1.
Al-Biruni
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Abū Rayḥān Muḥammad ibn Aḥmad Al-Bīrūnī, known as Al-Biruni in English, was an Iranian scholar and polymath from Khwarezm. He studied almost all fields of science and was compensated for his research, royalty and powerful members of society sought out Al-Biruni to conduct research and study in order to uncover certain findings. He lived during the Islamic Golden Age, in which scholarly thought went hand in hand with the thinking and methodology of the Islamic religion. In addition to type of influence, Al-Biruni was also influenced by other nations, such as the Greek. He was conversant in Khwarezmian, Persian, Arabic, Sanskrit and he spent a large part of his life in Ghazni in modern-day Afghanistan, capital of the Ghaznavid dynasty which was based in what is now central-eastern Afghanistan. In 1017 he traveled to the Indian subcontinent and authored Tarikh Al-Hind after exploring the Hindu faith practised in India and he was given the title founder of Indology. He was a writer on customs and creeds of various nations. He also made contributions to Earth sciences, and is regarded as the father of geodesy for his important contributions to that field and he was born in the outer district of Kath, the capital of the Afrighid dynasty of Khwarezm. The word Biruni means from the outer-district in Persian, and so became his nisba. Al-Birunis relatives also took interest in the studies of science as well and he even had ties to royalty as there are links in his family to the families of prestigious elites. In order to conduct his research, Al-Biruni used different types of methods to tackle the different fields he studied, people consider Al-Biruni to be one of the greatest scientists in history and especially of Islam because of his discoveries and methodology. He lived during the Islamic Golden Age, which promoted astronomy and he was sympathetic to the Afrighids, who were overthrown by the rival dynasty of Mamunids in 995. He left his homeland for Bukhara, then under the Samanid ruler Mansur II the son of Nuh, there he corresponded with Avicenna and there are extant exchanges of views between these two scholars. In 998, he went to the court of the Ziyarid amir of Tabaristan and he also visited the court of the Bavandid ruler Al-Marzuban. Accepting the definite demise of the Afrighids at the hands of the Mamunids and their court at Gorganj was gaining fame for its gathering of brilliant scientists. In 1017, Mahmud of Ghazni took Rey, most scholars, including al-Biruni, were taken to Ghazni, the capital of the Ghaznavid dynasty. Biruni was made court astrologer and accompanied Mahmud on his invasions into India and he was forty-four years old when he went on the journeys with Mahmud of Ghazni. Biruni became acquainted with all related to India

2.
History of mathematics
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Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322, the Rhind Mathematical Papyrus, All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. Greek mathematics greatly refined the methods and expanded the subject matter of mathematics, Chinese mathematics made early contributions, including a place value system. Islamic mathematics, in turn, developed and expanded the known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, from ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, the origins of mathematical thought lie in the concepts of number, magnitude, and form. Modern studies of cognition have shown that these concepts are not unique to humans. Such concepts would have part of everyday life in hunter-gatherer societies. The idea of the number concept evolving gradually over time is supported by the existence of languages which preserve the distinction between one, two, and many, but not of numbers larger than two. Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. The Ishango bone, found near the headwaters of the Nile river, may be more than 20,000 years old, common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers or a six-month lunar calendar. He also writes that no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10, predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian, Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity. The majority of Babylonian mathematical work comes from two widely separated periods, The first few hundred years of the second millennium BC, and it is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics, in contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient Sumerians and they developed a complex system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises, the earliest traces of the Babylonian numerals also date back to this period

3.
