131-810: Naveen or Navin ( Sanskrit : नवीन, IAST : navīna ; / n ə ˈ v iː n / na- VEEN ) is an Indian and Pakistani male given name and surname . The word means "new", "young", "bright", "creative". Naveen is chiefly used in Indian languages, and its origin is Sanskrit . It is derived from the element 'Navina' meaning new. The name 'Navina' is the female form of Naveen. Navean, Naveane, Naveene, Navin (English and Indian), Navine, Navyn, and Navyne are variants of Naveen. Notable people with that name include: Sanskrit Sanskrit ( / ˈ s æ n s k r ɪ t / ; attributively 𑀲𑀁𑀲𑁆𑀓𑀾𑀢𑀁 , संस्कृत- , saṃskṛta- ; nominally संस्कृतम् , saṃskṛtam , IPA: [ˈsɐ̃skr̩tɐm] )
262-587: 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
393-451: A dead language in the most common usage of the term. Pollock's notion of the "death of Sanskrit" remains in this unclear realm between academia and public opinion when he says that "most observers would agree that, in some crucial way, Sanskrit is dead." Indian mathematics#Styles of memorisation Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of
524-433: A focus on Indian philosophies and Sanskrit. Though written in a number of different scripts, the dominant language of Hindu texts has been Sanskrit. It or a hybrid form of Sanskrit became the preferred language of Mahayana Buddhism scholarship; for example, one of the early and influential Buddhist philosophers, Nagarjuna (~200 CE), used Classical Sanskrit as the language for his texts. According to Renou, Sanskrit had
655-478: 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
786-469: A highly compressed mnemonic form, the sūtra (literally, "thread"): The knowers of the sūtra know it as having few phonemes, being devoid of ambiguity, containing the essence, facing everything, being without pause and unobjectionable. 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
917-538: A language competed with numerous, less exact vernacular Indian languages called Prakritic languages ( prākṛta - ). The term prakrta literally means "original, natural, normal, artless", states Franklin Southworth . The relationship between Prakrit and Sanskrit is found in Indian texts dated to the 1st millennium CE. Patañjali acknowledged that Prakrit is the first language, one instinctively adopted by every child with all its imperfections and later leads to
1048-658: A limited role in the Theravada tradition (formerly known as the Hinayana) but the Prakrit works that have survived are of doubtful authenticity. Some of the canonical fragments of the early Buddhist traditions, discovered in the 20th century, suggest the early Buddhist traditions used an imperfect and reasonably good Sanskrit, sometimes with a Pali syntax, states Renou. The Mahāsāṃghika and Mahavastu, in their late Hinayana forms, used hybrid Sanskrit for their literature. Sanskrit
1179-673: A mathematical text called Tiloyapannati ; and Umasvati (c. 150 BCE), who, although better known for his influential writings on Jain philosophy and metaphysics , composed a mathematical work called the Tattvārtha Sūtra . Mathematicians of ancient and early medieval India were almost all Sanskrit pandits ( paṇḍita "learned man"), who were trained in Sanskrit language and literature, and possessed "a common stock of knowledge in grammar ( vyākaraṇa ), exegesis ( mīmāṃsā ) and logic ( nyāya )." Memorisation of "what
1310-454: A natural part of the earliest Vedic language, and that these developed in the centuries after the composition had been completed, and as a gradual unconscious process during the oral transmission by generations of reciters. The primary source for this argument is internal evidence of the text which betrays an instability of the phenomenon of retroflexion, with the same phrases having sandhi-induced retroflexion in some parts but not other. This
1441-479: A negative evidence to Pollock's hypothesis, but it is not positive evidence. A closer look at Sanskrit in the Indian history after the 12th century suggests that Sanskrit survived despite the odds. According to Hanneder, On a more public level the statement that Sanskrit is a dead language is misleading, for Sanskrit is quite obviously not as dead as other dead languages and the fact that it is spoken, written and read will probably convince most people that it cannot be
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#17328015844471572-546: A pan-Indo-Aryan accessibility to information and knowledge in the ancient and medieval times, in contrast to the Prakrit languages which were understood just regionally. It created a cultural bond across the subcontinent. As local languages and dialects evolved and diversified, Sanskrit served as the common language. It connected scholars from distant parts of South Asia such as Tamil Nadu and Kashmir, states Deshpande, as well as those from different fields of studies, though there must have been differences in its pronunciation given
1703-457: A part of a "methodological reflexion" on the sacred Vedas , which took the form of works called Vedāṇgas , or, "Ancillaries of the Veda" (7th–4th century BCE). The need to conserve the sound of sacred text by use of śikṣā ( phonetics ) and chhandas ( metrics ); to conserve its meaning by use of vyākaraṇa ( grammar ) and nirukta ( etymology ); and to correctly perform
1834-590: 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
1965-545: A precursor of the Backus–Naur form (used in the description programming languages ). Among the scholars of the post-Vedic period who contributed to mathematics, the most notable is Pingala ( piṅgalá ) ( fl. 300–200 BCE), a music theorist who authored the Chhandas Shastra ( chandaḥ-śāstra , also Chhandas Sutra chhandaḥ-sūtra ), a Sanskrit treatise on prosody . Pingala's work also contains
2096-522: A refined and standardized grammatical form that emerged in the mid-1st millennium BCE and was codified in the most comprehensive of ancient grammars, the Aṣṭādhyāyī ('Eight chapters') of Pāṇini . The greatest dramatist in Sanskrit, Kālidāsa , wrote in classical Sanskrit, and the foundations of modern arithmetic were first described in classical Sanskrit. The two major Sanskrit epics, the Mahābhārata and
