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Indian mathematics—which here is the mathematics that emerged in South Asia[1] from ancient times until the end of the 18th century—had its beginnings in the Bronze Age Indus Valley civilization (2600-1900 BCE) and the Iron Age Vedic culture (1500-500 BCE). Science is a body of empirical, theoretical, and practical knowledge about the natural world, produced by a global community of researchers The sociology and Philosophy of science, as well as the entire field of Science studies, have in the 20th century been preoccupied with the question of The Historiography of Science usually refers to the study of History of Science in its disciplinary aspects and practices (methods theories schools and Note The contents of this page are expected to change as consensus is reached The History of science in early cultures refers to the study of Protoscience in Ancient history, prior to the development of Science in the Middle The history of science in Classical Antiquity begins with the search for practical knowledge In the Middle Ages, Science progressed dramatically from the time of antiquity in areas as diverse as Astronomy, Medicine, and Mathematics During the Renaissance, the rediscovery of ancient scientific texts was accelerated after the Fall of Constantinople in 1453, and the invention of Printing The period which many historians of science call the Scientific Revolution can be roughly dated as having begun in 1543 the year in which Nicolaus Copernicus published For the current in the 19th century German idealism see Naturphilosophie Natural philosophy or the philosophy of nature (from Astronomy is the oldest of the Natural sciences dating back to antiquity, with its origins in the religious, Mythological, and Astrological The history of biology traces the study of the living world from ancient to modern times The history of Chemistry begins with the discovery of Fire, then Metallurgy which allowed purification of metals and the making of alloys as well as the exploitation Ecology is generally spoken of as a new science having only become prominent in the second half of the 20th Century This article explores the History of Geography. Ancient geography See also Ancient Greek geography Ancient Greeks environment The history of geology is concerned with the development of the natural science of geology The history of Paleontology traces the effort to understand the history of life on Earth by studying the Fossil record left behind by living organisms The modern discipline of Physics emerged in the 17th century following in traditions of inquiry established by Galileo Galilei, René Descartes, Isaac For more see Social sciences#History of the social sciences In ancient philosophy there was no difference between the Liberal arts of mathematics The history of economic thought deals with different thinkers and theories in the field of Political economy and Economics from the ancient world to the present See also History of grammar Linguistics as a study endeavors to describe and explain the human faculty of Language. While the study of politics is first found in Ancient Greece and ancient India, political science is a late arrival in terms of Social sciences. The History of Psychology as a scholarly study of the mind and behavior dates back to the Middle Ages. Sociology is a relatively new academic discipline among other Social sciences including Economics, Political science, Anthropology, and The history of technology is the history of the Invention of Tools and techniques Agronomy and the related disciplines of Agricultural science today are very different from what they were before about 1950 The history of computer science began long before the modern discipline of Computer science that emerged in the twentieth century The history of Materials science is the study of how different materials were used as influenced by the History of Earth and the Culture of the All human societies have medical beliefs that provide explanations for birth, Death, and Disease. This is a list of Timelines. Types of timelines Living graph Logarithmic timeline The term Bronze Age refers to a period in human cultural development when the most advanced Metalworking (at least in systematic and widespread use included techniques for The Indus Valley Civilization (Mature period 2600&ndash1900 BCE abbreviated IVC, was an ancient Civilization that flourished in the Indus River basin This article is about the archaeological period known as the Iron Age for the mythological Iron Age see Ages of Man. The Vedic Period (or Vedic Age) is the period in the History of India during which the Vedas, the oldest sacred texts of Hinduism, were being In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhatta, Brahmagupta, and Bhaskara II. Events By Place Western Roman Empire Italy is first invaded by Alaric (probable date Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics Brahmagupta ( (598–668 was an Indian mathematician and astronomer. Bhaskara (1114 &ndash 1185 also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher" was an Indian mathematician Indian mathematicians made early contributions to the study of the decimal number system,[2] zero,[3] negative numbers,[4] arithmetic, and algebra. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. A negative number is a Number that is less than zero, such as −2 Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. [5] In addition, trigonometry, having evolved in the Hellenistic world and having been introduced into ancient India through the translation of Greek works,[6] was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. This article focuses on the cultural aspects of the Hellenistic age for the historical aspects see Hellenistic period. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly [7] These mathematical concepts were transmitted to the Middle East, China, and Europe[5] and led to further developments that now form the foundations of many areas of mathematics. The Middle East is a Subcontinent with no clear boundaries often used as a synonym to Near East, in opposition to Far East. China ( Wade-Giles ( Mandarin) Chung¹kuo² is a cultural region, an ancient Civilization, and depending on perspective a National

Ancient and medieval Indian mathematical works, all composed in Sanskrit, usually consisted of a section of sutras in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student. Sanskrit (sa संस्कृता वाक् saṃskṛtā vāk, for short sa संस्कृतम् saṃskṛtam) is a historical This was followed by a second section consisting of a prose commentary (sometimes multiple commentaries by different scholars) that explained the problem in more detail and provided justification for the solution. In the prose section, the form (and therefore its memorization) was not considered as important as the ideas involved. [8][1] All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form. The oldest extant mathematical document produced on the Indian subcontinent is the birch bark Bakhshali Manuscript, discovered in 1881 in the village of Bakhshali, near Peshawar (modern day Pakistan) and is likely from the seventh century CE. The Bakhshali Manuscript is a mathematical manuscript written on birch bark which was found near the village of Bakhshali in 1881 in what was ( پښور; Urdu: پشاور) is the capital of the North-West Frontier Province and the administrative centre for the Federally Administered Pakistan () officially the Islamic Republic of Pakistan, is a country located in South Asia, Southwest Asia, Middle East and [9][10]

A later landmark in Indian mathematics was the development of the series expansions for trigonometric functions (sine, cosine, and arc tangent) by mathematicians of the Kerala School in the fifteenth century CE. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with 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 (apart from geometric series). Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + [11] However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Kerala ( Malayalam: {{Kerala in Malayalam}}; [12]

Contents

Fields of Indian mathematics

Some of the areas of mathematics studied in ancient and medieval India include the following:

Indian mathematics show many different ways of Indian culture. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone A negative number is a Number that is less than zero, such as −2 Brahmagupta ( (598–668 was an Indian mathematician and astronomer. The Hindu-Arabic numeral system is a Positional Decimal Numeral system first documented in the ninth century A positional notation or place-value notation system is a Numeral system in which each position is related to the next by a Constant multiplier a A numeral system (or system of numeration) is a Mathematical notation for representing numbers of a given set by symbols in a consistent manner In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness The Yajurveda ( Sanskrit यजुर्वेदः, a Tatpurusha compound of yajus "sacrificial formula' + veda Transfinite numbers are Cardinal numbers or Ordinal numbers that are larger than all finite numbers yet not necessarily absolutely infinite. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose This article presents and explains several methods which can be used to calculate Square roots Exponential identity Pocket calculators typically implement good In Mathematics, a cube root of a number denoted \sqrt{x} or x1/3 is a number a such that a 3 =  x Mahavira was a 9th century Indian Mathematician from Gulbarga who asserted that the Square root of a Negative number did A Pythagorean triple consists of three positive Integers a, b, and c, such that a 2 +  b 2 =  The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the Baudhāyana, (fl ca 800 BCE was an Indian mathematician whowas most likely also a priest In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry In Mathematics a transform is an Operator applied to a function so that under the transform certain operations are simplified Pāṇini ( IAST: Pāṇini Dēvanāgarī: sa पाणिनि a Patronymic meaning "descendant of {{IAST|Paṇi}} " was an ancient \begin{matrix}&&&&&1\\&&&&1&&1\\&&&1&&2&&1\\&&1&&3&&3&&1\\&1&&4&&6&&4&&1\end{matrix Pingala ( पिङ्गल piṅgalá) was an ancient Indian writer famous for his work the Chandas Shastra ( chandaḥ-śāstra Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics Brahmagupta ( (598–668 was an Indian mathematician and astronomer. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. Mahavira was a 9th century Indian Mathematician from Gulbarga who asserted that the Square root of a Negative number did In Mathematics, a quartic equation is one which can be expressed as a Quartic function equalling zero Mahavira was a 9th century Indian Mathematician from Gulbarga who asserted that the Square root of a Negative number did Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. In Formal semantics, Computer science and Linguistics, a formal grammar (also called formation rules) is a precise description of a Formal A formal language is a set of words, ie finite strings of letters, or symbols. In Computer science, Backus–Naur Form ( BNF) is a Metasyntax used to express Context-free grammars that is a formal way to describe Formal Pāṇini ( IAST: Pāṇini Dēvanāgarī: sa पाणिनि a Patronymic meaning "descendant of {{IAST|Paṇi}} " was an ancient Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition Pāṇini ( IAST: Pāṇini Dēvanāgarī: sa पाणिनि a Patronymic meaning "descendant of {{IAST|Paṇi}} " was an ancient In Mathematics, the Fibonacci numbers are a Sequence of numbers named after Leonardo of Pisa, known as Fibonacci Pingala ( पिङ्गल piṅgalá) was an ancient Indian writer famous for his work the Chandas Shastra ( chandaḥ-śāstra Morse code is a Character encoding for transmitting telegraphic information using standardized sequences of short and long elements to represent the letters numerals Pingala ( पिङ्गल piṅgalá) was an ancient Indian writer famous for his work the Chandas Shastra ( chandaḥ-śāstra In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce The word index is used in variety of senses in Mathematics. In perhaps the most frequent sense an index is a Superscript In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Algorism is the technique of performing basic Arithmetic by writing numbers in Place value form and applying a set of memorized rules and facts to the digits Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics Brahmagupta ( (598–668 was an Indian mathematician and astronomer. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. The Surya Siddhanta is a treatise of Indian astronomy. Later Indian mathematicians and astronomers such as Aryabhata and Varahamihira Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics In Mathematics, a trigonometric series is any series of the form \frac{1}{2}A_{o}+\displaystyle\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} Mādhava of Sangamagrama (born as Irinjaatappilly Madhavan Namboodiri) (c

