Islamic_mathematics

Islamic mathematics

In the history of mathematics, Islamic mathematics or Arabic mathematics refers to the mathematics developed in the Islamic world between 622 and 1600, in the part of the world where Islam was the dominant religious and cultural influence, and Arabic was the dominant language of scholarship. Islamic science and mathematics flourished under the Islamic caliphate (also known as the Islamic Empire) established across the Middle East, Central Asia, North Africa, Sicily, the Iberian Peninsula, and in parts of France and India in the 8th century. The center of Islamic mathematics was located in present-day Iraq and Iran, but at its greatest extent stretched from Turkey, North Africa and Spain in the west, to India in the east.^[1]

While most scientists in this period were Muslims and Arabic was the dominant language—much like Latin in Medieval Europe, Arabic was used as the written language of scholars throughout the Islamic world at the time—contributions were made by people of different ethnic groups (Arabs, Persians, Berbers, Moors, Turks) and sometimes different religions (Muslims, Christians, Jews, Sabians, Zoroastrians, irreligious).^[2]

Use of the term "Islam"

Bernard Lewis writes the following on the historical usage of the term "Islam" in What Went Wrong? Western Impact and Middle Eastern Response:^[3]

"There have been many civilizations in human history, almost all of which were local, in the sense that they were defined by a region and an ethnic group. This applied to all the ancient civilizations of the Middle East—Egypt, Babylon, Persia; to the great civilizations of Asia—India, China; and to the civilizations of Pre-Columbian America. There are two exceptions: Christendom and Islam. These are two civilizations defined by religion, in which religion is the primary defining force, not, as in India or China, a secondary aspect among others of an essentially regional and ethnically defined civilization. Here, again, another word of explanation is necessary."

"In English we use the word “Islam” with two distinct meanings, and the distinction is often blurred and lost and gives rise to considerable confusion. In the one sense, Islam is the counterpart of Christianity; that is to say, a religion in the strict sense of the word: a system of belief and worship. In the other sense, Islam is the counterpart of Christendom; that is to say, a civilization shaped and defined by a religion, but containing many elements apart from and even hostile to that religion, yet arising within that civilization."

In this article, "Islam" is used in the meaning of a civilization.

Origins and influences

The first century of the Islamic Arab Empire saw almost no scientific or mathematical achievements since the Arabs, with their newly conquered empire, had not yet gained any intellectual drive and research in other parts of the world had faded. In the second half of the eighth century Islam had a cultural awakening, and research in mathematics and the sciences increased.^[4] The Muslim Abbasid caliph al-Mamun (809-833) is said to have had a dream where Aristotle appeared to him, and as a consequence al-Mamun ordered that Arabic translation be made of as many Greek works as possible, including Ptolemy's Almagest and Euclid's Elements. Greek works would be given to the Muslims by the Byzantine Empire in exchange for treaties, as the two empires held an uneasy peace.^[4] Many of these Greek works were translated by Thabit ibn Qurra (826-901), who translated books written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.^[5] Historians are in debt to many Islamic translators, for it is through their work that many ancient Greek texts have survived only through Arabic translations.

Greek, Indian and Babylonian all played an important role in the development of early Islamic mathematics. The works of mathematicians such as Euclid, Apollonius, Archimedes, Diophantus, Aryabhata and Brahmagupta were all acquired by the Islamic world and incorporated into their mathematics. Perhaps the most influential mathematical contribution from India was the decimal place-value Indo-Arabic numeral system, also known as the Hindu numerals.^[6] The Persian historian al-Biruni (c. 1050) in his book Tariq al-Hind states that the Abbasid caliph al-Ma'mun had an embassy in India from which was brought a book to Baghdad that was translated into Arabic as Sindhind. It is generally assumed that Sindhind is none other than Brahmagupta's Brahmasphuta-siddhanta.^[7] The earliest translations from Sanskrit inspired several astronomical and astrological Arabic works, now mostly lost, some of which were even composed in verse.^[8]

Indian influences were later overwhelmed by Greek mathematical and astronomical texts. It is not clear why this occurred but it may have been due to the greater availability of Greek texts in the region, the larger number of practitioners of Greek mathematics in the region, or because Islamic mathematicians favored the deductive exposition of the Greeks over the elliptic Sanskrit verse of the Indians. Regardless of the reason, Indian mathematics soon became mostly eclipsed by or merged with the "Graeco-Islamic" science founded on Hellenistic treatises.^[8] Another likely reason for the declining Indian influence in later periods was due to Sindh achieving independance from the Caliphate, thus limiting access to Indian works. Nevertheless, Indian methods continued to play an important role in algebra, arithmetic and trigonometry.^[9]

Importance

J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive:

"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 16th, 17th, and 18th 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 Greek mathematics."

R. Rashed wrote in The development of Arabic mathematics: between arithmetic and algebra:

"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."

