Mathematics in the medieval Islamic world

In the history of mathematics, Arabic mathematics or Islamic mathematics refers to the mathematics developed by the Islamic civilisation between 622 and 1600. While most scientist in this period where Muslim and Arabic was the dominant language contributions where made by people of many religions (Christian, Jewish, Zoroastrian) and ethnicities (Persian, Turkish, Tajik). The centre of Arabic mathematics was located in present day Iraq and Iran, at it's greatest extent stretched from Turkey, North Africa and Spain in the west, to the border of China in the east.

Islamic science and mathematics flourished under the Islamic Caliphate (also known as the Arab Empire or Islamic Empire) established across the Middle East, Central Asia, North Africa, Sicily, the Iberian Peninsula, and in parts of France and Pakistan (known as India at the time) in the 8th century. Although most Islamic texts on mathematics were written in Arabic, they were not all written by Arabs, since — much like the status of Greek in the Hellenistic world — Arabic was used as the written language of non-Arab scholars throughout the Islamic world at the time. Many of the most important Islamic mathematicians were Persians.

Recent research paints a new picture of the debt that we owe to Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects, the mathematics studied today is far closer in style to that of Islamic mathematics than to that of Hellenistic mathematics.

Influences

Hellenistic mathematics and Indian mathematics had an important role in the development of early Islamic mathematics, especially works such as Euclid's classic geometry, Aryabhata's trigonometry and Brahmagupta's arithmetic, and it is thought that they contributed to the era of Islamic scientific innovation that lasted until the 14th century. Many ancient Greek texts have survived only as Arabic translations by Islamic scholars. Perhaps the most important mathematical contribution from India was the decimal place-value Indo-Arabic numeral system, also known as the Hindu numerals. The Persian historian al-Biruni (c. 1050) in his book Tariq al-Hind states that the great king al-Ma'mun had an embassy from India and with them brought a book which was translated to Arabic as Sindhind. It is assumed that Sindhind is none other than Brahmagupta's Brahmasphuta-siddhanta.

Famous Islamic mathematicians

Muhammad ibn Musa al-Khwarizmi

Main article: Muhammad ibn Musa al-Khwarizmi

File:Al-Kitab al-mukhtasar fi hisab al-jabr wa-l-muqabala.jpg

Page from the Al-Jabr wa-al-Muqabilah

An important figure of Islamic mathematics was Muḥammad ibn Mūsā al-Ḵwārizmī (c. 780-850), also known as al-Khwarizmi, the Persian mathematician and astronomer of the caliph of Baghdad. He wrote several important books and is today known for introducing the Indo-Arabic place-value decimal system, which we use today. The system was developed in India in the 6th century, but it was known to Europeans only in the 13th century, from a Latin translation of al-Khwarizmi. Medieval European mathematical works used the phrase "dixit Algorismi" ("so says al-Khwarizmi") when they used the decimal system; from this is derived the word "algorithm". Also the word "algebra" is derived from one of his works, Al-Jabr wa-al-Muqabilah, which dealt with equations, polynomials, reductions, etc. – specifically it explained how to reveal unknown quantities in equations by executing balancing procedures which preserve the equality. Though some claim that his personal religion was Zoroastrianism, it is much more likely that he was Muslim given that he was named after Muhammad, the prophet of Islam and that the name ibn Musa means "son of Moses", who is an important figure in the Jewish and Muslim religions but not in Zoroastrianism. Regardless, his work has always been and remains in the mainstream of Islamic intellectual history. Al-Khwarizmi is often considered the father of algebra and algorithms for his important works in these fields. But algebra actually existed before Al-Khwarizmi, it was written about centuries before he was born by the ancient Greek mathematician Diophantus.

Perhaps one of the most significant advances made by Islamic mathematics began 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 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.

Al-Khwarizmi's successors undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose.

Though Al-Khwarizmi's approach to mathematics was mostly algebraic, he did contribute to the study of practical geometry.

Mahani

Persian mathematician Mahani (known as al-Mahani) (b. 820) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.

Al-Kindi

Arab mathematician (also dealt with other sciences and philosophy) Al-Kindi wrote many books on the subject of mathematics (geometry, harmony of numbers, measuring proportions and time, Hindu-Arabic numerals). He has made Hindu-Arabic numerals very popular.

