Mathematics in the medieval Islamic world
The medieval Islamic world underwent significant developments in mathematics. Muhammad ibn Musa al-Khwārizmī played a key role in this transformation, introducing algebra as a distinct field in the 9th century. Al-Khwārizmī's approach, departing from earlier arithmetical traditions, laid the groundwork for the arithmetization of algebra, influencing mathematical thought for an extended period. Successors like al-Karaji expanded on his work, contributing to advancements in various mathematical domains. The practicality and broad applicability of these mathematical methods facilitated the dissemination of Arabic mathematics to the West, contributing substantially to the evolution of Western mathematics.[2]
Arabic mathematical knowledge spread through various channels during the medieval era, driven by the practical applications of al-Khwārizmī's methods. This dissemination was influenced not only by economic and political factors but also by cultural exchanges, exemplified by events such as the Crusades and the translation movement. The Islamic Golden Age, spanning from the 8th to the 14th century, marked a period of considerable advancements in various scientific disciplines, attracting scholars from medieval Europe seeking access to this knowledge. Trade routes and cultural interactions played a crucial role in introducing Arabic mathematical ideas to the West. The translation of Arabic mathematical texts, along with Greek and Roman works, during the 14th to 17th century, played a pivotal role in shaping the intellectual landscape of the Renaissance.
Origin and spread of Arab-Islamic mathematics
Arabic mathematics, particularly algebra, developed significantly during the
Al-Khwārizmī's approach was groundbreaking in that it did not arise from any previous "arithmetical" tradition, including that of Diophantus. He developed a new vocabulary for algebra, distinguishing between purely algebraic terms and those shared with arithmetic. Al-Khwārizmī noticed that the representation of numbers is crucial in daily life. Thus, he wanted to find or summarize a way to simplify the mathematical operation, so-called later, the algebra.[3] His algebra was initially focused on linear and quadratic equations and the elementary arithmetic of binomials and trinomials. This approach, which involved solving equations using radicals and related algebraic calculations, influenced mathematical thinking long after his death.
Al-Khwārizmī's proof of the rule for solving quadratic equations of the form (ax^2 + bx = c), commonly referred to as "squares plus roots equal numbers," was a monumental achievement in the history of algebra. This breakthrough laid the groundwork for the systematic approach to solving quadratic equations, which became a fundamental aspect of algebra as it developed in the Western world.[4] Al-Khwārizmī's method, which involved completing the square, not only provided a practical solution for equations of this type but also introduced an abstract and generalized approach to mathematical problems. His work, encapsulated in his seminal text "Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala" (The Compendious Book on Calculation by Completion and Balancing), was translated into Latin in the 12th century. This translation played a pivotal role in the transmission of algebraic knowledge to Europe, significantly influencing mathematicians during the Renaissance and shaping the evolution of modern mathematics.[4] Al-Khwārizmī's contributions, especially his proof for quadratic equations, are a testament to the rich mathematical heritage of the Islamic world and its enduring impact on Western mathematics.
The spread of Arabic mathematics to the West was facilitated by several factors. The practicality and general applicability of al-Khwārizmī's methods were significant. They were designed to convert numerical or geometrical problems into equations in normal form, leading to canonical solution formulae. His work and that of his successors like al-Karaji laid the foundation for advances in various mathematical fields, including number theory, numerical analysis, and rational Diophantine analysis.[5]
Al-Khwārizmī's algebra was an autonomous discipline with its historical perspective, eventually leading to the "arithmetization of algebra". His successors expanded on his work, adapting it to new theoretical and technical challenges and reorienting it towards a more arithmetical direction for abstract algebraic calculation.
Arabic mathematics, epitomized by al-Khwārizmī's work, was crucial in shaping the mathematical landscape. Its spread to the West was driven by its practical applications, the expansion of mathematical concepts by his successors, and the translation and adaptation of these ideas into the Western context. This spread was a complex process involving economics, politics, and cultural exchange, greatly influencing Western mathematics.
