1.
Arabic
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Arabic is a Central Semitic language that was first spoken in Iron Age northwestern Arabia and is now the lingua franca of the Arab world. Arabic is also the language of 1.7 billion Muslims. It is one of six languages of the United Nations. The modern written language is derived from the language of the Quran and it is widely taught in schools and universities, and is used to varying degrees in workplaces, government, and the media. The two formal varieties are grouped together as Literary Arabic, which is the language of 26 states. Modern Standard Arabic largely follows the standards of Quranic Arabic. Much of the new vocabulary is used to denote concepts that have arisen in the post-Quranic era, Arabic has influenced many languages around the globe throughout its history. During the Middle Ages, Literary Arabic was a vehicle of culture in Europe, especially in science, mathematics. As a result, many European languages have borrowed many words from it. Many words of Arabic origin are found in ancient languages like Latin. Balkan languages, including Greek, have acquired a significant number of Arabic words through contact with Ottoman Turkish. Arabic has also borrowed words from languages including Greek and Persian in medieval times. Arabic is a Central Semitic language, closely related to the Northwest Semitic languages, the Ancient South Arabian languages, the Semitic languages changed a great deal between Proto-Semitic and the establishment of the Central Semitic languages, particularly in grammar. Innovations of the Central Semitic languages—all maintained in Arabic—include, The conversion of the suffix-conjugated stative formation into a past tense, the conversion of the prefix-conjugated preterite-tense formation into a present tense. The elimination of other prefix-conjugated mood/aspect forms in favor of new moods formed by endings attached to the prefix-conjugation forms, the development of an internal passive. These features are evidence of descent from a hypothetical ancestor. In the southwest, various Central Semitic languages both belonging to and outside of the Ancient South Arabian family were spoken and it is also believed that the ancestors of the Modern South Arabian languages were also spoken in southern Arabia at this time. To the north, in the oases of northern Hijaz, Dadanitic and Taymanitic held some prestige as inscriptional languages, in Najd and parts of western Arabia, a language known to scholars as Thamudic C is attested
2.
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
3.
Gilan Province
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Gilan Province is one of the 31 provinces of Iran. It lies along the Caspian Sea, in Irans Region 3, west of the province of Mazandaran, east of the province of Ardabil and it also borders the Republic of Azerbaijan in the north, as well as Russia across the Caspian Sea. The northern part of the province is part of territory of South Talysh, at the center of the province is the main city of Rasht. Other towns in the province include Astara, Astaneh-e Ashrafiyyeh, Fuman, Lahijan, Langrud, Masouleh, Manjil, Rudbar, Roudsar, Shaft, Talesh, the main harbor port is Bandar-e Anzali. In antiquity, this area was a province of Persia known as Daylam, the Daylam region corresponds to the modern region of Gīlān. It seems that the Gelae entered the south of the Caspian coast and west of the Amardos River in the second or first century B. C. E. Pliny identifies them with the Cadusii who were living there previously and it is more likely that they were a separate people, had come from the region of Dagestan, and taken the place of the Kadusii. Their languages shares certain features with Caucasian languages. It was the place of origin of the Buyid dynasty, the people of the province had a prominent position during the Sassanid dynasty, so that their political power extended to Mesopotamia. The first recorded encounter between Gilanis and Deylamite warlords and invading Muslim Arab armies was at the Battle of Jalula in 637 AD, Deylamite commander Muta led an army of Gils, Deylamites, Persians and people of the Rey region. Muta was killed in the battle, and his army managed to retreat in an orderly manner. However, this appears to have been a Pyrrhic victory for the Arabs, unlike the Russians, Muslim Arabs never managed to conquer Gilan as they did with other provinces in Iran. Gilanis and Deylamites successfully repulsed all Arab attempts to occupy their land or to them to Islam. In fact, it was the Deylamites under the Buyid king Muizz al-Dawla who finally shifted the balance of power by conquering Baghdad in 945, Muizz al-Dawla, however, allowed the Abbasid caliphs to remain in comfortable, secluded captivity in their palaces. In the 9th and 10th centuries AD, Deylamites and later Gilanis gradually converted to Zaidite Shiism, Muslim chronicles of Varangian invasions of the littoral Caspian region in the 9th century record Deylamites as non-Muslim. These chronicles also show that the Deylamites were the warriors in the Caspian region who could fight the fearsome Varangian vikings as equals. Deylamite infantrymen had a very similar to the Swiss Reisläufer of the Late Middle Ages in Europe. Deylamite mercenaries served as far away as Egypt, Islamic Spain, buyids established the most successful of the Deylamite dynasties of Iran
4.
