1.
Up to
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In mathematics, the phrase up to appears in discussions about the elements of a set, and the conditions under which subsets of those elements may be considered equivalent. The statement elements a and b of set S are equivalent up to X means that a and b are equivalent if criterion X is ignored and that is, a and b can be transformed into one another if a transform corresponding to X is applied. Looking at the entire set S, when X is ignored the elements can be arranged in subsets whose elements are equivalent, such subsets are called equivalence classes. If X is some property or process, the phrase up to X means disregarding a possible difference in X, for instance the statement an integers prime factorization is unique up to ordering, means that the prime factorization is unique if we disregard the order of the factors. Further examples concerning up to isomorphism, up to permutations and up to rotations are described below, in informal contexts, mathematicians often use the word modulo for similar purposes, as in modulo isomorphism. The Tetris game does not allow reflections, so the former notation is likely to more natural. To add in the count, there is no formal notation. However, it is common to write there are seven reflecting tetrominos up to rotations, in this, Tetris provides an excellent example, as a reader might simply count 7 pieces ×4 rotations as 28, where some pieces have fewer than four rotation states. In the eight queens puzzle, if the eight queens are considered to be distinct, the regular n-gon, for given n, is unique up to similarity. In other words, if all similar n-gons are considered instances of the same n-gon, then there is only one regular n-gon. In group theory, for example, we may have a group G acting on a set X, another typical example is the statement that there are two different groups of order 4 up to isomorphism, or modulo isomorphism, there are two groups of order 4. This means that there are two classes of groups of order 4, assuming we consider groups to be equivalent if they are isomorphic. A hyperreal x and its standard part st are equal up to an infinitesimal difference, adequality All other things being equal Modulo Quotient set Quotient group Synecdoche Abuse of notation Up-to Techniques for Weak Bisimulation
2.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
3.
Magic constant
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The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a constant of 15. In general M = n ⋅ n 2 +12, the term magic constant or magic sum is similarly applied to other magic figures such as magic stars and magic cubes. The magic constant of a normal magic star is M =4 n +2. In 2013 Dirk Kinnaes found the magic series polytope, the number of unique sequences that form the magic constant is now known up to n =1000. In the mass model the value in each cell specifies the mass for that cell and this model has two notable properties. First it demonstrates the nature of all magic squares. If such a model is suspended from the cell the structure balances. The second property that can be calculated is the moment of inertia, summing the individual moments of inertia gives the moment of inertia for the magic square
4.
Word square
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A word square is a special type of acrostic. It consists of a set of written out in a square grid. The number of words, which is equal to the number of letters in each word, is known as the order of the square, the Sator Square is a famous word square in Latin. Thus the square consists of a palindrome, a reversal, if the words in a word square need not be true words, arbitrarily large squares of pronounceable combinations can be constructed. The following 12×12 array of letters appears in a Hebrew manuscript of The Book of the Sacred Magic of Abramelin the Mage of 1458, said to have given by God. An English edition appeared in 1898 and this is square 7 of Chapter IX of the Third Book, which is full of incomplete and complete squares. No source or explanation is given for any of the words, modern research indicates that a 12-square would be essentially impossible to construct from indexed words and phrases, even using a large number of languages. A specimen of the square was first published in English in 1859, the 7-square in 1877, the 8-square in 1884. It has been called the Holy Grail of logology, various methods have produced partial results to the 10-square problem, Tautonyms Since 1921, 10-squares have been constructed from reduplicated words and phrases like Alala. Each such square contains five words appearing twice, which in effect constitutes four identical 5-squares, 80% solution In 1976, Frank Rubin produced an incomplete ten-square containing two nonsense phrases at the top and eight dictionary words. If two words could be found containing the patterns SCENOOTL and HYETNNHY, this would become a complete ten-square, the largest source was the United States Board on Geographic Names National Imagery and Mapping Agency. In Word Ways in August and November 2002, he published several squares found in this wordlist, the square below has been held by some word square experts as essentially solving the 10-square problem, while others anticipate higher-quality 10-squares in the future. There are a few imperfections, Echeneidae is capitalized, Dioumabana and Adaletabat are places, many new large word squares and new species have arisen recently. However, modern combinatorics has demonstrated why the 10-square has taken so long to find, however, 11-squares are possible if words from a number of languages are allowed. It is possible to estimate the degree of difficulty of constructing word squares, 5-squares can be constructed with as little as a 250-word vocabulary. Roughly, for each step upwards, one four times the number of words. For a 9-square, one needs over 60,000 9-letter words, for large squares, the vocabulary prevents selecting more desirable words, and any resulting word squares use exotic words. The opposite problem occurs with small squares, a search will produce millions of examples
5.
On-Line Encyclopedia of Integer Sequences
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The On-Line Encyclopedia of Integer Sequences, also cited simply as Sloanes, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs, Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to professional mathematicians and amateurs, and is widely cited. As of 30 December 2016 it contains nearly 280,000 sequences, the database is searchable by keyword and by subsequence. Neil Sloane started collecting integer sequences as a student in 1965 to support his work in combinatorics. The database was at first stored on punched cards and he published selections from the database in book form twice, A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and these books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, as a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. The database continues to grow at a rate of some 10,000 entries a year, Sloane has personally managed his sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100, 000th sequence to the database, A100000, in 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS. org was created to simplify the collaboration of the OEIS editors and contributors, besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences, the sequence of numerators and the sequence of denominators, important irrational numbers such as π =3.1415926535897. are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, yet it still uses a form of conventional mathematical notation. Greek letters are represented by their full names, e. g. mu for μ. Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces, a represents the nth term of the sequence. Zero is often used to represent non-existent sequence elements, for example, A104157 enumerates the smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists. The value of a is 2, a is 1480028129, but there is no such 2×2 magic square, so a is 0
6.
Rotation (mathematics)
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Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a space that preserves at least one point. It can describe, for example, the motion of a body around a fixed point. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude, mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group, for example, in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are called active and passive transformations, the rotation group is a Lie group of rotations about a fixed point. This fixed point is called the center of rotation and is identified with the origin. The rotation group is a point stabilizer in a group of motions. For a particular rotation, The axis of rotation is a line of its fixed points and they exist only in n >2. The plane of rotation is a plane that is invariant under the rotation, unlike the axis, its points are not fixed themselves. The axis and the plane of a rotation are orthogonal, a representation of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. This meaning is somehow inverse to the meaning in the group theory, rotations of spaces of points and of respective vector spaces are not always clearly distinguished. The former are sometimes referred to as affine rotations, whereas the latter are vector rotations, see the article below for details. A motion of a Euclidean space is the same as its isometry, but a rotation also has to preserve the orientation structure. The improper rotation term refers to isometries that reverse the orientation, in the language of group theory the distinction is expressed as direct vs indirect isometries in the Euclidean group, where the former comprise the identity component. Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point, there are no non-trivial rotations in one dimension. In two dimensions, only a single angle is needed to specify a rotation about the origin – the angle of rotation that specifies an element of the circle group. The rotation is acting to rotate an object counterclockwise through an angle θ about the origin, composition of rotations sums their angles modulo 1 turn, which implies that all two-dimensional rotations about the same point commute
7.
Reflection (mathematics)
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In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points, this set is called the axis or plane of reflection. The image of a figure by a reflection is its image in the axis or plane of reflection. For example the image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b, a reflection is an involution, when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state. The term reflection is used for a larger class of mappings from a Euclidean space to itself. Such isometries have a set of fixed points that is an affine subspace, for instance a reflection through a point is an involutive isometry with just one fixed point, the image of the letter p under it would look like a d. This operation is known as a central inversion, and exhibits Euclidean space as a symmetric space. In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation, other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term reflection means reflection in a hyperplane, a figure that does not change upon undergoing a reflection is said to have reflectional symmetry. Some mathematicians use flip as a synonym for reflection, in a plane geometry, to find the reflection of a point drop a perpendicular from the point to the line used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure, step 2, construct circles centered at A′ and B′ having radius r. P and Q will be the points of intersection of two circles. Point Q is then the reflection of point P through line AB, the matrix for a reflection is orthogonal with determinant −1 and eigenvalues −1,1,1. The product of two matrices is a special orthogonal matrix that represents a rotation. Every rotation is the result of reflecting in an number of reflections in hyperplanes through the origin. Thus reflections generate the group, and this result is known as the Cartan–Dieudonné theorem. Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes, in general, a group generated by reflections in affine hyperplanes is known as a reflection group. The finite groups generated in this way are examples of Coxeter groups, note that the second term in the above equation is just twice the vector projection of v onto a
8.
