Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like

Aryabhata Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the '' Aryabhatiya'' (which ...

Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tr ...

, Bhaskara II, and Varāhamihira. The decimal number system in use today: "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." was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of

zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...

as a number,: "...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." negative numbers,: Modern arithmetic was known during medieval times as "Modus Indorum" or method of the Indians. Leonardo of Pisa wrote that compared to method of the Indians all other methods is a mistake. This method of the Indians is none other than our very simple arithmetic of addition, subtraction, multiplication and division. Rules for these four simple procedures was first written down by

during 7th century AD. "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."

arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...

, and

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

. In addition,

trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...

was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe"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." and led to further developments that now form the foundations of many areas of mathematics. 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 so important as the ideas involved. All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form. The oldest extant mathematical ''document'' produced on the Indian subcontinent is the birch bark Bakhshali Manuscript, discovered in 1881 in the village of

Bakhshali Bakhshali ( ur, بخشالی) is a village and union council in Mardan District, Khyber-Pakhtunkhwa, Pakistan. It is located at 34°17'0N 72°9'0E and has an altitude of 307 metres (1010 feet). History The village is notable for being the lo ...

, near Peshawar (modern day Pakistan) and is likely from the 7th century CE. A later landmark in Indian mathematics was the development of the

series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...

expansions for trigonometric functions (sine, cosine, and

arc tangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spe ...

) by mathematicians of the Kerala school in the 15th century CE. Their remarkable work, completed two centuries before the invention of

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

in Europe, provided what is now considered the first example of a power series (apart from geometric series). However, they did not formulate a systematic theory of differentiation and

integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...

, nor is there any ''direct'' evidence of their results being transmitted outside Kerala.

Prehistory

Excavations at

Harappa Harappa (; Urdu/ pnb, ) is an archaeological site in Punjab, Pakistan, about west of Sahiwal. The Bronze Age Harappan civilisation, now more often called the Indus Valley Civilisation, is named after the site, which takes its name from a mode ...

Mohenjo-daro Mohenjo-daro (; sd, موئن جو دڙو'', ''meaning 'Mound of the Dead Men';Indus Valley civilisation have uncovered evidence of the use of "practical mathematics". The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4:2:1, considered favourable for the stability of a brick structure. They used a standardised 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 Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

shapes, which included hexahedra,

barrel A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden staves and bound by wooden or metal hoops. The word vat is often used for large containers for liquids, ...

cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines conn ...

s, and cylinders, thereby demonstrating knowledge of basic geometry. The inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the ''Mohenjo-daro ruler''—whose unit of length (approximately 1.32 inches or 3.4 centimetres) was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length. Hollow cylindrical objects made of shell and found at Lothal (2200 BCE) and Dholavira are demonstrated to have the ability to measure angles in a plane, as well as to determine the position of stars for navigation.

Vedic period

Samhitas and Brahmanas

The religious texts of the

Vedic Period The Vedic period, or the Vedic age (), is the period in the late Bronze Age and early Iron Age of the history of India when the Vedic literature, including the Vedas (ca. 1300–900 BCE), was composed in the northern Indian subcontinent, betwe ...

provide evidence for the use of

large numbers Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical ...

. By the time of the '' '' (1200–900 BCE), numbers as high as were being included in the texts. For example, the '' mantra'' (sacred recitation) at the end of the ''annahoma'' ("food-oblation rite") performed during the ''aśvamedha'', and uttered just before-, during-, and just after sunrise, invokes powers of ten from a hundred to a trillion: The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta (RV 10.90.4): The Satapatha Brahmana ( 7th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.

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. Most mathematical problems considered in the ''Śulba Sūtras'' spring from "a single theological requirement," 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. According to , the ''Śulba Sūtras'' contain "the earliest extant verbal expression of the

Pythagorean Theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

in the world, although it had already been known to the Old Babylonians."

The diagonal rope (') of an oblong (rectangle) produces both which the flank (''pārśvamāni'') and the horizontal (') produce separately."

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. They contain lists of Pythagorean triples, which are particular cases of Diophantine equations.: "The arithmetic content of the ''Śulva Sūtras'' consists of rules for finding Pythagorean triples such as , , , and . 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." They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square.": "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 of 18 (3 − 2), which is about 3.088."

Baudhayana The (Sanskrit: बौधायन) are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics and is one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. Th ...

(c. 8th century BCE) composed the ''Baudhayana Sulba Sutra'', the best-known ''Sulba Sutra'', which contains examples of simple Pythagorean triples, such as: , , , , and , 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." 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." Baudhayana gives an expression for the square root of two: ::

\sqrt \approx 1 + \frac + \frac - \frac = 1.4142156 \ldots

The expression is accurate up to five decimal places, the true value being 1.41421356... This expression is similar in structure to the expression found on a Mesopotamian tablet from the Old Babylonian period (1900–1600

BCE Common Era (CE) and Before the Common Era (BCE) are year notations for the Gregorian calendar (and its predecessor, the Julian calendar), the world's most widely used calendar era. Common Era and Before the Common Era are alternatives to the o ...