Ibn al-Haytham
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Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham, also known by the Latinization Alhazen or Alhacen, was an Arab Muslim scientist, mathematician, astronomer, and philosopher. Ibn al-Haytham made significant contributions to the principles of optics, astronomy, mathematics and he was the first to explain that vision occurs when light bounces on an object and then is directed to ones eyes. He spent most of his close to the court of the Fatimid Caliphate in Cairo and earned his living authoring various treatises. In medieval Europe, Ibn al-Haytham was honored as Ptolemaeus Secundus or simply called The Physicist and he is also sometimes called al-Baṣrī after his birthplace Basra in Iraq, or al-Miṣrī. Ibn al-Haytham was born c.965 in Basra, which was part of the Buyid emirate. Alhazen arrived in Cairo under the reign of Fatimid Caliph al-Hakim, Alhazen continued to live in Cairo, in the neighborhood of the famous University of al-Azhar, until his death in 1040. Legend has it that after deciding the scheme was impractical and fearing the caliphs anger, during this time, he wrote his influential Book of Optics and continued to write further treatises on astronomy, geometry, number theory, optics and natural philosophy. Among his students were Sorkhab, a Persian from Semnan who was his student for three years, and Abu al-Wafa Mubashir ibn Fatek, an Egyptian prince who learned mathematics from Alhazen. Alhazen made significant contributions to optics, number theory, geometry, astronomy, Alhazens work on optics is credited with contributing a new emphasis on experiment. In al-Andalus, it was used by the prince of the Banu Hud dynasty of Zaragossa and author of an important mathematical text. A Latin translation of the Kitab al-Manazir was made probably in the twelfth or early thirteenth century. His research in catoptrics centred on spherical and parabolic mirrors and spherical aberration and he made the observation that the ratio between the angle of incidence and refraction does not remain constant, and investigated the magnifying power of a lens. His work on catoptrics also contains the known as Alhazens problem. Alhazen wrote as many as 200 books, although only 55 have survived, some of his treatises on optics survived only through Latin translation. During the Middle Ages his books on cosmology were translated into Latin, Hebrew, the crater Alhazen on the Moon is named in his honour, as was the asteroid 59239 Alhazen. In honour of Alhazen, the Aga Khan University named its Ophthalmology endowed chair as The Ibn-e-Haitham Associate Professor, Alhazen, by the name Ibn al-Haytham, is featured on the obverse of the Iraqi 10, 000-dinar banknote issued in 2003, and on 10-dinar notes from 1982. The 2015 International Year of Light celebrated the 1000th anniversary of the works on optics by Ibn Al-Haytham, Alhazens most famous work is his seven-volume treatise on optics Kitab al-Manazir, written from 1011 to 1021. Optics was translated into Latin by a scholar at the end of the 12th century or the beginning of the 13th century

4.
Islamic Golden Age
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This period is traditionally said to have ended with the collapse of the Abbasid caliphate due to Mongol invasions and the Sack of Baghdad in 1258 AD. A few contemporary scholars place the end of the Islamic Golden Age as late as the end of 15th to 16th centuries, the metaphor of a golden age began to be applied in 19th-century literature about Islamic history, in the context of the western aesthetic fashion known as Orientalism. There is no definition of term, and depending on whether it is used with a focus on cultural or on military achievement. During the early 20th century, the term was used only occasionally, the Muslim government heavily patronized scholars. The money spent on the Translation Movement for some translations is estimated to be equivalent to twice the annual research budget of the United Kingdom’s Medical Research Council. The best scholars and notable translators, such as Hunayn ibn Ishaq, had salaries that are estimated to be the equivalent of professional athletes today, the House of Wisdom was a library established in Abbasid-era Baghdad, Iraq by Caliph al-Mansur. During this period, the Muslims showed a strong interest in assimilating the knowledge of the civilizations that had been conquered. They also excelled in fields, in particular philosophy, science. For a long period of time the personal physicians of the Abbasid Caliphs were often Assyrian Christians, among the most prominent Christian families to serve as physicians to the caliphs were the Bukhtishu dynasty. Throughout the 4th to 7th centuries, Christian scholarly work in the Greek, the House of Wisdom was founded in Baghdad in 825, modelled after the Academy of Gondishapur. It was led by Christian physician Hunayn ibn Ishaq, with the support of Byzantine medicine, many of the most important philosophical and scientific works of the ancient world were translated, including the work of Galen, Hippocrates, Plato, Aristotle, Ptolemy and Archimedes. Many scholars of the House of Wisdom were of Christian background, the use of paper spread from China into Muslim regions in the eighth century, arriving in Al-Andalus on the Iberian peninsula, present-day Spain in the 10th century. It was easier to manufacture than parchment, less likely to crack than papyrus, Islamic paper makers devised assembly-line methods of hand-copying manuscripts to turn out editions far larger than any available in Europe for centuries. It was from countries that the rest of the world learned to make paper from linen. Ibn Rushd and Ibn Sina played a role in saving the works of Aristotle, whose ideas came to dominate the non-religious thought of the Christian. Ibn Sina and other such as al-Kindi and al-Farabi combined Aristotelianism and Neoplatonism with other ideas introduced through Islam. Arabic philosophic literature was translated into Latin and Ladino, contributing to the development of modern European philosophy, during this period, non-Muslims were allowed to flourish relative to treatment of religious minorities in the Christian Byzantine Empire. The Jewish philosopher Moses Maimonides, who lived in Andalusia, is an example, in epistemology, Ibn Tufail wrote the novel Hayy ibn Yaqdhan and in response Ibn al-Nafis wrote the novel Theologus Autodidactus

5.
