Mathematics during the
Golden Age of Islam, especially during the 9th and 10th centuries, was built on
Greek mathematics
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek math ...
(
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ...
,
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scienti ...
,
Apollonius) and
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupt ...
(
Aryabhata,
Brahmagupta). Important progress was made, such as full development of the decimal
place-value system to include
decimal fractions, the first systematised study of
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
, and advances in
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
trigonometry.
Arabic works played an important role in the transmission of mathematics to Europe during the 10th—12th centuries.
Concepts
Algebra
The study of
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
, the name of which is derived from the
Arabic
Arabic (, ' ; , ' or ) is a Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C. E.Watson; Walte ...
word meaning completion or "reunion of broken parts", flourished during the
Islamic golden age
The Islamic Golden Age was a period of cultural, economic, and scientific flourishing in the history of Islam, traditionally dated from the 8th century to the 14th century. This period is traditionally understood to have begun during the reign ...
.
Muhammad ibn Musa al-Khwarizmi, a Persian scholar in the
House of Wisdom
The House of Wisdom ( ar, بيت الحكمة, Bayt al-Ḥikmah), also known as the Grand Library of Baghdad, refers to either a major Abbasid public academy and intellectual center in Baghdad or to a large private library belonging to the Abba ...
in
Baghdad
Baghdad (; ar, بَغْدَاد , ) is the capital of Iraq and the second-largest city in the Arab world after Cairo. It is located on the Tigris near the ruins of the ancient city of Babylon and the Sassanid Persian capital of Ctesipho ...
was the founder of algebra, is along with the
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
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 root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
s of first and second degree (linear and quadratic)
polynomial equations
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equati ...
. He introduces the method of
reduction, and unlike Diophantus, also 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:
Several other mathematicians during this time period expanded on the algebra of Al-Khwarizmi. ]Abu Kamil Shuja'
Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ ( Latinized as Auoquamel, ar, أبو كامل شجاع بن أسلم بن محمد بن شجاع, also known as ''Al-ḥāsib al-miṣrī''—lit. "the Egyptian reckoner") (c. 850 – ...
wrote a book of algebra accompanied with geometrical illustrations and proofs. He also enumerated all the possible solutions to some of his problems. Abu al-Jud Abū al-Jūd Muḥammad b. Aḥmad b. al-Layth was an Iranian peoples, Iranian mathematician. He lived during 10th century and was a contemporary of Al-Biruni. Not much is known about his life. He seems to have lived in the east of Khurasan, within ...
, 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
Omar Khayyam (c. 1038/48 in Iran
Iran, officially the Islamic Republic of Iran, and also called Persia, is a country located in Western Asia. It is bordered by Iraq and Turkey to the west, by Azerbaijan and Armenia to the northwest, by the Caspian Sea and Turkm ...
– 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 section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
s. 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 , with ''a'' and ''b'' positive, he would note that the maximum point of the curve occurs at , 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
The earliest implicit traces of mathematical induction can be found in Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ...
'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
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a con ...
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
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, althoug ...
.
Irrational numbers
The Greeks had discovered irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s, 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 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 (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power ...
translations of Al-Khwarizmi's Arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...
on the Indian numerals introduced the decimal positional number system
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
to the Western world
The Western world, also known as the West, primarily refers to the various nations and states in the regions of Europe, North America, and Oceania. . His ''Compendious Book on Calculation by Completion and Balancing'' presented the first systematic solution of linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and quadratic equations. In Renaissance
The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass id ...
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
Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of import ...
's ''Geography
Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, a ...
'' and wrote on astronomy and astrology. However, C.A. Nallino
Carlo Alfonso Nallino (18 February 1872 – 25 July 1938) was an Italian orientalist.
Biography
Nallino was born in Turin, and studied literature under Italo Pizzi at the University of Turin. From 1896 he taught in the Istituto Universit ...
suggests that al-Khwarizmi's original work was not based on Ptolemy but on a derivative world map, presumably in Syriac or Arabic
Arabic (, ' ; , ' or ) is a Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C. E.Watson; Walte ...
.
Spherical 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.
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. Al-Khwarizmi did not use negative numbers or negative coefficients. But within fifty years, Abu Kamil illustrated the rules of signs for expanding the multiplication . Al-Karaji wrote in his book ''al-Fakhrī'' that "negative quantities must be counted as terms". In the 10th century, Abū al-Wafā' al-Būzjānī considered debts as negative numbers in ''A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen''.
