (the book on sundials) and Kitab al-Tarikh (the book of history) but the two have been lost.

See also

• Al-Khwarizmi (crater) — A crater on the far side of the moon named after al-Khwārizmī.
• Khwarizmi International Award

References and further reading

1. al-Khwārizmī, Muhammad ibn Mūsā, Encarta.
2. Al-Khwarizmi, Encyclopedia Britannica.
3. Khwarizmi, al-, The American Heritage® Dictionary of the English Language: Fourth Edition. 2000.
4. Al-Khowarizmi, The Columbia Encyclopedia, Sixth Edition. 2001-05.
5. Daffa, Ali (1977). The Muslim contribution to mathematics. London: Croom Helm. ISBN 0-85664-464-1.
6. Daunicht, Hubert (1968–1970). Der Osten nach der Erdkarte al-Ḫuwārizmīs : Beiträge zur historischen Geographie und Geschichte Asiens (in German). Bonner orientalistische Studien. N.S. ; Bd. 19. LCCN 71-0. {{cite book}}: Check date values in: |year= (help)CS1 maint: year (link)
7. Folkerts, Menso (1997). Die älteste lateinische Schrift über das indische Rechnen nach al-Ḫwārizmī (in German and Latin). München: Bayerische Akademie der Wissenschaften. ISBN 3-7696-0108-4.{{cite book}}: CS1 maint: unrecognized language (link)
8. Gandz, Solomon (1926). "The Origin of the Term "Algebra"". The American Mathematical Monthly. 33 (9): 437–440. ISSN 0002-9890. {{cite journal}}: Unknown parameter |month= ignored (help)
9. Gandz, Solomon (1936). "The Sources of al-Khowārizmī's Algebra". Osiris. 1: 263–277. ISSN 0369-7827.
10. Gandz, Solomon (1938). "The Algebra of Inheritance: A Rehabilitation of Al-Khuwārizmī". Osiris. 5: 319–391. ISSN 0369-7827.
11. Hogendijk (1991). "Al-Khwārizmī's Table of the "Sine of the Hours" and the Underlying Sine Table". Historia Scientiarum. 42: 1–12. {{cite journal}}: Unknown parameter |First= ignored (|first= suggested) (help)
12. Hogendijk, Jan P. (1998). "al-Khwarzimi". Pythagoras. 38 (2): 4–5. ISSN 0033-4766.
13. Hughes, Barnabas B. (1986). "Gererd of Cremona's Translation of al-Khwārizmī's al-Jabr: A Critical Edition". Mediaeval Studies. 48: 211–263.
14. Barnabas Hughes. Robert of Chester's Latin translation of al-Khwarizmi's al-Jabr: A new critical edition. In Latin. F. Steiner Verlag Wiesbaden (1989). ISBN 3-515-04589-9.
15. Kennedy, E. S. (1964). "Al-Khwārizmī on the Jewish Calendar". Scripta Mathematica. 27: 55–59.
16. King, David A. (1983). Al-Khwārizmī 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 85-0. {{cite book}}: Italic or bold markup not allowed in: |publisher= (help)
17. Donald E. Knuth, Algorithms in Modern Mathematics and Computer Science. Springer-Verlag. 1979. ISBN 0-387-11157-3.
18. Donald E. Knuth, The Art of Computer Programming: Fundamental Algorithms 3rd edition. Addison-Wesley. 1997. ISBN 0-201-89683-4.
19. Mžik, Hanz von (1926). Das Kitāb Ṣūrat al-Arḍ des Abū Ǧa‘far Muḥammad ibn Mūsā al-Ḫuwārizmī. Leipzig.{{cite book}}: CS1 maint: location missing publisher (link)
20. Neugebauer, Otto (1962). The Astronomical Tables of al-Khwarizmi. Historisk-filosofiske Skrifter. Bind 4, nr. 2.
21. Jeffrey A. Oaks. Was al-Khwarizmi an applied algebraist?. The University of Indianapolis.
22. O'Connor, John J.; Robertson, Edmund F., "Abu Ja'far Muhammad ibn Musa Al-Khwarizmi", MacTutor History of Mathematics Archive, University of St Andrews
23. O'Connor, John J.; Robertson, Edmund F., "Abraham bar Hiyya Ha-Nasi", MacTutor History of Mathematics Archive, University of St Andrews
24. O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics Archive, University of St Andrews
25. Roshdi Rashed, The development of Arabic mathematics: between arithmetic and algebra, London, 1994.
26. Rosen, Fredrick (2004-09-01). The Algebra of Mohammed Ben Musa. Kessinger Publishing. ISBN 1-4179-4914-7.
27. Rosenfeld, Boris A. (1993). Menso Folkerts and J. P. Hogendijk (ed.). ""Geometric trigonometry" in treatises of al-Khwārizmī, al-Māhānī and Ibn al-Haytham". Vestiga mathematica: Studies in Medieval and Early Modern Mathematics in Honour of H. L. L. Busard. Amsterdam: Rodopi. ISBN 90-5183-536-1.
28. Julius Ruska. Zur ältesten arabischen Algebra und Rechenkunst. ISBN 3-533-03817-3.
29. Fuat Sezgin. Geschichte des arabischen Schrifttums. 1974, E. J. Brill, Leiden, the Netherlands.
30. Sezgin, F., ed., Islamic Mathematics and Astronomy, Frankfurt: Institut für Geschichte der arabisch-islamischen Wissenschaften, 1997-9.
31. Suter, H. : Die astronomischen Tafeln des Muhammed ibn Mûsâ al-Khwârizmî in der Bearbeitung des Maslama ibn Ahmed al-Madjrîtî und der latein. Übersetzung des Athelhard von Bath auf Grund der Vorarbeiten von A. Bjørnbo und R. Besthorn in Kopenhagen. Hrsg. und komm. Kopenhagen 1914. 288 pp. Repr. 1997 (Islamic Mathematics and Astronomy. 7). ISBN 3-8298-4008-X.
32. Toomer, Gerald (1970–1990). "Al-Khwārizmī, Abu Jaʿfar Muḥammad ibn Mūsā". In Charles Coulston Gillispie (ed.). Dictionary of Scientific Biography. Vol. 7. New York: Charles Scribner's Sons. pp. 358–365. ISBN 0-684-16962-2. {{cite encyclopedia}}: Check date values in: |year= (help)CS1 maint: year (link)
33. Van Dalen, B. Al-Khwarizmi's Astronomical Tables Revisited: Analysis of the Equation of Time.

1. possibly because it is mistaken with that of Ǧaʿfar Muḥammad ibn Mūsā ibn Šākir. M. Dunlop. Muḥammad b. Mūsā al-Khwārizmī. JRAS 1943 p. 248-250).
2. Encyclopedia Britannica. al-Khwarizmi.
3. http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Cartography.html
4. In al-Khwārizmī's opinion, "the Claudian" indicated that Ptolemy was a descendent of the emperor Claudius.

• 8th century births
• 9th century deaths
• Muslim astrologers
• Muslim astronomers
• Muslim mathematicians
• Muslim geographers
• Persian astrologers
• Persian astronomers
• Persian mathematicians
• Persian geographers
• Medieval astronomers
• People with craters of the Moon named after them