Mathematics during the Golden Age of Islam, especially during the 9th
and 10th centuries, was built on
Greek mathematics (Euclid,
Archimedes, Apollonius) and
Indian mathematics (Aryabhata,
Brahmagupta). Important progress was made, such as the full
development of the decimal place-value system to include decimal
fractions, the first systematised study of algebra (named for The
Compendious Book on Calculation by Completion and Balancing by scholar
Al-Khwarizmi), and advances in geometry and trigonometry.
Arabic works also played an important role in the transmission of
mathematics to Europe during the 10th to 12th centuries.
1.2 Cubic equations
1.4 Irrational numbers
1.5 Spherical trigonometry
2 Other major figures
4 See also
7 Further reading
8 External links
Omar Khayyám's "Cubic equations and intersections of conic sections"
the first page of the two-chaptered manuscript kept in Tehran
Further information: History of algebra
The study of algebra, the name of which is derived from the Arabic
word meaning completion or "reunion of broken parts", flourished
during the Islamic golden age. Muhammad ibn Musa al-Khwarizmi, a
scholar in the
House of Wisdom
House of Wisdom in Baghdad, is along with the Greek
mathematician Diophantus, known as the father of algebra. In his book
The Compendious Book on Calculation by Completion and Balancing,
Al-Khwarizmi deals with ways to solve for the positive roots of first
and second degree (linear and quadratic) polynomial equations. He also
introduces the method of reduction, and unlike Diophantus, gives
general solutions for the equations he deals with.
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
Ibn al-Banna' al-Marrakushi and Abū
al-Ḥasan ibn ʿAlī al-Qalaṣādī.
On the work done by Al-Khwarizmi, J. J. O'Connor and Edmund F.
"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 the 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
— MacTutor History of
Several other mathematicians during this time period expanded on the
algebra of Al-Khwarizmi. Omar Khayyam, along with Sharaf al-Dīn
al-Tūsī, found several solutions of the cubic equation. Omar Khayyam
found the general geometric solution of a cubic equation.
To solve the third-degree equation x3 + a2x = b
Khayyám constructed the parabola x2 = ay, a circle with
diameter b/a2, and a vertical line through the intersection point. The
solution is given by the length of the horizontal line segment from
the origin to the intersection of the vertical line and the x-axis.
Further information: Cubic equation
Omar Khayyam (c. 1038/48 in
Iran – 1123/24) 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. Khayyám obtained the solutions of these
equations by finding the intersection points of two conic sections.
This method had been used by the Greeks, but they did not
generalize the method to cover all equations with positive roots.
Sharaf al-Dīn al-Ṭūsī (? in Tus,
Iran – 1213/4) developed a
novel approach to the investigation of cubic equations—an approach
which entailed finding the point at which a cubic polynomial obtains
its maximum value. For example, to solve the equation
displaystyle x^ 3 +a=bx
, with a and b positive, he would note that the maximum point of the
displaystyle y=bx-x^ 3
displaystyle x=textstyle sqrt frac b 3
, and that the equation would have no solutions, one solution or two
solutions, depending on whether the height of the curve at that point
was less than, equal to, or greater than a. His surviving works give
no indication of how he discovered his formulae for the maxima of
these curves. Various conjectures have been proposed to account for
his discovery of them.
See also: Mathematical induction § History
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 proof by induction for arithmetic sequences was
introduced by al-Karaji (c. 1000) and continued by al-Samaw'al, who
used it for special cases of the binomial theorem and properties of
The Greeks had discovered irrational numbers, but were not happy with
them and only able to cope by drawing a distinction between magnitude
and number. In the Greek view, magnitudes varied continuously and
could be used for entities such as line segments, whereas numbers were
discrete. Hence, irrationals could only be handled geometrically; and
Greek mathematics was mainly geometrical. Islamic
Abū Kāmil Shujāʿ ibn Aslam and Ibn Tahir
al-Baghdadi slowly removed the distinction between magnitude and
number, allowing irrational quantities to appear as coefficients in
equations and to be solutions of algebraic equations. They
worked freely with irrationals as mathematical objects, but they did
not examine closely their nature.
In the twelfth century,
Latin translations of Al-Khwarizmi's
Arithmetic on the
Indian numerals introduced the decimal positional
number system to the Western world. His Compendious Book on
Calculation by Completion and Balancing presented the first systematic
solution of linear and quadratic equations. In
Renaissance Europe, he
was considered the original inventor of algebra, although it is now
known that his work is based on older Indian or Greek sources. He
revised Ptolemy's Geography and wrote on astronomy and astrology.