Equivalence class
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In mathematics, when the elements of some set S have a notion of equivalence defined on them, then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the equivalence class if. Formally, given a set S and an equivalence relation ~ on S and it may be proven from the defining properties of equivalence relations that the equivalence classes form a partition of S. This partition – the set of equivalence classes – is sometimes called the quotient set or the quotient space of S by ~ and is denoted by S / ~. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories. If X is the set of all cars, and ~ is the relation has the same color as. X/~ could be identified with the set of all car colors. Let X be the set of all rectangles in a plane, for each positive real number A there will be an equivalence class of all the rectangles that have area A. Consider the modulo 2 equivalence relation on the set Z of integers, x ~ y if and this relation gives rise to exactly two equivalence classes, one class consisting of all even numbers, and the other consisting of all odd numbers. Under this relation, and all represent the element of Z/~. Let X be the set of ordered pairs of integers with b not zero, the same construction can be generalized to the field of fractions of any integral domain. In this situation, each equivalence class determines a point at infinity, the equivalence class of an element a is denoted and is defined as the set = of elements that are related to a by ~. An alternative notation R can be used to denote the class of the element a. This is said to be the R-equivalence class of a, the set of all equivalence classes in X with respect to an equivalence relation R is denoted as X/R and called X modulo R. The surjective map x ↦ from X onto X/R, which each element to its equivalence class, is called the canonical surjection or the canonical projection map. When an element is chosen in each class, this defines an injective map called a section. If this section is denoted by s, one has = c for every equivalence class c, the element s is called a representative of c. Any element of a class may be chosen as a representative of the class, sometimes, there is a section that is more natural than the other ones
9.
Arabic numerals
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In this numeral system, a sequence of digits such as 975 is read as a single number, using the position of the digit in the sequence to interpret its value. The symbol for zero is the key to the effectiveness of the system, the system was adopted by Arab mathematicians in Baghdad and passed on to the Arabs farther west. There is some evidence to suggest that the numerals in their current form developed from Arabic letters in the Maghreb, the current form of the numerals developed in North Africa, distinct in form from the Indian and eastern Arabic numerals. The use of Arabic numerals spread around the world through European trade, books, the term Arabic numerals is ambiguous. It most commonly refers to the widely used in Europe. Arabic numerals is also the name for the entire family of related numerals of Arabic. It may also be intended to mean the numerals used by Arabs and it would be more appropriate to refer to the Arabic numeral system, where the value of a digit in a number depends on its position. The decimal Hindu–Arabic numeral system was developed in India by AD700, the development was gradual, spanning several centuries, but the decisive step was probably provided by Brahmaguptas formulation of zero as a number in AD628. The system was revolutionary by including zero in positional notation, thereby limiting the number of digits to ten. It is considered an important milestone in the development of mathematics, one may distinguish between this positional system, which is identical throughout the family, and the precise glyphs used to write the numerals, which varied regionally. The glyphs most commonly used in conjunction with the Latin script since early modern times are 0123456789. The first universally accepted inscription containing the use of the 0 glyph in India is first recorded in the 9th century, in an inscription at Gwalior in Central India dated to 870. Numerous Indian documents on copper plates exist, with the symbol for zero in them, dated back as far as the 6th century AD. Inscriptions in Indonesia and Cambodia dating to AD683 have also been found and their work was principally responsible for the diffusion of the Indian system of numeration in the Middle East and the West. In the 10th century, Middle-Eastern mathematicians extended the decimal system to include fractions. The decimal point notation was introduced by Sind ibn Ali, who wrote the earliest treatise on Arabic numerals. Ghubar numerals themselves are probably of Roman origin, some popular myths have argued that the original forms of these symbols indicated their numeric value through the number of angles they contained, but no evidence exists of any such origin. In 825 Al-Khwārizmī wrote a treatise in Arabic, On the Calculation with Hindu Numerals, Algoritmi, the translators rendition of the authors name, gave rise to the word algorithm
10.
Yuan dynasty
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The Yuan dynasty, officially the Great Yuan, was the empire or ruling dynasty of China established by Kublai Khan, leader of the Mongolian Borjigin clan. His realm was, by point, isolated from the other khanates and controlled most of present-day China and its surrounding areas. Some of the Mongolian Emperors of the Yuan mastered the Chinese language, while others used their native language. The Yuan dynasty is considered both a successor to the Mongol Empire and an imperial Chinese dynasty and it was the khanate ruled by the successors of Möngke Khan after the division of the Mongol Empire. In official Chinese histories, the Yuan dynasty bore the Mandate of Heaven, following the Song dynasty, the dynasty was established by Kublai Khan, yet he placed his grandfather Genghis Khan on the imperial records as the official founder of the dynasty as Taizu. In addition to Emperor of China, Kublai Khan also claimed the title of Great Khan, supreme over the other khanates, the Chagatai, the Golden Horde. As such, the Yuan was also referred to as the Empire of the Great Khan. However, while the claim of supremacy by the Yuan emperors was at times recognized by the khans, their subservience was nominal. In 1271, Kublai Khan imposed the name Great Yuan, establishing the Yuan dynasty, dà Yuán is from the clause 大哉乾元 in the Commentaries on the Classic of Changes section regarding Qián. The counterpart in Mongolian language was Dai Ön Ulus, also rendered as Ikh Yuan Üls or Yekhe Yuan Ulus, in Mongolian, Dai Ön is often used in conjunction with the Yeke Mongghul Ulus, resulting in Dai Ön Yeke Mongghul Ulus, meaning Great Mongol State. Nevertheless, both terms can refer to the khanate within the Mongol Empire directly ruled by Great Khans before the actual establishment of the Yuan dynasty by Kublai Khan in 1271. Genghis Khan united the Mongol and Turkic tribes of the steppes and he and his successors expanded the Mongol empire across Asia. Under the reign of Genghis third son, Ögedei Khan, the Mongols destroyed the weakened Jin dynasty in 1234, Ögedei offered his nephew Kublai a position in Xingzhou, Hebei. Kublai was unable to read Chinese but had several Han Chinese teachers attached to him since his early years by his mother Sorghaghtani and he sought the counsel of Chinese Buddhist and Confucian advisers. Möngke Khan succeeded Ögedeis son, Güyük, as Great Khan in 1251 and he granted his brother Kublai control over Mongol held territories in China. Kublai built schools for Confucian scholars, issued paper money, revived Chinese rituals and he adopted as his capital city Kaiping in Inner Mongolia, later renamed Shangdu. Many Han Chinese and Khitan defected to the Mongols to fight against the Jin, two Han Chinese leaders, Shi Tianze, Liu Heima, and the Khitan Xiao Zhala defected and commanded the 3 Tumens in the Mongol army. Liu Heima and Shi Tianze served Ogödei Khan, Liu Heima and Shi Tianxiang led armies against Western Xia for the Mongols
11.
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
12.
Baghdad
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Baghdad is the capital of the Republic of Iraq. The population of Baghdad, as of 2016, is approximately 8,765,000 making it the largest city in Iraq, the second largest city in the Arab world, and the second largest city in Western Asia. Located along the Tigris River, the city was founded in the 8th century, within a short time of its inception, Baghdad evolved into a significant cultural, commercial, and intellectual center for the Islamic world. This, in addition to housing several key institutions, garnered the city a worldwide reputation as the Centre of Learning. Throughout the High Middle Ages, Baghdad was considered to be the largest city in the world with a population of 1,200,000 people. The city was destroyed at the hands of the Mongol Empire in 1258, resulting in a decline that would linger through many centuries due to frequent plagues. With the recognition of Iraq as an independent state in 1938, in contemporary times, the city has often faced severe infrastructural damage, most recently due to the 2003 invasion of Iraq, and the subsequent Iraq War that lasted until December 2011. In recent years, the city has been subjected to insurgency attacks. As of 2012, Baghdad was listed as one of the least hospitable places in the world to live, the site where the city of Baghdad developed has been populated for millennia. By the 8th century AD, several villages had developed there, including a Persian hamlet called Baghdad, the name is of Indo-European origin and a Middle Persian compound of Bagh god and dād given by, translating to Bestowed by God or Gods gift. In Old Persian the first element can be traced to boghu and is related to Slavic bog god, a similar term in Middle Persian is the name Mithradāt, known in English by its Hellenistic form Mithridates, meaning gift of Mithra. There are a number of locations in the wider region whose names are compounds of the word bagh, including Baghlan. The name of the town Baghdati in Georgia shares the same etymological origins, when the Abbasid caliph, al-Mansur, founded a completely new city for his capital, he chose the name Madinat al-Salaam or City of Peace. This was the name on coins, weights, and other official usage. By the 11th century, Baghdad became almost the exclusive name for the world-renowned metropolis, after the fall of the Umayyads, the first Muslim dynasty, the victorious Abbasid rulers wanted their own capital whence they could rule. They chose a site north of the Sassanid capital of Ctesiphon, on 30 July 762, the caliph Al-Mansur commissioned the construction of the city, mansur believed that Baghdad was the perfect city to be the capital of the Islamic empire under the Abbasids. Mansur loved the site so much he is quoted saying, This is indeed the city that I am to found, where I am to live, and where my descendants will reign afterward. The citys growth was helped by its excellent location, based on at least two factors, it had control over strategic and trading routes along the Tigris, the abundance of water in a dry climate
13.
