Indian mathematics: Difference between revisions

Content deleted Content added

Inline

Revision as of 22:55, 24 February 2006

The chronology of Indian mathematics spans from the Indus Valley civilization (3300-1500 BC) and Vedic civilization (1500-500 BC) to modern India.

Indian mathematicians have made outstanding contributions to the development of mathematics as we know it today. The Indian decimal notation of numbers, negative numbers and concept of zero have probably provided some of the biggest impetuses to advances in the field. Concepts from ancient and medieval India were carried to China and the Middle East, where they were studied extensively. From there they made their way to Europe and other parts of the world.

It would probably be impossible to examine the whole range of subjects covered by the history of Indian mathematics over a period of 5000 years in a single article, so only a general summary of Indian mathematics is given here, with links to other wikipedia articles for more detailed information.

Indian contributions to mathematics

A handful of Indian contributions to mathematics include the following:

Decimal system — goes back as far as the Indus Valley civilization
Infinity — see Yajur Veda
Geometry — see Sulba Sutras
Square roots and cube roots — see Baudhayana, Sulba Sutras, Bakhshali approximation
Pythagorean triples — see Sulba Sutras
Earliest statement and numerical proof of the Pythagorean theorem — see Sulba Sutras, Baudhayana, Apastamba
Irrational numbers — see Sulba Sutras
Quadratic equations — see Sulba Sutras
Metarules, transformations, recursions — see Panini
Formal language theory — see Panini
Turing machine — see Panini
The Panini-Backus Normal Form — see Panini
Binary number system — see Pingala
Pascal's triangle — see Pingala
The series popularly known as Fibonacci series — see Pingala
Transfinite numbers — see Jaina mathematics, Ancient Jaina Mathematics: an Introduction
Logarithms — see Jaina mathematics
Cubic equations and quartic equations — see Jaina mathematics
Negative numbers — see Brahmagupta, The Bakhshali manuscript
Zero — see Hindu-Arabic numeral system, Chapter 6 - The Bakhshali manuscript
Hindu-Arabic numeral system, the modern positional notation numeral system now used universally — see Hindu-Arabic numerals
Trigonometric functions (sine, cosine, versine, and inverse sine) and trigonometric tables — see Aryabhata, Aryabhata the Elder, Varahamihira
Algebra — see for example Aryabhata, Brahmagupta
Algorithms — see Brahmagupta, Ancient India - Mathematics
Differential calculus — see Bhaskara, Kerala School
Mathematical analysis, including discoveries foundational to the development of calculus — see Madhava, Kerala School, Bhaskara, Jyeshtadeva
"James Gregory" series expansion — see Madhava
Infinite series and Trigonometric series — see Kerala School
Floating point numbers — Kerala School

Harappan Mathematics (3300 BC - 1500 BC)

The first appearance of evidence of the use of mathematics in the Indian subcontinent was in the Indus Valley Civilization, which dates back to around 3300 BC. Excavations at Harrapa, Mohenjo-daro and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The mathematics used by this early Harrapan civilisation was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced brick technology, which utilised ratios. The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding. [1]

The people of the Indus Valley Civilization achieved great accuracy in measuring length, mass, and time. They were the first to develop a system of uniform weights and measures. Their measurements were extremely precise. Their smallest division, which is marked on an ivory scale found in Lothal, was approximately 1.704mm, the smallest division ever recorded on a scale of the Bronze Age. Harappan engineers followed the decimal division of measurement for all practical purposes, including the measurement of mass as revealed by their hexahedron weights.

Brick sizes were in a perfect ratio of 4:2:1. Decimal weights were based on ratios of 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with each unit weighing approximately 28 grams, similar to the English ounce or Greek uncia.

Also, many of the weights uncovered have been produced in definite geometrical shapes (cuboid, barrel, cone, and cylinder to name a few) which present knowledge of basic geometry, including the circle.

This culture also produced artistic designs of a mathematical nature and there is evidence on carvings that these people could draw concentric and intersecting circles and triangles.

