History of India: Scientific & Mathematical Discoveries Part 1 Do you know India is one of the Some perennial contributions of Indian Hindu intellectuals in the field of science and mathematics: greatest contributors of Scientific

knowledge? Beyond 7000 BCE – 1500 BCE (Harappan Indus Civilization)

‐ Earliest of knowledge of mathematics: geometry, weight & measurement system. (ruler designed to accuracy of 1.32 inches per unit) Science and philosophy were both highly developed disciplines in ancient 1500 BCE – 400 BCE (Vedic Period) India. However, because Indian ‐ Early contributions to the study of the decimal number system, zero, negative numbers, arithmetic philosophic thought was considerably and algebra. Use of large number, i.e. high as 1012 is used in Yajurveda Samhita. more mature and found particular ‐ Methodological memorization techniques to transmit literary texts of philosophy and science. favor amongst intellectuals, the notion ‐ Advanced geometric calculations provided in Sulba sutras. Yajnavalkya, Apastamba, Manava, persists that any early scientific Hiranyakesin, Varaha and Vadhula are some of those who made great contributions in Geometry. contribution came solely from the West, In astronomy Yajnavalkya (1800 BCE) knew of a 95‐year cycle to harmonize the motions of the sun Greece in particular. Because of this and the moon and he also knew that the sun's circuit was asymmetric. Baudhayana is first to prove erroneous belief, which is perpetuated in his Shulva Sutra, (c. 700 BCE) the formula, now known as the so‐called "Pythagorean theorem". by a wide variety of scholars, it is ‐ Charaka, wrote his famous Ayurvedic treatise Charaka Samhita, contains many remarks which are necessary to briefly examine the held in reverence even today. Some of them are in the fields of physiology, etiology and embryology. history of Indian scientific Charaka was the first physician to present the concept of digestion, metabolism and immunity. thought. Jawaharlal Nehru wrote in his Charaka also knew the fundamentals of genetics, factors determining the sex of a child, and mentions book The Discovery of India: "Till that genetic defect in a child, like lameness or blindness, was not due parents, but due to defects in recently many European thinkers the ovum or sperm of the parents which is today an accepted fact. Agnivesa, under the guidance of imagined that everything that was the great ancient physician Atreya, he wrote an encyclopedic treatise in the c.700 BCE. worthwhile had its origins in Greece or ‐ Sushruta (c. 600 BCE) Hindu surgeon, who lived in Kasi, of "India today is estimated to have about Rome." From the very earliest times, whom much of modern medicine can be traced to. Surgery thirty million manuscripts, the largest India had made its contribution to the (Shastrakarma) is one of the eight branches of Ayurveda. body of handwritten reading material texture of Western thought and living. anywhere in the world. The literate Sushruta, best known for (rhinoplasty) plastic surgery, Michael Edwardes author of British include cosmetic surgery, treatise on medical ethics, culture of Indian science goes back to at least the fifth century B.C.” ‐ Pingree 1988 India, writes that throughout the definitions for 121 surgical implements, listed the diagnosis literatures of Europe, tales of Indian of 1,120 diseases, control of infection through antiseptics, use of drugs to control bleeding, origin can be discovered. European toxicology, psychiatry, midwifery, cataract operations and classification of burns. Sushruta was the mathematics and, through them, the earliest one to study the human anatomy and to prescribe surgical anesthesia. full range of European technical ‐ Pā ini (c. 520­460 BCE). His work on Sanskrit algebraic grammar includes early use of Boolean achievement, could hardly have logic, null operator, context free grammars and a precursor of Backus–Naur form. ‐ Pakudha Katyayana (c.500 BCE) was among Indian philosopher who also propounded ideas existed without Indian numerals and mathematical concepts. But until the about atomic constitution of the material world. Famous historian of mathematics, Florian Cajori (1893), suggested that “In algebra, beginning of European colonization in 400 BCE – 400 CE