Shams al-Ma'arif
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Shams al-Maarif or Shams al-Maarif wa Lataif al-Awarif is a 13th-century grimoire written on Arabic magic and a manual for achieving esoteric spirituality. It was written by The Sufi Sheikh Ahmad bin Ali Al-buni in Egypt, in contemporary form the book consists of two volumes, Shams al-Maarif al-Kubra and Shams al-Maarif al-Sughra, the former being the larger of the two. The table of contents that were introduced in the printed editions of the work contain a list of unnumbered chapters. However, prior to the press and various other standardisations. While being popular, it carries a reputation for being suppressed and banned for much of Islamic history. Many Sufi orders, such as the Naqshbandi-Haqqani order have recognised its legitimacy and use as a compendium for the occult, another title by the same author, namely Manba Usool al-Hikmah, is considered its companion text. Some of these rituals have had degrees of notability. Outside of the Arab and Western world, several editions of the book have been published in the Urdu, alchemy and chemistry in Islam Islamic astrology Ruhaniyya Shams al-Maarif al-Kubra wa Lataifu al-Avarif Partial Translation in Spanish and First Comparative Edition by Jaime Coullaut Cordero
14.
Magic (illusion)
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Magic is one of the oldest performing arts in the world in which audiences are entertained by staged tricks or illusions of seemingly impossible or supernatural feats using natural means. These feats are called magic tricks, effects, or illusions, the term magic etymologically derives from the Greek word mageia. In ancient times, Greeks and Persians had been at war for centuries, ritual acts of Persian priests came to be known as mageia, and then magika—which eventually came to mean any foreign, unorthodox, or illegitimate ritual practice. The first book containing explanations of magic tricks appeared in 1584, during the 17th century, many similar books were published that described magic tricks. Until the 18th century, magic shows were a source of entertainment at fairs. A founding figure of modern entertainment magic was Jean Eugène Robert-Houdin, John Henry Anderson was pioneering the same transition in London in the 1840s. Towards the end of the 19th century, large magic shows permanently staged at big theatre venues became the norm, as a form of entertainment, magic easily moved from theatrical venues to television magic specials. Performances that modern observers would recognize as conjuring have been practiced throughout history, for many recorded centuries, magicians were associated with the devil and the occult. During the 19th and 20th centuries, many stage magicians even capitalized on this notion in their advertisements. The same level of ingenuity that was used to produce famous ancient deceptions such as the Trojan Horse would also have used for entertainment. They were also used by the practitioners of various religions and cults from ancient times onwards to frighten uneducated people into obedience or turn them into adherents, however, the profession of the illusionist gained strength only in the 18th century, and has enjoyed several popular vogues since. Opinions vary among magicians on how to categorize a given effect, Magicians may pull a rabbit from an empty hat, make something seem to disappear, or transform a red silk handkerchief into a green silk handkerchief. Magicians may also destroy something, like cutting a head off, other illusions include making something appear to defy gravity, making a solid object appear to pass through another object, or appearing to predict the choice of a spectator. Many magical routines use combinations of effects, one of the earliest books on the subject is Gantzionys work of 1489, Natural and Unnatural Magic, which describes and explains old-time tricks. Among the tricks discussed were sleight-of-hand manipulations with rope, paper, at the time, fear and belief in witchcraft was widespread and the book tried to demonstrate that these fears were misplaced. All obtainable copies were burned on the accession of James I in 1603 and it began to reappear in print in 1651. In the early 18th century, as belief in witchcraft was waning, a notable figure in this transition was the English showman, Isaac Fawkes, who began to promote his act in advertisements from the 1720s – he even claimed to have performed for King George II. He throws up a Pack of Cards, and causes them to be living birds flying about the room and he causes living Beasts, Birds, and other Creatures to appear upon the Table
15.
Lo Shu Square
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Lo Shu Square, or the Nine Halls Diagram, is the unique normal magic square of order three. Chinese legends concerning the pre-historic Emperor Yu tell of the Lo Shu, in ancient China there was a huge deluge, the people offered sacrifices to the god of one of the flooding rivers, the Luo river, to try to calm his anger. Early records are ambiguous, referring to a map, and date to 650 BCE, but clearly refer to a magic square by 80 CE. The odd and even numbers alternate in the periphery of the Lo Shu pattern, the 4 even numbers are at the four corners, the sums in each of the 3 rows, in each of the 3 columns, and in both diagonals, are all 15. Since 5 is in the cell, the sum of any two other cells that are directly through the 5 from each other is 10. The Lo Shu is sometimes connected numerologically with the Ba Gua 八卦8 trigrams, because north is placed at the bottom of maps in China, the 3x3 magic square having number 1 at the bottom and 9 at the top is used in preference to the other rotations/reflections. As seen in the Later Heaven arrangement,1 and 9 correspond with ☵ Kǎn 水 Water, in the Early Heaven arrangement, they would correspond with ☷ Kūn 地 Earth and ☰ Qián 天 Heaven respectively. Like the Ho Tu, the Lo Shu square, in conjunction with the 8 trigrams, is used as a mandalic representation important in Feng Shui geomancy. Associative magic square Sator Square Tetractys Yellow River Map Camunian rose Yoshio Mikami, The Development of Mathematics in China and Japan,1913, Library of Congress 61-13497. Frank J. Swetz, The Legacy of the Luoshu, A. K. Peters / CRC Press, 2nd Rev Ed edition, media related to Luoshu at Wikimedia Commons Lo Shu Square, Definition, Nature and History
16.
Ming dynasty
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The Ming dynasty was the ruling dynasty of China – then known as the Empire of the Great Ming – for 276 years following the collapse of the Mongol-led Yuan dynasty. The Ming, described by some as one of the greatest eras of orderly government, although the primary capital of Beijing fell in 1644 to a rebellion led by Li Zicheng, regimes loyal to the Ming throne – collectively called the Southern Ming – survived until 1683. He rewarded his supporters and employed them as a counterweight against the Confucian scholar-bureaucrats. One, Zheng He, led seven enormous voyages of exploration into the Indian Ocean as far as Arabia, the rise of new emperors and new factions diminished such extravagances, the capture of the Zhengtong Emperor during the 1449 Tumu Crisis ended them completely. The imperial navy was allowed to fall into disrepair while forced labor constructed the Liaodong palisade, haijin laws intended to protect the coasts from Japanese pirates instead turned many into smugglers and pirates themselves. The growth of Portuguese, Spanish, and Dutch trade created new demand for Chinese products and produced an influx of Japanese. This abundance of specie remonetized the Ming economy, whose money had suffered repeated hyperinflation and was no longer trusted. While traditional Confucians opposed such a prominent role for commerce and the newly rich it created, combined with crop failure, floods, and epidemic, the dynasty collapsed before the rebel leader Li Zicheng, who was defeated by the Manchu-led Eight Banner armies who founded the Qing dynasty. The Mongol-led Yuan dynasty ruled before the establishment of the Ming dynasty, consequently, agriculture and the economy were in shambles, and rebellion broke out among the hundreds of thousands of peasants called upon to work on repairing the dykes of the Yellow River. A number of Han Chinese groups revolted, including the Red Turbans in 1351, the Red Turbans were affiliated with the White Lotus, a Buddhist secret society. Zhu Yuanzhang was a peasant and Buddhist monk who joined the Red Turbans in 1352. In 1356, Zhus rebel force captured the city of Nanjing, with the Yuan dynasty crumbling, competing rebel groups began fighting for control of the country and thus the right to establish a new dynasty. In 1363, Zhu Yuanzhang eliminated his archrival and leader of the rebel Han faction, Chen Youliang, in the Battle of Lake Poyang, arguably the largest naval battle in history. Known for its ambitious use of ships, Zhus force of 200,000 Ming sailors were able to defeat a Han rebel force over triple their size, claimed to be 650. The victory destroyed the last opposing rebel faction, leaving Zhu Yuanzhang in uncontested control of the bountiful Yangtze River Valley, Zhu Yuanzhang took Hongwu, or Vastly Martial, as his era name. Hongwu made an effort to rebuild state infrastructure. He built a 48 km long wall around Nanjing, as well as new palaces, Hongwu organized a military system known as the weisuo, which was similar to the fubing system of the Tang dynasty. With a growing suspicion of his ministers and subjects, Hongwu established the Jinyiwei, some 100,000 people were executed in a series of purges during his rule
17.
