In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics, is the mathematics developed in the Islamic world between 622 and 1600, during what is known as the Islamic Golden Age, in that part of the world where Islam was the dominant religion. Islamic science and mathematics flourished under the Islamic caliphate (also known as the Islamic Empire) established across the Middle East, Central Asia, North Africa, Southern Italy, the Iberian Peninsula, and, at its peak, parts of France and India as well. Islamic activity in mathematics was largely centered around modernday Iraq and Persia, but at its greatest extent stretched from North Africa and Spain in the west to India in the east.^{[1]}
While most scientists in this period were Muslims and wrote in Arabic,^{[2]} many of the best known contributors were Persians^{[3]}^{[4]} as well as Arabs,^{[4]} in addition to Berber, Moorish and Turkic contributors, as well as some from other religions (Christians, Jews, Sabians, Zoroastrians, and the irreligious).^{[2]} Arabic was the dominant language—much like Latin in Medieval Europe, Arabic was the written lingua franca of most scholars throughout the Islamic world.
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Bernard Lewis writes the following on the historical usage of the term "Islam" in What Went Wrong? Western Impact and Middle Eastern Response:^{[5]}
"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 PreColumbian 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" and the adjective "Islamic" is used in the meaning described above; that is, of a civilization.
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.^{[6]} The Muslim Abbasid caliph alMamun (809833) is said to have had a dream where Aristotle appeared to him, and as a consequence alMamun 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.^{[6]} Many of these Greek works were translated by Thabit ibn Qurra (826901), who translated books written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.^{[7]} 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 placevalue IndoArabic numeral system, also known as the Hindu numerals.^{[8]} The Persian historian alBiruni (c. 1050) in his book Tariq alHind states that alMa'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 Brahmasphutasiddhanta.^{[9]} The earliest translations from Sanskrit inspired several astronomical and astrological Arabic works, now mostly lost, some of which were even composed in verse.^{[10]} Biruni described Indian mathematics as a "mix of common pebbles and costly crystals".^{[11]}
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 "GraecoIslamic" science founded on Hellenistic treatises.^{[10]} Another likely reason for the declining Indian influence in later periods was due to Sindh achieving independence from the Caliphate, thus limiting access to Indian works. Nevertheless, Indian methods continued to play an important role in algebra, arithmetic and trigonometry.^{[12]}
Besides the Greek and Indian tradition, a third tradition which had a significant influence on mathematics in medieval Islam was the "mathematics of practitioners", which included the applied mathematics of "surveyors, builders, artisans, in geometric design, tax and treasury officials, and some merchants." This applied form of mathematics transcended ethnic divisions and was a common heritage of the lands incorporated into the Islamic world.^{[8]} This tradition also includes the religious observances specific to Islam, which served as a major impetus for the development of mathematics as well as astronomy.^{[13]}
A major impetus for the flowering of mathematics as well as astronomy in medieval Islam came from religious observances, which presented an assortment of problems in astronomy and mathematics, specifically in trigonometry, spherical geometry,^{[13]} algebra^{[14]} and arithmetic.^{[15]}
The Islamic law of inheritance served as an impetus behind the development of algebra (derived from the Arabic aljabr) by Muhammad ibn Mūsā alKhwārizmī and other medieval Islamic mathematicians. AlKhwārizmī's Hisab aljabr w’almuqabala devoted a chapter on the solution to the Islamic law of inheritance using algebra. He formulated the rules of inheritance as linear equations, hence his knowledge of quadratic equations was not required.^{[14]} Later mathematicians who specialized in the Islamic law of inheritance included AlHassār, who developed the modern symbolic mathematical notation for fractions in the 12th century,^{[15]} and Abū alHasan ibn Alī alQalasādī, who developed an algebraic notation which took "the first steps toward the introduction of algebraic symbolism" in the 15th century.^{[16]}
In order to observe holy days on the Islamic calendar in which timings were determined by phases of the moon, astronomers initially used Ptolemy's method to calculate the place of the moon and stars. The method Ptolemy used to solve spherical triangles, however, was a clumsy one devised late in the first century by Menelaus of Alexandria. It involved setting up two intersecting right triangles; by applying Menelaus' theorem it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the sun's altitude, for instance, repeated applications of Menelaus' theorem were required. For medieval Islamic astronomers, there was an obvious challenge to find a simpler trigonometric method.^{[13]}
Regarding the issue of moon sighting, Islamic months do not begin at the astronomical new moon, defined as the time when the moon has the same celestial longitude as the sun and is therefore invisible; instead they begin when the thin crescent moon is first sighted in the western evening sky.^{[13]} The Qur'an says: "They ask you about the waxing and waning phases of the crescent moons, say they are to mark fixed times for mankind and Hajj."^{[17]}^{[18]} This led Muslims to find the phases of the moon in the sky, and their efforts led to new mathematical calculations.^{[19]}
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).^{[13]}^{[20]}
Muslims are also expected to pray towards the Kaaba in Mecca and orient their mosques in that direction. Thus they need to determine the direction of Mecca (Qibla) from a given location.^{[21]}^{[22]} Another problem is the time of Salah. Muslims need to determine from celestial bodies the proper times for the prayers at sunrise, at midday, in the afternoon, at sunset, and in the evening.^{[13]}^{[20]}
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:
"AlKhwarizmi'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."
