1.
Bhinmal
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Bhinmal is a town in the Jalore District of Rajasthan, India. It is 72 kilometres south of Jalore town, Bhinmal was the capital of Gurjaradesa, comprising southern Rajasthan and northern Gujarat of modern times. The town is the birthplace of the Sanskrit poet Magha and mathematician-astronomer Brahmagupta, the original name of Bhinmal was Bhillamala, the plateau of Bhils. It was the capital of the kingdom of Gurjaradesa, a name derived from the Gurjara people. The kingdom is first attested in Banas Harshacharita and its king is said to have been subdued by Harshas father Prabhakaravardhana. The surrounding kingdoms were mentioned as Sindha, Lāta and Malava, indicating that the region included northern Gujarat and he distinguished it from the neighbouring kingdoms of Bharukaccha, Ujjayini, Malava, Valabhi and Surashtra. The Gurjara kingdom was said to have measured 833 miles in circuit and its ruler was a 20-year old kshatriya, the Arab chroniclers of Sindh, narrated the campaigns of Arab governors on Jurz, the Arabic term for Gurjara. They mentioned it jointly with Mermad and Al Baylaman, the country was first conquered by Mohammad bin Qasim and, for a second time, by Junayd. Upon bin Qasims victory, Al-Baladhuri mentioned that the Indian rulers, including that of Bhinmal, accepted Islam and they presumably recanted after bin Qasims departure, which made Junayds attack necessary. After Junayds reconquest, the kingdom at Bhinmal appears to have been annexed by the Arabs, a new dynasty was founded by Nagabhata I at Jalore, in the vicinity of Bhinmal, in about 730 CE, soon after Junayds end of term in Sindh. Nagabhata is said to have defeated the invincible Gurjaras, presumably those of Bhinmal, another account credits him for having defeated a Muslim ruler. Nagabhata is also known to have repelled the Arabs during a later raid and his dynasty later expanded to Ujjain and called itself Pratihara. Nagabhatas successor Vatsaraja lost Ujjain to the Rashtrakuta prince Dhruva, who claimed to have him into trackless desert. An inscription in Daulatpura from 843 AD mentions Vatsaraja having made grants near Didwana, in due course, the Pratiharas became the dominant force of the entire Rajasthan and Gujarat regions, establishing a powerful empire centered at Kannauj, the former capital of Harshavardhana. Ala ud din Khilji as the ruler of the Khilji dynasty also destroyed and looted Srimala when he conquered Jalore in 1310 AD. Prior to that, Srimala was a city of northwestern India. The city was out in the shape of a square. The mid-15th-century chronicle Kanhadade Prabandha provides descriptions of indiscriminate attacks by Muslims on Bhinmal, the city of Bhinmal had four gates
2.
Ujjain
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Ujjain is the largest city in Ujjain district of the Indian state of Madhya Pradesh. It is the fifth largest city in Madhya Pradesh by population and is the centre of Ujjain district. An ancient city situated on the bank of the Kshipra River. It emerged as the centre of central India around 600 BCE. It was the capital of the ancient Avanti kingdom, one of the sixteen mahajanapadas and it remained an important political, commercial and cultural centre of central India until the early 19th century, when the British administrators decided to develop Indore as an alternative to it. Ujjain continues to be an important place of pilgrimage for Shaivites, Vaishnavites, Ujjain has been selected as one of the hundred Indian cities to be developed as a smart city under PM Narendra Modis flagship Smart Cities Mission. Excavations at Kayatha have revealed chalcolithic agricultural settlements dating to around 2000 BCE, chalcolithic sites have also been discovered at other areas around Ujjain, including Nagda, but excavations at Ujjain itself have not revealed any chalcolithic settlements. H. D. Sankalia theorized that the settlements at Ujjain were probably destroyed by the Iron Age settlers. According to Hermann Kulke and Dietmar Rothermund, Avanti, whose capital was Ujjain, was one of the earliest outposts in central India, around 600 BCE, Ujjain emerged as the political, commercial and cultural centre of Malwa plateau. The ancient walled city of Ujjain was located around the Garh Kalika hill on the bank of river Kshipra and this city covered an irregular pentagonal area of 0.875 km2. It was surrounded by a 12 m high mud rampart, the archaeological investigations have also indicated the presence of a 45 m wide and 6.6 m deep moat around the city. According to F. R. Allchin and George Erdosy, these city defences were constructed between 6th and 4th centuries BCE, dieter Schlingloff believes that these were built before 600 BCE. This period is characterised by structures made of stone and burnt-brick, tools and weapons made of iron, according to the Puranic texts, a branch of the legendary Haihaya dynasty ruled over Ujjain. In the Mauryan period, Ujjain remained the centre of the region. From this period, Northern Black Polished Ware, copper coins, terracotta ring wells, during the reign of his father Bindusara, Ashoka served as the viceroy of Ujjain. Ujjain was subsequently controlled by a number of empires and dynasties, including the Shungas, the Western Satraps, the Satavahanas, the Guptas, the Paramaras shifted the regions capital from Ujjain to Dhar. Raja Bharthari wrote his epics, Virat Katha, Neeti Sataka. The writings of Bhasa are set in Ujjain, and he lived in the city
3.
Zero
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Gonzalo Barrios, known by his gamertag ZeRo, is a Chilean professional Super Smash Bros. player. He is considered one of the best Super Smash Bros. for Wii U players in the world, ZeRo had a record-breaking 56-tournament winning streak in 2015, in which he also won several high-profile tournaments like EVO2015 and The Big House 5. In the past has also been a top ranked Super Smash Bros and he mains Diddy Kong in Super Smash Bros. for Wii U, and mained Pit in Project M, Meta Knight in Brawl, and Fox in Melee. Barrios has had ZeRo as his gamertag since 2005 and he has been playing Smash since Super Smash Bros. in 1999. He started to travel and play in Melee tournaments in a local Akiba Game Store in early 2007, ZeRo quit Smash completely until December 2010 and then focused only on Brawl. ZeRo placed second in Brawl at Apex 2014, losing to Nairo and was the champion of the Smash Wii U at Apex 2015 and he defeated Dabuz, who was playing Captain Olimar, in the finals. ZeRo qualified for the MLG Anaheim 2014 championship bracket and finished 17th, ZeRo was ranked in 2014 by Melee it on Me as the 35th best Melee player in the world. On November 25,2014, he criticized Diddy Kongs repetitive playstyle in Smash Wii U, however, ZeRo later retracted this statement, and now says Diddy Kong is his favorite character to play with. Barrios streams on twitch. tv Mondays through Thursdays at 1PM PST, ZeRo attributes much of his success to training with Mew2King. He was sponsored by CLASH Tournaments for part of 2014 until resigning in November, e-Sports Earnings estimates that ZeRo has earned a career total of US$31,484.22 from tournaments. ZeRo was considered the third best Brawl player in the world by CLASH Tournaments in the 2014 SSBBRank, in early 2014, he picked up Melee again, as well as the mod Project M. Since November 2014 when he placed third at Skys Smash 4 Invitational, ZeRo won EVO2015, the largest Smash for Wii U. tournament at the time beating Mr. R in the finals. On the August 1,2015, Team SoloMid announced ZeRo as the player in their Super Smash Bros. division. At The Big House 5, ZeRo was knocked into Losers bracket very early and he qualified for the top 32. He led off by defeating Sonic main StaticManny, Sheik main top Melee Captain Falcon main Wizzrobe, Mario main Ally, in top 8, he beat Mario main 2Scoops Zenyou, Rosalina and Luma main Raquayza07, and beat Mario main ANTi in a very close set. In Losers finals, he handily beat Dabuz, who now plays Rosalina and Luma, after easily beating Nairo the first set of Grand Finals, ZeRo clutched out a win in the second set, 3-2, to win TBH5, despite suffering an early upset. In MLG World Finals 2015, ZeRo defeated Ness main Nakat, StaticManny, Ally, there, Nairo took two sets off of ZeRo, ending ZeRos reign at 56 tournaments. Kotaku named ZeRo, The Smash Bros, champ as one of the gamers of 2015, honoring his 56 tournament streak
4.