History of China
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Written records of the history of China can be found from as early as 1500 BC under the Shang dynasty. Ancient historical texts such as the Records of the Grand Historian and the Bamboo Annals describe a Xia dynasty, with thousands of years of continuous history, China is one of the worlds oldest civilizations, and is regarded as one of the cradles of civilization. Much of Chinese culture, literature and philosophy developed during the Zhou dynasty. This is one of multiple periods of failed statehood in Chinese history, between eras of multiple kingdoms and warlordism, Chinese dynasties have ruled parts or all of China, in some eras control stretched as far as Xinjiang and Tibet, as at present. In 221 BC Qin Shi Huang united the warring kingdoms and created for himself the title of emperor of the Qin dynasty. Successive dynasties developed bureaucratic systems that enabled the emperor to control vast territories directly, in the 21 centuries from 206 BC until AD1912, routine administrative tasks were handled by a special elite, the Scholar-officials. Young men were selected through difficult examinations and were well-versed in calligraphy and philosophy. What is now China was inhabited by Homo erectus more than a million years ago, recent study shows that the stone tools found at Xiaochangliang site are magnetostratigraphically dated to 1.36 million years ago. The archaeological site of Xihoudu in Shanxi Province is the earliest recorded use of fire by Homo erectus, the excavations at Yuanmou and later Lantian show early habitation. Perhaps the most famous specimen of Homo erectus found in China is the so-called Peking Man discovered in 1923–27, fossilised teeth of Homo sapiens dating to 125, 000–80,000 BC have been discovered in Fuyan Cave in Dao County in Hunan. The Neolithic age in China can be traced back to about 10,000 BC, Early evidence for proto-Chinese millet agriculture is radiocarbon-dated to about 7000 BC. The earliest evidence of cultivated rice, found by the Yangtze River, is carbon-dated to 8,000 years ago, farming gave rise to the Jiahu culture. At Damaidi in Ningxia,3,172 cliff carvings dating to 6000–5000 BC have been discovered, featuring 8,453 individual characters such as the sun, moon, stars, gods and these pictographs are reputed to be similar to the earliest characters confirmed to be written Chinese. Chinese proto-writing existed in Jiahu around 7000 BC, Dadiwan from 5800 BC to 5400 BC, Damaidi around 6000 BC, some scholars have suggested that Jiahu symbols were the earliest Chinese writing system. With agriculture came increased population, the ability to store and redistribute crops, Later, Yangshao culture was superseded by the Longshan culture, which was also centered on the Yellow River from about 3000 BC to 2000 BC. Bronze artifacts have been found at the Majiayao culture site, The Bronze Age is also represented at the Lower Xiajiadian culture site in northeast China. Sanxingdui located in what is now Sichuan province is believed to be the site of a ancient city. The site was first discovered in 1929 and then re-discovered in 1986, Chinese archaeologists have identified the Sanxingdui culture to be part of the ancient kingdom of Shu, linking the artifacts found at the site to its early legendary kings
18.
Yu the Great
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The dates proposed for Yus reign predate the oldest known written records in China, the oracle bones of the late Shang dynasty, by nearly a millennium. No inscriptions on artifacts from the era of Yu, nor the later oracle bones, make any mention of Yu. The lack of anything remotely close to documentary evidence has led to some controversy over the historicity of Yu. Many of the stories about Yu were collected in Sima Qians famous Records of the Grand Historian, Yu and other sage-kings of Ancient China were lauded for their virtues and morals by Confucius and other Chinese teachers. Yu is one of the few Chinese rulers posthumously honored with the epithet the Great, Yu was said to have been born at Mount Wen, in modern-day Beichuan County, Sichuan Province, though there are debates as to whether he was born in Shifang instead. Yus mother was of the Youxin clan named either Nüzhi or Nüxi, when Yu was a child, his father Gun moved the people east toward the Central Plain. King Yao enfeoffed Gun as lord of Chong, usually identified as the peak of Mount Song. Yu is thus believed to have grown up on the slopes of Mount Song and he later married a woman from Mount Tu who is generally referred to as Tushan-shi. They had a son named Qi, a literally meaning revelation. The location of Mount Tu has always been disputed, the two most probable locations are Mount Tu in Anhui Province and the Tu Peak of the Southern Mountain in Chongqing Municipality. During the reign of king Yao, the Chinese heartland was frequently plagued by floods that prevented further economic, Yus father, Gun, was tasked with devising a system to control the flooding. As an adult, Yu continued his fathers work and made a study of the river systems in an attempt to learn why his fathers great efforts had failed. Instead of directly damming the rivers flow, Yu made a system of canals which relieved floodwater into fields. The dredging and irrigation were successful, and allowed ancient Chinese culture to flourish along the Yellow River, Wei River, the project earned Yu renown throughout Chinese history, and is referred to in Chinese history as Great Yu Controls the Waters. In particular, Mount Longmen along the Yellow River had a narrow channel which blocked water from flowing freely east toward the ocean. Yu is said to have brought a number of workers to open up this channel. In a mythical version of story, presented in Wang Jias 4th century AD work Shi Yi Ji, Yu is assisted in his work by a yellow dragon. Another local myth says that Yu created the Sanmenxia Three Passes Gorge of the Yangzi River by cutting a mountain ridge with a divine battle-axe to control flooding, traditional stories say that Yu sacrificed a great deal of his body to control the floods
19.
Turtle
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Turtles are reptiles of the order Testudines characterised by a special bony or cartilaginous shell developed from their ribs and acting as a shield. Turtle may refer to the order as a whole or to fresh-water, the order Testudines includes both extant and extinct species. The earliest known members of this date from 157 million years ago, making turtles one of the oldest reptile groups. Of the 327 known species alive today, some are highly endangered, turtles are ectotherms—animals commonly called cold-blooded—meaning that their internal temperature varies according to the ambient environment. However, because of their metabolic rate, leatherback sea turtles have a body temperature that is noticeably higher than that of the surrounding water. Turtles are classified as amniotes, along with reptiles, birds. Like other amniotes, turtles breathe air and do not lay eggs underwater, Chelonia is based on the Greek word χελώνη chelone tortoise, turtle, also denoting armor or interlocking shields, testudines, on the other hand, is based on the Latin word testudo tortoise. Turtle may either refer to the order as a whole, or to particular turtles that make up a form taxon that is not monophyletic, the meaning of the word turtle differs from region to region. In North America, all chelonians are commonly called turtles, including terrapins, in Great Britain, the word turtle is used for sea-dwelling species, but not for tortoises. The term tortoise usually refers to any land-dwelling, non-swimming chelonian, most land-dwelling chelonians are in the Testudinidae family, only one of the 14 extant turtle families. Terrapin is used to describe several species of small, edible, hard-shell turtles, typically found in brackish waters. Some languages do not have this distinction, as all of these are referred to by the same name, for example, in Spanish, the word tortuga is used for turtles, tortoises, and terrapins. A sea-dwelling turtle is tortuga marina, a freshwater species tortuga de río, the largest living chelonian is the leatherback sea turtle, which reaches a shell length of 200 cm and can reach a weight of over 900 kg. Freshwater turtles are generally smaller, but with the largest species, the Asian softshell turtle Pelochelys cantorii, a few individuals have been reported up to 200 cm. This dwarfs even the better-known alligator snapping turtle, the largest chelonian in North America and they became extinct at the same time as the appearance of man, and it is assumed humans hunted them for food. The only surviving giant tortoises are on the Seychelles and Galápagos Islands and can grow to over 130 cm in length, the largest ever chelonian was Archelon ischyros, a Late Cretaceous sea turtle known to have been up to 4.6 m long. The smallest turtle is the speckled padloper tortoise of South Africa and it measures no more than 8 cm in length and weighs about 140 g. Two other species of turtles are the American mud turtles
20.