Biographies

Al-Ḥajjāj ibn Yūsuf ibn Maṭar (786 – 833): Al-Ḥajjāj translated Euclid's Elements into Arabic.
Muḥammad ibn Mūsā al-Khwārizmī (c. 780 Khwarezm/Baghdad – c. 850 Baghdad): Al-Khwārizmī was a Persian mathematician, astronomer, astrologer and geographer. He worked most of his life as a scholar in the House of Wisdom in Baghdad. His Algebra was the first book on the systematic solution of linear and quadratic equations. Latin translations of his Arithmetic, on the Indian numerals, introduced the decimal positional number system to the Western world in the 12th century. He revised and updated Ptolemy's Geography as well as writing several works on astronomy and astrology.
Al-ʿAbbās ibn Saʿid al-Jawharī (c. 800 Baghdad? – c. 860 Baghdad?): Al-Jawharī was a mathematician who worked at the House of Wisdom in Baghdad. His most important work was his Commentary on Euclid's Elements which contained nearly 50 additional propositions and an attempted proof of the parallel postulate.
ʿAbd al-Hamīd ibn Turk (fl. 830 Baghdad): Ibn Turk wrote a work on algebra of which only a chapter on the solution of quadratic equations has survived.
Yaʿqūb ibn Isḥāq al-Kindī (c. 801 Kufah – 873 Baghdad): Al-Kindī (or Alkindus) was a philosopher and scientist who worked as the House of Wisdom in Baghdad where he wrote commentaries on many Greek works. His contributions to mathematics include many works on arithmetic and geometry.
Hunayn ibn Ishaq (808 Al-Hirah – 873 Baghdad): Hunayn (or Johannitus) was a translator who worked at the House of Wisdom in Baghdad. Translated many Greek works including those by Plato, Aristotle, Galen, Hippocrates, and the Neoplatonists.
Banū Mūsā (c. 800 Baghdad – 873+ Baghdad): The Banū Mūsā where three brothers who worked at the House of Wisdom in Baghdad. Their most famous mathematical treatise is The Book of the Measurement of Plane and Spherical Figures, which considered similar problems as Archimedes did in his On the Measurement of the Circle and On the sphere and the cylinder. They contributed individually as well. The eldest, Jaʿfar Muḥammad (c. 800) specialised in geometry and astronomy. He wrote a critical revision on Apollonius' Conics called Premises of the book of conics. Aḥmad (c. 805) specialised in mechanics and wrote a work on pneumatic devices called On mechanics. The youngest, al-Ḥasan (c. 810) specialised in geometry and wrote a work on the ellipse called The elongated circular figure.
Al-Mahani
Ahmed ibn Yusuf
Thabit ibn Qurra (Syria-Iraq, 835-901)
Al-Hashimi (Iraq? ca. 850-900)
Muḥammad ibn Jābir al-Ḥarrānī al-Battānī (c. 853 Harran – 929 Qasr al-Jiss near Samarra)
Abu Kamil (Egypt? ca. 900)
Sinan ibn Tabit (ca. 880 - 943)
Al-Nayrizi
Ibrahim ibn Sinan (Iraq, 909-946)
Al-Khazin (Iraq-Iran, ca. 920-980)
Al-Karabisi (Iraq? 10th century?)
Ikhwan al-Safa' (Iraq, first half of 10th century): The Ikhwan al-Safa' ("brethren of purity") were a (mystical?) group in the city of Basra in Irak. The group authored a series of more than 50 letters on science, philosophy and theology. The first letter is on arithmetic and number theory, the second letter on geometry.
Al-Uqlidisi (Iraq-Iran, 10th century)
Al-Saghani (Iraq-Iran, ca. 940-1000)
Abū Sahl al-Qūhī (Iraq-Iran, ca. 940-1000)
Al-Khujandi
Abū al-Wafāʾ al-Būzjānī (Iraq-Iran, ca. 940-998)
Ibn Sahl (Iraq-Iran, ca. 940-1000)
Al-Sijzi (Iran, ca. 940-1000)
Labana of Cordoba (Spain, ca. 10th century): One of the few Islamic female mathematicians known by name, and the secretary of the Umayyad Caliph al-Hakem II. She was well-versed in the exact sciences, and could solve the most complex geometrical and algebraic problems known in her time.^[10]
Ibn Yunus (Egypt, ca. 950-1010)
Abu Nasr ibn `Iraq (Iraq-Iran, ca. 950-1030)
Kushyar ibn Labban (Iran, ca. 960-1010)
Al-Karaji (Iran, ca. 970-1030)
Ibn al-Haytham (Iraq-Egypt, ca. 965-1040)
Abū al-Rayḥān al-Bīrūnī (September 15, 973 in Kath, Khwarezm – December 13, 1048 in Gazna)
Ibn Sina
al-Baghdadi
Al-Nasawi
Al-Jayyani (Spain, ca. 1030-1090)
Ibn al-Zarqalluh (Azarquiel, al-Zarqali) (Spain, ca. 1030-1090)
Al-Mu'taman ibn Hud (Spain, ca. 1080)
al-Khayyam (Iran, ca. 1050-1130)
Ibn Yaḥyā al-Maghribī al-Samawʾal (ca. 1130, Baghdad – c. 1180, Maragha)
Al-Hassār (ca. 1100s, Maghreb): Developed the modern mathematical notation for fractions and the digits he uses for the ghubar numerals also cloesly resembles modern Western Arabic numerals.
Ibn al-Yāsamīn (ca. 1100s, Maghreb): The son of a Berber father and black African mother, he was the first to develop a mathematical notation for algebra since the time of Brahmagupta.
Sharaf al-Dīn al-Ṭūsī (Iran, ca. 1150-1215)
Ibn Mun`im (Maghreb, ca. 1210)
al-Marrakushi (Morocco, 13th century)
Naṣīr al-Dīn al-Ṭūsī (18 February 1201 in Tus, Khorasan – 26 June 1274 in Kadhimain near Baghdad)
Muḥyi al-Dīn al-Maghribī (c. 1220 Spain – c. 1283 Maragha)
Shams al-Dīn al-Samarqandī (c. 1250 Samarqand – c. 1310)
Ibn Baso (Spain, ca. 1250-1320)
Ibn al-Banna' (Maghreb, ca. 1300)
Kamal al-Din Al-Farisi (Iran, ca. 1300)
Al-Khalili (Syria, ca. 1350-1400)
Ibn al-Shatir (1306-1375)
Qāḍī Zāda al-Rūmī (1364 Bursa – 1436 Samarkand)
Jamshīd al-Kāshī (Iran, Uzbekistan, ca. 1420)
Ulugh Beg (Iran, Uzbekistan, 1394-1449)
Al-Umawi
Abū al-Hasan ibn Alī al-Qalasādī (Maghreb, 1412-1482): Last major medieval Arab mathematician. Pioneer of symbolic algebra.