Number system
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A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1,2,3, a notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are used for labels, for ordering. In common usage, number may refer to a symbol, a word, calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, the same term may also refer to number theory, the study of the properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world, for example, in Western society the number 13 is regarded as unlucky, and a million may signify a lot. Though it is now regarded as pseudoscience, numerology, the belief in a significance of numbers, permeated ancient. Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of problems in number theory which are still of interest today. During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Numbers should be distinguished from numerals, the used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system, Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by digit 5 or by the Roman numeral Ⅴ, notations used to represent numbers are discussed in the article numeral systems. The Roman numerals require extra symbols for larger numbers, different types of numbers have many different uses. Numbers can be classified into sets, called number systems, such as the natural numbers, the same number can be written in many different ways. For different methods of expressing numbers with symbols, such as the Roman numerals, each of these number systems may be considered as a proper subset of the next one. This is expressed, symbolically, by writing N ⊂ Z ⊂ Q ⊂ R ⊂ C, the most familiar numbers are the natural numbers,1,2,3, and so on. Traditionally, the sequence of numbers started with 1 However, in the 19th century, set theorists. Today, different mathematicians use the term to both sets, including 0 or not
5.
Brahmagupta's theorem
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It is named after the Indian mathematician Brahmagupta. More specifically, let A, B, C and D be four points on a such that the lines AC. Denote the intersection of AC and BD by M. Drop the perpendicular from M to the line BC, let F be the intersection of the line EM and the edge AD. Then, the states that F is the midpoint AD. We need to prove that AF = FD and we will prove that both AF and FD are in fact equal to FM. To prove that AF = FM, first note that the angles FAM and CBM are equal, furthermore, the angles CBM and CME are both complementary to angle BCM, and are therefore equal. Finally, the angles CME and FMA are the same, hence, AFM is an isosceles triangle, and thus the sides AF and FM are equal. The proof that FD = FM goes similarly, the angles FDM, BCM, BME and DMF are all equal, so DFM is an isosceles triangle and it follows that AF = FD, as the theorem claims. Brahmaguptas formula for the area of a cyclic quadrilateral Brahmaguptas Theorem at cut-the-knot Weisstein, Eric W. Brahmaguptas theorem
6.
Brahmagupta's interpolation formula
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Brahmagupatas interpolation formula is a second-order polynomial interpolation formula developed by the Indian mathematician and astronomer Brahmagupta in the early 7th century CE. The Sanskrit couplet describing the formula can be found in the part of Khandakadyaka a work of Brahmagupta completed in 665 CE. The same couplet appears in Dhyana-graha-adhikara an earlier work of Brahmagupta, however internal evidences suggest that Dhyana-graha-adhikara could be dated earlier than Brahmasphuta-siddhanta a work of Brahmagupta composed in 628 CE. Hence the invention of the interpolation formula by Brahmagupta should be placed near the beginning of the second quarter of the 7th century CE. Brahmagupta was the first to invent and use a formula using second-order differences in the history of mathematics. Brahmagupas interpolation formula is equivalent to modern-day second-order Newton–Stirling interpolation formula, given a set of tabulated values of a function f in the table below, let it be required to compute the value of f, xr < a < xr+1. Assuming that the tabulated values of x are equally spaced with a common spacing of h. Writing D r = f r +1 − f r the following table can be formed, the linear interpolation formula to compute f is f = f r + t D r where t = a − x r h. For the computation of f, Brahmagupta replaces Dr with another expression which gives more accurate values, vikala is the amount in minutes by which the interval has been covered at the point where we want to interpolate. In the present notations it is a − xr, the new expression which replaces fr+1 − fr is called sphuta-bhogyakhanda. Add the result to half the sum of the gatakhanda and the bhogyakhanda if their half-sum is less than the bhogyakhanda, subtract if greater. This formula was originally stated for the computation of the values of the function for which the common interval in the underlying base table was 900 minutes or 15 degrees. So the reference to 900 is in fact a reference to the common interval h. Brahmaguptas method computation of shutabhogyakhanda can be formulated in modern notation as follows, the ± sign is to be taken according to whether 1/2 is less than or greater than Dr+1, or equivalently, according to whether Dr < Dr+1 or Dr > Dr+1. Brahmaguptas expression can be put in the form, sphuta-bhogyakhanda = D r + D r +12 + t D r +1 − D r 2. This is Stirlings interpolation formula truncated at the second-order differences and it is not known how Brahmagupta arrived at his interpolation formula. It is also interesting to note that Brahmagupta has given a formula for the case where the values of the independent variable are not equally spaced. Brahmaguptas identity Brahmagupta matrix Brahmagupta–Fibonacci identity
7.
Brahmagupta's formula
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In Euclidean geometry, Brahmaguptas formula finds the area of any cyclic quadrilateral given the lengths of the sides. Brahmaguptas formula gives the area K of a quadrilateral whose sides have lengths a, b, c, d as K = where s. This formula generalizes Herons formula for the area of a triangle, a triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle, if the semiperimeter is not used, Brahmaguptas formula is K =14. Another equivalent version is K =2 +8 a b c d −24 ⋅ Here the notations in the figure to the right are used. The area K of the cyclic quadrilateral equals the sum of the areas of △ADB and △BDC, but since ABCD is a cyclic quadrilateral, ∠DAB = 180° − ∠DCB. Therefore, K =12 p q sin A +12 r s sin A K2 =142 sin 2 A4 K2 =2 =2 −2 cos 2 A. Solving for common side DB, in △ADB and △BDC, the law of cosines gives p 2 + q 2 −2 p q cos A = r 2 + s 2 −2 r s cos C. Substituting cos C = −cos A and rearranging, we have 2 cos A = p 2 + q 2 − r 2 − s 2. Substituting this in the equation for the area,4 K2 =2 −14216 K2 =42 −2. The right-hand side is of the form a2 − b2 = and hence can be written as which, upon rearranging the terms in the square brackets, introducing the semiperimeter S = p + q + r + s/2,16 K2 =16. Taking the square root, we get K =, an alternative, non-trigonometric proof utilizes two applications of Herons triangle area formula on similar triangles. This more general formula is known as Bretschneiders formula and it is a property of cyclic quadrilaterals that opposite angles of a quadrilateral sum to 180°. It follows from the equation that the area of a cyclic quadrilateral is the maximum possible area for any quadrilateral with the given side lengths. A related formula, which was proved by Coolidge, also gives the area of a convex quadrilateral. It is K = −14 where p and q are the lengths of the diagonals of the quadrilateral, in a cyclic quadrilateral, pq = ac + bd according to Ptolemys theorem, and the formula of Coolidge reduces to Brahmaguptas formula. Herons formula for the area of a triangle is the case obtained by taking d =0. The relationship between the general and extended form of Brahmaguptas formula is similar to how the law of cosines extends the Pythagorean theorem, increasingly complicated closed-form formulas exist for the area of general polygons on circles, as described by Maley et al
8.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
9.
Astronomy
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Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, galaxies, and comets, while the phenomena include supernovae explosions, gamma ray bursts, more generally, all astronomical phenomena that originate outside Earths atmosphere are within the purview of astronomy. A related but distinct subject, physical cosmology, is concerned with the study of the Universe as a whole, Astronomy is the oldest of the natural sciences. The early civilizations in recorded history, such as the Babylonians, Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, during the 20th century, the field of professional astronomy split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the results and observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can play an active role, especially in the discovery. Amateur astronomers have made and contributed to many important astronomical discoveries, Astronomy means law of the stars. Astronomy should not be confused with astrology, the system which claims that human affairs are correlated with the positions of celestial objects. Although the two share a common origin, they are now entirely distinct. Generally, either the term astronomy or astrophysics may be used to refer to this subject, however, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Few fields, such as astrometry, are purely astronomy rather than also astrophysics, some titles of the leading scientific journals in this field includeThe Astronomical Journal, The Astrophysical Journal and Astronomy and Astrophysics. In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye, in some locations, early cultures assembled massive artifacts that possibly had some astronomical purpose. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye, most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon, the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the model of the Universe, or the Ptolemaic system. The Babylonians discovered that lunar eclipses recurred in a cycle known as a saros
10.