Yang Hui
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Yang Hui, courtesy name Qianguang, was a late-Song dynasty Chinese mathematician from Qiantang. Yang worked on magic squares, magic circles and the binomial theorem and this triangle was the same as Pascals Triangle, discovered by Yangs predecessor Jia Xian. Yang was also a contemporary to the famous mathematician Qin Jiushao. In his book known as Rújī Shìsuǒ or Piling-up Powers and Unlocking Coefficients, Jia described the method used as li cheng shi suo. It appeared again in a publication of Zhu Shijies book Jade Mirror of the Four Unknowns of 1303 AD, around 1275 AD, Yang finally had two published mathematical books, which were known as the Xugu Zhaiqi Suanfa and the Suanfa Tongbian Benmo. In his writing, he criticized the earlier works of Li Chunfeng and Liu Yi. In his written work, Yang provided theoretical proof for the proposition that the complements of the parallelograms which are about the diameter of any given parallelogram are equal to one another. This was the idea expressed in the Greek mathematician Euclids forty-third proposition of his first book, only Yang used the case of a rectangle. There were also a number of other problems and theoretical mathematical propositions posed by Yang that were strikingly similar to the Euclidean system. However, the first books of Euclid to be translated into Chinese was by the effort of the Italian Jesuit Matteo Ricci. History of mathematics List of mathematicians Chinese mathematics Needham, Joseph, science and Civilization in China, Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Encyclopedia of China, 1st ed. Yang Hui at MacTutor
21.
Eastern Arabic numerals
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These numbers are known as أرقام هندية in Arabic. They are sometimes also called Indic numerals in English, however, that is sometimes discouraged as it can lead to confusion with Indian numerals, used in Brahmic scripts of India. Each numeral in the Persian variant has a different Unicode point even if it looks identical to the Eastern Arabic numeral counterpart, however the variants used with Urdu, Sindhi and other South Asian languages are not encoded separately from the Persian variants. See U+0660 through U+0669 and U+06F0 through U+06F9, written numerals are arranged with their lowest-value digit to the right, with higher value positions added to the left. That is identical to the arrangement used by Western texts using Hindu-Arabic numerals even though Arabic script is read from right to left. There is no conflict unless numerical layout is necessary, as is the case for arithmetic problems and lists of numbers, Eastern Arabic numerals remain strongly predominant vis-à-vis Western Arabic numerals in many countries to the East of the Arab world, particularly in Iran and Afghanistan. In Pakistan, Western Arabic numerals are more used as a considerable majority of the population is anglophone. Eastern numerals still continue to see use in Urdu publications and newspapers, in North Africa, only Western Arabic numerals are now commonly used. In medieval times, these used a slightly different set
22.
Islamic Golden Age
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This period is traditionally said to have ended with the collapse of the Abbasid caliphate due to Mongol invasions and the Sack of Baghdad in 1258 AD. A few contemporary scholars place the end of the Islamic Golden Age as late as the end of 15th to 16th centuries, the metaphor of a golden age began to be applied in 19th-century literature about Islamic history, in the context of the western aesthetic fashion known as Orientalism. There is no definition of term, and depending on whether it is used with a focus on cultural or on military achievement. During the early 20th century, the term was used only occasionally, the Muslim government heavily patronized scholars. The money spent on the Translation Movement for some translations is estimated to be equivalent to twice the annual research budget of the United Kingdom’s Medical Research Council. The best scholars and notable translators, such as Hunayn ibn Ishaq, had salaries that are estimated to be the equivalent of professional athletes today, the House of Wisdom was a library established in Abbasid-era Baghdad, Iraq by Caliph al-Mansur. During this period, the Muslims showed a strong interest in assimilating the knowledge of the civilizations that had been conquered. They also excelled in fields, in particular philosophy, science. For a long period of time the personal physicians of the Abbasid Caliphs were often Assyrian Christians, among the most prominent Christian families to serve as physicians to the caliphs were the Bukhtishu dynasty. Throughout the 4th to 7th centuries, Christian scholarly work in the Greek, the House of Wisdom was founded in Baghdad in 825, modelled after the Academy of Gondishapur. It was led by Christian physician Hunayn ibn Ishaq, with the support of Byzantine medicine, many of the most important philosophical and scientific works of the ancient world were translated, including the work of Galen, Hippocrates, Plato, Aristotle, Ptolemy and Archimedes. Many scholars of the House of Wisdom were of Christian background, the use of paper spread from China into Muslim regions in the eighth century, arriving in Al-Andalus on the Iberian peninsula, present-day Spain in the 10th century. It was easier to manufacture than parchment, less likely to crack than papyrus, Islamic paper makers devised assembly-line methods of hand-copying manuscripts to turn out editions far larger than any available in Europe for centuries. It was from countries that the rest of the world learned to make paper from linen. Ibn Rushd and Ibn Sina played a role in saving the works of Aristotle, whose ideas came to dominate the non-religious thought of the Christian. Ibn Sina and other such as al-Kindi and al-Farabi combined Aristotelianism and Neoplatonism with other ideas introduced through Islam. Arabic philosophic literature was translated into Latin and Ladino, contributing to the development of modern European philosophy, during this period, non-Muslims were allowed to flourish relative to treatment of religious minorities in the Christian Byzantine Empire. The Jewish philosopher Moses Maimonides, who lived in Andalusia, is an example, in epistemology, Ibn Tufail wrote the novel Hayy ibn Yaqdhan and in response Ibn al-Nafis wrote the novel Theologus Autodidactus
23.
Chess
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Chess is a two-player strategy board game played on a chessboard, a checkered gameboard with 64 squares arranged in an eight-by-eight grid. Chess is played by millions of people worldwide, both amateurs and professionals, each player begins the game with 16 pieces, one king, one queen, two rooks, two knights, two bishops, and eight pawns. Each of the six piece types moves differently, with the most powerful being the queen, the objective is to checkmate the opponents king by placing it under an inescapable threat of capture. To this end, a players pieces are used to attack and capture the opponents pieces, in addition to checkmate, the game can be won by voluntary resignation by the opponent, which typically occurs when too much material is lost, or if checkmate appears unavoidable. A game may result in a draw in several ways. Chess is believed to have originated in India, some time before the 7th century, chaturanga is also the likely ancestor of the Eastern strategy games xiangqi, janggi and shogi. The pieces took on their current powers in Spain in the late 15th century, the first generally recognized World Chess Champion, Wilhelm Steinitz, claimed his title in 1886. Since 1948, the World Championship has been controlled by FIDE, the international governing body. There is also a Correspondence Chess World Championship and a World Computer Chess Championship, online chess has opened amateur and professional competition to a wide and varied group of players. There are also many variants, with different rules, different pieces. FIDE awards titles to skilled players, the highest of which is grandmaster, many national chess organizations also have a title system. However, these are not recognised by FIDE, the term master may refer to a formal title or may be used more loosely for any skilled player. Until recently, chess was a sport of the International Olympic Committee. Chess was included in the 2006 and 2010 Asian Games, since the 1990s, computer analysis has contributed significantly to chess theory, particularly in the endgame. The computer IBM Deep Blue was the first machine to overcome a reigning World Chess Champion in a match when it defeated Garry Kasparov in 1997, the rise of strong computer programs that can be run on hand-held devices has led to increasing concerns about cheating during tournaments. The official rules of chess are maintained by FIDE, chesss international governing body, along with information on official chess tournaments, the rules are described in the FIDE Handbook, Laws of Chess section. Chess is played on a board of eight rows and eight columns. The colors of the 64 squares alternate and are referred to as light, the chessboard is placed with a light square at the right-hand end of the rank nearest to each player
24.
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
25.
Arabs
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Arabs are an ethnic group inhabiting the Arab world. They primarily live in the Arab states in Western Asia, North Africa, the Horn of Africa, the Arabs are first mentioned in the mid-ninth century BCE as a tribal people dwelling in the central Arabian Peninsula. The Arabs appear to have been under the vassalage of the Neo-Assyrian Empire, tradition holds that Arabs descend from Ishmael, the son of Abraham. The Arabian Desert is the birthplace of Arab, there are other Arab groups as well that spread in the land and existed for millennia. Before the expansion of the Caliphate, Arab referred to any of the largely nomadic Semitic people from the northern to the central Arabian Peninsula and Syrian Desert. Presently, Arab refers to a number of people whose native regions form the Arab world due to spread of Arabs throughout the region during the early Arab conquests of the 7th and 8th centuries. The Arabs forged the Rashidun, Umayyad and the Abbasid caliphates, whose borders reached southern France in the west, China in the east, Anatolia in the north, and this was one of the largest land empires in history. The Great Arab Revolt has had as big an impact on the modern Middle East as the World War I, the war signaled the end of the Ottoman Empire. They are modern states and became significant as distinct political entities after the fall and defeat, following adoption of the Alexandria Protocol in 1944, the Arab League was founded on 22 March 1945. The Charter of the Arab League endorsed the principle of an Arab homeland whilst respecting the sovereignty of its member states. Beyond the boundaries of the League of Arab States, Arabs can also be found in the global diaspora, the ties that bind Arabs are ethnic, linguistic, cultural, historical, identical, nationalist, geographical and political. The Arabs have their own customs, language, architecture, art, literature, music, dance, media, cuisine, dress, society, sports, the total number of Arabs are an estimated 450 million. This makes them the second largest ethnic group after the Han Chinese. Arabs are a group in terms of religious affiliations and practices. In the pre-Islamic era, most Arabs followed polytheistic religions, some tribes had adopted Christianity or Judaism, and a few individuals, the hanifs, apparently observed monotheism. Today, Arabs are mainly adherents of Islam, with sizable Christian minorities, Arab Muslims primarily belong to the Sunni, Shiite, Ibadi, Alawite, Druze and Ismaili denominations. Arab Christians generally follow one of the Eastern Christian Churches, such as the Maronite, Coptic Orthodox, Greek Orthodox, Greek Catholic, or Chaldean churches. Listed among the booty captured by the army of king Shalmaneser III of Assyria in the Battle of Qarqar are 1000 camels of Gi-in-di-buu the ar-ba-a-a or Gindibu belonging to the Arab
26.