Harappan Mathematics (2600 BCE – 1700 BCE)

See also: Indus Valley Civilization

The earliest evidence of the use of mathematics in South Asia is in the artifacts of the Indus Valley Civilization (IVC), also called the Harappan civilization. The Indus Valley Civilization (Mature period 2600&ndash1900 BCE abbreviated IVC, was an ancient Civilization that flourished in the Indus River basin The Indus Valley Civilization (Mature period 2600&ndash1900 BCE abbreviated IVC, was an ancient Civilization that flourished in the Indus River basin Excavations at Harappa, Mohenjo-daro(Pakistan) and other locations in the Indus river valley have uncovered evidence of the use of practical mathematics. Harappa ( Urdu:, Hindi: हड़प्पा) is a City in Punjab, northeast Pakistan, about 35km (22 miles southwest Mohenjo-daro (موئن جودڑو موئن جو دڙو मोहन जोदड़ो Mound of the Dead was one of the largest city-settlements of the Indus Valley Civilization The Indus River { Sanskrit: सिन्धु Sindhu; Urdu: urd {{Nastaliq سندھ}} Sindh; Sindhi: snd The people of the IVC manufactured bricks whose dimensions were in the proportion 4:2:1, considered favorable for the stability of a brick structure. They used a standardized system of weights based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with the unit weight equaling approximately 28 grams (and approximately equal to the English ounce or Greek uncia). They mass produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, and cylinders, thereby demonstrating knowledge of basic geometry. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position A hexahedron (plural hexahedra is a Polyhedron with six faces A barrel or cask is a hollow cylindrical container traditionally made of Wood Staves and bound with Iron Hoops The A cone is a three-dimensional Geometric shape that tapers smoothly from a flat round base to a point called the apex or vertex A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position

The inhabitants of Indus civilization also tried to standardize measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length (approximately 1. 32 inches or 3. 4 centimetres) 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 Oral Mathematical Tradition

Mathematicians of ancient and early medieval India were almost all Sanskrit pandits (paṇḍita "learned man"),[13] who were trained in Sanskrit language and literature, and possessed "a common stock of knowledge in grammar (vyākaraṇa), exegesis (mīmāṃsā) and logic (nyāya). Sanskrit (sa संस्कृता वाक् saṃskṛtā vāk, for short sa संस्कृतम् saṃskṛtam) is a historical A paṇḍit or pundit ( Devanagari: पण्डित is a scholar a teacher particularly one skilled in Sanskrit and Hindu Law, Religion The Sanskrit grammatical tradition of vyākaraṇa is one of the six Vedanga disciplines Exegesis (from the Greek 'to lead out' involves an extensive and critical interpretation of an authoritative text, especially of a Holy Mīmāṃsā, a Sanskrit word meaning "investigation" (compare Greek ἱστορία) is the name of an Astika ("orthodox" school Nyāya ( Sanskrit ni-āyá, literally "recursion" used in the sense of " Syllogism, inference" is the name given to one of the six orthodox "[13] Memorization of "what is heard" (śruti in Sanskrit) through recitation played a major role in the transmission of sacred texts in ancient India. If you are looking for the singer see Shruti Haasan. For other meanings see Śruti (disambiguation. Memorization and recitation was also used to transmit philosophical and literary works, as well as treatises on ritual and grammar. Modern scholars of ancient India have noted the "truly remarkable achievements of the Indian pandits who have preserved enormously bulky texts orally for millennia. "[14]

Styles of Memorization

Prodigous energy was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity. [15] For example, memorization of the sacred Vedas included up to eleven forms of recitation of the same text. "Veda" redirects here For other uses see Veda (disambiguation. The texts were subsequently "proof-read" by comparing the different recited versions. Forms of recitation included the jaṭā-pāṭha (literally "mesh recitation") in which every two adjacent words in the text were first recited in their original order, then repeated in the reverse order, and finally repeated again in the original order. [16] The recitation thus proceeded as:

word1word2, word2word1, word1word2; word2word3, word3word2, word2word3; . . .

In another form of recitation, dvaja-pāṭha[16] (literally "flag recitation") a sequence of N words were recited (and memorized) by pairing the first two and last two words and then proceeding as:

word1word2, word(N-1)wordN; word2word3, word(N-3)word(N-2); . . . ; word(N-1)wordN, word1word2;

The most complex form of recitation, ghana-pāṭha (literally "dense recitation"), according to (Filliozat 2004, p.  139), took the form:

word1word2, word2word1, word1word2word3, word3word2word1, word1word2word3; word2word3, word3word2, word2word3word4, word4word3word2, word2word3word4; . . .

That these methods have been effective, is testified to by the preservation of the most ancient Indian religious text, the Ṛgveda (ca. 1500 BCE), as a single text, without any variant readings. The Rigveda ( Sanskrit sa ऋग्वेद ṛgveda, a compound of ṛc "praise verse" and veda "knowledge" Circa (often abbreviated c, ca, ca or cca and sometimes Italicized to show it is Latin) means "about" [16] Similar methods were used for memorizing mathematical texts, whose transmission remained exclusively oral until the end of the Vedic period (ca. The Vedic Period (or Vedic Age) is the period in the History of India during which the Vedas, the oldest sacred texts of Hinduism, were being 500 BCE).

The Sūtra Genre

Mathematical activity in ancient India began as a part of a "methodological reflexion" on the sacred Vedas, which took the form of works called Vedāṇgas, or, "Ancillaries of the Veda" (7th-4th century BCE). "Veda" redirects here For other uses see Veda (disambiguation. The Vedanga ( vedāṅga, "member of the Veda" are six auxiliary disciplines for the understanding and tradition of the Vedas. [17] The need to conserve the sound of sacred text by use of śikṣā (phonetics) and chandas (metrics); to conserve its meaning by use of vyākaraṇa (grammar) and nirukta (etymology); and to correctly perform the rites at the correct time by the use of kalpa (ritual) and jyotiṣa (astronomy), gave rise to the six disciplines of the Vedāṇgas. See Shiksha (NGO for the Indian non-governmental organization Phonetics (from the Greek φωνή ( phonê) "sound" or "voice" is the study of the physical sounds of human speech The main principle of Vedic meter is measurement by the number of syllables In Poetry, the meter or metre is the basic rhythmic structure of a verse. The Sanskrit grammatical tradition of vyākaraṇa is one of the six Vedanga disciplines Grammar is the field of Linguistics that covers the Rules governing the use of any given natural language. Nirukta ("explanation etymological interpretation" is one of the six {{IAST|Vedānga}} disciplines of Hinduism, treating Etymology, particularly Etymology is the study of the History of Words &mdash when they entered a language from what source and how their form and meaning have changed over time The KalPa is an Ice hockey team in the SM-liiga. They play in Kuopio, Finland at the Niiralan monttu. A ritual is a set of actions often thought to have Symbolic value the performance of which is usually prescribed by a Religion or by the Traditions Jyotiṣa ( Sanskrit jyotiṣa, from jyótis- "light heavenly body" also spelled Jyotish and Jyotisha in English Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study [17] Mathematics arose as a part of the last two disciplines, ritual and astronomy (which also included astrology). Since the Vedāṇgas immediately preceded the use of writing in ancient India, they formed the last of the exclusively oral literature. They were expressed in a highly compressed mnemonic form, the sūtra (literally, "thread"):

The knowers of the sūtra know it as having few phonemes, being devoid of ambiguity, containing the essence, facing everything, being without pause and unobjectionable. [17]

Extreme brevity was achieved through multiple means, which included using ellipsis "beyond the tolerance of natural language,"[17] using technical names instead of longer descriptive names, abridging lists by only mentioning the first and last entries, and using markers and variables. Ellipsis (plural ellipses; from Greek 'omission' in Printing and Writing refers to a mark or series of marks that usually indicate an intentional [17] The sūtras create the impression that communication through the text was "only a part of the whole instruction. The rest of the instruction must have been transmitted by the so-called Guru-shishya parampara, 'uninterrupted succession from teacher (guru) to the student (śisya),' and it was not open to the general public" and perhaps even kept secret. The guru-shishya tradition lineage or Parampara, is a spiritual relationship in traditional Hinduism where teachings are transmitted from a [18] The brevity achieved in a sūtra is demonstrated in the following example from the Baudhāyana Śulba Sūtra (700 BCE).

