FACTOID # 22: South Dakota has the highest employment ratio in America, but the lowest median earnings of full-time male employees.
 
 Home   Encyclopedia   Statistics   States A-Z   Flags   Maps   FAQ   About 
 
WHAT'S NEW
 

SEARCH ALL

FACTS & STATISTICS    Advanced view

Search encyclopedia, statistics and forums:

 

 

(* = Graphable)

 

 


Encyclopedia > Indian mathematics
History of science
Background
Theories/sociology
Historiography
Pseudoscience
By era
In early cultures
in Classical Antiquity
In the Middle Ages
In the Renaissance
Scientific Revolution
By topic
Natural sciences
Astronomy
Biology
Chemistry
Ecology
Geography
Geology
Physics
Social sciences
Economics
Linguistics
Political science
Psychology
Sociology
Technology
Agricultural science
Computer science
Materials science
Medicine
Navigational pages
Timelines
Portal
Categories

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). In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. Indian mathematicians made early contributions to the study of the decimal number system,[2] zero,[3] negative numbers,[4] arithmetic, and algebra.[5] In addition, trigonometry, having evolved in the Hellenistic world and 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.[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. Science is a body of empirical and theoretical knowledge, produced by a global community of researchers, making use of specific techniques for the observation and explanation of real phenomena, this techne summed up under the banner of scientific method. ... Image File history File links Download high resolution version (1020x1508, 359 KB) Book cover Frontispiece of : Tabulae Rudolphinae : quibus astronomicae . ... 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 large-scale patterns and trends in the development of science, and asking questions about how science works both in a philosophical and practical sense. ... The historiography of science is the historical study of the history of science (which often overlaps the history of technology, the history of medicine, and the history of mathematics). ... A pseudoscience is any body of knowledge purported to be scientific or supported by science but which fails to comply with the scientific method. ... In prehistoric times, advice and knowledge was passed from generation to generation in an oral tradition. ... The Ptolemaic system of celestial motion, from Harmonia Macrocosmica, 1661. ... The history of science in the Middle Ages refers to the discoveries in the field of natural philosophy throughout the Middle Ages - the middle period in a traditional schematic division of European history. ... Leonardo da Vincis Vitruvian Man, an example of the blend of art and science during the Renaissance. ... The event which most historians of science call the scientific revolution can be dated roughly as having begun in 1543, the year in which Nicolaus Copernicus published his De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres) and Andreas Vesalius published his De humani corporis fabrica (On the... Natural philosophy or the philosophy of nature, known in Latin as philosophia naturalis, is a term applied to the objective study of nature and the physical universe before the development of modern science. ... Astronomy is the oldest of the natural sciences, dating back to antiquity, with its origins in the religious, mythological, and astrological practices of pre-history: vestiges of these are still found in astrology, a discipline long interwoven with public and governmental astronomy, and not completely disentangled from it until a... The history of biology dates as far back as the rise of various civilization as classic philosophers did their own ways of biology as a system of understanding life. ... Portrait of Monsieur Lavoisier and his Wife, by Jacques-Louis David The history of chemistry may be said to begin with the distinction of chemistry from alchemy by Robert Boyle in his work The Sceptical Chymist (1661). ... ÛEcology is generally spoken of as a new science, having only become prominent in the second half of the 20th Century. ... Wikipedia does not yet have an article with this exact name. ... The known history of physics is thought to have begun around 2400 BC, when members of the Harappan civilization used shell objects to serve as compasses for measuring the angles of the sky. ... For more, see: Social science#History In ancient philosophy, there was no difference between the liberal arts of mathematics and the study of history, poetry or politics—only with the development of mathematical proof did there gradually arise a perceived difference between scientific disciplines and others, the humanities or liberal... The term economics was coined around 1870 and popularized by Alfred Marshall, as a substitute for the earlier term political economy which has been used through the 18th-19th centuries, with Adam Smith, David Ricardo and Karl Marx as its main thinkers and which today is frequently referred to as... Efforts to describe and explain the human language faculty have been undertaken throughout recorded history. ... Antecedents of political science While the study of politics is first found in the Western tradition in Ancient Greece, 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, in Europe, back to the Late Middle Ages. ... Sociology is a relatively new academic discipline among other social sciences including economics, political science, anthropology, and psychology. ... The wheel was invented circa 4000 BC, and has become one of the worlds most famous, and most useful technologies. ... Agronomy today is very different from what it was 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 rooted in the history of the Earth and the culture of the peoples of the Earth. ... This article does not adequately cite its references or sources. ... Chronologies or timelines are important in understanding history. ... Image File history File links Trafficcones2. ... Shortcut: WP:CSD Current list: Category:Candidates for speedy deletion There are a few, limited, cases where admins can delete Wikipedia pages on sight. Non-admins can ask for an admin to delete such a page, either by listing it on speedy deletions, or by adding either a {{delete}} or... Map of South Asia (see note on Kashmir). ... The Bronze Age is a period in a civilizations development when the most advanced metalworking has developed the techniques of smelting copper from natural outcroppings and alloys it to cast bronze. ... Excavated ruins of Mohenjo-daro. ... BCE is a TLA that may stand for: Before the Common Era, date notation equivalent to BC (e. ... Iron Age Axe found on Gotland This article is about the archaeological period known as the Iron Age, for the mythological Iron Age see Iron Age (mythology). ... The Vedic Civilization is the Indo-Aryan culture associated with the Vedas. ... ... (Redirected from 1200 CE) Events University of Paris receives charter from Philip II of France Births Matthew Paris, English Benedictine monk and chronicler (approximate date). ... Statue of Aryabhata on the grounds of IUCAA, Pune. ... Brahmagupta (ब्रह्मगुप्त) (598-668) was an Indian mathematician and astronomer. ... Bhāskara (1114-1185), also called Bhāskara II and Bhāskarācārya (Bhaskara the teacher) was an Indian mathematician. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ... For other uses, see zero or 0. ... A negative number is a number that is less than zero, such as −3. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... Wikibooks has a book on the topic of Trigonometry Trigonometry (from Greek trigōnon triangle + metron measure[1]) is a branch of mathematics that deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). ... The term Hellenistic (derived from HéllÄ“n, the Greeks traditional self-described ethnic name) was established by the German historian Johann Gustav Droysen to refer to the spreading of Greek culture over the non-Greek people that were conquered by Alexander the Great. ... Ancient India may refer to: the ancient History of India, which generally includes the ancient history of the whole Indian subcontinent the legendary Kingdoms of Ancient India in Sanskrit literature the Iron Age Mahajanapadas the Middle kingdoms of India of Antiquity and the Early Middle Ages Category: ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... A map showing countries commonly considered to be part of the Middle East The Middle East is a region comprising the lands around the southern and eastern parts of the Mediterranean Sea, a territory that extends from the eastern Mediterranean Sea to the Persian Gulf. ... World map showing the location of Europe. ...


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. 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); the manuscript is likely from the seventh century CE.[9][10] The Sanskrit language ( , for short ) is a classical language of India, a liturgical language of Hinduism, Buddhism, Sikhism, and Jainism, and one of the 23 official languages of India. ... Sūtra (sex) (Sanskrit) or Sutta (Pāli) literally means a rope or thread that holds things together, and more metaphorically refers to an aphorism (or line, rule, formula), or a collection of such aphorisms in the form of a manual. ... Map of South Asia (see note) This article deals with the geophysical region in Asia. ... The Bakhshali Manuscript is a mathematical manuscript written on birch bark which was found near the village of Bakhshali in what is now Pakistan in 1881. ... Peshāwar (Urdu: پشاور; Pashto: پښور) literally means City on the Frontier in Persian and is known as Pekhawar in Pashto. ...


A later landmark in Indian mathematics was the development of the series expansion for trigonometric functions (sine, cosine, and arc tangent) by mathematicians of the Kerala School in the fifteenth century CE. Their remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series).[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.[12] (See Charges of Eurocentrism below for recent research in this area.) In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. ... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... For a non-technical overview of the subject, see Calculus. ... In calculus, the integral of a function is an extension of the concept of a sum. ... , Kerala ( ; Malayalam: കേരളം; ) is a state on the Malabar Coast of southwestern India. ...