Indian mathematics
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Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Mahāvīra, Bhaskara II, Madhava of Sangamagrama, the decimal number system in worldwide use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, in addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China and this was followed by a second section consisting of a prose commentary that explained the problem in more detail and provided justification for the solution. In the prose section, the form was not considered so important as the ideas involved, all mathematical works were orally transmitted until approximately 500 BCE, thereafter, they were transmitted both orally and in manuscript form. A later landmark in Indian mathematics was the development of the series expansions for functions by mathematicians of the Kerala school in the 15th century CE. Their remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series. However, they did not formulate a theory of differentiation and integration. Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilisation have uncovered evidence of the use of practical mathematics. The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4,2,1, considered favourable for the stability of a brick structure. They used a system of weights based on the ratios, 1/20, 1/10, 1/5, 1/2,1,2,5,10,20,50,100,200. They mass-produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, the inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length was divided into ten equal parts, bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length. The religious texts of the Vedic Period provide evidence for the use of large numbers, by the time of the Yajurvedasaṃhitā-, numbers as high as 1012 were being included in the texts. The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta, With three-fourths Puruṣa went up, the Satapatha Brahmana contains rules for ritual geometric constructions that are similar to the Sulba Sutras. The Śulba Sūtras list rules for the construction of fire altars. Most mathematical problems considered in the Śulba Sūtras spring from a single theological requirement, according to, the Śulba Sūtras contain the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. The diagonal rope of an oblong produces both which the flank and the horizontal <ropes> produce separately and they contain lists of Pythagorean triples, which are particular cases of Diophantine equations
11.
Indian astronomy
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Indian astronomy has a long history stretching from pre-historic to modern times. Some of the earliest roots of Indian astronomy can be dated to the period of Indus Valley Civilization or earlier, astronomy later developed as a discipline of Vedanga or one of the auxiliary disciplines associated with the study of the Vedas, dating 1500 BCE or older. The oldest known text is the Vedanga Jyotisha, dated to 1400–1200 BCE, as with other traditions, the original application of astronomy was thus religious. Indian astronomy flowered in the 5th-6th century, with Aryabhata, whose Aryabhatiya represented the pinnacle of astronomical knowledge at the time, Later the Indian astronomy significantly influenced Muslim astronomy, Chinese astronomy, European astronomy, and others. Other astronomers of the era who further elaborated on Aryabhatas work include Brahmagupta, Varahamihira. Some of the earliest forms of astronomy can be dated to the period of Indus Valley Civilization or earlier, some cosmological concepts are present in the Vedas, as are notions of the movement of heavenly bodies and the course of the year. Thus, the Shulba Sutras, texts dedicated to altar construction, discusses advanced mathematics, Vedanga Jyotisha is another of the earliest known Indian texts on astronomy, it includes the details about the sun, moon, nakshatras, lunisolar calendar. Greek astronomical ideas began to enter India in the 4th century BCE following the conquests of Alexander the Great, by the early centuries of the Common Era, Indo-Greek influence on the astronomical tradition is visible, with texts such as the Yavanajataka and Romaka Siddhanta. Later astronomers mention the existence of various siddhantas during this period and these were not fixed texts but rather an oral tradition of knowledge, and their content is not extant. The text today known as Surya Siddhanta dates to the Gupta period and was received by Aryabhata, the classical era of Indian astronomy begins in the late Gupta era, in the 5th to 6th centuries. The Pañcasiddhāntikā by Varāhamihira approximates the method for determination of the direction from any three positions of the shadow using a gnomon. By the time of Aryabhata the motion of planets was treated to be rather than circular. The divisions of the year were on the basis of religious rites, in the Vedānga Jyotiṣa, the year begins with the winter solstice. Hindu calendars have several eras, The Hindu calendar, counting from the start of the Kali Yuga, has its epoch on 18 February 3102 BCE Julian, the Vikrama Samvat calendar, introduced about the 12th century, counts from 56–57 BCE. The Saka Era, used in some Hindu calendars and in the Indian national calendar, has its epoch near the equinox of year 78. The Saptarshi calendar traditionally has its epoch at 3076 BCE and this device finds mention in the works of Varāhamihira, Āryabhata, Bhāskara, Brahmagupta, among others. The Cross-staff, known as Yasti-yantra, was used by the time of Bhaskara II and this device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale. The clepsydra was used in India for astronomical purposes until recent times, Ōhashi notes that, Several astronomers also described water-driven instruments such as the model of fighting sheep
12.
Sanskrit
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Sanskrit is the primary liturgical language of Hinduism, a philosophical language of Hinduism, Buddhism, and Jainism, and a literary language and lingua franca of ancient and medieval South Asia. As a result of transmission of Hindu and Buddhist culture to Southeast Asia and parts of Central Asia, as one of the oldest Indo-European languages for which substantial written documentation exists, Sanskrit holds a prominent position in Indo-European studies. The body of Sanskrit literature encompasses a rich tradition of poetry and drama as well as scientific, technical, philosophical, the compositions of Sanskrit were orally transmitted for much of its early history by methods of memorization of exceptional complexity, rigor, and fidelity. Thereafter, variants and derivatives of the Brahmi script came to be used, Sanskrit is today one of the 22 languages listed in the Eighth Schedule of the Constitution of India, which mandates the Indian government to develop the language. It continues to be used as a ceremonial language in Hindu religious rituals and Buddhist practice in the form of hymns. The Sanskrit verbal adjective sáṃskṛta- may be translated as refined, elaborated, as a term for refined or elaborated speech, the adjective appears only in Epic and Classical Sanskrit in the Manusmṛti and the Mahabharata. The pre-Classical form of Sanskrit is known as Vedic Sanskrit, with the language of the Rigveda being the oldest and most archaic stage preserved, Classical Sanskrit is the standard register as laid out in the grammar of Pāṇini, around the fourth century BCE. Sanskrit, as defined by Pāṇini, evolved out of the earlier Vedic form, the present form of Vedic Sanskrit can be traced back to as early as the second millennium BCE. Scholars often distinguish Vedic Sanskrit and Classical or Pāṇinian Sanskrit as separate dialects, although they are quite similar, they differ in a number of essential points of phonology, vocabulary, grammar and syntax. Vedic Sanskrit is the language of the Vedas, a collection of hymns, incantations and theological and religio-philosophical discussions in the Brahmanas. Modern linguists consider the metrical hymns of the Rigveda Samhita to be the earliest, for nearly 2000 years, Sanskrit was the language of a cultural order that exerted influence across South Asia, Inner Asia, Southeast Asia, and to a certain extent East Asia. A significant form of post-Vedic Sanskrit is found in the Sanskrit of Indian epic poetry—the Ramayana, the deviations from Pāṇini in the epics are generally considered to be on account of interference from Prakrits, or innovations, and not because they are pre-Paninian. Traditional Sanskrit scholars call such deviations ārṣa, meaning of the ṛṣis, in some contexts, there are also more prakritisms than in Classical Sanskrit proper. There were four principal dialects of classical Sanskrit, paścimottarī, madhyadeśī, pūrvi, the predecessors of the first three dialects are attested in Vedic Brāhmaṇas, of which the first one was regarded as the purest. In the 2001 Census of India,14,035 Indians reported Sanskrit to be their first language, in India, Sanskrit is among the 14 original languages of the Eighth Schedule to the Constitution. The state of Uttarakhand in India has ruled Sanskrit as its official language. In October 2012 social activist Hemant Goswami filed a petition in the Punjab. More than 3,000 Sanskrit works have been composed since Indias independence in 1947, much of this work has been judged of high quality, in comparison to both classical Sanskrit literature and modern literature in other Indian languages
13.