there was probably a mutual giving and Asia, India’s contribution was usually ­ Jaina mathematicians flourished during this period and receiving [between Greece and India].” And filtered through other cultures. contributed in enumeration of very large numbers and he "suspects that Diophantus got his first infinities, devised notations for simple powers (and glimpse of algebraic knowledge from India.” Grant Duff, British Historian of India: exponents) of numbers like squares and cubes to define simple algebraic equations. Apparently also "Many of the advances in the the first to use the word shunya (literally void in Sanskrit) to refer to zero, after which more than a sciences that we consider today to millennium later, their appellation became the English word "zero". Jaina mathematicians included have been made in Europe were in Bhadrabahu (c. 298BCE), Yativrisham Acharya (c. 176 BCE) and Umasvati (c. 150 BCE). fact made in India centuries ago." ‐ Kanada (300 BCE) founder of the Vaiseshika (Distinctionism) philosophy, among the first Indian philosophers who formulated ideas of atom in a systematic manner in the field of Atomic Physics. Dr. Vincent Smith: ‐ Ancient India possessed advanced medical knowledge. Doctors knew about metabolism, the "India suffers today, in the circulatory system, genetics, and the nervous system as well The great picture of Indian medicine is one estimation of the world, more as the transmission of specific characteristics by heredity. of rapid development in the Vedic and through the world's ignorance of Vedic physicians understood medical ways to counteract Buddhist period, followed by centuries of the achievements of the heroes of the effects of poison gas, performed Caesarean sections and slow and cautious improvement. In the time of Alexander, says Garrison, "Hindu Indian history than through the brain operations, and used anesthetics. Vaccination, physicians and surgeons enjoyed a well‐ absence or insignificance of such th unknown to Europe before the 18 century, was known in deserved reputation for superior knowledge achievement." India as early as 550 A.D, mention in a text attributed to and skill," and even Greeks like Aristotle is Dhanwantari, one of the earliest Hindu physicians. believed to have been indebted to them. So Indians of current times, born in ­ Katyayana (c. 200 BCE) wrote about geometry in too with the Chinese, Persians and Arabs. independent India to a very large Katyayana Sulba Sutra, including the general We find Persians and Arabs translating into their languages, in the eighth century A.D., extend still continues to suffer the Pythagorean theorem and a computation of the square root “colonized mindset” or Macaulayism. of 2 correct to five decimal places. the thousand‐year‐old compendia of Sushrata and Charaka. The great Caliph Unfortunately there those who believe ­ Patanjali (c. 100BCE) compiled systematically the Haroun‐al‐Rashid accepted the all scientific advancement was made fundamentals of Yoga (Raja Yoga). Patanjali was the earliest preeminence of Indian medicine and in West though the truth is that both metal expert. His Loha Sastra, describes metal preparation. scholarship, and imported Indian physicians East and West share an equal ­ Pingala (c.1st century BCE) a musical theorist, to organize hospitals and medical schools in developed a system of binary enumeration convertible to Baghdad. Lord Amphill concludes that estimation. Some ignorantly attribute decimal numerals, described in his Chandahsaastra. His medieval and modern Europe owes its any modern scientific contribution of system is quite similar to that of Leibniz, who lived roughly system of medicine directly to the Arabs, Indians to British colonialism and fourteen hundred years later! and through them to India. believe that by expelling Hinduism, ‐ Lagadha (c. 100 A.D.) text Vedanga Jyotisha of Lagadha went beyond the earlier calendrical India would make greater leap in astronomy to develop a theory for the mean motions of the sun and the moon. This marked the scientific advancement. Yet ‘Hindu beginnings of the application of mathematics to the motions of the heavenly bodies. India’ enjoyed the prestige of being the world leader in the scientific front for over hundreds of years till the nation was plundered by Muslims invaders and later British colonization. History of India: Scientific & Mathematical Discoveries Part 2

