List of Indian inventions and discoveries: Difference between revisions

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This list of Indian inventions and discoveries details the inventions, scientific discoveries and contributions of ancient and modern India, including both the ancient and medieval nations in the subcontinent historically referred to as India and the modern Indian state. It draws from the whole cultural and technological history of India, during which architecture, astronomy, cartography, metallurgy, logic, mathematics, metrology and mineralogy were among the branches of study pursued by its scholars. During recent times science and technology in the Republic of India has also focused on automobile engineering, information technology, communications as well as research into space and polar technology.

For the purposes of this list, inventions are regarded as technological firsts developed in India, and as such does not include foreign technologies which India acquired through contact. It also does not include technologies or discoveries developed elsewhere and later invented separately in India, nor inventions by Indian emigres in other places. Changes in minor concepts of design or style and artistic innovations do not appear on in the lists.

• Iron pillar of Delhi:The world's first iron pillar was the Iron pillar of Delhi—erected at the time of Chandragupta II Vikramaditya (375–413).^[1] The pillar has attracted attention of archaeologists and materials scientists and has been called "a testament to the skill of ancient Indian blacksmiths" because of its high resistance to corrosion.^[2]
• Stepwell: Earliest clear evidence of the origins of the stepwell is found in the Indus Valley Civilization's archaeological site at Mohenjodaro in Pakistan^[3] and Dholavira in India.^[4] The three features of stepwells in the subcontinent are evident from one particular site, abandoned by 2500 BCE, which combines a bathing pool, steps leading down to water, and figures of some religious importance into one structure.^[3] The early centuries immediately before the common era saw the Buddhists and the Jains of India adapt the stepwells into their architecture.^[3] Both the wells and the form of ritual bathing reached other parts of the world with Buddhism.^[3] Rock-cut step wells in the subcontinent date from 200 to 400 CE.^[5] Subsequently, the wells at Dhank (550-625 CE) and stepped ponds at Bhinmal (850-950 CE) were constructed.^[5]
• Stupa: The origin of the stupa can be traced to 3rd-century BCE India.^[6] It was used as a commemorative monument associated with storing sacred relics.^[6] The stupa architecture was adopted in Southeast and East Asia, where it evolved into the pagoda, a Buddhist monument used for enshrining sacred relics.^[6]

• Squat toilet: toilets platforms above drains, in the proximity of wells, are found in several houses of the cities of Mohenjodaro and Harappa from the 3rd millennium B.C.^[7]

• Ruler: Rulers made from Ivory were in use by the Indus Valley Civilization in what today is Pakistan and Northwestern India prior to 1500 BCE.^[8] Excavations at Lothal (2400 BCE) have yielded one such ruler calibrated to about 1/16 of an inch—less than 2 millimeters.^[8] Ian Whitelaw (2007) holds that 'The Mohenjodaro ruler is divided into units corresponding to 1.32 inches (33.5 mm) and these are marked out in decimal subdivisions with amazing accuracy—to within 0.005 of an inch. They correspond closely with the "hasta" increments of 1 3/8 inches traditionally used in South India in ancient architecture. Ancient bricks found throughout the region have dimensions that correspond to these units.'^[9] Shigeo Iwata (2008) further writes 'The minimum division of graduation found in the segment of an ivory-made linear measure excavated in Lothal was 1.79 mm (that corresponds to 1/940 of a fathom), while that of the fragment of a shell-made one from Mohenjodaro was 6.72 mm (1/250 of a fathom), and that of bronze-made one from Harappa was 9.33 mm (1/180 of a fathom).'^[10] The weights and measures of the Indus civilization also reached Persia and Central Asia, where they were further modified.^[10]
• Weighing scale: The earliest evidence for the existence of weighing scale dates to 2400 BC-1800 BC in the Indus valley civilization prior to which no banking was performed due to lack of scales.^[11]
• Crescograph: The crescograph, a device for measuring growth in plants, was invented in the early 20th century by the Bengali scientist Sir Jagadish Chandra Bose.^[12][13]
• Incense clock: The incense clock is a timekeeping device used to measure minutes, hours, or days, incense clocks were commonly used at homes and temples in dynastic times. Although popularly associated with China the incense clock is believed to have originated in India, at least in its fundamental form if not function.^[14][15] Early incense clocks found in China between the 6th and 8th centuries CE—the period it appeared in China all seem to have Devanāgarī carvings on them instead of Chinese seal characters.^[14][15] Incense itself was introduced to China from India in the early centuries CE, along with the spread of Buddhism by traveling monks.^[16][17][18] Edward Schafer asserts that incense clocks were probably an Indian invention, transmitted to China, which explains the Devanāgarī inscriptions on early incense clocks found in China.^[14] Silvio Bedini on the other hand asserts that incense clocks were derived in part from incense seals mentioned in Tantric Buddhist scriptures, which first came to light in China after those scriptures from India were translated into Chinese, but holds that the time-telling function of the seal was incorporated by the Chinese.^[15]

• Crucible steel: Perhaps as early as 300 BC—although certainly by 200 BC—high quality steel was being produced in southern India, by what Europeans would later call the crucible technique.^[19] In this system, high-purity wrought iron, charcoal, and glass were mixed in a crucible and heated until the iron melted and absorbed the carbon.^[19]

• Wootz steel: Wootz steel is an ultra-high carbon steel and the first form of crucible steel manufactured by the applications and use of nanomaterials in its microstructure and is characterised by its ultra-high carbon content exhibiting properties such as superplasticity,high impact hardness and held sway for over a millennium in three continents - a feat unlikely to be surpassed by advanced materials of the current era.^[20] Archaeological and Tamil language literary evidence suggests that this manufacturing process was already in existence in South India well before the common era, with wootz steel exported from the Chera dynasty and called Seric Iron in Rome, and later known as Damascus steel in Europe.^{[21][22][23][24]} Reproduction research is currently being undertaken by various scientists like Dr. Oleg Sherby and Dr. Jeff Wadsworth and Lawrence Livermore National Laboratory have all done research, attempting to create steels with characteristics similar to Wootz, but without success J.D Verhoeven and Al Pendray attained some success in the reconstruction methods of production, proved the role of impurities of ore in the pattern creation, and reproduced Wootz steel with patterns microscopically and visually identical to one of the ancient blade patterns.^[25]
• Seamless celestial globe: Considered one of the most remarkable feats in metallurgy, it was invented in India in between 1589 and 1590 CE.^[26][27] Before they were rediscovered in the 1980s, it was believed by modern metallurgists to be technically impossible to produce metal globes without any seams, even with modern technology.^[27][27]