'Abd al-Hamid ibn Turk

'Abd al-Hamid ibn Turk contributed to the study of quadratic equations.

Thabit ibn Qurra

Arab mathematician and geometer Thabit ibn Qurra (b. 836) made many contributions to mathematics, particularly geometry. In his work on number theory, he discovered an important theorem which allowed pairs of amicable numbers to be found, that is two numbers such that each is the sum of the proper divisors of the other. Amicable numbers later played a large role in Islamic mathematics.

Astronomy, time-keeping and geography provided other motivations for geometrical and trigonometrical research. Thabit ibn Qurra studied curves required in the construction of sundials. Thabit ibn Qurra also undertook both theoretical and observational work in astronomy.

Abu Kamil

Egyptian mathematician Abu Kamil ibn Aslam (850) forms an important link in the development of algebra between al-Khwarizmi and al-Karaji. He had begun to understand what we would write in symbols as $\ x^{n}.x^{m}=x^{m+n}$. He also studied algebra using irrational numbers.

Al-Battani

Arab mathematician and astronomer Abu Abdallah Muhammad ibn Jabir al-Battani (868-929) made accurate astronomical observations which allowed him to improve on Ptolemy's data for the Sun and the Moon. He also produced a number of trigonometrical relationships:

$\tan a={\frac {\sin a}{\cos a}}$

$\sec a={\sqrt {1+\tan ^{2}a}}$

He also solved the equation sin x = a cos x discovering the formula:

$\sin x={\frac {a}{\sqrt {1+a^{2}}}}$

and used al-Marwazi's idea of tangents ("shadows") to develop equations for calculating tangents and cotangents, compiling tables of them.

Sinan ibn Thabit

Arab scientist Sinan ibn Thabit ibn Qurra (c. 880-943) was the son of Thabit ibn Qurra and the father of Ibrahim ibn Sinan. He wrote the mathematical treatise On the elements of geometry, a commentary on Archimedes' On triangles, and several other astronomical and political treatises. He studied medicine, the science of Euclid, the Almagest, astronomy, the theories of meteorological phenomena, logic and metaphysics.

Ibrahim ibn Sinan

Although Islamic mathematicians are most famed for their work on algebra, number theory and numeral systems, they also made considerable contributions to geometry, trigonometry and mathematical astronomy. Ibrahim ibn Sinan ibn Thabit ibn Qurra (b. 908), son of Sinan ibn Thabit and grandson of Thabit ibn Qurra, introduced a method of integration more general than that of Archimedes, and was a leading figure in a revival and continuation of Greek higher geometry in the Islamic world. He studied optics and investigated the optical properties of mirrors made from conic sections.

Ibrahim ibn Sinan, like his grandfather, also studied curves required in the construction of sundials, for the purposes of astronomy, time-keeping and geography, which provided motivations for geometrical and trigonometrical research.

Abu'l-Hasan al-Uqlidisi

The Indian methods of arithmetic with the Indo-Arabic numerals were originally used with a dust board similar to a blackboard. Arab mathematician Abu'l-Hasan al-Uqlidisi (b. 920) showed how to modify the Indian methods of arithmetic for pen and paper use.

Abul Wáfa

Persian mathematician Abu'l-Wáfa (940-998) invented the tangent function. The Indo-Arabic system of calculating allowed the extraction of roots by Abu'l-Wáfa.

Abu'l-Wáfa applied spherical geometry to astronomy and also used formulas involving sine and tangent.

Abu Bakr Karaji

Algebra was further developed by Persian mathematician Karaji (known as Abu Bakr al-Karaji in Arabic) (953-1029) in his treatise al-Fakhri, where he extends the methodology to incorporate integral powers and integral roots of unknown quantities.

Karaji is seen by many as the first person to completely free algebra from geometrical operations and to replace them with the arithmetical type of operations which are at the core of algebra today. He was first to define the monomials $\ x,x^{2},x^{3},...$ and $\ 1/x,1/x^{2},1/x^{3},...$ and to give rules for products of any two of these. He started a school of algebra which flourished for several hundreds of years.

The discovery of the binomial theorem for integer exponents by al-Karaji was a major factor in the development of numerical analysis based on the decimal system.

