al-Khwārizmī
Al-Khwārizmī, also known as Muhammad ibn Mūsā al-Khwārizmī, was a prominent scholar of the early ninth century, primarily associated with Baghdad during the caliphates of al-Ma'mūn and al-Mu'taṣim. His contributions to mathematics and astronomy laid foundational principles that would influence future scholars. Notably, he is credited with being the first mathematician to systematically write about algebra, with his seminal work titled "Kitāb al-jabr wa al-muqābalah," which translates to "The Book of Integration and Equation." This text provided rules for solving linear and quadratic equations, drawing from earlier Babylonian mathematical traditions.
In addition to algebra, al-Khwārizmī made significant advances in astronomy, producing tables that were crucial for understanding celestial movements and determining prayer times in Islam. His works also included studies on arithmetic, geometry, and geography, influencing not only the Islamic world but also later European mathematics when translated into Latin. The term "algorithm" is derived from his name, highlighting his lasting impact on mathematical practices. Overall, al-Khwārizmī's legacy is recognized as a bridge between ancient and modern mathematical thought, showcasing the importance of cross-cultural scholarship in the development of science and mathematics.
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al-Khwārizmī
Arabic mathematician and astronomer
- Born: c. 780
- Birthplace: Probably Khwārizm (now in Khiva, Uzbekistan)
- Died: c. 850
- Place of death: Possibly Baghdad (now in Iraq)
Al-Khwārizmī is the author of several important mathematical works. The Latin translations of his writings introduced the concepts of algebra and Hindu-Arabic numerals into the mathematics of medieval Europe. He also compiled a set of astronomical tables used widely in the Islamic Near East.
Early Life
Very little is known of the life of al-Khwārizmī (ahl-KWAHR-ihz-mee). The name al-Khwārizmī means literally “the man from Khwārizm”; the epithet may also, however, be interpreted to indicate the origin of one’s “stock.” The historian al-Ṭabarī asserts that al-Khwārizmī actually came from Qutrubull, a district not far from Baghdad, between the Tigris and Euphrates Rivers. Some sources even give his place of birth as Baghdad. Historians do agree that he lived at Baghdad in the early ninth century under the caliphates of al-Ma՚mūn (r. 813-833) and al-Muՙtaṣim (r. 833-842).
In Kitāb al-Fihrist (c. 987; book of chronicles), Ibn al-Nadīm’s entry on al-Khwārizmī reads,
al-Khwārizmī. His name was Muhammad ibn Mūsā and his family origin was from Khwārazm. He was temporarily associated with the Treasury of the “House of Wisdom” of al-Ma՚mūn. He was one of the leading scholars in astronomy. People both before and after the observations [conducted under al-Ma՚mūn] used to rely on his first and second zījes [astronomical tables] which were both known by the name Sindhind. His books are (as follows): (1) the Zīj, in two [editions], the first and the second; (2) the book on sundials; (3) the book on the use of the astrolabe; (4) the book on the construction of the astrolabe; and (5) the [chronicle].
Al-Nadim’s list is, however, incomplete. He mentions only the astronomical studies and omits an algebra, an arithmetic, a study of the quadrivium, and an adaptation of Ptolemy’s geography. Al-Khwārizmī was apparently well known in Baghdad for his scholarly works on astronomy and mathematics . His inheritance tables on the distribution of money were widely used.
Life’s Work
Al-Khwārizmī is credited by early Arab scholars Ibn Khaldūn (1332-1406) and Kâtib Çelebî (1609-1657) with being the first mathematician to write about algebra. The word “algebra” comes from the second word of the title, Kitāb al-jabr wa al-muqābalah (c. 820). It is his best-known work. Literally, the title means “the book of integration and equation.” It contained rules for arithmetical solutions of linear and quadratic equations, for elementary geometry, and for inheritance problems concerning the distribution of wealth according to proportions. The algebra was based on a long tradition originating in Babylonian mathematics of the early second millennium b.c.e. When it was first translated into Latin in the twelfth century, the rules for the distribution of wealth, which had been so popular in the Near East, were omitted. Translated into English from a Latin version in 1915 by Louis Charles Karpinski, the book opens with a pious exhortation that reveals al-Khwārizmī’s belief in an ordered universe.
