Khwārizmī

Khwarizmi
Abu Abdullah Muhammad bin Musa al-Khwarizmi.jpg
A stamp issued September 6, 1983 in the Soviet Union, commemorating al-Khwārizmī's (approximate) 1200th birthday.
Born c. 780
Died c. 850

Al-Khwarizmi (Mohammad ebne Mūsā Khwārazmī محمد بن موسی خوارزمی) was a mathematician, astronomer, astrologer and geographer of Persian[1][2][3] heritage. He was born around 780 in Khwārizm[2][4][5], in contemporary Khiva, Uzbekistan, which was then part of the Persian Empire, and died around 850. He worked most of his life as a scholar in the House of Wisdom in Baghdad.

His Algebra, written around 820, was the first book on the systematic solution of linear and quadratic equations. Consequently he is considered by many to be the father of algebra,[6] a title some scholars assign to Diophantus. In the twelfth century, Latin translations of his Arithmetic, which explained Indian numerals, introduced decimal positional number system to the Western world.[5] He was among the first to use zero as a place holder in positional base notation.[7] The word algorithm derives from his name.[7] He revised and updated Ptolemy's Geography as well as writing several works on astronomy and astrology.

His contributions not only made a great impact on mathematics, but on language as well. The word algebra is derived from al-jabr, one of the two operations used to solve quadratic equations, as described in his book.

Contents

Etymology

Statue of Khwarizmi in front of the Faculty of Mathematics of Amirkabir University of Technology in Tehran, Persia/Iran.

The words algorism and algorithm stem from Algoritmi, the Latinization of his name.[8] (In Finnish the term for algorithm is just that, "algoritmi".) His name is also the origin of the Spanish word guarismo[9] and of the Portuguese word algarismo, [2] both meaning digit.

Yaqut al-Hamawi in his Mu'jem al-baladan wrote that the name "Khwarezm" is a compound name (in Persian) of "Khwar" (خور), and "-razm" (زم), referring to the abundance of cooked fish meat as a main diet of the peoples of this area.[10] C. E. Bosworth, however, believes the Persian name to be made up of خور, meaning "the sun," and زم, meaning "Earth", designating "the land from which the sun rises."[11] More correctly, however, the Iranic compound stands for "lowland" from khwar/khar, "low" and zam/zem, "earth, land."[12]. In the Avestan language it is "Xvairizem", in Old Persian "Huwarazmish", and in Modern Persian it is Khwārazm. The name also appears in Achaemenid inscriptions as "Huvarazmish", and declared to be part of the Persian Empire.

Life

Few details about al-Khwārizmī's life are known; it is not even certain where he was born. His name indicates he might have come from Khwarezm (Khiva), then part of Greater Khorasan, the eastern part of the territory of Persia, in the Abbasid empire, now Xorazm Province of Uzbekistan. Abu Rayhan Biruni (a native Chorasmian) explicitly states: "The people of Khwarizm are a branch of the Persian tree".[13]

His kunya is given as either Abū ʿAbd Allāh (Arabic: أبو عبد الله) or Abū Jaʿfar (أبو جعفر in Arabic).[14] The historian Tabari gave his name as Muhammad ibn Musa al-Khwārizmī al-Majousi al-Katarbali (Arabic: محمد بن موسى الخوارزميّ المجوسيّ القطربّليّ). The epithet al-Qutrubbulli indicates he might instead have come from Qutrubbull, a small town near Baghdad. However, Rashed[15] points out that

there is no need to be an expert on the period or a philologist to see that al-Tabari's second citation should read “Muhammad ibn Mūsa al-Khwārizmī and al-Majūsi al-Qutrubbulli,” and that there are two people (al-Khwārizmī and al-Majūsi al-Qutrubbulli) 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 al-Khwārizmī, occasionally even the origins of his knowledge, had not been made. Recently, G. J. Toomer … with naive confidence constructed an entire fantasy on the error which cannot be denied the merit of amusing the reader.

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.[1]

In Ibn al-Nadīm's Kitāb al-Fihrist we find a short biography on al-Khwārizmī, together with a list of 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 the sciences and mathematics, which included the translation of Greek and Sanskrit scientific manuscripts.

