1.
Science and technology in Iran
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Iran has made considerable advances in science and technology through education and training, despite international sanctions in almost all aspects of research during the past 30 years. Irans university population swelled from 100,000 in 1979 to 2 million in 2006, 70% of its science and engineering students are women. Irans scientific progress is reported to be the fastest in the world, Iran has made great strides in different sectors, including aerospace, nuclear science, medical development, as well as stem cell and cloning research. Persia was a cradle of science in earlier times, contributing to medicine, mathematics, science, trying to revive the golden time of Persian science, Irans scientists now are cautiously reaching out to the world. Many individual Iranian scientists, along with the Iranian Academy of Medical Sciences, Science in Persia evolved in two main phases separated by the arrival and widespread adoption of Islam in the region. References to scientific subjects such as science and mathematics occur in books written in the Pahlavi languages. The Qanat originated in pre-Achaemenid Persia, the oldest and largest known qanat is in the Iranian city of Gonabad, which, after 2,700 years, still provides drinking and agricultural water to nearly 40,000 people. Persian philosophers and inventors may have created the first batteries in the Parthian or Sassanid eras, some have suggested that the batteries may have been used medicinally. Windwheels were developed by the Babylonians ca.1700 BC to pump water for irrigation, in the 7th century, Persian engineers in Greater Iran developed a more advanced wind-power machine, the windmill, building upon the basic model developed by the Babylonians. The 9th century mathematician Muhammad Ibn Musa-al-Kharazmi created the Logarithm table, developed algebra and his writings were translated into Latin by Gerard of Cremona under the title, De jebra et almucabola. Robert of Chester also translated it under the title Liber algebras et almucabala, the works of Kharazmi exercised a profound influence on the development of mathematical thought in the medieval West. The practice and study of medicine in Iran has a long, situated at the crossroads of the East and West, Persia was often involved in developments in ancient Greek and Indian medicine, pre- and post-Islamic Iran have been involved in medicine as well. For example, the first teaching hospital where medical students methodically practiced on patients under the supervision of physicians was the Academy of Gundishapur in the Persian Empire. Some experts go so far as to claim that, to a large extent. The idea of xenotransplantation dates to the days of Achaemenidae, as evidenced by engravings of many mythologic chimeras still present in Persepolis, several documents still exist from which the definitions and treatments of the headache in medieval Persia can be ascertained. These documents give detailed and precise information on the different types of headaches. The medieval physicians listed various signs and symptoms, apparent causes, the medieval writings are both accurate and vivid, and they provide long lists of substances used in the treatment of headaches. Many of the approaches of physicians in medieval Persia are accepted today, however, although largely mythical in content, the passage illustrates working knowledge of anesthesia in ancient Persia
2.
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
3.
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
4.
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
5.
Persian language
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Persian, also known by its endonym Farsi, is one of the Western Iranian languages within the Indo-Iranian branch of the Indo-European language family. It is primarily spoken in Iran, Afghanistan, and Tajikistan and it is mostly written in the Persian alphabet, a modified variant of the Arabic script. Its grammar is similar to that of many contemporary European languages, Persian gets its name from its origin at the capital of the Achaemenid Empire, Persis, hence the name Persian. A Persian-speaking person may be referred to as Persophone, there are approximately 110 million Persian speakers worldwide, with the language holding official status in Iran, Afghanistan, and Tajikistan. For centuries, Persian has also been a cultural language in other regions of Western Asia, Central Asia. It also exerted influence on Arabic, particularly Bahrani Arabic. Persian is one of the Western Iranian languages within the Indo-European family, other Western Iranian languages are the Kurdish languages, Gilaki, Mazanderani, Talysh, and Balochi. Persian is classified as a member of the Southwestern subgroup within Western Iranian along with Lari, Kumzari, in Persian, the language is known by several names, Western Persian, Parsi or Farsi has been the name used by all native speakers until the 20th century. Since the latter decades of the 20th century, for reasons, in English. Tajiki is the variety of Persian spoken in Tajikistan and Uzbekistan by the Tajiks, according to the Oxford English Dictionary, the term Persian as a language name is first attested in English in the mid-16th century. Native Iranian Persian speakers call it Fārsi, Farsi is the Arabicized form of Pārsi, subsequent to Muslim conquest of Persia, due to a lack of the phoneme /p/ in Standard Arabic. The origin of the name Farsi and the place of origin of the language which is Fars Province is the Arabicized form of Pārs, in English, this language has historically been known as Persian, though Farsi has also gained some currency. Farsi is encountered in some literature as a name for the language. In modern English the word Farsi refers to the language while Parsi describes Zoroastrians, some Persian language scholars such as Ehsan Yarshater, editor of Encyclopædia Iranica, and University of Arizona professor Kamran Talattof, have also rejected the usage of Farsi in their articles. The international language-encoding standard ISO 639-1 uses the code fa, as its system is mostly based on the local names. The more detailed standard ISO 639-3 uses the name Persian for the dialect continuum spoken across Iran and Afghanistan and this consists of the individual languages Dari and Iranian Persian. Currently, Voice of America, BBC World Service, Deutsche Welle, Radio Free Europe/Radio Liberty also includes a Tajik service and an Afghan service. This is also the case for the American Association of Teachers of Persian, The Centre for Promotion of Persian Language and Literature, Persian is an Iranian language belonging to the Indo-Iranian branch of the Indo-European family of languages
6.