400 CE – 1200 CE (Classical Period) This period is often known as the golden age of Indian science and mathematics, their contributions would spread to Asia, the Middle East, and eventually to Europe. ‐ Ganesa Datvajna. His book Graha Laghavam and Surya Siddhanta (disputed ­ its authorship is unknown) describe astronomical procedures and algorithms. Surya Siddhanta (believed by Hindus to be handed down from 3000 BC by aid of complex mnemonic) computed the earth's diameter to be 7,840 miles, the distance earth ‐ moon as 253,000 miles, close to modern measurements. ­ Aryabhata I (475 ­550 A.D.) well known Hindu mathematician and astronomer, (born in possibly Kerala, while some indicate Deccan South, some believe it to be central India). Aryabhatta's Magnum Opus, the Aryabhattiya (499 A.D) was translated into Latin in the 13th century. His work was translated in Arabic and played an important role in Islamic civilization. Through this translation, European mathematicians got to know methods for calculating the areas of triangles, volumes of spheres as well as square and cube root. Aryabhata discovers about eclipses, heliocentric theory of gravitation and the sun being the source of moonlight predating 1500 years ago and 1000 years before Copernicus and Galileo. He made the fundamental advance in finding the lengths of chords of circles, gave the value of π as 3.1416, gave methods for extracting square roots, summing arithmetic series, solving indeterminate equations of the type ax ‐by = c, and also gave what later came to be known as the table of Sines. Among the first one to propounded the theory that earth was a sphere, with the planets spinning on their axes and following an elliptical orbit around the Sun. In addition to the physical sciences, ‐Varahamihira (505 ­ 587 A.D) Claim for the first time that there should be a force, which is keeping very interesting and modern concepts bodies stuck to the earth, and also keeping heavenly bodies in their determined places. Thus the concept of botany and biology, including the of the existence of some tractive force that governs the falling of objects to the earth and their remaining concepts of micro‐organisms, are also stationary after having once fallen; as also determining the positions which heavenly bodies occupy, was encountered in these ancient texts, for recognised. It was also recognised that this force is a tractive force. The Sanskrit term for gravity is example, in the Mahabharata. The ancient text of Agastya Samhita Gurutvakarshan, which is an amalgam of Guru‐tva‐akarshan. Akarshan means to be attracted thus the describes the method of making electric fact that the character of this force was of attraction was also recognised. battery, and that water can be split into ‐ Numerous alchemical & chemistry texts were written between the 9th and 14th centuries AD, some of oxygen and hydrogen. the known alchemists are Govinda Bhagwatpada, Siddha Nityanatha, Somadeva, Vagbhatta, Yasodhara, Ramesvara Bhatta, Dhundukanatha, Ramacandra Guha, Sarvajnacandra, Surya Pandita, Camunda, Devanatha and Suresvara. There still many texts whose authorship is unknown. ‐ Practitioners of the Siddha system of medicine wrote a number of alchemical texts known as Mappu texts in the Tamil language. The more prominent Siddhas were Agastya, Bogar, Ramdevar and Karuvurar. ‐ Brahmagupta (598 ­665 A.D.) Born in Bhillamala (today's Bhinmal) in Gujarat is renowned for introduction of negative numbers and operations on zero into arithmetic. His main work was later translated into Arabic. He formulated the rule of three and proposed rules for the solution of quadratic and simultaneous equations and gave the formula for the area of a cyclic quadrilateral. He was the first mathematician to treat algebra and arithmetic as two different branches of mathematics. He gave the solution of the indeterminate equation Nx²+1 = y². He is also the founder of the branch of higher mathematics known as "Numerical Analysis". Brahmagupta estimated that the circumference of the earth as 36,000 kms which comes quite close to the actual circumference known today. Many Indian