The design of the domestic fire altar in the Śulba Sūtra
The design of the domestic fire altar in the Śulba Sūtra

The domestic fire-altar in the Vedic period was required by ritual to have a square base and be constituted of five layers of bricks with 21 bricks in each layer. The Vedic Period (or Vedic Age) is the period in the History of India during which the Vedas, the oldest sacred texts of Hinduism, were being One method of constructing the altar was to divide one side of the square into three equal parts using a cord or rope, to next divide the transverse (or perpendicular) side into seven equal parts, and thereby sub-divide the square into 21 congruent rectangles. The bricks were then designed to be of the shape of the constituent rectangle and the layer was created. To form the next layer, the same formula was used, but the bricks were arranged transversely. [19] The process was then repeated three more times (with alternating directions) in order to complete the construction. In the Baudhāyana Śulba Sūtra, this procedure is described in the following words:

"II. 64. After dividing the quadri-lateral in seven, one divides the transverse [cord] in three.
II. 65. In another layer one places the [bricks] North-pointing. "[19]

According to (Filliozat 2004, p.  144), the officiant constructing the altar has only a few tools and materials at his disposal: a cord (Sanskrit, rajju, f. ), two pegs (Sanskrit, śanku, m. ), and clay to make the bricks (Sanskrit, iṣṭakā, f. ). Concision is achieved in the sūtra, by not explicitly mentioning what the adjective "transverse" qualifies; however, from the feminine form of the (Sanskrit) adjective used, it is easily inferred to qualify "cord. " Similarly, in the second stanza, "bricks" are not explicitly mentioned, but inferred again by the feminine plural form of "North-pointing. " Finally, the first stanza, never explicitly says that the first layer of bricks are oriented in the East-West direction, but that too is implied by the explicit mention of "North-pointing" in the second stanza; for, if the orientation was meant to be the same in the two layers, it would either not be mentioned at all or be only mentioned in the first stanza. All these inferences are made by the officiant as he recalls the formula from his memory. [19]

Vedic Period (1500 BCE - 400 BCE)

See also: Vedanga and Vedas

The religious texts of the Vedic Period provide evidence for the use of large numbers. The Vedanga ( vedāṅga, "member of the Veda" are six auxiliary disciplines for the understanding and tradition of the Vedas. "Veda" redirects here For other uses see Veda (disambiguation. The Vedic Period (or Vedic Age) is the period in the History of India during which the Vedas, the oldest sacred texts of Hinduism, were being Different Cultures used different traditional Numeral systems for naming large numbers. By the time of the last Veda, the Yajurvedasaṃhitā (1200-900 BCE), numbers as high as 1012 were being included in the texts. The Yajurveda ( Sanskrit यजुर्वेदः, a Tatpurusha compound of yajus "sacrificial formula' + veda [20] For example, the mantra (sacrificial formula) at the end of the annahoma ("food-oblation rite") performed during the aśvamedha ("horse sacrifice"), and uttered just before-, during-, and just after sunrise, invokes powers of ten from a hundred to a trillion:[20]

"Hail to śata ("hundred," 102), hail to sahasra ("thousand," 103), hail to ayuta ("ten thousand," 104), hail to niyuta ("hundred thousand," 105), hail to prayuta ("million," 106), hail to arbuda ("ten million," 107), hail to nyarbuda ("hundred million," 108), hail to samudra ("billion," 109, literally "ocean"), hail to madhya ("ten billion," 1010, literally "middle"), hail to anta ("hundred billion," 1011, lit. A mantra ( Devanāgarī मन्त्र (or mantram is a religious or mystical syllable or poem typically from the Sanskrit language The Ashvamedha ( Sanskrit: sa अश्वमेध aśvamedhá; " Horse sacrifice " was one of the most important royal Rituals , "end"), hail to parārdha ("one trillion," 1012 lit. , "beyond parts"), hail to the dawn (uśas), hail to the twilight (vyuṣṭi), hail to the one which is going to rise (udeṣyat), hail to the one which is rising (udyat), hail to the one which has just risen (udita), hail to the heaven (svarga), hail to the world (loka), hail to all. "[20]

The Satapatha Brahmana (9th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras. The Shatapatha Brahmana (sa शतपथ ब्राह्मण śatapatha brāhmaṇa, " Brahmana of one-hundred paths" abbreviated ŚB [21]

Śulba Sūtras

Main article: Śulba Sūtras

The Śulba Sūtras (literally, "Aphorisms of the Chords" in Vedic Sanskrit) (c. The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the Vedic Sanskrit is an ancient Indian language, the language of the Vedas, the oldest Shruti texts of Hinduism. 700-400 BCE) list rules for the construction of sacrificial fire altars. [22] Most mathematical problems considered in the Śulba Sūtras spring from "a single theological requirement,"[23] that of constructing fire altars which have different shapes but occupy the same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks. [23]

According to (Hayashi 2005, p.  363), 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 (akṣṇayā-rajju) of an oblong (rectangle) produces both which the flank (pārśvamāni) and the horizontal (tiryaṇmānī) <ropes> produce separately. "[24]

Since the statement is a sūtra, it is necessarily compressed and what the ropes produce is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by the teacher to the student. [24]

They contain lists of Pythagorean triples,[25] which are particular cases of Diophantine equations. A Pythagorean triple consists of three positive Integers a, b, and c, such that a 2 +  b 2 =  In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only [26] They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square. Squaring the circle is a problem proposed by ancient Geometers. "[27]

Baudhayana (c. Baudhāyana, (fl ca 800 BCE was an Indian mathematician whowas most likely also a priest 8th century BCE) composed the Baudhayana Sulba Sutra, the best-known Sulba Sutra, which contains examples of simple Pythagorean triples, such as: (3,4,5), (5,12,13), (8,15,17), (7,24,25), and (12,35,37)[28] as well as a statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square. "[28] It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together. "[28] Baudhayana gives a formula for the square root of two,[29]

\sqrt{2} = 1 + \frac{1}{3} + \frac{1}{3\cdot4} - \frac{1}{3\cdot 4\cdot 34} \approx 1.4142156 \cdots

The formula is accurate up to five decimal places, the true value being 1.41421356 \cdots [30] This formula is similar in structure to the formula found on a Mesopotamian tablet[31] from the Old Babylonian period (1900-1600 BCE):[29]

\sqrt{2} = 1 + \frac{24}{60} + \frac{51}{60^2} + \frac{10}{60^3} = 1.41421297.

which expresses \sqrt{2} in the sexagesimal system, and which too is accurate up to 5 decimal places (after rounding). The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2}   or   √2

According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322 written ca. Of the approximately half million Babylonian Clay tablets excavated since the beginning of the 19th century several thousand are of a mathematical nature 1850 BCE[32] "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple,[33] indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BCE. "Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India. "[34] Dani goes on to say:

"As the main objective of the Sulvasutras was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the Sulvasutras. The occurrence of the triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily. "[34]

In all three Sulba Sutras were composed. The remaining two, the Manava Sulba Sutra composed by Manava (fl. Manava (c 750 BC - 690 BC) is the author of the Indian geometric text of Sulba Sutras. 750-650 BCE) and the Apastamba Sulba Sutra, composed by Apastamba (c. 600 BCE), contained results similar to the Baudhayana Sulba Sutra.

Vyakarana

An important landmark of the Vedic period was the work of Sanskrit grammarian, Pāṇini (c. The Sanskrit grammatical tradition of vyākaraṇa is one of the six Vedanga disciplines Pāṇini ( IAST: Pāṇini Dēvanāgarī: sa पाणिनि a Patronymic meaning "descendant of {{IAST|Paṇi}} " was an ancient 520-460 BCE). His grammar includes early use of Boolean logic, of the null operator, and of context free grammars, and includes a precursor of the Backus–Naur form (used in the description programming languages). Boolean logic is a complete system for Logical operations It was named after George Boole, who first defined an algebraic system of In Formal language theory, a context-free grammar ( CFG) is a grammar in which every production rule is of the form V &rarr In Computer science, Backus–Naur Form ( BNF) is a Metasyntax used to express Context-free grammars that is a formal way to describe Formal A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer.

Jaina Mathematics (400 BCE - 200 CE)

Although Jainism as a religion and philosophy predates its most famous exponent, Mahavira (6th century BC), who was a contemporary of Gautama Buddha, most Jaina texts on mathematical topics were composed after the 6th century BCE. Jainism, traditionally known as Jain Dharma / Shraman Dharma (जैन धर्म is an ancient religion of India. Mahavira (महावीर lit Great Hero) (599 – 527 BCE is the name most commonly used to refer to the Indian sage Vardhamana ( Sanskrit: वर्धमान The 6th century BC started the first day of 600 BC and ended the last day of 501 BC. Siddhārtha Gautama ( Sanskrit; Pali: Siddhattha Gotama) was a spiritual Teacher from Ancient India and the founder Jaina mathematicians are important historically as crucial links between the mathematics of the Vedic period and that of the "Classical period. Jainism, traditionally known as Jain Dharma / Shraman Dharma (जैन धर्म is an ancient religion of India. "

A significant historical contribution of Jaina mathematicians lay in their freeing Indian mathematics from its religious and ritualistic constraints. In particular, their fascination with the enumeration of very large numbers and infinities, led them to classify numbers into three classes: enumerable, innumerable and infinite. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness Not content with a simple notion of infinity, they went on to define five different types of infinity: the infinite in one direction, the infinite in two directions, the infinite in area, the infinite everywhere, and the infinite perpetually. In addition, Jaina mathematicians devised notations for simple powers (and exponents) of numbers like squares and cubes, which enabled them to define simple algebraic equations (beezganit samikaran). Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Jaina mathematicians were apparently also the first to use the word shunya (literally void in Sanskrit) to refer to zero. Sanskrit (sa संस्कृता वाक् saṃskṛtā vāk, for short sa संस्कृतम् saṃskṛtam) is a historical More than a millennium later, their appellation became the English word "zero" after a tortuous journey of translations and transliterations from India to Europe . (See Zero: Etymology. )

In addition to Surya Prajnapti, important Jaina works on mathematics included the Vaishali Ganit (c. 3rd century BCE); the Sthananga Sutra (fl. 300 BCE - 200 CE); the Anoyogdwar Sutra (fl. 200 BCE - 100 CE); and the Satkhandagama (c. 2nd century CE). Important Jaina mathematicians included Bhadrabahu (d. Acharya Bhadrabahu (433 BC - 357 BC ? was a Jain monk He is more famously known as a spiritual teacher of Chandragupta Maurya and author of several texts related 298 BCE), the author of two astronomical works, the Bhadrabahavi-Samhita and a commentary on the Surya Prajinapti; Yativrisham Acharya (c. 176 BCE), who authored a mathematical text called Tiloyapannati; and Umasvati (c. Acharya Umaswati is the author of Tattvartha Sutra, the best known Jaina text 150 BCE), who, although better known for his influential writings on Jaina philosophy and metaphysics, composed a mathematical work called Tattwarthadhigama-Sutra Bhashya. Metaphysics is the branch of Philosophy investigating principles of reality transcending those of any particular science

