Indian mathematics: Difference between revisions

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'''Indian mathematics''' emerged in the [[Indian subcontinent]]<ref name=plofker/> from 1200 BCE<ref name=hayashi2005-p360-361>{{Harv|Hayashi|2005|pp=360–361}}</ref> 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]], [[Brahmagupta]], [[Bhaskara II]], and [[Varāhamihira]]. The [[Decimal|decimal number system]] in use today<ref name=irfah346>{{Harv|Ifrah|2000|p=346}}: "The measure of the genius of Indian civilisation, to which we owe our modern (number) system, is all the greater in that it was the only one in all history to have achieved this triumph.  Some cultures succeeded, earlier than the Indian, in discovering one or at best two of the characteristics of this intellectual feat.  But none of them managed to bring together into a complete and coherent system the necessary and sufficient conditions for a number-system with the same potential as our own. "</ref> was first recorded in Indian mathematics.<ref>{{Harv|Plofker|2009|pp=44–47}}</ref> Indian mathematicians made significant early contributions to the study of the concept of [[0 (number)|zero]] as a number,<ref name=bourbaki46>{{Harv|Bourbaki|1998|p=46}}: "...our decimal system, which (by the agency of the Arabs) is derived from Hindu mathematics, where its use is attested already from the first centuries of our era.  It must be noted moreover that the conception of zero as a number and not as a simple symbol of separation) and its introduction into calculations, also count amongst the original contribution of the Hindus."</ref> [[negative numbers]],<ref name=bourbaki49>{{Harv|Bourbaki|1998|p=49}}: 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 [[Brahmagupta]] 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."</ref> [[arithmetic]], and [[algebra]].<ref name=concise-britannica/> In addition, [[trigonometry]]<ref>{{Harv|Pingree|2003|p=45}} Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. Greek mathematicians used the full chord and never imagined the half chord that we use today.  Half chord was first used by Aryabhata which made trigonometry much more simple.  In fact, the Indian astronomers in the third or fourth century, using a pre-Ptolemaic Greek table of chords, produced tables of sines and versines, from which it was trivial to derive cosines. This new system of trigonometry, produced in India, was transmitted to the Arabs in the late eighth century and by them, in an expanded form, to the Latin West and the Byzantine East in the twelfth century."</ref>

was further advanced in India, and, in particular, the modern definitions of [[sine]] and [[cosine]] were developed there.<ref>{{Harv|Bourbaki|1998|p=126}}: "As for trigonometry, it is disdained by geometers and abandoned to surveyors and astronomers; it is these latter ([[Aristarchus of Samos|Aristarchus]], [[Hipparchus]], [[Ptolemy]]) who establish the fundamental relations between the sides and angles of a right angled triangle (plane or spherical) and draw up the first tables (they consist of tables giving the ''chord'' of the arc cut out by an angle <math>\theta < \pi</math> on a circle of radius ''r'', in other words the number <math> 2r\sin\left(\theta/2\right)</math>; the introduction of the sine, more easily handled, is due to Hindu mathematicians of the Middle Ages)."</ref> These mathematical concepts were transmitted to the Middle East, China, and Europe<ref name=concise-britannica>"algebra"  2007.  [https://www.britannica.com/ebc/article-231064 ''Britannica Concise Encyclopedia''] {{Webarchive|url=https://web.archive.org/web/20070929134632/http://www.britannica.com/ebc/article-231064 |date=29 September 2007 }}.  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."</ref> and led to further developments that now form the foundations of many areas of mathematics.