Persian people
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The Persians are an Iranian ethnic group that make up over half the population of Iran. They share a cultural system and are native speakers of the Persian language. The ancient Persians were originally a branch of the ancient Iranian population who entered modern-day Iran by the early 10th century BC. The English term Persian derives from Latin Persia, itself deriving from Greek Persís, in the Bible, it is referred to as Parás —sometimes Paras uMadai —within the books of Esther, Daniel, Ezra and Nehemya. Although Persis was originally one of the provinces of ancient Iran, varieties of this term were adopted through Greek sources, thus, in the Western world, the term Persian came to refer to all inhabitants of the country. 10th-century Iraqi historian Al-Masudi refers to Pahlavi, Dari and Azari as dialects of the Persian language, in 1333, medieval Moroccan traveler and scholar Ibn Battuta, referred to the people of Kabul as a specific sub-tribe of Persians. Lady Mary Sheil, in her observation of Iran during the Qajar era, describes Persians, Kurds and Leks to identify themselves as descendants of the ancient Persians. On March 21,1935, the king of Iran, Reza Shah Pahlavi, issued a decree asking the international community to use the term Iran. However, the term Persian is still used to designate the predominant population of the Iranian peoples living in the Iranian cultural continent. The earliest known written record attributed to the Persians is from the Black Obelisk of Shalmaneser III, the inscription mentions Parsua as a tribal chiefdom in modern-day western Iran. The ancient Persians were originally a branch of the Iranian population that, in the early 10th century BC. They were initially dominated by the Assyrians for much of the first three centuries after arriving in the region, however, they played a role in the downfall of the Neo-Assyrian Empire. The Medes, another branch of population, founded the unified empire of Media as the regions dominant cultural and political power in c.625 BC. Meanwhile, the Persian dynasty of the Achaemenids formed a state to the central Median power. In c.552 BC, the Achaemenids began a revolution which led to the conquest of the empire by Cyrus II in c.550 BC. They spread their influence to the rest of what is called the Iranian Plateau, at its greatest extent, the Achaemenid Empire stretched from parts of Eastern Europe in the west, to the Indus Valley in the east, making it the largest empire the world had yet seen. The Achaemenids developed the infrastructure to support their growing influence, including the creation of Pasargadae and its legacy and impact on the kingdom of Macedon was also notably huge, even for centuries after the withdrawal of the Persians from Europe following the Greco-Persian Wars. The empire collapsed in 330 BC following the conquests of Alexander the Great, until the Parthian era, the Iranian identity had an ethnic, linguistic, and religious value, however, it did not yet have a political import
7.
Muslim
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A Muslim is someone who follows or practices Islam, a monotheistic Abrahamic religion. Muslims consider the Quran, their book, to be the verbatim word of God as revealed to the Islamic prophet. They also follow the teachings and practices of Muhammad as recorded in traditional accounts, Muslim is an Arabic word meaning one who submits. Most Muslims will accept anyone who has publicly pronounced Shahadah as a Muslim, the shahadah states, There is no god but the God and Muhammad is the last messenger of the God. The testimony authorized by God in the Quran that can found in Surah 3,18 states, There is no god except God, which in Arabic, is the exact testimony which God Himself utters, as well as the angels and those who possess knowledge utter. The word muslim is the active participle of the verb of which islām is a verbal noun, based on the triliteral S-L-M to be whole. A female adherent is a muslima, the plural form in Arabic is muslimūn or muslimīn, and its feminine equivalent is muslimāt. The Arabic form muslimun is the stem IV participle of the triliteral S-L-M, the ordinary word in English is Muslim. It is sometimes transliterated as Moslem, which is an older spelling, the word Mosalman is a common equivalent for Muslim used in Central Asia. Until at least the mid-1960s, many English-language writers used the term Mohammedans or Mahometans, although such terms were not necessarily intended to be pejorative, Muslims argue that the terms are offensive because they allegedly imply that Muslims worship Muhammad rather than God. Other obsolete terms include Muslimite and Muslimist, musulmán/Mosalmán is a synonym for Muslim and is modified from Arabic. In English it was sometimes spelled Mussulman and has become archaic in usage, the Muslim philosopher Ibn Arabi said, A Muslim is a person who has dedicated his worship exclusively to God. Islam means making ones religion and faith Gods alone. The Quran states that men were Muslims because they submitted to God, preached His message and upheld His values. Thus, in Surah 3,52 of the Quran, Jesus disciples tell him, We believe in God, and you be our witness that we are Muslims. In Muslim belief, before the Quran, God had given the Tawrat to Moses, the Zabur to David and the Injil to Jesus, who are all considered important Muslim prophets. The most populous Muslim-majority country is Indonesia, home to 12. 7% of the worlds Muslims, followed by Pakistan, Bangladesh, and Egypt. About 20% of the worlds Muslims lives in the Middle East and North Africa, Sizable minorities are found in India, China, Russia, Ethiopia. The country with the highest proportion of self-described Muslims as a proportion of its population is Morocco
8.