Contents

Fields of Indian mathematics

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

Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus... A negative number is a number that is less than zero, such as −3. ... Brahmagupta (ब्रह्मगुप्त) (598-668) was an Indian mathematician and astronomer. ... For other uses, see zero or 0. ... The Hindu-Arabic numeral system (also called Algorism) is a positional decimal numeral system documented from the 9th 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 common ratio, called the base or radix of that numeral system. ... A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... 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 of problems that arise from their study. ... The infinity symbol ∞ in several typefaces. ... The Yajur Veda यजुर्वेद is one of the four Hindu Vedas; it contains religious texts focussing on liturgy and ritual. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ... 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 m/n, where m and n are integers. ... The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... This article presents and explains several methods which can be used to calculate square roots. ... Plot of y = In mathematics, the cube root of a number, denoted or x1/3, is the number a such that a3 = x. ... Mahavira was a 10th century Indian mathematician from Gulbarga who asserted that the square root of a negative number did not exist. ... The Pythagorean theorem: a2 + b2 = c2 A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. ... The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ... Baudhāyana, (fl. ... Apastamba (c. ... In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry among the three sides of a right triangle. ... In mathematics, a transformation in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. ... Indian postage stamp depicting (2004), with the implication that he used (पाणिनि; IPA ) was an ancient Indian grammarian from Gandhara (traditionally 520–460 BC, but estimates range from the 7th to 4th centuries BC). ... 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 The first six rows of Pascals triangle In mathematics, Pascals triangle is a geometric arrangement of the binomial coefficients in a triangle. ... Pingala (पिङ्गल ) is the supposed author of the Chandas shastra (, also Chandas sutra ), a Sanskrit treatise on prosody considered one of the Vedanga. ... 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 Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ... Statue of Aryabhata on the grounds of IUCAA, Pune. ... Brahmagupta (ब्रह्मगुप्त) (598-668) was an Indian mathematician and astronomer. ... Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ... Mahavira was a 10th century Indian mathematician from Gulbarga who asserted that the square root of a negative number did not exist. ... Bhaskara (1114-1185), also known as Bhaskara II and Bhaskara Achārya (Bhaskara the teacher), was an Indian mathematician-astronomer. ... In mathematics, a quartic equation is the result of setting a quartic function equal to zero. ... Mahavira was a 10th century Indian mathematician from Gulbarga who asserted that the square root of a negative number did not exist. ... Bhaskara (1114-1185), also known as Bhaskara II and Bhaskara Achārya (Bhaskara the teacher), was an Indian mathematician-astronomer. ... Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ... In computer science and linguistics, a formal grammar, or sometimes simply grammar, is a precise description of a formal language — that is, of a set of strings. ... In mathematics, logic, and computer science, a formal language is a set of finite-length words (i. ... The Backus-Naur form (BNF) (also known as Backus normal form) is a metasyntax used to express context-free grammars: that is, a formal way to describe formal languages. ... Indian postage stamp depicting (2004), with the implication that he used (पाणिनि; IPA ) was an ancient Indian grammarian from Gandhara (traditionally 520–460 BC, but estimates range from the 7th to 4th centuries BC). ... A visual form of recursion known as the Droste effect. ... Indian postage stamp depicting (2004), with the implication that he used (पाणिनि; IPA ) was an ancient Indian grammarian from Gandhara (traditionally 520–460 BC, but estimates range from the 7th to 4th centuries BC). ... A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above – see golden spiral. ... Pingala (पिङ्गल ) is the supposed author of the Chandas shastra (, also Chandas sutra ), a Sanskrit treatise on prosody considered one of the Vedanga. ... 1922 Chart of the Morse Code Letters and Numerals Morse code is a method for transmitting telegraphic information, using standardized sequences of short and long elements to represent the letters, numerals, punctuation and special characters of a message. ... Pingala (पिङ्गल ) is the supposed author of the Chandas shastra (, also Chandas sutra ), a Sanskrit treatise on prosody considered one of the Vedanga. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... In mathematics, an index is a superscript or subscript to a symbol. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... Algorism comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in which numbers written using ten symbols having the values 0 through 9 are combined using a place-value system (positional notation), where each symbol has ten times the weight of the one... Statue of Aryabhata on the grounds of IUCAA, Pune. ... Brahmagupta (ब्रह्मगुप्त) (598-668) was an Indian mathematician and astronomer. ... Wikibooks has a book on the topic of Trigonometry Trigonometry (from Greek trigōnon triangle + metron measure[1]) is a branch of mathematics that deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). ... All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other... This article aims at providing a thorough (but not verse by verse) exposition of most important topics of and problems related to Surya Siddhanta and its comparison with ancient and modern astronomy, together with its use in astrology. ... Statue of Aryabhata on the grounds of IUCAA, Pune. ... In mathematics, a Fourier series of a periodic function, named in honor of Joseph Fourier (1768-1830), represents the function as a sum of periodic functions of the form where e is Eulers number and i the imaginary unit. ... Madhavan (മാധവന്) of Sangamagramam (1350–1425) was a prominent mathematician-astronomer from Kerala, India. ... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ...

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. Excavations at Harappa, Mohenjo-daro and other locations in the Indus river valley have uncovered evidence of the use of practical mathematics. 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. Excavated ruins of Mohenjo-daro. ... Map of South Asia (see note on Kashmir). ... Excavated ruins of Mohenjo-daro. ... Harappa (Urdu: ہڑپا) is a city in Punjab, northeast Pakistan, about 35km (22 miles) southwest of Sahiwal. ... Mohenjo-daro (literally, mound of the dead), like Harappa, was a city of the Indus Valley civilization. ... Satellite image of the Indus River basin. ... Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ... A hexahedron is a polyhedron with 6 faces. ... Traditional wooden barrels in Cutchogue Modern stainless steel beer barrels—also called casks or kegs—outside the Castle Rock microbrewery in Nottingham, England For other uses, see Barrel (disambiguation). ... This article is about the geometric object, for other uses see Cone. ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ...


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) 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)."[13] Memorization of "what is heard" (śruti in Sanskrit) through recitation played a major role in the transmission of sacred texts in ancient India. 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] The Sanskrit language ( , for short ) is a classical language of India, a liturgical language of Hinduism, Buddhism, Sikhism, and Jainism, and one of the 23 official languages of India. ... A pandit or pundit(पन्दित् in Devanagari) is a Hindu Brahmin who has memorized a substantial portion of the Vedas, along with the proper rhythms and melodies for chanting or singing them. ... The Sanskrit grammatical tradition of , 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 scripture, such as of the Old and New Testaments of the Bible, the Talmud, the Midrash, the Quran, etc. ... The main objective of the Purva (earlier) Mimamsa school was to establish the authority of the Vedas. ... Nyaya (pronounced as nyα:yÉ™) is the name given to one of the six orthodox or astika schools of Hindu philosophy - specifically the school of logic. ... The Å›ruti (Sanskrit thing heard, sound) is the smallest interval of the tuning system of Indian classical music. ...

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. 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: The Vedas are part of the Hindu Shruti; these religious scriptures form part of the core of the Brahminical and Vedic traditions within Hinduism and are the inspirational, metaphysical and mythological foundation for later Vedanta, Yoga, Tantra and even Bhakti forms of Hinduism. ...

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.[16] Similar methods were used for memorizing mathematical texts, whose transmission remained exclusively oral until the end of the Vedic period (ca. 500 BCE). The Rigveda (Sanskrit: , a tatpurusha compound of praise, verse and knowledge) is a collection of Vedic Sanskrit hymns dedicated to the gods. ... Look up Circa on Wiktionary, the free dictionary The Latin word circa, literally meaning about, is often used to describe various dates (often birth and death dates) that are uncertain. ... The Vedic period (or Vedic Age) is the period in the history of India when the sacred Vedic Sanskrit texts such as the Vedas were composed. ...

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).[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.[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 Vedas are part of the Hindu Shruti; these religious scriptures form part of the core of the Brahminical and Vedic traditions within Hinduism and are the inspirational, metaphysical and mythological foundation for later Vedanta, Yoga, Tantra and even Bhakti forms of Hinduism. ... The Vedanga (IAST , member of the Veda) are six auxiliary disciplines for the understanding and tradition of the Vedas. ... Shiksha is an NGO devoted to improving the standards of education in New Delhi and its neighbouring regions. ... Phonetics (from the Greek word φωνή, phone meaning sound, voice) is the study of the sounds of human speech. ... The verses of the Vedas have a variety of different meters. ... In literature, meter or metre (sometimes known as prosody) is a term used in the scansion (analysis into metrical patterns) of poetry, usually indicated by the kind of feet and the number of them. ... The Sanskrit grammatical tradition of , is one of the six Vedanga disciplines. ... For the surname, see Grammer. ... Nirukta is Vedic glossary of difficult words. ... Not to be confused with Entomology, the study of insects. ... Kalevan Pallo is a professional Finnish ice hockey team. ... A ritual is a set of actions, performed mainly for their symbolic value, which is prescribed by a religion or by the traditions of a community. ... Jyotisha (, in Hindi and English usage Jyotish; sometimes called Hindu astrology, Indian astrology, and/or Vedic astrology) is the Hindu system of astrology, one of the six disciplines of Vedanga, and regarded as one of the oldest schools of ancient astrology to have had an independent origin, affecting all other... A giant Hubble mosaic of the Crab Nebula, a supernova remnant Astronomy (also frequently referred to as astrophysics) is the scientific study of celestial objects (such as stars, planets, comets, and galaxies) and phenomena that originate outside the Earths atmosphere (such as the cosmic background radiation). ... SÅ«tra (sex) (Sanskrit) or Sutta (Pāli) literally means a rope or thread that holds things together, and more metaphorically refers to an aphorism (or line, rule, formula), or a collection of such aphorisms in the form of a manual. ...