A later landmark in Indian mathematics was the development of the [[Series (mathematics)|series]] expansions for [[trigonometric function]]s (sine, cosine, and [[arc tangent]]) by mathematicians of the [[Kerala school of astronomy and mathematics|Kerala school]] in the 15th century CE.  Their remarkable work, completed two centuries before the invention of [[calculus]] in Europe, provided what is now considered the first example of a [[power series]] (apart from geometric series).<ref>{{Harv|Stillwell|2004|p=173}}</ref> However, they did not formulate a systematic theory of [[derivative|differentiation]] and [[integral|integration]], nor is there any ''direct'' evidence of their results being transmitted outside [[Kerala]].<ref>{{Harv|Bressoud|2002|p=12}} Quote: "There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use."</ref><ref>{{Harv|Plofker|2001|p=293}} Quote: "It is not unusual to encounter in discussions of Indian mathematics such assertions as that "the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)” [Joseph 1991, 300], or that "we may consider Madhava to have been the founder of mathematical analysis" (Joseph 1991, 293), or that Bhaskara II may claim to be "the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus" (Bag 1979, 294). ... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). ... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian "discovery of the principle of the differential calculus" somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context.  The differential "principle" was not generalised to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its

Excavations at [[Harappa]], [[Mohenjo-daro]] and other sites of the [[Indus Valley civilisation]] have uncovered evidence of the use of "practical mathematics". The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4:2:1, considered favourable for the stability of a brick structure. They used a 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&nbsp;grams (and approximately equal to the English ounce or Greek uncia).  They mass-produced weights in regular [[geometrical]] shapes, which included [[hexahedron|hexahedra]], [[barrel]]s, [[cone (geometry)|cone]]s, and [[cylinder (geometry)|cylinder]]s, thereby demonstrating knowledge of basic [[geometry]].<ref>{{Citation|last=Sergent|first=Bernard|title=Genèse de l'Inde|year=1997|page=113|language=fr|isbn=978-2-228-89116-5|publisher=Payot|location=Paris}}</ref>

The religious texts of the [[Vedic Period]] provide evidence for the use of [[History of large numbers|large numbers]]. By the time of the ''[[Yajurveda|{{IAST|Yajurvedasaṃhitā-}}]]'' (1200–900 BCE), numbers as high as {{math|10¹²}} were being included in the texts.<ref name="hayashi2005-p360-361"/> For example, the ''[[mantra]]'' (sacred recitation) at the end of the ''annahoma'' ("food-oblation rite") performed during the [[Ashvamedha|''aśvamedha'']], and uttered just before-, during-, and just after sunrise, invokes powers of ten from a hundred to a trillion:<ref name=hayashi2005-p360-361/>

The ''[[Shulba Sutras|Śulba Sūtras]]'' (literally, "Aphorisms of the Chords" in [[Vedic Sanskrit]]) (c. 700–400 BCE) list rules for the construction of sacrificial fire altars.<ref>{{Harv|Staal|1999}}</ref> Most mathematical problems considered in the  ''Śulba Sūtras'' spring from "a single theological requirement,"<ref name=hayashi2003-p118>{{Harv|Hayashi|2003|p=118}}</ref> 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.<ref name=hayashi2003-p118/>

[[Baudhayana]] (c. 8th century BCE) composed the ''Baudhayana Sulba Sutra'', the best-known ''Sulba Sutra'', which contains examples of simple Pythagorean triples, such as: {{math|(3, 4, 5)}}, {{math|(5, 12, 13)}}, {{math|(8, 15, 17)}}, {{math|(7, 24, 25)}}, and {{math|(12, 35, 37)}},<ref name=joseph229>{{Harv|Joseph|2000|p=229}}</ref> 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."<ref name=joseph229/><ref>{{Cite web|url=https://www.theintellibrain.com/vedicmaths/|title=Vedic Maths Complete Detail|website= ALLEN IntelliBrain|access-date=22 October 2022}}</ref> 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."<ref name=joseph229/> Baudhayana gives an expression for the [[square root of two]]:<ref name=cooke200>{{Harv|Cooke|2005|p=200}}</ref>

