Thābit ibn Qurra
Al-Ṣābiʾ Thābit ibn Qurrah al-Ḥarrānī was a Arab Sabian mathematician, physician and translator who lived in Baghdad in the second half of the ninth century during the time of Abbasid Caliphate. Thābit ibn Qurrah made important discoveries in algebra and astronomy. In astronomy, Thābit is considered one of the first reformers of the Ptolemaic system, in mechanics he was a founder of statics. Thābit was born in Harran in Upper Mesopotamia, which at the time was part of the Diyar Mudar subdivision of the al-Jazira region of the Abbasid Caliphate; the city of Harran was never Christianized. By the early Muslim conquests, the people of Harran were still adhering to the cult of Sin. Thābit and his pupils lived in the midst of the most intellectually vibrant, the largest, city of the time, Baghdad, he occupied himself with mathematics, astrology, mechanics and philosophy. In his life, Thābit's patron was the Abbasid Caliph al-Mu'tadid. Thābit became courtier. Thābit died in Baghdad. After him, the greatest Sabean name was al-Battani.
Thābit's native language was Syriac, the eastern Aramaic variety from Edessa, he was fluent in both Greek and Arabic. Thābit translated from Greek into Arabic works by Apollonius of Perga, Archimedes and Ptolemy, he revised the translation of Euclid's Elements of Hunayn ibn Ishaq. He rewrote Hunayn's translation of Ptolemy's Almagest and translated Ptolemy's Geography. Thābit's translation of a work by Archimedes which gave a construction of a regular heptagon was discovered in the 20th century, the original having been lost; the medieval astronomical theory of the trepidation of the equinoxes is attributed to Thābit. But it had been described by Theon of Alexandria in his comments of the Handy Tables of Ptolemy. According to Copernicus, Thābit determined the length of the sidereal year as 365 days, 6 hours, 9 minutes and 12 seconds. Copernicus based his claim on the Latin text attributed to Thābit. Thābit published his observations of the Sun. In mathematics, Thābit discovered an equation for determining amicable numbers.
He wrote on the theory of numbers, extended their use to describe the ratios between geometrical quantities, a step which the Greeks did not take. He is known for having calculated the solution to a chessboard problem involving an exponential series, he computed the volume of the paraboloid. He described a generalization of Pythagoras' theorem. In physics, Thābit rejected the Peripatetic and Aristotelian notions of a "natural place" for each element, he instead proposed a theory of motion in which both the upward and downward motions are caused by weight, that the order of the universe is a result of two competing attractions: one of these being "between the sublunar and celestial elements", the other being "between all parts of each element separately". and in mechanics he was a founder of statics. Only a few of Thābit's works are preserved in their original form. On the Sector-Figure which deals with Menelaus' theorem. On the Composition of Ratios Thabit number Thebit Roshdi Rashed, Thābit ibn Qurra.
Science and Philosophy in Ninth-Century Baghdad, Walter de Gruyter, 2009. Francis J. Carmody: The astronomical works of Thābit b. Qurra. 262 pp. Berkeley and Los Angeles: University of California Press, 1960. Rashed, Roshdi. Les Mathématiques Infinitésimales du IXe au XIe Siècle 1: Fondateurs et commentateurs: Banū Mūsā, Ibn Qurra, Ibn Sīnān, al-Khāzin, al-Qūhī, Ibn al-Samḥ, Ibn Hūd. London. Reviews: Seyyed Hossein Nasr in Isis 89 pp. 112-113. Churton, Tobias; the Golden Builders: Alchemists and the First Freemasons. Barnes and Noble Publishing, 2006. Hakim S Ayub Ali. Zakhira-i Thābit ibn Qurra, India, 1987. Palmeri, JoAnn. "Thābit ibn Qurra". In Thomas Hockey; the Biographical Encyclopedia of Astronomers. New York: Springer. Pp. 1129–30. ISBN 978-0-387-31022-0. O'Connor, John J.. Rosenfeld, B. A.. T.. "Thābit Ibn Qurra, Al-Ṣābiʾ Al-Ḥarrānī". Complete Dictionary of Scientific Biography. Encyclopedia.com. Thabit ibn Qurra on Astrology & Magic
In the history of Europe, the Middle Ages lasted from the 5th to the 15th century. It began with the fall of the Western Roman Empire and merged into the Renaissance and the Age of Discovery; the Middle Ages is the middle period of the three traditional divisions of Western history: classical antiquity, the medieval period, the modern period. The medieval period is itself subdivided into the Early and Late Middle Ages. Population decline, counterurbanisation, collapse of centralized authority and mass migrations of tribes, which had begun in Late Antiquity, continued in the Early Middle Ages; the large-scale movements of the Migration Period, including various Germanic peoples, formed new kingdoms in what remained of the Western Roman Empire. In the 7th century, North Africa and the Middle East—once part of the Byzantine Empire—came under the rule of the Umayyad Caliphate, an Islamic empire, after conquest by Muhammad's successors. Although there were substantial changes in society and political structures, the break with classical antiquity was not complete.