The term algebra is derived from the Arabic term aljabr in the title of AlKhwarizmi's Aljabr wa'l muqabalah. He originally used the term aljabr 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.^{[24]}
There are three theories about the origins of Islamic algebra. The first emphasizes Hindu influence, the second emphasizes Mesopotamian or PersianSyriac influence, and the third emphasizes Greek influence. Many scholars believe that it is the result of a combination of all three sources.^{[25]}
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. twentytwo) with Arabic numerals (eg. 22), but the Arabs never adopted or developed a syncopated or symbolic algebra,^{[7]} until the work of Ibn alBanna alMarrakushi in the 13th century and Abū alHasan ibn Alī alQalasādī in the 15th century.^{[16]}
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:^{[26]}
The Muslim^{[27]} Persian mathematician Muhammad ibn Mūsā alKhwārizmī (c. 780850) was a faculty member of the "House of Wisdom" (Bait alhikma) in Baghdad, which was established by AlMamun. AlKhwarizmi, 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.^{[6]} One of alKhwarizmi's most famous books is entitled Aljabr 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.^{[28]} 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 AlKhwarizmi originally described as aljabr.^{[24]}
AlJabr is divided into six chapters, each of which deals with a different type of formula. The first chapter of AlJabr 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).^{[29]}
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 alKhwarizmi, 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"^{[30]}^{[31]} but debate now exists as to whether or not AlKhwarizmi deserves this title instead.^{[30]} Those who support Diophantus point to the fact that the algebra found in AlJabr is more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while AlJabr is fully rhetorical.^{[30]} Those who support AlKhwarizmi point to the fact that he gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,^{[32]} introduced the fundamental methods of reduction and balancing,^{[24]} and 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.^{[33]} In addition, R. Rashed and Angela Armstrong write:
"AlKhwarizmi'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."^{[34]}
'Abd alHamīd ibn Turk (fl. 830) authored a manuscript entitled Logical Necessities in Mixed Equations, which is very similar to alKhwarzimi's AlJabr and was published at around the same time as, or even possibly earlier than, AlJabr.^{[35]} The manuscript gives the exact same geometric demonstration as is found in AlJabr, and in one case the same example as found in AlJabr, and even goes beyond AlJabr by giving a geometric proof that if the determinant is negative then the quadratic equation has no solution.^{[35]} The similarity between these two works has led some historians to conclude that Islamic algebra may have been well developed by the time of alKhwarizmi and 'Abd alHamid.^{[35]}
Arabic mathematicians were also the first to treat irrational numbers as algebraic objects.^{[36]} The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850930) 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.^{[37]} He was also the first to solve three nonlinear simultaneous equations with three unknown variables.^{[38]}
AlKarkhi (9531029), also known as AlKaraji, was the successor of Abū alWafā' alBūzjānī (940998) and he was the first to discover the solution to equations of the form ax^{2n} + bx^{n} = c.^{[39]} AlKarkhi only considered positive roots.^{[39]} AlKarkhi 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 AlKaraji for being "the first who introduced the theory of algebraic calculus". Stemming from this, AlKaraji investigated binomial coefficients and Pascal's triangle.^{[40]}
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 alSafa (Encyclopedia of the Brethren of Purity); simpler magic squares were known to several earlier Arab mathematicians.^{[41]}
The Arab mathematician Ahmad alBuni, 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.^{[41]}
Omar Khayyám (c. 10501123) wrote a book on Algebra that went beyond AlJabr to include equations of the third degree.^{[42]} 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.^{[42]} 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.^{[42]} He only considered positive roots and he did not go past the third degree.^{[42]} He also saw a strong relationship between Geometry and Algebra.^{[42]}
In the 12th century, Sharaf alDīn alTūsī found algebraic and numerical solutions to cubic equations and was the first to discover the derivative of cubic polynomials.^{[43]} His Treatise on Equations dealt with equations up to the third degree. The treatise does not follow AlKaraji'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.^{[44]} He understood the importance of the discriminant of the cubic equation and used an early version of Cardano's formula^{[45]} to find algebraic solutions to certain types of cubic equations.^{[43]}
Sharaf alDin also developed the concept of a function. In his analysis of the equation for example, he begins by changing the equation's form to . 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 . To determine this, he finds a maximum value for the function. He proves that the maximum value occurs when , which gives the functional value . Sharaf alDin then states that if this value is less than , there are no positive solutions; if it is equal to , then there is one solution at ; and if it is greater than , then there are two solutions, one between and and one between and . This was the earliest form of dynamic functional algebra.^{[46]}
In numerical analysis, the essence of Viète's method was known to Sharaf alDīn alTūsī in the 12th century, and it is possible that the algebraic tradition of Sharaf alDīn, as well as his predecessor Omar Khayyám and successor Jamshīd alKāshī, was known to 16th century European algebraists, or whom François Viète was the most important.^{[47]}
A method algebraically equivalent to Newton's method was also known to Sharaf alDīn. In the 15th century, his successor alKashi later used a form of Newton's method to numerically solve to find roots of . In western Europe, a similar method was later described by Henry Biggs in his Trigonometria Britannica, published in 1633.^{[48]}
AlHassār, a mathematician from the Maghreb (North Africa) specializing in Islamic inheritance jurisprudence during the 12th century, developed the modern symbolic mathematical notation for fractions, where the numerator and denominator are separated by a horizontal bar. This same fractional notation appeared soon after in the work of Fibonacci in the 13th century.^{[15]}
Abū alHasan ibn Alī alQalasādī (1412–1482) was the last major medieval Arab algebraist, who improved on the algebraic notation earlier used in the Maghreb by Ibn alBanna in the 13th century^{[16]} and by Ibn alYāsamīn in the 12th century.^{[15]} In contrast to the syncopated notations of their predecessors, Diophantus and Brahmagupta, which lacked symbols for mathematical operations,^{[49]} alQalasadi'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.^{[16]}
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 AlJabr, 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” () is often used to avoid potential confusion with the multiplication symbol.