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
27.
Ahmad al-Buni
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Al-Buni lived in Egypt and learned from many eminent Sufi masters of his time. It was to be banned soon after as heretical by followers of the Islamic orthodoxy, instead of sihr, this kind of magic was called Ilm al-Hikmah, Ilm al-simiyah and Ruhaniyat. Most of the so-called mujarrabât books on sorcery in the Muslim world are simplified excerpts from the Shams al-maârif, the book remains the seminal work on Theurgy and esoteric arts to this day. 1200, Ahmad al-Buni showed how to construct magic squares using a simple bordering technique, al-Buni wrote about Latin squares and constructed, for example,4 x 4 Latin squares using letters from one of the 99 names of Allah. His works on traditional healing remains a point of reference among Yoruba Muslim healers in Nigeria, Ahmad al-Buni also left a list of other titles that he wrote. Unfortunately, very few of them have survived and it is stated in his work Manba’ Usul al-Hikmah that he acquired his knowledge of the esoteric properties of the letters from his personal teacher Abu Abdillah Shams al-Din al-Asfahâni. Al-Buni also made mention in his work of Plato, Aristotle, Hermes, Alexander the Great. In one of his works, he recounted a story of his discovery of a cache of manuscripts buried under the pyramids and his work is said to have influenced the Hurufis and the New Lettrist International. He may also have influenced the late Shii movement of Babism. Ahmad al-Buni Shams al-Maairf al-Kubra, Cairo,1928, Ahmad al-Buni, Sharh Ism Allah al-azam fi al-Ruhani, printed in 1357 AH or in Egypt al-Matbaat al-Mahmudiyyat al-Tujjariyyat bil-Azhar. Ahmad al-Buni, Kabs al-iktida, Oriental Manuscripts in Durham University Library Edgar W. Francis, the Names of God in the Writings of Ahmad ibn Ali al-Buni Shams al-Maarif al-Kubra - شمس المعارف ولطائف العوارف Kabs al-iktida - قبس الإقتداء
28.
Kuberakolam
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Kuberakolam is a certain magic square of order 3 constructed using rice flour and drawn on the floors of several houses in South India. In Hindu mythology, Kubera is the god of riches and wealth, a Kubera kolam is also a creation in the form of a kolam which is a form of painting that is drawn using rice powder /chalk/chalk powder and colored powders in many places in South India. A kolam is basically a drawing composed of lines and loops, the Kubera kolam magic square is formed by the numbers 20,21,22,23,24,25,26,27,28. The arrangement of the square is depicted below, To construct the Kubera kolam, the numbers are then written in the following order, start from 24 and then the numbers 28,23,22,27,20,25,26,21 in that order. A coin and a flower are usually placed in each cell, in this magic square, the numbers in each row, and in each column, and the numbers in the forward and backward main diagonals, all add up to the same number, namely,72. The Lost Symbol, a 2009 novel written by American writer Dan Brown, attempts have been made to use the Kubera Kolam to introduce randomization in image steganography
29.
Pandiagonal magic square
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A pandiagonal magic square remains pandiagonally magic not only under rotation or reflection, but also if a row or column is moved from one side of the square to the opposite side. As such, an n×n pandiagonal magic square can be regarded as having 8n2 orientations and it is easily shown that non-trivial pandiagonal numerical magic squares of order 3 do not exist. However, if the magic square concept is generalized to include geometric shapes instead of numbers—the geometric magic squares discovered by Lee Sallows—a 3×3 panmagic square does exist, the smallest non-trivial pandiagonal magic squares consisting of numbers are 4×4 squares. g. There are only three distinct 4×4 pandiagonal magic squares, namely A above and the following, These three are closely related. B and C can be seen to differ only because the components of each semi-diagonal are reversed, consequently, no 4×4 panmagic squares are associative, though they all fulfil the further requirement for a 4×4 most-perfect magic square, that each 2×2 subsquare sums to 34. There are many 5×5 pandiagonal magic squares, unlike 4×4 panmagic squares, these can be associative. This leads to 100 quincunx sums, including broken quincunxes analogous to broken diagonals, the quincunx sums can be proved by taking linear combinations of the row, column, and diagonal sums. Consider the panmagic square with magic sum Z, the net result is 5A+5E+5M+5U+5Y = 5Z, which divided by 5 gives the quincunx sum. Similar linear combinations can be constructed for the other quincunx patterns H+L+M+N+R, C+K+M+O+W, no panmagic square exists of order 4n+2 if consecutive integers are used. But certain sequences of nonconsecutive integers do admit order- panmagic squares, consider the sum 1+2+3+5+6+7 =24. A × panmagic square can be built by the following algorithm, set up the first column of the square with the first 6n±1 natural numbers. Example, Copy the first column into the column but shift it ring-wise by 2 rows. Example, Continue copying the current column into the column with ring-wise shift by 2 rows until the square is filled completely. Example, Build a second square and copy the first square into it, so you have to exchange rows and columns. Build the final square by multiplying the second square by 6n±1, adding the first square, example, A + ×AT - A 4n×4n panmagic square can be built by the following algorithm. Put the first 2n natural numbers into the first row and the first 2n columns of the square, example, Put the next 2n natural numbers beneath the first 2n natural numbers in inverse sequence. Each vertical pair must have the same sum, example, Copy that 2×2n rectangle 2n-1 times beneath the first rectangle. Example, Copy the left 4n×2n rectangle into the right 4n×2n rectangle, example, Build a second 4n×4n square and copy the first square into it but turn it by 90°
30.
Most-perfect magic square
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A most-perfect magic square of order n is a magic square containing the numbers 1 to n2 with two additional properties, Each 2×2 subsquare sums to 2s, where s = n2 +1. All pairs of integers distant n/2 along a diagonal sum to s, specific examples of most-perfect magic squares that begin with the 2015 date demonstrate how theory and computer science are able to define this group of magic squares. Only 16 of the 49 2x2 cell blocks that sum to 130 are accented by the different colored fonts in the 8x8 example and these squares were then scanned for squares with 20,15 in the proper cells for any of the 8 rotations. The 2015 squares all originated with principal reversible square number #31 and this square has values that sum to 35 on opposite sides of the vertical midline in the first two rows. The image below shows numbers completely surrounded by numbers with a blue background. There are 108 of these 2x2 subsquares that have the sum for the 4x4x4 most-perfect cube. All most-perfect magic squares are panmagic squares, apart from the trivial case of the first order square, most-perfect magic squares are all of order 4n. In their book, Kathleen Ollerenshaw and David S. Brée give a method of construction and enumeration of all magic squares. They also show there is a one-to-one correspondence between reversible squares and most-perfect magic squares. For n =36, there are about 2.7 ×1044 essentially different most-perfect magic squares. The second property implies that each pair of the integers with the same background colour in the 4×4 square below have the same sum
31.