Badakhshan
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Badakhshan is a historic region comprising parts of what is now northeastern Afghanistan and southeastern Tajikistan. The name is retained in Badakhshan Province, which is one of the 34 provinces of Afghanistan and is located in North-East Afghanistan, much of historic Badakhshan lies within Tajikistans Gorno-Badakhshan Autonomous Region located in the south-eastern part of the country. The music of Badakhshan is an important part of the cultural heritage. Badakhshan has a diverse ethno-linguistic and religious community, tajiks and Pamiris are the majority while a tiny minority of Kyrgyzs and Uzbeks also are found in their own villages. There are also groups of speakers of several Pamir languages of the Eastern Iranian language group, during the 20th century within Gorno-Badakhshan Autonomous Region in Tajikistan the speakers of Pamir languages formed their own separate ethnic identity as Pamiris. The Pamiri people were not officially recognized as an ethnic group in Tajikistan. The main religions of Badakhshan are Ismaili Islam and Sunni Islam, the people of this province have a rich cultural heritage and they have preserved unique ancient forms of music, poetry and dance. Badakhshan was an important trading center during antiquity, lapis lazuli was traded exclusively from there as early as the second half of the 4th millennium BC. Badakhshan was an important region when the Silk Road passed through and its significance is its geo-economic role in trades of silk and ancient commodities transactions between the East and West. According to Marco Polo, Badashan/ Badakshan was a province where Balas ruby could be found under the mountain Syghinan, the region was ruled over by the mirs of Badakhshan. Sultan Muhammad of Badakhshan was the last of a series of kings who traced their descent to Alexander the Great, when Mahmud died, Amir Khusroe Khan, one of his nobles, blinded Baysinghar Mirza, killed the second prince, and ruled as usurper. He submitted to Mughal Emperor Babur in 1504 CE, when Babur took Kandahar in 1506 CE, from Shah Beg Arghun, he sent Khan Mirza as governor to Badakhshan. A son was born to Khan Mirza by the name of Mirza Sulaiman in 1514 CE and they were released by Emperor Humayun in 1545, and took again possession of Badakhshan. Bent on making conquests, he invaded Balkh in 1560, but had to return and his son, Mirza Ibrahim, was killed in battle. When Akbar became Mughal Emperor, his stepbrother Mirza Muhammad Hakims mother had killed by Shah Abul Maali. He returned to Kabul in 1566, when Akbars troops had left that country, Mirza Sulaimans wife was Khurram Begum, of the Kipchak tribe. She was clever and had her husband so much in her power and her enemy was Muhtarim Khanum, the widow of Prince Kamran Mirza. Mirza Sulaiman wanted to marry her, but Khurram Begum got her married, against her will, to Mirza Ibrahim, by whom she had a son, Mirza Shahrukh
5.
Algebra
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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols, as such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine, abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are unknown or allowed to take on many values. For example, in x +2 =5 the letter x is unknown, in E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of object in abstract algebra is called an algebra. A mathematician who does research in algebra is called an algebraist, the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wal-muḳābala by Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the century, from either Spanish, Italian. It originally referred to the procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century, the word algebra has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, algebra names a broad part of mathematics, as a single word with an article or in plural, an algebra or algebras denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication, when some authors use the term algebra, they make a subset of the following additional assumptions, associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word refers to a generalization of the above concept. With a qualifier, there is the distinction, Without an article, it means a part of algebra, such as linear algebra, elementary algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, sometimes both meanings exist for the same qualifier, as in the sentence, Commutative algebra is the study of commutative rings, which are commutative algebras over the integers
6.