Algebra

A page from the Al-jabr wa'l muqabalah by Al-Khwarizmi.

The term algebra is derived from the Arabic term al-jabr in the title of Al-Khwarizmi's Al-jabr wa'l muqabalah. He originally used the term al-jabr to describe the method of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation.^[11]

There are three theories about the origins of Arabic algebra. The first emphasizes Hindu influence, the second emphasizes Mesopotamian or Persian-Syriac influence, and the third emphasizes Greek influence. Many scholars believe that it is the result of a combination of all three sources.^[12]

Throughout their time in power, before the fall of Islamic civilization, the Arabs used a fully rhetorical algebra, where sometimes even the numbers were spelled out in words. The Arabs would eventually replace spelled out numbers (eg. twenty-two) with Arabic numerals (eg. 22), but the Arabs never adopted or developed a syncopated or symbolic algebra,^[5] until the work of Ibn al-Banna al-Marrakushi in the 13th century and Abū al-Hasan ibn Alī al-Qalasādī in the 15th century.^[13]

There were four conceptual stages in the development of algebra, three of which either began in, or were significantly advanced in, the Islamic world. These four stages were as follows:^[14]

Geometric stage, where the concepts of algebra are largely geometric. This dates back to the Babylonians and continued with the Greeks, and was revived by Omar Khayyam.
Static equation-solving stage, where the objective is to find numbers satisfying certain relationships. The move away from geometric algebra dates back to Diophantus and Brahmagupta, but algebra didn't decisively move to the static equation-solving stage until Al-Khwarizmi's Al-Jabr.
Dynamic function stage, where motion is an underlying idea. The idea of a function began emerging with Sharaf al-Dīn al-Tūsī, but algebra didn't decisively move to the dynamic function stage until Gottfried Leibniz.
Abstract stage, where mathematical structure plays a central role. Abstract algebra is largely a product of the 19th and 20th centuries.

Static equation-solving algebra

Al-Khwarizmi and Al-jabr wa'l muqabalah

The Muslim^[15] Persian mathematician Muhammad ibn Mūsā al-Khwārizmī (c. 780-850) was a faculty member of the "House of Wisdom" (Bait al-hikma) in Baghdad, which was established by Al-Mamun. Al-Khwarizmi, who died around 850 A.D., wrote more than half a dozen mathematical and astronomical works; some of which were based on the Indian Sindhind.^[4] One of al-Khwarizmi's most famous books is entitled Al-jabr wa'l muqabalah or The Compendious Book on Calculation by Completion and Balancing, and it gives an exhaustive account of solving polynomials up to the second degree.^[16] The book also introduced the fundamental method of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which Al-Khwarizmi originally described as al-jabr.^[11]

Al-Jabr is divided into six chapters, each of which deals with a different type of formula. The first chapter of Al-Jabr deals with equations whose squares equal its roots (ax² = bx), the second chapter deals with squares equal to number (ax² = c), the third chapter deals with roots equal to a number (bx = c), the fourth chapter deals with squares and roots equal a number (ax² + bx = c), the fifth chapter deals with squares and number equal roots (ax² + c = bx), and the sixth and final chapter deals with roots and number equal to squares (bx + c = ax²).^[17]

J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive:

"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 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."