Shaivite
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Shaivism is one of the major traditions within Hinduism that reveres Shiva as the Supreme Being or its metaphysical concept of Brahman. The followers of Shaivism are called Shaivas or Saivas, like much of Hinduism, the Shaiva have many sub-traditions, ranging from devotional dualistic theism such as Shaiva Siddhanta to yoga-oriented monistic non-theism such as Kashmiri Shaivism. It considers both the Vedas and the Agama texts as important sources of theology, Shaivism has ancient roots, traceable in the Vedic literature of 2nd millennium BCE, but this is in the form of the Vedic deity Rudra. In the early centuries of the era is the first clear evidence of Pāśupata Shaivism. Both devotional and monistic Shaivism became popular in the 1st millennium CE and it arrived in Southeast Asia shortly thereafter, leading to thousands of Shaiva temples on the islands of Indonesia as well as Cambodia and Vietnam, co-evolving with Buddhism in these regions. In the contemporary era, Shaivism is one of the aspects of Hinduism. Shaivism theology ranges from Shiva being the creator, preserver, destroyer to being the same as the Atman within oneself and it is closely related to Shaktism, and some Shaiva worship in Shiva and Shakti temples. It is the Hindu tradition that most accepts ascetic life and emphasizes yoga, Shaivism is one of the largest traditions within Hinduism. Shiva literally means kind, friendly, gracious, or auspicious, as a proper name, it means The Auspicious One. The word Shiva is used as an adjective in the Rig Veda, as an epithet for several Rigvedic deities, the term Shiva also connotes liberation, final emancipation and the auspicious one, this adjective sense of usage is addressed to many deities in Vedic layers of literature. The term evolved from the Vedic Rudra-Shiva to the noun Shiva in the Epics, the Sanskrit word śaiva or Shaiva means relating to the god Shiva, while the related beliefs, practices, history, literature and sub-traditions constitute Shaivism. The reverence for Shiva is one of the traditions, found widely across India, Sri Lanka. While Shiva is revered broadly, Hinduism itself is a complex religion, Shaivism is a major tradition within Hinduism, with a theology that is predominantly related to the Hindu god Shiva. Shaivism has many different sub-traditions with regional variations and differences in philosophy, Shaivism has a vast literature with different philosophical schools, ranging from nondualism, dualism, and mixed schools. The origins of Shaivism are unclear and a matter of debate among scholars, some trace the origins to the Indus Valley civilization, which reached its peak around 2500–2000 BCE. Archeological discoveries show seals that suggest a deity that appears like Shiva. Of these is the Pashupati seal, which scholars interpreted as someone seated in a meditating yoga pose surrounded by animals. This Pashupati seal has been interpreted by scholars as a prototype of Shiva
14.
Mount Abu
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Mount Abu is a popular hill station in the Aravalli Range in Sirohi district of Rajasthan state in western India near the border with Gujarat. The mountain forms a rocky plateau 22 km long by 9 km wide. The highest peak on the mountain is Guru Shikhar at 1,722 m above sea level and it is referred to as an oasis in the desert as its heights are home to rivers, lakes, waterfalls and evergreen forests. The nearest train station is Abu Road railway station,27 km away, the ancient name of Mount Abu is Arbudaanchal. In the Puranas, the region has been referred to as Arbudaranya and it is believed that sage Vashistha retired to the southern spur at Mount Abu following his differences with sage Vishvamitra. There is another mythology according to which a serpent named Arbuda saved the life of Nandi, the incident happened on the mountain that is currently known as Mount Abu and so the mountain is named Arbudaranya after that incident which gradually became Abu. The conquest of Mount Abu in 1311 CE by Rao Lumba of Deora-Chauhan dynasty brought to an end the reign of the Parmars and he shifted the capital city to Chandravati in the plains. After the destruction of Chandravati in 1405, Rao Shasmal made Sirohi his headquarters, later it was leased by the British government from the then Maharaja of Sirohi for use as the headquarters. The Arbuda Mountains region is said to be abode of the famous Gurjars. The association of the Gurjars with the mountain is noticed in many inscriptions and these Gurjars migrated from the Arbuda mountain region. As early as sixth century CE, they set up one or more principalities in Rajasthan, almost all or a larger part of Rajasthan and Gujarat had been known as Gurjaratra or Gurjarabhumi for centuries before the Mughal period. According to a legend, sage Vasishta performed a yajna at the peak of Mount Abu. In answer to his prayer, a youth arose from the Agnikunda — the first Agnivansha Rajput, Mount Abu town, the only hill station in Rajasthan, is at an elevation of 1,220 m. It has been a retreat from the heat of Rajasthan. The Mount Abu Wildlife Sanctuary was established in 1960 and covers 290 km² of the mountain, the oldest of these is the Vimal Vasahi temple, built in 1021 AD by Vimal Shah and dedicated to the first of the Jain Tirthankaras. They include the Achaleswar Mahadev Temple and the Kantinath Temple and it is the location of the headquarters of the Brahma Kumaris. The Achalgarh Fort, built in the 14th century by Rana Kumbha of Mewar, is nearby, the Toad Rock is on a hill near the lake. Close to the fort is the Achaleshwar Mahadev Temple, a popular Shiva temple, the Durga Ambika Mata Temple lies in a cleft of rock in Jagat, just outside Mount Abu town
15.
Xuanzang
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602–664, was a Chinese Buddhist monk, scholar, traveller, and translator who described the interaction between China and India in the early Tang dynasty. Born in what is now Henan province around 602, from boyhood he took to reading books, including the Chinese classics. While residing in the city of Luoyang, Xuanzang was ordained as a śrāmaṇera at the age of thirteen. Due to the political and social unrest caused by the fall of the Sui dynasty, he went to Chengdu in Sichuan and he later travelled throughout China in search of sacred books of Buddhism. At length, he came to Changan, then under the rule of Emperor Taizong of Tang. He knew about Faxians visit to India and, like him, was concerned about the incomplete, hsüan, Hüan, Huan and Chuang are also found. Another form of his style was Yuanzang, written 元奘. Tang Monk is also transliterated /Thang Seng/, another of Xuanzangs standard aliases is Sanzang Fashi, 法 being a Chinese translation for Sanskrit Dharma or Pali/Pakrit Dhamma, the implied meaning being Buddhism. Sanzang is the Chinese term for the Buddhist canon, or Tripiṭaka, in some sources Xuanzangs is said to have been born Chen Hui or Chen Yig. Xuanzang was born Chen Hui around 602 in Chenhe Village, Goushi Town, Luozhou and his family was noted for its erudition for generations, and Xuanzang was the youngest of four children. His ancestor was Chen Shi, a minister of the Eastern Han dynasty and his great-grandfather Chen Qin served as the prefect of Shangdang during the Eastern Wei, his grandfather Chen Kang was a professor in the Taixue during the Northern Qi. According to traditional biographies, Xuanzang displayed a superb intelligence and earnestness, although his household was essentially Confucian, at a young age, Xuanzang expressed interest in becoming a Buddhist monk like one of his elder brothers. After the death of his father in 611, he lived with his older brother Chén Sù for five years at Jingtu Monastery in Luoyang, during this time he studied Mahayana as well as various early Buddhist schools, preferring the former. In 618, the Sui Dynasty collapsed and Xuanzang and his brother fled to Changan, which had proclaimed as the capital of the Tang dynasty. Here the two spent two or three years in further study in the monastery of Kong Hui, including the Abhidharma-kośa Śāstra. When Xuanzang requested to take Buddhist orders at the age of thirteen, Xuanzang was fully ordained as a monk in 622, at the age of twenty. The myriad contradictions and discrepancies in the texts at that time prompted Xuanzang to decide to go to India and he subsequently left his brother and returned to Changan to study foreign languages and to continue his study of Buddhism. He began his mastery of Sanskrit in 626, and probably also studied Tocharian, during this time, Xuanzang also became interested in the metaphysical Yogacara school of Buddhism
16.
Gurjaratra
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Gurjaradesa or Gurjaratra is a historical region in India comprising the eastern Rajasthan and northern Gujarat during the period of 6th -12th century CE. Its name is believed to derive from the dominance of the Gurjara tribes in this region, the predominant power of the region, the Gurjara-Pratiharas eventually controlled a major part of North India centered at Kannauj. The modern state of Gujarat derives its name from the ancient Gurjaratra, gurjaradēśa, or Gurjara country, is first attested in Banas Harshacharita. Its king is said to have been subdued by Harshas father Prabhakaravardhana, the bracketing of the country with Sindha, Lāta and Malava indicates that the region including the northern Gujarat and Rajasthan is meant. He distinguished it from the kingdoms of Bharukaccha, Ujjayini, Malava, Valabhi. The Gurjara kingdom was said to have measured 833 miles in circuit and its ruler was a 20-year old kshatriya, who was distinguished for his wisdom and courage. It is known that, in 628 CE, the kingdom at Bhinmal was ruled by a Chapa dynasty ruler Vyāgrahamukha and it is believed that the young ruler mentioned by Hieun Tsang must have been his immediate successor. It appears that the Gurjara country at that time comprised modern Rajasthan, following the death of Harsha, his empire split up into small kingdoms. Gurjaradesa is believed to have become independent, the Arab chroniclers of Sindh, narrated the campaigns of Arab governors on Jurz, the Arabic term for Gurjara. They mentioned it jointly with Mermad and Al Baylaman, the country was first conquered by Mohammad bin Qasim and, for a second time, by Junayd. Upon bin Qasims victory, Al-Baladhuri mentioned that the Indian rulers, including that of Bhinmal, accepted Islam and they presumably recanted after bin Qasims departure, which made Junayds attack necessary. After Junayds reconquest, the kingdom at Bhinmal appears to have been annexed by the Arabs, a Gurjara kingdom was founded by Harichandra Rohilladhi at Mandore in about 600 CE. This is expected to have been a small kingdom and his descendant, Nagabhata, shifted the capital to Merta in about 680 CE. Eventually, this dynasty adopted the designation of Pratihara in line with the Imperial Pratiharas and they are often referred to as Mandore Pratiharas by historians. The Broach line of Gurjaras was founded by Dadda I, who is identified with Harichandras youngest son of the name by many historians. These Gurjaras were always recognized as though their allegiance might have varied over time. They are believed to have wrested a fair portion of the Lata province of the Chaulukyas, a final line of Gurjaras was founded by Nagabhata I at Jalore, in the vicinity of Bhinmal, in about 730 CE, soon after Junayds end of term in Sindh. Nagabhata is said to have defeated the invincible Gurjaras, presumably those of Bhinmal, another account credits him for having defeated a Muslim ruler
17.