Contributions

Khwarizmi, the great Persian mathematician, astronomer, astrologer and geographer.
The frontispiece of Frederic Rosen's The Algebra of Mohammed ben Musa (1831)

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 in the Arabic language on the subject, al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala (Arabic 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 Persian 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".[16]

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

Main article: The Compendious Book on Calculation by Completion and Balancing
A page from al-Khwārizmī's 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 CE. The term 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. A unique Arabic copy is kept at Oxford and was translated in 1831 by F. Rosen. A Latin translation is kept in Cambridge.[17]

The al-jabr is considered the foundational text of modern algebra. It provided an exhaustive account of solving polynomial equations up to the second degree,[18] 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.[19]

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)

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, x² = 40x - 4x² is reduced to 5x² = 40x. Al-muqābala is the process of bringing quantities of the same type to the same side of the equation. For example, x²+14 = x+5 is reduced to x²+9 = x.

Several authors have also published texts under the name of Kitāb al-ğabr wa-l-muqābala, including Abū Ḥanīfa al-Dīnawarī, Abū Kāmil Shujā ibn Aslam, Abū Muḥammad al-ʿAdlī, Abū Yūsuf al-Miṣṣīṣī, 'Abd al-Hamīd ibn Turk, Sind ibn ʿAlī, Sahl ibn Bišr, and Šarafaddīn al-Ṭūsī.

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 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."[20]

R. Rashed and Angela Armstrong write:

"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 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."[21]

Page from a Latin translation, beginning with "Dixit algorizmi"

Arithmetic

Al-Khwārizmī'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 al-Khwārizmī"), or Algoritmi de numero Indorum ("al-Khwārizmī on the Hindu Art of Reckoning"), a name given to the work by Baldassarre Boncompagni in 1857. The original Arabic title was possibly Kitāb al-Jamʿ wa-l-tafrīq bi-ḥisāb al-Hind[22] ("The Book of Addition and Subtraction According to the Hindu Calculation")[23]

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-Khwarizmi. Both "algorithm" and "algorism" are derived from the Latinized forms of al-Khwarizmi's name, Algoritmi and Algorismi, respectively.

Astronomy

Corpus Christi College MS 283

Al-Khwārizmī's Zīj al-Sindhind[1] (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.[24] 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. Al-Khwarizmi's work marked the beginning of non-traditional methods of study and calculations.[25]

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

Al-Khwarizmi 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.[27] 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 al-Khwārizmī in ninth-century Baghdad.[28]

The first quadrants and mural instruments were invented by al-Khwarizmi in ninth century Baghdad.[29] The sine quadrant, invented by al-Khwārizmī, was used for astronomical calculations.[30] The first horary quadrant for specific latitudes, was also invented by al-Khwārizmī in Baghdad, then center of the development of quadrants.[30] It was used to determine time (especially the times of prayer) by observations of the Sun or stars.[31] The Quadrans Vetus was a universal horary quadrant, an ingenious mathematical device invented by al-Khwarizmi 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.[30]

Geography

A fifteenth century map based on Ptolomy's Geography for comparison.

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.[32]

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.[33]

Al-Khwārizmī corrected Ptolemy's gross overestimate for the length of the Mediterranean Sea[34] (from the Canary Islands to the eastern shores of the Mediterranean); Ptolemy overestimated it at 63 degrees of longitude, while al-Khwarizmi almost correctly estimated it at nearly 50 degrees of longitude. He "also depicted the Atlantic and Indian Oceans as open bodies of water, not land-locked seas as Ptolemy had done."[35] Al-Khwarizmi 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 al-Khwarizmi's prime meridian.[34]

Jewish calendar

Al-Khwārizmī wrote several other works including a treatise on the Hebrew 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.[1]

Other works

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, 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