Optics
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Optics is the branch of physics which involves the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light, because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties. Most optical phenomena can be accounted for using the classical description of light. Complete electromagnetic descriptions of light are, however, often difficult to apply in practice, practical optics is usually done using simplified models. The most common of these, geometric optics, treats light as a collection of rays that travel in straight lines, physical optics is a more comprehensive model of light, which includes wave effects such as diffraction and interference that cannot be accounted for in geometric optics. Historically, the model of light was developed first, followed by the wave model of light. Progress in electromagnetic theory in the 19th century led to the discovery that waves were in fact electromagnetic radiation. Some phenomena depend on the fact that light has both wave-like and particle-like properties, explanation of these effects requires quantum mechanics. When considering lights particle-like properties, the light is modelled as a collection of particles called photons, quantum optics deals with the application of quantum mechanics to optical systems. Optical science is relevant to and studied in many related disciplines including astronomy, various engineering fields, photography, practical applications of optics are found in a variety of technologies and everyday objects, including mirrors, lenses, telescopes, microscopes, lasers, and fibre optics. Optics began with the development of lenses by the ancient Egyptians and Mesopotamians, the earliest known lenses, made from polished crystal, often quartz, date from as early as 700 BC for Assyrian lenses such as the Layard/Nimrud lens. The ancient Romans and Greeks filled glass spheres with water to make lenses, the word optics comes from the ancient Greek word ὀπτική, meaning appearance, look. Greek philosophy on optics broke down into two opposing theories on how vision worked, the theory and the emission theory. The intro-mission approach saw vision as coming from objects casting off copies of themselves that were captured by the eye, plato first articulated the emission theory, the idea that visual perception is accomplished by rays emitted by the eyes. He also commented on the parity reversal of mirrors in Timaeus, some hundred years later, Euclid wrote a treatise entitled Optics where he linked vision to geometry, creating geometrical optics. Ptolemy, in his treatise Optics, held a theory of vision, the rays from the eye formed a cone, the vertex being within the eye. The rays were sensitive, and conveyed back to the observer’s intellect about the distance. He summarised much of Euclid and went on to describe a way to measure the angle of refraction, during the Middle Ages, Greek ideas about optics were resurrected and extended by writers in the Muslim world
9.
Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers
10.
Astronomy in the medieval Islamic world
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Islamic astronomy comprises the astronomical developments made in the Islamic world, particularly during the Islamic Golden Age, and mostly written in the Arabic language. These developments mostly took place in the Middle East, Central Asia, Al-Andalus, and North Africa and these included Greek, Sassanid, and Indian works in particular, which were translated and built upon. Islamic astronomy also had an influence on Chinese astronomy and Malian astronomy, a significant number of stars in the sky, such as Aldebaran, Altair and Deneb, and astronomical terms such as alidade, azimuth, and nadir, are still referred to by their Arabic names. A large corpus of literature from Islamic astronomy remains today, numbering approximately 10,000 manuscripts scattered throughout the world, even so, a reasonably accurate picture of Islamic activity in the field of astronomy can be reconstructed. These observations were based on the rising and setting of stars. 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. The first astronomical texts that were translated into Arabic were of Indian and Persian origin, 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 function in place of the chords of arc used in Greek trigonometry. The House of Wisdom was an established in Baghdad under Abbasid caliph Al-Mamun in the early 9th century. From this time, independent investigation into the Ptolemaic system became possible, Astronomical research was greatly 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 supported this work financially, but 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 as it introduced Ptolemaic concepts into Islamic sciences and this work also marks the turning point in Islamic astronomy. Hitherto, Muslim astronomers had adopted a primarily research approach to the field, translating works of others, al-Khwarizmis work marked the beginning of nontraditional methods of study and calculations. In 850, al-Farghani wrote Kitab fi Jawani, the book primarily gave a summary of Ptolemic cosmography. However, it also 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, and the circumference of the earth. The book was circulated through the Muslim world, and even translated into Latin. 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
11.
Book of Optics
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The Book of Optics is a seven-volume treatise on optics and other fields of study composed by the medieval Arab scholar Ibn al-Haytham, known in the West as Alhazen or Alhacen. Alhazens work extensively affected the development of optics in Europe between 1260 and 1650, before the Book of Optics was written, two theories of vision existed. The extramission or emission theory was forwarded by the mathematicians Euclid and Ptolemy, when these rays reached the object they allowed the viewer to perceive its color, shape and size. The intromission theory, held by the followers of Aristotle and Galen, argued that sight was caused by agents, al-Haytham offered many reasons against the extramission theory, pointing to the fact that eyes can be damaged by looking directly at bright lights, such as the sun. He claimed the low probability that the eye can fill the entirety of space as soon as the eyelids are opened as an observer looks up into the night sky. According to this theory, the object being viewed is considered to be a compilation of an amount of points. In the Book of Optics, al-Haytham claimed the existence of primary and secondary light, the book describes how the essential form of light comes from self-luminous bodies and that accidental light comes from objects that obtain and emit light from those self-luminous bodies. According to Ibn al-Haytham, primary light comes from self-luminous bodies, accidental light can only exist if there is a source of primary light. Both primary and secondary light travel in straight lines, transparency is a characteristic of a body that can transmit light through them, such as air and water, although no body can completely transmit light or be entirely transparent. Opaque objects are those through which light cannot pass through directly, opaque objects are struck with light and can become luminous bodies themselves which radiate secondary light. Light can be refracted by going through partially transparent objects and can also be reflected by striking smooth objects such as mirrors, al-Haytham presented many experiments in Optics that upheld his claims about light and its transmission. He also claimed that acts much like light, being a distinct quality of a form. Through experimentation he concluded that color cannot exist without air, as objects radiate light in straight lines in all directions, the eye must also be hit with this light over its outer surface. This idea presented a problem for al-Haytham and his predecessors, as if this was the case, al-Haytham solved this problem using his theory of refraction. According to al-Haytham, this causes them to be refracted and weakened and he claimed that all the rays other than the one that hits the eye perpendicularly are not involved in vision. Other parts of the eye are the aqueous humor in front of the crystalline humor and these, however, do not play as critical of a role in vision as the crystalline humor. The crystalline humor transmits the image it perceives to the brain through an optic nerve, Book I - Book I deals with al-Haythams theories on light, colors, and vision. Book II - Book II is where al-Haytham presents his theory of visual perception, Book III and Book VI - Book III and Book VI present al-Haythams ideas on the errors in visual perception with Book VI focusing on errors related to reflection
12.