Rajasthan
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Rajasthan is Indias largest state by area. Elsewhere it is bordered by the other Indian states, Punjab to the north, Haryana and Uttar Pradesh to the northeast, Madhya Pradesh to the southeast, and Gujarat to the southwest. Rajasthan is also home to two national reserves, the Ranthambore National Park in Sawai Madhopur and Sariska Tiger Reserve in Alwar. The state was formed on 30 March 1949 when Rajputana – the name adopted by the British Raj for its dependencies in the region – was merged into the Dominion of India. Its capital and largest city is Jaipur, also known as Pink City, other important cities are Jodhpur, Udaipur, Bikaner, Kota and Ajmer. Parts of what is now Rajasthan were partly part of the Vedic Civilisation, kalibangan, in Hanumangarh district, was a major provincial capital of the Indus Valley Civilization. Matsya Kingdom of the Vedic civilisation of India, is said to roughly corresponded to the state of Jaipur in Rajasthan. The capital of Matsya was at Viratanagar, which is said to have named after its founder king Virata. Bhargava identifies the two districts of Jhunjhunu and Sikar and parts of Jaipur district along with Haryana districts of Mahendragarh, bhargava also locates the present day Sahibi River as the Vedic Drishadwati River, which along with Saraswati River formed the borders of the Vedic state of Brahmavarta. Manu and Bhrigu narrated the Manusmriti to a congregation of seers in this area only, the Indo-Scythians invaded the area of Ujjain and established the Saka era, marking the beginning of the long-lived Saka Western Satraps state. Gurjars ruled for many dynasties in this part of the country, up to the tenth century almost the whole of North India, acknowledged the supremacy of the Gurjars with their seat of power at Kannauj. The Gurjar Pratihar Empire acted as a barrier for Arab invaders from the 8th to the 11th century, the chief accomplishment of the Gurjara Pratihara empire lies in its successful resistance to foreign invasions from the west, starting in the days of Junaid. Majumdar says that this was acknowledged by the Arab writers. He further notes that historians of India have wondered at the progress of Muslim invaders in India. Traditionally the Rajputs, Jats, Meenas, REBARI, Gurjars, Bhils, Rajpurohit, Charans, Yadavs, Bishnois, Sermals, PhulMali, all these tribes suffered great difficulties in protecting their culture and the land. Millions of them were killed trying to protect their land, a number of Gurjars had been exterminated in Bhinmal and Ajmer areas fighting with the invaders. Meenas were rulers of Bundi, Hadoti and the Dhundhar region, hem Chandra Vikramaditya, the Hindu Emperor, was born in the village of Machheri in Alwar District in 1501. Hem Chandra was killed in the battlefield at Second Battle of Panipat fighting against Mughals on 5 November 1556, maharana Pratap of Mewar resisted Akbar in the famous Battle of Haldighati and later operated from hilly areas of his kingdom
18.
Gujarat
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Gujarat is a state in Western India, sometimes referred to as the Jewel of Western India. It has an area of 196,024 km2 with a coastline of 1,600 km, most of which lies on the Kathiawar peninsula, and a population in excess of 60 million. The state is bordered by Rajasthan to the north, Maharashtra to the south, Madhya Pradesh to the east, and the Arabian Sea and its capital city is Gandhinagar, while its largest city is Ahmedabad. Gujarat is home to the Gujarati-speaking people of India, the state encompasses some sites of the ancient Indus Valley Civilization, such as Lothal and Dholavira. Lothal is believed to be one of the worlds first seaports, Gujarat was known to the ancient Greeks, and was familiar in other Western centres of civilisation through the end of the European Middle Ages. Modern-day Gujarat is derived from Sanskrit term Gurjaradesa, the Gurjar nation, parts of modern Rajasthan and Gujarat have been known as Gurjaratra or Gurjarabhumi for centuries before the Mughal period. Gujarat was one of the centres of the Indus Valley Civilization. It contains ancient metropolitan cities from the Indus Valley such as Lothal, Dholavira, the ancient city of Lothal was where Indias first port was established. The ancient city of Dholavira is one of the largest and most prominent archaeological sites in India, the most recent discovery was Gola Dhoro. Altogether, about 50 Indus Valley settlement ruins have been discovered in Gujarat, the ancient history of Gujarat was enriched by the commercial activities of its inhabitants. There is clear evidence of trade and commerce ties with Egypt, Bahrain. The early history of Gujarat reflects the grandeur of Chandragupta Maurya who conquered a number of earlier states in what is now Gujarat. Pushyagupta, a Vaishya, was appointed governor of Saurashtra by the Mauryan regime and he ruled Giringer and built a dam on the Sudarshan lake. Between the decline of Mauryan power and Saurashtra coming under the sway of the Samprati Mauryas of Ujjain, in the first half of the 1st century AD there is the story of a merchant of King Gondaphares landing in Gujarat with Apostle Thomas. The incident of the cup-bearer killed by a lion might indicate that the city described is in Gujarat. For nearly 300 years from the start of the 1st century AD, the weather-beaten rock at Junagadh gives a glimpse of the ruler Rudradaman I of the Saka satraps known as Western Satraps, or Kshatraps. Mahakshatrap Rudradaman I founded the Kardamaka dynasty which ruled from Anupa on the banks of the Narmada up to the Aparanta region which bordered Punjab, in Gujarat several battles were fought between the south Indian Satavahana dynasty and the Western Satraps. The greatest ruler of the Satavahana Dynasty was Gautamiputra Satakarni who defeated the Western Satraps, the Kshatrapa dynasty was replaced by the Gupta Empire with the conquest of Gujarat by Chandragupta Vikramaditya
19.
Aryabhata I
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Aryabhata or Aryabhata I was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the Āryabhaṭīya and the Arya-siddhanta, furthermore, in most instances Aryabhatta would not fit the metre either. Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga and this corresponds to 499 CE, and implies that he was born in 476. Aryabhata called himself a native of Kusumapura or Pataliputra, Bhāskara I describes Aryabhata as āśmakīya, one belonging to the Aśmaka country. During the Buddhas time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers in central India. It has been claimed that the aśmaka where Aryabhata originated may be the present day Kodungallur which was the capital city of Thiruvanchikkulam of ancient Kerala. This is based on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr, however, K. Chandra Hari has argued for the Kerala hypothesis on the basis of astronomical evidence. Aryabhata mentions Lanka on several occasions in the Aryabhatiya, but his Lanka is an abstraction and it is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time. Both Hindu and Buddhist tradition, as well as Bhāskara I, identify Kusumapura as Pāṭaliputra, Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar. Aryabhata is the author of treatises on mathematics and astronomy. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and it also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, a third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of work is not known. Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, direct details of Aryabhatas work are known only from the Aryabhatiya. The name Aryabhatiya is due to later commentators, Aryabhata himself may not have given it a name. His disciple Bhaskara I calls it Ashmakatantra and it is also occasionally referred to as Arya-shatas-aShTa, because there are 108 verses in the text. It is written in the terse style typical of sutra literature
20.
