Mathematics
Mathematics during the
Golden Age of Islam , especially during the
9th and 10th centuries, was built on
Greek mathematics
Greek mathematics (
Euclid
Euclid ,
Archimedes
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
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.
CONTENTS
* 1 History
* 1.1
Algebra
Algebra
* 1.2 Cubic equations
* 1.3 Induction
* 1.4 Irrational numbers
* 1.5 Spherical trigonometry
* 2 Other major figures
* 3 Gallery
* 4 See also
* 5 References
* 6 Sources
* 7 Further reading
* 8 External links
HISTORY
Omar Khayyám
Omar Khayyám 's "Cubic equations and intersections of conic
sections" the first page of the two-chaptered manuscript kept in
Tehran University
ALGEBRA
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
Islamic golden age .
Muhammad ibn Musa al-Khwarizmi , a
scholar in the
House of Wisdom
House of Wisdom in
Baghdad
Baghdad , is along with the Greek
mathematician
Diophantus
Diophantus , known as the father of algebra. In his book
The Compendious Book on Calculation by Completion and Balancing ,
Al-Khwarizmi
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.
Robertson said:
"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
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." — MacTutor History of
Mathematics
Mathematics archive
Several other mathematicians during this time period expanded on the
algebra of Al-Khwarizmi.
Omar Khayyam
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.
CUBIC EQUATIONS
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
Omar Khayyam (c. 1038/48 in
Iran
Iran – 1123/24) wrote the Treatise on
Demonstration of Problems of
Algebra
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
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 x 3 + a
= b x {displaystyle x^{3}+a=bx} , with a and b positive, he would
note that the maximum point of the curve y = b x x 3
{displaystyle y=bx-x^{3}} occurs at x = b 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.
INDUCTION
See also: Mathematical induction § History
The earliest implicit traces of mathematical induction can be found
in
Euclid
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
Pascal\'s triangle .
IRRATIONAL NUMBERS
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 indeed
Greek mathematics
Greek mathematics was mainly
geometrical. Islamic mathematicians including 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
Latin translations of
Al-Khwarizmi
Al-Khwarizmi 's
Arithmetic on the
Indian numerals introduced the decimal positional
number system to the
Western world
Western world . His Compendious Book on
Calculation by Completion and Balancing presented the first systematic
solution of linear and quadratic equations . In
Renaissance
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
Ptolemy 's Geography and wrote on astronomy and astrology.
However,
C.A. Nallino suggests that al-Khwarizmi's original work was
not based on
Ptolemy
Ptolemy but on a derivative world map, presumably in
Syriac or Arabic .
SPHERICAL TRIGONOMETRY
Further information:
Law of sines
Law of sines and
History of trigonometry
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 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 (826–901)
*
Sind ibn Ali (d. after 864)
*
Ismail al-Jazari (1136–1206)
*
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
Ibn al-Haytham (ca. 965–1040)
*
Abū al-Rayḥān al-Bīrūnī (973–1048) (trigonometry)
*
Ibn Maḍāʾ (c. 1116–1196)
*
Jamshīd al-Kāshī (c. 1380–1429) (decimals and estimation of
the circle constant)
GALLERY
*
Engraving of
Abū Sahl al-Qūhī 's perfect compass to draw conic
sections.
*
The theorem of Ibn Haytham .
SEE ALSO
*
Arabic numerals
Arabic numerals
* Indian influence on Islamic mathematics in medieval Islam
*
History of calculus
*
History of geometry
History of geometry
*
Science in the medieval Islamic world
Science in the medieval Islamic world
*
Timeline of Islamic science and technology
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."
* ^
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
Online Etymology Dictionary .
* ^ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of
Mathematics
Mathematics (Second ed.). John Wiley & Sons. p. 228. ISBN
0-471-54397-7 .
* ^ Swetz, Frank J. (1993). Learning Activities from the History of
Mathematics. Walch Publishing. p. 26. ISBN 978-0-8251-2264-4 .
* ^ 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
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
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.
JSTOR
JSTOR 604533 . doi :10.2307/604533 .
* ^ A B Sesiano, Jacques (2000). Helaine, Selin; Ubiratan,
D'Ambrosio, eds. Islamic mathematics.
Mathematics
Mathematics Across Cultures: The
History of Non-western Mathematics. Springer. pp. 137–157. ISBN
1-4020-0260-2 .
* ^ O\'Connor, John J. ; Robertson, Edmund F. , "Abu Mansur ibn
Tahir Al-Baghdadi", MacTutor History of
Mathematics
Mathematics archive ,
University of St Andrews
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
University of St Andrews .
* ^ Berggren, J. Lennart (2007). "
Mathematics
Mathematics in Medieval Islam".
The
Mathematics
Mathematics of Egypt, Mesopotamia, China, India, and Islam: A
Sourcebook. Princeton University Press. p. 518. ISBN 978-0-691-11485-9
.
SOURCES
* Boyer, Carl B. (1991), "Greek
Trigonometry
Trigonometry and Mensuration, and
The Arabic Hegemony", A History of
Mathematics
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
Mathematics (4th rev.
ed.), Dover Publications, ISBN 0-486-60255-9
FURTHER READING
Books on Islamic mathematics
* Berggren, J. Lennart (1986). Episodes in the
Mathematics
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
Mathematics of Medieval Islam".
American Mathematical Monthly .
Mathematical Association of America. 95 (6): 567.
JSTOR
JSTOR 2322777 . doi
:10.2307/2322777 .
* Review: Hogendijk, Jan P.; Berggren, J. L. (1989). "Episodes in
the
Mathematics
Mathematics of Medieval Islam by J. Lennart Berggren". Journal of
the American Oriental Society. American Oriental Society. 109 (4):
697–698.
JSTOR
JSTOR 604119 . doi :10.2307/604119 .
* 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. ISBN 0-521-25844-8 .
* Smith, David E. (1958). History of Mathematics. Dover
Publications. ISBN 0-486-20429-4 .
* Rashed, Roshdi (2001). The Development of Arabic Mathematics:
Between Arithmetic and Algebra. Translated by A. F. W. Armstrong.
Springer. ISBN 0-7923-2565-6 .
* Rosen, Fredrick (1831). The
Algebra
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. ISBN 0-684-16962-2 .
* 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. ISBN 978-2-7116-0734-1 .
Book chapters on Islamic mathematics
* Berggren, J. Lennart (2007). "
Mathematics
Mathematics in Medieval Islam". In
Victor J. Katz. The
Mathematics
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. 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
.
* 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
Mathematics (2nd ed.). Princeton University
Press. ISBN 0-691-00659-8 . (Reviewed: Katz, Victor J.; Joseph,
George Gheverghese