Book of Algebra and Almucabola, concerning arithmetical and Geometrical problems.In the name of God, tender and compassionate, begins the book of Restoration and Opposition of number put forth by Mohammed Al-Khowarizmi, the son of Moses. Mohammed said, Praise God the creator who has bestowed on man the power to discover the significance of numbers. Indeed, reflecting that all things which men need require computation, I discovered that all things involve number and I discovered that number is nothing other than that which is composed of units. Unity therefore is implied in every number. Moreover I discovered all numbers to be so arranged that they proceed from unity up to ten.
In the same introduction, al-Khwārizmī describes three kinds of numbers, “roots, squares, and numbers.” He sums up the relationships among them in the following way:
Squares equal to roots,
Karpinski explains that these three types, designated as “simple” by Omar Khayyám and other Arab mathematicians, “correspond in modern algebraic notation to the following: ax2 = bx, ax2 = n, and bx = n.
The first six chapters of al-Khwārizmī’s algebra deal with the following mathematical relationships, those relationships concerning “squares equal to roots,” “squares equal to numbers,” “roots equal to numbers,” “squares and roots equal to numbers,” “squares and numbers equal to roots,” and “roots and numbers equal to a square.” These chapters are followed by illustrative geometrical demonstrations and then many problems with their solutions.
Some of his problems are purely formal, whereas others appear in practical contexts. One of his formal problems states,
If from a square I subtract four of its roots and then take one-third of the remainder, finding this equal to four of the roots, the square will be 256.
His explanation is simple:
Since one-third of the remainder is equal to four roots, one knows that the remainder itself will equal 12 roots. Therefore, add this to the four, giving 16 roots. This (16) is the root of the square.
This relationship can also be stated 1/3 (x2 – 4x) = 4x.
An interesting chapter on mercantile transactions asserts that “mercantile transactions and all things pertaining thereto involve two ideas and four numbers.” Karpinski explains,
The two ideas appear to be the notions of quantity and cost; the four numbers represent unit of measure and price per unit, quantity desired and cost of the same.
Al-Khwārizmī’s last mercantile problem is,
A man is hired to work in a vineyard 30 days for 10 pence. He works six days. How much of the agreed price should he receive?Explanation. It is evident that six days are one-fifth of the whole time; and it is also evident that the man should receive pay having the same relation to the agreed price that the time he works bears to the whole time, 30 days. What we have proposed, is explained as follows. The month, i.e., 30 days, represents the measure, and ten represents the price. Six days represents the quantity, and in asking what part of the agreed price is due to the worker you ask the cost. Therefore multiply the price 10 by the quantity 6, which is inversely proportional to it. Divide the product 60 by the measure 30, giving 2 pence. This will be the cost, and will represent the amount due to the worker.
For Muslims, al-Khwārizmī’s astronomical works are perhaps even more important than his algebra. His astronomical tables were used for accurate timekeeping. In Islam, the times of the five daily prayers are determined by the apparent position of the sun in the sky and vary naturally throughout the year. In al-Khwārizmī’s work on the construction and use of the astrolabe, the times of midday and afternoon prayers are determined by measuring shadow lengths. These timekeeping techniques were widely used for centuries.
Al-Khwārizmī also created tables to compute the local direction of Mecca. This is fundamental to Muslims because it is the direction they face when they pray, bury their dead, and perform various ritual acts. It is no wonder that in Islamic texts, al-Khwārizmī is referred to as “the astronomer.”
Al-Khwārizmī’s book on arithmetic has been preserved in only one version. Translated into Latin and published in Rome in 1857 by Prince Baldassare Boncompagni, al-Khwārizmī’s Algoritmi de numero indorum appears as part 1 of a volume entitled Tratti d’aritmetica. The title means “al-Khwārizmī concerning the Hindu art of reckoning.” This is the derivation of the word “algorithm.” The arithmetic introduced Arabic numerals and the art of calculating by decimal notation. The only copy of this work is in the Cambridge University library.
His study of the quadrivium the medieval curriculum of arithmetic, music, astronomy, and geometry is entitled Liber ysagogarum Alchorismi in artem astronomicam a magistro A. compositus (1126). It was the first of al-Khwārizmī’s writings to appear in Europe. The identity of the writer “A” is not certain, but he is assumed to be the English mathematician and scholar Adelard of Bath (c. 1075-after 1142-1146), who is known as the translator of al-Khwārizmī’s tables. These trigonometric tables were among the first of the Arabic studies in mathematics to appear in Europe.
Al-Khwārizmī enjoyed an excellent reputation among his fellow Arab scholars. Some of his numerical examples were repeated for centuries, becoming so standardized that many subsequent mathematicians did not consider it necessary to acknowledge al-Khwārizmī as the source. Karpinski observes that “the equation x2 + 10x = 39 runs like a thread of gold through the algebras for several centuries.”