): ::

\sqrt \approx 1 + \frac + \frac + \frac = 1.41421297 \ldots

which expresses in the sexagesimal system, and which is also accurate up to 5 decimal places. According to mathematician S. G. Dani, the Babylonian cuneiform tablet

Plimpton 322 Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University. This tablet, believed to have been written about 1800 BC, has a tabl ...

written c. 1850 BCE "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple, 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." Dani goes on to say: In all, three ''Sulba Sutras'' were composed. The remaining two, the ''Manava Sulba Sutra'' composed by

Manava Manava (c. 750 BC – 690 BC) is an author of the Hindu geometric text of '' Sulba Sutras.'' The Manava Sulbasutra is not the oldest (the one by Baudhayana is older), nor is it one of the most important, there being at least three Sulbas ...

(fl. 750–650 BCE) and the ''Apastamba Sulba Sutra'', composed by

Apastamba ''Āpastamba Dharmasūtra'' (Sanskrit: आपस्तम्ब धर्मसूत्र) is a Sanskrit text and one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. It is one of ...

(c. 600 BCE), contained results similar to the ''Baudhayana Sulba Sutra''. ;Vyakarana An important landmark of the Vedic period was the work of Sanskrit grammarian, (c. 520–460 BCE). His grammar includes early use of

Boolean logic In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...

, of the

null Null may refer to: Science, technology, and mathematics Computing * Null (SQL) (or NULL), a special marker and keyword in SQL indicating that something has no value *Null character, the zero-valued ASCII character, also designated by , often used ...

operator, and of

context free grammar In formal language theory, a context-free grammar (CFG) is a formal grammar whose production rules are of the form :A\ \to\ \alpha with A a ''single'' nonterminal symbol, and \alpha a string of terminals and/or nonterminals (\alpha can be emp ...

s, and includes a precursor of the

Backus–Naur form In computer science, Backus–Naur form () or Backus normal form (BNF) is a metasyntax notation for context-free grammars, often used to describe the syntax of languages used in computing, such as computer programming languages, document formats ...

(used in the description programming languages).

Pingala (300 BCE – 200 BCE)

Among the scholars of the post-Vedic period who contributed to mathematics, the most notable is Pingala (') ( fl. 300–200 BCE), a music theorist who authored the Chhandas

Shastra ''Shastra'' (, IAST: , ) is a Sanskrit word that means "precept, rules, manual, compendium, book or treatise" in a general sense.Monier Williams, Monier Williams' Sanskrit-English Dictionary, Oxford University Press, Article on 'zAstra'' The wo ...

(', also Chhandas Sutra '), a Sanskrit treatise on prosody. There is evidence that in his work on the enumeration of syllabic combinations, Pingala stumbled upon both

Pascal's triangle In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ...

and

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the te ...

s, although he did not have knowledge of the

binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...

itself. 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: The text also indicates that Pingala was aware of the

combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...

identity: ::

+  +  + \cdots +  +  = 2^n

;Kātyāyana

Kātyāyana Kātyāyana (कात्यायन) also spelled as Katyayana (est. c. 6th to 3rd century BCE) was a Sanskrit grammarian, mathematician and Vedic priest who lived in ancient India. पतञ्जलीमहर्षिः Patanjali Maharsh ...

(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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

and a computation of the square root of 2 correct to five decimal places.

Jain mathematics (400 BCE – 200 CE)

Although Jainism as a religion and philosophy predates its most famous exponent, the great

Mahavira Mahavira (Sanskrit: महावीर) also known as Vardhaman, was the 24th ''tirthankara'' (supreme preacher) of Jainism. He was the spiritual successor of the 23rd ''tirthankara'' Parshvanatha. Mahavira was born in the early part of the 6 ...

swami (6th century BCE), most Jain texts on mathematical topics were composed after the 6th century BCE.

Jain Jainism ( ), also known as Jain Dharma, is an Indian religion. Jainism traces its spiritual ideas and history through the succession of twenty-four tirthankaras (supreme preachers of ''Dharma''), with the first in the current time cycle being ...

mathematicians are important historically as crucial links between the mathematics of the Vedic period and that of the "classical period." A significant historical contribution of Jain 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, their texts 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, Jain mathematicians devised notations for simple powers (and exponents) of numbers like squares and cubes, which enabled them to define simple

algebraic equations In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...

(''beejganita samikaran''). Jain 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.) In addition to ''Surya Prajnapti'', important Jain works on mathematics included the ''

Sthananga Sutra Sthananga Sutra (Sanskrit: Sthānāṅgasūtra Prakrit: Ṭhāṇaṃgasutta) (c. 3rd-4th century CE) forms part of the first eleven Angas of the Jaina Canon which have survived despite the bad effects of this Hundavasarpini kala as per the � ...

'' (c. 300 BCE – 200 CE); the ''Anuyogadwara Sutra'' (c. 200 BCE – 100 CE), which includes the earliest known description of

factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) ...

s in Indian mathematics; and the ''

Satkhandagama The (Sanskrit: "Scripture in Six Parts") is the foremost and oldest Digambara Jain sacred text. According to Digambara tradition, the original canonical scriptures of the Jains were totally lost within a few centuries of ''Nirvana'' of Mah ...