According to mathematician S. G. Dani, the Babylonian cuneiform tablet [[Plimpton 322]] written c. 1850 BCE<ref>Mathematics Department, University of British Columbia, [http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.html ''The Babylonian tabled Plimpton 322''] {{Webarchive|url=https://web.archive.org/web/20200617151320/http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.html |date=17 June 2020 }}.</ref> "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple,<ref>Three positive integers <math>(a, b, c) </math> form a ''primitive'' Pythagorean triple if {{math|1=c² = a²+b²}} and if the highest common factor of {{math|a, b, c}} is 1. In the particular Plimpton322 example, this means that {{math|1=13500²+12709² = 18541²}} and that the three numbers do not have any common factors. However some scholars have disputed the Pythagorean interpretation of this tablet; see Plimpton 322 for details.</ref> 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."<ref name=dani>{{Harv|Dani|2003}}</ref> Dani goes on to say:

An important landmark of the Vedic period was the work of [[Sanskrit grammarian]], {{IAST|[[Pāṇini]]}} (c. 520–460 BCE). His grammar includes early use of [[Boolean logic]], of the [[Null function|null]] operator, and of [[context free grammar]]s, and includes a precursor of the [[Backus–Naur form]] (used in the description [[programming languages]]).<ref>{{cite journal|last1=Ingerman|first1=Peter Zilahy|title="Pānini-Backus Form" suggested|journal=Communications of the ACM|date=1 March 1967|volume=10|issue=3|pages=137|doi=10.1145/363162.363165|s2cid=52817672|issn=0001-0782|doi-access=free}}</ref><ref>{{cite web|title=Panini-Backus|url=http://www.infinityfoundation.com/mandala/t_es/t_es_rao-t_syntax.htm|website=infinityfoundation.com|access-date=16 March 2018}}</ref>

Among the scholars of the post-Vedic period who contributed to mathematics, the most notable is [[Pingala]] (''{{IAST|piṅgalá}}'') ([[Floruit|fl.]] 300–200 BCE), a [[music theory|music theorist]] who authored the [[Chhandas]] [[Shastra]] (''{{IAST|chandaḥ-śāstra}}'', also Chhandas Sutra ''{{IAST|chhandaḥ-sūtra}}''), a [[Sanskrit]] treatise on [[Sanskrit prosody|prosody]].  There is evidence that in his work on the enumeration of syllabic combinations, Pingala stumbled upon both [[Pascal's triangle]] and [[binomial coefficient]]s, although he did not have knowledge of the [[binomial theorem]] itself.<ref name=fowler96>{{Harv|Fowler|1996|p=11}}</ref><ref name=singh36>{{Harv|Singh|1936|pp=623–624}}</ref> Pingala's work also contains the basic ideas of [[Fibonacci number]]s (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 ''[[Mount Meru (mythology)|Meru]]-prastāra'' (literally "the staircase to Mount Meru"), has this to say:

Prodigious energy was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity.<ref>{{Harv|Staal|1986}}</ref> For example, memorisation of the sacred ''[[Veda]]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 ''{{IAST|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 in the original order.<ref name=filliozat-p139>{{Harv|Filliozat|2004|p=139}}</ref> The recitation thus proceeded as:

That these methods have been effective, is testified to by the preservation of the most ancient Indian religious text, the ''[[Rigveda|{{IAST|Ṛgveda}}]]'' (c. 1500 BCE), as a single text, without any variant readings.<ref name=filliozat-p139/> Similar methods were used for memorising mathematical texts, whose transmission remained exclusively oral until the end of the [[Vedic period]] (c. 500 BCE).

