Al-Khwarizmi

"al-Khwārizmī" redirects here. For other uses, see al-Khwārizmī (disambiguation).

Muhammad ibn Mūsā al-Khwārizmī
A stamp issued September 6, 1983 in the Soviet Union, commemorating al-Khwārizmī's (approximate) 1200th birthday.
Born	c. 780
Died	c. 850
Known for	Contributions to mathematics

Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī (c. 780, Khwārizm – c. 850) was a Persian mathematician, astronomer and geographer, a scholar in the House of Wisdom in Baghdad.

His Kitab al-Jabr wa-l-Muqabala presented the first systematic solution of linear and quadratic equations. He is considered the founder of algebra, a credit he shares with Diophantus. In the twelfth century, Latin translations of his work on the Indian numerals, introduced the decimal positional number system to the Western world. He revised Ptolemy's Geography and wrote on astronomy and astrology.

His contributions had a great impact on language. "Algebra" is derived from al-jabr, one of the two operations he used to solve quadratic equations. Algorism and algorithm stem from Algoritmi, the Latin form of his name. His name is the origin of (Spanish) guarismo and of (Portuguese) algarismo, both meaning digit.

Life

File:Muhammad ibn Musa al-Khwarizmi (statue).jpg

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

Few details of al-Khwārizmī'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 Persian Empire, now Xorazm Province in Uzbekistan. Abu Rayhan Biruni calls the people of Khwarizm "a branch of the Persian tree".

Al-Tabari gave his name as Muhammad ibn Musa al-Khwārizmī al-Majousi al-Katarbali (Arabic: محمد بن موسى الخوارزميّ المجوسـيّ القطربّـليّ). The epithet al-Qutrubbulli could indicate he might instead have come from Qutrubbul (Qatrabbul), a viticulture district near Baghdad. However, Rashed 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.

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 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

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 830 book on the subject, "The Compendious Book on Calculation by Completion and Balancing" (al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabalaالكتاب 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. Al-Khwārizmī, 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.

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.

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 al-Ma'mun, the caliph, overseeing 70 geographers.

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.

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 book was written with the encouragement of the Caliph Al-Ma'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. 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.

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, 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.

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 (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 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.

The above discussion uses modern mathematical notation for the types of problems which the book discusses. However, in Al-Khwārizmī'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 eighty-one times." Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts."

In modern notation this process, with 'x' the "thing" (shay') or "root", is given by the steps,

$(10-x)^{2}=81x$

$x^{2}+100=101x$

Let the roots of the equation be 'p' and 'q'. Then ${\tfrac {p+q}{2}}=50{\tfrac {1}{2}}$, $pq=100$ and

${\tfrac {p-q}{2}}={\sqrt {{\tfrac {p+q}{2}}-pq}}={\sqrt {2550{\tfrac {1}{4}}-100}}=49{\tfrac {1}{2}}$

So a root is given by

$x=50{\tfrac {1}{2}}-49{\tfrac {1}{2}}=1$

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."

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."

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 ("The Book of Addition and Subtraction According to the Hindu Calculation")

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 (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. 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.

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). 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. 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.

The first quadrants and mural instruments were invented by al-Khwarizmi in ninth century Baghdad. The sine quadrant, invented by al-Khwārizmī, was used for astronomical calculations. The first horary quadrant for specific latitudes, was also invented by al-Khwārizmī in Baghdad, then center of the development of quadrants. It was used to determine time (especially the times of prayer) by observations of the Sun or stars. 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.

Geography

Hubert Daunicht's reconstruction of al-Khwārizmī's planisphere.

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.

Al-Khwārizmī corrected Ptolemy's gross overestimate for the length of the Mediterranean Sea (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." 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.

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.

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.