The geography Kitāb surat al-ard (book of the form of the earth) differs in several respects from Ptolemy’s geography. Like Ptolemy’, it is a description of a world map and contains a list of the coordinates of the principal places on it, but al-Khwārizmī’s arrangement is radically different, and it is clear that the map to which it refers is not the same as the map Ptolemy described. It is supposed that al-Khwārizmī’s world map was the one constructed by al-Ma՚mūn. This map was an improvement over Ptolemy’, correcting distortions in the supposed length of the Mediterranean. It was far more accurate, too, in its description of the areas under Islamic rule. Because it contained errors of its own, however, the geography written by al-Khwārizmī failed to replace the Ptolemaic geography used in Europe.
Significance
Al-Khwārizmī’s importance in the history of mathematics is inarguable. Two notable arithmetic books, Alexander de Villa Dei’s Carmen de Algorismo (twelfth century) and Johannes de Sacrobosco’s Algorismus vulgaris (thirteenth century), owe much to al-Khwārizmī’s arithmetic and were widely used for several hundred years. In the ninth century, Abū Kāmil drew on al-Khwārizmī’s works for his own writings on algebra. In turn, mathematician and scholar Leonardo of Pisa (c. 1170-c. 1240) was influenced by Abū Kamīl. Numerous commentaries on Abū Kamīl’s work kept al-Khwārizmī’s influence alive in the Middle Ages and during the Renaissance.
Karpinski states concisely what appears to be the consensus of opinion among historians.
Mathematical science was more vitally influenced by Moḥhammed ibn Mūsā than by any other writer from the time of the Greeks to Regiomontanus (1436-1476).
Bibliography
Bell, Eric T. The Development of Mathematics. New York: McGraw-Hill, 1945. Begins with a historical review of the field of mathematics from the first-known texts through successive stages of discoveries, ending at midpoint in the twentieth century. The chapter of most interest to students of Islamic science is entitled“Detour Through India, Arabia, and Spain, 400-1300.”
Cajori, Florian. A History of Mathematics. 1931. 5th ed. Providence, R.I.: AMS Chelsea, 2000. This classic work has several important characteristics that still merit mention. It covers standard non-Western mathematical traditions (Hindu and Islamic). The author manages to give detailed information on individual mathematicians’ original findings while keeping information accessible to the general reader.
Hogendijk, Jan P., and Abdelhamid I. Sabra, eds. The Enterprise of Science in Islam: New Perspectives. Cambridge, Mass.: MIT Press, 2003. A collection surveying the history of Islamic science, including mathematics and astronomy. Illustrations, bibliography, index.
Karpinski, Louis Charles. Robert of Chester’s Latin Translation of the Algebra of Al-Khowarizmi. New York: Macmillan, 1915. Dated, but an admirable work of scholarship, with useful commentary. Contains Latin and English translations on facing pages and pages from selected works by al-Khwārizmī in the original Islamic text.
Kennedy, E. S., ed. Studies in the Islamic Exact Sciences. Edited by David A. King and Mary Helen Kennedy. Beirut: American University of Beirut Press, 1983. Provides a rather technical treatment of several scientific disciplines that flourished in early Islamic times, including the development, through trigonometry, of accurate astronomical calculations. Written especially for those with a substantial background in mathematics.
King, D. A. Al-Khwārizmī and New Trends in Mathematical Astronomy in the Ninth Century. New York: Hagop Kevorkian Center for Near Eastern Studies, New York University, 1983. Discusses some newly discovered works of al-Khwārizmī. While it presupposes a background in mathematics, this work contains interesting charts and graphs that offer a taste of al-Khwārizmī’s methods.
Nasr, Seyyed Hossein. Islamic Science: An Illustrated Study. London: World of Islam Festival, 1976. A carefully researched photographic record of the tools of Islamic science. Textual treatment of historical figures is more limited than in the author’s book below. Illustrations from Islamic astronomy.
Nasr, Seyyed Hossein. Science and Civilization in Islam. Cambridge, Mass.: Harvard University Press, 1968. Contains a broad historical setting against which to view al-Khwārizmī.
Usmanov, Z. D., and I. Hodjiev. “The Legacy of al-Khwārizmī.” Quantum 8, no. 6 (July-August, 1998). A brief overview of al-Khwārizmī’s foundational work in algebra. Presents several algebraic figures as examples.