'' (c. 2nd century CE). Important Jain mathematicians included

Bhadrabahu Ācārya Bhadrabāhu (c. 367 - c. 298 BC) was, according to the ''Digambara'' sect of Jainism, the last '' Shruta Kevalin'' (all knowing by hearsay, that is indirectly) in Jainism . He was the last ''acharya'' of the undivided Jain ''sangha''. ...

(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 ''Tiloya Panatti'' or ''Trilokaprajnapati'' is one of the earlier Prakrit texts on Jain cosmology composed by Acharya Yativrshabha. The subject matter Jain cosmology has a unique perception of the Universe. It perceives different solar and lu ...

''; and

Umasvati Umaswati, also spelled as Umasvati and known as Umaswami, was an Indian scholar, possibly between 2nd-century and 5th-century CE, known for his foundational writings on Jainism. He authored the Jain text ''Tattvartha Sutra'' (literally '"All Th ...

(c. 150 BCE), who, although better known for his influential writings on Jain philosophy and metaphysics, composed a mathematical work called '' Tattwarthadhigama-Sutra Bhashya''.

Oral tradition

Mathematicians of ancient and early medieval India were almost all Sanskrit pandits (' "learned man"), who were trained in Sanskrit language and literature, and possessed "a common stock of knowledge in grammar ( '), exegesis ( ') and logic ( ''nyāya'')." Memorisation of "what is heard" (''

śruti ''Shruti'' ( sa, श्रुति, , ) in Sanskrit means "that which is heard" and refers to the body of most authoritative, ancient religious texts comprising the central canon of Hinduism. Manusmriti states: ''Śrutistu vedo vijñeyaḥ'' ...

'' in Sanskrit) through recitation played a major role in the transmission of sacred texts in ancient India. Memorisation 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."

Styles of memorisation

Prodigious energy was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity. For example, memorisation of the sacred ''

Veda FIle:Atharva-Veda samhita page 471 illustration.png, upright=1.2, The Vedas are ancient Sanskrit texts of Hinduism. Above: A page from the ''Atharvaveda''. The Vedas (, , ) are a large body of religious texts originating in ancient India. Co ...

s'' 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 ' (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 in the original order. The recitation thus proceeded as:

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

In another form of recitation, ' (literally "flag recitation") a sequence of ''N'' words were recited (and memorised) by pairing the first two and last two words and then proceeding as:

word₁word₂, word_{''N'' − 1}word_''N''; word₂word₃, word_{''N'' − 2}word_{''N'' − 1}; ..; word_{''N'' − 1}word_''N'', word₁word₂;

The most complex form of recitation, ' (literally "dense recitation"), according to , 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 '' '' (c. 1500 BCE), as a single text, without any variant readings. Similar methods were used for memorising mathematical texts, whose transmission remained exclusively oral until the end of the

Vedic period The Vedic period, or the Vedic age (), is the period in the late Bronze Age and early Iron Age of the history of India when the Vedic literature, including the Vedas (ca. 1300–900 BCE), was composed in the northern Indian subcontinent, betwe ...

(c. 500 BCE).

The ''Sutra'' genre

Mathematical activity in ancient India began as a part of a "methodological reflexion" on the sacred

s, which took the form of works called ', or, "Ancillaries of the Veda" (7th–4th century BCE). The need to conserve the sound of sacred text by use of ' ( phonetics) and '' chhandas'' ( metrics); to conserve its meaning by use of ' ( grammar) and ''

nirukta ''Nirukta'' ( sa, निरुक्त, , "explained, interpreted") is one of the six ancient Vedangas, or ancillary science connected with the Vedas – the scriptures of Hinduism.James Lochtefeld (2002), "Nirukta" in The Illustrated Encycl ...

'' ( etymology); and to correctly perform the rites at the correct time by the use of ''

kalpa Kalevan Pallo (KalPa) is a professional ice hockey team which competes in the Finnish Liiga. They play in Kuopio, Finland at the Olvi Areena. Team history Established in 1929 as ''Sortavalan Palloseura'' in Sortavala, the club relocated to Kuop ...

'' ( ritual) and ' (

astrology Astrology is a range of divinatory practices, recognized as pseudoscientific since the 18th century, that claim to discern information about human affairs and terrestrial events by studying the apparent positions of celestial objects. Dif ...

), gave rise to the six disciplines of the '. Mathematics arose as a part of the last two disciplines, ritual and astronomy (which also included astrology). Since the ' 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"): Extreme brevity was achieved through multiple means, which included using ellipsis "beyond the tolerance of natural language," using technical names instead of longer descriptive names, abridging lists by only mentioning the first and last entries, and using markers and variables. The ''sūtras'' create the impression that communication through the text was "only a part of the whole instruction. The rest of the instruction must have been transmitted by the so-called ''Guru-shishya parampara'', 'uninterrupted succession from teacher (''guru'') to the student (''śisya''),' and it was not open to the general public" and perhaps even kept secret. The brevity achieved in a ''sūtra'' is demonstrated in the following example from the Baudhāyana ''Śulba Sūtra'' (700 BCE). Domestic fire altar

The domestic fire-altar in the

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. 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: According to , 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, ', 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.