The Indian numeral system came to be known to both the Persian mathematician AlKhwarizmi, whose book On the Calculation with Hindu Numerals written circa 825, and the Arab mathematician AlKindi, who wrote four volumes, On the Use of the Indian Numerals (Ketab fi Isti'mal al'Adad alHindi) circa 830, are principally responsible for the diffusion of the Indian system of numeration in the MiddleEast and the West [3]. In the 10th century, MiddleEastern mathematicians extended the decimal numeral system to include fractions using decimal point notation, as recorded in a treatise by Syrian mathematician Abu'lHasan alUqlidisi in 952953.
In the Arab world—until early modern times—the Arabic numeral system was often only used by mathematicians. Muslim astronomers mostly used the Babylonian numeral system, and merchants mostly used the Abjad numerals. A distinctive "Western Arabic" variant of the symbols begins to emerge in ca. the 10th century in the Maghreb and AlAndalus, called the ghubar ("sandtable" or "dusttable") numerals, which is the direct ancestor to the modern Western Arabic numerals now used throughout the world.^{[50]}
The first mentions of the numerals in the West are found in the Codex Vigilanus of 976 [4]. From the 980s, Gerbert of Aurillac (later, Pope Silvester II) began to spread knowledge of the numerals in Europe. Gerbert studied in Barcelona in his youth, and he is known to have requested mathematical treatises concerning the astrolabe from Lupitus of Barcelona after he had returned to France.
AlKhwārizmī, the Persian scientist, wrote in 825 a treatise On the Calculation with Hindu Numerals, which was translated into Latin in the 12th century, as Algoritmi de numero Indorum, where "Algoritmi", the translator's rendition of the author's name gave rise to the word algorithm (Latin algorithmus) with a meaning "calculation method".
AlHassār, a mathematician from the Maghreb (North Africa) specializing in Islamic inheritance jurisprudence during the 12th century, developed the modern symbolic mathematical notation for fractions, where the numerator and denominator are separated by a horizontal bar. The "dust ciphers he used are also nearly identical to the digits used in the current Western Arabic numerals. These same digits and fractional notation appear soon after in the work of Fibonacci in the 13th century.^{[15]}
In discussing the origins of decimal fractions, Dirk Jan Struik states that (p. 7):^{[51]}
"The introduction of decimal fractions as a common computational practice can be dated back to the Flemish pamphelet De Thiende, published at Leyden in 1585, together with a French translation, La Disme, by the Flemish mathematician Simon Stevin (15481620), then settled in the Northern Netherlands. It is true that decimal fractions were used by the Chinese many centuries before Stevin and that the Persian astronomer AlKāshī used both decimal and sexagesimal fractions with great ease in his Key to arithmetic (Samarkand, early fifteenth century).^{[52]}"
While the Persian mathematician Jamshīd alKāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggrenn notes that he was mistaken, as decimal fractions were first used five centuries before him by the Baghdadi mathematician Abu'lHasan alUqlidisi as early as the 10th century.^{[38]}
The Middle Ages saw the acceptance of zero, negative, integral and fractional numbers, first by Indian mathematicians and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects,^{[36]} which was made possible by the development of algebra. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers, and they criticized Euclid's idea of ratios, developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude.^{[53]} In his commentary on Book 10 of the Elements, the Persian mathematician AlMahani (d. 874/884) examined and classified quadratic irrationals and cubic irrationals. He provided definitions for rational and irrational magnitudes, which he treated as irrational numbers. He dealt with them freely but explains them in geometric terms as follows:^{[54]}
"It will be a rational (magnitude) when we, for instance, say 10, 12, 3%, 6%, etc., because its value is pronounced and expressed quantitatively. What is not rational is irrational and it is impossible to pronounce and represent its value quantitatively. For example: the roots of numbers such as 10, 15, 20 which are not squares, the sides of numbers which are not cubes etc."
In contrast to Euclid's concept of magnitudes as lines, AlMahani considered integers and fractions as rational magnitudes, and square roots and cube roots as irrational magnitudes. He also introduced an arithmetical approach to the concept of irrationality, as he attributes the following to irrational magnitudes:^{[54]}
"their sums or differences, or results of their addition to a rational magnitude, or results of subtracting a magnitude of this kind from an irrational one, or of a rational magnitude from it."
The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form of square roots, cube roots and fourth roots.^{[37]} In the 10th century, the Iraqi mathematician AlHashimi provided general proofs (rather than geometric demonstrations) for irrational numbers, as he considered multiplication, division, and other arithmetical functions.^{[55]} Abū Ja'far alKhāzin (900971) provides a definition of rational and irrational magnitudes, stating that if a definite quantity is:^{[56]}
"contained in a certain given magnitude once or many times, then this (given) magnitude corresponds to a rational number. . . . Each time when this (latter) magnitude comprises a half, or a third, or a quarter of the given magnitude (of the unit), or, compared with (the unit), comprises three, five, or three fifths, it is a rational magnitude. And, in general, each magnitude that corresponds to this magnitude (i.e. to the unit), as one number to another, is rational. If, however, a magnitude cannot be represented as a multiple, a part (l/n), or parts (m/n) of a given magnitude, it is irrational, i.e. it cannot be expressed other than by means of roots."