• J Sharp: Visual J# (pronounced "jay-sharp") programming language was a transitional language for programmers of Java and Visual J++ languages, so they could use their existing knowledge and applications on .NET Framework. It was developed by the Hyderabad-based Microsoft India Development Center at HITEC City in India.^[28][29]
• Kojo: Kojo is a programming language and integrated development environment (IDE) for computer programming and learning. Kojo is an open-source software. It was created, and is actively developed, by Lalit Pant, a computer programmer and teacher living in Dehradun, India.^[30][31]

• Plough: The earliest known instance of a ploughed field was found at Kalibangan ^[32]
• India ink: Known in Asia since the third millennia BCE, and used in India since at least the 4th century BCE.^[33] Masi, an early ink in India was an admixture of several chemical components.^[33], with the carbon black from which India ink is produced obtained by burning bones, tar, pitch, and other substances.^{[34][35][35][36]} Documents dating to the 3rd century CE, written in Kharosthi, with ink have been unearthed in East Turkestan, Xinjiang.^[37] The practice of writing with ink and a sharp pointed needle was common in ancient South India.^[38] Several Jain sutras in India were compiled in ink.^[39]
• Iron and mercury coherer: In 1899, the Bengali physicist Sir Jagdish Chandra Bose announced the development of an "iron-mercury-iron coherer with telephone detector" in a paper presented at the Royal Society, London.^[40] He also later received U.S. patent 755,840, "Detector for electrical disturbances" (1904), for a specific electromagnetic receiver.
• Microwave Communication: The first public demonstration of microwave transmission was made by Jagadish Chandra Bose, in Calcutta, in 1895, two years before a similar demonstration by Marconi in England, and just a year after Oliver Lodge's commemorative lecture on Radio communication, following Hertz's death. Bose's revolutionary demonstration forms the foundation of the technology used in mobile telephony, radars, satellite communication, radios, television broadcast, WiFi, remote controls and countless other applications.^[41][42]
• Murty Shearing Interferometer: Invented by M.V.R.K. Murty, a type of Lateral Shearing Interferometer utilizes a laser source for measuring refractive index.^[43][44]
• Mysorean rockets: The first iron-cased and metal-cylinder rockets were developed by Tipu Sultan, ruler of the South Indian Kingdom of Mysore, and his father Haither Ali, in the 1780s. He successfully used these iron-cased rockets against the larger forces of the British East India Company during the Anglo-Mysore Wars. The Mysore Rockets of this period were much more advanced than what the British had seen, chiefly because of the use of iron tubes for holding the propellant; this enabled higher thrust and longer range for the missile (up to 2 km range). After Tipu's eventual defeat in the Fourth Anglo-Mysore War and the capture of the Mysore iron rockets, they were influential in British rocket development, inspiring the Congreve rocket, and were soon put into use in the Napoleonic Wars.^[45]

• Reversible inhibition of sperm under guidance: RISUG, formerly referred to as the synthetic polymer styrene maleic anhydride (SMA), is the development name of a male contraceptive injection developed at IIT Kharagpur in India by the team of Dr. Sujoy K Guha. Phase III clinical trials are underway in India, slowed by insufficient volunteers.^[46] It has been patented in India, China, Bangladesh, and the United States.^[46] A method based on RISUG, Vasalgel, is currently under development in the US.^[47]
• Shampoo: The word shampoo in English is derived from Hindustani chāmpo (चाँपो [tʃãːpoː]),^[48] and dates to 1762.^[49] A variety of herbs and their extracts were used as shampoos since ancient times in India. A very effective early shampoo was made by boiling Sapindus with dried Indian gooseberry (aamla) and a few other herbs, using the strained extract. Sapindus, also known as soapberries or soapnuts, is called Ksuna (Sanskrit: क्षुण)^[50] in ancient Indian texts and its fruit pulp contain saponins,a natural surfactant. The extract of Ksuna, creates a lather which Indian texts identify as phenaka (Sanskrit: फेनक),^[51] leaves the hair soft, shiny and manageable. Other products used for hair cleansing were shikakai (Acacia concinna), soapnuts (Sapindus), hibiscus flowers,^[52][53] ritha (Sapindus mukorossi) and arappu (Albizzia amara).^[54] Guru Nanak, the founding prophet and the first Guru of Sikhism, made references to soapberry tree and soap in 16th century.^[55] Washing of hair and body massage (champu) during a daily strip wash was an indulgence of early colonial traders in India. When they returned to Europe, they introduced their newly learnt habits, including the hair treatment they called shampoo.^[56]
• Toe stirrup: The earliest known manifestation of the stirrup, which was a toe loop that held the big toe was used in India in as early as 500 BCE^[57] or perhaps by 200 BCE according to other sources.^[58][59] This ancient stirrup consisted of a looped rope for the big toe which was at the bottom of a saddle made of fibre or leather.^[59] Such a configuration made it suitable for the warm climate of most of India where people used to ride horses barefoot.^[59] A pair of megalithic double bent iron bars with curvature at each end, excavated in Junapani in the central Indian state of Madhya Pradesh have been regarded as stirrups although they could as well be something else.^[60] Buddhist carvings in the temples of Sanchi, Mathura and the Bhaja caves dating back between the 1st and 2nd century BCE figure horsemen riding with elaborate saddles with feet slipped under girths.^[61][62][63] Sir John Marshall described the Sanchi relief as "the earliest example by some five centuries of the use of stirrups in any part of the world".^[63] In the 1st century CE horse riders in northern India, where winters are sometimes long and cold, were recorded to have their booted feet attached to hooked stirrups.^[58] However the form, the conception of the primitive Indian stirrup spread west and east, gradually evolving into the stirrup of today.^[59][62]