Quadratic equation
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If a =0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and may be distinguished by calling them, respectively, the coefficient, the linear coefficient. Because the quadratic equation involves only one unknown, it is called univariate, solutions to problems equivalent to the quadratic equation were known as early as 2000 BC. A quadratic equation with real or complex coefficients has two solutions, called roots and these two solutions may or may not be distinct, and they may or may not be real. It may be possible to express a quadratic equation ax2 + bx + c =0 as a product =0. In some cases, it is possible, by inspection, to determine values of p, q, r. If the quadratic equation is written in the form, then the Zero Factor Property states that the quadratic equation is satisfied if px + q =0 or rx + s =0. Solving these two linear equations provides the roots of the quadratic, for most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. As an example, x2 + 5x +6 factors as, the more general case where a does not equal 1 can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection. Except for special cases such as where b =0 or c =0 and this means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection. The process of completing the square makes use of the identity x 2 +2 h x + h 2 =2. Starting with an equation in standard form, ax2 + bx + c =0 Divide each side by a. Subtract the constant term c/a from both sides, add the square of one-half of b/a, the coefficient of x, to both sides. This completes the square, converting the left side into a perfect square, write the left side as a square and simplify the right side if necessary. Produce two linear equations by equating the square root of the side with the positive and negative square roots of the right side. Completing the square can be used to derive a formula for solving quadratic equations. The mathematical proof will now be briefly summarized and it can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation,2 = b 2 −4 a c 4 a 2. Taking the square root of both sides, and isolating x, gives, x = − b ± b 2 −4 a c 2 a and these result in slightly different forms for the solution, but are otherwise equivalent
7.
Discriminant
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In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them. The discriminant is widely used in theory, either directly or through its generalization as the discriminant of a number field. For factoring a polynomial with coefficients, the standard method consists in factoring first its reduction modulo a prime number not dividing the discriminant. In algebraic geometry, the discriminant with respect to one of the variables characterizes the points of a hypersurface where the implicit function theorem does not apply, the term discriminant was coined in 1851 by the British mathematician James Joseph Sylvester. The nonzero entries of the first column of the Sylvester matrix are a n and n a n, the division by a n may be not well defined if the ring of the coefficients contains zero divisors. Such a problem may be avoided by replacing a n by 1 in the first column of the Sylvester matrix before computing the determinant, in any case, the discriminant is a polynomial in a 0, …, a n with integer coefficients. When the polynomial is defined over a field, the theorem of algebra implies that it has n roots, r1. Rn, not necessarily all distinct, in an algebraically closed extension of the field and this expression of the discriminant is often taken as a definition. It makes immediate that if the polynomial has a root, then its discriminant is zero. The discriminant of a polynomial is rarely considered. If needed, it is defined to be equal to 1. There is no convention for the discriminant of a constant polynomial. For small degrees, the discriminant is rather simple, but for higher degrees, the discriminant of a general quartic has 16 terms, that of a quintic has 59 terms, and that of a sextic has 246 terms. The quadratic polynomial a x 2 + b x + c has discriminant b 2 −4 a c. The square root of the discriminant appears in the formula for the roots of the quadratic polynomial. The discriminant is zero if and only if the two roots are equal, if a, b, c are real numbers, the polynomial has two distinct real roots if the discriminant is positive, and two complex conjugate roots if it is negative. If a, b, c are rational numbers, then the discriminant is the square of a number, if. In particular, the polynomial x 3 + p x + q has discriminant −4 p 3 −27 q 2, the discriminant is zero if and only if at least two roots are equal
8.