Notes

  1. ↑ 1.0 1.1 1.2 1.3 Toomer 1990
  2. ↑ 2.0 2.1 Hogendijk, Jan P. (1998). "al-Khwarzimi". Pythagoras 38 (2): 4–5. ISSN 0033–4766. http://www.kennislink.nl/web/show?id=116543. 
  3. ↑ Oaks, Jeffrey A.. "Was al-Khwarizmi an applied algebraist?". University of Indianapolis. Retrieved on 2008-05-30.
  4. ↑ Berggren 1986
  5. ↑ 5.0 5.1 Struik 1987, p. 93
  6. ↑ Gandz, Solomon (1936). "The Sources of al-Khowārizmī's Algebra". Osiris 1: 263–277. doi:10.1086/368426. ISSN 0369–7827. http://links.jstor.org/sici?sici=0369-7827%28193601%291%3A1%3C263%3ATSOAA%3E2.0.CO%3B2–3. 
  7. ↑ 7.0 7.1 Khwarizmi summary
  8. ↑ Daffa 1977
  9. ↑ Knuth, Donald (1979). Algorithms in Modern Mathematics and Computer Science. Springer-Verlag. ISBN 0-387-11157-3. http://historical.ncstrl.org/litesite-data/stan/CS-TR-80-786.pdf. 
  10. ↑ Yaqut al-Hamawi, Mu'jem al-baladan, Vol2, p395
  11. ↑ C. E. Bosworth, The Encyclopedia of Islam, Vol. IV, 1978. p. 1061
  12. ↑ Encyclopedia Iranica, Chorasmia, Yuri Aleksandrovich Rapoport [1]
  13. ↑ Abu Rahyan Biruni, "Athar al-Baqqiya 'an al-Qurun al-Xaliyyah"(Vestiges of the past : the chronology of ancient nations), Tehran, Miras-e-Maktub, 2001. Original Arabic of the quote: "و أما أهل خوارزم، و إن کانوا غصنا ً من دوحة الفُرس" (pg. 56)
  14. ↑ Dunlop, M. (1943). Muḥammad b. Mūsā al-Khwārizmī. JRAS. pp. pp. 248–250. 
  15. ↑ Rashed, Roshdi (1988), "al-Khwārizmī'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 
  16. ↑ "al-Khwarizmi". Encyclopædia Britannica. Retrieved on 2008-05-30.
  17. ↑ Karpinski, L. C. (1912). "History of Mathematics in the Recent Edition of the Encyclopædia Britannica". American Association for the Advancement of Science. 
  18. ↑ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc.. pp. 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."

  19. ↑ (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."
  20. ↑ O'Connor, John J.; Robertson, Edmund F., "Khwārizmī", MacTutor History of Mathematics archive 
  21. ↑ Rashed, R.; Armstrong, Angela (1994), The Development of Arabic Mathematics, Springer, pp. 11-2, ISBN 0792325656, OCLC 29181926 
  22. ↑ Ruska, Julius. "Zur ältesten arabischen Algebra und Rechenkunst". Isis. 
  23. ↑ Berggren 1986, p. 7
  24. ↑ Kennedy 1956, pp. 26–9
  25. ↑ (Dallal 1999, p. 163)
  26. ↑ Neugebauer, Otto (1962). The Astronomical Tables of al-Khwarizmi. 
  27. ↑ (King 1999a, pp. 168-9)
  28. ↑ David A. King (2002), "A Vetustissimus Arabic Text on the Quadrans Vetus", Journal for the History of Astronomy 33: 237-255 [238-9]
  29. ↑ David A. King, "Islamic Astronomy", in Christopher Walker (1999), ed., Astronomy before the telescope, p. 167-168. British Museum Press. ISBN 0-7141-2733-7.
  30. ↑ 30.0 30.1 30.2 (King 2002, pp. 237-238)
  31. ↑ (King 1999a, pp. 167-8)
  32. ↑ "The history of cartography". GAP computer algebra system. Retrieved on 2008-05-30.
  33. ↑ Daunicht, Hubert (1968–1970) (in German). Der Osten nach der Erdkarte al-Ḫuwārizmīs : Beiträge zur historischen Geographie und Geschichte Asiens. Bonner orientalistische Studien. N.S.; Bd. 19. LCCN 71-468286. 
  34. ↑ 34.0 34.1 Edward S. Kennedy, Mathematical Geography, p. 188, in (Rashed & Morelon 1996, pp. 185-201)
  35. ↑ Covington, Richard (2007), , Saudi Aramco World, May-June 2007: 17-21, http://www.saudiaramcoworld.com/issue/200703/the.third.dimension.htm, retrieved on 2008-07-06 

References

  • Berggren, J. Lennart (1986), Episodes in the Mathematics of Medieval Islam, New York: Springer Science+Business Media, ISBN 0-387-96318-9 
  • Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc.. ISBN 0471543977. 
  • Daffa, Ali Abdullah al- (1977), The Muslim contribution to mathematics, London: Croom Helm, ISBN 0-85664-464-1 
  • 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, 143-174, ISBN 0-7141-2733-7
  • 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 
  • Toomer, Gerald (1990), Gillispie, Charles Coulston, ed., Al-Khwārizmī, Abu Jaʿfar Muḥammad ibn Mūsā, 7, New York: Charles Scribner's Sons, ISBN 0-684-16962-2 