Al-Haytham

Al-Haytham (b. 965), also known as Alhazen, in his work on number theory, seems to have been the first to attempt to classify all even perfect numbers (numbers equal to the sum of their proper divisors) as those of the form $\ 2^{k-1}(2^{k}-1)$ where $\ 2^{k}-1$ is prime.

Al-Haytham is also the first person that we know to state Wilson's theorem, namely that if $\ p$ is prime then $\ 1+(p-1)!$ is divisible by $\ p$. It is unclear whether he knew how to prove this result. It is called Wilson's theorem because of a comment made by Edward Waring in 1770 that John Wilson had noticed the result. There is no evidence that John Wilson knew how to prove it and most certainly Waring did not. Joseph Louis Lagrange gave the first proof in 1771 and it should be noticed that it is more than 750 years after al-Haytham before number theory surpasses this achievement of Islamic mathematics.

Al-Haytham also studied optics and investigated the optical properties of mirrors made from conic sections.

Abu Nasr Mansur

Abu Nasr Mansur ibn Ali ibn Iraq (970-1036) applied spherical geometry to astronomy and also used formulas involving sine and tangent. He is well known for discovering the sine law.

Abu Sahl Kuhi (al-Kuhi)

Persian mathematician Abu Sahl Waijan ibn Rustam al-Quhi (10th century), also known as Abu Sahl al-Kuhi or just Kuhi, was a leading figure in a revival and continuation of Greek higher geometry in the Islamic world. He studied optics and investigated the optical properties of mirrors made from conic sections. He also did some important work on the centers of gravity.

Biruni

Persian mathematician Biruni (known in Arabic as al-Biruni) (b. 973) used the sine formula in both astronomy and in the calculation of longitudes and latitudes of many cities. Both astronomy and geography motivated al-Biruni's extensive studies of projecting a hemisphere onto the plane.

His contributions to mathematics include:

• theoretical and practical arithmetic
• summation of series
• combinatorial analysis
• the rule of three
• irrational numbers
• ratio theory
• algebraic definitions
• method of solving algebraic equations
• geometry
• Archimedes' theorems

Some of his notable achievements included:

• At the age of seventeen, he calculated the latitude of Kath, Khwarazm, using the maximum altitude of the sun.
• By the age of twenty-two, he had written several short works, including a study of map projections, "Cartography", which included a methodology for projecting a hemisphere on a plane.
• By the age of twenty-seven, he had written a book called "Chronology" which referred to other work he had completed (now lost) that included one book about the astrolabe, one about the decimal system, four about astrology, and two about history.
• He attempted to refine the calculations of Eratosthenes using his methods, but his calculations were off by approximately 400 miles.

Al-Baghdadi

Arab mathematician al-Baghdadi (b. 980) looked at a slight variant of Thabit ibn Qurra's theorem of amicable numbers. There were three different types of arithmetic used around this period and, by the end of the 10th century, authors such as al-Baghdadi were writing texts comparing the three numeral systems: Finger-reckoning arithmetic (a system derived from counting on the fingers with the numerals written entirely in words), the sexagesimal numeral system (developed by the Babylonians), and the Indo-Arabic numerals. This third system of calculating allowed most of the advances in numerical methods. Al-Baghdadi also contributed to improvements in the Indo-Arabic decimal system.he had quite an irrgeular choice in his verb usage.

Omar Khayyam

The Persian poet Omar Khayyam (b. 1048) was also a mathematician, and wrote Discussions of the Difficulties in Euclid, a book about flaws in Euclid's Elements. He gave a geometric solution to cubic equations, one of the most original discoveries in Islamic mathematics. He was also very influential in calendar reform. He also wrote influential work on Euclid's parallel postulate.

Omar Khayyam gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. Khayyam also wrote that he hoped to give a full description of the algebraic solution of cubic equations in a later work:

"If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements which are greatly useful in this art will be prepared."

The Indo-Arabic system of calculating also allowed the extraction of roots by Omar Khayyam. Omar Khayyam also combined the use of trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means.

Al-Samawal

Moroccan mathematician Al-Samawal (b. 1130) was an important member of al-Karaji's school of algebra. Al-Samawal was the first to give the new topic of algebra a precise description when he wrote that it was concerned "with operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known."

Sharaf al-Din al-Tusi

Persian mathematician Sharaf al-Din al-Tusi (b. 1135), although almost exactly the same age as al-Samawal, did not follow the general development that came through al-Karaji's school of algebra but rather followed Khayyam's application of algebra to geometry. He wrote a treatise on cubic equations, which represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry.