Avicenna
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Avicenna or Ibn Sīnā was a Persian polymath who is regarded as one of the most significant thinkers and writers of the Islamic Golden Age. Of the 450 works he is known to have written, around 240 have survived, in 1973, Avicennas Canon Of Medicine was reprinted in New York. Besides philosophy and medicine, Avicennas corpus includes writings on astronomy, alchemy, geography and geology, psychology, Islamic theology, logic, mathematics, physics and poetry. Avicenna is a Latin corruption of the Arabic patronym Ibn Sīnā, meaning Son of Sina, however, Avicenna was not the son, but the great-great-grandson of a man named Sina. His full name was Abū ʿAlī al-Ḥusayn ibn ʿAbd Allāh ibn al-Ḥasan ibn ʿAlī ibn Sīnā, Ibn Sina created an extensive corpus of works during what is commonly known as the Islamic Golden Age, in which the translations of Greco-Roman, Persian, and Indian texts were studied extensively. Under the Samanids, Bukhara rivaled Baghdad as a capital of the Islamic world. The study of the Quran and the Hadith thrived in such a scholarly atmosphere, philosophy, Fiqh and theology were further developed, most noticeably by Avicenna and his opponents. Al-Razi and Al-Farabi had provided methodology and knowledge in medicine and philosophy, Avicenna had access to the great libraries of Balkh, Khwarezm, Gorgan, Rey, Isfahan and Hamadan. Various texts show that he debated philosophical points with the greatest scholars of the time, aruzi Samarqandi describes how before Avicenna left Khwarezm he had met Al-Biruni, Abu Nasr Iraqi, Abu Sahl Masihi and Abu al-Khayr Khammar. Avicenna was born c. 980 in Afshana, a village near Bukhara, the capital of the Samanids, a Persian dynasty in Central Asia and Greater Khorasan. His mother, named Setareh, was from Bukhara, his father, Abdullah, was a respected Ismaili scholar from Balkh and his father worked in the government of Samanid in the village Kharmasain, a Sunni regional power. After five years, his brother, Mahmoud, was born. Avicenna first began to learn the Quran and literature in such a way that when he was ten years old he had learned all of them. According to his autobiography, Avicenna had memorised the entire Quran by the age of 10 and he learned Indian arithmetic from an Indian greengrocer, ءMahmoud Massahi and he began to learn more from a wandering scholar who gained a livelihood by curing the sick and teaching the young. He also studied Fiqh under the Sunni Hanafi scholar Ismail al-Zahid, Avicenna was taught some extent of philosophy books such as Introduction s Porphyry, Euclids Elements, Ptolemys Almagest by an unpopular philosopher, Abu Abdullah Nateli, who claimed philosophizing. As a teenager, he was troubled by the Metaphysics of Aristotle. For the next year and a half, he studied philosophy, in such moments of baffled inquiry, he would leave his books, perform the requisite ablutions, then go to the mosque, and continue in prayer till light broke on his difficulties. Deep into the night, he would continue his studies, and even in his dreams problems would pursue him and work out their solution
13.
Rainbow
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A rainbow is a meteorological phenomenon that is caused by reflection, refraction and dispersion of light in water droplets resulting in a spectrum of light appearing in the sky. It takes the form of a multicoloured arc, Rainbows caused by sunlight always appear in the section of sky directly opposite the sun. However, the observer sees only an arc formed by illuminated droplets above the ground. In a primary rainbow, the arc shows red on the outer part and this rainbow is caused by light being refracted when entering a droplet of water, then reflected inside on the back of the droplet and refracted again when leaving it. In a double rainbow, an arc is seen outside the primary arc. A rainbow is not located at a distance from the observer. Thus, a rainbow is not an object and cannot be physically approached, indeed, it is impossible for an observer to see a rainbow from water droplets at any angle other than the customary one of 42 degrees from the direction opposite the light source. Even if an observer sees another observer who seems under or at the end of a rainbow, Rainbows span a continuous spectrum of colours. Rainbows can be caused by many forms of airborne water and these include not only rain, but also mist, spray, and airborne dew. Rainbows can be observed there are water drops in the air. Because of this, rainbows are seen in the western sky during the morning. The most spectacular rainbow displays happen when half the sky is dark with raining clouds. The result is a rainbow that contrasts with the darkened background. During such good visibility conditions, the larger but fainter secondary rainbow is often visible and it appears about 10° outside of the primary rainbow, with inverse order of colours. The rainbow effect is commonly seen near waterfalls or fountains. In addition, the effect can be created by dispersing water droplets into the air during a sunny day. Rarely, a moonbow, lunar rainbow or nighttime rainbow, can be seen on strongly moonlit nights, as human visual perception for colour is poor in low light, moonbows are often perceived to be white. It is difficult to photograph the complete semicircle of a rainbow in one frame, for a 35 mm camera, a wide-angle lens with a focal length of 19 mm or less would be required
14.