• Pseudomonas putida:Indian (Bengali) inventor and microbiologist Ananda Mohan Chakrabarty created a species of man made micro organism to break down crude oil.^[64] He genetically engineered^{[65][66][67][68][69][70]} a new species of Pseudomonas bacteria ("the oil-eating bacteria") in 1971.^[71] United States Supreme Court granted Chakrabarty's invention patent even though it was a living species. The court ruling decreed that Chakrabarty's discovery was "not nature's handiwork, but his own..." The inventor Chakrabarty secured his patent in 1980^[72](see Diamond v. Chakrabarty)

• Chaturanga: The precursor of chess originated in India during the Gupta dynasty (c. 280-550 CE).^{[73][74][75][76]} Both the Persians and Arabs ascribe the origins of the game of Chess to the Indians.^[75][77][78] The words for "chess" in Old Persian and Arabic are chatrang and shatranj respectively — terms derived from caturaṅga in Sanskrit,^[79][80] which literally means an army of four divisions or four corps.^[81][82] Chess spread throughout the world and many variants of the game soon began taking shape.^[83] This game was introduced to the Near East from India and became a part of the princely or courtly education of Persian nobility.^[81] Buddhist pilgrims, Silk Road traders and others carried it to the Far East where it was transformed and assimilated into a game often played on the intersection of the lines of the board rather than within the squares.^[83] Chaturanga reached Europe through Persia, the Byzantine empire and the expanding Arabian empire.^[82][84] Muslims carried Shatranj to North Africa, Sicily, and Spain by the 10th century where it took its final modern form of chess.^[83]
• Kabaddi: The game of kabaddi originated in India during prehistory.^[85] Suggestions on how it evolved into the modern form range from wrestling exercises, military drills, and collective self-defense but most authorities agree that the game existed in some form or the other in India during the period between 1500 and 400 BCE.^[85]
• Ludo: Pachisi originated in India by the 6th century.^[86] The earliest evidence of this game in India is the depiction of boards on the caves of Ajanta.^[86] This game was played by the Mughal emperors of India; a notable example being that of Akbar, who played living Pachisi using girls from his harem.^[86][87] A variant of this game, called Luodo, made its way to England during the British Raj.^[86]
• Snakes and ladders: Vaikunta pali Snakes and ladders originated in India as a game based on morality.^[88] During British rule of India, this game made its way to England, and was eventually introduced in the United States of America by game-pioneer Milton Bradley in 1943.^[88]
• Suits game: Kridapatram is an early suits game, made of painted rags, invented in Ancient India. The term kridapatram literally means "painted rags for playing."^{[89][90][91][92][93]} Paper playing cards first appeared in East Asia during the 9th century.^[89][94] The medieval Indian game of ganjifa, or playing cards, is first recorded in the 16th century.^[95]

• Button: Ornamental buttons—made from seashell—were used in the Indus Valley Civilization for ornamental purposes by 2000 BCE.^[96] Some buttons were carved into geometric shapes and had holes pierced into them so that they could be attached to clothing by using a thread.^[96] Ian McNeil (1990) holds that: "The button, in fact, was originally used more as an ornament than as a fastening, the earliest known being found at Mohenjo-daro in the Indus Valley. It is made of a curved shell and about 5000 years old."^[97]

• Calico: Calico had originated in the subcontinent by the 11th century and found mention in Indian literature, by the 12th-century writer Hemachandra. He has mentioned calico fabric prints done in a lotus design.^[98] The Indian textile merchants traded in calico with the Africans by the 15th century and calico fabrics from Gujarat appeared in Egypt.^[98] Trade with Europe followed from the 17th century onwards.^[98] Within India, calico originated in Kozhikode.^[98]
• Carding devices: Historian of science Joseph Needham ascribes the invention of bow-instruments used in textile technology to India.^[99] The earliest evidence for using bow-instruments for carding comes from India (2nd century CE).^[99] These carding devices, called kaman and dhunaki would loosen the texture of the fiber by the means of a vibrating string.^[99]
• Charkha (Spinning wheel): invented in India, between 500 and 1000 C.E.^[100]
• Chintz: The origin of Chintz is from the printed all cotton fabric of calico in India.^[101] The origin of the word chintz itself is from the Hindi language word चित्र् (chitr), which means an image.^[101][102]
• Muslin: The fabric was named after the city where Europeans first encountered it, Mosul, in what is now Iraq, but the fabric actually originated from Dhaka in what is now Bangladesh.^[103][104] In the 9th century, an Arab merchant named Sulaiman makes note of the material's origin in Bengal (known as Ruhml in Arabic).^[104]

• Palampore: पालमपोर् (Hindi language) of Indian origin^[105] was imported to the western world—notable England and Colonial America—from India.^[106][107] In 17th-century England these hand painted cotton fabrics influenced native crewel work design.^[106] Shipping vessels from India also took palampore to colonial America, where it was used in quilting.^[107]
• Prayer flags: The Buddhist sūtras, written on cloth in India, were transmitted to other regions of the world.^[108] These sutras, written on banners, were the origin of prayer flags.^[108] Legend ascribes the origin of the prayer flag to the Shakyamuni Buddha, whose prayers were written on battle flags used by the devas against their adversaries, the asuras.^[109] The legend may have given the Indian bhikku a reason for carrying the 'heavenly' banner as a way of signyfying his commitment to ahimsa.^[110] This knowledge was carried into Tibet by 800 CE, and the actual flags were introduced no later than 1040 CE, where they were further modified.^[110] The Indian monk Atisha (980-1054 CE) introduced the Indian practice of printing on cloth prayer flags to Tibet.^[109]
• Single roller cotton gin: The Ajanta caves of India yield evidence of a single roller cotton gin in use by the 5th century.^[111] This cotton gin was used in India until innovations were made in form of foot powered gins.^[112] The cotton gin was invented in India as a mechanical device known as charkhi, more technically the "wooden-worm-worked roller". This mechanical device was, in some parts of India, driven by water power.^[99]