George Sarton
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George Alfred Leon Sarton, a Belgian-American chemist and historian, is considered the founder of the discipline of history of science. He has a significant importance in the history of science and his most influential work was the Introduction to the History of Science, which consists of three volumes and 4,296 pages. Sarton ultimately aimed to achieve an integrated philosophy of science provided a connection between the sciences and the humanities, which he referred to as the new humanism. George Alfred Leon Sarton was born in Ghent, Belgium on August 31,1884 and his parents were Alfred Sarton and Léonie Van Halmé, his mother died when he was less than a year old. He graduated from the University of Ghent in 1906 and two years won a gold medal for one of his papers on chemistry. He received his PhD in mathematics at the University of Ghent in 1911 and he emigrated to the United States from Belgium due to First World War, and worked there the rest of his life, researching and writing about the history of science. In 1911, he married Mabel Eleanor Elwes, an English artist and their daughter Eleanore Marie was born the following year in 1912. Although he and his emigrated to England after World War I broke out, they immigrated to the United States in 1915. He worked for the Carnegie Foundation for International Peace and lectured at Harvard University, at Harvard, he became a lecturer in 1920, and a professor of the history of science from 1940 until his retirement in 1951. He was also an associate of the Carnegie Institution of Washington from 1919 until 1948. By the time of his death, he had completed only the first three volumes, Sarton had been inspired for his project by his study of Leonardo da Vinci, but he had not reached this period in history before dying. After his death, a selection of his papers was edited by Dorothy Stimson. It was published by Harvard University Press in 1962, in honor of Sartons achievements, the History of Science Society created the award known as the George Sarton Medal. It is the most prestigious award of the History of Science Society and it has been awarded annually since 1955 to an outstanding historian of science selected from the international scholarly community. The medal honors a scholar for lifetime scholarly achievement, Sarton was the founder of this society and of its journals, Isis and Osiris, which publish articles on science and culture. Sarton, George, Introduction to the History of Science, Carnegie Institution of Washington Publication no.376, baltimore, Williams and Wilkins, Co. George Sarton, The Incubation of Western Culture in the Middle East, a George C. Keiser Foundation Lecture, March 29,1950, Washington, D. C, Introduction to the History of Science. Ancient science through the Golden Age of Greece, Cambridge, Mass, hellenistic science and culture in the last three centuries B. C
21.
Sindh
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Sindh /sɪnd/ is one of the four provinces of Pakistan, in the southeast of the country. Historically home to the Sindhi people, it is locally known as the Mehran. It was formerly known as Sind until 1956, Sindh is the third largest province of Pakistan by area, and second largest province by population after Punjab. Sindh is bordered by Balochistan province to the west, and Punjab province to the north, Sindh also borders the Indian states of Gujarat and Rajasthan to the east, and Arabian Sea to the south. Sindhs climate is noted for hot summers and mild winters, the provincial capital of Sindh is Pakistans largest city and financial hub, Karachi. Sindh has Pakistans second largest economy with Karachi being its capital hosts the headquarters of several multinational banks. Sindh is home to a portion of Pakistans industrial sector. The remainder of Sindh has an agriculture based economy, and produces fruit, food consumer items, Sindh is also the centre of Pakistans pharmaceutical industry. Sindh is known for its culture which is strongly influenced by Sufism. Several important Sufi shrines are located throughout the province which attract millions of annual devotees, Sindh also has Pakistans highest percentage of Hindu residents. Karachi and other centres of Sindh have seen ethnic tensions between the native Sindhis and the Muhajirs boil over into violence on several occasions. Sindh is home to two UNESCO World Heritage Sites - the Historical Monuments at Makli, and the Archaeological Ruins at Moenjodaro, the word Sindh is derived from the Sanskrit language and is adapted from the Sanskrit term Sindhu which literally means river hence a reference to Indus River. Spelling of its name as Sind was discontinued in 1988 by an amendment passed in Sindh Assembly. The Greeks who conquered Sindh in 325 BC under the command of Alexander the Great rendered it as Indós, the ancient Iranians referred to everything east of the river Indus as hind from the word Sindh. When the British arrived in the 17th century in India, then ruled by the Maratha Empire, they applied the Greek version of the name Sindh to all of South Asia, calling it India. The name of Pakistan is actually an acronym in which the letter s is derived from the first letter in Sindh, Sindhs first known village settlements date as far back as 7000 BCE. Permanent settlements at Mehrgarh, currently in Balochistan, to the west expanded into Sindh and this culture blossomed over several millennia and gave rise to the Indus Valley Civilization around 3000 BCE. The primitive village communities in Balochistan were still struggling against a difficult highland environment and this was one of the most developed urban civilizations of the ancient world
22.
Al-Mansur
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Al-Mansur or Abu Jafar Abdallah ibn Muhammad al-Mansur was the second Abbasid Caliph reigning from 136 AH to 158 AH and succeeding his brother Abu al-Abbas al-Saffah. Al-Mansur is generally regarded as the founder of the Abbasid Caliphate, one of the largest polities in world history. He is also known for founding the city of Madinat al-Salam which was to become the core of imperial Baghdad. Al-Mansur was born at the home of the Abbasid family after their emigration from the Hejaz in 95 AH and he reigned from Dhu al-Hijjah 136 AH until Dhu al-Hijjah 158 AH. In 762 he founded as new residence and palace city Madinat as-Salam. Al-Mansur was concerned with the solidity of his regime after the death of his brother Abul Abbas, in 754 he defeated Abdallah ibn Alis bid for the Caliphate, and in 755 he arranged the assassination of Abu Muslim. When Isa ibn Musa, al-Mansurs intended successor, fell under suspicion of corruption, al-Mahdi was appointed in his stead, like his elder brother Saffah he wanted to unite the land, so he got rid of all of his opposition. It was under al-Mansur that a committee, mostly made up of Syriac-speaking Christians, was set up in Baghdad with the purpose of translating extant Greek works into Arabic. Due to the Abbasids orientation toward the East, many Persians came to play a role in the Empire. This was in contrast to the preceding Umayyad era, in which non-Arabs were kept out of these affairs. In 756, al-Mansur sent over 4,000 Arab mercenaries to assist the Chinese in the An Shi Rebellion against An Lushan, after the war, al-Mansur was referred to as A-pu-cha-fo in the Chinese Tang Annals. Al-Mansur died in 775 on his way to Mecca to make hajj and he was buried somewhere along the way in one of hundreds of graves dug in order to hide his body from the Umayyads. He was succeeded by his son, al-Mahdi, according to a number of sources, Abu Hanifa an-Numan was imprisoned by al-Mansur. There is also an account of foreboding verses al-Mansur saw written on the wall just before his death, when al-Mansur died, the caliphates treasury contained 600,000 dirhams and fourteen million dinars. In 2008, MBC1 depicted the life and leadership of al-Mansur in a series aired during the holy month of Ramadan. Masudi, The Meadows of Gold, The Abbasids, transl, Paul Lunde and Caroline Stone, Kegan Paul, London and New York,1989 Kennedy, Hugh, When Baghdad Ruled The Muslim World, Cambridge, Da Capo Press,2004
23.
Muhammad al-Fazari
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Muhammad ibn Ibrahim al-Fazari was a Muslim philosopher, mathematician and astronomer. He is not to be confused with his father Ibrāhīm al-Fazārī, also an astronomer, some sources refer to him as an Arab, other sources state that he was a Persian. Al-Fazārī translated many books into Arabic and Persian. He is credited to have built the first astrolabe in the Islamic world, along with Yaʿqūb ibn Ṭāriq and his father he helped translate the Indian astronomical text by Brahmagupta, the Brāhmasphuṭasiddhānta, into Arabic as Az-Zīj ‛alā Sinī al-‛Arab. or the Sindhind. This translation was possibly the vehicle by means of which the Hindu numerals were transmitted from India to Islam, Hindu and Buddhist contribution to science in medieval Islam List of Iranian scientists and scholars List of Arab scientists List of Iranian scientists zij Plofker, Kim
24.