Pingala

Among other scholars of this period who contributed to mathematics, the most notable is Pingala (piṅgalá) (fl. 300-200 BCE), a musical theorist who authored the Chandas Shastra (chandaḥ-śāstra, also Chandas Sutra chandaḥ-sūtra), a Sanskrit treatise on prosody. Pingala ( पिङ्गल piṅgalá) was an ancient Indian writer famous for his work the Chandas Shastra ( chandaḥ-śāstra Music theory is the field of study that deals with the Mechanics of music and how Music works The main principle of Vedic meter is measurement by the number of syllables Śāstra (anglicized either shastra or sastra) is a Sanskrit word used to denote Education /knowledge in a general sense Sanskrit (sa संस्कृता वाक् saṃskṛtā vāk, for short sa संस्कृतम् saṃskṛtam) is a historical There is evidence that in his work on the enumeration of syllabic combinations, Pingala stumbled upon both the Pascal triangle and Binomial coefficients, although he did not have knowledge of the Binomial theorem itself. \begin{matrix}&&&&&1\\&&&&1&&1\\&&&1&&2&&1\\&&1&&3&&3&&1\\&1&&4&&6&&4&&1\end{matrix In Mathematics, the binomial coefficient \tbinom nk is the Coefficient of the x   k term in the Polynomial In Mathematics, the binomial theorem is an important Formula giving the expansion of powers of Sums Its simplest version says [35][36] Pingala's work also contains the basic ideas of Fibonacci numbers (called maatraameru). In Mathematics, the Fibonacci numbers are a Sequence of numbers named after Leonardo of Pisa, known as Fibonacci Although the Chandah sutra hasn't survived in its entirety, a 10th century commentary on it by Halāyudha has. Halāyudha, who refers to the Pascal triangle as Meru-prastāra (literally "the staircase to Mount Meru"), has this to say:

"Draw a square. For the Mountain in Tanzania, see Mount Meru (Tanzania. Mount Meru ( Sanskrit: मेरु (also called For the Mountain in Tanzania, see Mount Meru (Tanzania. Mount Meru ( Sanskrit: मेरु (also called Beginning at half the square, draw two other similar squares below it; below these two, three other squares, and so on. The marking should be started by putting 1 in the first square. Put 1 in each of the two squares of the second line. In the third line put 1 in the two squares at the ends and, in the middle square, the sum of the digits in the two squares lying above it. In the fourth line put 1 in the two squares at the ends. In the middle ones put the sum of the digits in the two squares above each. Proceed in this way. Of these lines, the second gives the combinations with one syllable, the third the combinations with two syllables, . . . "[35]

The text also indicates that Pingala was aware of the combinatorial identity:[36]

{n \choose 0} + {n \choose 1} + {n \choose 2} + \cdots + {n \choose n-1} + {n \choose n} = 2^n
Katyayana

Though not a Jaina mathematician, Katyayana (c. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects Kātyāyana (c 3rd century BC was a Sanskrit grammarian, mathematician and Vedic priest who lived in ancient India. 3rd century BCE) is notable for being the last of the Vedic mathematicians. He wrote the Katyayana Sulba Sutra, which presented much geometry, including the general Pythagorean theorem and a computation of the square root of 2 correct to five decimal places. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry

The Written Tradition: Prose Commentary

With the increasing complexity of mathematics and other exact sciences, both writing and computation were required. Consequently, many mathematical works began to be written down in manuscripts that were then copied and re-copied from generation to generation.

"India today is estimated to have about thirty million manuscripts, the largest body of handwritten reading material anywhere in the world. The literate culture of Indian science goes back to at least the fifth century B. C. . . . as is shown by the elements of Mesopotamian omen literature and astronomy that entered India at that time and (were) definitely not . . . preserved orally. "[37]

The earliest mathematical prose commentary was that on the work, Āryabhaṭīya (written 499 CE), a work on astronomy and mathematics. Āryabhatīya, an astronomical treatise is the Magnum opus and only extant work of the 5th century Indian mathematician Aryabhata. The mathematical portion of the Āryabhaṭīya was composed of 33 sūtras (in verse form) consisting of mathematical statements or rules, but without any proofs. [38] However, according to (Hayashi 2003, p.  123), "this does not necessarily mean that their authors did not prove them. It was probably a matter of style of exposition. " From the time of Bhaskara I (600 CE onwards), prose commentaries increasingly began to include some derivations (upapatti). Bhāskara (commonly called Bhāskara I to avoid confusion with the 12th century mathematician Bhāskara II) (c Bhaskara I's commentary on the Āryabhaṭīya, had the following structure:[38]

Typically, for any mathematical topic, students in ancient India first memorized the sūtras, which, as explained earlier, were "deliberately inadequate"[37] in explanatory details (in order to pithily convey the bare-bone mathematical rules). The students then worked through the topics of the prose commentary by writing (and drawing diagrams) on chalk- and dust-boards (i. e. boards covered with dust). The latter activity, a staple of mathematical work, was to later prompt mathematician-astronomer, Brahmagupta (fl. 7th century CE), to characterize astronomical computations as "dust work" (Sanskrit: dhulikarman). Brahmagupta ( (598–668 was an Indian mathematician and astronomer. [39]

Numerals and the Decimal Number System

The earliest extant script used in India was the Kharoṣṭhī script used in the Gandhara culture of the north-west. A writing system is a type of Symbolic system used to represent elements or statements expressible in Language. The Kharoṣṭhī script, also known as the Gāndhārī script, is an ancient Abugida (an alphasyllabary based on consonants with graphical variations to express Gandhāra ( Sanskrit: गन्धार Urdu: گندھارا Gandḥārā; also known as Waihind in Persian is the name of an ancient It is thought to be of Aramaic origin and it was in use from the fourth century BCE to the fourth century CE. Aramaic is a Semitic language with Almost contemporaneously, another script, the Brahmi, appeared on much of the sub-continent, and would later become the foundation of many scripts of South Asia and South-east Asia. Both scripts had numeral symbols and numeral systems, which were initially not based on a place-value system. [40] The first datable evidence of the use of the decimal place-value system in India is found in the Yavanajātaka (ca. 270 CE) of Sphujidhvaja, a versification of an earlier (ca. The Yavanajataka ( Sanskrit for "Saying ( Jataka) of the Greeks ( Yavanas)" is the earliest writing of Indian astrology Circa (often abbreviated c, ca, ca or cca and sometimes Italicized to show it is Latin) means "about" 150 CE) Indian prose adaptation of a lost work of Hellenistic astrology. [41]

Bakhshali Manuscript

The oldest extant mathematical manuscript in South Asia is the Bakhshali Manuscript, a birch bark manuscript written in "Buddhist hybrid Sanskrit"[10] in the Śāradā script, which was used in the northwestern region of the Indian subcontinent between the 8th and 12th centuries CE. The Bakhshali Manuscript is a mathematical manuscript written on birch bark which was found near the village of Bakhshali in 1881 in what was [42] The manuscript was discovered in 1881 by a farmer while digging in a stone enclosure in the village of Bakhshali, near Peshawar (then in British India and now in Pakistan). ( پښور; Urdu: پشاور) is the capital of the North-West Frontier Province and the administrative centre for the Federally Administered For usage see British rule in India British Raj ( rāj, lit "reign" in Hindustani) primarily refers to the British Pakistan () officially the Islamic Republic of Pakistan, is a country located in South Asia, Southwest Asia, Middle East and Of unknown authorship and now preserved in the Bodleian Library in Oxford University, the manuscript has been variously dated—as early as the "early centuries of the Christian era"[43] and as late as between the 9th and 12th century CE. The Bodleian Library ( the main Research library of the University of Oxford, is one of the oldest libraries in Europe, and in England The University of Oxford (informally "Oxford University" or simply "Oxford" located in the city of Oxford, Oxfordshire, England is the [44] The 7th century CE is now considered a plausible date,[45] albeit with the likelihood that the "manuscript in its present-day form constitutes a commentary or a copy of an anterior mathematical work. "[46]

The surviving manuscript has seventy leaves, some of which are in fragments. Its mathematical content consists of rules and examples, written in verse, together with prose commentaries, which include solutions to the examples. [42] The topics treated include arithmetic (fractions, square roots, profit and loss, simple interest, the rule of three, and regula falsi) and algebra (simultaneous linear equations and quadratic equations), and arithmetic progressions. In Numerical analysis, the false position method or regula falsi method is a Root-finding algorithm that combines features from the Bisection method In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. In addition, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero. "[42] Many of its problems are the so-called equalization problems that lead to systems of linear equations. One example from Fragment III-5-3v is the following:

"One merchant has seven asava horses, a second has nine haya horses, and a third has ten camels. They are equally well off in the value of their animals if each gives two animals, one to each of the others. Find the price of each animal and the total value for the animals possessed by each merchant. "[47]

The prose commentary accompanying the example solves the problem by converting it to three (under-determined) equations in four unknowns and assuming that the prices are all integers. [47]

Classical Period (400 - 1200)