The Hellenistic mathematician Diophantus was traditionally known as "the father of algebra"^[18]^[19] but debate now exists as to whether or not Al-Khwarizmi deserves this title instead.^[18] Those who support Diophantus point to the fact that the algebra found in Al-Jabr is more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical.^[18] Those who support Al-Khwarizmi point to the fact that he gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,^[20] was the first to teach algebra in an elementary form and for its own sake, whereas Diophantus was primarily concerned with the theory of numbers.^[21] R. Rashed and Angela Armstrong write:

"Al-Khwarizmi's text can be seen to be distinct not only from the Babylonian tablets, but also from Diophantus' Arithmetica. It no longer concerns a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."^[22]

Logical Necessities in Mixed Equations

'Abd al-Hamīd ibn Turk (fl. 830) authored a manuscript entitled Logical Necessities in Mixed Equations, which is very similar to al-Khwarzimi's Al-Jabr and was published at around the same time as, or even possibly earlier than, Al-Jabr.^[23] The manuscript gives the exact same geometric demonstration as is found in Al-Jabr, and in one case the same example as found in Al-Jabr, and even goes beyond Al-Jabr by giving a geometric proof that if the determinant is negative then the quadratic equation has no solution.^[23] The similarity between these two works has led some historians to conclude that Arabic algebra may have been well developed by the time of al-Khwarizmi and 'Abd al-Hamid.^[23]

Abū Kāmil and al-Karkhi

Arabic mathematicians were also the first to treat irrational numbers as algebraic objects.^[24] The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850-930) was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation.^[25]

Al-Karkhi (953-1029), also known as Al-Karaji, was the successor of Abū al-Wafā' al-Būzjānī (940-998) and he was the first to discover the solution to equations of the form ax²ⁿ + bxⁿ = c.^[26] Al-Karkhi only considered positive roots.^[26] Al-Karkhi is also regarded as the first person to free algebra from geometrical operations and replace them with the type of arithmetic operations which are at the core of algebra today. His work on algebra and polynomials, gave the rules for arithmetic operations to manipulate polynomials. The historian of mathematics F. Woepcke, in Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi (Paris, 1853), praised Al-Karaji for being "the first who introduced the theory of algebraic calculus". Stemming from this, Al-Karaji investigated binomial coefficients and Pascal's triangle.^[27]

Arabic manuscript from the 12th century depicting the Brethren of Purity.

Linear algebra

In linear algebra and recreational mathematics, magic squares were known to Arab mathematicians, possibly as early as the 7th century, when the Arabs got into contact with Indian or South Asian culture, and learned Indian mathematics and astronomy, including other aspects of combinatorial mathematics. It has also been suggested that the idea came via China. The first magic squares of order 5 and 6 appear in an encyclopedia from Baghdad circa 983 AD, the Rasa'il Ihkwan al-Safa (Encyclopedia of the Brethren of Purity); simpler magic squares were known to several earlier Arab mathematicians.^[28]

The Arab mathematician Ahmad al-Buni, who worked on magic squares around 1200 AD, attributed mystical properties to them, although no details of these supposed properties are known. There are also references to the use of magic squares in astrological calculations, a practice that seems to have originated with the Arabs.^[28]

Geometric algebra

Omar Khayyám (c. 1050-1123) wrote a book on Algebra that went beyond Al-Jabr to include equations of the third degree.^[29] Omar Khayyám provided both arithmetic and geometric solutions for quadratic equations, but he only gave geometric solutions for general cubic equations since he mistakenly believed that arithmetic solutions were impossible.^[29] His method of solving cubic equations by using intersecting conics had been used by Menaechmus, Archimedes, and Alhazen, but Omar Khayyám generalized the method to cover all cubic equations with positive roots.^[29] He only considered positive roots and he did not go past the third degree.^[29] He also saw a strong relationship between Geometry and Algebra.^[29]

Dynamic functional algebra

In the 12th century, Sharaf al-Dīn al-Tūsī found algebraic and numerical solutions to cubic equations and was the first to discover the derivative of cubic polynomials.^[30] His Treatise on Equations dealt with equations up to the third degree. The treatise does not follow Al-Karaji's school of algebra, but instead represents "an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry." The treatise dealt with 25 types of equations, including twelve types of linear equations and quadratic equations, eight types of cubic equations with positive solutions, and five types of cubic equations which may not have positive solutions.^[31] He understood the importance of the discriminant of the cubic equation and used an early version of Cardano's formula^[32] to find algebraic solutions to certain types of cubic equations.^[30]

Sharaf al-Din also developed the concept of a function. In his analysis of the equation $\ x^3 + d = bx^2$ for example, he begins by changing the equation's form to $\ x^2 (b - x) = d$ . He then states that the question of whether the equation has a solution depends on whether or not the “function” on the left side reaches the value $\ d$ . To determine this, he finds a maximum value for the function. He proves that the maximum value occurs when $x = \frac{2b}{3}$ , which gives the functional value $\frac{4b^3}{27}$ . Sharaf al-Din then states that if this value is less than $\ d$ , there are no positive solutions; if it is equal to $\ d$ , then there is one solution at $x = \frac{2b}{3}$ ; and if it is greater than $\ d$ , then there are two solutions, one between $\ 0$ and $\frac{2b}{3}$ and one between $\frac{2b}{3}$ and $\ b$ . This was the earliest form of dynamic functional algebra.^[33]

A stamp issued 1979 in Iran commemorating Jamshīd al-Kāshī.