Further to the use of circles in decorative design there is indication of the use of bullock carts, the wheels of which may have had a metallic band wrapped round the rim. This clearly points to the possession of knowledge of the ratio of the length of the circumference of the circle and its diameter, and thus values of π.

Also of great interest is a remarkably accurate decimal ruler known as the Mohenjo-daro ruler. Subdivisions on the ruler have a maximum error of just 0.005 inches and, at a length of 1.32 inches, have been named the Indus inch. Furthermore, a correspondence has been noted between the Indus scale and brick size. Bricks (found in various locations) were found to have dimensions that were integral multiples of the graduations of their respective scales, which suggests advanced mathematical thinking.

Some historians believe the Harappan civilization may have used a base 8 numeral system.

Unique Harappan inventions include an instrument which was used to measure whole sections of the horizon and the tidal dock. The engineering skill of the Harappans was remarkable, especially in building docks after a careful study of tides, waves, and currents.

In Lothal, a thick ring-like shell object found with four slits each in two margins served as a compass to measure angles on plane surfaces or in horizon in multiples of 40–360 degrees. Such shell instruments were probably invented to measure 8–12 whole sections of the horizon and sky, explaining the slits on the lower and upper margins. Archaeologists consider this as evidence the Lothal experts had achieved something 2,000 years before the Greeks are credited with doing: an 8–12 fold division of horizon and sky, as well as an instrument to measure angles and perhaps the position of stars, and for navigation purposes. Lothal contributes one of three measurement scales that are integrated and linear (others found in Harappa and Mohenjodaro). An ivory scale from Lothal has the smallest-known decimal divisions in Indus civilization. The scale is 6mm thick, 15 mm broad and the available length is 128 mm, but only 27 graduations are visible over 146 mm, the distance between graduation lines being 1.704 mm (the small size indicate use for finer purposes). The sum total of ten graduations from Lothal is approximate to the angula in the Arthashastra. The Lothal craftsmen took care to ensure durability and accuracy of stone weights by blunting edges before polishing. The Lothal weight of 12.184 gm is almost equal to the Egyptian Oedet of 13.792 gm.

Vedic Mathematics (1500 BC - 500 BC)

Note: The article on Vedic mathematics is based on a system of mental calculation developed by Shri Bharati Krishna Tirthaji, which may be based on a lost appendix of Atharva-Veda

As a result of the mathematics required for the construction of religious altars, many rules and developments of geometry are found in Vedic works, along with many astronomical developments for religious purposes. These include the use of geometric shapes, including triangles, rectangles, squares, trapezia and circles, equivalence through numbers and area, squaring the circle and vice versa, the Pythagorean theorem and a list of Pythagorean triples discovered algebraically, and computations of π.

Vedic works also contain all four arithmetical operators (addition, subtraction, multiplication and division), a definite system for denoting any number up to 10⁵⁵, the existence of zero, prime numbers, the rule of three, and a number of other discoveries. Of all the mathematics contained in the Vedic works, it is the definite appearance of decimal symbols for numerals and a place value system that should perhaps be considered the most phenomenal.

Vedas (1500-500 BC)

The Rig-Veda (1500-1200 BC) contains some rules for the construction of great fire altars. [2]

The Yajur-Veda (1200-900 BC) contains sacrificial formulae for ceremonial occasions, and the earliest known use of numbers up to a trillion (parardha) and numbers even larger upto 10⁵⁵. It also contains the earliest evidence of numeric infinity (purna "fullness"), stating that if you subtract purna from purna, you are still left with purna.

The Atharva-Veda (1200-900 BC) contains arithmetical sequences and a collection of magical formulae and spells. According to Shri Bharati Krishna Tirthaji, his system of mental calculation also known as Vedic mathematics is based on a lost appendix of the Atharva-Veda.

Samhitas (1200-500 BC)

The Samhitas contain fractions, as well as equations, such as 972x² = 972 + m for example, along with rules implying knowledge of the Pythagorean theorem.

The Taittiriya Samhita (1200-900 BC) contains rules for the construction of great fire altars, and gives a rule implying knowledge of the Pythagorean theorem.