Further reading

  • Dunlop, Douglas Morton (1943). "Muhammad ibn-Musa al-Khwarizmi". Journal of the Royal Asiatic Society of Great Britain & Ireland: 248–250. 
  • Folkerts, Menso (1997) (in German and Latin). Die älteste lateinische Schrift über das indische Rechnen nach al-Ḫwārizmī. München: Bayerische Akademie der Wissenschaften. ISBN 3-7696-0108-4. 
  • Gandz, Solomon (November 1926). "The Origin of the Term "Algebra"". The American Mathematical Monthly 33 (9): 437–440. doi:10.2307/2299605. ISSN 0002–9890. http://links.jstor.org/sici?sici=0002-9890%28192611%2933%3A9%3C437%3ATOOTT%22%3E2.0.CO%3B2–0. 
  • Gandz, Solomon (1938). "The Algebra of Inheritance: A Rehabilitation of Al-Khuwārizmī". Osiris 5 (5): 319–391. doi:10.1086/368492. ISSN 0369–7827. http://links.jstor.org/sici?sici=0369-7827%281938%291%3A5%3C319%3ATAOIAR%3E2.0.CO%3B2–2. 
  • Hogendijk, Jan P. (1991). "Al-Khwārizmī's Table of the "Sine of the Hours" and the Underlying Sine Table". Historia Scientiarum 42: 1–12. 
  • Hughes, Barnabas B. (1986). "Gererd of Cremona's Translation of al-Khwārizmī's al-Jabr: A Critical Edition". Mediaeval Studies 48: 211–263. 
  • 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.
  • Karpinski, L. C. (1915). Robert of Chester's Latin Translation of the Algebra of Al-Khowarizmi, with an Introduction, Critical Notes and English Version. The Macmillan Company. 
  • Kennedy, E. S. (1964). "Al-Khwārizmī on the Jewish Calendar". Scripta Mathematica 27: 55–59. 
  • 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 85-150177. 
  • Mžik, Hanz von (1926). Das Kitāb Ṣūrat al-Arḍ des Abū Ǧa‘far Muḥammad ibn Mūsā al-Ḫuwārizmī. Leipzig. 
  • O'Connor, John J.; Robertson, Edmund F., "Abu Ja'far Muhammad ibn Musa Al-Khwarizmi", MacTutor History of Mathematics archive 
  • O'Connor, John J.; Robertson, Edmund F., "Abraham bar Hiyya Ha-Nasi", MacTutor History of Mathematics archive 
  • O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive 
  • Roshdi Rashed, The development of Arabic mathematics: between arithmetic and algebra, London, 1994.
  • Rosen, Fredrick (2004-09-01). The Algebra of Mohammed Ben Musa. Kessinger Publishing. ISBN 1-4179-4914-7. 
  • 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. 
  • 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 arabisch-islamischen Wissenschaften, 1997–9.
  • Suter, H. [Ed.]: 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.
  • Van Dalen, B. Al-Khwarizmi's Astronomical Tables Revisited: Analysis of the Equation of Time.
Islamic mathematics
Mathematicians
'Abd al-Hamīd ibn Turk · Abd al-Rahman al-Sufi · Abū al-Hasan ibn Alī al-Qalasādī · Abū al-Wafā' al-Būzjānī · Abū Ishāq Ibrāhīm al-Zarqālī · Abū Ja'far al-Khāzin · Abū Kāmil Shujā ibn Aslam · Abu'l-Hasan al-Uqlidisi · Abu-Mahmud al-Khujandi · Abu Nasr Mansur · Abū Rayhān al-Bīrūnī · Abū Sahl al-Qūhī · Ahmed ibn Yusuf · Al-Abbās ibn Said al-Jawharī · Al-Birjandi · Al-Hajjāj ibn Yūsuf ibn Matar · Al-Jayyani · Al-Karaji · Al-Khazini · Al-Kindi · Al-Mahani · Al-Nayrizi · Al-Saghani · Al-Sijzi · Alī ibn Ahmad al-Nasawī · Ali Kuşçu · Avicenna · Banū Mūsā · Brethren of Purity · Hunayn ibn Ishaq · Ibn al-Banna al-Marrakushi · Ibn al-Haytham · Ibn al-Shatir  · Ibn Sahl · Ibn Tahir al-Baghdadi · Ibn Yahyā al-Maghribī al-Samaw'al · Ibn Yunus · Ibrahim ibn Sinan · Jamshīd al-Kāshī · Kamāl al-Dīn al-Fārisī · Kushyar ibn Labban · Muhammad Baqir Yazdi · Muhammad ibn Jābir al-Harrānī al-Battānī · Muhammad ibn Mūsā al-Khwārizmī · Muhyi al-Dīn al-Maghribī · Nasīr al-Dīn al-Tūsī · Omar Khayyám · Qāḍī Zāda al-Rūmī · Al-Khalili · Shams al-Dīn al-Samarqandī  · Sharaf al-Dīn al-Tūsī · Sinan ibn Thabit · Taqi al-Din · Thābit ibn Qurra · Ulugh Beg · Yusuf al-Mu'taman ibn Hud