Nasir al-Din al-Tusi

Spherical trigonometry was largely developed by Muslims, and systematized (along with plane trigonometry) by Persian Shi'a mathematician Nasir al-Din al-Tusi (1201–1274). He also wrote influential work on Euclid's parallel postulate.

Nasir al-Din al-Tusi, like many other Muslim mathematicians, based his theoretical astronomy on Ptolemy's work, but al-Tusi made the most significant development in the Ptolemaic planetary system until the development of the Nicolaus Copernicus. One of these developments is the Tusi-couple, which was later used in the Copernican model.

Ibn Al-Banna

Moroccan mathematician ibn al-Banna (b. 1256) used symbols in algebra, though symbols were used by other Islamic mathematicians at least a century before this.

Al-Farisi

Persian mathematician Al-Farisi (b. 1260) gave a new proof of Thabit ibn Qurra's theorem of amicable numbers, introducing important new ideas concerning factorisation and combinatorial methods. He also gave the pair of amicable numbers 17,296 and 18,416 which have been attributed to Leonhard Euler, but we know that these were known earlier than al-Farisi, perhaps even by Thabit ibn Qurra himself. Apart from number theory, his other major contribution to mathematics was on light.

Ghiyath al-Kashi

Persian mathematician Ghiyath al-Kashi (1380-1429) contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real numbers such as π, which he computed to 16 decimal place of accuracy. His contribution to decimal fractions is so major that for many years he was considered as their inventor. Kashi also developed an algorithm for calculating nth roots, which was a special case of the methods given many centuries later by Ruffini and Horner.

Al-Kashi also produced tables of trigonometric functions as part of his studies of astronomy. His sine tables were correct to 4 sexagesimal digits, which corresponds to approximately 8 decimal places of accuracy. The construction of astronomical instruments such as the astrolabe, invented by Mohammad al-Fazari, was also a speciality of Muslim mathematicians.

Ulugh Beg

Timurid mathematician Ulugh Beg (1393 or 1394 – 1449), also ruler of the Timurid Empire, produced tables of trigonometric functions as part of his studies of astronomy. His sine and tangent tables were correct to 8 decimal places of accuracy.

Al-Qalasadi

Moorish mathematician Abu'l Hasan ibn Ali al Qalasadi (b. 1412) used symbols in algebra, though symbols were used by other Islamic mathematicians much earlier.

In the time of the Ottoman Empire (from 15th century onwards) the development of Islamic mathematics became stagnant. This parallels the stagnation of mathematics when the Romans conquerored the Hellenistic world.

Muhammad Baqir Yazdi

In the 17th century, Muhammad Baqir Yazdi gave the pair of amicable numbers 9,363,584 and 9,437,056 many years before Euler's contribution to amicable numbers.

Translations

Many Arabic texts on Islamic mathematics were translated into Latin and had an important role in the evolution of later European mathematics. A list of translations, from Greek and Sanskrit to Arabic, and from Arabic to Latin, is given below.

Greek to Arabic

The following mathematical Greek texts on Hellenistic mathematics were translated into Arabic, and subsequently into Latin:

• Euclid's Data, Optics, Phaenomena and On Divisions.
• Euclid's Elements by al-Hajjaj (c. 8th century)
• Revision of Euclid's Elements by Thabit ibn Qurra.
• Apollonius' Conics by Thabit ibn Qurra.
• Ptolemy's Almagest by Thabit ibn Qurra.
• Archimedes' Sphere and Cylinder and Measurement of the Circle by Thabit ibn Qurra.
• Archimedes' On triangles by Sinan ibn Thabit.
• Diophantus' Arithmetica by Abu'l-Wáfa.
• Menelaus of Alexandria's Sphaerica.
• Theodosius of Bithynia's Spherics.
• Diocles' treatise on mirrors.
• Pappus of Alexandria's work on mechanics.