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.[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.[18] The brevity achieved in a sūtra is demonstrated in the following example from the Baudhāyana Śulba Sūtra (700 BCE). Distinguish from ellipse. ... The guru-shishya tradition (also guru-shishya parampara or lineage, or teacher-disciple relationship) is a spiritual relationship found within traditional Hinduism which is centered around the transmission of teachings from a guru (teacher, ) to a (disciple, ). The term shishya roughly equates to the western term disciple, and in some...

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. 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: Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... The Vedic period (or Vedic Age) is the period in the history of India when the sacred Vedic Sanskrit texts such as the Vedas were composed. ...

"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. By the time of the last Veda, the Yajurvedasaṃhitā (1200-900 BCE), numbers as high as 1012 were being included in the texts.[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] The Vedanga (IAST , member of the Veda) are six auxiliary disciplines for the understanding and tradition of the Vedas. ... The Vedas (Sanskrit: वेद) are a large corpus of texts originating in Ancient India. ... The Vedic period (or Vedic Age) is the period in the history of India when the sacred Vedic Sanskrit texts such as the Vedas were composed. ... Different cultures used different traditional numeral systems for naming large numbers. ... The Yajurveda (Sanskrit , a tatpurusha compound of sacrifice + veda knowledge) is one of the four Hindu Vedas. ... In Tibet, many Buddhists carve mantras into rocks as a form of devotion. ... The Ashvamedha (Sanskrit horse sacrifice) was one of the most important royal rituals of Vedic religion, described in detail in the Yajurveda (YV TS 7. ...

"Hail to śata ("hundred," 102), hail to sahasra ("thousand," 103), hail to ayuta ("ten thousand," 104), hail to niyuta ("hundred thousand," 104), 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., "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.[21] Shatapatha Brahmana (Brahmana of one-hundred paths) is one of the prose texts describing the Vedic ritual. ...

Śulba Sūtras

The Śulba Sūtras (literally, "Aphorisms of the Chords" in Vedic Sanskrit) (c. 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] The Shulba Sutras (Sanskrit : string, cord, rope) are sutra texts belonging to the Åšrauta ritual and containing geometry related to altar construction, including the problem of squaring the circle. ... Vedic Sanskrit is the language of the Vedas, which are the earliest sacred texts of India,. The Vedas were first passed down orally and therefore have no known date. ...


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 term Old Babylonian is a period in Mesopotamian history that refers, roughly, to the period between the end of the Third Dynasty of Ur (c. ...

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] and there is some evidence that they consider simple Diophantine equations.[26] They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square."[27] The Pythagorean theorem: a2 + b2 = c2 A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. ... In mathematics, a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. ... This square and circle have the same area. ...


Baudhayana (c. 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] Baudhāyana, (fl. ... The square root of two is the positive real number which, when multiplied by itself, gives a product of two. ...

sqrt{2} = 1 + frac{1}{3} + frac{1}{3cdot4} - frac{1}{3cdot 4cdot 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] BCE is a TLA that may stand for: Before the Common Era, date notation equivalent to BC (e. ...

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).


According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322 written ca. 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: Of the approximately half million clay tablets excavated at the beginning of the 19th century, about 400 are of a mathematical nature. ... BCE is a TLA that may stand for: Before the Common Era, date notation equivalent to BC (e. ... BCE is a TLA that may stand for: Before the Common Era, date notation equivalent to BC (e. ...

"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 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. 750-650 BCE) and the Apastamba Sulba Sutra, composed by Apastamba (c. 600 BCE), contained results similar to the Baudhayana Sulba Sutra. Manava (c. ... Apastamba (c. ...

Vyakarana

An important landmark of the Vedic period was the work of Sanskrit grammarian, Pāṇini (c. 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). The Sanskrit grammatical tradition of is one of the six Vedanga disciplines. ... Indian postage stamp depicting (2004), with the implication that he used (पाणिनि; IPA ) was an ancient Indian grammarian from Gandhara (traditionally 520–460 BC, but estimates range from the 7th to 4th centuries BC). ... Boolean logic is a complete system for logical operations. ... KK Null, a Japanese musician Null, a special value in computer programming. ... In linguistics and computer science, a context-free grammar (CFG) is a formal grammar in which every production rule is of the form V &#8594; w where V is a non-terminal symbol and w is a string consisting of terminals and/or non-terminals. ... The Backus–Naur form (also known as BNF, the Backus–Naur formalism, Backus normal form, or Panini–Backus Form) is a metasyntax used to express context-free grammars: that is, a formal way to describe formal languages. ... Other listings of programming languages are: Categorical list of programming languages Generational list of programming languages Chronological list of programming languages Note: Esoteric programming languages have been moved to the separate List of esoteric programming languages. ...


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 topcs were composed after the 6th century BCE. Jaina mathematicians are important historically as crucial links between the mathematics of the Vedic period and that of the "Classical period." Jain and Jaina redirect here. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... (2nd millennium BC - 1st millennium BC - 1st millennium) The 6th century BC started on January 1, 600 BC and ended on December 31, 501 BC. // Monument 1, an Olmec colossal head at La Venta The 5th and 6th centuries BC were a time of empires, but more importantly, a time... Standing Buddha sculpture, ancient region of Gandhara, northern Pakistan, 1st century CE, Musée Guimet. ... JAIN is an activity within the Java Community Process, developing APIs for the creation of telephony (voice and data) services. ...


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. 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). Jaina mathematicians were apparently also the first to use the word shunya (literally void in Sanskrit) to refer to zero. 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.) The infinity symbol ∞ in several typefaces. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... The Sanskrit language ( , for short ) is a classical language of India, a liturgical language of Hinduism, Buddhism, and Jainism, and one of the 23 official languages of India. ... For other uses, see zero or 0. ...


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. 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. 150 BCE), who, although better known for his influential writings on Jaina philosophy and metaphysics, composed a mathematical work called Tattwarthadhigama-Sutra Bhashya. Vaishali is one of the districts of Bihar state, India. ... Bhadrabahu was a Jain saint. ... Acharya Umasvati is the author of Tatvartha Sutra, the best known Jain text. ... Plato (Left) and Aristotle (right), by Raphael (Stanza della Segnatura, Rome) Metaphysics is the branch of philosophy concerned with explaining the ultimate nature of reality, being, and the world. ...

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. 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.[35][36] Pingala's work also contains the basic ideas of Fibonacci numbers (called maatraameru). 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: Pingala (पिङ्गल ) is the supposed author of the Chandas shastra (, also Chandas sutra ), a Sanskrit treatise on prosody considered one of the Vedanga. ... Floruit (or fl. ... Music theory is a set of systems for analyzing, classifying, and composing music and the elements of music. ... The verses of the Vedas have a variety of different meters. ... Shastra is a Sanskrit word used (to be pronoucned (shaastra) to denote education/knowledge in a general sense. ... The Sanskrit language ( , for short ) is a classical language of India, a liturgical language of Hinduism, Buddhism, Sikhism, and Jainism, and one of the 23 official languages of India. ... Prosody may mean several things: Prosody consists of distinctive variations of stress, tone, and timing in spoken language. ... 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 The first six rows of Pascals triangle In mathematics, Pascals triangle is a geometric arrangement of the binomial coefficients in a triangle. ... In mathematics, in particular in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here m! denotes the factorial of m). ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ... A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above – see golden spiral. ... For the mountain in Tanzania, see Mount Meru, Tanzania. ... For the mountain in Tanzania, see Mount Meru, Tanzania. ...

"Draw a square. 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] Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ...

 {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. 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. Kātyāyana (c. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry among the three sides of a right triangle. ...