Al-Khwarizmi
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Muḥammad ibn Mūsā al-Khwārizmī, formerly Latinized as Algoritmi, was a Persian mathematician, astronomer, and geographer during the Abbasid Caliphate, a scholar in the House of Wisdom in Baghdad. In the 12th century, Latin translations of his work on the Indian numerals introduced the decimal number system to the Western world. Al-Khwārizmīs The Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and he is often considered one of the fathers of algebra. He revised Ptolemys Geography and wrote on astronomy and astrology, some words reflect the importance of al-Khwārizmīs contributions to mathematics. Algebra is derived from al-jabr, one of the two operations he used to solve quadratic equations, algorism and algorithm stem from Algoritmi, the Latin form of his name. His name is also the origin of guarismo and of algarismo, few details of al-Khwārizmīs life are known with certainty. He was born in a Persian family and Ibn al-Nadim gives his birthplace as Khwarezm in Greater Khorasan, muhammad ibn Jarir al-Tabari gives his name as Muḥammad ibn Musá al-Khwārizmiyy al-Majūsiyy al-Quṭrubbaliyy. The epithet al-Qutrubbulli could indicate he might instead have come from Qutrubbul and this would not be worth mentioning if a series of errors concerning the personality of al-Khwārizmī, occasionally even the origins of his knowledge, had not been made. Recently, G. J. Toomer. with naive confidence constructed an entire fantasy on the error which cannot be denied the merit of amusing the reader. Regarding al-Khwārizmīs religion, Toomer writes, Another epithet given to him by al-Ṭabarī, al-Majūsī, Ibn al-Nadīms Kitāb al-Fihrist includes a short biography on al-Khwārizmī together with a list of the books he wrote. Al-Khwārizmī accomplished most of his work in the period between 813 and 833, douglas Morton Dunlop suggests that it may have been possible that Muḥammad ibn Mūsā al-Khwārizmī was in fact the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the three Banū Mūsā. Al-Khwārizmīs contributions to mathematics, geography, astronomy, and cartography established the basis for innovation in algebra, on the Calculation with Hindu Numerals written about 825, was principally responsible for spreading the Hindu–Arabic numeral system throughout the Middle East and Europe. It was translated into Latin as Algoritmi de numero Indorum, al-Khwārizmī, rendered as Algoritmi, led to the term algorithm. Some of his work was based on Persian and Babylonian astronomy, Indian numbers, al-Khwārizmī systematized and corrected Ptolemys data for Africa and the Middle East. Another major book was Kitab surat al-ard, presenting the coordinates of places based on those in the Geography of Ptolemy but with improved values for the Mediterranean Sea, Asia and he also wrote on mechanical devices like the astrolabe and sundial. He assisted a project to determine the circumference of the Earth and in making a map for al-Mamun. When, in the 12th century, his works spread to Europe through Latin translations, the Compendious Book on Calculation by Completion and Balancing is a mathematical book written approximately 830 CE. The term algebra is derived from the name of one of the operations with equations described in this book
25.
Linear equation
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A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. A simple example of an equation with only one variable, x, may be written in the form, ax + b =0, where a and b are constants. The constants may be numbers, parameters, or even functions of parameters. Linear equations can have one or more variables. An example of an equation with three variables, x, y, and z, is given by, ax + by + cz + d =0, where a, b, c, and d are constants and a, b. Linear equations occur frequently in most subareas of mathematics and especially in applied mathematics, an equation is linear if the sum of the exponents of the variables of each term is one. Equations with exponents greater than one are non-linear, an example of a non-linear equation of two variables is axy + b =0, where a and b are constants and a ≠0. It has two variables, x and y, and is non-linear because the sum of the exponents of the variables in the first term and this article considers the case of a single equation for which one searches the real solutions. All its content applies for complex solutions and, more generally for linear equations with coefficients, a linear equation in one unknown x may always be rewritten a x = b. If a ≠0, there is a solution x = b a. The origin of the name comes from the fact that the set of solutions of such an equation forms a straight line in the plane. Linear equations can be using the laws of elementary algebra into several different forms. These equations are referred to as the equations of the straight line. In what follows, x, y, t, and θ are variables, in the general form the linear equation is written as, A x + B y = C, where A and B are not both equal to zero. The equation is written so that A ≥0, by convention. The graph of the equation is a line, and every straight line can be represented by an equation in the above form. If A is nonzero, then the x-intercept, that is, if B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis, is C/B, and the slope of the line is −A/B. The general form is written as, a x + b y + c =0
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Quadratic equation
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If a =0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and may be distinguished by calling them, respectively, the coefficient, the linear coefficient. Because the quadratic equation involves only one unknown, it is called univariate, solutions to problems equivalent to the quadratic equation were known as early as 2000 BC. A quadratic equation with real or complex coefficients has two solutions, called roots and these two solutions may or may not be distinct, and they may or may not be real. It may be possible to express a quadratic equation ax2 + bx + c =0 as a product =0. In some cases, it is possible, by inspection, to determine values of p, q, r. If the quadratic equation is written in the form, then the Zero Factor Property states that the quadratic equation is satisfied if px + q =0 or rx + s =0. Solving these two linear equations provides the roots of the quadratic, for most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. As an example, x2 + 5x +6 factors as, the more general case where a does not equal 1 can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection. Except for special cases such as where b =0 or c =0 and this means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection. The process of completing the square makes use of the identity x 2 +2 h x + h 2 =2. Starting with an equation in standard form, ax2 + bx + c =0 Divide each side by a. Subtract the constant term c/a from both sides, add the square of one-half of b/a, the coefficient of x, to both sides. This completes the square, converting the left side into a perfect square, write the left side as a square and simplify the right side if necessary. Produce two linear equations by equating the square root of the side with the positive and negative square roots of the right side. Completing the square can be used to derive a formula for solving quadratic equations. The mathematical proof will now be briefly summarized and it can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation,2 = b 2 −4 a c 4 a 2. Taking the square root of both sides, and isolating x, gives, x = − b ± b 2 −4 a c 2 a and these result in slightly different forms for the solution, but are otherwise equivalent
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Diophantus
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Diophantus of Alexandria, sometimes called the father of algebra, was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica, many of which are now lost. These texts deal with solving algebraic equations and this led to tremendous advances in number theory, and the study of Diophantine equations and of Diophantine approximations remain important areas of mathematical research. Diophantus coined the term παρισότης to refer to an approximate equality and this term was rendered as adaequalitas in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers, thus he allowed positive rational numbers for the coefficients, in modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation, little is known about the life of Diophantus. He lived in Alexandria, Egypt, probably from between AD200 and 214 to 284 or 298, much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus. One of the states, Here lies Diophantus, the wonder behold. Alas, the child of master and sage After attaining half the measure of his fathers life chill fate took him. After consoling his fate by the science of numbers for four years and this puzzle implies that Diophantus age x can be expressed as x = x/6 + x/12 + x/7 +5 + x/2 +4 which gives x a value of 84 years. However, the accuracy of the information cannot be independently confirmed, the Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations, of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arab books discovered in 1968 are also by Diophantus. Some Diophantine problems from Arithmetica have been found in Arabic sources and it should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus, “Our author not the slightest trace of a general, comprehensive method is discernible, each problem calls for some special method which refuses to work even for the most closely related problems. The portion of the Greek Arithmetica that survived, however, was, like all ancient Greek texts transmitted to the modern world, copied by. In addition, some portion of the Arithmetica probably survived in the Arab tradition. ”Arithmetica was first translated from Greek into Latin by Bombelli in 1570, however, Bombelli borrowed many of the problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander, the best known Latin translation of Arithmetica was made by Bachet in 1621 and became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it, and made notes in the margins. I have a marvelous proof of this proposition which this margin is too narrow to contain. ”Fermats proof was never found
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History of algebra