John Wiley & Sons
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Founded in 1807, Wiley is also known for publishing For Dummies. As of 2015, the company had 4,900 employees, Wiley was established in 1807 when Charles Wiley opened a print shop in Manhattan. Wiley later shifted its focus to scientific, technical, and engineering subject areas, Charles Wileys son John took over the business when his father died in 1826. The firm was successively named Wiley, Lane & Co. then Wiley & Putnam, the company acquired its present name in 1876, when Johns second son William H. Wiley joined his brother Charles in the business. Through the 20th century, the company expanded its activities, the sciences. Since the establishment of the Nobel Prize in 1901, Wiley and its companies have published the works of more than 450 Nobel Laureates. Wiley in December 2010 opened an office in Dubai, to build on its business in the Middle East more effectively, the company has had an office in Beijing, China, since 2001, and China is now its sixth-largest market for STEM content. Wiley established publishing operations in India in 2006, and has established a presence in North Africa through sales contracts with academic institutions in Tunisia, Libya, and Egypt. On April 16,2012, the announced the establishment of Wiley Brasil Editora LTDA in São Paulo, Brazil. Wileys scientific, technical, and medical business was expanded by the acquisition of Blackwell Publishing in February 2007. Through a backfile initiative completed in 2007,8.2 million pages of content have been made available online. Other major journals published include Angewandte Chemie, Advanced Materials, Hepatology, International Finance, launched commercially in 1999, Wiley InterScience provided online access to Wiley journals, major reference works, and books, including backfile content. Journals previously from Blackwell Publishing were available online from Blackwell Synergy until they were integrated into Wiley InterScience on June 30,2008, in December 2007, Wiley also began distributing its technical titles through the Safari Books Online e-reference service. On February 17,2012, Wiley announced the acquisition of Inscape Holdings Inc. which provides DISC assessments and training for interpersonal business skills. On August 13,2012, Wiley announced it entered into an agreement to sell all of its travel assets, including all of its interests in the Frommers brand. On October 2,2012, Wiley announced it would acquire Deltak edu, LLC, Deltak is expected to contribute solid growth to both Wileys Global Education business and Wiley overall. Seventh-generation members Jesse and Nate Wiley work in the companys Professional/Trade and Scientific, Technical, Medical, and Scholarly businesses, respectively. Wiley has been owned since 1962, and listed on the New York Stock Exchange since 1995, its stock is traded under the symbols NYSE, JW. A and NYSE
9.
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
10.
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
11.
Diophantus
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Diophantus of Alexandria, sometimes called the father of algebra, was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica, many of which are now lost. These texts deal with solving algebraic equations and this led to tremendous advances in number theory, and the study of Diophantine equations and of Diophantine approximations remain important areas of mathematical research. Diophantus coined the term παρισότης to refer to an approximate equality and this term was rendered as adaequalitas in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers, thus he allowed positive rational numbers for the coefficients, in modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation, little is known about the life of Diophantus. He lived in Alexandria, Egypt, probably from between AD200 and 214 to 284 or 298, much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus. One of the states, Here lies Diophantus, the wonder behold. Alas, the child of master and sage After attaining half the measure of his fathers life chill fate took him. After consoling his fate by the science of numbers for four years and this puzzle implies that Diophantus age x can be expressed as x = x/6 + x/12 + x/7 +5 + x/2 +4 which gives x a value of 84 years. However, the accuracy of the information cannot be independently confirmed, the Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations, of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arab books discovered in 1968 are also by Diophantus. Some Diophantine problems from Arithmetica have been found in Arabic sources and it should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus, “Our author not the slightest trace of a general, comprehensive method is discernible, each problem calls for some special method which refuses to work even for the most closely related problems. The portion of the Greek Arithmetica that survived, however, was, like all ancient Greek texts transmitted to the modern world, copied by. In addition, some portion of the Arithmetica probably survived in the Arab tradition. ”Arithmetica was first translated from Greek into Latin by Bombelli in 1570, however, Bombelli borrowed many of the problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander, the best known Latin translation of Arithmetica was made by Bachet in 1621 and became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it, and made notes in the margins. I have a marvelous proof of this proposition which this margin is too narrow to contain. ”Fermats proof was never found
12.