The period known as the Islamic Golden Age (8th to 14th century) was characterized by significant advancements in various fields, including mathematics. Scholars in the Islamic world made substantial contributions to mathematics, astronomy, medicine, and other sciences. As a result, the intellectual achievements of Islamic scholars attracted the attention of scholars in medieval Europe who sought to access this wealth of knowledge. Trade routes, such as the Silk Road, facilitated the movement of goods, ideas, and knowledge between the East and West. Cities like Baghdad, Cairo, and Cordoba became centers of learning and attracted scholars from different cultural backgrounds.Therefore, mathematical knowledge from the Islamic world found its way to Europe through various channels. Meanwhile, the Crusades connected Western Europeans with the Islamic world. While the primary purpose of the Crusades was military, there was also cultural exchange and exposure to Islamic knowledge, including mathematics. European scholars who traveled to the Holy Land and other parts of the Islamic world gained access to Arabic manuscripts and mathematical treatises. During the 14th to 17th century, the translation of Arabic mathematical texts, along with Greek and Roman ones, played a crucial role in shaping the intellectual landscape of the Renaissance. Figures like Fibonacci, who studied in North Africa and the Middle East, helped introduce and popularize Arabic numerals and mathematical concepts in Europe.
Concepts
Algebra
The study of
Al-Khwarizmi's algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the algebraic work of Diophantus, which was syncopated, meaning that some symbolism is used. The transition to symbolic algebra, where only symbols are used, can be seen in the work of
On the work done by Al-Khwarizmi, J. J. O'Connor and Edmund F. Robertson said:[11]
"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
MacTutor History of Mathematics archive
Several other mathematicians during this time period expanded on the algebra of Al-Khwarizmi.
Cubic equations
Omar Khayyam (c. 1038/48 in Iran – 1123/24)[12] wrote the Treatise on Demonstration of Problems of Algebra containing the systematic solution of cubic or third-order equations, going beyond the Algebra of al-Khwārizmī.[13] Khayyám obtained the solutions of these equations by finding the intersection points of two conic sections. This method had been used by the Greeks,[14] but they did not generalize the method to cover all equations with positive roots.[13]
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Induction
The earliest implicit traces of mathematical induction can be found in Euclid's proof that the number of primes is infinite (c. 300 BCE). The first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique (1665).
In between, implicit
Irrational numbers
The Greeks had discovered
In the twelfth century,
Spherical trigonometry
The spherical
Negative numbers
In the 9th century, Islamic mathematicians were familiar with negative numbers from the works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid.[25] Al-Khwarizmi did not use negative numbers or negative coefficients.[25] But within fifty years, Abu Kamil illustrated the rules of signs for expanding the multiplication .
By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve
the product of a negative number—al-nāqiṣ—by a positive number—al-zāʾid—is negative, and by a negative number is positive. If we subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if we subtract a negative number from a lower negative number. If we subtract a negative number from a positive number, the remainder is their positive sum. If we subtract a positive number from an empty power (martaba khāliyya), the remainder is the same negative, and if we subtract a negative number from an empty power, the remainder is the same positive number.[25]
Double false position
Between the 9th and 10th centuries, the
Influences
The influence of medieval Arab-Islamic mathematics to the rest of the world is wide and profound, in both the realm of science and mathematics. The knowledge of the Arabs went into the western world through Spain and Sicily during the translation movement. "The Moors (western Mohammedans from that part of North Africa once known as Mauritania) crossed over into Spain early in the seventh century, bringing with them the cultural resources of the Arab world".[28] In the 13th century, King Alfonso X of Castile established the Toledo School of Translators, in the Kingdom of Castile, where scholars translated numerous scientific and philosophical works from Arabic into Latin. The translations included Islamic contributions to trigonometry, which helps European mathematicians and astronomers in their studies. European scholars such as Gerard of Cremona (1114–1187) played a key role in translating and disseminating these works, thus making them accessible to a wider audience. Cremona is said to have translated into Latin "no fewer than 90 complete Arabic texts."[28] European mathematicians, building on the foundations laid by Islamic scholars, further developed practical trigonometry for applications in navigation, cartography, and celestial navigation, thus pushing forward the age of discovery and scientific revolution. The practical applications of trigonometry for navigation and astronomy became increasingly important during the Age of Exploration.