C.A. Nallino suggests that al-Khwarizmi's original work was
not based on
Ptolemy but on a derivative world map, presumably in
Syriac or Arabic.
Law of sines
Law of sines and History of trigonometry
The spherical law of sines was discovered in the 10th century: it has
been attributed variously to Abu-Mahmud Khojandi, Nasir al-Din al-Tusi
and Abu Nasr Mansur, with
Abu al-Wafa' Buzjani
Abu al-Wafa' Buzjani as a contributor.
Ibn Muʿādh al-Jayyānī's The book of unknown arcs of a sphere in
the 11th century introduced the general law of sines. The plane
law of sines was described in the 13th century by Nasīr al-Dīn
al-Tūsī. In his On the Sector Figure, he stated the law of sines for
plane and spherical triangles, and provided proofs for this law.
Other major figures
'Abd al-Hamīd ibn Turk (fl. 830) (quadratics)
Thabit ibn Qurra
Thabit ibn Qurra (826–901)
Sind ibn Ali (d. after 864)
Ismail al-Jazari (1136–1206)
Abū Sahl al-Qūhī
Abū Sahl al-Qūhī (c. 940–1000) (centers of gravity)
Abu'l-Hasan al-Uqlidisi (952–953) (arithmetic)
'Abd al-'Aziz al-Qabisi (d. 967)
Ibn al-Haytham (ca. 965–1040)
Abū al-Rayḥān al-Bīrūnī
Abū al-Rayḥān al-Bīrūnī (973–1048) (trigonometry)
Ibn Maḍāʾ (c. 1116–1196)
Jamshīd al-Kāshī (c. 1380–1429) (decimals and estimation of the
Engraving of Abū Sahl al-Qūhī's perfect compass to draw conic
The theorem of Ibn Haytham.
Indian influence on Islamic mathematics in medieval Islam
History of calculus
History of geometry
Science in the medieval Islamic world
Timeline of Islamic science and technology
^ 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."
Adolph P. Yushkevich
Adolph P. Yushkevich Sertima, Ivan Van (1992), Golden age of the
Moor, Volume 11, Transaction Publishers, p. 394,
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."
^ "algebra". Online Etymology Dictionary.
^ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of
Mathematics (Second ed.). John Wiley & Sons. p. 228.
^ Swetz, Frank J. (1993). Learning Activities from the History of
Mathematics. Walch Publishing. p. 26.
^ a b Gullberg, Jan (1997). Mathematics: From the Birth of Numbers. W.
W. Norton. p. 298. 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
^ O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics:
forgotten brilliance?", MacTutor History of
University of St Andrews .
^ Struik 1987, p. 96.
^ a b Boyer 1991, pp. 241–242.
^ Struik 1987, p. 97.
^ Berggren, J. Lennart; Al-Tūsī, Sharaf Al-Dīn; Rashed, Roshdi
(1990). "Innovation and Tradition in Sharaf al-Dīn al-Ṭūsī's
al-Muʿādalāt". Journal of the American Oriental Society. 110 (2):
304–309. doi:10.2307/604533. JSTOR 604533.
^ a b Sesiano, Jacques (2000). Helaine, Selin; Ubiratan, D'Ambrosio,
eds. Islamic mathematics.
Mathematics Across Cultures: The History of
Non-western Mathematics. Springer. pp. 137–157.
^ 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; Toomer 1990
^ 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, J. Lennart (2007). "
Mathematics in Medieval Islam". The
Mathematics of Egypt, Mesopotamia, China, India, and Islam: A
Sourcebook. Princeton University Press. p. 518.
Boyer, Carl B. (1991), "Greek
Trigonometry and Mensuration, and The
Arabic Hegemony", A History of
Mathematics (2nd ed.), New York City:
John Wiley & Sons, ISBN 0-471-54397-7
Nallino, C.A. (1939), "Al-Ḥuwārismī e il suo rifacimento della
Geografia di Tolomeo", Raccolta di scritti editi e inediti, V, Rome:
Istituto per l'Oriente, pp. 458–532 . (in Italian)
Struik, Dirk J. (1987), A Concise History of
Mathematics (4th rev.
ed.), Dover Publications, ISBN 0-486-60255-9
Books on Islamic mathematics
Berggren, J. Lennart (1986). Episodes in the
Mathematics of Medieval
Islam. New York: Springer-Verlag. ISBN 0-387-96318-9.
Review: Toomer, Gerald J.; Berggren, J. L. (1988). "Episodes in the
Mathematics of Medieval Islam". American Mathematical Monthly.
Mathematical Association of America. 95 (6): 567. doi:10.2307/2322777.