The still-sizeable Byzantine Empire, Rome's direct continuation, survived in the Eastern Mediterranean and remained a major power. The empire's law code, the Corpus Juris Civilis or "Code of Justinian", was rediscovered in Northern Italy in 1070 and became admired in the Middle Ages. In the West, most kingdoms incorporated the few extant Roman institutions. Monasteries were founded; the Franks, under the Carolingian dynasty established the Carolingian Empire during the 8th and early 9th century. It covered much of Western Europe but succumbed to the pressures of internal civil wars combined with external invasions: Vikings from the north, Magyars from the east, Saracens from the south. During the High Middle Ages, which began after 1000, the population of Europe increased as technological and agricultural innovations allowed trade to flourish and the Medieval Warm Period climate change allowed crop yields to increase. Manorialism, the organisation of peasants into villages that owed rent and labour services to the nobles, feudalism, the political structure whereby knights and lower-status nobles owed military service to their overlords in return for the right to rent from lands and manors, were two of the ways society was organised in the High Middle Ages.
The Crusades, first preached in 1095, were military attempts by Western European Christians to regain control of the Holy Land from Muslims. Kings became the heads of centralised nation-states, reducing crime and violence but making the ideal of a unified Christendom more distant. Intellectual life was marked by scholasticism, a philosophy that emphasised joining faith to reason, by the founding of universities; the theology of Thomas Aquinas, the paintings of Giotto, the poetry of Dante and Chaucer, the travels of Marco Polo, the Gothic architecture of cathedrals such as Chartres are among the outstanding achievements toward the end of this period and into the Late Middle Ages. The Late Middle Ages was marked by difficulties and calamities including famine and war, which diminished the population of Europe. Controversy and the Western Schism within the Catholic Church paralleled the interstate conflict, civil strife, peasant revolts that occurred in the kingdoms. Cultural and technological developments transformed European society, concluding the Late Middle Ages and beginning the early modern period.
The Middle Ages is one of the three major periods in the most enduring scheme for analysing European history: classical civilisation, or Antiquity. The "Middle Ages" first appears in Latin in 1469 as media tempestas or "middle season". In early usage, there were many variants, including medium aevum, or "middle age", first recorded in 1604, media saecula, or "middle ages", first recorded in 1625; the alternative term "medieval" derives from medium aevum. Medieval writers divided history into periods such as the "Six Ages" or the "Four Empires", considered their time to be the last before the end of the world; when referring to their own times, they spoke of them as being "modern". In the 1330s, the humanist and poet Petrarch referred to pre-Christian times as antiqua and to the Christian period as nova. Leonardo Bruni was the first historian to use tripartite periodisation in his History of the Florentine People, with a middle period "between the fall of the Roman Empire and the revival of city life sometime in late eleventh and twelfth centuries".