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. The earliest mathematical prose commentary was that on the work, '' '' (written 499 CE), a work on astronomy and mathematics. The mathematical portion of the ' was composed of 33 ''sūtras'' (in verse form) consisting of mathematical statements or rules, but without any proofs. However, according to , "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 ', had the following structure: *Rule ('sūtra') in verse by *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 (', literally "to make conviction") of the answer. These became rare by the 13th century, derivations or proofs being favoured by then. Typically, for any mathematical topic, students in ancient India first memorised the ''sūtras'', which, as explained earlier, were "deliberately inadequate" 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,

( fl. 7th century CE), to characterise astronomical computations as "dust work" (Sanskrit: ''dhulikarman'').

Numerals and the decimal number system

It is well known that the decimal place-value system ''in use today'' was first recorded in India, then transmitted to the Islamic world, and eventually to Europe. The Syrian bishop Severus Sebokht wrote in the mid-7th century CE about the "nine signs" of the Indians for expressing numbers. However, how, when, and where the first decimal place value system was invented is not so clear. The earliest extant

script Script may refer to: Writing systems * Script, a distinctive writing system, based on a repertoire of specific elements or symbols, or that repertoire * Script (styles of handwriting) ** Script typeface, a typeface with characteristics of ha ...

used in India was the 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 4th century BCE to the 4th century CE. Almost contemporaneously, another script, the

Brāhmī script Brahmi (; ; ISO: ''Brāhmī'') is a writing system of ancient South Asia. "Until the late nineteenth century, the script of the Aśokan (non-Kharosthi) inscriptions and its immediate derivatives was referred to by various names such as 'lath' o ...

, 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. The earliest surviving evidence of decimal place value numerals in India and southeast Asia is from the middle of the first millennium CE. A copper plate from Gujarat, India mentions the date 595 CE, written in a decimal place value notation, although there is some doubt as to the authenticity of the plate. Decimal numerals recording the years 683 CE have also been found in stone inscriptions in Indonesia and Cambodia, where Indian cultural influence was substantial. There are older textual sources, although the extant manuscript copies of these texts are from much later dates. Probably the earliest such source is the work of the Buddhist philosopher Vasumitra dated likely to the 1st century CE. Discussing the counting pits of merchants, Vasumitra remarks, "When he sameclay counting-piece is in the place of units, it is denoted as one, when in hundreds, one hundred." Although such references seem to imply that his readers had knowledge of a decimal place value representation, the "brevity of their allusions and the ambiguity of their dates, however, do not solidly establish the chronology of the development of this concept." A third decimal representation was employed in a verse composition technique, later labelled '' Bhuta-sankhya'' (literally, "object numbers") used by early Sanskrit authors of technical books. Since many early technical works were composed in verse, numbers were often represented by objects in the natural or religious world that correspondence to them; this allowed a many-to-one correspondence for each number and made verse composition easier. According to , the number 4, for example, could be represented by the word "

" (since there were four of these religious texts), the number 32 by the word "teeth" (since a full set consists of 32), and the number 1 by "moon" (since there is only one moon). So, Veda/teeth/moon would correspond to the decimal numeral 1324, as the convention for numbers was to enumerate their digits from right to left. The earliest reference employing object numbers is a c. 269 CE Sanskrit text, ''Yavanajātaka'' (literally "Greek horoscopy") of Sphujidhvaja, a versification of an earlier (c. 150 CE) Indian prose adaptation of a lost work of Hellenistic astrology. Such use seems to make the case that by the mid-3rd century CE, the decimal place value system was familiar, at least to readers of astronomical and astrological texts in India. It has been hypothesized that the Indian decimal place value system was based on the symbols used on Chinese counting boards from as early as the middle of the first millennium BCE. According to ,

These counting boards, like the Indian counting pits, ..., had a decimal place value structure ... Indians may well have learned of these decimal place value "rod numerals" from Chinese Buddhist pilgrims or other travelers, or they may have developed the concept independently from their earlier non-place-value system; no documentary evidence survives to confirm either conclusion."

Bakhshali Manuscript

The oldest extant mathematical manuscript in India is the '' Bakhshali Manuscript'', a birch bark manuscript written in "Buddhist hybrid Sanskrit" in the ''Śāradā'' script, which was used in the northwestern region of the Indian subcontinent between the 8th and 12th centuries CE. 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 The provinces of India, earlier presidencies of British India and still earlier, presidency towns, were the administrative divisions of British governance on the Indian subcontinent. Collectively, they have been called British India. In one ...

and now in Pakistan). Of unknown authorship and now preserved in the

Bodleian Library The Bodleian Library () is the main research library of the University of Oxford, and is one of the oldest libraries in Europe. It derives its name from its founder, Sir Thomas Bodley. With over 13 million printed items, it is the second ...

in the University of Oxford, the manuscript has been dated recently as 224 AD- 383 AD. 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. The topics treated include arithmetic (fractions, square roots, profit and loss, simple interest, the

rule of three Rule of three or Rule of Thirds may refer to: Science and technology *Rule of three (aeronautics), a rule of descent in aviation *Rule of three (C++ programming), a rule of thumb about class method definitions * Rule of three (computer programming ...