2227-538: A restrained language from which archaisms and unnecessary formal alternatives were excluded". The Classical form of the language simplified the sandhi rules but retained various aspects of the Vedic language, while adding rigor and flexibilities, so that it had sufficient means to express thoughts as well as being "capable of responding to the future increasing demands of an infinitely diversified literature", according to Renou. Pāṇini included numerous "optional rules" beyond
2358-424: 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
2489-439: A similar phonetic structure to Tamil. Hock et al. quoting George Hart state that there was influence of Old Tamil on Sanskrit. Hart compared Old Tamil and Classical Sanskrit to arrive at a conclusion that there was a common language from which these features both derived – "that both Tamil and Sanskrit derived their shared conventions, metres, and techniques from a common source, for it is clear that neither borrowed directly from
2620-562: 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 shapes, which included hexahedra , barrels , cones , and cylinders , thereby demonstrating knowledge of basic geometry . The inhabitants of Indus civilisation also tried to standardise measurement of length to
2751-518: 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
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#17328015844472882-711: Is a classical language belonging to the Indo-Aryan branch of the Indo-European languages . It arose in South Asia after its predecessor languages had diffused there from the northwest in the late Bronze Age . Sanskrit is the sacred language of Hinduism , the language of classical Hindu philosophy , and of historical texts of Buddhism and Jainism . It was a link language in ancient and medieval South Asia, and upon transmission of Hindu and Buddhist culture to Southeast Asia, East Asia and Central Asia in
3013-403: 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
3144-464: Is akin to that of Latin and Ancient Greek in Europe. Sanskrit has significantly influenced most modern languages of the Indian subcontinent , particularly the languages of the northern, western, central and eastern Indian subcontinent. Sanskrit declined starting about and after the 13th century. This coincides with the beginning of Islamic invasions of South Asia to create, and thereafter expand
3275-451: Is described in the following words: II.64. After dividing the quadri-lateral in seven, one divides the transverse [cord] in three. II.65. In another layer one places the [bricks] North-pointing. According to Filliozat, the officiant constructing the altar has only a few tools and materials at his disposal: a cord (Sanskrit, rajju , f.), two pegs (Sanskrit, śanku , m.), and clay to make the bricks (Sanskrit, iṣṭakā , f.). Concision
3406-452: Is found in the writing of Bharata Muni , the author of the ancient Natya Shastra text. The early Jain scholar Namisādhu acknowledged the difference, but disagreed that the Prakrit language was a corruption of Sanskrit. Namisādhu stated that the Prakrit language was the pūrvam ('came before, origin') and that it came naturally to children, while Sanskrit was a refinement of Prakrit through "purification by grammar". Sanskrit belongs to
3537-503: Is heard" ( śruti 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." Prodigious energy
3668-532: Is rare in the later version of the language. The Homerian Greek, like Ṛg-vedic Sanskrit, deploys simile extensively, but they are structurally very different. The early Vedic form of the Sanskrit language was far less homogenous compared to the Classical Sanskrit as defined by grammarians by about the mid-1st millennium BCE. According to Richard Gombrich—an Indologist and a scholar of Sanskrit, Pāli and Buddhist Studies—the archaic Vedic Sanskrit found in
3799-479: Is taken along with evidence of controversy, for example, in passages of the Aitareya-Āraṇyaka (700 BCE), which features a discussion on whether retroflexion is valid in particular cases. The Ṛg-veda is a collection of books, created by multiple authors. These authors represented different generations, and the mandalas 2 to 7 are the oldest while the mandalas 1 and 10 are relatively the youngest. Yet,
3930-589: Is the predominant language of one of the largest collection of historic manuscripts. The earliest known inscriptions in Sanskrit are from the 1st century BCE, such as the Ayodhya Inscription of Dhana and Ghosundi-Hathibada (Chittorgarh) . Though developed and nurtured by scholars of orthodox schools of Hinduism, Sanskrit has been the language for some of the key literary works and theology of heterodox schools of Indian philosophies such as Buddhism and Jainism. The structure and capabilities of
4061-566: Is there any direct evidence of their results being transmitted outside Kerala . 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
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4192-543: 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 used in India
4323-522: The uṣas (dawn) , hail to the vyuṣṭi (twilight), hail to udeṣyat (the one which is going to rise), hail to udyat (the one which is rising), hail udita (to the one which has just risen), hail to svarga (the heaven), hail to martya (the world), hail to all. The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta (RV 10.90.4): With three-fourths Puruṣa went up: one-fourth of him again
4454-540: The Bhagavata Purana , the Panchatantra and many other texts are all in the Sanskrit language. The Classical Sanskrit with its exacting grammar was thus the language of the Indian scholars and the educated classes, while others communicated with approximate or ungrammatical variants of it as well as other natural Indian languages. Sanskrit, as the learned language of Ancient India, thus existed alongside
4585-711: The Sthānāṅga Sūtra (c. 300 BCE – 200 CE); the Anuyogadwara Sutra (c. 200 BCE – 100 CE), which includes the earliest known description of factorials in Indian mathematics; and the Ṣaṭkhaṅḍāgama (c. 2nd century CE). Important Jain mathematicians included Bhadrabahu (d. 298 BCE), the author of two astronomical works, the Bhadrabahavi-Samhita and a commentary on the Surya Prajinapti ; Yativrisham Acharya (c. 176 BCE), who authored