Many of these concepts were eventually accepted by European mathematicians some time after the Latin translations of the 12th century. AlHassār, an Arabic mathematician from the Maghreb (North Africa) specializing in Islamic inheritance jurisprudence during the 12th century, developed the modern symbolic mathematical notation for fractions, where the numerator and denominator are separated by a horizontal bar. This same fractional notation appears soon after in the work of Fibonacci in the 13th century.^{[15]}
In number theory, Ibn alHaytham solved problems involving congruences using what is now called Wilson's theorem. In his Opuscula, Ibn alHaytham considers the solution of a system of congruences, and gives two general methods of solution. His first method, the canonical method, involved Wilson's theorem, while his second method involved a version of the Chinese remainder theorem. Another contribution to number theory is his work on perfect numbers. In his Analysis and synthesis, Ibn alHaytham was the first to discover that every even perfect number is of the form 2^{n−1}(2^{n} − 1) where 2^{n} − 1 is prime, but he was not able to prove this result successfully (Euler later proved it in the 18th century).^{[57]}
In the early 14th century, Kamāl alDīn alFārisī made a number of important contributions to number theory. His most impressive work in number theory is on amicable numbers. In Tadhkira alahbab fi bayan altahabb ("Memorandum for friends on the proof of amicability") introduced a major new approach to a whole area of number theory, introducing ideas concerning factorization and combinatorial methods. In fact, alFarisi's approach is based on the unique factorization of an integer into powers of prime numbers.
The successors of Muhammad ibn Mūsā alKhwārizmī (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.
AlMahani (born 820) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. AlKaraji (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.
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 nonEuclidean 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 nonnumerical, 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.^{[58]}
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.
Ibrahim ibn Sinan ibn Thabit (born 908), who introduced a method of integration more general than that of Archimedes, and alQuhi (born 940) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. These mathematicians, and in particular Ibn alHaytham (Alhazen), studied optics and investigated the optical properties of mirrors made from conic sections (see Mathematical physics).
Astronomy, timekeeping 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'lWafa 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.^{[13]}^{[20]}
In the early 11th century, Ibn alHaytham (Alhazen) was able to solve by purely algebraic means certain cubic equations, and then to interpret the results geometrically.^{[59]} Subsequently, Omar Khayyám discovered the general method of solving cubic equations by intersecting a parabola with a circle.^{[60]}
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 nonEuclidean 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^{[36]}^{[61]} and analytic geometry.^{[62]}
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 x^{3} + 200x = 20x^{2} + 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 alMahani and alKhazin were to translate geometric problems into algebraic equations (something which was essentially impossible before the work of Muḥammad ibn Mūsā alḴwārizmī). 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^{[42]} with his geometric solution of the general cubic equations,^{[62]} but the decisive step in analytic geometry came later with René Descartes.^{[42]}
Persian mathematician Sharafeddin Tusi (born 1135) did not follow the general development that came through alKaraji'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.^{[44]}
In the early 11th century, Ibn alHaytham (Alhazen) made the first attempt at proving the Euclidean parallel postulate, the fifth postulate in Euclid's Elements, using a proof by contradiction,^{[63]} where he introduced the concept of motion and transformation into geometry.^{[64]} He formulated the Lambert quadrilateral, which Boris Abramovich Rozenfeld names the "Ibn alHaytham–Lambert quadrilateral",^{[65]} and his attempted proof also shows similarities to Playfair's axiom.^{[66]}
In the late 11th century, Omar Khayyám made the first attempt at formulating a nonEuclidean postulate as an alternative to the Euclidean parallel postulate,^{[67]} and he was the first to consider the cases of elliptical geometry and hyperbolic geometry, though he excluded the latter.^{[68]}
In Commentaries on the difficult postulates of Euclid's book Khayyam made a contribution to nonEuclidean geometry, although this was not his intention. In trying to prove the parallel postulate he accidentally proved properties of figures in nonEuclidean 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 alMahani 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.
The KhayyamSaccheri quadrilateral was first considered by Omar Khayyam in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid.^{[65]} Unlike many commentators on Euclid before and after him (including of course Saccheri), Khayyam was not trying to prove the parallel postulate as such but to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle):
Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he (correctly) refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid. It wasn't until 600 years later that Giordano Vitale made an advance on the understanding of this quadrilateral in his book Euclide restituo (1680, 1686), when he used it to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. Saccheri himself based the whole of his long, heroic and ultimately flawed proof of the parallel postulate around the quadrilateral and its three cases, proving many theorems about its properties along the way.
In 1250, Nasīr alDīn alTūsī, in his Alrisala alshafiya'an alshakk fi'lkhutut almutawaziya (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 alDin attempted to derive a proof by contradiction of the parallel postulate.^{[70]} He was one of the first to consider the cases of elliptical geometry and hyperbolic geometry, though he ruled out both of them.^{[68]}
His son, Sadr alDin (sometimes known as "PseudoTusi"), wrote a book on the subject in 1298, based on alTusi's later thoughts, which presented one of the earliest arguments for a nonEuclidean hypothesis equivalent to the parallel postulate.^{[70]}^{[71]} Sadr alDin'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 nonEuclidean geometry.^{[70]} A proof from Sadr alDin'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 alDin's work, while Saccheri also derived his Saccheri quadrilateral from Sadr alDin, who himself based it on his father's work.^{[72]}
The theorems of Ibn alHaytham (Alhazen), Omar Khayyam and Nasir alDin alTusi 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 nonEuclidean 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.^{[73]}
The early Indian works on trigonometry were translated and expanded in the Muslim world by Arab and Persian mathematicians, who 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."^{[74]} Another important development was the subject's separation from astronomy. All works on trigonometry up until the 12th century treated it mainly as an adjunct to astronomy; the first treatment of trigonometry as a subject in its own right was by Nasīr alDīn alTūsī in the 13th century.^{[75]}
In the early 9th century, Muhammad ibn Mūsā alKhwārizmī (c. 780850) produced tables for the trigonometric functions of sines and cosine,^{[76]} and the first tables for tangents.^{[77]} In 830, Habash alHasib alMarwazi produced the first tables of cotangents as well as tangents.^{[75]}^{[78]} Muhammad ibn Jābir alHarrānī alBattānī (853929) discovered the reciprocal functions of 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°.^{[75]} By the 10th century, in the work of Abū alWafā' alBūzjānī (959998), 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 accurate tables of tangent values.