This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, and Bhaskara II give broader and clearer shape to many branches of mathematics. Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics Daivajna Varāhamihira ( Devanagari: वराहमिहिर 505 &ndash 587 also called Varaha or Mihira was an Indian Astronomer, Mathematician Brahmagupta ( (598–668 was an Indian mathematician and astronomer. Bhāskara (commonly called Bhāskara I to avoid confusion with the 12th century mathematician Bhāskara II) (c Mahavira was a 9th century Indian Mathematician from Gulbarga who asserted that the Square root of a Negative number did Bhaskara (1114 &ndash 1185 also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher" was an Indian mathematician Their contributions would spread to Asia, the Middle East, and eventually to Europe. The Middle East is a Subcontinent with no clear boundaries often used as a synonym to Near East, in opposition to Far East. Unlike Vedic mathematics, their works included both astronomical and mathematical contributions. In fact, mathematics of that period was included in the 'astral science' (jyotiḥśāstra) and consisted of three sub-disciplines: mathematical sciences (gaṇita or tantra), horoscope astrology (horā or jātaka) and divination (saṃhitā). [48] This tripartite division is seen in Varāhamihira's sixth century compilation—Pancasiddhantika[49] (literally panca, "five," siddhānta, "conclusion of deliberation", dated 575 CE)—of five earlier works, Surya Siddhanta, Romaka Siddhanta, Paulisa Siddhanta, Vasishtha Siddhanta and Paitamaha Siddhanta, which were adaptations of still earlier works of Mesopotamian, Greek, Egyptian, Roman and Indian astronomy. Daivajna Varāhamihira ( Devanagari: वराहमिहिर 505 &ndash 587 also called Varaha or Mihira was an Indian Astronomer, Mathematician Events By Place Europe The Kingdom of East Anglia is founded by the Angle groups "North Folk" and "South The Surya Siddhanta is a treatise of Indian astronomy. Later Indian mathematicians and astronomers such as Aryabhata and Varahamihira The Romaka Siddhanta (literally "Doctrine of the Romans" is an Indian astronomical treatise based on the works of the ancient Romans. The Paulisa Siddhanta (literally "Doctrine of Paul" is an Indian astronomical treatise based on the works of the Western scholar Paul of Alexandria (c Vasishtha Siddhanta is one of the earliest astronomical systems in use in India which is summarized in Varahamihira 's Pancha-Siddhantika (6th century As explained earlier, the main texts were composed in Sanskrit verse, and were followed by prose commentaries. [48]

Fifth and Sixth Centuries

Surya Siddhanta

Though its authorship is unknown, the Surya Siddhanta (c. The Surya Siddhanta is a treatise of Indian astronomy. Later Indian mathematicians and astronomers such as Aryabhata and Varahamihira 400) contains the roots of modern trigonometry. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Some authors consider that its was written under influence of Mesopotamia and Greece. [50] But according Flavius Filostratus records Pythagoras in 5th century BC and Apollonius of Tyana in the 1st century CE went to study in India. [51]Furthermore there is no hard evidence to prove that Greek mathematicians had strong influence on Greek astronomy. [52] [53]

This ancient text uses the following as trigonometric functions for the first time:

It also contains the earliest uses of:

Later Indian mathematicians such as Aryabhata made references to this text, while later Arabic and Latin translations were very influential in Europe and the Middle East. Arabic (ar الْعَرَبيّة (informally ar عَرَبيْ) in terms of the number of speakers is the largest living member of the Semitic language Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome.

Chhedi calendar

This Chhedi calendar (594) contains an early use of the modern place-value Hindu-Arabic numeral system now used universally (see also Hindu-Arabic numerals). A positional notation or place-value notation system is a Numeral system in which each position is related to the next by a Constant multiplier a The Hindu-Arabic numeral system is a Positional Decimal Numeral system first documented in the ninth century The arabic numerals (often capitalized are the ten Digits (0 1 2 3 4 5 6 7 8 9 which—along with the system

Aryabhata I

Aryabhata (476-550) wrote the Aryabhatiya. Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics He described the important fundamental principles of mathematics in 332 shlokas. A Sanskrit term shloka (श्लोक also spelt sloka specifically denotes a metered and often rhymed poetic verse or phrase The treatise contained:

Aryabhata also wrote the Arya Siddhanta, which is now lost. Aryabhata's contributions include:

Trigonometry:

Arithmetic:

Algebra:

Mathematical astronomy:

Calculus:

Varahamihira

Varahamihira (505-587) produced the Pancha Siddhanta (The Five Astronomical Canons). Daivajna Varāhamihira ( Devanagari: वराहमिहिर 505 &ndash 587 also called Varaha or Mihira was an Indian Astronomer, Mathematician He made important contributions to trigonometry, including sine and cosine tables to 4 decimal places of accuracy and the following formulas relating sine and cosine functions:

Seventh and Eighth Centuries

Brahmagupta's theorem states that AF = FD.
Brahmagupta's theorem states that AF = FD. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.

In the seventh century, two separate fields, arithmetic (which included mensuration) and algebra, began to emerge in Indian mathematics. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone Measurement is the process of estimating the magnitude of some attribute of an object such as its length or weight relative to some standard ( unit of measurement) such as Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. The two fields would later be called pāṭī-gaṇita (literally "mathematics of algorithms") and bīja-gaṇita (lit. "mathematics of seeds," with "seeds"—like the seeds of plants—representing unknowns with the potential to generate, in this case, the solutions of equations). [56] Brahmagupta, in his astronomical work Brāhma Sphuṭa Siddhānta (628 CE), included two chapters (12 and 18) devoted to these fields. Brahmagupta ( (598–668 was an Indian mathematician and astronomer. The main work of Brahmagupta, Brahmasphuta-siddhanta (The Opening of the Universe, written in the year 628, contains some remarkably advanced ideas including Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). [57] In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral:[57]

Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are perpendicular to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side. In Geometry, a cyclic quadrilateral is a Quadrilateral whose vertices all lie on a single Circle. In Geometry, a cyclic quadrilateral is a Quadrilateral whose vertices all lie on a single Circle. In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent

Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula), as well as a complete description of rational triangles (i. In Geometry, Heron's (or Hero's formula states that the Area (A of a Triangle whose sides have lengths a, b, and In Geometry, a Heronian triangle is a Triangle whose sidelengths and area are all Rational numbers It is named after Hero of Alexandria e. triangles with rational sides and rational areas).

Brahmagupta's formula: The area, A, of a cyclic quadrilateral with sides of lengths a, b, c, d, respectively, is given by

A = \sqrt{(s-a)(s-b)(s-c)(s-d)}

where s, the semiperimeter, given by: s=\frac{a+b+c+d}{2}.

Brahmagupta's Theorem on rational triangles: A triangle with rational sides a,b,c and rational area is of the form:

a = \frac{u^2}{v}+v, \ \ b=\frac{u^2}{w}+w, \ \ c=\frac{u^2}{v}+\frac{u^2}{w} - (v+w)

for some rational numbers u,v, and w. In Geometry, a cyclic quadrilateral is a Quadrilateral whose vertices all lie on a single Circle. In Geometry, the semiperimeter of a polygon is half its Perimeter. In Geometry, a Heronian triangle is a Triangle whose sidelengths and area are all Rational numbers It is named after Hero of Alexandria [58]

Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers[57] and is considered the first systematic treatment of the subject. The rules (which included a + 0 = \ a and a \times 0 = 0 ) were all correct, with one exception: \frac{0}{0} = 0 . [57] Later in the chapter, he gave the first explicit (although still not completely general) solution of the quadratic equation:

\ ax^2+bx=c
“ To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. (Brahmasphutasiddhanta (Colebrook translation, 1817, page 346)[59] ”

This is equivalent to:

x = \frac{\sqrt{4ac+b^2}-b}{2a}

Also in chapter 18, Brahmagupta was able to make progress in finding (integral) solutions of Pell's equation,[60]

\ x^2-Ny^2=1,

where N is a nonsquare integer. Pell's equation is any Diophantine equation of the form x^2-ny^2=1\ where n is a nonsquare integer and x He did this by discovering the following identity:[60]

Brahmagupta's Identity: \ (x^2-Ny^2)(x'^2-Ny'^2) = (xx'+Nyy')^2 - N(xy'+x'y)^2 which was a generalization of an earlier identity of Diophantus:[60] Brahmagupta used his identity to prove the following lemma:[60]

Lemma (Brahmagupta): If x=x_1,\ \ y=y_1 \ \ is a solution of \ \ x^2 - Ny^2 = k_1, and, x=x_2, \ \ y=y_2 \ \ is a solution of \ \ x^2 - Ny^2 = k_2, , then:

x=x_1x_2+Ny_1y_2,\ \ y=x_1y_2+x_2y_1 \ \ is a solution of \ x^2-Ny^2=k_1k_2

He then used this lemma to both generate infinitely many (integral) solutions of Pell's equation, given one solution, and state the following theorem:

Theorem (Brahmagupta): If the equation \ x^2 - Ny^2 =k has an integer solution for any one of \ k=\pm 4, \pm 2, -1 then Pell's equation:

\ x^2 -Ny^2 = 1

also has an integer solution. Diophantus of Alexandria ( Greek: b between 200 and 214 d between 284 and 298 AD sometimes called "the father of Algebra " a title some claim should Pell's equation is any Diophantine equation of the form x^2-ny^2=1\ where n is a nonsquare integer and x [61]

Brahmagupta did not actually prove the theorem, but rather worked out examples using his method. The first example he presented was:[60]

Example (Brahmagupta): Find integers \ x,\ y\ such that:

\ x^2 - 92y^2=1

In his commentary, Brahmagupta added, "a person solving this problem within a year is a mathematician. "[60] The solution he provided was:

\ x=1151, \ y=120
Bhaskara I

Bhaskara I (c. Bhāskara (commonly called Bhāskara I to avoid confusion with the 12th century mathematician Bhāskara II) (c 600-680) expanded the work of Aryabhata in his books titled Mahabhaskariya, Aryabhattiya Bhashya and Laghu Bhaskariya. He produced:

Ninth to Twelfth Centuries

Virasena

Virasena (9th century) was a Jaina mathematician in the court of Rashtrakuta King Amoghavarsha of Manyakheta, Karnataka. Virasena was an 8th century mathematician in India who gave the derivation of the Volume of a Frustum by a sort of infinite procedure The Rashtrakuta Dynasty ( Sanskrit: राष्ट्रकूट rāṣṭrakūṭa, Kannada: ರಾಷ್ಟ್ರಕೂಟ was a royal Amoghavarsha I (ಅಮೋಘವರ್ಷ ನೃಪತುಂಗ (800–878 C WikipediaWikiProject Indian cities for details --> Manyakheta (modern Malkhed on the banks of Kagini River in Gulbarga district, Karnataka He wrote the Dhavala, a commentary on Jaina mathematics, which:

Virasena also gave:

Mahavira

Mahavira Acharya (c. Mahavira was a 9th century Indian Mathematician from Gulbarga who asserted that the Square root of a Negative number did 800-870) from Karnataka, the last of the notable Jaina mathematicians, lived in the 9th century and was patronised by the Rashtrakuta king Amoghavarsha. Karnataka (ಕರ್ನಾಟಕ pronounced) is a state in the southern part of India The 9th century is the period from 801 to 900 in accordance with the Julian calendar in the Christian / Common Era. The Rashtrakuta Dynasty ( Sanskrit: राष्ट्रकूट rāṣṭrakūṭa, Kannada: ರಾಷ್ಟ್ರಕೂಟ was a royal Amoghavarsha I (ಅಮೋಘವರ್ಷ ನೃಪತುಂಗ (800–878 C He wrote a book titled Ganit Saar Sangraha on numerical mathematics, and also wrote treatises about a wide range of mathematical topics. These include the mathematics of:

Mahavira also:

Shridhara

Shridhara (c. Sridhara (c 870 Bengal ? India &ndash c 930 India was an Indian Mathematician known for two treatises Trisatika (sometimes called the 870-930), who lived in Bengal, wrote the books titled Nav Shatika, Tri Shatika and Pati Ganita. Etymology and ethnology The exact origin of the word Bangla or Bengal is unknown though it is believed to be derived from the Dravidian-speaking tribe Bang He gave:

The Pati Ganita is a work on arithmetic and mensuration. Measurement is the process of estimating the magnitude of some attribute of an object such as its length or weight relative to some standard ( unit of measurement) such as It deals with various operations, including:

Manjula

Aryabhata's differential equations were elaborated in the 10th century by Manjula (also Munjala), who realised that the expression[55]

\ \sin w' - \sin w

could be approximately expressed as

\ (w' - w)\cos w

He understood the concept of differentiation after solving the differential equation that resulted from substituting this expression into Aryabhata's differential equation. [55]

Aryabhata II

Aryabhata II (c. Aryabhata II (c 920 &ndash c 1000) was an Indian Mathematician and Astronomer, and the author of the Maha-Siddhanta 920-1000) wrote a commentary on Shridhara, and an astronomical treatise Maha-Siddhanta. Maha-Siddhanta was a work created by Indian mathematician and astronomer Aryabhata II in the tenth century A The Maha-Siddhanta has 18 chapters, and discusses:

Shripati

Shripati Mishra (1019-1066) wrote the books Siddhanta Shekhara, a major work on astronomy in 19 chapters, and Ganit Tilaka, an incomplete arithmetical treatise in 125 verses based on a work by Shridhara. Sripati (1019-1066 was an Indian Astronomer and Mathematician, the author of Dhikotidakarana (written in 1039 a work of twenty verses Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone Sridhara (c 870 Bengal ? India &ndash c 930 India was an Indian Mathematician known for two treatises Trisatika (sometimes called the He worked mainly on:

He was also the author of Dhikotidakarana, a work of twenty verses on:

The Dhruvamanasa is a work of 105 verses on:

Nemichandra Siddhanta Chakravati

Nemichandra Siddhanta Chakravati (c. 1100) authored a mathematical treatise titled Gome-mat Saar.

Bhaskara II

Bhāskara II (1114-1185) was a mathematician-astronomer who wrote a number of important treatises, namely the Siddhanta Shiromani, Lilavati, Bijaganita, Gola Addhaya, Griha Ganitam and Karan Kautoohal. Bhaskara (1114 &ndash 1185 also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher" was an Indian mathematician Lilavati (also Leelavati) was Indian mathematician Bhāskara II 's treatise on Mathematics in the twelfth century A number of his contributions were later transmitted to the Middle East and Europe. His contributions include:

Arithmetic:

Algebra:

Geometry:

Calculus:

Trigonometry:

Kerala Mathematics (1300 - 1600)

Main article: Kerala school of astronomy and mathematics

The Kerala school of astronomy and mathematics was founded by Madhava of Sangamagrama in Kerala, South India and included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. Spherical trigonometry is a part of Spherical geometry that deals with Polygons (especially Triangles on the Sphere and explains how to find relations Mādhava of Sangamagrama (born as Irinjaatappilly Madhavan Namboodiri) (c Kerala ( Malayalam: {{Kerala in Malayalam}}; South India is the area encompassing India 's states of Andhra Pradesh, Karnataka, Kerala and Tamil Nadu as well as the union Vatasseri Parameshvara (വടശ്ശേരി പരമേശ്വരന്‍ (1360-1425 was a major Indian Mathematician of Madhava of Sangamagrama Nilakantha Somayaji ( Malayalam: നീലകണ്ഠ സോമയാജി hindi नीलकण्ठ सोमयाजि (1444-1544 from Kerala, was a major Jyestadeva (ജ്യേഷ്ഠദേവ(ന് (1500 &ndash 1575 was an astronomer of the Kerala school founded by Madhava of Sangamagrama and Thrikkandiyoor Achyuta Pisharati (1550 &ndash 1621 was a renowned Sanskrit grammarian astrologer, astronomer and mathematician of his Melpathur Narayana Bhattathiri ( Malayalam: മേല്‍പ്പതൂര്‍ നാരായണ ഭട്ടതിരി (1559-1632 third student of Achyuta It flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559-1632). Melpathur Narayana Bhattathiri ( Malayalam: മേല്‍പ്പതൂര്‍ നാരായണ ഭട്ടതിരി (1559-1632 third student of Achyuta In attempting to solve astronomical problems, the Kerala school astronomers independently created a number of important mathematics concepts. The most important results, series expansion for trigonometric functions, were given in Sanskrit verse in a book by Neelakanta called Tantrasangraha and a commentary on this work called Tantrasangraha-vakhya of unknown authorship. Sanskrit (sa संस्कृता वाक् saṃskṛtā vāk, for short sa संस्कृतम् saṃskṛtam) is a historical The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhasa (c. 1500-c. 1610), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha. Not to be confused with the Malay language. Malayalam (മലയാളം malayāḷaṁ) is a Dravidian language used [62]

Their discovery of these three important series expansions of calculus—several centuries before calculus was developed in Europe by Isaac Newton and Gottfried Leibniz—was a landmark achievement in mathematics. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements However, the Kerala School cannot be said to have invented calculus,[63] because, while they were able to develop Taylor series expansions for the important trigonometric functions, they developed neither a comprehensive theory of differentiation or integration, nor the fundamental theorem of calculus. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. [64] The results obtained by the Kerala school include:

r\arctan(\frac{y}{x}) = \frac{1}{1}\cdot\frac{ry}{x} -\frac{1}{3}\cdot\frac{ry^3}{x^3} + \frac{1}{5}\cdot\frac{ry^5}{x^5} - \cdots , where y/x \leq 1.
\sin x = x - x\cdot\frac{x^2}{(2^2+2)r^2} + x\cdot \frac{x^2}{(2^2+2)r^2}\cdot\frac{x^2}{(4^2+4)r^2} - \cdot
r - \cos x = r\cdot \frac{x^2}{(2^2-2)r^2} - r\cdot \frac{x^2}{(2^2-2)r^2}\cdot \frac{x^2}{(4^2-4)r^2} + \cdots , where, for r = 1, the series reduce to the standard power series for these trigonometric functions, for example:
\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots + \infty
\frac{\pi}{4} \approx 1 - \frac{1}{3}+ \frac{1}{5} - \cdots (-1)^{(n-1)/2}\frac{1}{n} + (-1)^{(n+1)/2}f_i(n+1)
where f_1(n) = \frac{1}{2n}, \ f_2(n) = \frac{n/2}{n^2+1}, \ f_3(n) = \frac{(n/2)^2+1}{(n^2+5)n/2}.
\frac{\pi}{4} = \frac{3}{4} + \frac{1}{3^3-3} - \frac{1}{5^3-5} + \frac{1}{7^3-7} - \cdots \infty

The works of the Kerala school were first written up for the Western world by Englishman C. M. Whish in 1835. According to Whish, the Kerala mathematicians had "laid the foundation for a complete system of fluxions" and these works abounded "with fluxional forms and series to be found in no work of foreign countries. "[67] However, Whish's results were almost completely neglected, until over a century later, when the discoveries of the Kerala school were investigated again by C. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series in Yuktibhasa given in two papers,[68][69] a commentary on the Yuktibhasa's proof of the sine and cosine series[70] and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary). Sanskrit (sa संस्कृता वाक् saṃskṛtā vāk, for short sa संस्कृतम् saṃskṛtam) is a historical [71][72]

The Kerala mathematicians included Narayana Pandit (c. Narayana Pandit (नारायण पण्डित (1340 &ndash 1400 was a major Mathematician of the Kerala school. 1340-1400), who composed two works, an arithmetical treatise, Ganita Kaumudi, and an algebraic treatise, Bijganita Vatamsa. Narayana is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavati, titled Karmapradipika (or Karma-Paddhati). Bhaskara (1114 &ndash 1185 also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher" was an Indian mathematician Lilavati (also Leelavati) was Indian mathematician Bhāskara II 's treatise on Mathematics in the twelfth century Madhava of Sangamagramma (c. Mādhava of Sangamagrama (born as Irinjaatappilly Madhavan Namboodiri) (c 1340-1425) was the founder of the Kerala School. Although it is possible that he wrote Karana Paddhati a work written sometime between 1375 and 1475, all we really know of his work comes from works of later scholars.