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. 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). Bhaskara I's commentary on the Āryabhaṭīya, had the following structure:[38] Ä€ryabhatÄ«ya, an astronomical treastise, is the Magnum Opus and only extant work of the 5th century Indian Mathematician, Aryabhatta. ... Bhāskara, or Bhāskara I, (c. ...

  • Rule ('sūtra') in verse by Āryabhaṭa
  • Commentary by Bhāskara I, consisting of:
    • Elucidation of rule (derivations were still rare then, but became more common later)
    • Example (uddeśaka) usually in verse.
    • Setting (nyāsa/sthāpanā) of the numerical data.
    • Working (karana) of the solution.
    • Verification (pratyayakaraṇa, literally "to make conviction") of the answer. These became rare by the 13th century, derivations or proofs being favored by then.[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).[39] Statue of Aryabhata on the grounds of IUCAA, Pune. ... Bhāskara, or Bhāskara I, (c. ... Brahmagupta (ब्रह्मगुप्त) (598-668) was an Indian mathematician and astronomer. ... Floruit (or fl. ...


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. It is thought to be of Aramaic origin and it was in use from the fourth century BCE to the fourth century CE. 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. 150 CE) Indian prose adaptation of a lost work of Hellenistic astrology.[41] Writing systems of the world today. ... The KharoṣṭhÄ« script, also known as the GāndhārÄ« script, is an ancient alphabetic script used by the Gandhara culture of historic northwest India to write the Gandhari and Sanskrit languages (the Gandhara kingdom was located along the present-day border between Afghanistan and Pakistan between the Indus... Gandhāra (Sanskrit: गन्धार, Persian; Gandara, Waihind) (Urdu: گندھارا) is the name of an ancient Indian Mahajanapada, currently in northern Pakistan (the North-West Frontier Province and parts of northern Punjab and Kashmir) and eastern Afghanistan. ... Aramaic is a Semitic language with a four-thousand year history. ... BrāhmÄ« refers to the pre-modern members of the Brahmic family of scripts, attested from the 3rd century BC. The best known and earliest dated inscriptions in Brahmi are the rock-cut edicts of Ashoka. ... The Yavanajataka (Sanskrit for Saying (Jataka) of the Greeks (Yavanas)) is the earliest writing of Indian astrology. ... Look up Circa on Wiktionary, the free dictionary The Latin word circa, literally meaning about, is often used to describe various dates (often birth and death dates) that are uncertain. ...


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.[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). 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.[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 Bakhshali Manuscript is a mathematical manuscript written on birch bark which was found near the village of Bakhshali in what is now Pakistan in 1881. ... Peshāwar (Urdu: پشاور; Pashto: پښور) literally means City on the Frontier in Persian and is known as Pekhawar in Pashto. ... Anthem God Save The Queen/King British India, circa 1860 Capital Calcutta (1858-1912), New Delhi (1912-1947) Language(s) Hindi, Urdu, English and many others Government Monarchy Emperor of India  - 1877-1901 Victoria  - 1901-1910 Edward VII  - 1910-1936 George V  - January-December 1936 Edward VIII  - 1936-1947 George... Entrance to the Library, with the coats-of-arms of several Oxford colleges The Bodleian Library, the main research library of the University of Oxford, is one of the oldest libraries in Europe, and in England is second in size only to the British Library. ... The University of Oxford, located in the city of Oxford in England, is the oldest university in the English-speaking world. ...


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 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: The rule of three (or threefold law) is an important tenet in Wicca. ... In numerical analysis, the false position method or regula falsi method is a root-finding algorithm that combines features from the bisection method and the secant method. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

"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. Their contributions would spread to Asia, the Middle East, and eventually to Europe. 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. As explained earlier, the main texts were composed in Sanskrit verse, and were followed by prose commentaries.[48] Statue of Aryabhata on the grounds of IUCAA, Pune. ... Varahamihira (505 – 587) was an Indian astronomer, mathematician, and astrologer born in Ujjain. ... Brahmagupta (ब्रह्मगुप्त) (598-668) was an Indian mathematician and astronomer. ... Bhāskara, or Bhāskara I, (c. ... Mahavira was a 10th century Indian mathematician from Gulbarga who asserted that the square root of a negative number did not exist. ... Bhāskara (1114-1185), also called Bhāskara II and Bhāskarācārya (Bhaskara the teacher) was an Indian mathematician. ... World map showing the location of Asia. ... A map showing countries commonly considered to be part of the Middle East The Middle East is a region comprising the lands around the southern and eastern parts of the Mediterranean Sea, a territory that extends from the eastern Mediterranean Sea to the Persian Gulf. ... World map showing the location of Europe. ... Varahamihira (505 – 587) was an Indian astronomer, mathematician, and astrologer born in Ujjain. ... Events June 2 - Benedict succeeds John III as Pope The Kingdom of East Anglia founded by the Angle groups North Folk and South Folk, naming the places of Norfolk and Suffolk, respectively. ... “Era Vulgaris” redirects here. ... This article aims at providing a thorough (but not verse by verse) exposition of most important topics of and problems related to Surya Siddhanta and its comparison with ancient and modern astronomy, together with its use in astrology. ... 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 Varahamihiras Pancha-Siddhantika. ... Paitamaha Siddhanta is one of the earliest astronomical systems in use in India, which is summarized in Varahamihiras Pancha-Siddhantika. ...


Fifth and Sixth Centuries

Surya Siddhanta

Though its authorship is unknown, the Surya Siddhanta (c. 400) contains the roots of modern trigonometry. Due to the large number of foreign words in the documents, Historians have concluded that its roots are in Mesopotamia and Greece.[50] It uses the following as trigonometric functions for the first time: This article aims at providing a thorough (but not verse by verse) exposition of most important topics of and problems related to Surya Siddhanta and its comparison with ancient and modern astronomy, together with its use in astrology. ... Wikibooks has a book on the topic of Trigonometry Trigonometry (from Greek trigōnon triangle + metron measure[1]) is a branch of mathematics that deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). ...

It also contains the earliest uses of: In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

  • The Hindu cosmological time cycles explained in the text, which was copied from an earlier work, gives:
    • The average length of the sidereal year as 365.2563627 days, which is only 1.4 seconds longer than the modern value of 365.2563627 days.
    • The average length of the tropical year as 365.2421756 days, which is only 2 seconds shorter than the modern value of 365.2421988 days.

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. In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ... Secant is a term in mathematics. ... The sidereal year is the time for the Sun to return to the same position in respect to the stars of the celestial sphere. ... A tropical year is the length of time that the Sun, as viewed from the Earth, takes to return to the same position along the ecliptic (its path among the stars on the celestial sphere). ... Arabic can mean: From or related to Arabia From or related to the Arabs The Arabic language; see also Arabic grammar The Arabic alphabet, used for expressing the languages of Arabic, Persian, Malay ( Jawi), Kurdish, Panjabi, Pashto, Sindhi and Urdu, among others. ... Latin is an ancient Indo-European language originally spoken in Latium, the region immediately surrounding 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). The place value system is a method of writing numbers with a base 10 numerical system. ... The Hindu-Arabic numeral system (also called Algorism) is a positional decimal numeral system documented from the 9th century. ... Numerals sans-serif Arabic numerals, known formally as Hindu-Arabic numerals, and also as Indian numerals, Hindu numerals, Western Arabic numerals, European numerals, or Western numerals, are the most common symbolic representation of numbers around the world. ...

Aryabhata I

Aryabhata (476-550) wrote the Aryabhatiya. He described the important fundamental principles of mathematics in 332 shlokas. The treatise contained: Statue of Aryabhata on the grounds of IUCAA, Pune. ... Shloka is a verse, phrase, proverb or hymn of praise, usually composed in a specified meter. ...

Aryabhata also wrote the Arya Siddhanta, which is now lost. Aryabhata's contributions include: In mathematics, a quadratic equation is a polynomial equation of the second degree. ... Wikibooks has a book on the topic of Trigonometry Trigonometry (from Greek trigōnon triangle + metron measure[1]) is a branch of mathematics that deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). ... Lower-case pi The mathematical constant π is a real number which may be defined as the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ...


Trigonometry:

  • Introduced the trigonometric functions.
  • Defined the sine (jya) as the modern relationship between half an angle and half a chord.
  • Defined the cosine (kojya).
  • Defined the versine (ukramajya).
  • Defined the inverse sine (otkram jya).
  • Gave methods of calculating their approximate numerical values.
  • Contains the earliest tables of sine, cosine and versine values, in 3.75° intervals from 0° to 90°, to 4 decimal places of accuracy.
  • Contains the trigonometric formula sin (n + 1) x - sin nx = sin nx - sin (n - 1) x - (1/225)sin nx.
  • Spherical trigonometry.

