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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, Moors, Turks) and religions (Muslims, Christians, Jews, Sabians, Zoroastrians).[2]
Origins and influencesThe 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.[3] 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.[3] Many of these Greek works were translated by Thabit ibn Qurra (826-901), who translated books written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.[4] 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 Mesopotamian mathematics 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.[5] 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.[6] The earliest translations from Sanskrit inspired several astronomical and astrological Arabic works, now mostly lost, some of which were even composed in verse.[7] 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.[7] 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.[8] ImportanceJ. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive: R. Rashed wrote in The development of Arabic mathematics: between arithmetic and algebra: Biographies
FieldsAlgebra
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.[10] 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.[4]
The Muslim[11] Persian mathematician (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.[3] 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.[12] 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.[13] 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).[14] J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive: The Hellenistic mathematician Diophantus was traditionally known as "the father of algebra"[15][16] but debate now exists as to whether or not Al-Khwarizmi deserves this title instead.[15] 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,[17] 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.[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.[15]
'Abd al-Ham?d 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.[19] 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.[19] 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.[19]
Arabic mathematicians were also the first to treat irrational numbers as algebraic objects.[20] The Egyptian mathematician Ab? K?mil Shuj? 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.[21] Al-Karkhi (953-1029) was the successor of Ab? al-Waf?' al-B?zj?n? (940-998) and he was the first to discover the solution to equations of the form ax2n + bxn = c.[22] Al-Karkhi only considered positive roots.[22]
Omar Khayyám (c. 1050-1123) wrote a book on Algebra that went beyond Al-Jabr to include equations of the third degree.[23] 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.[23] 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.[23] He only considered positive roots and he did not go past the third degree.[23] He also saw a strong relationship between Geometry and Algebra.[23] In the 12th century, Sharaf al-D?n al-T?s? found algebraic and numerical solutions to cubic equations and was the first to discover the derivative of cubic polynomials.[24] 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.[25]
Ab? al-Hasan ibn Al? al-Qalas?d? (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[26] and by Ibn al-Y?sam?n in the 12th century.[27] In contrast to the syncopated notations of their predecessors, Diophantus and Brahmagupta, which lacked symbols for mathematical operations,[28] 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.[26] Applied mathematics
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-Abb?s ibn Said al-Jawhar? (ca. 800-860) in his Commentary on Euclid's Elements, the trigonometry of Aryabhata and Brahmagupta as elaborated on by Muhammad ibn M?s? al-Khw?rizm? (ca. 780-850), and the development of spherical geometry[29] by Ab? al-Waf?' al-B?zj?n? (940?998) and spherical trigonometry by Al-Jayyani (989-1079)[30] for determining the Qibla and times of Salah and Ramadan,[29] 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.[31][32]
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.[29] 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).[29][33] 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 principle contributions to mathematical astronomy reflected improved trigonometrical, computational and observational techniques.[34][35] 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.[36] Some z?jes go beyond this traditional content to explain or prove the theory or report the observations from which the tables were computed.[37] In observational astronomy, Muhammad ibn M?s? al-Khw?rizm?'s Zij al-Sindh (830) contains trigonometric tables for the movements of the sun, the moon and the five planets known at the time.[38] Al-Farghani'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.[39] Muhammad ibn J?bir al-Harr?n? al-Batt?n? (853-929) discovered that the direction of the Sun's eccentric was changing,[40] and studied the times of the new moon, lengths for the solar year and sidereal year, prediction of eclipses, and the phenomenon of parallax.[41] Around the same time, Yahya Ibn Abi Mansour wrote the Al-Zij al-Mumtahan, in which he completely revised the Almagest values.[42] In the 10th century, Abd al-Rahman al-Sufi (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.[43] In the late 10th century, Abu-Mahmud al-Khujandi accurately computed the axial tilt to be 23°32'19" (23.53°),[44] which was a significant improvement over the Greek and Indian estimates of 23°51'20" (23.86°) and 24°,[45] 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 one-quarter the brightness of the Moon, and that the star was low on the southern horizon. In 1031, al-Biruni'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. Al-Biruni also discovered that the distance between the Earth and the Sun is larger than Ptolemy's estimate, on the basis that Ptolemy disregarded the annual solar eclipses.[46][47] 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 Tusi-couple showed that linear motion could also be produced by applying circular motions only.[48] 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,[49] which gradually evolved from a reality based on Aristotelian physics to one based on an empirical and mathematical physics after the work of Ibn al-Shatir. 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 al-Shatir, al-Qushji, al-Birjandi and al-Khafri.[50][51][52] In particular, Ibn al-Shatir's geocentric model was mathematically identical to the later heliocentric Copernical model.[53]