• Indian clubs: The Indian club—which appeared in Europe during the 18th century—was used long by India's native soldiery before its introduction to Europe.^[113] During the British Raj the British officers in India performed calisthenic exercises with clubs to keep in physical condition.^[113] From Britain the use of club swinging spread to the rest of the world.^[113]
• Yoga: Yoga as a physical, mental, and spiritual practice originated in ancient India.^[114]

The fiber is also known as pashm or pashmina for its use in the handmade shawls of Kashmir, India.^[115] The woolen shawls made from wool in Kashmir region of India find written mention between the 3rd century BCE and the 11th century CE.^[116] However, the founder of the cashmere wool industry is traditionally held to be the 15th-century ruler of Kashmir, Zayn-ul-Abidin, who employed weavers from Central Asia.^[116]

• Cotton cultivation: Cotton was cultivated by the inhabitants of the Indus Valley Civilization by the 5th millennium BCE - 4th millennium BCE.^[117] The Indus cotton industry was well developed and some methods used in cotton spinning and fabrication continued to be practiced till the modern Industrialization of India.^[118] Well before the Common Era, the use of cotton textiles had spread from India to the Mediterranean and beyond.^[119]
• Indigo dye: Indigo, a blue pigment and a dye, was used in India, which was also the earliest major center for its production and processing.^[120] The Indigofera tinctoria variety of Indigo was domesticated in India.^[120] Indigo, used as a dye, made its way to the Greeks and the Romans via various trade routes, and was valued as a luxury product.^[120]
• Jute cultivation: Jute has been cultivated in India since ancient times.^[121] Raw jute was exported to the western world, where it was used to make ropes and cordage.^[121] The Indian jute industry, in turn, was modernized during the British Raj in India.^[121] The region of Bengal was the major center for Jute cultivation, and remained so before the modernization of India's jute industry in 1855, when Kolkata became a center for jute processing in India.^[121]
• Sugar refinement: Sugarcane was originally from tropical South Asia and Southeast Asia,^[122] with different species originating in India, and S. edule and S. officinarum from New Guinea.^[122] The process of producing crystallized sugar from sugarcane was discovered by the time of the Imperial Guptas,^[123] and the earliest reference of candied sugar comes from India.^[124] The process was soon transmitted to China with traveling Buddhist monks.^[124] Chinese documents confirm at least two missions to India, initiated in 647 CE, for obtaining technology for sugar-refining.^[125] Each mission returned with results on refining sugar.^[125]