Lagadha (1350-1200 BC)

Lagadha composed the Jyotisha Vedanga, which describes rules for tracking the motions of the sun and the moon. The 49 verses of the Jyotisha Vedanga gave procedures for calculating the time and position of the Sun and Moon in various naksatras (signs of the zodiac). Lagadha is the first known mathematician to have used geometry and trigonometry for astronomy, much of whose works were destroyed by foreign invaders of India.

Kalpa Vedanga (1200-900 BC)

The Kalpa Vedanga contains mathematical rules for rituals and ceremonials.

Yajnavalkya (900-800 BC)

Yajnavalkya composed the Shatapatha Brahmana, which contains geometric, constructional, algebraic and computational aspects. It contains several computations of π, with the closest being correct to 2 decimal places (the most accurate value of π upto that time), and gives a rule implying knowledge of the Pythagorean theorem, while the work also contains references to the motions of the sun and the moon. Yajnavalkya also advanced a 95-year cycle to synchronize the motions of the sun and the moon.

Sulba Sutras (800-500 BC)

Sulba Sutra means "Rule of Chords" in Vedic Sanskrit, which were appendices to the Vedas giving rules for the construction of religious altars. The Sulba Sutras contain the first use of irrational numbers, quadratic equations of the form ax² = c and ax² + bx = c, unarguable evidence for the use of the Pythagorean theorem and a list of Pythagorean triples discovered algebraically predating Pythagoras (572 BC - 497 BC), and evidence of a number of geometrical proofs. These discoveries are mostly a result of altar construction, which also led to the first known calculations for the square root of 2 found in three of the Sulba Sutras, which were remarkably accurate.

Baudhayana (800-700 BC)

Baudhayana composed the Baudhayana Sulba Sutra, which contains the Pythagorean theorem, geometric solutions of a linear equation in a single unknown, several approximations of π (the closest value being 3.114), along with the first use of irrational numbers and quadratic equations of the forms ax² = c and ax² + bx = c, and the first known calculation for the square root of 2, which was correct to a remarkable five decimal places.

Manava (750-650 BC)

Manava composed the Manava Sulba Sutra, which contains approximate constructions of circles from rectangles, and squares from circles, which give approximate values of π, with the closest value being 3.125.

Apastamba (600 BC)

Apastamba composed the Apastamba Sulba Sutra, which makes an attempt at squaring the circle and also considers the problem of dividing a segment into 7 equal parts. It also calculates the square root of 2 correct to five decimal places, and solves the general linear equation. The Apastamba Sulba Sutra also contains a numerical proof of the Pythagorean theorem, using an area computation. According to historian Albert Burk, this is the original proof of the theorem, and Pythagoras copied it on his visit to India.

Post-Vedic Mathematics (500 BC - 100 BC)

From around the 5th century BC, Vedic Sanskrit evolved into classical Sanskrit (due to the grammar written by Panini) while the Vedic religion evolved into classical Hinduism. The most important Hindu contributions to mathematics from this period came from the linguist Panini and musician Pingala, who were also responsible for important contributions to linguistics, music and computing. Other famous Hindu texts from this period include the Sathanang Sutra, Bhagvati Sutra and Anoyogdwar Sutra, while the Tiloyapannati by Yativrisham Acharya is also a famous writing of this time. Hindu mathematicians during this period used notations for squares, cubes and other exponents of numbers and gave shape to beezganit samikaran (algebraic equations). The word shunya meaning void is used by various Indian mathematicians from this period to refer to zero.

Panini (500-400 BC)

Panini was a Sanskrit grammarian (the world's earliest known linguist) who gave a comprehensive and scientific theory of phonetics, phonology, and morphology. He formulated the 3959 rules of Sanskrit morphology known as the Astadhyayi. The construction of sentences, compound nouns etc. is explained as ordered rules operating on underlying structures in a manner similar to modern theory. In many ways Panini's constructions are similar to the way that a mathematical function is defined today. Panini uses metarules, transformations, and recursions with such sophistication that his grammar has the computing power equivalent to a Turing machine. In this sense Panini may be considered the father of computing machines. Paninian grammars have also been devised for non-Sanskrit languages. Panini's work is also the forerunner to modern formal language theory. The Panini-Backus form used by most modern programming languages is also significantly similar to Panini's grammar rules.