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As a branch of mathematics, algebra emerged at the end of the 16th century in Europe, with the work of François Viète. Algebra can essentially be considered as doing computations similar to those of arithmetic, however, until the 19th century, algebra consisted essentially of the theory of equations. For example, the theorem of algebra belongs to the theory of equations and is not, nowadays. This article describes the history of the theory of equations, called here algebra, the treatise provided for the systematic solution of linear and quadratic equations. According to one history, t is not certain just what the terms al-jabr and muqabalah mean, Arabic influence in Spain long after the time of al-Khwarizmi is found in Don Quixote, where the word algebrista is used for a bone-setter, that is, a restorer. Algebra did not always make use of the symbolism that is now ubiquitous in mathematics, instead, the stages in the development of symbolic algebra are approximately as follows, Rhetorical algebra, in which equations are written in full sentences. For example, the form of x +1 =2 is The thing plus one equals two or possibly The thing plus 1 equals 2. Rhetorical algebra was first developed by the ancient Babylonians and remained dominant up to the 16th century, syncopated algebra, in which some symbolism is used, but which does not contain all of the characteristics of symbolic algebra. For instance, there may be a restriction that subtraction may be used once within one side of an equation. Syncopated algebraic expression first appeared in Diophantus Arithmetica, followed by Brahmaguptas Brahma Sphuta Siddhanta, symbolic algebra, in which full symbolism is used. Early steps toward this can be seen in the work of several Islamic mathematicians such as Ibn al-Banna and al-Qalasadi, later, René Descartes introduced the modern notation and showed that the problems occurring in geometry can be expressed and solved in terms of algebra. Equally important as the use or lack of symbolism in algebra was the degree of the equations that were addressed, X2 + p x = q x 2 = p x + q x 2 + q = p x where p and q are positive. This trichotomy comes about because quadratic equations of the form x 2 + p x + q =0, for instance, an equation of the form x 2 = A was solved by finding the side of a square of area A. In addition to the three stages of expressing algebraic ideas, there were four stages in the development of algebra that occurred alongside the changes in expression. These four stages were as follows, Geometric stage, where the concepts of algebra are largely geometric and this dates back to the Babylonians and continued with the Greeks, and was later revived by Omar Khayyám. Static equation-solving stage, where the objective is to find numbers satisfying certain relationships, the move away from geometric algebra dates back to Diophantus and Brahmagupta, but algebra didnt decisively move to the static equation-solving stage until Al-Khwarizmis Al-Jabr. Dynamic function stage, where motion is an underlying idea, the idea of a function began emerging with Sharaf al-Dīn al-Tūsī, but algebra did not decisively move to the dynamic function stage until Gottfried Leibniz. Abstract stage, where mathematical structure plays a central role, abstract algebra is largely a product of the 19th and 20th centuries
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Babylonians
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Babylonia was an ancient Akkadian-speaking state and cultural area based in central-southern Mesopotamia. A small Amorite-ruled state emerged in 1894 BC, which contained at this time the city of Babylon. Babylon greatly expanded during the reign of Hammurabi in the first half of the 18th century BC, during the reign of Hammurabi and afterwards, Babylonia was called Māt Akkadī the country of Akkad in the Akkadian language. It was often involved in rivalry with its older fellow Akkadian-speaking state of Assyria in northern Mesopotamia and it retained the Sumerian language for religious use, but by the time Babylon was founded, this was no longer a spoken language, having been wholly subsumed by Akkadian. The earliest mention of the city of Babylon can be found in a tablet from the reign of Sargon of Akkad. During the 3rd millennium BC, a cultural symbiosis occurred between Sumerian and Akkadian-speakers, which included widespread bilingualism. The influence of Sumerian on Akkadian and vice versa is evident in all areas, from lexical borrowing on a scale, to syntactic, morphological. This has prompted scholars to refer to Sumerian and Akkadian in the millennium as a sprachbund. Traditionally, the religious center of all Mesopotamia was the city of Nippur. The empire eventually disintegrated due to decline, climate change and civil war. Sumer rose up again with the Third Dynasty of Ur in the late 22nd century BC and they also seem to have gained ascendancy over most of the territory of the Akkadian kings of Assyria in northern Mesopotamia for a time. The states of the south were unable to stem the Amorite advance, King Ilu-shuma of the Old Assyrian Empire in a known inscription describes his exploits to the south as follows, The freedom of the Akkadians and their children I established. I established their freedom from the border of the marshes and Ur and Nippur, Awal, past scholars originally extrapolated from this text that it means he defeated the invading Amorites to the south, but there is no explicit record of that. More recently, the text has been taken to mean that Asshur supplied the south with copper from Anatolia and these policies were continued by his successors Erishum I and Ikunum. During the first centuries of what is called the Amorite period and his reign was concerned with establishing statehood amongst a sea of other minor city states and kingdoms in the region. However Sumuabum appears never to have bothered to give himself the title of King of Babylon, suggesting that Babylon itself was only a minor town or city. He was followed by Sumu-la-El, Sabium, Apil-Sin, each of whom ruled in the same manner as Sumuabum. Sin-Muballit was the first of these Amorite rulers to be regarded officially as a king of Babylon, the Elamites occupied huge swathes of southern Mesopotamia, and the early Amorite rulers were largely held in vassalage to Elam
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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
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Ancient Rome
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In its many centuries of existence, the Roman state evolved from a monarchy to a classical republic and then to an increasingly autocratic empire. Through conquest and assimilation, it came to dominate the Mediterranean region and then Western Europe, Asia Minor, North Africa and it is often grouped into classical antiquity together with ancient Greece, and their similar cultures and societies are known as the Greco-Roman world. Ancient Roman civilisation has contributed to modern government, law, politics, engineering, art, literature, architecture, technology, warfare, religion, language and society. Rome professionalised and expanded its military and created a system of government called res publica, the inspiration for modern republics such as the United States and France. By the end of the Republic, Rome had conquered the lands around the Mediterranean and beyond, its domain extended from the Atlantic to Arabia, the Roman Empire emerged with the end of the Republic and the dictatorship of Augustus Caesar. 721 years of Roman-Persian Wars started in 92 BC with their first war against Parthia and it would become the longest conflict in human history, and have major lasting effects and consequences for both empires. Under Trajan, the Empire reached its territorial peak, Republican mores and traditions started to decline during the imperial period, with civil wars becoming a prelude common to the rise of a new emperor. Splinter states, such as the Palmyrene Empire, would divide the Empire during the crisis of the 3rd century. Plagued by internal instability and attacked by various migrating peoples, the part of the empire broke up into independent kingdoms in the 5th century. This splintering is a landmark historians use to divide the ancient period of history from the pre-medieval Dark Ages of Europe. King Numitor was deposed from his throne by his brother, Amulius, while Numitors daughter, Rhea Silvia, because Rhea Silvia was raped and impregnated by Mars, the Roman god of war, the twins were considered half-divine. The new king, Amulius, feared Romulus and Remus would take back the throne, a she-wolf saved and raised them, and when they were old enough, they returned the throne of Alba Longa to Numitor. Romulus became the source of the citys name, in order to attract people to the city, Rome became a sanctuary for the indigent, exiled, and unwanted. This caused a problem for Rome, which had a large workforce but was bereft of women, Romulus traveled to the neighboring towns and tribes and attempted to secure marriage rights, but as Rome was so full of undesirables they all refused. Legend says that the Latins invited the Sabines to a festival and stole their unmarried maidens, leading to the integration of the Latins, after a long time in rough seas, they landed at the banks of the Tiber River. Not long after they landed, the men wanted to take to the sea again, one woman, named Roma, suggested that the women burn the ships out at sea to prevent them from leaving. At first, the men were angry with Roma, but they realized that they were in the ideal place to settle. They named the settlement after the woman who torched their ships, the Roman poet Virgil recounted this legend in his classical epic poem the Aeneid
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Division by zero
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In mathematics, division by zero is division where the divisor is zero. Such a division can be expressed as a/0 where a is the dividend. In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by 0, gives a, and so division by zero is undefined. Since any number multiplied by zero is zero, the expression 0/0 also has no defined value, in computing, a program error may result from an attempt to divide by zero. When division is explained at the elementary level, it is often considered as splitting a set of objects into equal parts. As an example, consider having ten cookies, and these cookies are to be distributed equally to five people at a table, each person would receive 105 =2 cookies. Similarly, if there are ten cookies, and only one person at the table, so, for dividing by zero, what is the number of cookies that each person receives when 10 cookies are evenly distributed amongst 0 people at a table. Certain words can be pinpointed in the question to highlight the problem, the problem with this question is the when. There is no way to evenly distribute 10 cookies to nobody, in mathematical jargon, a set of 10 items cannot be partitioned into 0 subsets. So 100, at least in elementary arithmetic, is said to be either meaningless, similar problems occur if one has 0 cookies and 0 people, but this time the problem is in the phrase the number. A partition is possible, but since the partition has 0 parts, vacuously every set in our partition has a number of elements, be it 0,2,5. If there are, say,5 cookies and 2 people, in any integer partition of a 5-set into 2 parts, one of the parts of the partition will have more elements than the other. But the problem with 5 cookies and 2 people can be solved by cutting one cookie in half, the problem with 5 cookies and 0 people cannot be solved in any way that preserves the meaning of divides. Another way of looking at division by zero is that division can always be checked using multiplication. Considering the 10/0 example above, setting x = 10/0, if x equals ten divided by zero, then x times zero equals ten, but there is no x that, when multiplied by zero, gives ten. If instead of x=10/0 we have x=0/0, then every x satisfies the question what number x, multiplied by zero, the Brahmasphutasiddhanta of Brahmagupta is the earliest known text to treat zero as a number in its own right and to define operations involving zero. The author could not explain division by zero in his texts, according to Brahmagupta, A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is zero or is expressed as a fraction with zero as numerator