, 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." Many of its problems are of a category known as 'equalisation problems' that lead to systems of linear equations. One example from Fragment III-5-3v is the following: 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. In 2017, three samples from the manuscript were shown by radiocarbon dating to come from three different centuries: from 224 to 383 AD, 680-779 AD, and 885-993 AD. It is not known how fragments from different centuries came to be packaged together.

Classical period (400–1600)

This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as

, Varahamihira,

, Bhaskara I,

, Bhaskara II,

Madhava of Sangamagrama Iriññāttappiḷḷi Mādhavan known as Mādhava of Sangamagrāma () was an Indian mathematician and astronomer from the town believed to be present-day Kallettumkara, Aloor Panchayath, Irinjalakuda in Thrissur District, Kerala, India. He is ...

and

Nilakantha Somayaji Keļallur Nilakantha Somayaji (14 June 1444 – 1544), also referred to as Keļallur Comatiri, was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensi ...

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). This tripartite division is seen in Varāhamihira's 6th century compilation—''Pancasiddhantika'' (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.

Fifth and sixth centuries

;Surya Siddhanta Though its authorship is unknown, the '' Surya Siddhanta'' (c. 400) contains the roots of modern

. Because it contains many words of foreign origin, some authors consider that it was written under the influence of Mesopotamia and Greece. This ancient text uses the following as trigonometric functions for the first time: *Sine ('' Jya''). *Cosine ('' Kojya''). * Inverse sine (''Otkram jya''). It also contains the earliest uses of: * Tangent. * Secant. Later Indian mathematicians such as Aryabhata made references to this text, while later

Arabic Arabic (, ' ; , ' or ) is a Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C. E.Watson; Walte ...

and Latin translations were very influential in Europe and the Middle East. ;Chhedi calendar This Chhedi calendar (594) contains an early use of the modern place-value Hindu–Arabic numeral system now used universally. ;Aryabhata I

(476–550) wrote the ''Aryabhatiya.'' He described the important fundamental principles of mathematics in 332

shlokas Shloka or śloka ( sa, श्लोक , from the root , Macdonell, Arthur A., ''A Sanskrit Grammar for Students'', Appendix II, p. 232 (Oxford University Press, 3rd edition, 1927). in a broader sense, according to Monier-Williams's dictionary, is ...

. The treatise contained: * Quadratic equations *

Trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...

*The value of π, correct to 4 decimal places. Aryabhata also wrote the ''Arya Siddhanta'', which is now lost. Aryabhata's contributions include: Trigonometry: (See also : Aryabhata's sine table) *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 ('' utkrama-jya''). *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: *

Continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...

s. Algebra: *Solutions of simultaneous quadratic equations. *Whole number solutions of

linear equations In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficie ...

by a method equivalent to the modern method. *General solution of the indeterminate linear equation . Mathematical astronomy: *Accurate calculations for astronomical constants, such as the: ** Solar eclipse. ** Lunar eclipse. **The formula for the sum of the

cubes In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

, which was an important step in the development of integral calculus. ;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: *

\sin^2(x) + \cos^2(x) = 1

\sin(x)=\cos\left(\frac-x\right)

\frac=\sin^2(x)

Seventh and eighth centuries

In the 7th century, two separate fields,

(which included measurement) and

, began to emerge in Indian mathematics. The two fields would later be called ' (literally "mathematics of algorithms") and ' (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).

, in his astronomical work '' '' (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). In the latter section, he stated his famous theorem on the diagonals of a

cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...

: Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

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. Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalisation of Heron's formula), as well as a complete description of

rational triangle An integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers. A rational triangle can be defined as one having all sides with rational length; any such rational triangle can be integrally rescaled (ca ...

s (''i.e.'' triangles with rational sides and rational areas). Brahmagupta's formula: The area, ''A'', of a cyclic quadrilateral with sides of lengths ''a'', ''b'', ''c'', ''d'', respectively, is given by :

A = \sqrt \,

where ''s'', the

semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate ...

, given by

s=\frac.

Brahmagupta's Theorem on rational triangles: A triangle with rational sides

a, b, c

and rational area is of the form: :

a = \frac+v, \ \ b=\frac+w, \ \ c=\frac+\frac - (v+w)

for some rational numbers

u, v,

and

w

. Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers 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

. Later in the chapter, he gave the first explicit (although still not completely general) solution of the quadratic equation: :

\ ax^2+bx=c

This is equivalent to: :

x = \frac

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

\ x^2-Ny^2=1,

where

N

is a nonsquare integer. He did this by discovering the following identity: Brahmagupta's Identity:

\ (x^2-Ny^2)(x'^2-Ny'^2) = (xx'+Nyy')^2 - N(xy'+x'y)^2

which was a generalisation of an earlier identity of

Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...

: Brahmagupta used his identity to prove the following lemma: Lemma (Brahmagupta): If

x=x_1,\ \ y=y_1 \ \

is a solution of

\ \ x^2 - Ny^2 = k_1,

and,

x=x_2, \ \ y=y_2 \ \

is a solution of

\ \ x^2 - Ny^2 = k_2,

, then: :

x=x_1x_2+Ny_1y_2,\ \ y=x_1y_2+x_2y_1 \ \

is a solution of

\ x^2-Ny^2=k_1k_2

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

\ x^2 - Ny^2 =k

has an integer solution for any one of

\ k=\pm 4, \pm 2, -1

then Pell's equation: :

\ x^2 -Ny^2 = 1

also has an integer solution. Brahmagupta did not actually prove the theorem, but rather worked out examples using his method. The first example he presented was: 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." 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'', ''Aryabhatiya-bhashya'' and ''Laghu-bhaskariya''. He produced: *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 two decimal places.