Tripartite periodisation became standard after the 17th-century German historian Christoph Cellarius divided history into three periods: ancient and modern. The most given starting point for the Middle Ages is around 500, with the date of 476 first used by Bruni. Starting dates are sometimes used in the outer parts of Europe. For Europe as a whole, 1500 is considered to be the end of the Middle Ages, but there is no universally agreed upon end date. Depending on the context, events such as the conquest of Constantinople by the Turks in 1453, Christopher Columbus's first voyage to the Americas in 1492, or the Protestant Reformation in 1517 are sometimes used. English historians use the Battle of Bosworth Field in 1485 to mark the end of the period. For Spain, dates used are the death of King Ferdinand II in 1516, the death of Queen Isabella I of Castile in 1504, or the conquest of Granada in 1492. Historians from Romance-speaking countries tend to divide the Middle Ages into two parts: an earlier "High" and late
Astronomy in the medieval Islamic world
Islamic astronomy comprises the astronomical developments made in the Islamic world during the Islamic Golden Age, written in the Arabic language. These developments took place in the Middle East, Central Asia, Al-Andalus, North Africa, in the Far East and India, it parallels the genesis of other Islamic sciences in its assimilation of foreign material and the amalgamation of the disparate elements of that material to create a science with Islamic characteristics. These included Greek and Indian works in particular, which were translated and built upon. Islamic astronomy played a significant role in the revival of Byzantine and European astronomy following the loss of knowledge during the early medieval period, notably with the production of Latin translations of Arabic works during the 12th century. Islamic astronomy had an influence on Chinese astronomy and Malian astronomy. A significant number of stars in the sky, such as Aldebaran and Deneb, astronomical terms such as alidade and nadir, are still referred to by their Arabic names.
A large corpus of literature from Islamic astronomy remains today, numbering 10,000 manuscripts scattered throughout the world, many of which have not been read or catalogued. So, a reasonably accurate picture of Islamic activity in the field of astronomy can be reconstructed. Ahmad Dallal notes that, unlike the Babylonians and Indians, who had developed elaborate systems of mathematical astronomical study, the pre-Islamic Arabs relied on empirical observations; these observations were based on the rising and setting of particular stars, this area of astronomical study was known as anwa. Anwa continued to be developed after Islamization by the Arabs, where Islamic astronomers added mathematical methods to their empirical observations. According to David King, after the rise of Islam, the religious obligation to determine the qibla and prayer times inspired more progress in astronomy for centuries. Donald Hill divided Islamic Astronomy into the four following distinct time periods in its history: Following the Islamic conquests, under the early caliphate, Muslim scholars began to absorb Hellenistic and Indian astronomical knowledge via translations into Arabic.
The first astronomical texts that were translated into Arabic were of Persian origin. The most notable of the texts was Zij al-Sindhind, an 8th-century Indian astronomical work, translated by Muhammad ibn Ibrahim al-Fazari and Yaqub ibn Tariq after 770 CE with the assistance of Indian astronomers who visited the court of caliph Al-Mansur in 770. Another text translated was the Zij al-Shah, a collection of astronomical tables compiled in Sasanid Persia over two centuries. Fragments of texts during this period indicate that Arabs adopted the sine function in place of the chords of arc used in Greek trigonometry; the House of Wisdom was an academy established in Baghdad under Abbasid caliph Al-Ma'mun in the early 9th century. From this time, independent investigation into the Ptolemaic system became possible. According to Dallal, the use of parameters and calculation methods from different scientific traditions made the Ptolemaic tradition "receptive right from the beginning to the possibility of observational refinement and mathematical restructuring".
Astronomical research was supported by the Abbasid caliph al-Mamun through The House of Wisdom. Baghdad and Damascus became the centers of such activity; the caliphs not only endowed the work with formal prestige. The first major Muslim work of astronomy was Zij al-Sindh by al-Khwarizmi in 830; the work contains tables for the movements of the Sun, the Moon and the five planets known at the time. The work is significant; this work marks the turning point in Islamic astronomy. Hitherto, Muslim astronomers had adopted a research approach to the field, translating works of others and learning discovered knowledge. Al-Khwarizmi's work marked the beginning of nontraditional methods of study and calculations. In 850, al-Farghani wrote Kitab fi Jawani; the book gave a summary of Ptolemic cosmography. However, it corrected Ptolemy based on findings of earlier Arab astronomers. Al-Farghani gave revised values for the obliquity of the ecliptic, the precessional movement of the apogees of the Sun and the Moon, the circumference of the Earth.