Numerical analysis

In numerical analysis, the essence of Viète's method was known to Sharaf al-Dīn al-Tūsī in the 12th century, and it is possible that the algebraic tradition of Sharaf al-Dīn, as well as his predecessor Omar Khayyám and successor Jamshīd al-Kāshī, was known to 16th century European algebraists, or whom François Viète was the most important.^[34]

A method algebraically equivalent to Newton's method was also known to Sharaf al-Dīn. In the 15th century, his successor al-Kashi later used a form of Newton's method to numerically solve $\ x^P - N = 0$ to find roots of $\ N$ . In western Europe, a similar method was later described by Henry Biggs in his Trigonometria Britannica, published in 1633.^[35]

Symbolic algebra

Abū al-Hasan ibn Alī al-Qalasādī (1412-1482) was the last major medieval Arab algebraist, who improved on the algebraic notation earlier used in the Maghreb by Ibn al-Banna in the 13th century^[13] and by Ibn al-Yāsamīn in the 12th century.^[36] In contrast to the syncopated notations of their predecessors, Diophantus and Brahmagupta, which lacked symbols for mathematical operations,^[37] al-Qalasadi's algebraic notation was the first to have symbols for these functions and was thus "the first steps toward the introduction of algebraic symbolism." He represented mathematical symbols using characters from the Arabic alphabet.^[13]

The symbol $x$ now commonly denotes an unknown variable. Even though any letter can be used, $x$ is the most common choice. This usage can be traced back to the Arabic word šay' شيء = “thing,” used in Arabic algebra texts such as the Al-Jabr, and was taken into Old Spanish with the pronunciation “šei,” which was written xei, and was soon habitually abbreviated to $x$ . (The Spanish pronunciation of “x” has changed since). Some sources say that this $x$ is an abbreviation of Latin causa, which was a translation of Arabic شيء. This started the habit of using letters to represent quantities in algebra. In mathematics, an “italicized x” ( $x\!$ ) is often used to avoid potential confusion with the multiplication symbol.

Geometry

An engraving by Albrecht Dürer featuring Mashallah, from the title page of the De scientia motus orbis (Latin version with engraving, 1504). As in many medieval illustrations, the compass here is an icon of religion as well as science, in reference to God as the architect of creation.

The successors of Muhammad ibn Mūsā al-Khwārizmī (born 780) 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.

Al-Mahani (born 820) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Al-Karaji (born 953) completely freed algebra from geometrical operations and replaced them with the arithmetical type of operations which are at the core of algebra today.

Early Islamic geometry

See also Applied mathematics

Thabit ibn Qurra (known as Thebit in Latin) (born 836) contributed to a number of areas in mathematics, where he played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-Euclidean geometry. An important geometrical aspect of Thabit's work was his book on the composition of ratios. In this book, Thabit deals with arithmetical operations applied to ratios of geometrical quantities. The Greeks had dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of arithmetic could be applied. By introducing arithmetical operations on quantities previously regarded as geometric and non-numerical, Thabit started a trend which led eventually to the generalization of the number concept. Another important contribution Thabit made to geometry was his generalization of the Pythagorean theorem, which he extended from special right triangles to all right triangles in general, along with a general proof.^[38]

In some respects, Thabit is critical of the ideas of Plato and Aristotle, particularly regarding motion. It would seem that here his ideas are based on an acceptance of using arguments concerning motion in his geometrical arguments. Another important contribution Thabit made to geometry was his generalization of the Pythagorean theorem, which he extended from special right triangles to all triangles in general, along with a general proof.^[38]

Ibrahim ibn Sinan ibn Thabit (born 908), who introduced a method of integration more general than that of Archimedes, and al-Quhi (born 940) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. These mathematicians, and in particular Ibn al-Haytham (Alhazen), studied optics and investigated the optical properties of mirrors made from conic sections (see Mathematical physics).

Astronomy, time-keeping and geography provided other motivations for geometrical and trigonometrical research. For example Ibrahim ibn Sinan and his grandfather Thabit ibn Qurra both studied curves required in the construction of sundials. Abu'l-Wafa and Abu Nasr Mansur pioneered spherical geometry in order to solve difficult problems in Islamic astronomy. For example, to predict the first visibility of the moon, it was necessary to describe its motion with respect to the horizon, and this problem demands fairly sophisticated spherical geometry. Finding the direction of Mecca (Qibla) and the time for Salah prayers and Ramadan are what led to Muslims developing spherical geometry.^[39]^[40]

Trigonometry

The early Indian works on trigonometry were translated and expanded in the Muslim world by Arab and Persian mathematicians. They enunciated a large number of theorems which freed the subject of trigonometry from dependence upon the complete quadrilateral, as was the case in Hellenistic mathematics due to the application of Menelaus' theorem. According to E. S. Kennedy, it was after this development in Islamic mathematics that "the first real trigonometry emerged, in the sense that only then did the object of study become the spherical or plane triangle, its sides and angles."^[41]