4716-666: The aśvamedha , and uttered just before-, during-, and just after sunrise, invokes powers of ten from a hundred to a trillion: Hail to śata ("hundred," 10 ), hail to sahasra ("thousand," 10 ), hail to ayuta ("ten thousand," 10 ), hail to niyuta ("hundred thousand," 10 ), hail to prayuta ("million," 10 ), hail to arbuda ("ten million," 10 ), hail to nyarbuda ("hundred million," 10 ), hail to samudra ("billion," 10 , literally "ocean"), hail to madhya ("ten billion," 10 , literally "middle"), hail to anta ("hundred billion," 10 , lit., "end"), hail to parārdha ("one trillion," 10 lit., "beyond parts"), hail to
4847-580: The Dalai Lama , the Sanskrit language is a parent language that is at the foundation of many modern languages of India and the one that promoted Indian thought to other distant countries. In Tibetan Buddhism, states the Dalai Lama, Sanskrit language has been a revered one and called legjar lhai-ka or "elegant language of the gods". It has been the means of transmitting the "profound wisdom of Buddhist philosophy" to Tibet. The Sanskrit language created
4978-613: The Indo-European family of languages . It is one of the three earliest ancient documented languages that arose from a common root language now referred to as Proto-Indo-European : Other Indo-European languages distantly related to Sanskrit include archaic and Classical Latin ( c. 600 BCE–100 CE, Italic languages ), Gothic (archaic Germanic language , c. 350 CE ), Old Norse ( c. 200 CE and after), Old Avestan ( c. late 2nd millennium BCE ) and Younger Avestan ( c. 900 BCE). The closest ancient relatives of Vedic Sanskrit in
5109-642: The Manava Sulba Sutra composed by Manava (fl. 750–650 BCE) and the Apastamba Sulba Sutra , composed by Apastamba (c. 600 BCE), contained results similar to the Baudhayana Sulba Sutra . An important landmark of the Vedic period was the work of Sanskrit grammarian , Pāṇini (c. 520–460 BCE). His grammar includes early use of Boolean logic , of the null operator, and of context free grammars , and includes
5240-506: The Old Babylonians ." The diagonal rope ( akṣṇayā-rajju ) of an oblong (rectangle) produces both which the flank ( pārśvamāni ) and the horizontal ( tiryaṇmānī ) <ropes> produce separately." 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
5371-753: The Rigveda had already evolved in the Vedic period, as evidenced in the later Vedic literature. Gombrich posits that the language in the early Upanishads of Hinduism and the late Vedic literature approaches Classical Sanskrit, while the archaic Vedic Sanskrit had by the Buddha 's time become unintelligible to all except ancient Indian sages. The formalization of the Saṃskṛta language is credited to Pāṇini , along with Patañjali's Mahābhāṣya and Katyayana's commentary that preceded Patañjali's work. Panini composed Aṣṭādhyāyī ('Eight-Chapter Grammar'), which became
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5502-532: The Rigveda , a collection of 1,028 hymns composed between 1500 BCE and 1200 BCE by Indo-Aryan tribes migrating east from the mountains of what is today northern Afghanistan across northern Pakistan and into northwestern India. Vedic Sanskrit interacted with the preexisting ancient languages of the subcontinent, absorbing names of newly encountered plants and animals; in addition, the ancient Dravidian languages influenced Sanskrit's phonology and syntax. Sanskrit can also more narrowly refer to Classical Sanskrit ,
5633-531: The Rāmāyaṇa , however, were composed in a range of oral storytelling registers called Epic Sanskrit which was used in northern India between 400 BCE and 300 CE, and roughly contemporary with classical Sanskrit. In the following centuries, Sanskrit became tradition-bound, stopped being learned as a first language, and ultimately stopped developing as a living language. The hymns of the Rigveda are notably similar to
5764-406: The sandhi rules, both internal and external. Quite many words found in the early Vedic Sanskrit language are never found in late Vedic Sanskrit or Classical Sanskrit literature, while some words have different and new meanings in Classical Sanskrit when contextually compared to the early Vedic Sanskrit literature. Arthur Macdonell was among the early colonial era scholars who summarized some of
5895-599: The square root of 2 correct to five decimal places. Although Jainism as a religion and philosophy predates its most famous exponent, the great Mahaviraswami (6th century BCE), most Jain texts on mathematical topics were composed after the 6th century BCE. Jain 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
6026-473: The square root of two : 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 ): which expresses √ 2 in the sexagesimal system, and which is also accurate up to 5 decimal places. According to mathematician S. G. Dani,
6157-500: The verbal adjective sáṃskṛta- is a compound word consisting of sáṃ ('together, good, well, perfected') and kṛta - ('made, formed, work'). It connotes a work that has been "well prepared, pure and perfect, polished, sacred". According to Biderman, the perfection contextually being referred to in the etymological origins of the word is its tonal—rather than semantic—qualities. Sound and oral transmission were highly valued qualities in ancient India, and its sages refined
6288-414: The 13th century, a premier center of Sanskrit literary creativity, Sanskrit literature there disappeared, perhaps in the "fires that periodically engulfed the capital of Kashmir" or the "Mongol invasion of 1320" states Pollock. The Sanskrit literature which was once widely disseminated out of the northwest regions of the subcontinent, stopped after the 12th century. As Hindu kingdoms fell in the eastern and
6419-471: 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 , Varāhamihira , and Madhava . The decimal number system in use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers , arithmetic , and algebra . In addition, trigonometry
6550-532: The 7th century where he established a major center of learning and language translation under the patronage of Emperor Taizong. By the early 1st millennium CE, Sanskrit had spread Buddhist and Hindu ideas to Southeast Asia, parts of the East Asia and the Central Asia. It was accepted as a language of high culture and the preferred language by some of the local ruling elites in these regions. According to