Arithmetic: All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other... The versed sine, also called the versine and, in Latin, the sinus versus (flipped sine) or the sagitta (arrow), is a trigonometric function versin(&#952;) (sometimes further abbreviated vers) defined by the equation: versin(&#952;) = 1 &#8722; cos(&#952;) = 2 sin2(&#952; / 2) There are also three corresponding functions: the... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ...

Algebra: In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

  • Solutions of simultaneous quadratic equations.
  • Whole number solutions of linear equations by a method equivalent to the modern method.
  • General solution of the indeterminate linear equation .

Mathematical astronomy: A linear equation in algebra is an equation which is constructed by equating two linear functions. ...

  • Proposed for the first time, a heliocentric solar system with the planets spinning on their axes and following an elliptical orbit around the Sun.
  • Accurate calculations for astronomical constants, such as the:

Calculus: In astronomy, heliocentrism is the theory that the Sun is at the center of the Universe and/or the Solar System. ... Major features of the Solar System (not to scale; from left to right): Pluto, Neptune, Uranus, Saturn, Jupiter, the asteroid belt, the Sun, Mercury, Venus, Earth and its Moon, and Mars. ... The axis of rotation of a rotating body is a line such that the distance between any point on the line and any point of the body remains constant under the rotation. ... For other uses, see Ellipse (disambiguation). ... Photo taken during the 1999 eclipse. ... Time lapse movie of the 3 March 2007 lunar eclipse A lunar eclipse occurs whenever the Moon passes through some portion of the Earths shadow. ... y=x³, for integer values of 1≤x≤25. ...

  • Infinitesimals:
    • In the course of developing a precise mapping of the lunar eclipse, Aryabhatta was obliged to introduce the concept of infinitesimals (tatkalika gati) to designate the near instantaneous motion of the moon.[52]
  • Differential equations:
    • He expressed the near instantaneous motion of the moon in the form of a basic differential equation.[52]
  • Exponential function:
Varahamihira

Varahamihira (505-587) produced the Pancha Siddhanta (The Five Astronomical Canons). 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: In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ... A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... The exponential function is one of the most important functions in mathematics. ... The exponential function is one of the most important functions in mathematics. ... Varahamihira (505 – 587) was an Indian astronomer, mathematician, and astrologer born in Ujjain. ... Wikibooks has a book on the topic of Trigonometry Trigonometry (from Greek trigōnon triangle + metron measure[1]) is a branch of mathematics that deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

  • sin2(x) + cos2(x) = 1
  • sin(x)=cosleft(frac{pi}{2}-xright)
  • frac{1-cos(2x)}{2}=sin^2(x)

Seventh and Eighth Centuries

Brahmagupta's theorem states that AF = FD.
Brahmagupta's theorem states that AF = FD.

In the seventh century, two separate fields, arithmetic (which included mensuration) and algebra, began to emerge in Indian mathematics. 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).[53] Brahmagupta, in his astronomical work Brāhma Sphuṭa Siddhānta (628 CE), included two chapters (12 and 18) devoted to these fields. 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).[54] In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral:[54] Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... Measurement is the determination of the size or magnitude of something. ... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... Brahmagupta (ब्रह्मगुप्त) (598-668) was an Indian mathematician and astronomer. ... The main work of Brahmagupta, Brahmasphutasiddhanta (The Opening of the Universe), written in 628, contains some remarkably advanced ideas, including a good understanding of the mathematical role of zero, rules for manipulating both positive and negative numbers, a method for computing square roots, methods of solving linear and some quadratic... In geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. ...


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. ... Fig. ...


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.e. triangles with rational sides and rational areas). A triangle with sides a, b, and c. ... In geometry, a Heronian triangle is a triangle whose sidelengths and area are all rational numbers. ...


Brahmagupta's formula: The area, A, of a cyclic quadrilateral with sides of lengths a, b, c, d, respectively, is given by In geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. ...

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

where s, the semiperimeter, given by:  s=frac{a+b+c+d}{2}. The semiperimeter of a mathematical shape is defined as half of the shapes perimeter. ...


Brahmagupta's Theorem on rational triangles: A triangle with rational sides a,b,c and rational area is of the form: In geometry, a Heronian triangle is a triangle whose sidelengths and area are all rational numbers. ...

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.[55]


Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers[54] 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 .[54] Later in the chapter, he gave the first explicit (although still not completely general) solution of the quadratic equation: In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

 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. (Brahmasphutasiddhanta (Colebrook translation, 1817, page 346)[56]

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,[57] Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ...

 x^2-Ny^2=1,

where N is a nonsquare integer. He did this by discovering the following identity:[57]


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:[57] Brahmagupta used his identity to prove the following lemma:[57] Title page of the 1621 edition of Diophantus Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac. ...


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: Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ...

  x^2 -Ny^2 = 1

also has an integer solution.[58]


Brahmagupta did not actually prove the theorem, but rather worked out examples using his method. The first example he presented was:[57]


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."[57] The solution he provided was:

 x=1151,  y=120
Bhaskara I

Bhaskara I (c. 600-680) expanded the work of Aryabhata in his books titled Mahabhaskariya, Aryabhattiya Bhashya and Laghu Bhaskariya. He produced: Bhāskara, or Bhāskara I, (c. ...

  • Solutions of indeterminate equations.
  • A rational approximation of the sine function.
  • A formula for calculating the sine of an acute angle without the use of a table, correct to 2 decimal places.

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

Ninth to Twelfth Centuries

Virasena

Virasena (9th century) was a Jaina mathematician in the court of Rashtrakuta King Amoghavarsha of Manyakheta, Karnataka. He wrote the Dhavala, a commentary on Jaina mathematics, which: Virasena was a 9th century Indian mathematician who gave derivation of the volume of a frustrum by a sort of infinite procedure. ... Jain cave in Ellora The Rastrakutas (Sanskrit/Maharashtri Prakrit [1]/Marathi[2][3]:राष्ट्रकूट, Kannada: ರಾಷ್ಟ್ರಕೂಟ) were a dynasty which ruled the southern and the central parts or the Deccan, India during the 8th - 10th century. ... Amoghavarsha Nripathunga was the greatest of the Rashtrakuta kings. ... Modern Malkheda in Karnataka, once tha capital of Rashtrakutas ...

  • Deals with logarithms to base 2 (ardhaccheda) and describes its laws.
  • First uses logarithms to base 3 (trakacheda) and base 4 (caturthacheda).

Virasena also gave:

  • The derivation of the volume of a frustum by a sort of infinite procedure.
Mahavira

Mahavira Acharya (c. 800-870) from Karnataka, the last of the notable Jaina mathematicians, lived in the 9th century and was patronised by the Rashtrakuta king Amoghavarsha. 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: The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ... A frustum is the portion of a solid â€“ normally a cone or pyramid â€“ which lies between two parallel planes cutting the solid. ... Mahavira was a 10th century Indian mathematician from Gulbarga who asserted that the square root of a negative number did not exist. ... Karnātakā   (Kannada: ಕನಾ೯ಟಕ) (IPA: ) is one of the four southern states of India. ... As a means of recording the passage of time the 9th century was that century that lasted from 801 to 900. ... Jain cave in Ellora The Rastrakutas (Sanskrit/Maharashtri Prakrit [1]/Marathi[2][3]:राष्ट्रकूट, Kannada: ರಾಷ್ಟ್ರಕೂಟ) were a dynasty which ruled the southern and the central parts or the Deccan, India during the 8th - 10th century. ... Amoghavarsha Nripathunga was the greatest of the Rashtrakuta kings. ...

Mahavira also: For other uses, see zero or 0. ... y=x², for all integer values of 1≤x≤25. ... In arithmetic and algebra, the cube of a number n is its third power &#8212; the result of multiplying it by itself two times: n3 = n × n × n. ... In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... Plot of y = In mathematics, the cube root of a number, denoted or x1/3, is the number a such that a3 = x. ... In mathematics, a series is often represented as the sum of a sequence of terms. ... In mathematics, plane geometry may mean: geometry of the Euclidean plane; or sometimes geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others; or geometry of the hyperbolic plane or two-dimensional spherical geometry. ... In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. ... This article is about the optical phenomenon; for other meanings, see Shadow (disambiguation). ... For other uses, see Ellipse (disambiguation). ... In geometry, a quadrilateral is a polygon with four sides or edges and four vertices or corners. ... Circle illustration This article is about the shape and mathematical concept of circle. ...