Jamshīd alKāshī (13931449) 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°.^{[79]} In one of his numerical approximations of π, he correctly computed 2π to 9 sexagesimal digits.^{[80]} In order to determine sin 1°, alKashi discovered the following tripleangle formula often attributed to François Viète in the 16th century:^{[81]}
AlKashi, alongside his colleague Ulugh Beg (13941449), gave accurate tables of sines and tangents correct to 8 decimal places. Taqi alDin (15261585) contributed to trigonometry in his Sidrat alMuntaha, 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 alWafā, Ibn Yunus (ca. 1000), alKashi, Qāḍī Zāda alRūmī (13371412), Ulugh Beg and Mirim Chelebi improved on the value. Taqi alDin then solves the problem to obtain the value of sin 1° to a precision of 8 sexagesimals (the equivalent of 14 decimals):^{[82]}
Muhammad ibn Jābir alHarrānī alBattānī (853929) formulated a number of important trigonometrical relationships such as:
In the 10th century, Abū alWafā' alBūzjānī discovered the law of sines for spherical trigonometry:^{[78]}
Abū alWafā' also developed the following trigonometric formula:
Abū alWafā also established the angle addition identities, e.g. sin (a + b).^{[78]} Also in the late 10th and early 11th centuries, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the following formula:
Also in the 11th century, AlJayyani's The book of unknown arcs of a sphere introduced the general law of sines.^{[83]} In the 13th century, Nasīr alDīn alTūsī, in his On the Sector Figure, stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for these laws.^{[38]} Jamshīd alKāshī (13931449) provided the first explicit statement of the law of cosines in a form suitable for triangulation.^{[79]} As such, the law of cosines is known the théorème d'AlKashi in France.
Hellenistic methods dealing with spherical triangles were known, particularly the method of Menelaus of Alexandria, who developed Menelaus' theorem to deal with spherical problems.^{[84]}^{[85]} However, E. S. Kennedy points out that while it was possible in prelslamic 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.^{[86]} In order to observe holy days on the Islamic calendar in which timings were determined by phases of the moon, astronomers initially used Menelaus' method to calculate the place of the moon and stars, though this method proved to be clumsy and difficult. It involved setting up two intersecting right triangles; by applying Menelaus' theorem it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the sun's altitude, for instance, repeated applications of Menelaus' theorem were required. For medieval Islamic astronomers, there was an obvious challenge to find a simpler trigonometric method.^{[13]}
In the early 9th century, Muhammad ibn Mūsā alKhwārizmī was an early pioneer in spherical trigonometry and wrote a treatise on the subject.^{[77]} In the 10th century, Abū alWafā' alBūzjānī discovered the law of sines for spherical trigonometry.^{[78]} In the 11th century, AlJayyani (989–1079) of AlAndalus wrote The book of unknown arcs of a sphere, which is considered "the first treatise on spherical trigonometry" in its modern form.^{[83]} It "contains formulae for righthanded 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.^{[83]} In the 13th century, Nasīr alDīn alTūsī developed spherical trigonometry into its present form,^{[75]} and listed the six distinct cases of a rightangled triangle in spherical trigonometry.^{[38]} In his On the Sector Figure, he also stated the law of sines for plane and spherical triangles, and discovered the law of tangents for spherical triangles.^{[38]}
The method of triangulation, which was unknown in the GrecoRoman world, was also first developed by Muslim mathematicians, who applied it to practical uses such as surveying^{[87]} and Islamic geography, as described by Abu Rayhan Biruni in the early 11th century. Biruni employed triangulation techniques to measure the size of the Earth and the distances between places (see Mathematical geography and geodesy section).^{[88]}
In the late 11th century, Omar Khayyám (10481131) solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables (see Geometric algebra and Algebraic and analytic geometry sections). Jamshīd alKāshī (13931449) provided the first explicit statement of the law of cosines in a form suitable for triangulation.^{[79]}
Around 1000 AD, AlKaraji, using mathematical induction, found a proof for the sum of integral cubes.^{[89]} The historian of mathematics, F. Woepcke,^{[90]} praised AlKaraji for being "the first who introduced the theory of algebraic calculus." Shortly afterwards, Ibn alHaytham (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 AbulHasan ibn Haydur (d. 1413) and Abu Abdallah ibn Ghazi (14371514), by Jamshīd alKāshī (c. 13801429) in The Calculator's Key, and by the Indian mathematicians of the Kerala school of astronomy and mathematics in the 15th16th centuries.^{[70]}
In the 12th century, the Persian mathematician Sharaf alDīn alTūsī was the first to discover the derivative of cubic polynomials, an important result in differential calculus.^{[43]} 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 , alTusi finds the maximum point of the curve . He uses the derivative of the function to find that the maximum point occurs at , and then finds the maximum value for y at by substituting back into . He finds that the equation has a solution if , and alTusi thus deduces that the equation has a positive root if , where D is the discriminant of the equation.^{[44]}
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 AlAbbās ibn Said alJawharī (ca. 800860) in his Commentary on Euclid's Elements, the trigonometry of Aryabhata and Brahmagupta as elaborated on by Muhammad ibn Mūsā alKhwārizmī (ca. 780850), and the development of spherical geometry^{[13]} by Abū alWafā' alBūzjānī (940–998) and spherical trigonometry by AlJayyani (9891079)^{[83]} for determining the Qibla and times of Salah and Ramadan,^{[13]} 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 selfsimilar fractal quasicrystalline tilings such as the Penrose tilings, predating them by five centuries.^{[91]}^{[92]}
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.^{[13]} 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).^{[13]}^{[20]}
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.^{[93]}^{[94]} 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.^{[95]} Some zījes go beyond this traditional content to explain or prove the theory or report the observations from which the tables were computed.^{[96]}