The shaping of our mathematics can be attributed to Al-Khwarizmi, the chief librarian of the observatory, research center and library called the House of Wisdom in Baghdad. His treatise, "Hisab al-jabr w'al-muqabala" (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 also

• Al-Khwarizmi (crater) — A crater on the far side of the moon named after al-Khwārizmī.
• Khwarizmi International Award — An Iranian award named after al-Khwārizmī.
• Mathematics in medieval Islam
• Astronomy in medieval Islam

Further reading

Biographical

• Toomer, Gerald (1990). "Al-Khwārizmī, Abu Jaʿfar Muḥammad ibn Mūsā". In Gillispie, Charles Coulston (ed.). Dictionary of Scientific Biography. Vol. 7. New York: Charles Scribner's Sons. ISBN 0-684-16962-2.
• Dunlop, Douglas Morton (1943). "Muḥammad b. Mūsā al-Khwārizmī". Journal of the Royal Asiatic Society of Great Britain & Ireland. Cambridge University: 248–250.
• O'Connor, John J.; Robertson, Edmund F., "Abu Ja'far Muhammad ibn Musa Al-Khwarizmi", MacTutor History of Mathematics Archive, University of St Andrews
• 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.

Algebra

• Gandz, Solomon (1926). "The Origin of the Term "Algebra"". The American Mathematical Monthly. 33 (9): 437–440. doi:10.2307/2299605. ISSN 0002–9890. {{cite journal}}: Unknown parameter |month= ignored (help)
• Gandz, Solomon (1936). "The Sources of al-Khowārizmī's Algebra". Osiris. 1: 263–277. doi:10.1086/368426. ISSN 0369–7827.
• Gandz, Solomon (1938). "The Algebra of Inheritance: A Rehabilitation of Al-Khuwārizmī". Osiris. 5 (5): 319–391. doi:10.1086/368492. ISSN 0369–7827.
• Hughes, Barnabas (1986). "Gerard 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 an English Version. The Macmillan Company.
• Rosen, Fredrick (1831). The Algebra of Mohammed Ben Musa. Kessinger Publishing. ISBN 1-4179-4914-7.
• Ruska, Julius. "Zur ältesten arabischen Algebra und Rechenkunst". Isis.

Arithmetic

• 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)

Astronomy

• Goldstein, B. R. (1968). Commentary on the Astronomical Tables of Al-Khwarizmi: By Ibn Al-Muthanna. Yale University Press. ISBN 0300004982.
• Hogendijk, Jan P. (1991). "Al-Khwārizmī's Table of the "Sine of the Hours" and the Underlying Sine Table". Historia Scientiarum. 42: 1–12.
• 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)
• Neugebauer, Otto (1962). The Astronomical Tables of al-Khwarizmi.
• 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.
• 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.

Jewish calendar

• Kennedy, E. S. (1964). "Al-Khwārizmī on the Jewish Calendar". Scripta Mathematica. 27: 55–59.

Geography

• 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}}: CS1 maint: unrecognized language (link)
• Mžik, Hanz von (1915). "Ptolemaeus und die Karten der arabischen Geographen". Mitteil. d. k. k. Geogr. Ges. in Wien. 58: 152.
• Mžik, Hanz von (1916). "Afrika nach der arabischen Bearbeitung der γεωγραφικὴ ὑφήγησις des Cl. Ptolomeaus von Muh. ibn Mūsa al-Hwarizmi". Denkschriften d. Akad. d. Wissen. in Wien, Phil.-hist. Kl. 59.
• 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)
• Nallino, C. A. (1896), "Al-Ḫuwārizmī 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 references

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 0-387-96318-9
• Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. ISBN 0471543977. {{cite book}}: |edition= has extra text (help)
• Daffa, Ali Abdullah al- (1977), The Muslim contribution to mathematics, London: Croom Helm, ISBN 0-85664-464-1
• Template:Harvard reference
• Kennedy, E.S. (1956), A Survey of Islamic Astronomical Tables; Transactions of the American Philosophical Society, vol. 46, Philadelphia: American Philosophical Society
• Template:Harvard reference
• Template:Harvard reference
• 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 Ha-Nasi", MacTutor History of Mathematics Archive, University of St Andrews
• O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics Archive, University of St Andrews
• Roshdi Rashed, The development of Arabic mathematics: between arithmetic and algebra, London, 1994.