6681-579: The Babylonian cuneiform tablet Plimpton 322 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
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#17328015844476812-425: The Classical Sanskrit language launched ancient Indian speculations about "the nature and function of language", what is the relationship between words and their meanings in the context of a community of speakers, whether this relationship is objective or subjective, discovered or is created, how individuals learn and relate to the world around them through language, and about the limits of language? They speculated on
6943-532: The Dravidian languages borrowed from Sanskrit vocabulary, but they have also affected Sanskrit on deeper levels of structure, "for instance in the domain of phonology where Indo-Aryan retroflexes have been attributed to Dravidian influence". Similarly, Ferenc Ruzca states that all the major shifts in Indo-Aryan phonetics over two millennia can be attributed to the constant influence of a Dravidian language with
7074-521: The Dravidian words and forms, without modifying the word order; but the same thing is not possible in rendering a Persian or English sentence into a non-Indo-Aryan language. Shulman mentions that "Dravidian nonfinite verbal forms (called vinaiyeccam in Tamil) shaped the usage of the Sanskrit nonfinite verbs (originally derived from inflected forms of action nouns in Vedic). This particularly salient case of
7205-476: The Indo-Aryan language underwent rapid linguistic change and morphed into the Vedic Sanskrit language. The pre-Classical form of Sanskrit is known as Vedic Sanskrit . The earliest attested Sanskrit text is the Rigveda , a Hindu scripture from the mid- to late-second millennium BCE. No written records from such an early period survive, if any ever existed, but scholars are generally confident that
7336-519: The Indo-European languages are the Nuristani languages found in the remote Hindu Kush region of northeastern Afghanistan and northwestern Himalayas, as well as the extinct Avestan and Old Persian – both are Iranian languages . Sanskrit belongs to the satem group of the Indo-European languages. Colonial era scholars familiar with Latin and Greek were struck by the resemblance of
7467-532: The Muslim rule in the form of Sultanates, and later the Mughal Empire . Sheldon Pollock characterises the decline of Sanskrit as a long-term "cultural, social, and political change". He dismisses the idea that Sanskrit declined due to "struggle with barbarous invaders", and emphasises factors such as the increasing attractiveness of vernacular language for literary expression. With the fall of Kashmir around
7598-496: The Muslim rulers. Hindu rulers such as Shivaji of the Maratha Empire , reversed the process, by re-adopting Sanskrit and re-asserting their socio-linguistic identity. After Islamic rule disintegrated in South Asia and the colonial rule era began, Sanskrit re-emerged but in the form of a "ghostly existence" in regions such as Bengal. This decline was the result of "political institutions and civic ethos" that did not support
7729-499: The Saṃskṛta language, both in its vocabulary and grammar, to the classical languages of Europe. In The Oxford Introduction to Proto-Indo-European and the Proto-Indo-European World , Mallory and Adams illustrate the resemblance with the following examples of cognate forms (with the addition of Old English for further comparison): The correspondences suggest some common root, and historical links between some of
7860-583: The South India, such as the great Vijayanagara Empire , so did Sanskrit. There were exceptions and short periods of imperial support for Sanskrit, mostly concentrated during the reign of the tolerant Mughal emperor Akbar . Muslim rulers patronized the Middle Eastern language and scripts found in Persia and Arabia, and the Indians linguistically adapted to this Persianization to gain employment with
7991-447: The Vedic Sanskrit in these books of the Ṛg-veda "hardly presents any dialectical diversity", states Louis Renou – an Indologist known for his scholarship of the Sanskrit literature and the Ṛg-veda in particular. According to Renou, this implies that the Vedic Sanskrit language had a "set linguistic pattern" by the second half of the 2nd millennium BCE. Beyond the Ṛg-veda, the ancient literature in Vedic Sanskrit that has survived into
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#17328015844478122-451: The Vedic Sanskrit's bahulam framework, to respect liberty and creativity so that individual writers separated by geography or time would have the choice to express facts and their views in their own way, where tradition followed competitive forms of the Sanskrit language. The phonetic differences between Vedic Sanskrit and Classical Sanskrit, as discerned from the current state of the surviving literature, are negligible when compared to
8253-549: The ability to measure angles in a plane, as well as to determine the position of stars for navigation. The religious texts of the Vedic Period provide evidence for the use of large numbers . By the time of the Yajurvedasaṃhitā- (1200–900 BCE), numbers as high as 10 were being included in the texts. For example, the mantra (sacred recitation) at the end of the annahoma ("food-oblation rite") performed during
8384-407: The alphabet, the structure of words, and its exacting grammar into a "collection of sounds, a kind of sublime musical mold" as an integral language they called Saṃskṛta . From the late Vedic period onwards, state Annette Wilke and Oliver Moebus, resonating sound and its musical foundations attracted an "exceptionally large amount of linguistic, philosophical and religious literature" in India. Sound
8515-425: The bare-bone mathematical rules). The students then worked through the topics of the prose commentary by writing (and drawing diagrams) on chalk- and dust-boards ( i.e. boards covered with dust). The latter activity, a staple of mathematical work, was to later prompt mathematician-astronomer, Brahmagupta ( fl. 7th century CE), to characterise astronomical computations as "dust work" (Sanskrit: dhulikarman ). It