• AKS primality test: The AKS primality test is a deterministic primality-proving algorithm created and published by three Indian Institute of Technology Kanpur computer scientists, Manindra Agrawal, Neeraj Kayal, and Nitin Saxena on 6 August 2002 in a paper titled PRIMES is in P.^[126][127] Commenting on the impact of this discovery, Paul Leyland noted: "One reason for the excitement within the mathematical community is not only does this algorithm settle a long-standing problem, it also does so in a brilliantly simple manner. Everyone is now wondering what else has been similarly overlooked".^[127][128]
• Finite Difference Interpolation: The Indian mathematician Brahmagupta presented what is possibly the first instance^[129][130] of finite difference interpolation around 665 CE.^[131]
• Algebraic abbreviations: The mathematician Brahmagupta had begun using abbreviations for unknowns by the 7th century.^[132] He employed abbreviations for multiple unknowns occurring in one complex problem.^[132] Brahmagupta also used abbreviations for square roots and cube roots.^[132]
• Seshadri constant:In algebraic geometry, a Seshadri constant is an invariant of an ample line bundle L at a point P on an algebraic variety.The name is in honour of the Indian mathematician C. S. Seshadri.
• Basu's theorem: The Basu's theorem, a result of Debabrata Basu (1955) states that any complete sufficient statistic is independent of any ancillary statistic.^[133][134]
• Rolle's theorem: Bhāskara II is credited with knowledge of Rolle's theorem although its named after Michel Rolle who described with insufficient proof and was later proved by cauchy.^[135]
• Kosambi-Karhunen-Loève theorem: Also known as the Karhunen–Loève theorem. The Kosambi-Karhunen-Loève theorem is a representation of a stochastic process as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function on a bounded interval. Stochastic processes given by infinite series of this form were first^[136] considered by Damodar Dharmananda Kosambi.^[137]
• Brahmagupta–Fibonacci identity, Brahmagupta formula, Brahmagupta matrix, and Brahmagupta theorem: Discovered by the Indian mathematician, Brahmagupta (598–668 CE).^{[138][139][140][141]}
• Chakravala method: The Chakravala method, a cyclic algorithm to solve indeterminate quadratic equations is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE)^{[142][143][144]} although some attribute it to Jayadeva (c. 950~1000 CE).^[145] Jayadeva pointed out that Brahmagupta’s approach to solving equations of this type would yield infinitely large number of solutions, to which he then described a general method of solving such equations.^[146] Jayadeva's method was later refined by Bhāskara II in his Bijaganita treatise to be known as the Chakravala method, chakra (derived from cakraṃ चक्रं) meaning 'wheel' in Sanskrit, relevant to the cyclic nature of the algorithm.^[146][147] With reference to the Chakravala method, E. O. Selenuis held that no European performances at the time of Bhāskara, nor much later, came up to its marvellous height of mathematical complexity.^{[142][146][148]}
• Hindu number system: With decimal place-value and a symbol for zero, this system was the ancestor of the widely used Arabic numeral system. It was developed in the Indian subcontinent between the 1st and 6th centuries CE.^[149][150]
• Decimal mark:The practice of using a decimal mark is derived from the decimal system used in Indian mathematics.^[151]
• Fibonacci numbers: This sequence was first described by Virahanka (c. 700 AD), Gopāla (c. 1135), and Hemachandra (c. 1150),^[152] as an outgrowth of the earlier writings on Sanskrit prosody by Pingala (c. 200 BC).
• Zero, symbol: Indians were the first to use the zero as a symbol and in arithmetic operations, although Babylonians used zero to signify the 'absent'.^[153] In those earlier times a blank space was used to denote zero, later when it created confusion a dot was used to denote zero (could be found in Bakhshali manuscript).^[154] In 500 AD circa Aryabhata again gave a new symbol for zero (0).
• Law of signs in multiplication: The earliest use of notation for negative numbers, as subtrahend, is credited by scholars to the Chinese, dating back to the 2nd century BC.^[155] Like the Chinese, the Indians used negative numbers as subtrahend, but were the first to establish the "law of signs" with regards to the multiplication of positive and negative numbers, which did not appear in Chinese texts until 1299.^[155] Indian mathematicians were aware of negative numbers by the 7th century,^[155] and their role in mathematical problems of debt was understood.^[156] Mostly consistent and correct rules for working with negative numbers were formulated,^[157] and the diffusion of these rules led the Arab intermediaries to pass it on to Europe.,^[156] for example (+)×(-)=(-),(-)×(-)=(+) etc.
• Madhava series: The infinite series for π and for the trigonometric sine, cosine, and arctangent is now attributed to Madhava of Sangamagrama (c. 1340 – 1425) and his Kerala school of astronomy and mathematics.^[158][159] He made use of the series expansion of ${\displaystyle \arctan x}$ to obtain an infinite series expression for π.^[158] Their rational approximation of the error for the finite sum of their series are of particular interest. They manipulated the error term to derive a faster converging series for π.^[160] They used the improved series to derive a rational expression,^[160]${\displaystyle 104348/33215}$ for π correct up to eleven decimal places, i.e. ${\displaystyle 3.1415926539214}$.^[161][162] Madhava of Sangamagrama and his successors at the Kerala school of astronomy and mathematics used geometric methods to derive large sum approximations for sine, cosine, and arctangent. They found a number of special cases of series later derived by Brook Taylor series. They also found the second-order Taylor approximations for these functions, and the third-order Taylor approximation for sine.^{[163][164][165]}
• Pascal's triangle: Described in the 6th century CE by Varahamihira^[166] and in the 10th century by Halayudha,^[167] commenting on an obscure reference by Pingala (the author of an earlier work on prosody) to the "Meru-prastaara", or the "Staircase of Mount Meru", in relation to binomial coefficients. (It was also independently discovered in the 10th or 11th century in Persia and China.)
• Pell's equation, integral solution for: About a thousand years before Pell's time, Indian scholar Brahmagupta (598–668 CE) was able to find integral solutions to vargaprakṛiti (Pell's equation):^[168][169] ${\displaystyle \ x^{2}-Ny^{2}=1,}$ where N is a nonsquare integer, in his Brâhma-sphuṭa-siddhânta treatise.^[169]
• Ramanujan theta function, Ramanujan prime, Ramanujan summation, Ramanujan graph and Ramanujan's sum: Discovered by the Indian mathematician Srinivasa Ramanujan in the early 20th century.^[170]
• Shrikhande graph: Graph invented by the Indian mathematician S.S. Shrikhande in 1959.
• Sign convention: Symbols, signs and mathematical notation were employed in an early form in India by the 6th century when the mathematician-astronomer Aryabhata recommended the use of letters to represent unknown quantities.^[132] By the 7th century Brahmagupta had already begun using abbreviations for unknowns, even for multiple unknowns occurring in one complex problem.^[132] Brahmagupta also managed to use abbreviations for square roots and cube roots.^[132] By the 7th century fractions were written in a manner similar to the modern times, except for the bar separating the numerator and the denominator.^[132] A dot symbol for negative numbers was also employed.^[132] The Bakhshali Manuscript displays a cross, much like the modern '+' sign, except that it symbolized subtraction when written just after the number affected.^[132] The '=' sign for equality did not exist.^[132] Indian mathematics was transmitted to the Islamic world where this notation was seldom accepted initially and the scribes continued to write mathematics in full and without symbols.^[171]
• Modern elementary arithmetic: Modum indorum or the method of the Indians for arithmetic operations was popularized by al-khwarizmi and al-kindi by means of their respective works such as in Al-Khwarizmi's On the Calculation with Hindu Numerals (ca. 825), On the Use of the Indian Numerals (ca. 830)^[172] as early as the 8th and 9th centuries.They, amongst other works, contributed to the diffusion of the Indian system of arithmetic in the Middle-East and the West.The significance of the development of the positional number system is described by the French mathematician Pierre Simon Laplace (1749–1827) who wrote:

"It is India that gave us the ingenuous method of expressing all numbers by the means of ten symbols, each symbol receiving a value of position, as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit, but its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity."

• Trigonometric functions (adapted from Greek): * Trigonometric functions (adapted from Greek): The trigonometric functions sine and versine originated in Indian astronomy, adapted from the full-chord Greek versions (to the modern half-chord versions). They were described in detail by Aryabhata in the late 5th century, but were likely developed earlier in the Siddhantas, astronomical treatises of the 3rd or 4th century.^[173][174] Later, the 6th-century astronomer Varahamihira discovered a few basic trigonometric formulas and identities, such as sin^2(x) + cos^2(x) = 1.^[166]
• Formal systems: Panini is credited with the creation of the first Formal System in the world.