Mathematical activity in ancient India began as a part of a "methodological reflexion" on the sacred [[Veda]]s, which took the form of works called [[Vedanga|''{{IAST|Vedāṇgas}}'']], or, "Ancillaries of the Veda" (7th–4th century BCE).<ref name=filliozat2004-p140-141>{{Harv|Filliozat|2004|pp=140–141}}</ref> The need to conserve the sound of sacred text by use of [[shiksha|''{{IAST|śikṣā}}'']] ([[phonetics]]) and ''[[chhandas]]'' ([[Metre (poetry)|metric]]s); to conserve its meaning by use of [[vyakarana|''{{IAST|vyākaraṇa}}'']] ([[grammar]]) and ''[[nirukta]]'' ([[etymology]]); and to correctly perform the rites at the correct time by the use of ''[[Kalpa (aeon)|kalpa]]'' ([[ritual]]) and [[jyotisha|''{{IAST|jyotiṣa}}'']] ([[astrology]]), gave rise to the six disciplines of the ''{{IAST|Vedāṇgas}}''.<ref name=filliozat2004-p140-141/> Mathematics arose as a part of the last two disciplines, ritual and astronomy (which also included astrology).

Extreme brevity was achieved through multiple means, which included using [[ellipsis]] "beyond the tolerance of natural language,"<ref name=filliozat2004-p140-141/> using technical names instead of longer descriptive names, abridging lists by only mentioning the first and last entries, and using markers and variables.<ref name=filliozat2004-p140-141/> 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 tradition|''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.<ref>{{Harv|Yano|2006|p=146}}</ref> The brevity achieved in a ''sūtra'' is demonstrated in the following example from the Baudhāyana ''Śulba Sūtra'' (700 BCE).

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.<ref name=filliozat2004-p143-144>{{Harv|Filliozat|2004|pp=143–144}}</ref> 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:

The earliest mathematical prose commentary was that on the work, ''[[Aryabhatiya|{{IAST|Āryabhaṭīya}}]]'' (written 499 CE), a work on astronomy and mathematics.  The mathematical portion of the ''{{IAST|Āryabhaṭīya}}'' was composed of 33 ''sūtras'' (in verse form) consisting of mathematical statements or rules, but without any proofs.<ref name=hayashi03-p122-123>{{Harv|Hayashi|2003|pp=122–123}}</ref> However, according to {{Harv|Hayashi|2003|p=123}}, "this does not necessarily mean that their authors did not prove them.  It was probably a matter of style of exposition."  From the time of [[Bhaskara I]] (600 CE onwards), prose commentaries increasingly began to include some derivations (''upapatti'').  Bhaskara I's commentary on the ''{{IAST|Āryabhaṭīya}}'', had the following structure:<ref name=hayashi03-p122-123/>

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.<ref name=plofker2007-p395>{{Harv|Plofker|2007|p=395}}</ref> The Syrian bishop [[Severus Sebokht]] wrote in the mid-7th century CE about the "nine signs" of the Indians for expressing numbers.<ref name=plofker2007-p395/> However, how, when, and where the first decimal place value system was invented is not so clear.<ref>{{Harv|Plofker|2007|p=395}}; {{Harv|Plofker|2009|pp=47–48}}</ref>

The earliest surviving evidence of decimal place value numerals in India and southeast Asia is from the middle of the first millennium CE.<ref name=plofker2009-p45>{{Harv|Plofker|2009|p=45}}</ref> 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.<ref name=plofker2009-p45/> Decimal numerals recording the years 683 CE have also been found in stone inscriptions in Indonesia and Cambodia, where Indian cultural influence was substantial.<ref name=plofker2009-p45/>

There are older textual sources, although the extant manuscript copies of these texts are from much later dates.<ref name=plofker2009-p46>{{Harv|Plofker|2009|p=46}}</ref> Probably the earliest such source is the work of the Buddhist philosopher Vasumitra dated likely to the 1st century CE.<ref name=plofker2009-p46/> Discussing the counting pits of merchants, Vasumitra remarks, "When [the same] clay counting-piece is in the place of units, it is denoted as one, when in hundreds, one hundred."<ref name=plofker2009-p46/> 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."<ref name=plofker2009-p46/>