Nasir al-Din al-Tusi
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Khawaja Muhammad ibn Muhammad ibn al-Hasan al-Tūsī, better known as Nasīr al-Dīn Tūsī, was a Persian polymath, architect, philosopher, physician, scientist, theologian and Marja Taqleed. He was of the Twelver Shī‘ah Islamic belief, the Muslim scholar Ibn Khaldun considered Tusi to be the greatest of the later Persian scholars. Nasir al-Din Tusi was born in the city of Tus in medieval Khorasan in the year 1201, in Hamadan and Tus he studied the Quran, Hadith, Shia jurisprudence, logic, philosophy, mathematics, medicine and astronomy. He was apparently born into a Shī‘ah family and lost his father at a young age, at a young age he moved to Nishapur to study philosophy under Farid al-Din Damad and mathematics under Muhammad Hasib. He met also Farid al-Din Attar, the legendary Sufi master who was killed by Mongol invaders. In Mosul he studied mathematics and astronomy with Kamal al-Din Yunus and he was captured after the invasion of the Alamut castle by the Mongol forces. Tusi has about 150 works, of which 25 are in Persian and the remaining are in Arabic, here are some of his major works, Kitāb al-Shakl al-qattāʴ Book on the complete quadrilateral. A five volume summary of trigonometry, al-Tadhkirah fiilm al-hayah – A memoir on the science of astronomy. Many commentaries were written about this work called Sharh al-Tadhkirah - Commentaries were written by Abd al-Ali ibn Muhammad ibn al-Husayn al-Birjandi, akhlaq-i Nasiri – A work on ethics. Al-Risalah al-Asturlabiyah – A Treatise on astrolabe, Zij-i ilkhani – A major astronomical treatise, completed in 1272. Sharh al-isharat Awsaf al-Ashraf a short work in Persian Tajrīd al-iʿtiqād – A commentary on Shia doctrines. During his stay in Nishapur, Tusi established a reputation as an exceptional scholar, tusi’s prose writing, which number over 150 works, represent one of the largest collections by a single Islamic author. Writing in both Arabic and Persian, Nasir al-Din Tusi dealt with religious topics and non-religious or secular subjects. His works include the definitive Arabic versions of the works of Euclid, Archimedes, Ptolemy, Autolycus, Tusi convinced Hulegu Khan to construct an observatory for establishing accurate astronomical tables for better astrological predictions. Beginning in 1259, the Rasad Khaneh observatory was constructed in Azarbaijan, south of the river Aras, and to the west of Maragheh, the capital of the Ilkhanate Empire. Based on the observations in this for the time being most advanced observatory and this book contains astronomical tables for calculating the positions of the planets and the names of the stars. His model for the system is believed to be the most advanced of his time. Between Ptolemy and Copernicus, he is considered by many to be one of the most eminent astronomers of his time, for his planetary models, he invented a geometrical technique called a Tusi-couple, which generates linear motion from the sum of two circular motions

6.
Qutb al-Din al-Shirazi
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Mosleh Shirazi was a 13th-century Persian polymath and poet who made contributions to astronomy, mathematics, medicine, physics, music theory, philosophy and Sufism. He was born in Kazerun in October 1236 to a family with a tradition of Sufism and his father, Zia al-Din Masud Kazeruni was a physician by profession and also a leading Sufi of the Kazeruni order. Zia Al-Din received his Kherqa from Shahab al-Din Omar Suhrawardi, Qutb al-Din was garbed by the Kherqa as blessing by his father at age of ten. Later on, he received his own robe from the hands of Najib al-Din Bozgush Shirazni. Quṭb al-Din began studying medicine under his father and his father practiced and taught medicine at the Mozaffari hospital in Shiraz. After the passing away of his father, his uncle and other masters of the period trained him in medicine and he also studied the Qanun of the famous Persian scholar Avicenna and its commentaries. In particular he read the commentary of Fakhr al-Din Razi on the Canon of Medicine and this led to his own decision to write his own commentary, where he resolved many of the issues in the company of Nasir al-Din al-Tusi. Qutb al-Din lost his father at the age of fourteen and replaced him as the ophthalmologist at the Mozaffari hospital in Shiraz, at the same time, he pursued his education under his uncle Kamal al-Din Abul Khayr and then Sharaf al-Din Zaki Bushkani, and Shams al-Din Mohammad Kishi. All three were teachers of the Canon of Avicenna. He quit his medical profession ten years later and began to devote his time to education under the guidance of Nasir al-Din al-Tusi. When Nasir al-Din al-Tusi, the renowned scholar-vizier of the Mongol Holagu Khan established the observatory of Maragha and he left Shiraz sometime after 1260 and was in Maragha about 1262. In Maragha, Qutb