Shridhara

Shridhara (c. 870-930), who lived in Bengal, wrote the books titled Nav Shatika, Tri Shatika and Pati Ganita. He gave: In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... A negative number is a number that is less than zero, such as &#8722;3. ... In mathematics, a series is often represented as the sum of a sequence of terms. ... y=x², for all integer values of 1≤x≤25. ... In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. ... Area is a physical quantity expressing the size of a part of a surface. ... The perimeter is the distance around a given two-dimensional object. ... For other uses, see Ellipse (disambiguation). ... Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... Shridhara (Hindi: श्रीधर) was an 8th century Indian mathematician who gave a good rule for finding the volume of a sphere, and also the formula for solving quadratic equations. ... Bengal (Bengali: বঙ্গ Bôngo, বাংলা Bangla, বঙ্গদেশ Bôngodesh or বাংলাদেশ Bangladesh), is a historical and geographical region in the northeast of South Asia. ...

The Pati Ganita is a work on arithmetic and mensuration. It deals with various operations, including: The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ... A sphere is a perfectly symmetrical geometrical object. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... Measurement is the determination of the size or magnitude of something. ...

  • Elementary operations
  • Extracting square and cube roots.
  • Fractions.
  • Eight rules given for operations involving zero.
  • Methods of summation of different arithmetic and geometric series, which were to become standard references in later works.
Manjula

Aryabhata's differential equations were elaborated in the 10th century by Manjula (also Munjala), who realised that the expression[52] For evaluation of sums in closed form see evaluating sums. ...


 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.[52]

Aryabhata II

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

  • Numerical mathematics (Ank Ganit).
  • Algebra.
  • Solutions of indeterminate equations (kuttaka).
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. He worked mainly on: Sripati (1019-1066) was an Indian astronomer and mathematician and author of Dhikotidakarana written in 1039, a work of twenty verses on solar and lunar eclipses; Dhruvamanasa written in 1056, a work of 105 verses on calculating planetary longitudes, eclipses and planetary transits; Siddhantasekhara a major work on astronomy in... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... Shridhara (Hindi: श्रीधर) was an 8th century Indian mathematician who gave a good rule for finding the volume of a sphere, and also the formula for solving quadratic equations. ...

He was also the author of Dhikotidakarana, a work of twenty verses on: It has been suggested that this article or section be merged into Combination. ...

The Dhruvamanasa is a work of 105 verses on: Photo taken during the 1999 eclipse. ... Time lapse movie of the 3 March 2007 lunar eclipse A lunar eclipse occurs whenever the Moon passes through some portion of the Earths shadow. ...

Nemichandra Siddhanta Chakravati

Nemichandra Siddhanta Chakravati (c. 1100) authored a mathematical treatise titled Gome-mat Saar. Longitude, sometimes denoted by the Greek letter λ (lambda),[1][2] describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. ... “Total eclipse” redirects here. ... The word transit, when used alone, has several possible meanings in English means of transport, including mass transit, rapid transit, public transit, public transport Further information: transit (transportation) in astronomy an event involving two bodies along the same line of sight Further information: astronomical transit in navigational position lines when...

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. A number of his contributions were later transmitted to the Middle East and Europe. His contributions include: Bhaskara (1114-1185), also known as Bhaskara II and Bhaskara Achārya (Bhaskara the teacher), was an Indian mathematician-astronomer. ... Wikipedia does not yet have an article with this exact name. ...


Arithmetic:

Algebra: In mathematics, plane geometry may mean: geometry of the Euclidean plane; or sometimes geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others; or geometry of the hyperbolic plane or two-dimensional spherical geometry. ... In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. ... The cantilever spar of this cable-stay bridge, the Sundial Bridge at Turtle Bay, forms the gnomon of a large garden sundial The gnomon is the part of a sundial that casts the shadow. ... In combinatorial mathematics, a combination of members of a set is a subset. ... For other uses, see zero or 0. ... The infinity symbol ∞ in several typefaces. ...

  • The recognition of a positive number having two square roots.
  • Surds.
  • Operations with products of several unknowns.
  • The solutions of:
    • Quadratic equations.
    • Cubic equations.
    • Quartic equations.
    • Equations with more than one unknown.
    • Quadratic equations with more than one unknown.
    • The general form of Pell's equation using the chakravala method.
    • The general indeterminate quadratic equation using the chakravala method.
    • Indeterminate cubic equations.
    • Indeterminate quartic equations.
    • Indeterminate higher-order polynomial equations.

Geometry: In phonetics, surd is an older (and now rarely-used) alternate name for a voiceless consonant. ... Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ... The Chakravala method is a cyclic algorithm to solve quadratic integer equations. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

Calculus: In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry among the three sides of a right triangle. ...

Trigonometry: Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... For a non-technical overview of the subject, see Calculus. ... A differential can mean one of several things: Differential (mathematics) Differential (mechanics) Differential signaling is used to carry high speed digital signals. ... Differentiation can mean the following: In biology: cellular differentiation; evolutionary differentiation; In mathematics: see: derivative In cosmogony: planetary differentiation Differentiation (geology); Differentiation (logic); Differentiation (marketing). ... In calculus, Rolles theorem states that if a function f is continuous on a closed interval and differentiable on the open interval , and then there is some number c in the open interval such that . Intuitively, this means that if a smooth curve is equal at two points then... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ... Lower-case pi The mathematical constant π is a real number which may be defined as the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ...

Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ...

Kerala Mathematics (1300 - 1600)

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. It flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559-1632). 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. 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.[59] The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... Madhavan (മാധവന്) of Sangamagramam (1350–1425) was a prominent mathematician-astronomer from Kerala, India. ... , Kerala ( ; Malayalam: കേരളം; ) is a state on the Malabar Coast of southwestern India. ... South India is a linguistic-cultural region of India that comprises the four states of Andhra Pradesh, Karnataka, Kerala and Tamil Nadu and the two Union Territories of Lakshadweep and Puducherry, whose inhabitants are collectively referred to as South Indians. ... Parameshvara (&#2346;&#2352;&#2350;&#2375;&#2358;&#2381;&#2357;&#2352;) (1360-1425) was a major mathematician of the Kerala school. ... Neelakanta Somayaji (1444-1545) was a member of the Kerala school of astronomy and mathematics and was a student of Damodara. ... Jyestadeva (1500-1610), was an astronomer of the Kerala school founded by Madhava of Sangamagrama and a student of Damodara. ... Achyuta Pisharati (1550–1621) was a renowned Sanskrit grammarian, astrologer and mathematician of his time. ... Melpathur Narayana Bhattathiri (1559-1632), third student of Achyuta Pisharati, was a member of Madhava of Sangamagramas school of Astronomy and Mathematics. ... This 14th-century statue from south India depicts the gods Shiva (on the left) and Uma (on the right). ... (15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ... Melpathur Narayana Bhattathiri (1559-1632), third student of Achyuta Pisharati, was a member of Madhava of Sangamagramas school of Astronomy and Mathematics. ... January 15 - Elizabeth I of England is crowned in Westminster Abbey. ... See also: 1632 (novel) Events February 22 - Galileos Dialogue Concerning the Two Chief World Systems is published July 23 - 300 colonists for New France depart Dieppe November 8 - Wladyslaw IV Waza elected king of the Polish-Lithuanian Commonwealth after Zygmunt III Waza death November 16 - Battle of Lützen... All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other... The Sanskrit language ( , for short ) is a classical language of India, a liturgical language of Hinduism, Buddhism, Sikhism, and Jainism, and one of the 23 official languages of India. ... Malayalam (മലയാളം ) is the language spoken predominantly in the state of Kerala, in southern India. ...


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. However, the Kerala School cannot be said to have invented calculus,[60] 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.[61] The results obtained by the Kerala school include: Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... Sir Isaac Newton, (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist, regarded by many as the greatest figure in the history of science. ... It has been suggested that this article be split into multiple articles. ... For a non-technical overview of the subject, see Calculus. ... In calculus, the integral of a function is an extension of the concept of a sum. ... The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse operations: if a continuous function is first integrated and then differentiated, the original function is retrieved. ...