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.<ref name=plofker2009-p47>{{Harv|Plofker|2009|p=47}}</ref> 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.<ref name=plofker2009-p47/> According to {{Harv|Plofker|2009}}, the number 4, for example, could be represented by the word "[[Veda]]" (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).<ref name=plofker2009-p47/> 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.<ref name=plofker2009-p47/> The earliest reference employing object numbers is a c. 269 CE Sanskrit text, [[Yavanajataka|''Yavanajātaka'']] (literally "Greek horoscopy") of Sphujidhvaja, a versification of an earlier (c. 150 CE) Indian prose adaptation of a lost work of Hellenistic astrology.<ref>{{Harv|Pingree|1978|p=494}}</ref> 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.<ref name=plofker2009-p47/>

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.<ref name=plofker2009-p48>{{Harv|Plofker|2009|p=48}}</ref> According to {{Harv|Plofker|2009}}, <blockquote>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."<ref name=plofker2009-p48/></blockquote>

The oldest extant mathematical manuscript in India is the ''[[Bakhshali Manuscript]]'', a birch bark manuscript written in "Buddhist hybrid Sanskrit"<ref name=plofker-brit6/> in the ''Śāradā'' script, which was used in the northwestern region of the Indian subcontinent between the 8th and 12th centuries CE.<ref name=hayashi2005-371>{{Harv|Hayashi|2005|p=371}}</ref> The manuscript was discovered in 1881 by a farmer while digging in a stone enclosure in the village of Bakhshali, near [[Peshawar]] (then in [[British India]] and now in [[Pakistan]]).  Of unknown authorship and now preserved in the [[Bodleian Library]] in the [[University of Oxford]], the manuscript has been dated recently as 224 AD- 383 AD.<ref>{{cite web | url=https://blog.sciencemuseum.org.uk/illuminating-india-starring-oldest-recorded-origins-zero-bakhshali-manuscript/ | title=Illuminating India: Starring the oldest recorded origins of 'zero', the Bakhshali manuscript }}</ref>

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.<ref name=hayashi2005-371/> The topics treated include arithmetic (fractions, square roots, profit and loss, simple interest, the [[rule of three (mathematics)|rule of three]], and ''[[regula falsi]]'') and algebra (simultaneous linear equations and [[quadratic equations]]), and arithmetic progressions.  In addition, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."<ref name=hayashi2005-371/> 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:

This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as [[Aryabhata]], [[Varahamihira]], [[Brahmagupta]], [[Bhaskara I]], [[Mahavira (mathematician)|Mahavira]], [[Bhaskara II]], [[Madhava of Sangamagrama]] and [[Nilakantha Somayaji]] 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).<ref name=hayashi2003-p119>{{Harv|Hayashi|2003|p=119}}</ref> This tripartite division is seen in Varāhamihira's 6th century compilation—''Pancasiddhantika''<ref>{{Harv|Neugebauer|Pingree|1970}}</ref> (literally ''panca'', "five," ''siddhānta'', "conclusion of deliberation", dated 575 [[Common Era|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.<ref name=hayashi2003-p119/>

===Fourth to sixth centuries===

Though its authorship is unknown, the ''[[Surya Siddhanta]] ,'' [[Hinduism|a hindu treatise]] (c. 400) contains the roots of modern [[trigonometry]].{{Citation needed|date=March 2011}} Because it contains many words of foreign origin, some authors consider that it was written under the influence of [[Babylonian mathematics|Mesopotamia]] and Greece.<ref name="Origins of Sulva Sutras and Siddhanta">{{Citation|first=Roger|last=Cooke|author-link=Roger Cooke (mathematician)|title=The History of Mathematics: A Brief Course|publisher=Wiley-Interscience|year=1997|chapter=The Mathematics of the Hindus|isbn=978-0-471-18082-1|quote=The word ''Siddhanta'' means ''that which is proved or established''. The ''Sulva Sutras'' are of Hindu origin, but the ''Siddhantas'' contain so many words of foreign origin that they undoubtedly have roots in [[Mesopotamia]] and Greece.|page=[https://archive.org/details/historyofmathema0000cook/page/197 197]|chapter-url=https://archive.org/details/historyofmathema0000cook/page/197}}</ref>{{Better source needed|date=April 2017}}

Later Indian mathematicians such as Aryabhata made references to this text, while later [[Arabic]] and [[Latin]] translations were very influential in Europe and the Middle East. calendar.