Pingala (400-200 BC)

Pingala was the author of the Chhandah-shastra. He invented the binary number system while forming a matrix for musical purposes, and is also credited for the first use of the Fibonacci series and Pascal's triangle, which he refers to as Meru-prastaara. He also used a dot (.) to denote zero, and also described the formation of a matrix. His work, along with Panini's work, was foundational to the development of computing.

Vaychali Ganit (300-200 BC)

This book discusses the following in detail - the basic calculations of mathematics, the numbers based on 10, fraction, square, cube, rule of false position, interest methods, questions on purchase and sale. The book has given the answers of the problems and also described testing methods.

Katyayana (200 BC)

His Katyayana Sulba Sutra presented much geometry, including a general version of the Pythagorean theorem and an accurate calculation of the square root of 2 correct to 5 decimal places.

Yativrisham Acharya (176 BC)

He wrote a famous mathematical text called Tiloyapannati.

Jaina Mathematics (400 BC - 200 CE)

Jainism was a religion and philosophy founded in the 6th century BC by Mahavira around the time Gautama Buddha founded Buddhism. Followers of these religions played an important role in the future development of India. Jaina mathematicians were particularly important in bridging the gap between earlier Indian mathematics and the 'Classical period', which was heralded by the work of Aryabhata I from the late 5th century CE.

Regrettably there are few extant Jaina works, but in the limited material that exists, an incredible level of originality is demonstrated. Perhaps the most historically important Jaina contribution to mathematics as a subject is the progression of the subject from purely practical or religious requirements. During the Jaina period, mathematics became an abstract discipline to be cultivated "for its own sake".

The important developments of the Jainas include the theory of numbers and their fascination with the enumeration of very large numbers and infinity. All numbers were classified into three sets: enumerable, innumerable and infinite. Five different types of infinity are recognised in Jaina works: infinite in one and two directions, infinite in area, infinite everywhere and infinite perpetually. This theory was not realised in Europe until the late 19th century.

Jaina works also contained: knowledge of the fundamental laws of indices, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations (the Jaina contribution to algebra has been severely neglected), formula for π (root 10, comes up almost inadvertently in a problem about infinity), operations with logarithms, and sequences and progressions.

Finally of interest is the appearance of Permutations and Combinations in Jaina works, which was used in the formation of a Pascal triangle, called Meru-prastara, used a few centuries after Hindu mathematician Pingala but many centuries before Pascal 'invented' it.

Surya Prajinapti (400-300 BC)

A mathematical text which classifies all numbers into three sets: enumerable, innumerable and infinite. It also recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.

Bhadrabahu (350-298 BC)

Bhadrabahu was the author of two astronomical works, the Bhadrabahavi-Samhita and a commentary on the Surya Prajinapti.

Sathanang Sutra (300 BC - 200 CE)

The Sathanang Sutra mentioned five types of infinities.

Anoyogdwar Sutra (300 BC - 200 CE)

The Anoyogdwar Sutra mentioned four types of Pramaan (Measure). This Granth (book) also described permutations and combinations which were termed as Bhang and Vikalp.

Umaswati (150-135 BC)

Umasvati was famous for his influential writings on Jaina philosophy and metaphysics but also wrote a work called Tattwarthadhigama-Sutra Bhashya, which contains mathematics. This book contains mathematical formulae and two methods of multiplication and division: multiplication by factor (also later mentioned by Brahmagupta) and division by factor (also found in the Trisatika of Shridhara).

Satkhandagama (100-200 CE)

The Satkhandagama contains the first known use of logarithms. Jaina mathematicians by this time had produced a theory of sets. In the Satkhandagama, various sets are operated upon by logarithmic functions to base two, by squaring and extracting square roots, and by raising to finite or infinite powers. The operations are repeated to produce new sets.