Pappus of Alexandria
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Pappus of Alexandria was one of the last great Alexandrian mathematicians of Antiquity, known for his Synagoge or Collection, and for Pappuss hexagon theorem in projective geometry. Nothing is known of his life, other than, that he had a son named Hermodorus, Collection, his best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives. It covers a range of topics, including geometry, recreational mathematics, doubling the cube, polygons. Pappus flourished in the 4th century AD, in a period of general stagnation in mathematical studies, he stands out as a remarkable exception. In this respect the fate of Pappus strikingly resembles that of Diophantus, in his surviving writings, Pappus gives no indication of the date of the authors whose works he makes use of, or of the time at which he himself wrote. If no other information were available, all that could be known would be that he was later than Ptolemy, whom he quotes, and earlier than Proclus. The Suda states that Pappus was of the age as Theon of Alexandria. A different date is given by a note to a late 10th-century manuscript, which states, next to an entry on Emperor Diocletian. This works out as October 18,320 AD, and so Pappus must have flourished c.320 AD. The great work of Pappus, in eight books and titled Synagoge or Collection, has not survived in complete form, the first book is lost, and the rest have suffered considerably. The Suda enumerates other works of Pappus, Χωρογραφία οἰκουμενική, commentary on the 4 books of Ptolemys Almagest, Ποταμοὺς τοὺς ἐν Λιβύῃ, Pappus himself mentions another commentary of his own on the Ἀνάλημμα of Diodorus of Alexandria. Pappus also wrote commentaries on Euclids Elements, and on Ptolemys Ἁρμονικά and these discoveries form, in fact, a text upon which Pappus enlarges discursively. Heath considered the systematic introductions to the books as valuable, for they set forth clearly an outline of the contents. From these introductions one can judge of the style of Pappuss writing, heath also found his characteristic exactness made his Collection a most admirable substitute for the texts of the many valuable treatises of earlier mathematicians of which time has deprived us. The portions of Collection which has survived can be summarized as follows and we can only conjecture that the lost Book I, like Book II, was concerned with arithmetic, Book III being clearly introduced as beginning a new subject. The whole of Book II discusses a method of multiplication from a book by Apollonius of Perga. The final propositions deal with multiplying together the values of Greek letters in two lines of poetry, producing two very large numbers approximately equal to 2*1054 and 2*1038. Book III contains geometrical problems, plane and solid, on the arithmetic, geometric and harmonic means between two straight lines, and the problem of representing all three in one and the same geometrical figure
13.
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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Abd al-Rahman al-Sufi
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The lunar crater Azophi and the minor planet 12621 Alsufi are named after him. Al-Sufi published his famous Book of Fixed Stars in 964, describing much of his work, al-Biruni reports that his work on the ecliptic was carried out in Shiraz. He lived at the Buyid court in Isfahan, abd al-Rahman al-Sufi was one of the famous nine Muslim astronomers. His name implies that he was from a Sufi Muslim background and he lived at the court of Emir Adud ad-Daula in Isfahan, Persia, and worked on translating and expanding Greek astronomical works, especially the Almagest of Ptolemy. He contributed several corrections to Ptolemys star list and did his own brightness and he identified the Large Magellanic Cloud, which is visible from Yemen, though not from Isfahan, it was not seen by Europeans until Magellans voyage in the 16th century. He also made the earliest recorded observation of the Andromeda Galaxy in 964 AD and these were the first galaxies other than the Milky Way to be observed from Earth. He observed that the plane is inclined with respect to the celestial equator. He observed and described the stars, their positions, their magnitudes and their colour, for each constellation, he provided two drawings, one from the outside of a celestial globe, and the other from the inside. Since 2006, Astronomy Society of Iran – Amateur Committee hold an international Sufi Observing Competition in the memory of Sufi, the first competition was held in 2006 in the north of Semnan Province and the second was held in the summer of 2008 in Ladiz near the Zahedan. More than 100 attendees from Iran and Iraq participated in the event
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Abu al-Wafa' Buzjani