In observational astronomy, Muhammad ibn Mūsā alKhwārizmī's Zij alSindh (830) contains trigonometric tables for the movements of the sun, the moon and the five planets known at the time.^{[97]} AlFarghani's A compendium of the science of stars (850) corrected Ptolemy's Almagest and gave revised values for the obliquity of the ecliptic, the precessional movement of the apogees of the sun and the moon, and the circumference of the earth.^{[98]} Muhammad ibn Jābir alHarrānī alBattānī (853929) discovered that the direction of the Sun's eccentric was changing,^{[99]} and studied the times of the new moon, lengths for the solar year and sidereal year, prediction of eclipses, and the phenomenon of parallax.^{[100]} Around the same time, Yahya Ibn Abi Mansour wrote the AlZij alMumtahan, in which he completely revised the Almagest values.^{[101]} In the 10th century, Abd alRahman alSufi (Azophi) carried out observations on the stars and described their positions, magnitudes, brightness, and colour and drawings for each constellation in his Book of Fixed Stars (964). Ibn Yunus observed more than 10,000 entries for the sun's position for many years using a large astrolabe with a diameter of nearly 1.4 meters. His observations on eclipses were still used centuries later in Simon Newcomb's investigations on the motion of the moon, while his other observations inspired Laplace's Obliquity of the Ecliptic and Inequalities of Jupiter and Saturn's.^{[102]}
In the late 10th century, AbuMahmud alKhujandi accurately computed the axial tilt to be 23°32'19" (23.53°),^{[103]} which was a significant improvement over the Greek and Indian estimates of 23°51'20" (23.86°) and 24°,^{[104]} and still very close to the modern measurement of 23°26' (23.44°). In 1006, the Egyptian astronomer Ali ibn Ridwan observed SN 1006, the brightest supernova in recorded history, and left a detailed description of the temporary star. He says that the object was two to three times as large as the disc of Venus and about onequarter the brightness of the Moon, and that the star was low on the southern horizon. In 1031, alBiruni's Canon Mas’udicus introduced the mathematical technique of analysing the acceleration of the planets, and first states that the motions of the solar apogee and the precession are not identical. AlBiruni also discovered that the distance between the Earth and the Sun is larger than Ptolemy's estimate, on the basis that Ptolemy disregarded annular eclipses.^{[105]}^{[106]}
During the "Maragha Revolution" of the 13th and 14th centuries, Muslim astronomers realized that astronomy should aim to describe the behavior of physical bodies in mathematical language, and should not remain a mathematical hypothesis, which would only save the phenomena. The Maragha astronomers also realized that the Aristotelian view of motion in the universe being only circular or linear was not true, as the Tusicouple showed that linear motion could also be produced by applying circular motions only.^{[107]} Unlike the ancient Greek and Hellenistic astronomers who were not concerned with the coherence between the mathematical and physical principles of a planetary theory, Islamic astronomers insisted on the need to match the mathematics with the real world surrounding them,^{[108]} which gradually evolved from a reality based on Aristotelian physics to one based on an empirical and mathematical physics after the work of Ibn alShatir. The Maragha Revolution was thus characterized by a shift away from the philosophical foundations of Aristotelian cosmology and Ptolemaic astronomy and towards a greater emphasis on the empirical observation and mathematization of astronomy and of nature in general, as exemplified in the works of Ibn alShatir, Ali Qushji, alBirjandi and alKhafri.^{[109]}^{[110]}^{[111]} In particular, Ibn alShatir's geocentric model was mathematically identical to the later heliocentric Copernical model.^{[112]}
The Muslim scholars, who held to the spherical Earth theory, used it in an impeccably Islamic manner, to calculate the distance and direction from any given point on the earth to Mecca. This determined the Qibla, or Muslim direction of prayer. Muslim mathematicians developed spherical trigonometry which was used in these calculations.^{[113]}
Around 830, Caliph alMa'mun commissioned a group of astronomers to measure the distance from Tadmur (Palmyra) to alRaqqah, in modern Syria. They found the cities to be separated by one degree of latitude and the distance between them to be 66 2/3 miles and thus calculated the Earth's circumference to be 24,000 miles.^{[114]} Another estimate given by AlFarghānī was 56 2/3 Arabic miles per degree, which corresponds to 111.8 km per degree and a circumference of 40,248 km, very close to the currently modern values of 111.3 km per degree and 40,068 km circumference, respectively.^{[115]}
In mathematical geography, Abū Rayhān alBīrūnī, around 1025, was the first to describe a polar equiazimuthal equidistant projection of the celestial sphere.^{[116]} He was also regarded as the most skilled when it came to mapping cities and measuring the distances between them, which he did for many cities in the Middle East and western Indian subcontinent. He often combined astronomical readings and mathematical equations, in order to develop methods of pinpointing locations by recording degrees of latitude and longitude. He also developed similar techniques when it came to measuring the heights of mountains, depths of valleys, and expanse of the horizon, in The Chronology of the Ancient Nations. He also discussed human geography and the planetary habitability of the Earth. He hypothesized that roughly a quarter of the Earth's surface is habitable by humans, and also argued that the shores of Asia and Europe were "separated by a vast sea, too dark and dense to navigate and too risky to try" in reference to the Atlantic Ocean and Pacific Ocean.^{[117]}