  • The (infinite) geometric series:  frac{1}{1-x} = 1 + x + x^2 + x^3 + dots + infty for | x | < 1[62] This formula was already known, for example, in the work of the 10th century Arab mathematician Alhazen (the Latinized form of the name Ibn Al-Haytham (965-1039)).[63]
  • A semi-rigorous proof (see "induction" remark below) of the result: 1^p+ 2^p + cdots + n^p approx frac{n^{p+1}}{p+1} for large n. This result was also known to Alhazen.[59]
  • Intuitive use of mathematical induction, however, the inductive hypothesis was not formulated or employed in proofs.[59]
  • Applications of ideas from (what was to become) differential and integral calculus to obtain (Taylor-Maclaurin) infinite series for sinx, cosx, and arctanx[60] The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:[59]
rarctan(frac{y}{x}) = frac{1}{1}cdotfrac{ry}{x} -frac{1}{3}cdotfrac{ry^3}{x^3} + frac{1}{5}cdotfrac{ry^5}{x^t} - cdots , where y/x leq 1.
sin x = x - xcdotfrac{x^2}{(2^2+2)r^2} + xcdot frac{x^2}{(2^2+2)r^2}cdotfrac{x^2}{(4^2+4)r^2} - cdot
 r - cos x = rcdot frac{x^2}{(2^2-2)r^2} - rcdot 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:
    • sin x = x - frac{x^3}{3!} + frac{x^5}{5!} - frac{x^7}{7!} + cdots and
    • cos x = 1 - frac{x^2}{2!} + frac{x^4}{4!} - frac{x^6}{6!} + cdots
  • Use of rectification (computation of length) of the arc of a circle to give a proof of these results. (The later method of Leibniz, using quadrature (i.e. computation of area under the arc of the circle, was not used.)[59]
  • Use of series expansion of arctanx to obtain an infinite series expression (later known as Gregory series) for π:[59]
frac{pi}{4} = 1 - frac{1}{3} + frac{1}{5} - frac{1}{7} + ldots + infty
  • A rational approximation of error for the finite sum of their series of interest. For example, the error, fi(n + 1), (for n odd, and i = 1, 2, 3) for the series:
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}.
  • Manipulation of error term to derive a faster converging series for π:[59]
frac{pi}{4} = frac{3}{4} + frac{1}{3^3-3} - frac{1}{5^3-5} + frac{1}{7^3-7} - cdots infty
  • Using the improved series to derive a rational expression,[59] 104348 / 33215 for π correct up to nine decimal places, i.e. 3.141592653
  • Use of an intuitive notion of limit to compute these results.[59]
  • A semi-rigorous (see remark on limits above) method of differentiation of some trigonometric functions.[61] However, they did not formulate the notion of a function, or have knowledge of the exponential or logarithmic functions.

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."[64] 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,[65][66] a commentary on the Yuktibhasa's proof of the sine and cosine series[67] and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary).[68][69] In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ... Alhazen Abu Ali al-Hasan Ibn Al-Haitham (also: Ibn al Haitham) (965-1040) (Arabic: أبو علي الحسن بن الهيثم) was an Arab Muslim mathematician; he is sometimes called al-Basri (Arabic: البصري), after his birthplace Basra, Arab Islamic Caliphate (now Iraq). ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. ... Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ... | Come and take it, slogan of the Texas Revolution 1835 was a common year starting on Thursday (see link for calendar). ... The Sanskrit language ( , for short ) is a classical language of India, a liturgical language of Hinduism, Buddhism, Sikhism, and Jainism, and one of the 23 official languages of India. ...


The Kerala mathematicians included Narayana Pandit (c. 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). Madhava of Sangamagramma (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. Narayana Pandit (&#2344;&#2366;&#2352;&#2366;&#2351;&#2339; &#2346;&#2339;&#2381;&#2337;&#2367;&#2340;) (1340-1400) was a major mathematician of the Kerala school. ... Bhāskara (1114-1185), also called Bhāskara II and Bhāskarācārya (Bhaskara the teacher) was an Indian mathematician. ... Wikipedia does not yet have an article with this exact name. ... Madhava (माधव) of Sangamagrama (1350-1425) was a major mathematician from Kerala, in South India. ... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ...


Parameshvara (c. 1370-1460) wrote commentaries on the works of Bhaskara I, Aryabhata and Bhaskara II. His Lilavati Bhasya, a commentary on Bhaskara II's Lilavati, contains one of his important discoveries: a version of the mean value theorem. 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). He elaborated and extended the contributions of Madhava. Parameshvara (&#2346;&#2352;&#2350;&#2375;&#2358;&#2381;&#2357;&#2352;) (1360-1425) was a major mathematician of the Kerala school. ... Bhāskara, or Bhāskara I, (c. ... Statue of Aryabhata on the grounds of IUCAA, Pune. ... Bhāskara (1114-1185), also called Bhāskara II and Bhāskarācārya (Bhaskara the teacher) was an Indian mathematician. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ... Nilakantha Somayaji (नीलकण्ठ सोमयाजि) (1444-1544), from Kerala, was a major mathematician and astronomer. ... 1501 was a common year starting on Tuesday (see link for calendar) of the Gregorian calendar. ...


Citrabhanu (c. 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. These types are all the possible pairs of equations of the following seven forms: Citrabanu(1530) a mathematician from the Kerala school in the 16th century mathematician from Kerala who gave integer solutions to 21 types of systems of two simultaneous Diophantine equations in two unknowns. ... In mathematics, simultaneous equations are a set of equations where variables are shared. ...


 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. 1500-1575) was another member of the Kerala School. His key work was the Yukti-bhasa (written in Malayalam, a regional language of Kerala). Jyesthadeva presented proofs of most mathematical theorems and infinite series earlier discovered by Madhava and other Kerala School mathematicians. Jyesthadeva (1500-1575), born in Kerala, was a major mathematician, and author of the 1501 Yukti-bhasa, which was a survey of Kerala mathematics and astronomy that was unique at the time for its exacting proofs of the theorems it presented. ... Malayalam (മലയാളം ) is the language spoken predominantly in the state of Kerala, in southern India. ... , Kerala ( ; Malayalam: കേരളം; ) is a state on the Malabar Coast of southwestern India. ...


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. According to G. G. Joseph: Here is a chronology of the main Indian mathematicians: BC Yajnavalkya, 1800 BC, the author of the altar mathematics of the Shatapatha Brahmana. ... This article does not cite any references or sources. ... Eurocentrism is the practice, conscious or otherwise, of placing emphasis on European (and, generally, Western) concerns, culture and values at the expense of those of other cultures. ...

[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"[70]

The historian of mathematics, Florian Cajori, suggested that he "suspect[s] that Diophantus got his first glimpse of algebraic knowledge from India."[71] Florian Cajori at Colorado College Florian Cajori was born February 28, 1859 in St Aignan (near Thusis), Graubünden, Switzerland. ... Title page of the 1621 edition of Diophantus Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac. ...


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. 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.[72] Kerala was in continuous contact with China and Arabia, and, from around 1500, with Europe. 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.[72] 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."[60][73] Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... , Kerala ( ; Malayalam: കേരളം; ) is a state on the Malabar Coast of southwestern India. ... The Society of Jesus (Latin: Societas Iesu), commonly known as the Jesuits, is a Roman Catholic religious order. ... The Arabian Peninsula The Arabian Peninsula is a mainly desert peninsula in Southwest Asia at the junction of Africa and Asia and an important part of the greater Middle East. ... 1500 was a common year starting on Monday (see link for calendar) of the Gregorian calendar. ...


Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus.[61] 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."[61] The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own;[61] 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."[61] 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.[61] Sir Isaac Newton, (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist, regarded by many as the greatest figure in the history of science. ... It has been suggested that this article be split into multiple articles. ... For a non-technical overview of the subject, see Calculus. ... In calculus, the integral of a function is an extension of the concept of a sum. ... This article or section does not cite any references or sources. ... City flag City coat of arms Motto: Fluctuat nec mergitur (Latin: Tossed by the waves, she does not sink) The Eiffel Tower in Paris, as seen from the esplanade du Trocadéro. ...


See also

The chronology of Indian mathematics spans from the Indus valley civilization and the Vedas to Modern times. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... The development of logic in India dates back to the analysis of inference by Aksapada Gautama, founder of the Nyaya school of Hindu philosophy, probably in the first or second centuries BCE, and so stands as one of the three original traditions of logic, alongside the Greek and Chinese traditions. ... The astronomy and the astrology of Ancient India (Jyotisha) is based upon sidereal calculations. ... For a list of biographies of mathematicians, see list of mathematicians. ...