8646-593: 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: Draw a square. Beginning at half the square, draw two other similar squares below it; below these two, three other squares, and so on. The marking should be started by putting 1 in
8777-440: The capacity to understand the old Prakrit languages such as Ardhamagadhi . A section of European scholars state that Sanskrit was never a spoken language. However, evidences shows that Sanskrit was a spoken language, essential for oral tradition that preserved the vast number of Sanskrit manuscripts from ancient India. The textual evidence in the works of Yaksa, Panini, and Patanajali affirms that Classical Sanskrit in their era
8908-527: The close relationship between the Indo-Iranian tongues and the Baltic and Slavic languages , vocabulary exchange with the non-Indo-European Uralic languages , and the nature of the attested Indo-European words for flora and fauna. The pre-history of Indo-Aryan languages which preceded Vedic Sanskrit is unclear and various hypotheses place it over a fairly wide limit. According to Thomas Burrow, based on
9039-471: The combinations with one syllable, the third the combinations with two syllables, ... The text also indicates that Pingala was aware of the combinatorial identity: Kātyāyana (c. 3rd century BCE) is notable for being the last of the Vedic mathematicians. He wrote the Katyayana Sulba Sutra , which presented much geometry , including the general Pythagorean theorem and a computation of
9170-614: The context of a speech or language, is found in verses 5.28.17–19 of the Ramayana . Outside the learned sphere of written Classical Sanskrit, vernacular colloquial dialects ( Prakrits ) continued to evolve. Sanskrit co-existed with numerous other Prakrit languages of ancient India. The Prakrit languages of India also have ancient roots and some Sanskrit scholars have called these Apabhramsa , literally 'spoiled'. The Vedic literature includes words whose phonetic equivalent are not found in other Indo-European languages but which are found in
9301-494: 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: As the main objective of the Sulvasutras was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the Sulvasutras . The occurrence of
9432-653: The crystallization of Classical Sanskrit. As in this period the Indo-Aryan tribes had not yet made contact with the inhabitants of the South of the subcontinent, this suggests a significant presence of Dravidian speakers in North India (the central Gangetic plain and the classical Madhyadeśa) who were instrumental in this substratal influence on Sanskrit. Extant manuscripts in Sanskrit number over 30 million, one hundred times those in Greek and Latin combined, constituting
9563-467: The detailed and sophisticated treatise then transmitted it through his students. Modern scholarship generally accepts that he knew of a form of writing, based on references to words such as Lipi ('script') and lipikara ('scribe') in section 3.2 of the Aṣṭādhyāyī . The Classical Sanskrit language formalized by Pāṇini, states Renou, is "not an impoverished language", rather it is "a controlled and
9694-471: The differences between the Vedic and Classical Sanskrit. Louis Renou published in 1956, in French, a more extensive discussion of the similarities, the differences and the evolution of the Vedic Sanskrit within the Vedic period and then to the Classical Sanskrit along with his views on the history. This work has been translated by Jagbans Balbir. The earliest known use of the word Saṃskṛta (Sanskrit), in
9825-460: The distant major ancient languages of the world. The Indo-Aryan migrations theory explains the common features shared by Sanskrit and other Indo-European languages by proposing that the original speakers of what became Sanskrit arrived in South Asia from a region of common origin, somewhere north-west of the Indus region , during the early 2nd millennium BCE. Evidence for such a theory includes
9956-490: The early medieval era, it became a language of religion and high culture , and of the political elites in some of these regions. As a result, Sanskrit had a lasting impact on the languages of South Asia, Southeast Asia and East Asia, especially in their formal and learned vocabularies. Sanskrit generally connotes several Old Indo-Aryan language varieties. The most archaic of these is the Vedic Sanskrit found in
10087-632: 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 ( bījagaṇita samīkaraṇa ). Jain mathematicians were apparently also
10218-599: The fifth century B.C. ... as is shown by the elements of Mesopotamian omen literature and astronomy that entered India at that time and (were) definitely not ... preserved orally. The earliest mathematical prose commentary was that on the work, Āryabhaṭīya (written 499 CE), a work on astronomy and mathematics. The mathematical portion of the Āryabhaṭīya was composed of 33 sūtras (in verse form) consisting of mathematical statements or rules, but without any proofs. However, according to Hayashi, "this does not necessarily mean that their authors did not prove them. It
10349-449: 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
10480-548: The first language of the respective speakers. The Sanskrit language brought Indo-Aryan speaking people together, particularly its elite scholars. Some of these scholars of Indian history regionally produced vernacularized Sanskrit to reach wider audiences, as evidenced by texts discovered in Rajasthan, Gujarat, and Maharashtra. Once the audience became familiar with the easier to understand vernacularized version of Sanskrit, those interested could graduate from colloquial Sanskrit to
10611-407: 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. With
10742-403: The first square. Put 1 in each of the two squares of the second line. In the third line put 1 in the two squares at the ends and, in the middle square, the sum of the digits in the two squares lying above it. In the fourth line put 1 in the two squares at the ends. In the middle ones put the sum of the digits in the two squares above each. Proceed in this way. Of these lines, the second gives