Al-Battānī is one of the islamic mathematicians who made great contributions to the development of trigonometry. He "innovated new trigonometric functions, created a table of cotangents, and made some formulas in spherical trigonometry."[29] These discoveries, together with his astronomical works which are praised for their accuracy, greatly advanced astronomical calculations and instruments.
Al-Khayyām (1048–1131) was a Persian mathematician, astronomer, and poet, known for his work on algebra and geometry, particularly his investigations into the solutions of cubic equations. He was "the first in history to elaborate a geometrical theory of equations with degrees ≤ 3",[30] and has great influence on the work of Descartes, a French mathematician who is often regarded as the founder of analytical geometry. Indeed, "to read Descartes' s Géométrie is to look upstream towards al-Khayyām and al-Ṭūsī; and downstream towards Newton, Leibniz, Cramer, Bézout and the Bernoulli brothers".[30] Numerous problems that appear in "La Géométrie" (Geometry) have foundations that date back to al-Khayyām.
Abū Kāmil (Arabic: أبو كامل شجاع بن أسلم بن محمد بن شجاع, also known as Al-ḥāsib al-miṣrī—lit. "The Egyptian Calculator") (c. 850 – c. 930), was studied algebra following the author of Algebra, al-Khwārizmī. His Book of Algebra (Kitāb fī al-jabr wa al-muqābala) is "essentially a commentary on and elaboration of al-Khwārizmī's work; in part for that reason and in part for its own merit, the book enjoyed widespread popularity in the Muslim world".[31] It contains 69 problems, which is more than al-Khwārizmī who had 40 in his book.[31] Abū Kāmil's Algebra plays a significant role in shaping the trajectory of Western mathematics, particularly in its impact on the works of the Italian mathematician Leonardo of Pisa, widely recognized as Fibonacci. In his Liber Abaci (1202), Fibonacci extensively incorporated ideas from Arabic mathematicians, using approximately 29 problems from Book of Algebra with scarce modification.[31]
Western historians' perception of the contribution of Arab mathematicians
Despite the fundamental works Arabic mathematicians have done on the development of Algebra and algebraic geometry, Western historians in the 18th and early 19th century still regarded it as a fact that
In 18th century
Conclusion
The medieval Arab-Islamic world played a crucial role in shaping the trajectory of mathematics, with
Other major figures
Sally P. Ragep, a historian of science in Islam, estimated in 2019 that "tens of thousands" of Arabic manuscripts in mathematical sciences and philosophy remain unread, which give studies which "reflect individual biases and a limited focus on a relatively few texts and scholars".[33][full citation needed]
- 'Abd al-Hamīd ibn Turk(fl. 830) (quadratics)
- Sind ibn Ali(d. after 864)
- Thabit ibn Qurra(826–901)
- Al-Battānī (before 858 – 929)
- Abū Kāmil (c. 850 – c. 930)
- Abu'l-Hasan al-Uqlidisi (fl. 952) (arithmetic)
- 'Abd al-'Aziz al-Qabisi(d. 967)
- Abū Sahl al-Qūhī(c. 940–1000) (centres of gravity)
- Ibn al-Haytham (c. 965–1040)
- Abū al-Rayḥān al-Bīrūnī(973–1048) (trigonometry)
- Al-Khayyām (1048–1131)
- Ibn Maḍāʾ(c. 1116–1196)
- Ismail al-Jazari (1136–1206)
- Jamshīd al-Kāshī(c. 1380–1429) (decimals and estimation of the circle constant)
Gallery
-
Engraving ofAbū Sahl al-Qūhī's perfect compass to draw conic sections
See also
- Arabic numerals
- Indian influence on Islamic mathematics in medieval Islam
- History of calculus
- History of geometry
- Science in the medieval Islamic world
- Timeline of science and engineering in the Muslim world
References
- ^ Katz (1993): "A complete history of mathematics of medieval Islam cannot yet be written, since so many of these Arabic manuscripts lie unstudied... Still, the general outline... is known. In particular, Islamic mathematicians fully developed the decimal place-value number system to include decimal fractions, systematised the study of algebra and began to consider the relationship between algebra and geometry, studied and made advances on the major Greek geometrical treatises of Euclid, Archimedes, and Apollonius, and made significant improvements in plane and spherical geometry."