Ninth to twelfth centuries

;Virasena Virasena (8th century) was a Jain mathematician in the court of Rashtrakuta King

Amoghavarsha Amoghavarsha I (also known as Amoghavarsha Nrupathunga I) (r.814–878 CE) was the greatest emperor of the Rashtrakuta dynasty, and one of the most notable rulers of Ancient India. His reign of 64 years is one of the longest precisely dated mo ...

Manyakheta Malkhed originally known as Manyakheta (IAST: Mānyakheṭa, Prakrit: "Mannakheḍa"), and also known as Malkhed,Village code= 311400 Malkhed (J), Gulbarga, Karnataka is a town in Karnataka, India. It is located on the banks of Kagina river i ...

, Karnataka. He wrote the ''Dhavala'', a commentary on Jain mathematics, which: *Deals with the concept of ''ardhaccheda'', the number of times a number could be halved, and lists various rules involving this operation. This coincides with the

binary logarithm In mathematics, the binary logarithm () is the power to which the number must be raised to obtain the value . That is, for any real number , :x=\log_2 n \quad\Longleftrightarrow\quad 2^x=n. For example, the binary logarithm of is , the ...

when applied to powers of two, but differs on other numbers, more closely resembling the 2-adic order. *The same concept for base 3 (''trakacheda'') and base 4 (''caturthacheda''). Virasena also gave: *The derivation of the volume of a frustum by a sort of infinite procedure. It is thought that much of the mathematical material in the ''Dhavala'' can attributed to previous writers, especially Kundakunda, Shamakunda, Tumbulura, Samantabhadra and Bappadeva and date who wrote between 200 and 600 CE. ;Mahavira Mahavira Acharya (c. 800–870) from Karnataka, the last of the notable Jain 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: *

Zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...

* Squares *

Cubes In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

* square roots,

cube root In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. ...

s, and the

extending beyond these * Plane geometry * Solid geometry *Problems relating to the casting of

shadows A shadow is a dark area where light from a light source is blocked by an opaque object. It occupies all of the three-dimensional volume behind an object with light in front of it. The cross section of a shadow is a two-dimensional silhouette, ...

*Formulae derived to calculate the area of an ellipse and quadrilateral inside a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

. Mahavira also: *Asserted that the square root of a negative number did not exist *Gave the sum of a series whose terms are squares of an arithmetical progression, and gave empirical rules for area and perimeter of an ellipse. *Solved cubic equations. *Solved quartic equations. *Solved some quintic equations and higher-order polynomials. *Gave the general solutions of the higher order polynomial equations: **

\ ax^n = q

a \frac = p

*Solved indeterminate quadratic equations. *Solved indeterminate cubic equations. *Solved indeterminate higher order equations. ;Shridhara Shridhara (c. 870–930), who lived in

Bengal Bengal ( ; bn, বাংলা/বঙ্গ, translit=Bānglā/Bôngô, ) is a geopolitical, cultural and historical region in South Asia, specifically in the eastern part of the Indian subcontinent at the apex of the Bay of Bengal, predo ...

, wrote the books titled ''Nav Shatika'', ''Tri Shatika'' and ''Pati Ganita''. He gave: *A good rule for finding the volume of a sphere. *The formula for solving quadratic equations. The ''Pati Ganita'' is a work on arithmetic and measurement. It deals with various operations, including: *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 expressionJoseph (2000), p. 298–300. :

\ \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. ;Aryabhata II

Aryabhata II Āryabhaṭa (c. 920 – c. 1000) was an Indian mathematician and astronomer, and the author of the ''Maha-Siddhanta Āryabhaṭa (c. 920 – c. 1000) was an Indian mathematician and astronomer An astronomer is a scientist in the ...

(c. 920–1000) wrote a commentary on Shridhara, and an astronomical treatise '' Maha-Siddhanta''. The Maha-Siddhanta has 18 chapters, and discusses: *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

al treatise in 125 verses based on a work by Shridhara. He worked mainly on: * Permutations and combinations. *General solution of the simultaneous indeterminate linear equation. He was also the author of ''Dhikotidakarana'', a work of twenty verses on: * Solar eclipse. * Lunar eclipse. The ''Dhruvamanasa'' is a work of 105 verses on: *Calculating planetary longitudes * eclipses. *planetary transits. ;Nemichandra Siddhanta Chakravati Nemichandra Siddhanta Chakravati (c. 1100) authored a mathematical treatise titled ''Gome-mat Saar''. ;Bhaskara II

Bhāskara II Bhāskara II (c. 1114–1185), also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian mathematician and astronomer. From verses, in his main work, Siddhānta Shiroma ...