10873-540: The first to use the word shunya (literally void in Sanskrit ) to refer to zero. This word is the ultimate etymological origin of the English word "zero" , as it was calqued into Arabic as ṣifr and then subsequently borrowed into Medieval Latin as zephirum , finally arriving at English after passing through one or more Romance languages (c.f. French zéro , Italian zero ). In addition to Surya Prajnapti , important Jain works on mathematics included
11004-419: The form: That these methods have been effective is testified to by the preservation of the most ancient Indian religious text, the Ṛgveda (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 (c. 500 BCE). Mathematical activity in ancient India began as
11135-412: The foundation of Vyākaraṇa, a Vedānga . The Aṣṭādhyāyī was not the first description of Sanskrit grammar, but it is the earliest that has survived in full, and the culmination of a long grammatical tradition that Fortson says, is "one of the intellectual wonders of the ancient world". Pāṇini cites ten scholars on the phonological and grammatical aspects of the Sanskrit language before him, as well as
11266-537: The gods Varuna, Mitra, Indra, and Nasatya found in the earliest layers of the Vedic literature. O Bṛhaspati, when in giving names they first set forth the beginning of Language, Their most excellent and spotless secret was laid bare through love, When the wise ones formed Language with their mind, purifying it like grain with a winnowing fan, Then friends knew friendships – an auspicious mark placed on their language. — Rigveda 10.71.1–4 Translated by Roger Woodard The Vedic Sanskrit found in
11397-431: The historic Sanskrit literary culture and the failure of new Sanskrit literature to assimilate into the changing cultural and political environment. Sheldon Pollock states that in some crucial way, "Sanskrit is dead ". After the 12th century, the Sanskrit literary works were reduced to "reinscription and restatements" of ideas already explored, and any creativity was restricted to hymns and verses. This contrasted with
11528-458: 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 , near Peshawar (modern day Pakistan ) and is likely from the 7th century CE. A later landmark in Indian mathematics
11659-458: The increasing complexity of mathematics and other exact sciences, both writing and computation were required. Consequently, many mathematical works began to be written down in manuscripts that were then copied and re-copied from generation to generation. India today is estimated to have about thirty million manuscripts, the largest body of handwritten reading material anywhere in the world. The literate culture of Indian science goes back to at least
11790-486: The intense change that must have occurred in the pre-Vedic period between the Proto-Indo-Aryan language and Vedic Sanskrit. The noticeable differences between the Vedic and the Classical Sanskrit include the much-expanded grammar and grammatical categories as well as the differences in the accent, the semantics and the syntax. There are also some differences between how some of the nouns and verbs end, as well as
11921-432: The largest cultural heritage that any civilization has produced prior to the invention of the printing press. — Foreword of Sanskrit Computational Linguistics (2009), Gérard Huet, Amba Kulkarni and Peter Scharf Sanskrit has been the predominant language of Hindu texts encompassing a rich tradition of philosophical and religious texts, as well as poetry, music, drama , scientific , technical and others. It
12052-412: The linguistic expression and sets the standard for the Sanskrit language. Pāṇini made use of a technical metalanguage consisting of a syntax, morphology and lexicon. This metalanguage is organised according to a series of meta-rules, some of which are explicitly stated while others can be deduced. Despite differences in the analysis from that of modern linguistics, Pāṇini's work has been found valuable and
12183-514: The literary works. The Indian tradition, states Winternitz , has favored the learning and the usage of multiple languages from the ancient times. Sanskrit was a spoken language in the educated and the elite classes, but it was also a language that must have been understood in a wider circle of society because the widely popular folk epics and stories such as the Ramayana , the Mahabharata ,
12314-511: The modern age include the Samaveda , Yajurveda , Atharvaveda , along with the embedded and layered Vedic texts such as the Brahmanas , Aranyakas , and the early Upanishads . These Vedic documents reflect the dialects of Sanskrit found in the various parts of the northwestern, northern, and eastern Indian subcontinent. According to Michael Witzel, Vedic Sanskrit was a spoken language of
12445-429: The more advanced Classical Sanskrit. Rituals and the rites-of-passage ceremonies have been and continue to be the other occasions where a wide spectrum of people hear Sanskrit, and occasionally join in to speak some Sanskrit words such as namah . Classical Sanskrit is the standard register as laid out in the grammar of Pāṇini , around the fourth century BCE. Its position in the cultures of Greater India
12576-401: The most advanced analysis of linguistics until the twentieth century. Pāṇini's comprehensive and scientific theory of grammar is conventionally taken to mark the start of Classical Sanskrit. His systematic treatise inspired and made Sanskrit the preeminent Indian language of learning and literature for two millennia. It is unclear whether Pāṇini himself wrote his treatise or he orally created
12707-602: The most archaic poems of the Iranian and Greek language families, the Gathas of old Avestan and Iliad of Homer . As the Rigveda was orally transmitted by methods of memorisation of exceptional complexity, rigour and fidelity, as a single text without variant readings, its preserved archaic syntax and morphology are of vital importance in the reconstruction of the common ancestor language Proto-Indo-European . Sanskrit does not have an attested native script: from around
12838-597: The numbers are thought to signify a wish to be aligned with the prestige of the language. Sanskrit has been taught in traditional gurukulas since ancient times; it is widely taught today at the secondary school level. The oldest Sanskrit college is the Benares Sanskrit College founded in 1791 during East India Company rule . Sanskrit continues to be widely used as a ceremonial and ritual language in Hindu and Buddhist hymns and chants . In Sanskrit,