^ Smith (1958), Vol. 1, Chapter VII.4: "In a general way it may be said that the Golden Age of Arabian mathematics was confined largely to the 9th and 10th centuries; that the world owes a great debt to Arab scholars for preserving and transmitting to posterity the classics of Greek mathematics; and that their work was chiefly that of transmission, although they developed considerable originality in algebra and showed some genius in their work in trigonometry." - ISBN 1-56000-581-5. "The Islamic mathematicians exercised a prolific influence on the development of science in Europe, enriched as much by their own discoveries as those they had inherited by the Greeks, the Indians, the Syrians, the Babylonians, etc."
- ^ ISBN 978-1-108-05507-9.
- ^ a b Swetz, Frank J. (2012-08-15). Mathematical Treasures: Mesopotamian Accounting Tokens (Report). Washington, DC: The MAA Mathematical Sciences Digital Library.
- ^ "Extending al-Karaji's Work on Sums of Odd Powers of Integers - Introduction | Mathematical Association of America". maa.org. Retrieved 2023-12-15.
- ^ "algebra". Online Etymology Dictionary.
- ^ Boyer 1991, p. 228.
- ISBN 978-0-8251-2264-4.
- ^ ISBN 0-393-04002-X.
- ^ O'Connor, John J.; Robertson, Edmund F., "al-Marrakushi ibn Al-Banna", 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
- ^ Struik 1987, p. 96.
- ^ a b Boyer 1991, pp. 241–242.
- ^ Struik 1987, p. 97.
- JSTOR 604533.
- ^ )
- ^ O'Connor, John J.; Robertson, Edmund F., "Abu Mansur ibn Tahir Al-Baghdadi", MacTutor History of Mathematics Archive, University of St Andrews
- ^ Allen, G. Donald (n.d.). "The History of Infinity" (PDF). Texas A&M University. Retrieved 7 September 2016.
- ^ Struik 1987, p. 93
- ^ Rosen 1831, p. v–vi.
- ISBN 0-684-16962-2– via Encyclopedia.com.
- ^ Nallino 1939.
- ^ O'Connor, John J.; Robertson, Edmund F., "Abu Abd Allah Muhammad ibn Muadh Al-Jayyani", MacTutor History of Mathematics Archive, University of St Andrews
- ^ Berggren 2007, p. 518.
- ^ ISBN 9780792325659.
- ^ ISBN 9781402045592
- ^ Schwartz, R. K. (2004). Issues in the Origin and Development of Hisab al-Khata'ayn (Calculation by Double False Position) (PDF). Eighth North African Meeting on the History of Arab Mathematics. Radès, Tunisia. Archived from the original (PDF) on 2014-05-16. Retrieved 2012-06-08. "Issues in the Origin and Development of Hisab al-Khata'ayn (Calculation by Double False Position)". Archived from the original (.doc) on 2011-09-15.
- ^ ISSN 0721-832X.
- ^ "Edited by", Contributions to Non-Standard Analysis, Elsevier, pp. iii, 1972, retrieved 2023-12-15
- ^ ISBN 978-1-317-62239-0.
- ^ ISSN 0721-832X.
- ^ ISSN 0068-0346.
- ^ "Science Teaching in Pre-Modern Societies", in Film Screening and Panel Discussion, McGill University, 15 January 2019.