(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: Arithmetic: *Interest computation *Arithmetical and geometrical progressions *Plane geometry *Solid geometry *The shadow of the

gnomon A gnomon (; ) is the part of a sundial that casts a shadow. The term is used for a variety of purposes in mathematics and other fields. History A painted stick dating from 2300 BC that was excavated at the astronomical site of Taosi is the o ...

*Solutions of

combinations In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are t ...

*Gave a proof for division by zero being infinity. Algebra: *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: *Gave a proof of the

. Calculus: *Conceived of differential calculus. *Discovered the derivative. *Discovered the

differential coefficient {{Unreferenced, date=June 2019, bot=noref (GreenC bot) In physics, the differential coefficient of a function ''f''(''x'') is what is now called its derivative ''df''(''x'')/''dx'', the (not necessarily constant) multiplicative factor or ''coefficie ...

. *Developed differentiation. *Stated Rolle's theorem, a special case of the mean value theorem (one of the most important theorems of calculus and analysis). *Derived the differential of the sine function. *Computed π, correct to five decimal places. *Calculated the length of the Earth's revolution around the Sun to 9 decimal places. Trigonometry: *Developments of spherical trigonometry *The trigonometric formulas: **

\ \sin(a+b)=\sin(a) \cos(b) + \sin(b) \cos(a)

\ \sin(a-b)=\sin(a) \cos(b) - \sin(b) \cos(a)

Kerala mathematics (1300–1600)

The Kerala school of astronomy and mathematics was founded by

in Kerala, South India and included among its members:

Parameshvara Vatasseri Parameshvara Nambudiri ( 1380–1460) was a major Indian mathematician and astronomer of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama. He was also an astrologer. Parameshvara was a proponent of ...

, Neelakanta Somayaji, Jyeshtadeva,

Achyuta Pisharati Achyuta Pisharodi (c. 1550 at Thrikkandiyur (aka Kundapura), Tirur, Kerala, India – 7 July 1621 in Kerala) was a Sanskrit grammarian, astrologer, astronomer and mathematician who studied under Jyeṣṭhadeva and was a member of Madhava o ...

Melpathur Narayana Bhattathiri Melputtur Narayana Bhattatiri ( ml, മേല്പുത്തൂർ നാരായണ ഭട്ടതിരി Mēlputtūr Nārāyaṇa Bhaṭṭatiri; 1560–1646/1666), third student of Achyuta Pisharati, was a member of Madhava of Sangamagr ...

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 ''

Yuktibhāṣā ''Yuktibhāṣā'' ( ml, യുക്തിഭാഷ, lit=Rationale), also known as Gaṇita-yukti-bhāṣā and (''Compendium of Astronomical Rationale''), is a major treatise on mathematics and astronomy, written by the Indian astronomer Jy ...

'' (c.1500–c.1610), written in

Malayalam Malayalam (; , ) is a Dravidian language spoken in the Indian state of Kerala and the union territories of Lakshadweep and Puducherry (Mahé district) by the Malayali people. It is one of 22 scheduled languages of India. Malayalam was des ...

, by Jyesthadeva. Their discovery of these three important series expansions of

—several centuries before calculus was developed in Europe by Isaac Newton and

Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mat ...

—was an achievement. However, the Kerala School did not invent ''calculus'', because, while they were able to develop Taylor series expansions for the important trigonometric functions, differentiation, term by term

convergence tests In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series \sum_^\infty a_n. List of tests Limit of the summand If ...

, iterative methods for solutions of non-linear equations, and the theory that the area under a curve is its integral, they developed neither a theory of differentiation or

, nor the

fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, o ...

. The results obtained by the Kerala school include: *The (infinite)

geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each su ...

\frac = 1 + x + x^2 + x^3 + x^4+ \cdots\text, x, <1

*A semi-rigorous proof (see "induction" remark below) of the result:

1^p+ 2^p + \cdots + n^p \approx \frac

for large ''n''. *Intuitive use of

mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...

, however, the ''

inductive hypothesis Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...

'' was not formulated or employed in proofs. *Applications of ideas from (what was to become) differential and integral calculus to obtain (Taylor–Maclaurin) infinite series for sin x, cos x, and arctan x. The ''Tantrasangraha-vakhya'' gives the series in verse, which when translated to mathematical notation, can be written as: ::

r\arctan\left(\frac\right) = \frac\cdot\frac -\frac\cdot\frac + \frac\cdot\frac - \cdots ,\texty/x \leq 1.

r\sin x = x - x \frac + x \frac\cdot\frac - \cdots

r - \cos x = r \frac - r \frac \frac + \cdots,

: where, for ''r'' = 1, the series reduces to the standard power series for these trigonometric functions, for example: ::

\sin x = x - \frac + \frac - \frac + \cdots

: and ::

\cos x = 1 - \frac + \frac - \frac + \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.) *Use of the series expansion of

\arctan x

to obtain the

Leibniz formula for π In mathematics, the Leibniz formula for , named after Gottfried Leibniz, states that 1-\frac+\frac-\frac+\frac-\cdots=\frac, an alternating series. It is also called the Madhava–Leibniz series as it is a special case of a more general ser ...