Parameshvara (c. Vatasseri Parameshvara (വടശ്ശേരി പരമേശ്വരന്‍ (1360-1425 was a major Indian Mathematician of Madhava of Sangamagrama 1370-1460) wrote commentaries on the works of Bhaskara I, Aryabhata and Bhaskara II. Bhāskara (commonly called Bhāskara I to avoid confusion with the 12th century mathematician Bhāskara II) (c Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics Bhaskara (1114 &ndash 1185 also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher" was an Indian mathematician His Lilavati Bhasya, a commentary on Bhaskara II's Lilavati, contains one of his important discoveries: a version of the mean value theorem. In Calculus, the mean value theorem states roughly that given a section of a smooth curve there is at least one point on that section at which the Derivative Nilakantha Somayaji (1444-1544) composed the Tantra Samgraha (which 'spawned' a later anonymous commentary Tantrasangraha-vyakhya and a further commentary by the name Yuktidipaika, written in 1501). Nilakantha Somayaji ( Malayalam: നീലകണ്ഠ സോമയാജി hindi नीलकण्ठ सोमयाजि (1444-1544 from Kerala, was a major He elaborated and extended the contributions of Madhava.

Citrabhanu (c. Citrabanu (fl 1530 was a mathematician from the Kerala school in the 16th century 1530) was a 16th century mathematician from Kerala who gave integer solutions to 21 types of systems of two simultaneous algebraic equations in two unknowns. In Mathematics simultaneous equations are a set of Equations containing multiple variables These types are all the possible pairs of equations of the following seven forms:

\ x + y = a, x - y = b, xy = c, x^2 + y^2 = d, x^2 - y^2 = e, x^3 + y^3 = f, x^3 - y^3 = g

For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric. Jyesthadeva (c. Jyestadeva (ജ്യേഷ്ഠദേവ(ന് (1500 &ndash 1575 was an astronomer of the Kerala school founded by Madhava of Sangamagrama and 1500-1575) was another member of the Kerala School. His key work was the Yukti-bhasa (written in Malayalam, a regional language of Kerala). Not to be confused with the Malay language. Malayalam (മലയാളം malayāḷaṁ) is a Dravidian language used Kerala ( Malayalam: {{Kerala in Malayalam}}; Jyesthadeva presented proofs of most mathematical theorems and infinite series earlier discovered by Madhava and other Kerala School mathematicians.

Charges of Eurocentrism

It has been suggested that Indian contributions to mathematics have not been given due acknowledgement in modern history and that many discoveries and inventions by Indian mathematicians are presently culturally attributed to their Western counterparts, as a result of Eurocentrism. The chronology of Indian mathematics spans from the Indus valley civilization and the Vedas to Modern times The term Western world, the West or the Occident ( Latin: occidens -sunset -west as distinct from the Orient) can have multiple meanings Eurocentrism is the practice of viewing the world from a European perspective with an implied belief either consciously or subconsciously in the preeminence of European (and According to G. G. Joseph:

[Their work] takes on board some of the objections raised about the classical Eurocentric trajectory. The awareness [of Indian and Arabic mathematics] is all too likely to be tempered with dismissive rejections of their importance compared to Greek mathematics. The contributions from other civilizations - most notably China and India, are perceived either as borrowers from Greek sources or having made only minor contributions to mainstream mathematical development. An openness to more recent research findings, especially in the case of Indian and Chinese mathematics, is sadly missing"[73]

The historian of mathematics, Florian Cajori, suggested that he "suspect[s] that Diophantus got his first glimpse of algebraic knowledge from India. Florian Cajori ( February 28 1859 in St Aignan (near Thusis Graubünden, Switzerland &mdash August 15, 1930, Berkeley Diophantus of Alexandria ( Greek: b between 200 and 214 d between 284 and 298 AD sometimes called "the father of Algebra " a title some claim should "[74]

More recently, as discussed in the above section, the infinite series of calculus for trigonometric functions (rediscovered by Gregory, Taylor, and Maclaurin in the late 17th century) were described (with proofs) in India, by mathematicians of the Kerala School, remarkably some two centuries earlier. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Some scholars have recently suggested that knowledge of these results might have been transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries. Kerala ( Malayalam: {{Kerala in Malayalam}}; The Society of Jesus ( Latin: Societas Iesu, SJ and SI or SJ, SI) is a Catholic religious order [75] Kerala was in continuous contact with China and Arabia, and, from around 1500, with Europe. China ( Wade-Giles ( Mandarin) Chung¹kuo² is a cultural region, an ancient Civilization, and depending on perspective a National The Arabian Peninsula (in Arabic: شبه الجزيرة العربية šibh al-jazīra al-ʻarabīya or جزيرة العرب jazīrat al-ʻarab) The existence of communication routes and a suitable chronology certainly make such a transmission a possibility. However, there is no direct evidence by way of relevant manuscripts that such a transmission actually took place. [75] Indeed, according to David Bressoud, "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century. "[63][76]

Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus. [64] However, they were not able to, as Newton and Leibniz were, to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space "[64] The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own;[64] however, it is not known with certainty whether the immediate predecessors of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources we are not now aware. "[64] This is an active area of current research, especially in the manuscripts collections of Spain and Maghreb, research that is now being pursued, among other places, at the Centre National de Recherche Scientifique in Paris. Spain () or the Kingdom of Spain (Reino de España is a country located mostly in southwestern Europe on the Iberian Peninsula. The Maghreb (المغرب العربي al-Maġrib al-ʿArabī) also rendered Maghrib (or rarely Moghreb) meaning "place of Sunset Paris (ˈpærɨs in English; in French) is the Capital of France and the country's largest city [64]