12969-403: The oral transmission of the texts is reliable: they are ceremonial literature, where the exact phonetic expression and its preservation were a part of the historic tradition. However some scholars have suggested that the original Ṛg-veda differed in some fundamental ways in phonology compared to the sole surviving version available to us. In particular that retroflex consonants did not exist as
13100-431: The other." Reinöhl further states that there is a symmetric relationship between Dravidian languages like Kannada or Tamil, with Indo-Aryan languages like Bengali or Hindi, whereas the same relationship is not found for non-Indo-Aryan languages, for example, Persian or English: A sentence in a Dravidian language like Tamil or Kannada becomes ordinarily good Bengali or Hindi by substituting Bengali or Hindi equivalents for
13231-414: The possible influence of Dravidian on Sanskrit is only one of many items of syntactic assimilation, not least among them the large repertoire of morphological modality and aspect that, once one knows to look for it, can be found everywhere in classical and postclassical Sanskrit". The main influence of Dravidian on Sanskrit is found to have been concentrated in the timespan between the late Vedic period and
13362-439: The previous 1,500 years when "great experiments in moral and aesthetic imagination" marked the Indian scholarship using Classical Sanskrit, states Pollock. Scholars maintain that the Sanskrit language did not die, but rather only declined. Jurgen Hanneder disagrees with Pollock, finding his arguments elegant but "often arbitrary". According to Hanneder, a decline or regional absence of creative and innovative literature constitutes
13493-480: The problems of interpretation and misunderstanding. The purifying structure of the Sanskrit language removes these imperfections. The early Sanskrit grammarian Daṇḍin states, for example, that much in the Prakrit languages is etymologically rooted in Sanskrit, but involves "loss of sounds" and corruptions that result from a "disregard of the grammar". Daṇḍin acknowledged that there are words and confusing structures in Prakrit that thrive independent of Sanskrit. This view
13624-609: The regional Prakrit languages, which makes it likely that the interaction, the sharing of words and ideas began early in the Indian history. As the Indian thought diversified and challenged earlier beliefs of Hinduism, particularly in the form of Buddhism and Jainism , the Prakrit languages such as Pali in Theravada Buddhism and Ardhamagadhi in Jainism competed with Sanskrit in the ancient times. However, states Paul Dundas , these ancient Prakrit languages had "roughly
13755-497: The relationship between various Indo-European languages, the origin of all these languages may possibly be in what is now Central or Eastern Europe, while the Indo-Iranian group possibly arose in Central Russia. The Iranian and Indo-Aryan branches separated quite early. It is the Indo-Aryan branch that moved into eastern Iran and then south into South Asia in the first half of the 2nd millennium BCE. Once in ancient India,
13886-481: The rites at the correct time by the use of kalpa ( ritual ) and jyotiṣa ( astrology ), gave rise to the six disciplines of the Vedāṇgas . Mathematics arose as a part of the last two disciplines, ritual and astronomy (which also included astrology). Since the Vedāṇgas immediately preceded the use of writing in ancient India, they formed the last of the exclusively oral literature. They were expressed in
14017-562: The role of language, the ontological status of painting word-images through sound, and the need for rules so that it can serve as a means for a community of speakers, separated by geography or time, to share and understand profound ideas from each other. These speculations became particularly important to the Mīmāṃsā and the Nyaya schools of Hindu philosophy, and later to Vedanta and Mahayana Buddhism, states Frits Staal —a scholar of Linguistics with
14148-469: 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 Hayashi, the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to
14279-496: The same relationship to Sanskrit as medieval Italian does to Latin". The Indian tradition states that the Buddha and the Mahavira preferred the Prakrit language so that everyone could understand it. However, scholars such as Dundas have questioned this hypothesis. They state that there is no evidence for this and whatever evidence is available suggests that by the start of the common era, hardly anybody other than learned monks had
14410-556: The semi-nomadic Aryans . The Vedic Sanskrit language or a closely related Indo-European variant was recognized beyond ancient India as evidenced by the " Mitanni Treaty" between the ancient Hittite and Mitanni people, carved into a rock, in a region that now includes parts of Syria and Turkey. Parts of this treaty, such as the names of the Mitanni princes and technical terms related to horse training, for reasons not understood, are in early forms of Vedic Sanskrit. The treaty also invokes
14541-615: The social structures such as the role of the poet and the priests, the patronage economy, the phrasal equations, and some of the poetic metres. While there are similarities, state Jamison and Brereton, there are also differences between Vedic Sanskrit, the Old Avestan, and the Mycenaean Greek literature. For example, unlike the Sanskrit similes in the Ṛg-veda, the Old Avestan Gathas lack simile entirely, and it
14672-407: 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
14803-524: The teacher to the student. They contain lists of Pythagorean triples , which are particular cases of Diophantine equations . They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square." Baudhayana (c. 8th century BCE) composed the Baudhayana Sulba Sutra , the best-known Sulba Sutra , which contains examples of simple Pythagorean triples, such as: (3, 4, 5) , (5, 12, 13) , (8, 15, 17) , (7, 24, 25) , and (12, 35, 37) , as well as