astronomers had formulated ideas about gravity. Brahmagupta had said about gravity that "Bodies fall towards the earth as it is in the nature of the earth to attract bodies, just as it is in the nature of water to flow". ‐ Bhaskara I (c. 600­680 A.D.) expanded the work of Aryabhata. He produced solutions of indeterminate equations, a rational approximation of the sine function and a formula for calculating the sine of an acute angle without the use of a table, correct to 2 decimal places. ­ Mahavira (Jain prophet c. 800­870 A.D.) born in Karnataka, wrote on mathematics concepts of zero, squares, cubes, square‐roots, cube‐ roots, and the series extending beyond these. He also wrote on plane and solid geometry, as well as problems relating to the casting of shadows. He wrote Ganita Sara Sangraha in 850 A.D., which is the first text book on arithmetic in present day form. He is the only Indian mathematician who has briefly referred to the ellipse (which he called Ayatvrit). ‐ Virasena (c. 800 A.D.), Jain mathematician in court of Rashtrakuta Kingdom of Manyakheta, Karnataka, wrote the Dhavala, a commentary on Jain mathematics, deals logarithms to base 2 , base 3, base 4 and also derivation of the volume of a frustum by a sort of infinite procedure. ‐ Bhaskara II (1114­1185 A.D.) the most well known ancient Indian mathematician, born in Bijapur, Karnataka. He was the first to declare that any number divided by zero is infinity and that the sum of any number and infinity is also infinity. His work Siddhanta Siromani (1150 A.D.) contains many interesting concepts of arithmetic, algebra, planetary sphere ‐celestial globe and astronomical mathematics. Bhaskara introduced chakrawala, or the cyclic method, to solve algebraic equations which 6 centuries later, European mathematicians like Galois, Euler and Lagrange rediscovered this method and called it "inverse cyclic". Bhaskara can also be called the founder of differential calculus. He gave an example of what is now called "differential coefficient" and the basic idea of what is now called "Rolle's theorem". As an astronomer, Bhaskara is renowned for his concept of Tatkalikagati (instantaneous motion). ‐ Sridhara (c. 870­930 A.D.), who lived in Bengal, wrote works on algebra, giving rule for finding volume of sphere, solving quadratic equations and arithmetic operations. Manjula (c. 900 A.D.), Aryabhata II (c. 920­1000), Shripati Mishra (1019­1066), Nemichandra Siddhanta Chakravati (c. 1100) are among the other Indian mathematicians who made discoveries in arithmetic, differential equations, indeterminate equations, algebra, and astronomy (eclipse, planetary longitude, etc). Only a few decades after Nagarjuna ­ Nagarjuna (c. 931A.D) was born at Fort Daihak, in Gujarat. He was a chemist or alchemists. He deceased, Mahmud of Ghazni raided incorporate the ideas of earlier chemists and physicians. His efforts had been concentrated on and plundered Nagarjuna's hometown transforming the base metals into superior metal mainly gold. His treatise, Rasaratnakara dealt with the of Somnath in 1020 A.D. Nagarjuna's preparation of rasa (liquids, mainly mercury). Nagarjuna has discussed various combinations of liquids texts fell into the hands of Muslims. in this volume. He gave a survey of the status of metallurgy and alchemy as it existed in India in those days. Methods for theWhile the invaders ruthlessly destroyed extraction of metals like gold, silver, tin and copper from their ores and their purification were also mentioned. He discusses the architectural achievements of this about methods for the dissolution of diamonds, metals and pearls, listed the apparatus that was used by country and imposed their despotic rule, they also transmitted Indian earlier alchemists, the process of distillation,liquefaction, sublimation and roasting. Although he could not produce gold, thsciences to the outside world. (Persia, ese techniques did yield metals with gold like yellowish brilliance. Till today these methods are being used to manufacture Arabia and to Europe). imitation jewellery. Nagarjuna has also discussed methods for the preparation of mercury like calamine. He also wrote many treatises on organic chemistry and ayurvedic medicine, mainly with the preparation of medicinal drugs.