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Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī or Abū al-Wafā Būzhjānī was a Persian mathematician and astronomer who worked in Baghdad. He made important innovations in spherical trigonometry, and his work on arithmetics for businessmen contains the first instance of using numbers in a medieval Islamic text. He is also credited with compiling the tables of sines and tangents at 15 intervals and he also introduced the secant and cosecant functions, as well studied the interrelations between the six trigonometric lines associated with an arc. His Almagest was widely read by medieval Arabic astronomers in the centuries after his death and he is known to have written several other books that have not survived. He was born in Buzhgan, in Khorasan, at age 19, in 959 AD, he moved to Baghdad and remained there for the next forty years, and died there in 998. In Baghdad, he received patronage by members of the Buyid court, abu Al-Wafa was the first to build a wall quadrant to observe the sky. It has been suggested that he was influenced by the works of Al-Battani as the latter describes a quadrant instrument in his Kitāb az-Zīj, in 997, he participated in an experiment to determine the difference in local time between his location and that of al-Biruni. The result was close to present-day calculations, showing a difference of approximately 1 hour between the two longitudes. Abu al-Wafa is also known to have worked with Abū Sahl al-Qūhī, while what is extant from his works lacks theoretical innovation, his observational data were used by many later astronomers, including al-Biruni. Among his works on astronomy, only the first seven treatises of his Almagest are now extant, the work covers numerous topics in the fields of plane and spherical trigonometry, planetary theory, and solutions to determine the direction of Qibla. He established several trigonometric identities such as sin in their modern form, some sources suggest that he introduced the tangent function, although other sources give the credit for this innovation to al-Marwazi. A book of zij called Zīj al‐wāḍiḥ, no longer extant, a Book on Those Geometric Constructions Which Are Necessary for a Craftsman. This text contains over one hundred geometric constructions, including for a regular heptagon, the legacy of this text in Latin Europe is still debated. A Book on What Is Necessary from the Science of Arithmetic for Scribes and this is the first book where negative numbers have been used in the medieval Islamic texts. He also wrote translations and commentaries on the works of Diophantus, al-Khwārizmī. The crater Abul Wáfa on the Moon is named after him, oConnor, John J. Robertson, Edmund F. Mohammad Abul-Wafa Al-Buzjani, MacTutor History of Mathematics archive, University of St Andrews. Būzjānī, Abū al‐Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā al‐Būzjānī. A Study of Method, Historia Mathematica,39, 34–83, doi,10. 1016/j. hm.2011.09.001 Youschkevitch, A. P. Abūl-Wafāʾ Al-Būzjānī, Muḥammad Ibn Muḥammad Ibn Yaḥyā Ibn Ismāʿīl Ibn Al-ʿAbbās
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Al-Saghani
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Abu Hamid Ahmed ibn Mohammed al-Saghani al-Asturlabi was a Persian astronomer and historian of science. He flourished in Baghdad, where he died in 379-380 A. H/990 A. D, an inventor and maker of instruments, he worked in Sharaf al-Dawlas observatory and, perhaps, constructed the instruments which were used there. Worked on the trisection of the angle, al-Asturlabi wrote some of the earliest comments on the history of science. The ancients came to their achievements by virtue of their priority in time. Yet, how many things escaped them which then became the original inventions of modern scholars, list of Iranian scientists Puig, Roser. Ṣāghānī, Abū Ḥāmid Aḥmad ibn Muḥammad al‐Ṣāghānī al‐Asṭurlābī
19.
Brethren of Purity
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The Brethren of Purity were a secret society of Muslim philosophers in Basra, Iraq, in the 8th or 10th century CE. The structure of this organization and the identities of its members have never been clear. A good deal of Muslim and Western scholarship has been spent on just pinning down the identities of the Brethren, the story concerns a Barbary dove and its companions who get entangled in the net of a hunter seeking birds. Together, they themselves and the ensnaring net to a nearby rat, who is gracious enough to gnaw the birds free of the net, impressed by the rats altruistic deed. Soon a tortoise and gazelle also join the company of animals, in the final turn of events, the gazelle repays the tortoise by serving as a decoy and distracting the hunter while the rat and the others free the tortoise. After this, the animals are designated as the Ikwhan al-Safa, the Brethren regularly met on a fixed schedule. Compare the similar division of the Encyclopedia into four sections and the Jabirite symbolism of 4, the ranks were, The Craftsmen – a craftsman had to be at least 15 years