In the 7th century, two separate fields, [[arithmetic]] (which included [[measurement]]) and [[algebra]], began to emerge in Indian mathematics.  The two fields would later be called ''{{IAST|pāṭī-gaṇita}}'' (literally "mathematics of algorithms") and ''{{IAST|bīja-gaṇita}}'' (lit. "mathematics of seeds," with "seeds"—like the seeds of plants—representing unknowns with the potential to generate, in this case, the solutions of equations).<ref name=hayashi2005-p369>{{Harv|Hayashi|2005|p=369}}</ref> [[Brahmagupta]], in his astronomical work ''[[Brahmasphutasiddhanta|{{IAST|Brāhma Sphuṭa Siddhānta}}]]'' (628 CE), included two chapters (12 and 18) devoted to these fields.  Chapter 12, containing 66 Sanskrit verses, was divided into two sections:  "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).<ref name=hayashi2003-p121-122>{{Harv|Hayashi|2003|pp=121–122}}</ref> In the latter section, he stated his famous theorem on the diagonals of a [[cyclic quadrilateral]]:<ref name=hayashi2003-p121-122/>

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

Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers<ref name=hayashi2003-p121-122/> and is considered the first systematic treatment of the subject.  The rules (which included <math> a + 0 = \ a</math> and <math> a \times 0 = 0 </math>) were all correct, with one exception: <math> \frac{0}{0} = 0 </math>.<ref name=hayashi2003-p121-122/> Later in the chapter, he gave the first explicit (although still not completely general) solution of the '''[[quadratic equation]]''':

which was a generalisation of an earlier identity of [[Diophantus]]:<ref name=stillwell2004-p72-73/> Brahmagupta used his identity to prove the following lemma:<ref name=stillwell2004-p72-73/>

In his commentary, Brahmagupta added, "a person solving this problem within a year is a mathematician."<ref name=stillwell2004-p72-73/> The solution he provided was:

The [[Kerala school of astronomy and mathematics]] was founded by [[Madhava of Sangamagrama]] in Kerala, [[South India]] and included among its members: [[Parameshvara]], [[Neelakanta Somayaji]], [[Jyeshtadeva]], [[Achyuta Pisharati]], [[Melpathur Narayana Bhattathiri]] and Achyuta Panikkar. It flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559–1632).  In attempting to solve astronomical problems, the Kerala school astronomers ''independently'' created a number of important mathematics concepts.  The most important results, series expansion for [[trigonometric function]]s, 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]]'' (c.1500–c.1610), written in [[Malayalam]], by [[Jyesthadeva]].<ref name=roy>{{Harv|Roy|1990}}</ref>

Their discovery of these three important series expansions of [[calculus]]—several centuries before calculus was developed in Europe by [[Isaac Newton]] and [[Gottfried Leibniz]]—was an achievement.  However, the Kerala School did not invent ''calculus'',<ref name=bressoud/> because, while they were able to develop [[Taylor series]] expansions for the important [[trigonometric functions]], [[Derivative|differentiation]], term by term [[Integral|integration]], [[convergence tests]], [[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 [[Derivative|differentiation]] or [[Integral|integration]], nor the [[fundamental theorem of calculus]].<ref name=katz/> The results obtained by the Kerala school include:

*Applications of ideas from (what was to become) differential and integral calculus to obtain [[Taylor's theorem|(Taylor–Maclaurin) infinite series]] for sin x, cos x, and arctan x.<ref name=bressoud>{{Harv|Bressoud|2002}}</ref> The ''Tantrasangraha-vakhya'' gives the series in verse, which when translated to mathematical notation, can be written as:<ref name=roy/>