The book was circulated through the Muslim world, translated into Latin. In addition to Alfraganus's findings, Egyptian Astronomer Ibn Yunus was the first Astronomer to find valid fault in Ptolemy's calculations about the planet's movements and their peculiarity in the late 10th century. Ptolemy calculated that Earth's wobble, otherwise known as precession, varied 1 degree every 100 years. Ibn Yunus contradicted this finding by calculating; this was impossible to believe, since it was still thought that the Earth was the center of the universe. Ibn Yunus and Ibn al-Shatir's findings were part of Copernicus's calculations to figure out that the Sun was the center of the universe; the period when a distinctive Islamic system of astronomy flourished. The period began as the Muslim astronomers began questioning the framework of the Ptolemaic system of astronomy; these criticisms, remained within the geocentric framework and followed Ptolemy's astronomical paradigm.
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Numerical analysis finds application in all fields of engineering and the physical sciences, but in the 21st century the life sciences, social sciences, medicine and the arts have adopted elements of scientific computations; as an aspect of mathematics and computer science that generates and implements algorithms, the growth in power and the revolution in computing has raised the use of realistic mathematical models in science and engineering, complex numerical analysis is required to provide solutions to these more involved models of the world. Ordinary differential equations appear in celestial mechanics. Before the advent of modern computers, numerical methods depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead; these same interpolation formulas continue to be used as part of the software algorithms for solving differential equations.
One of the earliest mathematical writings is a Babylonian tablet from the Yale Babylonian Collection, which gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square. Being able to compute the sides of a triangle is important, for instance, in astronomy and construction. Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of the square root of 2, modern numerical analysis does not seek exact answers, because exact answers are impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors; the overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of, suggested by the following: Advanced numerical methods are essential in making numerical weather prediction feasible.
Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations. Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes; such simulations consist of solving partial differential equations numerically. Hedge funds use tools from all fields of numerical analysis to attempt to calculate the value of stocks and derivatives more than other market participants. Airlines use sophisticated optimization algorithms to decide ticket prices and crew assignments and fuel needs; such algorithms were developed within the overlapping field of operations research. Insurance companies use numerical programs for actuarial analysis; the rest of this section outlines several important themes of numerical analysis. The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.
To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. Using these tables calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve good numerical estimates of some functions; the canonical work in the field is the NIST publication edited by Abramowitz and Stegun, a 1000-plus page book of a large number of used formulas and functions and their values at many points. The function values are no longer useful when a computer is available, but the large listing of formulas can still be handy; the mechanical calculator was developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, it was found that these computers were useful for administrative purposes, but the invention of the computer influenced the field of numerical analysis, since now longer and more complicated calculations could be done.
Direct methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer. Examples include Gaussian elimination, the QR factorization method for solving systems of linear equations, the simplex method of linear programming. In practice, finite precision is used and the result is an approximation of the true solution. In contrast to direct methods, iterative methods are not expected to terminate in a finite number of steps. Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit. A convergence test involving the residual, is specified in order to decide when a sufficiently accurate solution has been found. Using infinite precision arithmetic these methods would not reach the solution within a finite number of steps. Examples include Newton's method, the bisection method, Jacobi iteration. In computational matrix algebra, iterative methods are generall
Mathematics in medieval Islam
Mathematics during the Golden Age of Islam during the 9th and 10th centuries, was built on Greek mathematics and Indian mathematics. Important progress was made, such as the full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra, advances in geometry and trigonometry. Arabic works played an important role in the transmission of mathematics to Europe during the 10th to 12th centuries; the study of algebra, the name of, derived from the Arabic word meaning completion or "reunion of broken parts", flourished during the Islamic golden age. Muhammad ibn Musa al-Khwarizmi, a scholar in the House of Wisdom in Baghdad, is along with the Greek mathematician Diophantus, known as the father of algebra. In his book The Compendious Book on Calculation by Completion and Balancing, Al-Khwarizmi deals with ways to solve for the positive roots of first and second degree polynomial equations, he introduces the method of reduction, unlike Diophantus, gives general solutions for the equations he deals with.