Sanskrit to Arabic

The following mathematical Sanskrit texts on Indian mathematics were translated into Arabic, and subsequently into Latin:

• The Sindhind by Ibrahim al-Fazari, Muhammad al-Fazari and Yaqub ibn Tāriq (c. 8th century).
• Surya Siddhanta by al-Fazari.
• Brahmagupta's Brahma Sphuta Siddhanta by al-Fazari.
• Brahmagupta's Khandakhayaka.
• Aryabhata's Aryabhatiya.
• Aryabhata's Arya Siddhanta.
• Varahamihira's Pancha Siddhanta.
• Bhaskara I's Lagu Bhaskariya.
• Bhaskara II's Lilavati (to Persian rather than Arabic).

Arabic to Latin

The following mathematical Arabic texts on Islamic mathematics were translated into Latin:

• Introduction to Astronomy by Adelard of Bath (fl. 1116-1142).
• Al-Khwarizmi's arithmetical work Liber ysagogarum Alchorismi and Astronomical Tables by Adelard of Bath.
• Al-Khwarizmi's trigonometrical tables which deal with the sine and tangent by Adelard of Bath (1126).
• Al-Khwarizmi's Zij al-Sindhind in Spain (1126).
• Liber alghoarismi de practica arismetrice, an ellaboration of al-Khwarizmi's Arithmetic, by John of Seville and Domingo Gundisalvo (fl. 1135-1153).
• Secretum Secretorum by John of Seville and Domingo Gundisalvo.
• Costa Ben Luca's De differentia spiritus et animae by John of Seville and Domingo Gundisalvo.
• al-Battani's De motu stellarum, which contains important material on trigonometry, by Plato of Tivoli (fl. 1134-1145).
• Abraham bar Hiyya's Liber embadorum by Plato of Tivoli.
• Liber de compositione alchimiae (The Book of the Composition of Alchemy) by Robert of Chester (f. 1141-1150).
• Al-Khwarizmi's Kitab al-Jabr wa-l-Muqabala (Algebra), Kitab al-Adad al-Hindi (Algoritmi de numero Indorum), and revised astronomical tables by Robert of Chester.
• Al-Khwarizmi's Kitab-ul Jama wat Tafriq by Bon Compagni (1157).
• Al-Khwarizmi's Algebra by Gerard of Cremona (fl. 1150-1185).
• Jabir ibn Aflah's Elementa astronomica by Gerard of Cremona.
• The Banu Musa's (Muhammad bin Musa, Ahmad bin Musa and Hasan bin Musa) works on geometry by Gerard of Cremona.
• Abdur Rahman's commentary on Euclid's Elements by Gerard of Cremona.
• Muhammad ibn Muhammad Baqi's commentary on Euclid's Elements by Gerard of Cremona.
• Abul Abbas Nairizi's commentaries on Euclid and Ptolemy by Gerard of Cremona.
• The works of Thabit ibn Qurra by Gerard of Cremona.
• Abu Kamil's Algebra.
• Al-Biruni's Tariq Al Hind.
• Al-Fazari's The Sindhind.
• The works of Omar Khayyam.
• The works of Nasir al-Din Tusi.
• The works of Mu'ayyad al-Din al-'Urdi (c. 1250).
• The works of Ibn al-Shatir (1304–1375).
• The works of Nasir al-Din Tusi (to Byzantine Greek rather than Latin).

References

• Berggren, J. L. Episodes in the Mathematics of Medieval Islam. Springer-Verlag: 1986.
• Berggren, J. L. Mathematics in medieval Islam. Encyclopaedia Britannica.
• Burton, David M. The History of Mathematics: An Introduction. McGraw Hill: 1997.
• Joseph, George Gheverghese. The Crest of the Peacock: Non-European Roots of Mathematics, 2nd Edition. Penguin Books: 2000.
• Katz, Victor J. A History of Mathematics: An Introduction, 2nd Edition. Addison-Wesley: 1998.
• Rashed, Roshdi. The Development of Arabic Mathematics: Between Arithmetic and Algebra. Transl. by A. F. W. Armstrong. Kluwer Academic Publishers: 1994.
• John J. O'Connor and Edmund F. Robertson. Arabic mathematics : forgotten brilliance?. MacTutor History of Mathematics archive. St Andrews University. 1999.
• John J. O'Connor and Edmund F. Robertson. Arabic/Islamic mathematics. MacTutor History of Mathematics archive. St Andrews University. 2004.
• George Saliba, Whose Science is Arabic Science in Renaissance Europe?, Columbia University, 1999.
• History of Trigonometry
• http://www.usc.edu/dept/MSA/introduction/woi_knowledge.html#28