al-din resumed his education under Nasir al-Din al-Tusi and he discussed the difficulties he had with Nasir al-Din al-Tusi on understanding the first book of the Canon of Avicenna. While working in the new observatory, studied astronomy under him, one of the important scientific projects was the completion of the new astronomical table. In his testament, Nasir al-Din al-Tusi advises his son ṣil-a-Din to work with Qutb al-Din in the completion of the Zij, qutb-al-Dins stay in Maragha was short. Subsequently, he traveled to Khorasan in the company of Nasir al-Din al-Tusi where he stayed to study under Najm al-Din Katebi Qazvini in the town of Jovayn, some time after 1268, he journeyed to Qazvin, Isfahan, Baghdad and later Konya in Anatolia. This was a time when the Persian poet Jalal al-Din Muhammad Balkhi was gaining fame there, in Konya, he studied the Jame al-Osul of Ibn Al-Athir with Sadr al-Din Qunawi. The governor of Konya, Moin al-Din Parvana appointed him as the judge of Sivas and it was during this time that he compiled the books the Meftāḥ al-meftāh, Ekhtiārāt al-moẓaffariya, and his commentary on Sakkāki. In the year 1282, he was envoy on behalf of the Ilkhanid Ahmad Takudar to Sayf al-Din Qalawun, in his letter to Qalawun, the Ilkhanid ruler mentions Qutb al-Din as the chief judge

7.
Mathematics in medieval Islam
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Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics and Indian mathematics. Arabic works also played an important role in the transmission of mathematics to Europe during the 10th to 12th centuries, the study of algebra, the name of which is 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 roots of first. He also introduces the method of reduction, and unlike Diophantus, Al-Khwarizmis algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the work of Diophantus, which was syncopated. The transition to symbolic algebra, where symbols are used, can be seen in the work of Ibn al-Banna al-Marrakushi. It is important to understand just how significant this new idea was and it was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a theory which allowed rational numbers, irrational numbers, geometrical magnitudes. It gave mathematics a whole new development path so much broader in concept to that which had existed before, 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 expanded on the algebra of Al-Khwarizmi. 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 Khayyám wrote the Treatise on Demonstration of Problems of Algebra containing the solution of cubic or third-order equations. Khayyám obtained the solutions of equations by finding the intersection points of two conic sections. This method had used by the Greeks, but they did not generalize the method to cover all equations with positive roots. Sharaf al-Dīn al-Ṭūsī developed an approach to the investigation of cubic equations—an approach which entailed finding the point at which a cubic polynomial obtains its maximum value. 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 Euclids 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

8.
Muhammad ibn Musa al-Khwarizmi
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Muḥammad ibn Mūsā al-Khwārizmī, formerly Latinized as Algoritmi, was a Persian mathematician, astronomer, and geographer during the Abbasid Caliphate, a scholar in the House of Wisdom in Baghdad. In the 12th century, Latin translations of his work on the Indian numerals introduced the decimal number system to the Western world. Al-Khwārizmīs The Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and he is often considered one of the fathers of algebra. He revised Ptolemys Geography and wrote on astronomy and astrology, some words reflect the importance of al-Khwārizmīs contributions to mathematics. Algebra is derived from al-jabr, one of the two operations he used to solve quadratic equations, algorism and algorithm stem from Algoritmi, the Latin form of his name. His name is also the origin of guarismo and of algarismo, few details of al-Khwārizmīs life are known with certainty. He was born in a Persian family and Ibn al-Nadim gives his birthplace as Khwarezm in Greater Khorasan, muhammad ibn Jarir al-Tabari gives his name as Muḥammad ibn Musá al-Khwārizmiyy al-Majūsiyy al-Quṭrubbaliyy. The epithet al-Qutrubbulli could indicate he might instead have come from Qutrubbul and this would not be worth mentioning if a series of errors concerning the personality of al-Khwārizmī, occasionally even the origins of his knowledge, had not been made. Recently, G. J. Toomer. with naive confidence constructed an entire fantasy on the error which cannot be denied the merit of amusing the reader. Regarding al-Khwārizmīs religion, Toomer writes, Another epithet given to him by al-Ṭabarī, al-Majūsī, Ibn al-Nadīms Kitāb al-Fihrist includes a short biography on al-Khwārizmī together with