Al-Khwarizmi's algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the algebraic work of Diophantus, syncopated, meaning that some symbolism is used; the transition to symbolic algebra, where only symbols are used, can be seen in the work of Ibn al-Banna' al-Marrakushi and Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī. On the work done by Al-Khwarizmi, J. J. O'Connor and Edmund F. Robertson said: "Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra, it is important to understand just. It was a revolutionary move away from the Greek concept of mathematics, 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, provided a vehicle for the 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." Several other mathematicians during this time period expanded on the algebra of Al-Khwarizmi. Abu Kamil Shuja' wrote a book of algebra accompanied with geometrical proofs, he enumerated all the possible solutions to some of his problems. Abu al-Jud, Omar Khayyam, along with Sharaf al-Dīn al-Tūsī, found several solutions of the cubic equation. Omar Khayyam found the general geometric solution of a cubic equation. Omar Khayyam wrote the Treatise on Demonstration of Problems of Algebra containing the systematic solution of cubic or third-order equations, going beyond the Algebra of al-Khwārizmī. Khayyám obtained the solutions of these equations by finding the intersection points of two conic sections; this method had been used by the Greeks, but they did not generalize the method to cover all equations with positive roots. Sharaf al-Dīn al-Ṭūsī developed a novel approach to the investigation of cubic equations—an approach which entailed finding the point at which a cubic polynomial obtains its maximum value.
For example, to solve the equation x 3 + a = b x, with a and b positive, he would note that the maximum point of the curve y = b x − x 3 occurs at x = b 3, that the equation would have no solutions, one solution or two solutions, depending on whether the height of the curve at that point was less than, equal to, or greater than a. His surviving works give no indication of how he discovered his formulae for the maxima of these curves. Various conjectures have been proposed to account for his discovery of them; the earliest implicit traces of mathematical induction can be found in Euclid's proof that the number of primes is infinite. The first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique. In between, implicit proof by induction for arithmetic sequences was introduced by al-Karaji and continued by al-Samaw'al, who used it for special cases of the binomial theorem and properties of Pascal's triangle; the Greeks had discovered irrational numbers, but were not happy with them and only able to cope by drawing a distinction between magnitude and number.
In the Greek view, magnitudes varied continuously and could be used for entities such as line segments, whereas numbers were discrete. Hence, irrationals could only be handled geometrically. Islamic mathematicians including Abū Kāmil Shujāʿ ibn Aslam and Ibn Tahir al-Baghdadi removed the distinction between magnitude and number, allowing irrational quantities to appear as coefficients in equations and to be solutions of algebraic equations, they worked with irrationals as mathematical objects, but they did not examine their nature. In the twelfth century, Latin translations of Al-Khwarizmi's Arithmetic on the Indian numerals introduced the decimal positional number system to the Western world, his Compendious Book on Calculation by Completion and Balancing presented the first systematic s
Tus spelled as Tous, Toos or Tūs, is an ancient city in Razavi Khorasan Province in Iran near Mashhad. To the ancient Greeks, it was known as Susia, it was known as Tusa. It was captured by Alexander the Great in 330 BCE. Tus was taken by the Umayyad caliph Abd al-Malik and remained under Umayyad control until 747, when a subordinate of Abu Muslim Khorasani defeated the Umayyad governor during the Abbasid Revolution. In 809, the Abbasid Caliph Harun al-Rashid fell ill and died in Tus, on his way to solve the unrest in Khorasan, his grave is located in the region. In 1220, Tus was sacked by the Mongol general, a year Tolui would kill most of its populace, destroying the tomb of Caliph Harun al-Rashid in the process. Decades Tus would be rebuilt under the governorship of Kuerguez; the most famous person who has emerged from that area is the poet Ferdowsi, author of the Persian epic Shahnameh, whose mausoleum, built in 1934 in time for the millennium of his birth, dominates the town. Other notable residents of Tus include the theologian, jurist and mystic al-Ghazali.