In the 9th century, Muhammad ibn Mūsā al-Khwārizmī (c. 780-850) produced tables of sines and tangents, and also developed spherical trigonometry. In circa 860, Habash al-Hasib al-Marwazi produced the first tables of cotangents as well as tangents. Muhammad ibn Jābir al-Harrānī al-Battānī (853-929) discovered the inverse trigonometric functions secant and cosecant, and produced the first table of cosecants, which he referred to as a "table of shadows" (in reference to the shadow of a gnomon), for each degree from 1° to 90°.^[42] He also formulated a number of important trigonometrical relationships such as:

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

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

By the 10th century, in the work of Abū al-Wafā' al-Būzjānī (959-998), Muslim mathematicians were using all six trigonometric functions, and had sine tables in 0.25° increments, to 8 decimal places of accuracy, as well as tables of tangent values. Abū al-Wafā' also developed the following trigonometric formula:

$\sin 2x = 2 \sin x \cos x \$

Abū al-Wafā also established the angle addition identities, e.g. sin (a + b), and discovered the sine formula for spherical trigonometry:

$\frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}.$

Also in the late 10th and early 11th centuries, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the following formula:

$\cos a \cos b = \frac{\cos(a+b) + \cos(a-b)}{2}$

Al-Jayyani (989–1079) of al-Andalus wrote The book of unknown arcs of a sphere, which is considered "the first treatise on spherical trigonometry" in its modern form,^[43] although spherical trigonometry in its ancient Hellenistic form was dealt with by earlier mathematicians such as Menelaus of Alexandria, who developed Menelaus' theorem to deal with spherical problems.^[44]^[45] However, E. S. Kennedy points out that while it was possible in pre-lslamic mathematics to compute the magnitudes of a spherical figure, in principle, by use of the table of chords and Menelaus' theorem, the application of the theorem to spherical problems was very difficult in practice.^[46] Al-Jayyani's work on spherical trigonometry "contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle." This treatise later had a "strong influence on European mathematics", and his "definition of ratios as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced Regiomontanus.^[43]

The method of triangulation, which was unknown in the Greco-Roman world, was also first developed by Muslim mathematicians, who applied it to practical uses such as surveying^[47] and Islamic geography, as described by Abū Rayhān al-Bīrūnī in the early 11th century.^[48] In the late 11th century, Omar Khayyám (1048-1131) solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables. All of these earlier works on trigonometry treated it mainly as an adjunct to astronomy; the first treatment as a subject in its own right was by Nasīr al-Dīn al-Tūsī in the 13th century. He also developed spherical trigonometry into its present form,^[42] and listed the six distinct cases of a right-angled triangle in spherical trigonometry. He also stated the law of sines and provided a proof for it.

Jamshīd al-Kāshī (1393-1449) gives trigonometric tables of values of the sine function to four sexagesimal digits (equivalent to 8 decimal places) for each 1° of argument with differences to be added for each 1/60 of 1°.^[49] In one of his numerical approximations of π, he correctly computed 2π to 9 sexagesimal digits.^[50] In order to determine sin 1°, al-Kashi discovered the following triple-angle formula often attributed to François Viète in the 16th century:^[51]

$\sin 3 \phi = 3 \sin \phi - 4 \sin^3 \phi\,.$

In French, the law of cosines is named Théorème d'Al-Kashi (Theorem of Al-Kashi), as al-Kashi was the first to provide an explicit statement of the law of cosines in a form suitable for triangulation. His colleague Ulugh Beg (1394-1449) gave accurate tables of sines and tangents correct to 8 decimal places.

Taqi al-Din (1526-1585) contributed to trigonometry in his Sidrat al-Muntaha, in which he was the first mathematician to compute a highly accurate numeric value for sin 1°. He discusses the values given by his predecessors, explaining how Ptolemy (ca. 150) used an approximate method to obtain his value of sin 1° and how Abū al-Wafā, Ibn Yunus (ca. 1000), al-Kashi, Qāḍī Zāda al-Rūmī (1337-1412), Ulugh Beg and Mirim Chelebi improved on the value. Taqi al-Din then solves the problem to obtain the value of sin 1° to a precision of 8 sexagesimals (the equivalent of 14 decimals):^[52]

$\sin 1^\circ = 1^P 2' 49'' 43''' 11'''' 14''''' 44''''''16''''''' \ (= 1/60 + 2/60^2 + 49/60^3 + \cdots)\,.$

Algebraic and analytic geometry

Statue of Omar Khayyám in Iran.

In the early 11th century, Ibn al-Haytham (Alhazen) was able to solve by purely algebraic means certain cubic equations, and then to interpret the results geometrically.^[53] Subsequently, Omar Khayyám discovered the general method of solving cubic equations by intersecting a parabola with a circle.^[54]

Omar Khayyám (1048-1122) was a Persian mathematician, as well as a poet. Along with his fame as a poet, he was also famous during his lifetime as a mathematician, well known for inventing the general method of solving cubic equations by intersecting a parabola with a circle. In addition he discovered the binomial expansion, and authored criticisms of Euclid's theories of parallels which made their way to England, where they contributed to the eventual development of non-Euclidean geometry. Omar Khayyam also combined the use of trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means. His work marked the beginnings of algebraic geometry^[55]^[56] and analytic geometry.^[57]

In a paper written by Khayyam before his famous algebra text Treatise on Demonstration of Problems of Algebra, he considers the problem: Find a point on a quadrant of a circle in such manner that when a normal is dropped from the point to one of the bounding radii, the ratio of the normal's length to that of the radius equals the ratio of the segments determined by the foot of the normal. Khayyam shows that this problem is equivalent to solving a second problem: Find a right triangle having the property that the hypotenuse equals the sum of one leg plus the altitude on the hypotenuse. This problem in turn led Khayyam to solve the cubic equation x³ + 200x = 20x² + 2000 and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables. Perhaps even more remarkable is the fact that Khayyam states that the solution of this cubic requires the use of conic sections and that it cannot be solved by compass and straightedge, a result which would not be proved for another 750 years.