Pre-Classical Period (200 BC - 400 CE)

The following texts bridged the gap between the earlier Jaina mathematics and the 'Classical period' of Indian mathematics. These texts played an important role in the Classical period that followed from 400 CE onwards. The authorship of these texts from is unknown.

Bakhshali Manuscript (200 BC - 400 CE)

There are eight principal topics discussed in the Bakhshali Manuscript: Examples of the rule of three (and profit and loss and interest), solution of linear equations with as many as five unknowns, the solution of the quadratic equation (development of remarkable quality), arithmetic (and geometric) progressions, compound series (some evidence that work begun by Jainas continued), quadratic indeterminate equations (origin of type ax/c = y), simultaneous equations, fractions and other advances in notation including the use of zero and negative sign. An improved method for calculating square roots allowed extremely accurate approximations for irrational numbers to be calculated, and can compute square roots of numbers as large as a million correct to at least 11 decimal places. [3]

Surya Siddhanta (300-400)

The Surya Siddhanta contains the roots of modern trignometry. It mentions Zya (Sine), Otkram Zya (Inverse Sine) and Kotizya (Cosine). Rules were laid down to determine the true motions of the luminaries, which conforms to their actual positions in the sky. Classical Indian mathematicians such as Aryabhata later made references to this text, while later Arabic and Latin translations were influential in Europe and the Islamic world.

Classical Period (400 CE - 1200 CE)

This period is often known as the golden age of Indian Mathematics. Although earlier Indian mathematics was also very significant, this period saw great mathematicians such as Aryabhata, Varahamihira, Brahmagupta, Mahavira Acharya and Bhaskara Acharya give a broad and clear shape to almost all the branches of mathematics. Their important contributions to mathematics would spread throughout Asia and the Middle East, and eventually Europe and other parts of the world.

Aryabhata I (Aryabhata the Elder) (476-550)

Aryabhata was a resident of Patna in the Indian state of Bihar. He described the important fundamental principles of mathematics in 332 shlokas. He produced the Aryabhatiya, a treatise on quadratic equations, trigonometry, the value of π, and various other scientific problems. He calculated the value of π correct to four decimal places. Aryabhata also wrote the Aryabhata-Siddhanta, which first introduced the trigonometric functions and methods of calculating their approximate numerical values. It defined the concepts of sine, cosine and versine, and also contains the earliest tables of sine and cosine values (in 3.75-degree intervals from 0 to 90 degrees). Aryabhata also obtained whole number solutions to linear equations by a method equivalent to the modern method. He also gave accurate calculations for astronomical constants, such as the solar eclipse and lunar eclipse, calculated the length of a day using integral calculus, and also proposed for the first time, a heliocentric solar system where the orbits of the planets around the sun are ellipses. The Aryabhatiya was translated into Arabic by the 10th century, and many developments from his book was later transmitted to Europe.

Varahamihira (575)

Varahamihira produced the Pancasiddhantika (The Five Astronomical Canons). He made important contributions to trigonometry, including the following formulas relating sine and cosine functions:

sin²x + cos²x = 1

sin x = cos(π/2 - x)

(1 - cos(2x))/2 = sin²x

Chhedi calendar (594)

This is the earliest known document which uses the modern place-value Hindu-Arabic numeral system now used universally (see also Hindu-Arabic numerals).

Bhaskara I (600-680)

Bhaskara I expanded the work of Aryabhata in his books titled Mahabhaskariya, Aryabhattiya Bhashya and Laghu Bhaskariya. He worked on indeterminate equations and also gave a rational approximation of the sine function.