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,<ref>{{Citation | last1 = Rajagopal | first1 = C. | last2 = Rangachari | first2 = M. S. | year = 1949 | title = A Neglected Chapter of Hindu Mathematics | journal = [[Scripta Mathematica]] | volume = 15 | pages = 201–209 | postscript = . }}</ref><ref>{{Citation | last1 = Rajagopal | first1 = C. | last2 = Rangachari | first2 = M. S. | year = 1951 | title = On the Hindu proof of Gregory's series | journal = [[Scripta Mathematica]] | volume = 17 | pages = 65–74 | postscript = . }}</ref> a commentary on the ''Yuktibhāṣā'''s proof of the sine and cosine series<ref>{{Citation | last1 = Rajagopal | first1 = C. | last2 = Venkataraman | first2 = A. | year = 1949 | title = The sine and cosine power series in Hindu mathematics | journal = Journal of the Royal Asiatic Society of Bengal (Science) | volume = 15 | pages = 1–13 | postscript = . }}</ref> and two papers that provide the Sanskrit verses of the ''Tantrasangrahavakhya'' for the series for arctan, sin, and cosine (with English translation and commentary).<ref>{{Citation | last1 = Rajagopal | first1 = C. | last2 = Rangachari | first2 = M. S. | year = 1977 | title = On an untapped source of medieval Keralese mathematics | doi = 10.1007/BF00348142 | journal = Archive for History of Exact Sciences | volume = 18 | issue = 2 | pages = 89–102 | s2cid = 51861422 | postscript = . }}</ref><ref>{{Citation | last1 = Rajagopal | first1 = C. | last2 = Rangachari | first2 = M. S. | year = 1986 | title = On Medieval Kerala Mathematics | journal = Archive for History of Exact Sciences | volume = 35 | issue = 2| pages = 91–99 | doi = 10.1007/BF00357622 | s2cid = 121678430 | postscript = . }}</ref>

</ref> However, he also wrote that "it is certain that portions of Hindu mathematics are of Greek origin".<ref>Florian Cajori (2010). "''[https://books.google.com/books?id=gZ2Us3F7dSwC&pg=PA94 A History of Elementary Mathematics – With Hints on Methods of Teaching]''".  p.94. {{ISBN|1-4460-2221-8}}</ref>

More recently, as discussed in the above section, the infinite series of [[calculus]] for trigonometric functions (rediscovered by Gregory, Taylor, and Maclaurin in the late 17th century) were described in India, by mathematicians of the [[Kerala school of astronomy and mathematics|Kerala school]], remarkably some two centuries earlier. Some scholars have recently suggested that knowledge of these results might have been transmitted to Europe through the trade route from [[Kerala]] by traders and [[Jesuit]] missionaries.<ref name=almeida/> Kerala was in continuous contact with China and [[Arabia]], and, from around 1500, with Europe.  The existence of communication routes and a suitable chronology certainly make such a transmission a possibility. However, there is no direct evidence by way of relevant manuscripts that such a transmission actually took place.<ref name=almeida>{{Citation | last1 = Almeida | first1 = D. F. | last2 = John | first2 = J. K. | last3 = Zadorozhnyy | first3 = A. | year = 2001 | title = Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications | journal = Journal of Natural Geometry | volume = 20 | pages = 77–104 | postscript = . }}</ref> 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."<ref name=bressoud/><ref name=gold>{{Citation | last1 = Gold | first1 = D. | last2 = Pingree | first2 = D. | year = 1991 | title = A hitherto unknown Sanskrit work concerning Madhava's derivation of the power series for sine and cosine | journal = Historia Scientiarum | volume = 42 | pages = 49–65 | postscript = . }}</ref>

Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus.<ref name=katz/> However, they did not, as [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]] 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."<ref name=katz/> The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own;<ref name=katz/> 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."<ref name=katz/> 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]].<ref name=katz/>

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