a list of the books he wrote. Al-Khwārizmī accomplished most of his work in the period between 813 and 833, douglas Morton Dunlop suggests that it may have been possible that Muḥammad ibn Mūsā al-Khwārizmī was in fact the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the three Banū Mūsā. Al-Khwārizmīs contributions to mathematics, geography, astronomy, and cartography established the basis for innovation in algebra, on the Calculation with Hindu Numerals written about 825, was principally responsible for spreading the Hindu–Arabic numeral system throughout the Middle East and Europe. It was translated into Latin as Algoritmi de numero Indorum, al-Khwārizmī, rendered as Algoritmi, led to the term algorithm. Some of his work was based on Persian and Babylonian astronomy, Indian numbers, al-Khwārizmī systematized and corrected Ptolemys data for Africa and the Middle East. Another major book was Kitab surat al-ard, presenting the coordinates of places based on those in the Geography of Ptolemy but with improved values for the Mediterranean Sea, Asia and he also wrote on mechanical devices like the astrolabe and sundial. He assisted a project to determine the circumference of the Earth and in making a map for al-Mamun. When, in the 12th century, his works spread to Europe through Latin translations, the Compendious Book on Calculation by Completion and Balancing is a mathematical book written approximately 830 CE. The term algebra is derived from the name of one of the operations with equations described in this book

9.
Al-Kindi
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Abu Yūsuf Yaʻqūb ibn ʼIsḥāq aṣ-Ṣabbāḥ al-Kindī, known as the Philosopher of the Arabs, was a Muslim Arab philosopher, polymath, mathematician, physician and musician. Al-Kindi was a descendant of the Kinda tribe and he was born in Basra and educated in Baghdad. In the field of mathematics, al-Kindi played an important role in introducing Indian numerals to the Islamic and he was a pioneer in cryptanalysis and devised several new methods of breaking ciphers. Using his mathematical and medical expertise, he was able to develop a scale that would allow doctors to quantify the potency of their medication, the central theme underpinning al-Kindis philosophical writings is the compatibility between philosophy and other orthodox Islamic sciences, particularly theology. And many of his works deal with subjects that theology had an immediate interest in and these include the nature of God, the soul and prophetic knowledge. Al-Kindi was born in Kufa to a family of the Kinda tribe, descended from the chieftain al-Ashath ibn Qays. His father Ishaq was the governor of Kufa, and al-Kindi received his education there. He later went to complete his studies in Baghdad, where he was patronized by the Abbasid caliphs al-Mamun and he was also well known for his beautiful calligraphy, and at one point was employed as a calligrapher by al-Mutawakkil. When al-Mamun died, his brother, al-Mutasim became Caliph, al-Kindis position would be enhanced under al-Mutasim, who appointed him as a tutor to his son. But on the accession of al-Wathiq, and especially of al-Mutawakkil, henry Corbin, an authority on Islamic studies, says that in 873, al-Kindi died a lonely man, in Baghdad during the reign of al-Mutamid. After his death, al-Kindis philosophical works quickly fell into obscurity and many of them were lost even to later Islamic scholars, felix Klein-Franke suggests a number of reasons for this, aside from the militant orthodoxy of al-Mutawakkil, the Mongols also destroyed countless libraries during their invasion. Al-Kindi was a master of different areas of thought and was held to be one of the greatest Islamic philosophers of his time. The Italian Renaissance scholar Geralomo Cardano considered him one of the twelve greatest minds of the Middle Ages, according to Ibn al-Nadim, al-Kindi wrote at least two hundred and sixty books, contributing heavily to geometry, medicine and philosophy, logic, and physics. His influence in the fields of physics, mathematics, medicine, philosophy and music were far-reaching and his greatest contribution to the development of Islamic philosophy was his efforts to make Greek thought both accessible and acceptable to a Muslim audience. Al-Kindi carried out this mission from the House of Wisdom, an institute of translation and learning patronized by the Abbasid Caliphs, in Baghdad. In his writings, one of al-Kindis central concerns was to demonstrate the compatibility between philosophy and natural theology on the one hand, and revealed or speculative theology on the other. Despite this, he did make clear that he believed revelation was a source of knowledge to reason because it guaranteed matters of faith that reason could not uncover. This was an important factor in the introduction and popularization of Greek philosophy in the Muslim intellectual world