Camera obscura
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The surroundings of the projected image have to be relatively dark for the image to be clear, so many historical camera obscura experiments were performed in dark rooms. The term camera obscura also refers to constructions or devices that use of the principle within a box. Camerae obscurae with a lens in the opening have been used since the second half of the 16th century, before the term camera obscura was first used in 1604, many other expressions were used including cubiculum obscurum, cubiculum tenebricosum, conclave obscurum and locus obscurus. Rays of light travel in straight lines and change when they are reflected and partly absorbed by an object, retaining information about the color, lit objects reflect rays of light in all directions. The human eye itself works much like a camera obscura with an opening, a biconvex lens, a camera obscura device consists of a box, tent or room with a small hole in one side. Light from a scene passes through the hole and strikes a surface inside, where the scene is reproduced, inverted and reversed. The image can be projected onto paper, and can then be traced to produce an accurate representation. In order to produce a reasonably clear projected image, the aperture has to be about 1/100th the distance to the screen, many camerae obscurae use a lens rather than a pinhole because it allows a larger aperture, giving a usable brightness while maintaining focus. As the pinhole is made smaller, the image gets sharper, with too small a pinhole, however, the sharpness worsens, due to diffraction. Using mirrors, as in an 18th-century overhead version, it is possible to project a right-side-up image, another more portable type is a box with an angled mirror projecting onto tracing paper placed on the glass top, the image being upright as viewed from the back. There are theories that occurrences of camera obscura effects inspired paleolithic cave paintings and it is also suggested that camera obscura projections could have played a role in Neolithic structures. Perforated gnomons projecting an image of the sun were described in the Chinese Zhoubi Suanjing writings. The location of the circle can be measured to tell the time of day. In Arab and European cultures its invention was later attributed to Egyptian astronomer. Some ancient sightings of gods and spirits, especially in worship, are thought to possibly have been conjured up by means of camera obscura projections. In these writings it is explained how the image in a collecting-point or treasure house is inverted by an intersecting point that collected the light. Light coming from the foot of a person would partly be hidden below. Rays from the head would partly be hidden above and partly form the part of the image
15.
Isaac Newton
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His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton also made contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus. Newtons Principia formulated the laws of motion and universal gravitation that dominated scientists view of the universe for the next three centuries. Newtons work on light was collected in his influential book Opticks. He also formulated a law of cooling, made the first theoretical calculation of the speed of sound. Newton was a fellow of Trinity College and the second Lucasian Professor of Mathematics at the University of Cambridge, politically and personally tied to the Whig party, Newton served two brief terms as Member of Parliament for the University of Cambridge, in 1689–90 and 1701–02. He was knighted by Queen Anne in 1705 and he spent the last three decades of his life in London, serving as Warden and Master of the Royal Mint and his father, also named Isaac Newton, had died three months before. Born prematurely, he was a child, his mother Hannah Ayscough reportedly said that he could have fit inside a quart mug. When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabas Smith, leaving her son in the care of his maternal grandmother, Newtons mother had three children from her second marriage. From the age of twelve until he was seventeen, Newton was educated at The Kings School, Grantham which taught Latin and Greek. He was removed from school, and by October 1659, he was to be found at Woolsthorpe-by-Colsterworth, Henry Stokes, master at the Kings School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a bully, he became the top-ranked student. In June 1661, he was admitted to Trinity College, Cambridge and he started as a subsizar—paying his way by performing valets duties—until he was awarded a scholarship in 1664, which guaranteed him four more years until he would get his M. A. He set down in his notebook a series of Quaestiones about mechanical philosophy as he found it, in 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton had obtained his B. A. degree in August 1665, in April 1667, he returned to Cambridge and in October was elected as a fellow of Trinity. Fellows were required to become ordained priests, although this was not enforced in the restoration years, however, by 1675 the issue could not be avoided and by then his unconventional views stood in the way. Nevertheless, Newton managed to avoid it by means of a special permission from Charles II. A and he was elected a Fellow of the Royal Society in 1672. Newtons work has been said to distinctly advance every branch of mathematics then studied and his work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newtons mathematical papers
16.
Amicable numbers
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Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. A pair of amicable numbers constitutes a sequence of period 2. A related concept is that of a number, which is a number that equals the sum of its own proper divisors. Numbers that are members of a sequence with period greater than 2 are known as sociable numbers. The smallest pair of numbers is. The first ten amicable pairs are, and, Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties. A general formula by some of these numbers could be derived was invented circa 850 by the Iraqi mathematician Thābit ibn Qurra. Other Arab mathematicians who studied amicable numbers are al-Majriti, al-Baghdadi, the Iranian mathematician Muhammad Baqir Yazdi discovered the pair, though this has often been attributed to Descartes. Much of the work of Eastern mathematicians in this area has been forgotten, Thābit ibn Qurras formula was rediscovered by Fermat and Descartes, to whom it is sometimes ascribed, and extended by Euler. It was extended further by Borho in 1972, Fermat and Descartes also rediscovered pairs of amicable numbers known to Arab mathematicians. Euler also discovered dozens of new pairs, the second smallest pair, was discovered in 1866 by a then teenage B. Paganini, having been overlooked by earlier mathematicians, by 1946 there were 390 known pairs, but the advent of computers has allowed the discovery of many thousands since then. Exhaustive searches have been carried out to find all pairs less than a bound, this bound being extended from 108 in 1970, to 1010 in 1986,1011 in 1993,1017 in 2015. As of April 2016, there are over 1,000,000,000 known amicable pairs, while these rules do generate some pairs of amicable numbers, many other pairs are known, so these rules are by no means comprehensive. The Thābit ibn Qurra theorem is a method for discovering amicable numbers invented in the century by the Arab mathematician Thābit ibn Qurra. This formula gives the pairs for n =2, for n =4, and for n =7, Numbers of the form 3×2n −1 are known as Thabit numbers. In order for Ibn Qurras formula to produce an amicable pair, to establish the theorem, Thâbit ibn Qurra proved nine lemmas divided into two groups. The first three deal with the determination of the aliquot parts of a natural integer
17.