Notes

  1. ^ a b (Encyclopaedia Britannica (Kim Plofker) 2007, p. 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  2rsinleft(theta/2right); the introduction of the sine, more easily handled, is due to Hindu mathematicians of the Middle ages)."
  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. 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.
  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.
  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. 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. ^ Victor J. Katz (1995). "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3), p. 163-174.
  52. ^ a b c d e Joseph (2000), p. 298-300.
  53. ^ (Hayashi 2005, p. 369)
  54. ^ a b c d (Hayashi 2003, pp. 121-122)
  55. ^ (Stillwell 2004, p. 77)
  56. ^ (Stillwell 2004, p. 87)
  57. ^ a b c d e f (Stillwell 2004, pp. 72-73)
  58. ^ (Stillwell 2004, pp. 74-76)
  59. ^ a b c d e f g h i (Roy 1990)
  60. ^ a b c (Bressoud 2002)
  61. ^ a b c d e f g (Katz 1995)
  62. ^ Singh, A. N. Singh. 1936. "On the Use of Series in Hindu Mathematics." Osiris 1:606-628.
  63. ^ Edwards, C. H., Jr. 1979. The Historical Development of the Calculus. New York: Springer-Verlag.
  64. ^ Charles Whish (1835). Transactions of the Royal Asiatic Society of Great Britain and Ireland. 
  65. ^ Rajagopal, C. and M. S. Rangachari. 1949. "A Neglected Chapter of Hindu Mathematics." Scripta Mathematica. 15:201-209.
  66. ^ Rajagopal, C. and M. S. Rangachari. 1951. "On the Hindu proof of Gregory's series." Ibid. 17:65-74.
  67. ^ 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.
  68. ^ 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.
  69. ^ Rajagopal, C. and M. S. Rangachari. 1986. "On Medieval Kerala Mathematics." Archive for the History of Exact Sciences. 35:91-99.
  70. ^ 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.
  71. ^ Cajori, Florian (1893). "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"” 
  72. ^ 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.
  73. ^ 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.

Statue of Aristarchus at Aristotle University in Thessalonica, Greece Aristarchus (310 BC - ca. ... For the Athenian tyrant, see Hipparchus (son of Pisistratus). ... A medieval artists rendition of Claudius Ptolemaeus Claudius Ptolemaeus (Greek: ; ca. ... Wikipedia does not yet have an article with this exact name. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... Of the approximately half million clay tablets excavated at the beginning of the 19th century, about 400 are of a mathematical nature. ... The Gupta Empire under Chandragupta II (ruled 375-415) The Gupta Empire was one of the largest political and military empires in ancient India. ... Florian Cajori at Colorado College Florian Cajori was born February 28, 1859 in St Aignan (near Thusis), Graubünden, Switzerland. ...

Source Books in Sanskrit

  • Keller, Agathe (2006), Expounding the Mathematical Seed. Vol. 1: The Translation: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya, Basel, Boston, and Berlin: Birkhäuser Verlag, 172 pages, ISBN 3764372915.
  • Keller, Agathe (2006), Expounding the Mathematical Seed. Vol. 2: The Supplements: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya, Basel, Boston, and Berlin: Birkhäuser Verlag, 206 pages, ISBN 3764372923.
  • Neugebauer, Otto & David Pingree (eds.) (1970), The Pañcasiddhāntikā of Varāhamihira, New edition with translation and commentary, (2 Vols.), Copenhagen.
  • Pingree, David (ed) (1978), The Yavanajātaka of Sphujidhvaja, edited, translated and commented by D. Pingree, Cambridge, MA: Harvard Oriental Series 48 (2 vols.).
  • Sarma, K. V. (ed) (1976), Āryabhaṭīya of Āryabhaṭa with the commentary of Sūryadeva Yajvan, critically edited with Introduction and Appendices, New Delhi: Indian National Science Academy.
  • Sen, S. N. & A. K. Bag (eds.) (1983), The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava, with Text, English Translation and Commentary, New Delhi: Indian National Science Academy.
  • Shukla, K. S. (ed) (1976), Āryabhaṭīya of Āryabhaṭa with the commentary of Bhāskara I and Someśvara, critically edited with Introduction, English Translation, Notes, Comments and Indexes, New Delhi: Indian National Science Academy.
  • Shukla, K. S. (ed) (1988), Āryabhaṭīya of Āryabhaṭa, critically edited with Introduction, English Translation, Notes, Comments and Indexes, in collaboration with K.V. Sarma, New Delhi: Indian National Science Academy.

The Sanskrit language ( , for short ) is a classical language of India, a liturgical language of Hinduism, Buddhism, Sikhism, and Jainism, and one of the 23 official languages of India. ... David Edwin Pingree (1933-2005), late University Professor and Professor of History of Mathematics and Classics at Brown University, was one of Americas foremost historians of the exact sciences in antiquity. ... David Edwin Pingree (1933-2005), late University Professor and Professor of History of Mathematics and Classics at Brown University, was one of Americas foremost historians of the exact sciences in antiquity. ...

References

Nicolas Bourbaki is the collective allonym under which a group of (mainly French) 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. ... The Springer-Verlag (pronounced SHPRING er FAIR lahk) was a worldwide publishing company base in Germany. ... David Fowler (1937 – 2004) was a historian of Greek mathematics who published work on pre-Eudoxian ratio theory (using the process he called anthyphaeresis). ... The Mathematical Association of America (MAA) is a professional society that focuses on undergraduate mathematics education. ... George Ifrah (1947-) was a professor of mathematics, and a historian of mathematics, especially numerals. ... David Edwin Pingree (1933-2005), late University Professor and Professor of History of Mathematics and Classics at Brown University, was one of Americas foremost historians of the exact sciences in antiquity. ... David Edwin Pingree (1933-2005), late University Professor and Professor of History of Mathematics and Classics at Brown University, was one of Americas foremost historians of the exact sciences in antiquity. ... David Edwin Pingree (1933-2005), late University Professor and Professor of History of Mathematics and Classics at Brown University, was one of Americas foremost historians of the exact sciences in antiquity. ... David Edwin Pingree (1933-2005), late University Professor and Professor of History of Mathematics and Classics at Brown University, was one of Americas foremost historians of the exact sciences in antiquity. ... The Mathematical Association of America (MAA) is a professional society that focuses on undergraduate mathematics education. ... Frits Staal (born 1930 in the Netherlands) is Emeritus Professor of Philosophy and South & Southeast Asian Studies at the University of California, Berkeley. ... Frits Staal (born 1930 in the Netherlands) is Emeritus Professor of Philosophy and South & Southeast Asian Studies at the University of California, Berkeley. ... Frits Staal (born 1930 in the Netherlands) is Emeritus Professor of Philosophy and South & Southeast Asian Studies at the University of California, Berkeley. ... Frits Staal (born 1930 in the Netherlands) is Emeritus Professor of Philosophy and South & Southeast Asian Studies at the University of California, Berkeley. ... Frits Staal (born 1930 in the Netherlands) is Emeritus Professor of Philosophy and South & Southeast Asian Studies at the University of California, Berkeley. ... George Frederick William Thibaut (March 20, 1848-1914) was an Indologist notable for his contributions to the understanding of ancient Indian mathematics and astronomy. ... Bartel Leendert van der Waerden (February 2, 1903 – January 12, 1996) was a Dutch mathematician who born in Amsterdam, Netherlands and died in Zürich, Switzerland. ... Bartel Leendert van der Waerden (February 2, 1903 – January 12, 1996) was a Dutch mathematician who born in Amsterdam, Netherlands and died in Zürich, Switzerland. ... Bartel Leendert van der Waerden (February 2, 1903 – January 12, 1996) was a Dutch mathematician who born in Amsterdam, Netherlands and died in Zürich, Switzerland. ...

External links


  Results from FactBites:
 
Indian Mathematics (6042 words)
This text bridged the gap between the earlier Jaina mathematics and the 'Classical period' of Indian mathematics, though the authorship of this text is unknown.
Although earlier Indian mathematics was also very significant, this period saw great mathematicians such as Aryabhata, Varahamihira, Brahmagupta, Mahavira and Bhaskara give a broader and clearer shape to almost all the branches of mathematics.
Moving on from the finite procedures of ancient mathematics to ''treat their limit passage to infinity'', which is considered to be the essence of modern classical analysis, and thus he is considered the father of Mathematical Analysis.
Indian mathematics (3631 words)
We shall examine the contributions of Indian mathematics in this article, but before looking at this contribution in more detail we should say clearly that the "huge debt" is the beautiful number system invented by the Indians on which much of mathematical development has rested.
Histories of Indian mathematics used to begin by describing the geometry contained in the Sulbasutras but research into the history of Indian mathematics has shown that the essentials of this geometry were older being contained in the altar constructions described in the Vedic mythology text the Shatapatha Brahmana and the Taittiriya Samhita.
The mathematics contained in the these texts is studied in some detail in the separate article on the Sulbasutras.
  More results at FactBites »

 
 

COMMENTARY     


Share your thoughts, questions and commentary here
Your name
Your comments

Want to know more?
Search encyclopedia, statistics and forums:

 


Press Releases |  Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms, 1022, m