Treatises
Almanac · Book of Fixed Stars · Book of Optics · De Gradibus · Encyclopedia of the Brethren of Purity · Tables of Toledo · Tabula Rogeriana · The Compendious Book on Calculation by Completion and Balancing · The Book of Healing · Zij · Zij-i Ilkhani · Zij-i-Sultani
Centers
Al-Azhar University · Al-Mustansiriya University · House of Knowledge · House of Wisdom · Istanbul observatory of Taqi al-Din · Madrasah · Maktab · Maragheh observatory · University of Al-Karaouine
Influences
Babylonian mathematics · Greek mathematics · Indian mathematics
Influenced
Byzantine mathematics · European mathematics · Indian mathematics
Islamic astronomy
Astronomers
Abd al-Rahman al-Sufi · Abū al-Wafā' al-Būzjānī · Abū Ishāq Ibrāhīm al-Zarqālī · Abū Ja'far al-Khāzin · Abu-Mahmud al-Khujandi · Abu Nasr Mansur · Abū Rayhān al-Bīrūnī · Abū Sahl al-Qūhī · Ahmed ibn Yusuf · Al-Birjandi · Al-Ghazali · Al-Hajjāj ibn Yūsuf ibn Matar · Al-Khazini · Al-Kindi · Al-Mahani · Al-Nayrizi · Al-Saghani · Al-Sijzi · Ali Kuşçu · Avicenna · Banū Mūsā · Brethren of Purity · Ibn al-Banna al-Marrakushi · Ibn al-Haytham · Ibn al-Shatir  · Ibn Yahyā al-Maghribī al-Samaw'al · Ibn Yunus · Ibrahim ibn Sinan · Ja'far al-Sadiq · Jamshīd al-Kāshī · Kamāl al-Dīn al-Fārisī · Kushyar ibn Labban · Muhammad ibn Jābir al-Harrānī al-Battānī · Muhammad ibn Mūsā al-Khwārizmī · Muhyi al-Dīn al-Maghribī · Nasīr al-Dīn al-Tūsī · Omar Khayyám · Qāḍī Zāda al-Rūmī · Shams al-Dīn Abū Abd Allāh al-Khalīlī · Shams al-Dīn al-Samarqandī  · Sharaf al-Dīn al-Tūsī · Sinan ibn Thabit · Taqi al-Din · Thābit ibn Qurra · Ulugh Beg · Zakariya al-Qazwini
Works
'Aja'ib al-makhluqat wa-ghara'ib al-mawjudat · Almanac · Arabic star names · Book of Optics · Encyclopedia of the Brethren of Purity · Star chart · Tabula Rogeriana · The Book of Healing · Zij (Astronomical catalog · Book of Fixed Stars · Star catalogue · Tables of Toledo · Trigonometry table · Zij-i Ilkhani · Zij-i-Sultani)
Instruments
Alidade · Analog computer · Aperture · Armillary sphere · Astrolabe · Astronomical clock · Celestial globe · Compass · Compass dial · Compass rose · Dioptra · Equatorial ring · Equitorium · Globe · Graph paper · Magnifying glass · Mural instrument · Navigational astrolabe · Nebula · Planisphere · Quadrant · Sextant · Spherical astrolabe · Sundial · Telescope · Triquetrum
Concepts
Almucantar · Apogee · Astrophysics · Azimuth · Celestial mechanics · Celestial spheres · Circular orbit · Deferent and epicycle · Earth's rotation · Eccentricity · Ecliptic · Elliptic orbit · Equant · Galaxy · Geocentrism · Gravitational potential energy · Gravity · Heliocentrism · Inertia · Islamic astrology · Moonlight · Obliquity · Parallax · Precession · Qibla · Specific gravity · Spherical Earth · Starlight · Sublunary sphere · Sunlight · Supernova · Temporal finitism · Trepidation · Triangulation · Tusi-couple
Centers
Al-Azhar University · House of Knowledge · House of Wisdom · Istanbul observatory of Taqi al-Din · Observatory · Madrasah · Maragheh observatory · Research institute · University of Al-Karaouine
Influences

Babylonian astronomy · Hellenistic astronomy · Indian astronomy
Influenced
Byzantine astronomy · Chinese astronomy · European astronomy · Indian astronomy