Khajuraho Group of Monuments
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The Khajuraho Group of Monuments is a group of Hindu and Jain temples in Madhya Pradesh, India, about 175 kilometres southeast of Jhansi. They are one of the UNESCO World Heritage Sites in India, the temples are famous for their nagara-style architectural symbolism and their erotic sculptures. Most Khajuraho temples were built between 950 and 1050 by the Chandela dynasty, historical records note that the Khajuraho temple site had 85 temples by the 12th century, spread over 20 square kilometers. Of these, only about 25 temples have survived, spread over 6 square kilometers, of the various surviving temples, the Kandariya Mahadeva Temple is decorated with a profusion of sculptures with intricate details, symbolism and expressiveness of ancient Indian art. The Khajuraho monuments are located in the Indian state of Madhya Pradesh, in Chhatarpur district, the temples are near a small town also known as Khajuraho, with a population of about 20,000 people. Khajuraho is served by Civil Aerodrome Khajuraho, with services to Delhi, Agra, Varanasi, the site is also linked by the Indian Railways service, with the railway station located approximately six kilometres from the entrance to the monuments. The 10th century Bhand Deva Temple in Rajasthan was built in the style of the Khajuraho monuments and is referred to as Little Khajuraho. The Khajuraho group of monuments was built during the rule of the Rajput Chandela dynasty, the building activity started almost immediately after the rise of their power, throughout their kingdom to be later known as Bundelkhand. Most temples were built during the reigns of the Hindu kings Yashovarman, yashovarmans legacy is best exhibited by The Lakshmana Temple. Vishvanatha temple best highlights King Dhangas reign, the largest and currently most famous surviving temple is Kandariya Mahadeva built in the reign of King Ganda from 1017-1029 CE. The temple inscriptions suggest many of the surviving temples were complete between 970 and 1030 CE, with further temples completed during the following decades. The Khajuraho temples were built about 35 miles from the city of Mahoba. In ancient and medieval literature, their kingdom has been referred to as Jijhoti, Jejahoti, Chih-chi-to, Khajuraho was mentioned by Abu Rihan-al-Biruni, the Persian historian who accompanied Mahmud of Ghazni in his raid of Kalinjar in 1022 CE, he mentions Khajuraho as the capital of Jajahuti. The raid was unsuccessful, and an accord was reached when the Hindu king agreed to pay a ransom to Mahmud of Ghazni to end the attack. Khajuraho temples were in use through the end of 12th century. This changed in the 13th century, after the army of Delhi Sultanate, under the command of the Muslim Sultan Qutb-ud-din Aibak, and on account of extreme asceticism they are all yellow in colour. Many Moslems attend these men in order to take lessons from them, central Indian region, where Khajuraho temples are, remained in the control of many different Muslim dynasties from 13th century through the 18th century. In this period, some temples were desecrated, followed by a period when they were left in neglect
32.
Philippe de La Hire
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Philippe de La Hire was a French painter, mathematician, astronomer, and architect. According to Bernard le Bovier de Fontenelle he was an academy unto himself and he was born in Paris, the son of Laurent de La Hire, a distinguished artist and Marguerite Coquin. In 1660, he moved to Venice for four years to study painting, upon his return to Paris, he became a disciple of Girard Desargues from whom he learned geometrical perspective and was received as a master painter on August 4,1670. His paintings have sometimes confused with those of his son, Jean Nicolas de La Hire. He also began to study science and showed an aptitude for mathematics, from 1679–1682 he made several observations and measurements of the French coastline, and in 1683 aided in mapping France by extending the Paris meridian to the north. In 1683 La Hire assumed the chair of mathematics at the Collège Royal, from 1687 onwards he taught at the Académie d’architecture. La Hire wrote on methods,1673, on conic sections,1685, a treatise on epicycloids,1694, one on roulettes,1702. His works on conic sections and epicycloids were based on the teaching of Desargues and he also translated the essay of Manuel Moschopulus on magic squares, and collected many of the theorems on them which were previously known, this was published in 1705. He also published a set of tables in 1702. La Hires work also extended to descriptive zoology, the study of respiration, two of his sons were also notable for their scientific achievements, Gabriel-Philippe de La Hire, mathematician, and Jean-Nicolas de La Hire, botanist. Mons La Hire, a mountain on the Moon, is named for him, unless otherwise stated La Hires works are in French. In, Histoire de lAcadémie royale des sciences, p, La Hire, Philippe de la, vol. 2, pp. 662–664, in The Dictionary of Seventeenth-Century French Philosophers, oConnor, John J. Robertson, Edmund F. Philippe de La Hire, MacTutor History of Mathematics archive, University of St Andrews. Philippe de La Hire at the Catholic Encyclopedia This text incorporates public domain material from the Rouse History of Mathematics
33.
Athanasius Kircher
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Athanasius Kircher, S. J. was a German Jesuit scholar and polymath who published around 40 major works, most notably in the fields of comparative religion, geology, and medicine. Kircher has been compared to fellow Jesuit Roger Boscovich and to Leonardo da Vinci for his enormous range of interests and he taught for more than forty years at the Roman College, where he set up a wunderkammer. A resurgence of interest in Kircher has occurred within the community in recent decades. Kircher claimed to have deciphered the writing of the ancient Egyptian language. He did, however, correctly establish the link between the ancient Egyptian and the Coptic languages, and some regard him as the founder of Egyptology. Kirchers work in geology included studies of volcanoes and fossils, Kircher also displayed a keen interest in technology and mechanical inventions, inventions attributed to him include a magnetic clock, various automatons and the first megaphone. The invention of the lantern is often misattributed to Kircher. A scientific star in his day, towards the end of his life he was eclipsed by the rationalism of René Descartes, in the late 20th century, however, the aesthetic qualities of his work again began to be appreciated. One modern scholar, Alan Cutler, described Kircher as a giant among seventeenth-century scholars, another scholar, Edward W. Schmidt, referred to Kircher as the last Renaissance man. Kircher was born on 2 May in either 1601 or 1602 in Geisa, Buchonia, near Fulda, currently Hesse, from his birthplace he took the epithets Bucho, Buchonius and Fuldensis which he sometimes added to his name. He attended the Jesuit College in Fulda from 1614 to 1618, the youngest of nine children, Kircher studied volcanoes owing to his passion for rocks and eruptions. He was taught Hebrew by a rabbi in addition to his studies at school and he studied philosophy and theology at Paderborn, but fled to Cologne in 1622 to escape advancing Protestant forces. On the journey, he escaped death after falling through the ice crossing the frozen Rhine — one of several occasions on which his life was endangered. Later, traveling to Heiligenstadt, he was caught and nearly hanged by a party of Protestant soldiers, from 1622 to 1624 Kircher was sent to begin his regency period in Koblenz as a teacher. He was ordained to the priesthood in 1628 and became professor of ethics and mathematics at the University of Würzburg, beginning in 1628, he also began to show an interest in Egyptian hieroglyphs. In 1631, while still at Würzburg, Kircher allegedly had a vision of a bright light. This was the year that Kircher published his first book, in 1633 he was called to Vienna by the emperor to succeed Kepler as Mathematician to the Habsburg court. On the intervention of Nicolas-Claude Fabri de Peiresc, the order was rescinded and he was sent instead to Rome to continue with his scholarly work, on the way, his ship was blown off course and he arrived in Rome before he knew of the changed decision
34.
Alfonso X of Castile
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Alfonso X, called the Wise, was the King of Castile, León and Galicia from 30 May 1252 until his death. During the Imperial election of 1257, a dissident faction chose him to be King of the Romans on 1 April and he renounced his imperial claim in 1275, and in creating an alliance with England in 1254, his claim on Gascony as well. Alfonso X fostered the development of a court that encouraged learning. Jews, Muslims, and Christians had prominent roles in his court, Alfonso was a prolific author of Galician poetry, such as the Cantigas de Santa Maria, which are equally notable for their musical notation as for their literary merit. Alfonsos scientific interests—he is sometimes nicknamed the Astrologer —led him to sponsor the creation of the Alfonsine tables, as a legislator he introduced the first vernacular law code in Spain, the Siete Partidas. He created the Mesta, an association of farmers in the central plain. He fought a war with Portugal, but a less successful one with Granada. The end of his reign was marred by a war with his eldest surviving son, the future Sancho IV. Born in Toledo, Kingdom of Castile, Alfonso was the eldest son of Ferdinand III of Castile and his mother was the paternal cousin of Holy Roman Emperor Frederick II, to whom Alfonso is often compared. His maternal grandparents were Philip of Swabia and Irene Angelina, little is known about his upbringing, but he was most likely raised in Toledo. For the first nine years of his life Alfonso was only heir to Castile until his paternal grandfather king Alfonso IX of Leon died and his father united the kingdoms of Castile and Leon. He began his career as a soldier, under the command of his father, after the election of Theobald I as king of Navarre, his father tried to arrange a marriage for Alfonso with Theobalds daughter, Blanche of Navarre, but the move was unsuccessful. At the same time, he had a relationship with Mayor Guillén de Guzmán. In 1240, he married Mayor Guillén de Guzmán, but the marriage was later annulled, in the same period he conquered several Muslim strongholds in Al-Andalus alongside his father, such as Murcia, Alicante and Cadiz. In 1249, Alfonso married Violante of Aragon, the daughter of King James I of Aragon and Yolande of Hungary, Alfonso succeeded his father as King of Castile and León in 1252. The following year he invaded Portugal, capturing the region of the Algarve, King Afonso III of Portugal had to surrender, but he gained an agreement by which, after he consented to marry Alfonso Xs daughter Beatrice of Castile, the land would be returned to their heirs. In 1263 he returned Algarve to the King of Portugal and signed the Treaty of Badajoz, in 1254 Alfonso X signed a treaty of alliance with the King of England and Duke of Aquitaine, Henry III, supporting him in the war against Louis IX of France. In 1256, at the death of William II of Holland, Alfonsos descent from the Hohenstaufen through his mother, Alfonsos election as King of the Romans by the imperial prince-electors misled him into complicated schemes that involved excessive expense but never succeeded
35.