: ::

\frac = 1 - \frac + \frac - \frac + \cdots

*A rational approximation of ''error'' for the finite sum of their series of interest. For example, the error,

f_i(n+1)

, (for ''n'' odd, and ''i'' = 1, 2, 3) for the series: ::

\frac \approx 1 - \frac+ \frac - \cdots + (-1)^\frac + (-1)^f_i(n+1)

\textf_1(n) = \frac, \ f_2(n) = \frac, \ f_3(n) = \frac.

*Manipulation of error term to derive a faster converging series for

\pi

: ::

\frac = \frac + \frac - \frac + \frac - \cdots

*Using the improved series to derive a rational expression, 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. *A semi-rigorous (see remark on limits above) method of differentiation of some trigonometric functions. 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.''" 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 ''Yuktibhāṣā'' given in two papers, a commentary on the ''Yuktibhāṣās proof of the sine and cosine series and two papers that provide the Sanskrit verses of the ''Tantrasangrahavakhya'' for the series for arctan, sin, and cosine (with English translation and commentary). Narayana Pandit is a 14th century mathematician who composed two important mathematical 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'').

(c. 1340–1425) was the founder of the Kerala School. Although it is possible that he wrote ''Karana Paddhati'' a work written sometime between 1375 and 1475, all we really know of his work comes from works of later scholars. Parameshvara (c. 1370–1460) wrote commentaries on the works of Bhaskara I,

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.

(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. 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: :

& x^2 - y^2 = e,\ x^3 + y^3 = f,\ x^3 - y^3 = g \end

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-bhāṣā'' (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.

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 Western may refer to: Places *Western, Nebraska, a village in the US *Western, New York, a town in the US *Western Creek, Tasmania, a locality in Australia *Western Junction, Tasmania, a locality in Australia *Western world, countries that id ...

counterparts, as a result of Eurocentrism. According to G. G. Joseph's take on "

Ethnomathematics In mathematics education, ethnomathematics is the study of the relationship between mathematics and culture. Often associated with "cultures without written expression", it may also be defined as "the mathematics which is practised among identifiab ...

heir worktakes on board some of the objections raised about the classical Eurocentric trajectory. The awareness f Indian and Arabic mathematicsis all too likely to be tempered with dismissive rejections of their importance compared to Greek mathematics. The contributions from other civilisations – 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"

The historian of mathematics, Florian Cajori, suggested that he and others "suspect that

got his first glimpse of algebraic knowledge from India." However, he also wrote that "it is certain that portions of Hindu mathematics are of Greek origin".Florian Cajori (2010). "
A History of Elementary Mathematics – With Hints on Methods of Teaching
'". p.94. More recently, as discussed in the above section, the infinite series of

for trigonometric functions (rediscovered by Gregory, Taylor, and Maclaurin in the late 17th century) were described 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 , image = Ihs-logo.svg , image_size = 175px , caption = ChristogramOfficial seal of the Jesuits , abbreviation = SJ , nickname = Jesuits , formation = , founders = ...

missionaries. Kerala was in continuous contact with China and

Arabia The Arabian Peninsula, (; ar, شِبْهُ الْجَزِيرَةِ الْعَرَبِيَّة, , "Arabian Peninsula" or , , "Island of the Arabs") or Arabia, is a peninsula of Western Asia, situated northeast of Africa on the Arabian Plat ...

, 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. 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." Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus. However, they did not, as Newton and

Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...

did, "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." The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; 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." This is an active area of current research, especially in the manuscript collections of Spain and Maghreb. This research is being pursued, among other places, at the

CNRS The French National Centre for Scientific Research (french: link=no, Centre national de la recherche scientifique, CNRS) is the French state research organisation and is the largest fundamental science agency in Europe. In 2016, it employed 31,637 ...

Notes

References

*. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. New edition with translation and commentary, (2 Vols.). *. *. *. *. * *. *. *. *. *. *. *. * *. *. *. *. *. *. *. * * * * * *

External links

Science and Mathematics in India
'' MacTutor History of Mathematics Archive'',

St Andrews University (Aien aristeuein) , motto_lang = grc , mottoeng = Ever to ExcelorEver to be the Best , established = , type = Public research university Ancient university , endowment ...

, 2000.
Indian Mathematicians
''MacTutor History of Mathematics Archive'', St Andrews University, 2004.
Indian Mathematics: Redressing the balanceStudent Projects in the History of Mathematics
Ian Pearce. ''MacTutor History of Mathematics Archive'', St Andrews University, 2002. *

a workshop on traditional Indian sciences for school children conducted by the Computer Science department of Anna University, Chennai, India.
Mathematics in ancient India by R. SridharanCombinatorial methods in ancient India

Mathematics before S. Ramanujan
{{DEFAULTSORT:Indian Mathematics Science and technology in India

Prehistory

Vedic period

Samhitas and Brahmanas

Śulba Sūtras

Pingala (300 BCE – 200 BCE)

Jain mathematics (400 BCE – 200 CE)

Oral tradition

Styles of memorisation

The ''Sutra'' genre

The written tradition: prose commentary

Numerals and the decimal number system

Bakhshali Manuscript

Classical period (400–1600)

Fifth and sixth centuries

Seventh and eighth centuries

Ninth to twelfth centuries

Kerala mathematics (1300–1600)

Charges of Eurocentrism

See also

Notes

References

Further reading

Source books in Sanskrit

External links