Review: Hogendijk, Jan P.; Berggren, J. L. (1989). "Episodes in the
Mathematics of Medieval Islam by J. Lennart Berggren". Journal of the
American Oriental Society. American Oriental Society. 109 (4):
697–698. doi:10.2307/604119. JSTOR 604119.
Daffa', Ali Abdullah al- (1977). The Muslim contribution to
mathematics. London: Croom Helm. ISBN 0-85664-464-1.
Katz, Victor J. (1993). A History of Mathematics: An Introduction.
HarperCollins college publishers. ISBN 0-673-38039-4.
Ronan, Colin A. (1983). The Cambridge Illustrated History of the
World's Science. Cambridge University Press.
Smith, David E. (1958). History of Mathematics. Dover Publications.
Rashed, Roshdi (2001). The Development of Arabic Mathematics: Between
Arithmetic and Algebra. Translated by A. F. W. Armstrong. Springer.
Rosen, Fredrick (1831). The
Algebra of Mohammed Ben Musa. Kessinger
Publishing. ISBN 1-4179-4914-7.
Toomer, Gerald (1990). "Al-Khwārizmī, Abu Ja'far Muḥammad ibn
Mūsā". In Gillispie, Charles Coulston. Dictionary of Scientific
Biography. 7. New York: Charles Scribner's Sons.
Youschkevitch, Adolf P.; Rozenfeld, Boris A. (1960). Die Mathematik
der Länder des Ostens im Mittelalter. Berlin. 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.
Book chapters on Islamic mathematics
Berggren, J. Lennart (2007). "
Mathematics in Medieval Islam". In
Victor J. Katz. The
Mathematics of Egypt, Mesopotamia, China, India,
and Islam: A Sourcebook (Second ed.). Princeton, New Jersey: Princeton
University. ISBN 978-0-691-11485-9.
Cooke, Roger (1997). "Islamic Mathematics". The History of
Mathematics: A Brief Course. Wiley-Interscience.
Books on Islamic science
Daffa, Ali Abdullah al-; Stroyls, J.J. (1984). Studies in the exact
sciences in medieval Islam. New York: Wiley.
Kennedy, E. S. (1984). Studies in the Islamic Exact Sciences. Syracuse
Univ Press. 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. ISBN 0-691-00659-8. (Reviewed: Katz, Victor J.;
Joseph, George Gheverghese (1992). "The Crest of the Peacock:
Non-European Roots of
Mathematics by George Gheverghese Joseph". The
Mathematics Journal. Mathematical Association of America. 23
(1): 82–84. doi:10.2307/2686206. 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.
Sezgin, Fuat (1997). Geschichte Des Arabischen Schrifttums (in
German). Brill Academic Publishers. 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.
Marcus du Sautoy
Marcus du Sautoy (presenter) (2008). "The Genius of the East". The
Story of Maths. BBC.
Jim Al-Khalili (presenter) (2010). Science and Islam. BBC.
Hogendijk, Jan P. (January 1999). "Bibliography of
Medieval Islamic Civilization".
O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics:
forgotten brilliance?", MacTutor History of
University of St Andrews .
Richard Covington, Rediscovering Arabic Science, 2007, Saudi Aramco
Mathematics in medieval Islam
'Abd al-Hamīd ibn Turk
Sind ibn Ali
Al-Ḥajjāj ibn Yūsuf
Hunayn ibn Ishaq
Na'im ibn Musa
Sahl ibn Bishr
Abu Said Gorgani
Ahmad ibn Yusuf
Brethren of Purity
Ibrahim ibn Sinan
Sinan ibn Thabit
Nazif ibn Yumn
Abu Nasr Mansur
Abu Mansur al-Baghdadi
Jabir ibn Aflah
Sharaf al-Dīn al-Ṭūsī
Muhyi al-Dīn al-Maghribī
Nasir al-Din al-Tusi
Shams al-Dīn al-Samarqandī
Ibn al‐Ha'im al‐Ishbili
Ibn Abi al-Shukr
Kamāl al-Dīn al-Fārisī
Qutb al-Din al-Shirazi
Muhammad Baqir Yazdi
Ibn Hamza al-Maghribi
Ibn Ghazi al-Miknasi
Ahmad Ibn al-Qadi
The Compendious Book on Calculation by Completion and Balancing
Principles of Hindu Reckoning
Book of Optics
The Book of Healing
Encyclopedia of the Brethren of Purity
Islamic geometric patterns
House of Knowledge
House of Wisdom
Constantinople observatory of Taqi al-Din
University of Al Quaraouiyine