Al-Tusi – a descriptor used for individuals associated with Tus Tus citadel Frye, R. N.. "The Sāmānids". In Frye, R. N; the Cambridge History of Iran, Volume 4: From the Arab Invasion to the Saljuqs. Cambridge: Cambridge University Press. Pp. 136–161. ISBN 978-0-521-20093-6. Litvinsky, Ahmad Hasan Dani. History of Civilizations of Central Asia: Age of Achievement, A. D. 750 to the end of the 15th-century. UNESCO. ISBN 9789231032110. Khaleghi-Motlagh, Djalal. "DAQĪQĪ, ABŪ MANṢŪR AḤMAD". Encyclopaedia Iranica, Vol. VI, Fasc. 6. Pp. 661–662. Media related to Tus, Iran at Wikimedia Commons Livius.org: Susia
Abu al-Wafa' Buzjani
Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī or Abū al-Wafā Būzhjānī was a Persian mathematician and astronomer who worked in Baghdad. He made important innovations in spherical trigonometry, his work on arithmetics for businessmen contains the first instance of using negative numbers in a medieval Islamic text, he is credited with compiling the tables of sines and tangents at 15' intervals. He introduced the secant and cosecant functions, as well studied the interrelations between the six trigonometric lines associated with an arc, his Almagest was read by medieval Arabic astronomers in the centuries after his death. He is known to have written several other books, he was born in Khorasan. At age 19, in 959 AD, he moved to Baghdad and remained there for the next forty years, died there in 998, he was a contemporary of the distinguished scientists Abū Sahl al-Qūhī and Al-Sijzi who were in Baghdad at the time and others like Abu Nasr ibn Iraq, Abu-Mahmud Khojandi, Kushyar ibn Labban and Al-Biruni.
In Baghdad, he received patronage by members of the Buyid court. Abu Al-Wafa' was the first to build a wall quadrant to observe the sky, it has been suggested that he was influenced by the works of Al-Battani as the latter describes a quadrant instrument in his Kitāb az-Zīj. His use of tangent helped to solve problems involving right-angled spherical triangles, developed a new technique to calculate sine tables, allowing him to construct more accurate tables than his predecessors. In 997, he participated in an experiment to determine the difference in local time between his location and that of al-Biruni; the result was close to present-day calculations, showing a difference of 1 hour between the two longitudes. Abu al-Wafa is known to have worked with Abū Sahl al-Qūhī, a famous maker of astronomical instruments. While what is extant from his works lacks theoretical innovation, his observational data were used by many astronomers, including al-Biruni. Among his works on astronomy, only the first seven treatises of his Almagest are now extant.
The work covers numerous topics in the fields of plane and spherical trigonometry, planetary theory, solutions to determine the direction of Qibla. He established several trigonometric identities such as sin in their modern form, where the Ancient Greek mathematicians had expressed the equivalent identities in terms of chords. Sin = sin α cos β ± cos α sin β sin = sin cos + cos sin cos = 1 − 2 sin 2 sin = 2 sin cos He discovered the law of sines for spherical triangles: sin A sin a = sin B sin b = sin C sin c where A, B, C are the sides and a, b, c are the opposing angles; some sources suggest that he introduced the tangent function, although other sources give the credit for this innovation to al-Marwazi. Almagest. A book of zij called Zīj al‐wāḍiḥ, no longer extant. "A Book on Those Geometric Constructions Which Are Necessary for a Craftsman". This text contains over one hundred geometric constructions, including for a regular heptagon, which have been reviewed and compared with other mathematical treatises.
The legacy of this text in Latin Europe is still debated. "A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen". This is the first book, he wrote translations and commentaries on the algebraic works of Diophantus, al-Khwārizmī, Euclid's Elements. The crater Abul Wáfa on the Moon is named after him. On June 2015 Google has changed its logo in memory of Abu al-Wafa' Buzjani. O'Connor, John J.. Hashemipour, Behnaz. "Būzjānī: Abū al‐Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā al‐Būzjānī". In Thomas Hockey; the Biographical Encyclopedia of Astronomers. New York: Springer. Pp. 188–9. ISBN 978-0-387-3