His Treatise on Demonstration of Problems of Algebra contained a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. In fact Khayyam gives an interesting historical account in which he claims that the Greeks had left nothing on the theory of cubic equations. Indeed, as Khayyam writes, the contributions by earlier writers such as al-Mahani and al-Khazin were to translate geometric problems into algebraic equations (something which was essentially impossible before the work of Muḥammad ibn Mūsā al-Ḵwārizmī). However, Khayyam himself seems to have been the first to conceive a general theory of cubic equations.

Omar Khayyám saw a strong relationship between geometry and algebra, and was moving in the right direction when he helped to close the gap between numerical and geometric algebra^[29] with his geometric solution of the general cubic equations,^[57] but the decisive step in analytic geometry came later with René Descartes.^[29]

Persian mathematician Sharafeddin Tusi (born 1135) 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, entitled Treatise on Equations, which represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the study of algebraic geometry.^[31]

Non-Euclidean geometry

Nasīr al-Dīn al-Tūsī commemorated on an Iranian stamp upon the 700th anniversary of his death.

In the early 11th century, Ibn al-Haytham (Alhazen) made the first attempt at proving the Euclidean parallel postulate, the fifth postulate in Euclid's Elements, using a proof by contradiction,^[58] where he introduced the concept of motion and transformation into geometry.^[59] He formulated the Lambert quadrilateral, which Boris Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert quadrilateral",^[60] and his attempted proof also shows similarities to Playfair's axiom.^[61]

In the late 11th century, Omar Khayyám made the first attempt at formulating a non-Euclidean postulate as an alternative to the Euclidean parallel postulate,^[62] and he was the first to consider the cases of elliptical geometry and hyperbolic geometry, though he excluded the latter.^[63]

In Commentaries on the difficult postulates of Euclid's book Khayyam made a contribution to non-Euclidean geometry, although this was not his intention. In trying to prove the parallel postulate he accidentally proved properties of figures in non-Euclidean geometries. Khayyam also gave important results on ratios in this book, extending Euclid's work to include the multiplication of ratios. The importance of Khayyam's contribution is that he examined both Euclid's definition of equality of ratios (which was that first proposed by Eudoxus) and the definition of equality of ratios as proposed by earlier Islamic mathematicians such as al-Mahani which was based on continued fractions. Khayyam proved that the two definitions are equivalent. He also posed the question of whether a ratio can be regarded as a number but leaves the question unanswered.

In 1250, Nasīr al-Dīn al-Tūsī, in his Al-risala al-shafiya'an al-shakk fi'l-khutut al-mutawaziya (Discussion Which Removes Doubt about Parallel Lines), wrote detailed critiques of the Euclidean parallel postulate and on Omar Khayyám's attempted proof a century earlier. Nasir al-Din attempted to derive a proof by contradiction of the parallel postulate.^[64] He was one of the first to consider the cases of elliptical geometry and hyperbolic geometry, though he ruled out both of them.^[63]

His son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), wrote a book on the subject in 1298, based on al-Tusi's later thoughts, which presented one of the earliest arguments for a non-Euclidean hypothesis equivalent to the parallel postulate.^[64]^[65] Sadr al-Din's work was published in Rome in 1594 and was studied by European geometers. This work marked the starting point for Giovanni Girolamo Saccheri's work on the subject, and eventually the development of modern non-Euclidean geometry.^[64] A proof from Sadr al-Din's work was quoted by John Wallis and Saccheri in the 17th and 18th centuries. They both derived their proofs of the parallel postulate from Sadr al-Din's work, while Saccheri also derived his Saccheri quadrilateral from Sadr al-Din, who himself based it on his father's work.^[66]

The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were the first theorems on elliptical geometry and hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works marked the beginning of non-Euclidean geometry and had a considerable influence on its development among later European geometers, including Witelo, Levi ben Gerson, Alfonso, John Wallis, and Giovanni Girolamo Saccheri.^[67]

Calculus

Ibn al-Haytham (Alhazen), author of the Book of Optics.