Sources
- Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". In Victor J. Katz (ed.). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (2nd ed.). Princeton, New Jersey: ISBN 978-0-691-11485-9.
- ISBN 0-471-54397-7
- Katz, Victor J. (1993). A History of Mathematics: An Introduction. HarperCollins college publishers. ISBN 0-673-38039-4.
- Nallino, C.A.(1939), "Al-Ḥuwārismī e il suo rifacimento della Geografia di Tolomeo", Raccolta di scritti editi e inediti (in Italian), vol. V, Rome: Istituto per l'Oriente, pp. 458–532
- Rosen, Fredrick (1831). The Algebra of Mohammed Ben Musa. Kessinger Publishing. ISBN 1-4179-4914-7.
- ISBN 0-486-20429-4.
- ISBN 0-486-60255-9
Further reading
- Books on Islamic mathematics
- Berggren, J. Lennart (1986). Episodes in the Mathematics of Medieval Islam. New York: Springer-Verlag. ISBN 0-387-96318-9.
- ISBN 0-85664-464-1.
- ISBN 0-521-25844-8.
- ISBN 0-7923-2565-6.
- ) Sowjetische Beiträge zur Geschichte der Naturwissenschaft pp. 62–160.
- Youschkevitch, Adolf P. (1976). Les mathématiques arabes: VIIIe–XVe siècles. translated by M. Cazenave and K. Jaouiche. Paris: Vrin. ISBN 978-2-7116-0734-1.
- Book chapters on Islamic mathematics
- Lindberg, D.C., and M. H. Shank, eds. The Cambridge History of Science. Volume 2: Medieval Science (Cambridge UP, 2013), chapters 2 and 3 mathematics in Islam.
- ISBN 0-471-18082-3.
- Books on Islamic science
- Daffa, Ali Abdullah al-; Stroyls, J.J. (1984). Studies in the exact sciences in medieval Islam. New York: Wiley. ISBN 0-471-90320-5.
- ISBN 0-8156-6067-7.
- Books on the history of mathematics
- Joseph, George Gheverghese (2000). The Crest of the Peacock: Non-European Roots of Mathematics (2nd ed.). Princeton University Press. JSTOR 2686206.)
- Youschkevitch, Adolf P. (1964). Gesichte der Mathematik im Mittelalter. Leipzig: BG Teubner Verlagsgesellschaft.
- Journal articles on Islamic mathematics
- Høyrup, Jens. “The Formation of «Islamic Mathematics»: Sources and Conditions”. Filosofi og Videnskabsteori på Roskilde Universitetscenter. 3. Række: Preprints og Reprints 1987 Nr. 1.
- Bibliographies and biographies
- Brockelmann, Carl. Geschichte der Arabischen Litteratur. 1.–2. Band, 1.–3. Supplementband. Berlin: Emil Fischer, 1898, 1902; Leiden: Brill, 1937, 1938, 1942.
- Sánchez Pérez, José A. (1921). Biografías de Matemáticos Árabes que florecieron en España. Madrid: Estanislao Maestre.
- ISBN 90-04-02007-1.
- Suter, Heinrich (1900). Die Mathematiker und Astronomen der Araber und ihre Werke. Abhandlungen zur Geschichte der Mathematischen Wissenschaften Mit Einschluss Ihrer Anwendungen, X Heft. Leipzig.
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- Television documentaries
- Marcus du Sautoy (presenter) (2008). "The Genius of the East". The Story of Maths. BBC.
- Science and Islam. BBC.
External links
- Hogendijk, Jan P. (January 1999). "Bibliography of Mathematics in Medieval Islamic Civilization".
- O'Connor, John J.; Robertson, Edmund F. (1999), "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics Archive, University of St Andrews
- Richard Covington, Rediscovering Arabic Science, 2007, Saudi Aramco World
- List of Inventions and Discoveries in Mathematics During the Islamic Golden Age