of age, their honorific was the pious and compassionate. There have been a number of theories as to the authors of the Brethren, though some members of the Ikhwan are known, it is not easy to work out exactly who, or how many, were part of this group of writers. The members referred to themselves as sleepers in the cave, an intellectual presence. In one passage they give as their reason for hiding their secrets from the people, not as fear of earthly violence, but as desire to protect their God-given gifts from the world. Yet they were aware that their esoteric teachings might provoke unrest. Some modern scholars have argued for an Ismaili origin to the writings, ian Richard Netton writes in Muslim Neoplatonists that, The Ikhwans concepts of exegesis of both Quran and Islamic tradition were tinged with the esoterism of the Ismailis. Ibn Qifti, reporting in the 7th/13th century in Tarikh-i Hukama that, some people attributed to an Alid Imam, proffering various names, whereas other put forward as author some early Mutazalite theologians. Among the Syrian Ismailis, the earliest reference of the Epistles and it implies the Epistles being the product of the joint efforts of the Ismaili dais. He charged him with the mission as was necessary and asked him to keep his identity concealed and this source not only asserts the connection of the Epistles with the Ismailis, but also indicates that the Imam himself was not the sole author, but only the issuer or presenter. Since the orthodox circles and the power had portrayed a wrong image of Ismailism. The prominent members of the secret association seem to be however, Abul Hasan al-Tirmizi, Abdullah bin Mubarak, Abdullah bin Hamdan, Abdullah bin Maymun, Said bin Hussain etc. The other Yamenite source connecting the Epistles with the Ismailis was the writing of Sayyadna Ibrahim bin al-Hussain al-Hamidi, among us too there are merchants, artisans, agriculturists and stock breeders
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Al-Battani
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Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī aṣ-Ṣābiʾ al-Battānī was an Arab astronomer, astrologer, and mathematician. He introduced a number of relations, and his Kitāb az-Zīj was frequently quoted by many medieval astronomers. Little is known about al-Battānīs life beside that he was born in Harran near Urfa, in Upper Mesopotamia, which is now in Turkey and his epithet aṣ-Ṣabi’ suggests that among his ancestry were members of the Sabian sect, however, his full name indicates that he was Muslim. Some western historians state that he is of noble origin, like an Arab prince and he lived and worked in Raqqa, a city in north central Syria. One of al-Battānīs best-known achievements in astronomy was the determination of the year as being 365 days,5 hours,46 minutes and 24 seconds which is only 2 minutes and 22 seconds off. He was able to some of Ptolemys results and compiled new tables of the Sun and Moon. Some of his measurements were more accurate than ones taken by Copernicus many centuries later. Researchers have ascribed this phenomenon to al-Battānī being in a location that is closer to the southern latitude. Al-Battānī discovered that the direction of the Suns apogee, as recorded by Ptolemy, was changing and he also introduced, probably independently of the 5th century Indian astronomer Aryabhata, the use of sines in calculation, and partially that of tangents. He also calculated the values for the precession of the equinoxes and he used a uniform rate for precession in his tables, choosing not to adopt the theory of trepidation attributed to his colleague Thabit ibn Qurra. Al-Battānīs work is considered instrumental in the development of science and astronomy, Al-Battānī was frequently quoted by Tycho Brahe, Riccioli, among others. Kepler and Galileo showed interest in some of his observations, and he also discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants, which he referred to as a table of shadows, for each degree from 1° to 90°. Al-Battānīs major work is Kitāb az-Zīj and it was largely based on Ptolemys theory, and other Greco-Syriac sources, while showing little Indian or Persian influence. In his zij, he provided descriptions of a quadrant instrument, a reprint appeared at Bologna in 1645. The original MS. is preserved at the Vatican, and the Escorial library possesses in MS, a treatise of some value by him on astronomical chronology. List of Arab scientists and scholars Zij Al-Battānī sive Albatenii, Opus Astronomicum, ad fidem codicis escurialensis arabice editum, ed. by Carlo Alfonso Nallino. Milan, Ulrico Hoepli, 1899-1907,412 +450 +288 pp. (anast, Al-Battānī, Abū ʿAbd Allāh Muḥammad Ibn Jābir Ibn Sinān al-Raqqī al-Ḥarrānī al–Ṣābi. OConnor, John J. Robertson, Edmund F. Abu Abdallah Mohammad ibn Jabir Al-Battani, MacTutor History of Mathematics archive, battānī, Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al‐Battānī al‐Ḥarrānī al‐Ṣābiʾ