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Indian mathematics—which here is the mathematics that emerged in South Asia^[1] from ancient times until the end of the 18th century—had its beginnings in the Bronze Age Indus Valley civilization (2600-1900 BCE) and the Iron Age Vedic culture (1500-500 BCE). 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. ...

[edit] Fields of Indian mathematics

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

• Arithmetic: Decimal system, Negative numbers (see Brahmagupta), Zero (see Hindu-Arabic numeral system), the modern positional notation numeral system, Floating point numbers (see Kerala School), Number theory, Infinity (see Yajur Veda), Transfinite numbers, Irrational numbers (see Sulba Sutras)
• Geometry: Square roots (see Bakhshali approximation), Cube roots (see Mahavira), Pythagorean triples (see Sulba Sutras; Baudhayana and Apastamba state the Pythagorean theorem without proof), Transformation (see Panini), Pascal's triangle (see Pingala)
• Algebra: Quadratic equations (see Sulba Sutras, Aryabhata, and Brahmagupta), Cubic equations (see Mahavira and Bhaskara), Quartic equations (biquadratic equations; see Mahavira and Bhaskara)
• Mathematical logic: Formal grammars, formal language theory, the Panini-Backus form (see Panini), Recursion (see Panini)
• General mathematics: Fibonacci numbers (see Pingala), Earliest forms of Morse code (see Pingala), Logarithms, indices (see Jaina mathematics), Algorithms, Algorism (see Aryabhata and Brahmagupta)
• Trigonometry: Trigonometric functions (see Surya Siddhanta and Aryabhata), Trigonometric series (see Madhava and Kerala School)

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. ...

[edit] 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.

[edit] 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. ...

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:

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

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 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]

[edit] 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. ...

The formula is accurate up to five decimal places, the true value being ^[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. ...

which expresses 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. ...

[edit] 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