• Ayurvedic and Siddha medicine: Ayurveda and Siddha are ancient systems of medicine practiced in South Asia. Ayurvedic ideas can be found in the Hindu text^[175] (mid-first millennium BCE). Ayurveda has evolved over thousands of years, and is still practiced today. In an internationalized form, it can be thought of as a complementary and alternative medicine. In village settings, away from urban centres, it is simply "medicine." The Sanskrit word आयुर्वेदः (āyur-vedaḥ) means "knowledge (veda) for longevity (āyur)".^[176] Siddha medicine is mostly prevalent in South India, and is transmitted in Tamil, not Sanskrit, texts. Herbs and minerals are basic raw materials of the Siddha therapeutic system whose origins may be dated to the early centuries CE.^[177][178]
• Nasal reconstruction using a paramedian forehead flap:The first to use this kind of rhinoplasty was the Indian physician susruta.
• Cataract surgery: Cataract surgery was known to the Indian physician Sushruta (3rd century CE).^[179] In India, cataract surgery was performed with a special tool called the Jabamukhi Salaka, a curved needle used to loosen the lens and push the cataract out of the field of vision.^[180] The eye would later be soaked with warm butter and then bandaged.^[180] Though this method was successful, Susruta cautioned that cataract surgery should only be performed when absolutely necessary.^[180] Greek philosophers and scientists traveled to India where these surgeries were performed by physicians.^[180] The removal of cataract by surgery was also introduced into China from India.^[181]
• Cure for Leprosy: Kearns & Nash (2008) state that the first mention of leprosy is described in the Indian medical treatise Sushruta Samhita (6th century BCE).^[182] However, The Oxford Illustrated Companion to Medicine holds that the mention of leprosy, as well as ritualistic cures for it, were described in the Atharva-veda (1500–1200 BCE), written before the Sushruta Samhita.^[183]
• Plastic surgery: Treatments for the plastic repair of a broken nose are first mentioned in the Edwin Smith Papyrus,^[184] a transcription of an Ancient Egyptian medical text, some of the oldest known surgical treatise, dated to the Old Kingdom from 3000 to 2500 BC.^[185] Plastic surgery was being carried out in India by 2000 BCE.^[186] The system of punishment by deforming a miscreant's body may have led to an increase in demand for this practice.^[186] The surgeon Sushruta contributed mainly to the field of plastic and cataract surgery.^[187] The medical works of both Sushruta and Charak were translated into Arabic language during the Abbasid Caliphate (750 CE).^[188] These translated Arabic works made their way into Europe via intermediaries.^[188] In Italy the Branca family of Sicily and Gaspare Tagliacozzi of Bologna became familiar with the techniques of Sushruta.^[188]
• Lithiasis treatment: The earliest operation for treating lithiasis, or the formations of stones in the body, is also given in the Sushruta Samhita (6th century BCE).^[189] The operation involved exposure and going up through the floor of the bladder.^[189]
• Visceral leishmaniasis, treatment of: The Indian (Bengali) medical practitioner Upendranath Brahmachari (19 December 1873 – 6 February 1946) was nominated for the Nobel Prize in Physiology or Medicine in 1929 for his discovery of 'ureastibamine (antimonial compound for treatment of kala azar) and a new disease, post-kalaazar dermal leishmanoid.'^[190] Brahmachari's cure for Visceral leishmaniasis was the urea salt of para-amino-phenyl stibnic acid which he called Urea Stibamine.^[191] Following the discovery of Urea Stibamine, Visceral leishmaniasis was largely eradicated from the world, except for some underdeveloped regions.^[191]
• Angina pectoris:The condition was named "hritshoola" in ancient India and was described by Sushruta (6th century BC).^[187]
• Low-cost sanitary pad machine: Arunachalam is the inventor of a low-cost sanitary pad making machine and has innovated grass-roots mechanisms for generating awareness about traditional unhygienic practices around menstruation in rural India.

• Diamond mining and diamond tools: Diamonds were first recognized and mined in central India,^{[192][193][194]} where significant alluvial deposits of the stone could then be found along the rivers Penner, Krishna and Godavari. It is unclear when diamonds were first mined in India, although estimated to be at least 5,000 years ago.^[195] India remained the world's only source of diamonds until the discovery of diamonds in Brazil in the 18th century.^{[196][197][198]} Golconda served as an important centre for diamonds in central India.^[199] Diamonds then were exported to other parts of the world, including Europe.^[199] Early references to diamonds in India come from Sanskrit texts.^[200] The Arthashastra of Kautilya mentions diamond trade in India.^[198] Buddhist works dating from the 4th century BCE mention it as a well-known and precious stone but don't mention the details of diamond cutting.^[192] Another Indian description written at the beginning of the 3rd century describes strength, regularity, brilliance, ability to scratch metals, and good refractive properties as the desirable qualities of a diamond.^[192] A Chinese work from the 3rd century BCE mentions: "Foreigners wear it [diamond] in the belief that it can ward off evil influences".^[192] The Chinese, who did not find diamonds in their country, initially used diamonds as a "jade cutting knife" instead of as a jewel.^[192]
• Zinc mining and medicinal zinc: Zinc was first smelted from zinc ore in India.^[201] Zinc mines of Zawar, near Udaipur, Rajasthan, were active during early Christian era.^[202][203] There are references of medicinal uses of zinc in the Charaka Samhita (300 BCE).^[204] The Rasaratna Samuccaya which dates back to the Tantric period (c. 5th - 13th century CE) explains the existence of two types of ores for zinc metal, one of which is ideal for metal extraction while the other is used for medicinal purpose.^[204][205]