Factorization
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In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viètes formulas relate the coefficients of a polynomial to its roots, the opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms. Integer factorization for large integers appears to be a difficult problem, there is no known method to carry it out quickly. Its complexity is the basis of the security of some public key cryptography algorithms. A matrix can also be factorized into a product of matrices of special types, One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types, QR decomposition, LQ, QL, RQ and this situation is generalized by factorization systems. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm, for very large numbers, no efficient classical algorithm is known. Modern techniques for factoring polynomials are fast and efficient, but use sophisticated mathematical ideas and these techniques are used in the construction of computer routines for carrying out polynomial factorization in Computer algebra systems. This article is concerned with classical techniques. While the general notion of factoring just means writing an expression as a product of simpler expressions, when factoring polynomials this means that the factors are to be polynomials of smaller degree. Thus, while x 2 − y = is a factorization of the expression, another issue concerns the coefficients of the factors. It is not always possible to do this, and a polynomial that can not be factored in this way is said to be irreducible over this type of coefficient, thus, x2 -2 is irreducible over the integers and x2 +4 is irreducible over the reals. In the first example, the integers 1 and -2 can also be thought of as real numbers, and if they are, then x 2 −2 = shows that this polynomial factors over the reals. Similarly, since the integers 1 and 4 can be thought of as real and hence complex numbers, x2 +4 splits over the complex numbers, i. e. x 2 +4 =. The fundamental theorem of algebra can be stated as, Every polynomial of n with complex number coefficients splits completely into n linear factors
18.
Combinatorics
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Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general methods were developed. One of the oldest and most accessible parts of combinatorics is graph theory, Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist or a combinatorist, basic combinatorial concepts and enumerative results appeared throughout the ancient world. Greek historian Plutarch discusses an argument between Chrysippus and Hipparchus of a rather delicate enumerative problem, which was shown to be related to Schröder–Hipparchus numbers. In the Ostomachion, Archimedes considers a tiling puzzle, in the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. The Indian mathematician Mahāvīra provided formulae for the number of permutations and combinations, later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J. J. Sylvester and Percy MacMahon helped lay the foundation for enumerative, graph theory also enjoyed an explosion of interest at the same time, especially in connection with the four color problem. In the second half of the 20th century, combinatorics enjoyed a rapid growth, in part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical science, but at the same time led to a partial fragmentation of the field. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, fibonacci numbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a framework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis, in contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Partition theory studies various enumeration and asymptotic problems related to integer partitions, originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field. It incorporates the bijective approach and various tools in analysis and analytic number theory, graphs are basic objects in combinatorics
19.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
20.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
21.
Euclid
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Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
22.
Fundamental theorem of arithmetic
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For example,1200 =24 ×31 ×52 =3 ×2 ×2 ×2 ×2 ×5 ×5 =5 ×2 ×3 ×2 ×5 ×2 ×2 = etc. The requirement that the factors be prime is necessary, factorizations containing composite numbers may not be unique. This theorem is one of the reasons why 1 is not considered a prime number, if 1 were prime. Book VII, propositions 30,31 and 32, and Book IX, proposition 14 of Euclids Elements are essentially the statement, proposition 30 is referred to as Euclids lemma. And it is the key in the proof of the theorem of arithmetic. Proposition 31 is proved directly by infinite descent, proposition 32 is derived from proposition 31, and prove that the decomposition is possible. Book IX, proposition 14 is derived from Book VII, proposition 30, indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case. Article 16 of Gauss Disquisitiones Arithmeticae is a modern statement. < pk are primes and the αi are positive integers and this representation is commonly extended to all positive integers, including one, by the convention that the empty product is equal to 1. This representation is called the representation of n, or the standard form of n. For example 999 = 33×37,1000 = 23×53,1001 = 7×11×13 Note that factors p0 =1 may be inserted without changing the value of n, allowing negative exponents provides a canonical form for positive rational numbers. However, as Integer factorization of large integers is much harder than computing their product, gcd or lcm, these formulas have, in practice, many arithmetical functions are defined using the canonical representation. In particular, the values of additive and multiplicative functions are determined by their values on the powers of prime numbers, the proof uses Euclids lemma, if a prime p divides the product of two natural numbers a and b, then p divides a or p divides b. We need to show that every integer greater than 1 is either prime or a product of primes, for the base case, note that 2 is prime. By induction, assume true for all numbers between 1 and n, if n is prime, there is nothing more to prove. Otherwise, there are integers a and b, where n = ab and 1 < a ≤ b < n, by the induction hypothesis, a = p1p2. pj and b = q1q2. qk are products of primes. But then n = ab = p1p2. pjq1q2. qk is a product of primes, assume that s >1 is the product of prime numbers in two different ways, s = p 1 p 2 ⋯ p m = q 1 q 2 ⋯ q n. We must show m = n and that the qj are a rearrangement of the pi, by Euclids lemma, p1 must divide one of the qj, relabeling the qj if necessary, say that p1 divides q1
23.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
24.