Indian contributions to mathematics, medicine and pure science have not been given due acknowledgement in modern history and that many discoveries and inventions by Indian mathematicians were known to their Western counterparts, copied by them, and presented as their own original work; and further, that this mass plagiarism has gone unrecognized due to Eurocentrism. According to G. G. Joseph: “[Their work] takes on board some of the objections raised about the classical Eurocentric trajectory. The awareness [of Indian and Arabic mathematics] is all too likely to be tempered with dismissive rejections of their importance compared to Greek mathematics. The contributions from other civilizations ‐ most notably China and India, are perceived either as borrowers from Greek sources or having made only minor contributions to mainstream mathematical development. An openness to more recent research findings, especially in the case of Indian and Chinese mathematics, is sadly missing”. Part 3

1300 CE – 1600 CE Transmission of Kerala School results ‐ Kerala School of Mathematics & Astronomy (c. 1300­1600) found by Madhava made landmark to Europe ‐ As discussed in this section, contribution to the world mathematics. Contributions made to the fields of infinite series and calculus the infinite series of calculus for (two centuries before calculus was known in Europe), trigonometric functions, advances in geometry, trigonometric functions (rediscovered arithmetic, algebra and astronomy. The school also had contributed in the field of linguistic and ayurveda by Gregory, Taylor, and Maclaurin in the late 17th century) were described . Some of the Keralan scientists are Parameshvara, Govindaswami, Neelakanta Somayaji, Jyeshtadeva, (with proofs) in India, by Achyuta Pisharati (1550­1621), Melpathur Narayana Bhattathiri (1559­1632 author of the mathematicians of the Kerala School, famous poem, Narayaneeyam) and Achyuta Panikkar. remarkably some two centuries earlier. ‐ Madhava (c. 1340­1425) discovered the Taylor series; the power series of π, usually attributed to Many historians such as A. K. Bag Leibniz but now known as the Madhava­Leibniz series; the solution of transcendental equations by (1979) suggested that knowledge of iteration; and the approximation of transcendental numbers by continued fractions. In astronomy, these results might have been Madhava discovered a procedure to determine the positions of the Moon every 36 minutes, and methods transmitted to Europe through the to estimate the motions of the planets. He also extended some earlier results of Bhaskara. trade route from Kerala by traders and Jesuit missionaries as Kerala was in ­ Sayana (c. 1300 A.D) was minister at the court of Vijayanagara Emperor Bukka. He explains a rucha in continuous contact with China and Rig Veda where the speed of light is calculated to be 2,202 yojanas in half a nimesha, which come to Arabia, and Europe. In fact it is known 186,536 miles per second. that the works of the Kerala School ­ Narayana Pandit (1340­1400), another mathematician of Kerala, wrote a number of treatises on were written up for the Western world arithmetic and algebra. Contribution include a rule to calculate approximate values of square roots, by many such as J.Warren in 1825 and investigations into the second order indeterminate equation nq2 + 1 = p2 (Pell's equation), solutions of C. M. Whish in 1835. indeterminate higher‐order equations, mathematical operations with zero, several geometrical rules, and a discussion of magic squares and similar figures. Narayana made minor contributions to the ideas of differential calculus found in Bhaskara II's work. He also made contributions to the topic of cyclic quadrilaterals. ‐ Parameshvara (1370­1460), of Kerala School, discovered Lhuilier's formula for the circumradius of a cyclic quadrilateral. Also decimal floating point numbers, the secant method and iterative methods for solution of non‐linear equations. An early version of the mean value theorem, is considered one of the most important results in differential calculus and one of the most important theorems in mathematical analysis, and was later essential in proving the fundamental theorem of calculus. Also founded of the Drigganita system of Astronomy. ­ Govindaswami (c.1400), of Kerala School, discovered the Newton‐Gauss interpolation formula. ‐ Neelakanta Somayaji (1444­1544) of Kerala School, wrote many notable works. Contribution include inductive mathematical proofs, a derivation and proof of the Madhava­Gregory series of the arctangent trigonometric function, improvements and proofs of other infinite series expansions by Madhava, an improved series expansion of π that converges more rapidly, and the relationship between the power series of π and arctangent. He also gave sophisticated explanations of the irrationality of π, the correct formulation for the equation of the center of the planets, and a heliocentric model of the solar system. ‐ Raghunatha (1475­1550) At its zenith during the time of Raghunatha, the School of New Logic (Navya