See also

Notes

  1. ^ a b (Encyclopaedia Britannica (Kim Plofker) 2007, p. The chronology of Indian mathematics spans from the Indus valley civilization and the Vedas to Modern times The development of Indian logic can be said to date back to the anviksiki of Medhatithi Gautama (c Indian astronomy —the earliest textual mention of which is given in the religious literature of India (2nd millennium BCE—became an established tradition by the 1st millennium BCE The area of study known as the history of mathematics is primarily an investigation into the origin of new discoveries in Mathematics and to a lesser extent an investigation In ancient Indian mathematics, a Vedic square is a special type of 9 × 9 Multiplication table.  1)
  2. ^ (Ifrah 2000, p.  346): "The measure of the genius of Indian civilisation, to which we owe our modern (number) system, is all the greater in that it was the only one in all history to have achieved this triumph. Some cultures succeeded, earlier than the Indian, in discovering one or at best two of the characteristics of this intellectual feat. But none of them managed to bring together into a complete and coherent system the necessary and sufficient conditions for a number-system with the same potential as our own. "
  3. ^ (Bourbaki 1998, p.  46): ". . . our decimal system, which (by the agency of the Arabs) is derived from Hindu mathematics, where its use is attested already from the first centuries of our era. It must be noted moreover that the conception of zero as a number and not as a simple symbol of separation) and its introduction into calculations, also count amongst the original contribution of the Hindus. "
  4. ^ (Bourbaki 1998, p.  49): "On this point, the Hindus are already conscious of the interpretation that negative numbers must have in certain cases (a debt in a commercial problem, for instance). In the following centuries, as there is a diffusion into the West (by intermediary of the Arabs) of the methods and results of Greek and Hindu mathematics, one becomes more used to the handling of these numbers, and one begins to have other "representation" for them which are geometric or dynamic. "
  5. ^ a b "algebra" 2007. Britannica Concise Encyclopedia. Encyclopædia Britannica Online. 16 May 2007. Quote: "A full-fledged decimal, positional system certainly existed in India by the 9th century (AD), yet many of its central ideas had been transmitted well before that time to China and the Islamic world. Indian arithmetic, moreover, developed consistent and correct rules for operating with positive and negative numbers and for treating zero like any other number, even in problematic contexts such as division. Several hundred years passed before European mathematicians fully integrated such ideas into the developing discipline of algebra. "
  6. ^ (Pingree 2003, p.  45) Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. In fact, the Indian astronomers in the third or fourth century, using a pre-Ptolemaic Greek table of chords, produced tables of sines and versines, from which it was trivial to derive cosines. This new system of trigonometry, produced in India, was transmitted to the Arabs in the late eighth century and by them, in an expanded form, to the Latin West and the Byzantine East in the twelfth century. "
  7. ^ (Bourbaki 1998, p.  126): "As for trigonometry, it is disdained by geometers and abandoned to surveyors and astronomers; it is these latter (Aristarchus, Hipparchus, Ptolemy) who establish the fundamental relations between the sides and angles of a right angled triangle (plane or spherical) and draw up the first tables (they consist of tables giving the chord of the arc cut out by an angle θ < π on a circle of radius r, in other words the number 2r\sin\left(\theta/2\right); the introduction of the sine, more easily handled, is due to Hindu mathematicians of the Middle Ages). Aristarchus (Ἀρίσταρχος 310 BC - ca 230 BC) was a Greek Astronomer and Mathematician, born on the island of Hipparchus ( Greek; ca 190 BC &ndash ca 120 BC was a Greek Astronomer, Geographer, and Mathematician of the Hellenistic Claudius Ptolemaeus ( Greek: Klaúdios Ptolemaîos; after 83 &ndash ca "
  8. ^ (Filliozat 2004, pp.  140-143)
  9. ^ (Hayashi 1995)
  10. ^ a b (Encyclopaedia Britannica (Kim Plofker) 2007, p.  6)
  11. ^ (Stillwell 2004, p.  173)
  12. ^ (Bressoud 2002, p.  12)
  13. ^ a b (Filliozat 2004, p.  137)
  14. ^ (Pingree 1988, p.  637)
  15. ^ (Staal 1986)
  16. ^ a b c (Filliozat 2004, p.  139)
  17. ^ a b c d e (Filliozat 2004, pp.  140-141)
  18. ^ (Yano 2006, p.  146)
  19. ^ a b c (Filliozat 2004, pp.  143-144)
  20. ^ a b c (Hayashi 2005, pp.  360-361)
  21. ^ A. Seidenberg, 1978. The origin of mathematics. Archive for the history of Exact Sciences, vol 18.
  22. ^ (Staal 1999)
  23. ^ a b (Hayashi 2003, p.  118)
  24. ^ a b (Hayashi 2005, p.  363)
  25. ^ Pythagorean triples are triples of integers (a,b,c) with the property: a2 + b2 = c2. Thus, 32 + 42 = 52, 82 + 152 = 172, 122 + 352 = 372 etc.
  26. ^ (Cooke 2005, p.  198): "The arithmetic content of the Śulva Sūtras consists of rules for finding Pythagorean triples such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 35, 37). It is not certain what practical use these arithmetic rules had. The best conjecture is that they were part of religious ritual. A Hindu home was required to have three fires burning at three different altars. The three altars were to be of different shapes, but all three were to have the same area. These conditions led to certain "Diophantine" problems, a particular case of which is the generation of Pythagorean triples, so as to make one square integer equal to the sum of two others. "
  27. ^ (Cooke 2005, pp.  199-200): "The requirement of three altars of equal areas but different shapes would explain the interest in transformation of areas. Among other transformation of area problems the Hindus considered in particular the problem of squaring the circle. The Bodhayana Sutra states the converse problem of constructing a circle equal to a given square. The following approximate construction is given as the solution. . . . this result is only approximate. The authors, however, made no distinction between the two results. In terms that we can appreciate, this construction gives a value for π of 18 (3 − 2√2), which is about 3. 088. "
  28. ^ a b c (Joseph 2000, p.  229)
  29. ^ a b (Cooke 2005, p.  200)
  30. ^ The value of this approximation, 577/408, is the seventh in a sequence of increasingly accurate approximations 3/2, 7/5, 17/12, . . . to √2, the numerators and denominators of which were known as "side and diameter numbers" to the ancient Greeks, and in modern mathematics are called the Pell numbers. In Mathematics, the Pell numbers and companion Pell numbers (Pell-Lucas numbers are both sequences of Integers that have been known since ancient If x/y is one term in this sequence of approximations, the next is (x+2y)/(x+y). These approximations may also be derived by truncating the continued fraction representation of √2. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}}
  31. ^ Neugebauer, O. and A. Sachs. 1945. Mathematical Cuneiform Texts, New Haven, CT, Yale University Press. p. 45.
  32. ^ Mathematics Department, University of British Columbia, The Babylonian tabled Plimpton 322.
  33. ^ Three positive integers (a,b,c) form a primitive Pythagorean triple if c2 = a2 + b2 and if the highest common factor of a,b,c is 1. In the particular Plimpton322 example, this means that 135002 + 127092 = 185412 and that the three numbers do not have any common factors. However some scholars have disputed the Pythagorean interpretation of this tablet; see Plimpton 322 for details. Of the approximately half million Babylonian Clay tablets excavated since the beginning of the 19th century several thousand are of a mathematical nature
  34. ^ a b (Dani 2003)
  35. ^ a b (Fowler 1996, p.  11)
  36. ^ a b (Singh 1936, pp.  623-624)
  37. ^ a b (Pingree 1988, p.  638)
  38. ^ a b c (Hayashi 2003, pp.  122-123)
  39. ^ (Hayashi 2003, p.  119)
  40. ^ (Hayashi 2005, p.  366)
  41. ^ (Pingree 1978, p.  494)
  42. ^ a b c (Hayashi 2005, p.  371)
  43. ^ (Datta 1931, p.  566)
  44. ^ (Ifrah 2000, p.  464) Quote: "To give the second or fourth century CE as the date of this document would be an evident contradiction; it would mean that a northern derivative of Gupta writing had been developed two or three centuries before the Gupta writing itself appeared. The Gupta Empire ( Hindi: गुप्त राजवंश was ruled by members of the Gupta dynasty from around 320 to 550 C Gupta only began to evolve into Shāradā style around the ninth century CE. In other words, the Bak(h)shali manuscript cannot have been written earlier than the ninth century CE. However, in the light of certain characteristic indications, it could not have been written any later than the twelfth century CE. "
  45. ^ (Hayashi 2005, p.  371) Quote:"The dates so far proposed for the Bakhshali work vary from the third to the twelfth centuries AD, but a recently made comparative study has shown many similarities, particularly in the style of exposition and terminology, between Bakhshalī work and Bhāskara I's commentary on the Āryabhatīya. This seems to indicate that both works belong to nearly the same period, although this does not deny the possibility that some of the rules and examples in the Bakhshālī work date from anterior periods. "
  46. ^ (Ifrah 2000, p.  464)
  47. ^ a b Anton, Howard and Chris Rorres. 2005. Elementary Linear Algebra with Applications. 9th edition. New York: John Wiley and Sons. 864 pages. ISBN 0471669598.
  48. ^ a b (Hayashi 2003, p.  119)
  49. ^ (Neugebauer & Pingree (eds.) 1970)
  50. ^ Cooke, Roger (1997). "The Mathematics of the Hindus", The History of Mathematics: A Brief Course. Wiley-Interscience, 197. ISBN 0471180823.  “The word Siddhanta means that which is proved or established. The Sulva Sutras are of Hindu origin, but the Siddhantas contain so many words of foreign origin that they undoubtedly have roots in Mesopotamia and Greece. ” 
  51. ^ (Betti 1996, p.  33)
  52. ^ (Phillimore 1912, p.  16)"We find ourselves here in an entirely new situation, because the influences of a later period have modified everything and given a vague and confused report of pre-history. This situation, especially tracing back to Ptolemy, does not offer any historical references, almost nothing is known about the astronomical knowledge of Hipparchus or Apollonius"
  53. ^ (Neugabauer 1975, p.  8)"We know that, from the Pahlavi translations of the astrological writings of the first and second century (in Persia) such as Teucer and Vettius Valens, and from the presence of Hindu texts as well as the Roman Almagest dated around 250 CE under king Shapur I. During the reign of Khosoro I…, it was revised, around 550 CE as the famous Zij ash-Shah, which has been demonstrated as heavily influenced by Hindu sources"
  54. ^ Victor J. Katz (1995). "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3), p. 163-174.
  55. ^ a b c d e Joseph (2000), p. 298-300.
  56. ^ (Hayashi 2005, p.  369)
  57. ^ a b c d (Hayashi 2003, pp.  121-122)
  58. ^ (Stillwell 2004, p.  77)
  59. ^ (Stillwell 2004, p.  87)
  60. ^ a b c d e f (Stillwell 2004, pp.  72-73)
  61. ^ (Stillwell 2004, pp.  74-76)
  62. ^ a b c d e f g h i (Roy 1990)
  63. ^ a b c (Bressoud 2002)
  64. ^ a b c d e f g (Katz 1995)
  65. ^ Singh, A. N. Singh. 1936. "On the Use of Series in Hindu Mathematics. " Osiris 1:606-628.
  66. ^ Edwards, C. H. , Jr. 1979. The Historical Development of the Calculus. New York: Springer-Verlag.
  67. ^ Charles Whish (1835). Transactions of the Royal Asiatic Society of Great Britain and Ireland.  
  68. ^ Rajagopal, C. and M. S. Rangachari. 1949. "A Neglected Chapter of Hindu Mathematics. " Scripta Mathematica. Scripta Mathematica was a quarterly journal published by Yeshiva University devoted to the philosophy history and expository treatment of mathematics 15:201-209.
  69. ^ Rajagopal, C. and M. S. Rangachari. 1951. "On the Hindu proof of Gregory's series. " Ibid. 17:65-74.
  70. ^ Rajagopal, C. and A. Venkataraman. 1949. "The sine and cosine power series in Hindu mathematics. " Journal of the Royal Asiatic Society of Bengal (Science). 15:1-13.
  71. ^ Rajagopal, C. and M. S. Rangachari. 1977. "On an untapped source of medieval Keralese mathematics. " Archive for the History of Exact Sciences. 18:89-102.
  72. ^ Rajagopal, C. and M. S. Rangachari. 1986. "On Medieval Kerala Mathematics. " Archive for the History of Exact Sciences. 35:91-99.
  73. ^ Joseph, G. G. 1997. "Foundations of Eurocentrism in Mathematics. " In Ethnomathematics: Challenging Eurocentrism in Mathematics Education (Eds. Powell, A. B. et al. ). SUNY Press. ISBN 0791433528. p. 67-68.
  74. ^ Cajori, Florian (1893). Florian Cajori ( February 28 1859 in St Aignan (near Thusis Graubünden, Switzerland &mdash August 15, 1930, Berkeley "The Hindoos", A History of Mathematics P 86 (in English). Macmillan & Co. .  “"In algebra, there was probably a mutual giving and receiving [between Greece and India]. We suspect that Diophantus got his first glimpse of algebraic knowledge from India"” 
  75. ^ a b Almeida, D. F. , J. K. John, and A. Zadorozhnyy. 2001. "Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications. " Journal of Natural Geometry, 20:77-104.
  76. ^ Gold, D. and D. Pingree. 1991. "A hitherto unknown Sanskrit work concerning Madhava's derivation of the power series for sine and cosine. " Historia Scientiarum. 42:49-65.

Source Books in Sanskrit

References

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