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.[54] Around 830, Caliph al-Ma'mun commissioned a group of astronomers to measure the distance from Tadmur (Palmyra) to al-Raqqah, 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.[55] Another estimate given 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.[56] 
Ab? Rayh?n al-B?r?n? was a polymath who is considered a pioneer in mathematical geography and geodesy. In mathematical geography, Ab? Rayh?n al-B?r?n?, around 1025, was the first to describe a polar equi-azimuthal equidistant projection of the celestial sphere.[57] 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 pin-pointing 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.[58] Ab? Rayh?n al-B?r?n? is considered the father of geodesy for his important contributions to the field,[59][60] along with his significant contributions to geography and geology. At the age of 17, al-Biruni calculated the latitude of Kath, Khwarazm, using the maximum altitude of the Sun. Al-Biruni 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.[46][61] 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, al-Biruni 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.[62]
Ibn al-Haytham'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 al-Haytham). 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 al-Haytham 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.[63] Ibn al-Haytham 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.[64] Mathematicians were not able to find an algebraic solution to the problem until the end of the 20th century.[65] Ibn al-Haytham 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.[66] Ab? Rayh?n al-B?r?n? (973-1048), and later al-Khazini (fl. 1115-1130), were the first to apply experimental scientific methods to the statics and dynamics fiels 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.[67] Abu 'Abd Allah Muhammad ibn Ma'udh, who lived in Al-Andalus 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.[68] In 1574, Taqi al-Din 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."[69][69] ArithmeticThe Indian numeral system came to be known to both the Persian mathematician Al-Khwarizmi, whose book On the Calculation with Hindu Numerals written circa 825, and the Arab mathematician Al-Kindi, who wrote four volumes, On the Use of the Indian Numerals (Ketab fi Isti'mal al-'Adad al-Hindi) circa 830, are principally responsible for the diffusion of the Indian system of numeration in the Middle-East and the West http://www-gap.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Indian_numerals.html. In the 10th century, Middle-Eastern mathematicians extended the decimal numeral system to include fractions using decimal point notation, as recorded in a treatise by Syrian mathematician Abu'l-Hasan al-Uqlidisi in 952-953. In the Arab world—until modern times—the Arabic numeral system was used only by mathematicians. Muslim scientists used the Babylonian numeral system, and merchants used the Abjad numerals. A distinctive "Western Arabic" variant of the symbols begins to emerge in ca. the 10th century in the Maghreb and Al-Andalus, called the ghubar ("sand-table" or "dust-table") numerals, which is the direct ancestor to the modern Western Arabic numerals now used throughout the world.[70] The first mentions of the numerals in the West are found in the Codex Vigilanus of 976 http://www.mathorigins.com/V.htm. 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. Al-Khw?rizm?, 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". Al-Hass?r, 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.[27] CalculusAround 1000 AD, Al-Karaji, using mathematical induction, found a proof for the sum of integral cubes.[71] The historian of mathematics, F. Woepcke,[72] 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 Jamsh?d al-K?sh? (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.[63] In the late 11th century, the Persian mathematician 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[23] with his geometric solution of the general cubic equations,[73] but the decisive step in analytic geometry came later with René Descartes.[23] In the 12th century, the Persian mathematician Sharaf al-D?n al-T?s? was the first to discover the derivative of cubic polynomials, an important result in differential calculus.[24] 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.[25] 
The first page of al-Kindi's manuscript On Deciphering Cryptographic Messages, containing the first descriptions of cryptanalysis and frequency analysis. CryptographyIn the 9th century, al-Kindi 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).[74] 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.[75] Ahmad al-Qalqashandi (1355-1418) wrote the Subh al-a 'sha, a 14-volume encyclopedia which included a section on cryptology. This information was attributed to Taj ad-Din Ali ibn ad-Duraihim ben Muhammad ath-Tha 'alibi al-Mausili 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 al-Duraihim 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. 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 M?s? al-Khw?rizm? (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.