14934-492: 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: In another form of recitation, dhvaja-pāṭha (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: The most complex form of recitation, ghana-pāṭha (literally "dense recitation"), according to Filliozat, took
15065-442: The triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily. In all, three Sulba Sutras were composed. The remaining two,
15196-653: The turn of the 1st-millennium CE, it has been written in various Brahmic scripts , and in the modern era most commonly in Devanagari . Sanskrit's status, function, and place in India's cultural heritage are recognized by its inclusion in the Constitution of India 's Eighth Schedule languages . However, despite attempts at revival, there are no first-language speakers of Sanskrit in India. In each of India's recent decennial censuses, several thousand citizens have reported Sanskrit to be their mother tongue, but
15327-408: The variants in the usage of Sanskrit in different regions of India. The ten Vedic scholars he quotes are Āpiśali, Kaśyapa , Gārgya, Gālava, Cakravarmaṇa, Bhāradvāja , Śākaṭāyana, Śākalya, Senaka and Sphoṭāyana. In the Aṣṭādhyāyī , language is observed in a manner that has no parallel among Greek or Latin grammarians. Pāṇini's grammar, according to Renou and Filliozat, is a classic that defines
15458-564: The vernacular Prakrits. Many Sanskrit dramas indicate that the language coexisted with the vernacular Prakrits. The cities of Varanasi , Paithan , Pune and Kanchipuram were centers of classical Sanskrit learning and public debates until the arrival of the colonial era. According to Lamotte , Sanskrit became the dominant literary and inscriptional language because of its precision in communication. It was, states Lamotte, an ideal instrument for presenting ideas, and as knowledge in Sanskrit multiplied, so did its spread and influence. Sanskrit
15589-502: The Ṛg-veda is distinctly more archaic than other Vedic texts, and in many respects, the Rigvedic language is notably more similar to those found in the archaic texts of Old Avestan Zoroastrian Gathas and Homer's Iliad and Odyssey . According to Stephanie W. Jamison and Joel P. Brereton – Indologists known for their translation of the Ṛg-veda – the Vedic Sanskrit literature "clearly inherited" from Indo-Iranian and Indo-European times
15720-408: Was a spoken language ( bhasha ) used by the cultured and educated. Some sutras expound upon the variant forms of spoken Sanskrit versus written Sanskrit. Chinese Buddhist pilgrim Xuanzang mentioned in his memoir that official philosophical debates in India were held in Sanskrit, not in the vernacular language of that region. According to Sanskrit linguist professor Madhav Deshpande, Sanskrit
15851-427: Was a spoken language in a colloquial form by the mid-1st millennium BCE which coexisted with a more formal, grammatically correct form of literary Sanskrit. This, states Deshpande, is true for modern languages where colloquial incorrect approximations and dialects of a language are spoken and understood, along with more "refined, sophisticated and grammatically accurate" forms of the same language being found in
15982-472: Was adopted voluntarily as a vehicle of high culture, arts, and profound ideas. Pollock disagrees with Lamotte, but concurs that Sanskrit's influence grew into what he terms a "Sanskrit Cosmopolis" over a region that included all of South Asia and much of southeast Asia. The Sanskrit language cosmopolis thrived beyond India between 300 and 1300 CE. Today, it is believed that Kashmiri is the closest language to Sanskrit. Reinöhl mentions that not only have
16113-738: Was also the language of some of the oldest surviving, authoritative and much followed philosophical works of Jainism such as the Tattvartha Sutra by Umaswati . The Sanskrit language has been one of the major means for the transmission of knowledge and ideas in Asian history. Indian texts in Sanskrit were already in China by 402 CE, carried by the influential Buddhist pilgrim Faxian who translated them into Chinese by 418 CE. Xuanzang , another Chinese Buddhist pilgrim, learnt Sanskrit in India and carried 657 Sanskrit texts to China in
16244-457: 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 Vedas included up to eleven forms of recitation of the same text. The texts were subsequently "proof-read" by comparing the different recited versions. Forms of recitation included the jaṭā-pāṭha (literally "mesh recitation") in which every two adjacent words in
16375-487: 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 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
16506-605: Was here. The Satapatha Brahmana ( c. 7th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras. 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
16637-466: Was probably a matter of style of exposition." From the time of Bhaskara I (600 CE onwards), prose commentaries increasingly began to include some derivations ( upapatti ). Bhaskara I's commentary on the Āryabhaṭīya , had the following structure: 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
16768-563: 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 [the same] clay 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
16899-618: Was the Kharoṣṭhī script used in the Gandhara culture of the north-west. It is thought to be of Aramaic origin and it was in use from the 4th century BCE to the 4th century CE. Almost contemporaneously, another script, the Brāhmī script , 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
17030-508: Was the development of the series expansions for trigonometric functions (sine, cosine, and arc tangent ) by mathematicians of the Kerala school in the 15th century CE. Their 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). However, they did not formulate a systematic theory of differentiation and integration , nor
17161-442: Was visualized as "pervading all creation", another representation of the world itself; the "mysterious magnum" of Hindu thought. The search for perfection in thought and the goal of liberation were among the dimensions of sacred sound, and the common thread that wove all ideas and inspirations together became the quest for what the ancient Indians believed to be a perfect language, the "phonocentric episteme" of Sanskrit. Sanskrit as
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