Integral calculus

Around 1000 AD, Al-Karaji, using mathematical induction, found a proof for the sum of integral cubes.^[68] The historian of mathematics, F. Woepcke,^[69] praised Al-Karaji for being "the first who introduced the theory of algebraic calculus." Shortly afterwards, Ibn al-Haytham (known as Alhazen in the West), an Iraqi mathematician working in Egypt, was the first mathematician to derive the formula for the sum of the fourth powers, and using an early proof by mathematical induction, he developed a method for determining the general formula for the sum of any integral powers. He used his result on sums of integral powers to perform an integration, in order to find the volume of a paraboloid. He was thus able to find the integrals for polynomials up to the fourth degree, and came close to finding a general formula for the integrals of any polynomials. This was fundamental to the development of infinitesimal and integral calculus. His results were repeated by the Moroccan mathematicians Abu-l-Hasan ibn Haydur (d. 1413) and Abu Abdallah ibn Ghazi (1437-1514), by Jamshīd al-Kāshī (c. 1380-1429) in The Calculator's Key, and by the Indian mathematicians of the Kerala school of astronomy and mathematics in the 15th-16th centuries.^[64]

Differential calculus

In the 12th century, the Persian mathematician Sharaf al-Dīn al-Tūsī was the first to discover the derivative of cubic polynomials, an important result in differential calculus.^[30] His Treatise on Equations developed concepts related to differential calculus, such as the derivative function and the maxima and minima of curves, in order to solve cubic equations which may not have positive solutions. For example, in order to solve the equation $\ x^3 + a = bx$ , al-Tusi finds the maximum point of the curve $\ bx - x^3 = a$ . He uses the derivative of the function to find that the maximum point occurs at $x = \sqrt{\frac{b}{3}}$ , and then finds the maximum value for y at $2(\frac{b}{3})^\frac{3}{2}$ by substituting $x = \sqrt{\frac{b}{3}}$ back into $\ y = bx - x^3$ . He finds that the equation $\ bx - x^3 = a$ has a solution if $a \le 2(\frac{b}{3})^\frac{3}{2}$ , and al-Tusi thus deduces that the equation has a positive root if $D = \frac{b^3}{27} - \frac{a^2}{4} \ge 0$ , where $D$ is the discriminant of the equation.^[31]

Applied mathematics

Geometric art and architecture

Main articles: Arabesque, Girih tiles, Islamic art, and Islamic architecture

Geometric artwork in the form of the Arabesque was not widely used in the Middle East or Mediterranean Basin until the golden age of Islam came into full bloom, when Arabesque became a common feature of Islamic art. Euclidean geometry as expounded on by Al-Abbās ibn Said al-Jawharī (ca. 800-860) in his Commentary on Euclid's Elements, the trigonometry of Aryabhata and Brahmagupta as elaborated on by Muhammad ibn Mūsā al-Khwārizmī (ca. 780-850), and the development of spherical geometry^[39] by Abū al-Wafā' al-Būzjānī (940–998) and spherical trigonometry by Al-Jayyani (989-1079)^[70] for determining the Qibla and times of Salah and Ramadan,^[39] all served as an impetus for the art form that was to become the Arabesque.

Recent discoveries have shown that geometrical quasicrystal patterns were first employed in the girih tiles found in medieval Islamic architecture dating back over five centuries ago. In 2007, Professor Peter Lu of Harvard University and Professor Paul Steinhardt of Princeton University published a paper in the journal Science suggesting that girih tilings possessed properties consistent with self-similar fractal quasicrystalline tilings such as the Penrose tilings, predating them by five centuries.^[71]^[72]

Mathematical astronomy

Main articles: Islamic astronomy and Zij

An impetus behind mathematical astronomy came from Islamic religious observances, which presented a host of problems in mathematical astronomy, particularly in spherical geometry. In solving these religious problems the Islamic scholars went far beyond the Greek mathematical methods.^[39] For example, predicting just when the crescent moon would become visible is a special challenge to Islamic mathematical astronomers. Although Ptolemy's theory of the complex lunar motion was tolerably accurate near the time of the new moon, it specified the moon's path only with respect to the ecliptic. To predict the first visibility of the moon, it was necessary to describe its motion with respect to the horizon, and this problem demands fairly sophisticated spherical geometry. Finding the direction of Mecca and the time of Salah are the reasons which led to Muslims developing spherical geometry. Solving any of these problems involves finding the unknown sides or angles of a triangle on the celestial sphere from the known sides and angles. A way of finding the time of day, for example, is to construct a triangle whose vertices are the zenith, the north celestial pole, and the sun's position. The observer must know the altitude of the sun and that of the pole; the former can be observed, and the latter is equal to the observer's latitude. The time is then given by the angle at the intersection of the meridian (the arc through the zenith and the pole) and the sun's hour circle (the arc through the sun and the pole).^[39]^[40]

The Zij treatises were astronomical books that tabulated the parameters used for astronomical calculations of the positions of the Sun, Moon, stars, and planets. Their principal contributions to mathematical astronomy reflected improved trigonometrical, computational and observational techniques.^[73]^[74] The Zij books were extensive, and typically included materials on chronology, geographical latitudes and longitudes, star tables, trigonometrical functions, functions in spherical astronomy, the equation of time, planetary motions, computation of eclipses, tables for first visibility of the lunar crescent, astronomical and/or astrological computations, and instructions for astronomical calculations using epicyclic geocentric models.^[75] Some zījes go beyond this traditional content to explain or prove the theory or report the observations from which the tables were computed.^[76]

In observational astronomy<

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