Paolo Dagomari di Prato
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Paolo Dagomari da Prato, known in Latin as Paulus Geometrus, was a noted Florentine mathematician and astronomer, such a maestro dellabbaco that he gained the epithet Paolo dellAbbaco. Franco Sacchetti called him Paolo Arismetra e Astrologo and Giorgio Vasari Paulo Strolago or Paolo Astrologo. He reputedly had 6, 000–10,000 pupils over the course of his life, being praised by contemporaries like Giovanni Gherardi da Prato, Filippo Villani, Paolo was born at Prato, the son of Piero Dagomari, who had moved to Florence. At Florence Paolo became the tutor of Jacopo Alighieri and a friend of Giovanni Boccaccio. The need for mathematics among the bankers and merchants of Florence led him to found a school of arithmetic at Santa Trinita, in 1363 he held the priorate of the quarter of S. Spirito from May–June. Paolo died in Florence and was buried in Santa Trinita under a now-lost epitaph and his portrait, in fresco, is painted on the vault of the Galleria degli Uffizi. In mathematics Paolo introduced the period or comma as a device for separating numbers into groups of three for easing calculations on the order of thousands and millions and he is most famous for his work on equations that fused geometry and arithmetic, which we today would recognise as algebra. His most important mathematical treatise was the Regoluzze, a manual of elementary arithmetic, some of the little rules are,1. If you wish to multiply numbers ending with a zero, multiply their figures, if you wish to multiply fraction by fraction, multiply the numerators with one another, and the denominators similarly. If you multiply the width of a circle by 22 and divide by 7, you will have the circumference
36.
Dante Alighieri
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Durante degli Alighieri, simply called Dante, was a major Italian poet of the Late Middle Ages. In the late Middle Ages, the majority of poetry was written in Latin. In De vulgari eloquentia, however, Dante defended use of the vernacular in literature, as a result, Dante played an instrumental role in establishing the national language of Italy. In addition, the first use of the interlocking three-line rhyme scheme, Dante has been called the Father of the Italian language and one of the greatest poets of world literature. In Italy, Dante is often referred to as il Sommo Poeta and il Poeta, he, Petrarch, Dante was born in Florence, Republic of Florence, present-day Italy. The exact date of his birth is unknown, although it is believed to be around 1265. This can be deduced from autobiographic allusions in the Divine Comedy, in 1265, the sun was in Gemini between approximately May 11 and June 11. Dante claimed that his family descended from the ancient Romans, but the earliest relative he could mention by name was Cacciaguida degli Elisei, born no earlier than about 1100. Dantes father, Alaghiero or Alighiero di Bellincione, was a White Guelph who suffered no reprisals after the Ghibellines won the Battle of Montaperti in the middle of the 13th century. Dantes family had loyalties to the Guelphs, an alliance that supported the Papacy and which was involved in complex opposition to the Ghibellines. The poets mother was Bella, likely a member of the Abati family and she died when Dante was not yet ten years old, and Alighiero soon married again, to Lapa di Chiarissimo Cialuffi. When Dante was 12, he was promised in marriage to Gemma di Manetto Donati, daughter of Manetto Donati, contracting marriages at this early age was quite common and involved a formal ceremony, including contracts signed before a notary. But by this time Dante had fallen in love with another, Beatrice Portinari, years after his marriage to Gemma he claims to have met Beatrice again, he wrote several sonnets to Beatrice but never mentioned Gemma in any of his poems. The exact date of his marriage is not known, the certain information is that, before his exile in 1301. Dante fought with the Guelph cavalry at the Battle of Campaldino and this victory brought about a reformation of the Florentine constitution. To take any part in life, one had to enroll in one of the citys many commercial or artisan guilds, so Dante entered the Physicians. In the following years, his name is recorded as speaking or voting in the various councils of the republic. A substantial portion of minutes from meetings in the years 1298–1300 was lost, however
37.
Luca Pacioli
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Fra Luca Bartolomeo de Pacioli was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and a seminal contributor to the field now known as accounting. He is referred to as The Father of Accounting and Bookkeeping in Europe and he was also called Luca di Borgo after his birthplace, Borgo Sansepolcro, Tuscany. Luca Pacioli was born between 1446 and 1448 in Sansepolcro where he received an abbaco education and this was education in the vernacular rather than Latin and focused on the knowledge required of merchants. His father was Bartolomeo Pacioli, however Luca Pacioli was said to have lived with the Befolci family as a child in his birth town Sansepolcro. He moved to Venice around 1464, where he continued his own education while working as a tutor to the three sons of a merchant and it was during this period that he wrote his first book, a treatise on arithmetic for the boys he was tutoring. Between 1472 and 1475, he became a Franciscan friar, in 1475, he started teaching in Perugia, first as a private teacher, from 1477 holding the first chair in mathematics. He wrote a textbook in the vernacular for his students. He continued to work as a tutor of mathematics and was, in fact. In 1494, his first book to be printed, Summa de arithmetica, proportioni et proportionalita, was published in Venice. In 1497, he accepted an invitation from Duke Ludovico Sforza to work in Milan, there he met, taught mathematics to, collaborated and lived with Leonardo da Vinci. In 1499, Pacioli and Leonardo were forced to flee Milan when Louis XII of France seized the city and their paths appear to have finally separated around 1506. Pacioli died at about the age of 70 in 1517, most likely in Sansepolcro where it is thought that he had spent much of his final years, the manuscript was written between December 1477 and 29 April 1478. It contains 16 sections on merchant arithmetic, such as barter, exchange, profit, mixing metals, one part of 25 pages is missing from the chapter on algebra. A modern transcription has been published by Calzoni and Cavazzoni along with a translation of the chapter on partitioning problems. Proportioni et proportionalita, a textbook for use in the schools of Northern Italy and it was a synthesis of the mathematical knowledge of his time and contained the first printed work on algebra written in the vernacular. It is also notable for including the first published description of the method that Venetian merchants used during the Italian Renaissance. The system he published included most of the cycle as we know it today. He described the use of journals and ledgers, and warned that a person should not go to sleep at night until the debits equaled the credits
38.
Ptolemy
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Claudius Ptolemy was a Greek writer, known as a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in the city of Alexandria in the Roman province of Egypt, wrote in Koine Greek, beyond that, few reliable details of his life are known. His birthplace has been given as Ptolemais Hermiou in the Thebaid in a statement by the 14th-century astronomer Theodore Meliteniotes. This is a very late attestation, however, and there is no reason to suppose that he ever lived elsewhere than Alexandria. Ptolemy wrote several treatises, three of which were of importance to later Byzantine, Islamic and European science. The first is the astronomical treatise now known as the Almagest, although it was entitled the Mathematical Treatise. The second is the Geography, which is a discussion of the geographic knowledge of the Greco-Roman world. The third is the treatise in which he attempted to adapt horoscopic astrology to the Aristotelian natural philosophy of his day. This is sometimes known as the Apotelesmatika but more known as the Tetrabiblos from the Greek meaning Four Books or by the Latin Quadripartitum. The name Claudius is a Roman nomen, the fact that Ptolemy bore it indicates he lived under the Roman rule of Egypt with the privileges and political rights of Roman citizenship. It would have suited custom if the first of Ptolemys family to become a citizen took the nomen from a Roman called Claudius who was responsible for granting citizenship, if, as was common, this was the emperor, citizenship would have been granted between AD41 and 68. The astronomer would also have had a praenomen, which remains unknown and it occurs once in Greek mythology, and is of Homeric form. All the kings after him, until Egypt became a Roman province in 30 BC, were also Ptolemies, abu Mashar recorded a belief that a different member of this royal line composed the book on astrology and attributed it to Ptolemy. The correct answer is not known”, Ptolemy wrote in Greek and can be shown to have utilized Babylonian astronomical data. He was a Roman citizen, but most scholars conclude that Ptolemy was ethnically Greek and he was often known in later Arabic sources as the Upper Egyptian, suggesting he may have had origins in southern Egypt. Later Arabic astronomers, geographers and physicists referred to him by his name in Arabic, Ptolemys Almagest is the only surviving comprehensive ancient treatise on astronomy. Ptolemy presented his models in convenient tables, which could be used to compute the future or past position of the planets. The Almagest also contains a catalogue, which is a version of a catalogue created by Hipparchus