• Gravity:Aryabhata first identified the force to explain why objects do not fall when the earth rotates, brahmagupta described gravity as an attractive force and used the term "gruhtvaakarshan" for gravity.^{[206][207][208]} Aryabhata developed a geocentric solar system of gravitation, and an eccentric elliptical model of the planets, where the planets spin on their axes and follow elliptical orbits,the Sun and the moon revolving around the earth in epicycles.
• Mitrogen Nalaxide, synthesis in pure form: Narendra Damodardas Modi synthesized MH₄NA₂ in its pure form, and became the first scientist to have done so.^[209] Prior to Narendra Modi's invention of Mitrogen Nalaxide it was thought that the untreated raw sewer releasing Mitrogen was harmful for living beings and planet earth, Modi's invention changed the world treated sewer waste. Sewer waste is now regarded as natural resource by UNESCO, Narednra Modi was also awarded with Honorary doctorate by the "Institute of eminence" JIO Institute..^[209]
• Ashtekar variables: In theoretical physics, Ashtekar (new) variables, named after Abhay Ashtekar who invented them, represent an unusual way to rewrite the metric on the three-dimensional spatial slices in terms of a SU(2) gauge field and its complementary variable. Ashtekar variables are the key building block of loop quantum gravity.
• Bhatnagar-Mathur Magnetic Interference Balance: Invented jointly by Shanti Swarup Bhatnagar and K.N. Mathur in 1928, the so-called 'Bhatnagar-Mathur Magnetic Interference Balance' was a modern instrument used for measuring various magnetic properties.^[210] The first appearance of this instrument in Europe was at a Royal Society exhibition in London, where it was later marketed by British firm Messers Adam Hilger and Co, London.^[210]
• Bhabha scattering: In 1935, Indian nuclear physicist Homi J. Bhabha published a paper in the Proceedings of the Royal Society, Series A, in which he performed the first calculation to determine the cross section of electron-positron scattering.^[211] Electron-positron scattering was later named Bhabha scattering, in honor of his contributions in the field.^[211]
• Bose–Einstein statistics, condensate : On 4 June 1924 the Indian physicist Satyendra Nath Bose mailed a short manuscript to Albert Einstein entitled Planck's Law and the Light Quantum Hypothesis seeking Einstein's influence to get it published after it was rejected by the prestigious journal Philosophical Magazine.^[212] The paper introduced what is today called Bose statistics, which showed how it could be used to derive the Planck blackbody spectrum from the assumption that light was made of photons.^[212][213] Einstein, recognizing the importance of the paper translated it into German himself and submitted it on Bose's behalf to the prestigious Zeitschrift für Physik.^[212][213] Einstein later applied Bose's principles on particles with mass and quickly predicted the Bose-Einstein condensate.^[213][214]
• Boson:The name boson was coined by Paul Dirac^[215] to commemorate the contribution of the Indian physicist Satyendra Nath Bose.^[216][217] In quantum mechanics, a boson (/ˈboʊsɒn/,^[218] /ˈboʊzɒn/^[219]) is a particle that follows Bose–Einstein statistics. Bosons make up one of the two classes of particles, the other being fermions.^[220]
• Braunstein-Ghosh-Severini Entropy: This modelling of entropy using network theory is used in the analysis of quantum gravity and is named after Sibasish Ghosh and his teammates, Samuel L. Braunstein and Simone Severini.
• Chandrasekhar limit and Chandrasekhar number: Discovered by and named after Subrahmanyan Chandrasekhar, who received the Nobel Prize in Physics in 1983 for his work on stellar structure and stellar evolution.^[221]
• Galena, applied use in electronics of: Bengali scientist Sir Jagadish Chandra Bose effectively used Galena crystals for constructing radio receivers.^[222] The Galena receivers of Bose were used to receive signals consisting of shortwave, white light and ultraviolet light.^[222] In 1904 Bose patented the use of Galena Detector which he called Point Contact Diode using Galena.^[223]
• Helium: The French astronomer, Pierre Janssen observed the Solar eclipse of 18 August 1868 from Guntur in Madras State, British India. He discovered the first evidence of helium as a bright yellow line in the chromograph of the Sun
• Mahalanobis distance: Introduced in 1936 by the Indian (Bengali) statistician Prasanta Chandra Mahalanobis (29 June 1893 – 28 June 1972), this distance measure, based upon the correlation between variables, is used to identify and analyze differing pattern with respect to one base.^[224]
• Mercurous Nitrite: The compound mercurous nitrite was discovered in 1896 by the Bengali chemist Prafulla Chandra Roy, who published his findings in the Journal of Asiatic Society of Bengal.^[209] The discovery contributed as a base for significant future research in the field of chemistry.^[209]
• Ramachandran plot, Ramachandran map, and Ramachandran angles: The Ramachandran plot and Ramachandran map were developed by Gopalasamudram Narayana Iyer Ramachandran, who published his results in the Journal of Molecular Biology in 1963. He also developed the Ramachandran angles, which serve as a convenient tool for communication, representation, and various kinds of data analysis.^[225]
• Raman effect: The Encyclopædia Britannica (2008) reports: "change in the wavelength of light that occurs when a light beam is deflected by molecules. The phenomenon is named for Sir Chandrasekhara Venkata Raman, who discovered it in 1928. When a beam of light traverses a dust-free, transparent sample of a chemical compound, a small fraction of the light emerges in directions other than that of the incident (incoming) beam. Most of this scattered light is of unchanged wavelength. A small part, however, has wavelengths different from that of the incident light; its presence is a result of the Raman effect."^[226]
• Raychaudhuri equation: Discovered by the Bengali physicist Amal Kumar Raychaudhuri in 1954. This was a key ingredient of the Penrose-Hawking singularity theorems of general relativity.^[227]

• Lunar water: Although the presence of water ice on the moon has been conjectured by various scientists since the 1960s, inconclusive evidence of free water ice had also been identified. The first incontrovertible evidence of water on the moon was provided by the payload Chace carried by the Moon Impact Probe released by Chandrayaan-1^{[228][229][230]} confirmed and established by NASA.^[231]
• Earth's orbit(Sidereal year):The Hindu cosmological time cycles explained in the Surya Siddhanta(c.600 CE), give the average length of the sidereal year (the length of the Earth's revolution around the Sun) as 365.2563627 days, which is only a negligible 1.4 seconds longer than the modern value of 365.256363004 days.^[232] This remains the most accurate estimate for the length of the sidereal year anywhere in the world for over a thousand years.
• Periodicity of comets: Indian astronomers by the 6th century believed that comets were celestial bodies that re-appeared periodically. This was the view expressed in the 6th century by the astronomers Varahamihira and Bhadrabahu, and the 10th-century astronomer Bhattotpala listed the names and estimated periods of certain comets, but it is unfortunately not known how these figures were calculated or how accurate they were.^[233]
• Saha ionization equation: The Saha equation, derived by the Bengali scientist Meghnad Saha (6 October 1893 – 16 February 1956) in 1920, conceptualizes ionizations in context of stellar atmospheres.^[234]
• Quasinormal modes of black holes: C. V. Vishveshwara discovered the quasinormal modes of black holes.^[235] These modes of black hole vibrations are one of the main targets of observation using the gravitational wave detectors.