By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve polynomial divisions. As al-Samaw'al writes:
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.
Double false position
Between the 9th and 10th centuries, the Egyptian mathematician Abu Kamil wrote a now-lost treatise on the use of double false position, known as the ''Book of the Two Errors'' (''Kitāb al-khaṭāʾayn''). The oldest surviving writing on double false position from the Middle East
The Middle East ( ar, الشرق الأوسط, ISO 233: ) is a geopolitical region commonly encompassing Arabia (including the Arabian Peninsula and Bahrain), Asia Minor (Asian part of Turkey except Hatay Province), East Thrace (Europ ...
is that of Qusta ibn Luqa (10th century), an Arab
The Arabs (singular: Arab; singular ar, عَرَبِيٌّ, DIN 31635: , , plural ar, عَرَب, DIN 31635, DIN 31635: , Arabic pronunciation: ), also known as the Arab people, are an ethnic group mainly inhabiting the Arab world in Wester ...
mathematician from Baalbek, Lebanon
Lebanon ( , ar, لُبْنَان, translit=lubnān, ), officially the Republic of Lebanon () or the Lebanese Republic, is a country in Western Asia. It is located between Syria to Lebanon–Syria border, the north and east and Israel to Blue ...
. He justified the technique by a formal, Euclidean-style geometric proof. Within the tradition of Golden Age Muslim mathematics, double false position was known as ''hisāb al-khaṭāʾayn'' ("reckoning by two errors"). It was used for centuries to solve practical problems such as commercial and juridical questions (estate partitions according to rules of Quranic inheritance), as well as purely recreational problems. The algorithm was often memorized with the aid of mnemonics, such as a verse attributed to Ibn al-Yasamin and balance-scale diagrams explained by al-Hassar and Ibn al-Banna, who were each mathematicians of Moroccan origin.
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"."Science Teaching in Pre-Modern Societies"
in Film Screening and Panel Discussion, ''McGill University'', 15 January 2019.
* 'Abd al-Hamīd ibn Turk
( fl. 830), known also as ( ar, ابومحمد عبدالحمید بن واسع بن ترک الجیلی) was a ninth-century Muslim mathematician. Not much is known about his life. The two records of him, one by Ibn Nadim and the other by al-Q ...
(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
Ḥasan Ibn al-Haytham, Latinized as Alhazen (; full name ; ), was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq.For the description of his main fields, see e.g. ("He is one of the prin ...
(c. 965–1040)
* Abū al-Rayḥān al-Bīrūnī
Abu Rayhan Muhammad ibn Ahmad al-Biruni (973 – after 1050) commonly known as al-Biruni, was a Khwarazmian Iranian in scholar and polymath during the Islamic Golden Age. He has been called variously the "founder of Indology", "Father of Co ...
(973–1048) (trigonometry)
* Ibn Maḍāʾ (c. 1116–1196)
* Jamshīd al-Kāshī (c. 1380–1429) (decimals and estimation of the circle constant)
Gallery
File:Gravure originale du compas parfait par Abū Sahl al-Qūhī.jpg, Engraving of Abū Sahl al-Qūhī's perfect compass to draw conic sections.
File:Theorem of al-Haitham.JPG, The theorem of Ibn Haytham.
See also
* Arabic numerals
Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as ...
* 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
Sources
*
*
*
*
*
*
*
Further reading
;Books on Islamic mathematics
*
** Review:
** Review:
*
*
*
* Sowjetische Beiträge zur Geschichte der Naturwissenschaft pp. 62–160.
*
; Book chapters on Islamic mathematics
*
; Books on Islamic science
*
*
; Books on the history of mathematics
* (Reviewed: )
*
;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.
*
*
*
; Television documentaries
* Marcus du Sautoy (presenter) (2008). "The Genius of the East". ''The Story of Maths
''The Story of Maths'' is a four-part British television series outlining aspects of the history of mathematics. It was a co-production between the Open University and the BBC and aired in October 2008 on BBC Four. The material was written and pre ...
''. BBC.
* Jim Al-Khalili (presenter) (2010). '' Science and Islam''. BBC.
External links
*
*
Richard Covington, ''Rediscovering Arabic Science'', 2007, Saudi Aramco World
List of Inventions and Discoveries in Mathematics During the Islamic Golden Age
{{DEFAULTSORT:Mathematics In Medieval Islam
Islamic Golden Age