Although 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.[76] 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.[76] 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. 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.[29][33]
Omar Khayyám (born 1048) 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[77][78] and analytic geometry.[73] Khayyam also made the first attempt at formulating a non-Euclidean postulate as an alternative to the Euclidean parallel postulate,[79] and he was the first to consider the cases of elliptical geometry and hyperbolic geometry, though he excluded the latter.[80] 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 Mu?ammad ibn M?s? al-?w?rizm?). However, Khayyam himself seems to have been the first to conceive a general theory of cubic equations. 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. 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.[25]
In 1250, Nas?r al-D?n al-T?s?, 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.[63] He was one of the first to consider the cases of elliptical geometry and hyperbolic geometry, though he ruled out both of them.[80] 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.[63][81] 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.[63] 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.[82] 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.[83] Mathematical inductionThe first known proof by mathematical induction was introduced in the al-Fakhri written by Al-Karaji 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.[84][85] 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 = 13) and the deriving of the truth for n = k from that of n = k - 1."[86] Shortly afterwards, Ibn al-Haytham (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.[87][88] Ibn Yahy? al-Maghrib? al-Samaw'al came closest to a modern proof by mathematical induction in pre-modern times, which he used to extend the proof of the binomial theorem and Pascal's triangle previously given by al-Karaji. Al-Samaw'al's inductive argument was only a short step from the full inductive proof of the general binomial theorem.[89] Number theoryIn number theory, Ibn al-Haytham solved problems involving congruences using what is now called Wilson's theorem. In his Opuscula, Ibn al-Haytham 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 al-Haytham was the first to discover that every even perfect number is of the form 2n−1(2n − 1) where 2n − 1 is prime, but he was not able to prove this result successfully (Euler later proved it in the 18th century).[90] Numerical analysisIn numerical analysis, the essence of Viète's method was known to al-Tusi, and it is possible that the algebraic tradition of al-Tusi, as well as his predecessor Omar Khayyám and successor Jamsh?d al-K?sh?, was known to 16th century European algebraists, or whom François Viète was the most important.[91] A method algebraically equivalent to Newton's method was also known to al-Tusi. His successor al-Kashi later used a form of Newton's method to 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.[92] Recreational mathematicsIn 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 (the Encyclopedia of the Brethern of Purity); simpler magic squares were known to several earlier Arab mathematicians.[93] 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.[93] TrigonometryThe 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."[94] In the 9th century, (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 J?bir al-Harr?n? al-Batt?n? (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°.[95] He also formulated a number of important trigonometrical relationships such as:
By the 10th century, in the work of Ab? al-Waf?' al-B?zj?n? (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. Ab? al-Waf?' also developed the following trigonometric formula:
Ab? al-Waf? also established the angle addition identities, e.g. sin (a + b), and discovered the sine formula for spherical trigonometry:
Also in the late 10th and early 11th centuries, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the following formula:
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,[96] 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.[97][98] 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.[99] 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.[96] 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[100] and Islamic geography, as described by Ab? Rayh?n al-B?r?n? in the early 11th century.[101] 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 Nas?r al-D?n al-T?s? in the 13th century. He also developed spherical trigonometry into its present form,[95] 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. Jamsh?d al-K?sh? (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°.[102] In one of his numerical approximations of ?, he correctly computed 2? to 9 sexagesimal digits.[103] In order to determine sin 1°, al-Kashi discovered the following triple-angle formula often attributed to François Viète in the 16th century:[104]
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 Ab? al-Waf?, Ibn Yunus (ca. 1000), al-Kashi, Q??? Z?da al-R?m? (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):[105]
See also
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