Abū Rayhān alBīrūnī is considered the father of geodesy for his important contributions to the field,^{[118]}^{[119]} along with his significant contributions to geography and geology. At the age of 17, alBiruni calculated the latitude of Kath, Khwarazm, using the maximum altitude of the Sun. AlBiruni also solved a complex geodesic equation in order to accurately compute the Earth's circumference, which were close to modern values of the Earth's circumference.^{[105]}^{[120]} His estimate of 6,339.9 km for the Earth radius was only 16.8 km less than the modern value of 6,356.7 km. In contrast to his predecessors who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, alBiruni developed a new method of using trigonometric calculations based on the angle between a plain and mountain top which yielded more accurate measurements of the Earth's circumference and made it possible for it to be measured by a single person from a single location.^{[121]}^{[122]}^{[123]}
Ibn alHaytham's work on geometric optics, particularly catoptrics, in "Book V" of the Book of Optics (1021) contains the important mathematical problem known as "Alhazen's problem" (Alhazen is the Latinized name of Ibn alHaytham). It comprises drawing lines from two points in the plane of a circle meeting at a point on the circumference and making equal angles with the normal at that point. This leads to an equation of the fourth degree. This eventually led Ibn alHaytham to derive the earliest 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, which was fundamental to the development of infinitesimal and integral calculus.^{[70]} Ibn alHaytham eventually solved "Alhazen's problem" using conic sections and a geometric proof, but Alhazen's problem remained influential in Europe, when later mathematicians such as Christiaan Huygens, James Gregory, Guillaume de l'Hôpital, Isaac Barrow, and many others, attempted to find an algebraic solution to the problem, using various methods, including analytic methods of geometry and derivation by complex numbers.^{[66]} Mathematicians were not able to find an algebraic solution to the problem until the end of the 20th century.^{[124]}
Ibn alHaytham also produced tables of corresponding angles of incidence and refraction of light passing from one medium to another show how closely he had approached discovering the law of constancy of ratio of sines, later attributed to Snell. He also correctly accounted for twilight being due to atmospheric refraction, estimating the Sun's depression to be 19 degrees below the horizon during the commencement of the phenomenon in the mornings or at its termination in the evenings.^{[125]}
Ibn alHaytham systematically endeavoured to mathematize physics in the context of his experimental research and controlled testing, which was oriented by geometric models of the structural mathematical principles that governed physical phenomena, particularly in relation to the explication of the behaviour and nature of vision and light.^{[126]} Ibn alHaytham also advanced in his Discourse on Place (Qawl fi almakan) a geometrical understanding of place as mathematical space that is akin to the 17th century conceptions of extensio by Descartes and analysis situs by Leibniz. Ibn alHaytham established his geometrical thesis about place as space in the context of his mathematical refutation of the Aristotelian physical definition of topos as a boundary surface of a containing body (as argued in Book delta [IV] of Aristotle's Physics).^{[127]}
Abū Rayhān alBīrūnī (9731048), and later alKhazini (fl. 11151130), were the first to apply experimental scientific methods to the statics and dynamics fields of mechanics, particularly for determining specific weights, such as those based on the theory of balances and weighing. Muslim physicists applied the mathematical theories of ratios and infinitesimal techniques, and introduced algebraic and fine calculation techniques into the field of statics.^{[128]}
Abu 'Abd Allah Muhammad ibn Ma'udh, who lived in AlAndalus during the second half of the 11th century, wrote a work on optics later translated into Latin as Liber de crepisculis, which was mistakenly attributed to Alhazen. This was a "short work containing an estimation of the angle of depression of the sun at the beginning of the morning twilight and at the end of the evening twilight, and an attempt to calculate on the basis of this and other data the height of the atmospheric moisture responsible for the refraction of the sun's rays." Through his experiments, he obtained the accurate value of 18°, which comes close to the modern value.^{[129]}
In 1574, Taqi alDin estimated that the stars are millions of kilometres away from the Earth and that the speed of light is constant, that if light had come from the eye, it would take too long for light "to travel to the star and come back to the eye. But this is not the case, since we see the star as soon as we open our eyes. Therefore the light must emerge from the object not from the eyes."^{[130]}^{[130]}
In the 9th century, alKindi was a pioneer in cryptanalysis and cryptology. He gave the first known recorded explanation of cryptanalysis in A Manuscript on Deciphering Cryptographic Messages. In particular, he is credited with developing the frequency analysis method whereby variations in the frequency of the occurrence of letters could be analyzed and exploited to break ciphers (i.e. crypanalysis by frequency analysis).^{[131]} This was detailed in a text recently rediscovered in the Ottoman archives in Istanbul, A Manuscript on Deciphering Cryptographic Messages, which also covers methods of cryptanalysis, encipherments, cryptanalysis of certain encipherments, and statistical analysis of letters and letter combinations in Arabic.^{[132]} AlKindi also had knowledge of polyalphabetic ciphers centuries before Leon Battista Alberti. AlKindi's book also introduced the classification of ciphers, developed Arabic phonetics and syntax, and described the use of several statistical techniques for cryptoanalysis. This book apparently antedates other cryptology references by several centuries, and it also predates writings on probability and statistics by Pascal and Fermat by nearly eight centuries.^{[133]}
Ahmad alQalqashandi (13551418) wrote the Subh ala 'sha, a 14volume encyclopedia which included a section on cryptology. This information was attributed to Taj adDin Ali ibn adDuraihim ben Muhammad athTha 'alibi alMausili who lived from 1312 to 1361, but whose writings on cryptology have been lost. The list of ciphers in this work included both substitution and transposition, and for the first time, a cipher with multiple substitutions for each plaintext letter. Also traced to Ibn alDuraihim is an exposition on and worked example of cryptanalysis, including the use of tables of letter frequencies and sets of letters which can not occur together in one word.