International Standard Serial Number
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An International Standard Serial Number is an eight-digit serial number used to uniquely identify a serial publication. The ISSN is especially helpful in distinguishing between serials with the same title, ISSN are used in ordering, cataloging, interlibrary loans, and other practices in connection with serial literature. The ISSN system was first drafted as an International Organization for Standardization international standard in 1971, ISO subcommittee TC 46/SC9 is responsible for maintaining the standard. When a serial with the content is published in more than one media type. For example, many serials are published both in print and electronic media, the ISSN system refers to these types as print ISSN and electronic ISSN, respectively. The format of the ISSN is an eight digit code, divided by a hyphen into two four-digit numbers, as an integer number, it can be represented by the first seven digits. The last code digit, which may be 0-9 or an X, is a check digit. Formally, the form of the ISSN code can be expressed as follows, NNNN-NNNC where N is in the set, a digit character. The ISSN of the journal Hearing Research, for example, is 0378-5955, where the final 5 is the check digit, for calculations, an upper case X in the check digit position indicates a check digit of 10. To confirm the check digit, calculate the sum of all eight digits of the ISSN multiplied by its position in the number, the modulus 11 of the sum must be 0. There is an online ISSN checker that can validate an ISSN, ISSN codes are assigned by a network of ISSN National Centres, usually located at national libraries and coordinated by the ISSN International Centre based in Paris. The International Centre is an organization created in 1974 through an agreement between UNESCO and the French government. The International Centre maintains a database of all ISSNs assigned worldwide, at the end of 2016, the ISSN Register contained records for 1,943,572 items. ISSN and ISBN codes are similar in concept, where ISBNs are assigned to individual books, an ISBN might be assigned for particular issues of a serial, in addition to the ISSN code for the serial as a whole. An ISSN, unlike the ISBN code, is an identifier associated with a serial title. For this reason a new ISSN is assigned to a serial each time it undergoes a major title change, separate ISSNs are needed for serials in different media. Thus, the print and electronic versions of a serial need separate ISSNs. Also, a CD-ROM version and a web version of a serial require different ISSNs since two different media are involved, however, the same ISSN can be used for different file formats of the same online serial
25.
Robert Grosseteste
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Robert Grosseteste was an English statesman, scholastic philosopher, theologian, scientist and Bishop of Lincoln. He was born of parents at Stradbroke in Suffolk. Upon his death, he was almost universally revered as a saint in England, a. C. Crombie calls him the real founder of the tradition of scientific thought in medieval Oxford, and in some ways, of the modern English intellectual tradition. There is very little evidence about Grossetestes education. He may have received an arts education at Hereford, in light of his connection with the Bishop of Hereford William de Vere in the 1190s. It is fairly certain that Grosseteste was a master by 1192, Grosseteste acquired a position in the bishops household, but at the death of this patron he disappears from the historical record for several years. He appears again in the thirteenth century as a judge-delegate in Hereford. By 1225, he had gained the benefice of Abbotsley in the diocese of Lincoln, on that period in his life, scholarship is divided. However, there is evidence that by 1229/30 he was teaching at Oxford, but on the periphery as the lector in theology to the Franciscans. He remained in this post until March 1235, Grosseteste may also have been appointed Chancellor of the University of Oxford. However, the evidence for this comes from a thirteenth century anecdote whose main claim is that Grosseteste was in fact entitled the master of students. However, after an illness in 1232, he resigned all his benefices. His reasons were due to changing attitudes about the plurality of benefices, as a master of the sacred page, Grosseteste trained the Franciscans in the standard curriculum of university theology. The Franciscan Roger Bacon was his most famous disciple, and acquired an interest in the method from him. Grosseteste lectured on the Psalter, the Pauline epistles, Genesis and he also led disputations on such subjects as the theological nature of truth and the efficacy of the Mosaic Law. Grosseteste also preached at the university and appears to have called to preach within the diocese as well. He collected some of those sermons, along with short notes and reflections, not long after he left Oxford. His theological writings reveal a continual interest in the world as a major resource for theological reflection
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University of St Andrews
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The University of St Andrews is a British public research university in St Andrews, Fife, Scotland. It is the oldest of the four ancient universities of Scotland, St Andrews was founded between 1410 and 1413, when the Avignon Antipope Benedict XIII issued a papal bull to a small founding group of Augustinian clergy. St Andrews is made up from a variety of institutions, including three constituent colleges and 18 academic schools organised into four faculties, the university occupies historic and modern buildings located throughout the town. The academic year is divided into two terms, Martinmas and Candlemas, in term time, over one-third of the towns population is either a staff member or student of the university. It is ranked as the third best university in the United Kingdom in national league tables, the Times Higher Education World Universities Ranking names St Andrews among the worlds Top 50 universities for Social Sciences, Arts and Humanities. St Andrews has the highest student satisfaction amongst all multi-faculty universities in the United Kingdom, St Andrews has many notable alumni and affiliated faculty, including eminent mathematicians, scientists, theologians, philosophers, and