• Iron working: Iron works were developed in India, around the same time as, but independently of, Anatolia and the Caucasus. Archaeological sites in India, such as Malhar, Dadupur, Raja Nala Ka Tila and Lahuradewa in present-day Uttar Pradesh show iron implements in the period between 1800 BCE—1200 BCE.^[236] Early iron objects found in India can be dated to 1400 BCE by employing the method of radiocarbon dating. Spikes, knives, daggers, arrow-heads, bowls, spoons, saucepans, axes, chisels, tongs, door fittings etc. ranging from 600 BCE to 200 BCE have been discovered from several archaeological sites of India.^[237] Some scholars believe that by the early 13th century BC, iron smelting was practiced on a bigger scale in India, suggesting that the date the technology's inception may be placed earlier.^[236] In Southern India (present day Mysore) iron appeared as early as 11th to 12th centuries BC; these developments were too early for any significant close contact with the northwest of the country.^[238] In the time of Chandragupta II Vikramaditya (375–413 CE), corrosion-resistant iron was used to erect the Iron pillar of Delhi, which has withstood corrosion for over 1,600 years.^[239]

• Simputer:The Simputer (acronym for "simple, inexpensive and multilingual people's computer") is a self-contained, open hardware handheld computer, designed for use in environments where computing devices such as personal computers are deemed inconvenient. It was developed in 1999 by 7 scientists of the Indian Institute of Science, Bangalore, led by Dr. Swami Manohar in collaboration with Encore India, a company based inBangalore.^[240] Originally envisaged to bring internet to the masses of India, the Simputer and its derivatives are today widely utilized by governments of several Indian states as part of their e-governance drive, the Indian Army, as well as by other public and private organizations.
• Backus-naur form:In computer science, Backus–Naur form or Backus normal form (BNF) or Panini Backus form^[241] is a notation technique for context-free grammars, often used to describe the syntax of languages used in computing, such as computer programming languages, document formats, instruction sets and communication protocols. They are applied wherever exact descriptions of languages are needed: for instance, in official language specifications, in manuals, and in textbooks on programming language theory.The idea of describing the structure of language using rewriting rules can be traced back to at least the work of Pāṇini (who lived sometime between the 7th and 4th century BCE).^[242][243] The name Pāṇini Backus form has also been suggested in view of the fact that the expansion Backus normal form may not be accurate, and that Pāṇini had independently developed a similar notation earlier.^[244] His notation to describe Sanskrit word structure notation is equivalent in power to that of Backus and has many similar properties.

• Formal grammar:Panini in his treatise,Astadyayi gives formal production rules and definitions to describe the formal grammar of Sanskrit.^[245] In formal language theory, a grammar (when the context is not given, often called a formal grammar for clarity) is a set of production rules for strings in a formal language. The rules describe how to form strings from the language's alphabet that are valid according to the language's syntax. A grammar does not describe the meaning of the strings or what can be done with them in whatever context—only their form.
• Same language subtitling:Same Language Subtitling (SLS) refers to the idea of subtitling in the same language as the audio, converse to the original idea of subtitling, which was to present a different language.This idea was struck upon by Brij Kothari, who believed that SLS makes reading practice an incidental, automatic,and subconscious part of popular TV entertainment, at a low per-person cost to shore up literacy rates in India.^[246] His idea was well received by the Government of India who now uses SLS on several national channels.For his idea, Kothari was adjudged a winner at the Development Marketplace— the World Bank’s Innovation Award which gave him enough funds to implement this programme nationally.The SLS innovation has been recognised by the Institute for Social Inventions, UK and the Tech Museum of Innovations, San Jose, USA and is the recipient of the International Literacy Prize from the Library of Congress (Washington, D.C.) and awards from the All Children Reading Grand Challenge (USAID), Tech Museum of Innovation (San Jose), the Institute for Social Inventions (London) and the NASSOM Foundation.^[247]

• Standardization:The oldest applications and evidence of standardisation come from the Indus valley civilization in the 5th millennium BC^[10] characterised by the existence of weights in various standards and categories as^[248] well as the Indus merchants usage of a centralised weight and measure system. Small weights were used to measure luxury goods, and larger weights were used for buying bulkier items, such as food grains etc.^[248] The weights and measures of the Indus civilisation also reached Persia and Central Asia, where they were further modified.^[249]
• Technical standards:Technical standards were being applied and used in the Indus valley civilization since the 5th millennium BC to enable gauging devices to be effectively used in angular measurement and measurement in construction.^[250] Uniform units of length were used in the planning and construction of towns such as Lothal, Surkotada, Kalibangan, Dholavira, Harappa, and Mohenjo-daro.^[10] The weights and measures of the Indus civilisation also reached Persia and Central Asia, where they were further modified.^[249] Shigeo Iwata describes the excavated weights unearthed from the Indus civilisation:

A total of 558 weights were excavated from Mohenjodaro, Harappa, and Chanhu-daro, not including defective weights. They did not find statistically significant differences between weights that were excavated from five different layers, each about 1.5 m in thickness. This was evidence that strong control existed for at least a 500-year period. The 13.7-g weight seems to be one of the units used in the Indus valley. The notation was based on the binary and decimal systems. 83% of the weights which were excavated from the above three cities were cubic, and 68% were made of chert.^[10]

• Timeline of Indian innovation
• Nalanda University
• History of science and technology in India
• List of Pakistani inventions and discoveries
• Timeline of historic inventions

• Ancient India’s Inventions In Science And Technology
• Essays on Indian Science and Technology.
• P. K. Ray, SCIENCE,CULTURE AND DEVELOPMENT — A CONNECTED PHENOMENA, Everyman’s Science Vol.
• History of Science in South Asia (hssa-journal.org). HSSA is a peer-reviewed, open-access, online journal for the history of science in India.