Nyaya) of Bengal and Bihar developed a methodology for a precise semantic analysis of language. Its formulations are equivalent to mathematical logic. ‐ Jyesthadeva (c. 1500­1600) of Kerala School. He wrote a very important work, the Yuktibhasa (written in Malayalam), is the world's first Calculus text. It contained proofs of theorems, derivations of rules and series, a derivation and proof of the Madhava‐Gregory series of the arctangent function, proofs of most mathematical theorems and infinite series earlier discovered by Madhava and other mathematicians of Kerala. It also contains a proof of the series expansion of the arctangent function (equivalent to Gregory's proof), and the sine and cosine functions. He also gave integer solutions of systems of first degree equations solved using kuttaka method, and rules of finding the sines and the cosines of the sum and difference of two angles. He gave earliest statement of Wallis' theorem, and geometrical derivations of infinite series. ­ Citrabhanu (c. 1530) another mathematician of Kerala, gave integer solutions to 21 types of systems of two simultaneous Diophantine equations in two unknowns, with explanation and justification of his rule. Some of his explanations are algebraic, while others are geometric. Under Christian dominance, Europe passes through dark ages 1600 CE – 1800 CE for over hundreds of years. By then Asian civilization such as of After this period, India was repeatedly raided by Muslims and other rulers and India and China had reach great scientific advancement and there was a lull in scientific research. The heinous barbarisms of the Islamic technological development. Europe’s scientific development conquests upto the recently repulsed British colonial rule ­ saw the nadir of only began in the mid 16th century (1500AD) onwards. The Hindu innovation. Industrial revolution and Renaissance passed India by. scientific knowledge from Indians, Chinese as well of Islamic Before Ramanujan, the only noteworthy of this period mathematician was world contributed to the scientific awakening of Europe beside Sawai Jai Singh II,who founded the present city of Jaipur in 1727 This Hindu king of those from Ancient Greece, Egypt and Rome (which was discarded as un‐Christian pagan earlier). Cultural Renaissance was a great patron of mathematicians and astronomers. He is known for building began in Italy in 14th century and spread to rest of Europe in observatories (Jantar Mantar) at Delhi, Jaipur, Ujjain, Varanasi and Mathura. 16th century. Europe’s Industrial revolution begins in the 18th century in England. 1800 CE – 1900 CE ‐ Sankara Varma (1800­1838) wrote Sadratnamala an astronomical treatise, that serves as a summary of most of the results of the Kerala School. He computed π correct to 17 decimal places. Interestingly the author stands out as the last notable name in the Kerala School of mathematics, though it was composed in the early 19th century. Under British occupation education and ‐ Some notable name appearing in the 19th and 20th centuries are Sanjeev Shah (1803­ 1896) in scientific potential of the Indian people mathematics, Jagadish Bose (1858­ 1937) in electromagnetics and plant life, pioneer of wireless was stifled. British education system, communication, Srinivasa Ramanujan (1887­ 1920) in mathematics, Chandrasekhar Venkata merely aimed at producing civil and Raman (1888‐ 1970) in physics, A. A. Krishnaswami Ayyangar (1892­1953) in mathematics, administrative services candidates. By Raghunath Purushottam Paranjape in mathematics, Meghnad Saha (1893­ 1956) in astrophysics, 19th century, only a few Indians were Satyendra Bose (1894­ 1974) in quantum theory, P.C. Mahalanobis (1893­1972)in statistics, exposed to foreign institutions and yet C.R. Rao in statistics, famous for his "theory of estimation"(1945). D.R. Kaprekar (1905­1988) in some of the greatest scientist in history emerged. mathematics. Well known for "Kaprekar Constant" 6174, Hargobind Khorana (1922–) in medicine, Harish Chandra (1923­1983) developed the branch of higher mathematics known as the infinite dimensional group representation theory, Subrahmanyan Chandrasekhar (1910–1995) in astrophysics and many more others. ‐ Many Indian scientists working abroad also made remarkable achievement in various scientific fields, such as India born Narendra Karmarkar, working at Bell Labs USA, stunned the world in 1984 with his new algorithm to solve linear programming problems. This made the complex calculations much faster, and had immediate applications in airports, warehouses, communication networks etc. ‐ 20th and 21st century are marking a resurgence as Hindus in India and in the diaspora, especially in the United States, strengthen their adopted lands with contributions in technology, medicine, engineering, fashion and the arts among many other disciplines.

The number of Indian scientists and their achievements (both ancient & modern) is too many to be listed here. More recent contributions of Indian sciences are part of the story of the contemporary world science.