The first known proof by mathematical induction was introduced in the alFakhri written by AlKaraji around 1000 AD, who used it to prove arithmetic sequences such as the binomial theorem, Pascal's triangle, and the sum formula for integral cubes.^{[134]}^{[135]} His proof was the first to make use of the two basic components of an inductive proof, "namely the truth of the statement for n = 1 (1 = 1^{3}) and the deriving of the truth for n = k from that of n = k  1."^{[136]}
Shortly afterwards, Ibn alHaytham (Alhazen) used the inductive method to prove the sum of fourth powers, and by extension, the sum of any integral powers, which was an important result in integral calculus. He only stated it for particular integers, but his proof for those integers was by induction and generalizable.^{[137]}^{[138]}
Ibn Yahyā alMaghribī alSamaw'al came closest to a modern proof by mathematical induction in premodern times, which he used to extend the proof of the binomial theorem and Pascal's triangle previously given by alKaraji. AlSamaw'al's inductive argument was only a short step from the full inductive proof of the general binomial theorem.^{[139]}
The astrolabe is a mathematical tool that could be used to solve all the standard problems of spherical astronomy in five different ways.
"AlKhwarizmi continued: "We have said enough so far as numbers are concerned, about the six types of equations. Now, however, it is necessary that we should demonsrate geometrically the truth of the same problems which we have explained in numbers." The ring of this passage is obviously Greek rather than Babylonian or Indian. There are, therefore, three main schools of thought on the origin of Arabic algebra: one emphasizes Hindu influence, another stresses the Mesopotamian, or SyriacPersian, tradition, and the third points to Greek inspiration. The truth is probably approached if we combine the three theories."
"the author's preface in Arabic gave fulsome praise to Mohammed, the prophet, and to alMamun, "the Commander of the Faithful"."
"The Arabs in general loved a good clear argument from premise to conclusion, as well as systematic organization  respects in which neither Diophantus nor the Hindus excelled."
Omar Khayyam (ca. 10501123), the "tentmaker," wrote an Algebra that went beyond that of alKhwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the sixteenth century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all thirddegree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved."
In effect, this method characterized parallel lines as lines always equidisant from one another and also introduced the concept of motion into geometry.
"In some sense, his treatment was better than ibn alHaytham's because he explicitly formulated a new postulate to replace Euclid's rather than have the latter hidden in a new definition."
"Khayyam's postulate had excluded the case of the hyperbolic geometry whereas alTusi's postulate ruled out both the hyperbolic and elliptic geometries."
"But in a manuscript probably written by his son Sadr alDin in 1298, based on Nasir alDin's later thoughts on the subject, there is a new argument based on another hypothesis, also equivalent to Euclid's, [...] The importance of this latter work is that it was published in Rome in 1594 and was studied by European geometers. In particular, it became the starting point for the work of Saccheri and ultimately for the discovery of nonEuclidean geometry."
"In PseudoTusi's Exposition of Euclid, [...] another statement is used instead of a postulate. It was independent of the Euclidean postulate V and easy to prove. [...] He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements."
"His book published in Rome considerably influenced the subsequent development of the theory of parallel lines. Indeed, J. Wallis (16161703) included a Latin translation of the proof of postulate V from this book in his own writing On the Fifth Postulate and the Fifth Definition from Euclid's Book 6 (De Postulato Quinto et Definitione Quinta lib. 6 Euclidis, 1663). Saccheri quited this proof in his Euclid Cleared of all Stains (Euclides ab omni naevo vindicatus, 1733). It seems possible that he borrowed the idea of considering the three hypotheses about the upper angles of the 'Saccheri quadrangle' from PseudoTusi. The latter inserted the exposition of this subject into his work, taking it from the writings of alTusi and Khayyam."
"Three scientists, Ibn alHaytham, Khayyam and alTusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the ninteenth century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between tthis postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investiagtions of their European couterparts. The first European attempt to prove the postulate on parallel lines  made by Witelo, the Polish scientists of the thirteenth century, while revising Ibn alHaytham's Book of Optics (Kitab alManazir)  was undoubtedly prompted by Arabic sources. The proofs put forward in the fourteenth century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the abovementioned Alfonso from Spain directly border on Ibn alHaytham's demonstration. Above, we have demonstrated that PseudoTusi's Exposition of Euclid had stimulated borth J. Wallis's and G. Saccheri's studies of the theory of parallel lines."
"Another important idea introduced by alKaraji and continued by alSamaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences. Thus alKaraji used such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] AlKaraji did not, however, state a general result for arbitrary n. He stated his theorem for the particular integer 10 [...] His proof, nevertheless, was clearly designed to be extendable to any other integer.
"AlKaraji also uses a form of mathematical induction in his arguments, although he certainly does not give a rigorous exposition of the principle."
"AlKaraji's argument includes in essence the two basic components of a modern argument by induction, namely the truth of the statement for n = 1 (1 = 1^{3}) and the deriving of the truth for n = k from that of n = k  1. Of course, this second component is not explicit since, in some sense, alKaraji's argument is in reverse; this is, he starts from n = 10 and goes down to 1 rather than proceeding upward. Nevertheless, his argument in alFakhri is the earliest extant proof of the sum formula for integral cubes."
"The central idea in ibn alHaytham's proof of the sum formulas was the derivation of the equation [...] Naturally, he did not state this result in general form. He only stated it for particular integers, [...] but his proof for each of those k is by induction on n and is immediately generalizable to any value of k."
"Like the proofs of alKaraji and ibn alHaytham, alSamaw'al's argument contains the two basic components of an inductive proof. He begins with a value for which the result is known, here n = 2, and then uses the result for a given integer to derive the result for the next. Although alSamaw'al did not have any way of stating, and therefore proving, the general binomial theorem, to modern readers there is only a short step from alSamaw'al's argument to a full inductive proof of the binomial theorem."