politicians. Six Nobel Laureates are among St Andrews alumni and former staff, a charter of privilege was bestowed upon the society of masters and scholars by the Bishop of St Andrews, Henry Wardlaw, on 28 February 1411. Wardlaw then successfully petitioned the Avignon Pope Benedict XIII to grant the university status by issuing a series of papal bulls. King James I of Scotland confirmed the charter of the university in 1432, subsequent kings supported the university with King James V confirming privileges of the university in 1532. A college of theology and arts called St Johns College was founded in 1418 by Robert of Montrose, St Salvators College was established in 1450, by Bishop James Kennedy. St Leonards College was founded in 1511 by Archbishop Alexander Stewart, St Johns College was refounded by Cardinal James Beaton under the name St Marys College in 1538 for the study of divinity and law. Some university buildings that date from this period are still in use today, such as St Salvators Chapel, St Leonards College Chapel, at this time, the majority of the teaching was of a religious nature and was conducted by clerics associated with the cathedral. During the 17th and 18th centuries, the university had mixed fortunes and was beset by civil. He described it as pining in decay and struggling for life, in the second half of the 19th century, pressure was building upon universities to open up higher education to women. In 1876, the University Senate decided to allow women to receive an education at St Andrews at a roughly equal to the Master of Arts degree that men were able to take at the time. The scheme came to be known as the L. L. A and it required women to pass five subjects at an ordinary level and one at honours level and entitled them to hold a degree from the university. In 1889 the Universities Act made it possible to admit women to St Andrews. Agnes Forbes Blackadder became the first woman to graduate from St Andrews on the level as men in October 1894
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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
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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
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Al-Mahani
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Abu-Abdullah Muhammad ibn Īsa Māhānī was a Persian Muslim mathematician and astronomer from Mahan, Kermān, Persia. A series of observations of lunar and solar eclipses and planetary conjunctions and he wrote commentaries on Euclid and Archimedes, and improved Ishaq ibn Hunayns translation of Menelaus of Alexandrias Spherics. He tried vainly to solve an Archimedean problem, to divide a sphere by means of a plane into two segments being in a ratio of volume. That problem led to an equation, x 3 + c 2 b = c x 2 which Muslim writers called al-Mahanis equation. List of Iranian scientists H. Suter, Die Mathematiker und Astronomen der Araber 26,1900 and his failure to solve the Archimedean problem is quoted by Omar al-Khayyami). Woepcke, Lalgebra dOmar Alkhayyami 2,96 sq. OConnor, John J. Robertson, Edmund F. Abu Abd Allah Muhammad ibn Isa Al-Mahani, MacTutor History of Mathematics archive, al-Māhānī, Abū Abd Allāh Muḥammad Ibn Īsā
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Hunayn ibn Ishaq
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Hunayn ibn Ishaq al-Ibadi was a famous and influential Nestorian Arab scholar, physician, and a scientist from Mesopotamia, what is now Iraq. He and his students transmitted their Syriac and Arabic translations of many classical Greek texts throughout the Islāmic world, Ḥunayn ibn Isḥaq was the most productive translator of Greek medical and scientific treatises in his day. He studied Greek and became known among the Arabs as the Sheikh of the translators and he mastered four languages, Arabic, Syriac, Greek and Persian. His translations did not require corrections, Hunayn’s method was followed by later translators. He was originally from southern Iraq but he spent his life in Baghdad. His fame went far beyond his own community, in the Abbasid era, a new interest in extending the study of Greek science had arisen. At that time, there was a vast amount of untranslated ancient Greek literature pertaining to philosophy, mathematics, natural science, and medicine. This valuable information was accessible to a very small minority of Middle Eastern scholars who knew the Greek language. In time, Hunayn ibn Ishaq became arguably the chief translator of the era, in his lifetime, ibn Ishaq translated 116 works, including Plato’s Timaeus, Aristotle’s Metaphysics, and the Old Testament, into Syriac and Arabic. Ibn Ishaq also produced 36 of his own books,21 of which covered the field of medicine and his son Ishaq, and his nephew Hubaysh, worked together with him at times to help translate. Hunayn ibn Ishaq is known for his translations, his method of translation and he has also been suggested by François Viré to be the true identity of the Arabic falconer Moamyn, author of De Scientia Venandi per Aves. Hunayn ibn Ishaq was a Nestorian Arab born in 809, during the Abbasid period, some sources describe him as an Assyrian. As a child, he learned the Syriac and Arabic languages, although al-Hira was known for commerce and banking, and his father was a pharmacist, Hunayn went to Baghdad in order to study medicine. Hunayn promised himself to return to Baghdad when he became a physician and he went abroad to master the Latin language. On his return to Baghdad, Hunayn displayed his newly acquired skills by reciting the works of Homer, in awe, ibn Masawayh reconciled with Hunayn, and the two started to work cooperatively. Hunayn was extremely motivated in his work to master Greek studies, the Abbasid Caliph al-Mamun noticed Hunayns talents and placed him in charge of the House of Wisdom, the Bayt al Hikmah. The House of Wisdom was an institution where Greek works were translated, the caliph also gave Hunayn the opportunity to travel to Byzantium in search of additional manuscripts, such as those of Aristotle and other prominent authors. In Hunayn ibn Ishaq’s lifetime, he devoted himself to working on a multitude of writings, Hunayn wrote on a variety of subjects that included philosophy, religion and medicine
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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