Brahmagupta (598-668)

Brahmagupta's famous work is his book titled Brahma-sphuta-siddhanta. Brahmgupta gave a method of calculating the volume of prisms and cones, described how to sum a geometric progression. He used an interpolation formula to compute values of sines, up to second order of the Newton-Stirling interpolation formula. He also invented the method of solving indeterminate equations of the second degree and was the first to use algebra to solve astronomical problems. In the Brahma-sphuta-siddhanta, zero is clearly explained for the first time, and the modern place-value Hindu-Arabic numeral system is fully developed. It also gives rules for manipulating both negative and positive numbers, methods for computing square roots, methods of solving linear and quadratic equations, and rules for summing series, along with Brahmagupta's identity, Brahmagupta's formula and the Brahmagupta theorem. He also developed methods for calculations of the motions and places of various planets, their rising and setting, conjunctions, and the calculation of eclipses of the sun and the moon. The Brahma-sphuta-siddhanta was translated into Arabic in 773, and many developments from his book was later transmitted to Europe.

Shridhara Acharya (650-850)

Shridhara wrote books titled Nav Shatika, Tri Shatika and Pati Ganit. He gave a good rule for finding the volume of a sphere, and also the formula for solving quadratic equations.

Mahavira Acharya (850)

Mahavira, the last of the notable Jaina mathematicians, lived in the 9th century. He wrote a book titled Ganit Saar Sangraha on numerical mathematics, and also wrote treatises about a wide range of mathematical topics. These include the mathematics of zero, squares, cubes, square roots, cube roots, and the series extending beyond these. He also wrote about plane and solid geometry, as well as problems relating to the casting of shadows. He derived formulae to calculate the area of an ellipse and quadrilateral inside a circle, asserted that the square root of a negative number did not exist, gave the sum of a series whose terms are squares of an arithmetical progression, and gave empirical rules for area and perimeter of an ellipse.

Aryabhata II (920-1000)

Aryabhata II wrote a book titled Maha-Siddhanta. This book discusses numerical mathematics (Ank Ganit) and algebra.

Shripati Mishra (1019-1066)

Shripati wrote the books Siddhanta Shekhara, a major work on astronomy in 19 chapters, and Ganit Tilaka, an incomplete arithmetical treatise in 125 verses based on a work by Shridhara. He worked mainly on permutations and combinations. He was also the author of Dhikotidakarana, a work of twenty verses on solar and lunar eclipses, and the Dhruvamanasa, a work of 105 verses on calculating planetary longitudes, eclipses and planetary transits.

Nemichandra Siddhanta Chakravati (1100)

He authored a mathematical treatise titled Gome-mat Saar.

Bhaskara Acharya (Bhaskara II) (1114-1185)

Bhaskara was a mathematician-astronomer who wrote a number of important treatises, namely the Siddhanta Shiromani, Lilavati, Bijaganita, Gola Addhaya, Griha Ganitam and Karan Kautoohal. The Lilavati covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations. The Bijaganita ("Algebra") was a work in twelve chapters. It was the first text to recognize that a positive number has two square roots. Other topics covered in the text were positive and negative numbers, zero, the unknown, surds, the kuttaka, indeterminate quadratic equations, simple equations, quadratic equations, equations with more than one unknown, quadratic equations with more than one unknown, and operations with products of several unknowns. Bhaskara also conceived differential calculus after discovering the derivative, and also developed Rolle's theorem, Pell's equation, a proof for the Pythagorean theorem, proof that division by zero is infinity, computed π correct to 5 decimal places, and calculated the time taken for the earth to orbit the sun to 9 decimal places. A number of his contributions were transmitted to Europe via the Arabs. His contibutions to trigonometry include spherical trigonometry and the formulas:

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

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

Keralese Mathematics (1300 CE -1600 CE)

The Kerala School was a school of mathematics and astronomy founded by Madhava in Kerala (in South India) which included as its prominent members Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. It flourished between the 14th and 16th centuries and has its intellectual roots with Aryabhatta who lived in the 5th century. The lineage continues down to modern times but the original research seems to have ended with Narayana Bhattathiri (1559-1632). These astronomers, in attempting to solve problems, invented revolutionary ideas of calculus. They discovered the theory of infinite series, tests of convergence (often attributed to Cauchy), differentiation, term by term integration, iterative methods for solution of non-linear equations, and the theory that the area under a curve is its integral. They achieved most of these results upto several centuries before European mathematicians.

Jyeshtadeva consolidated the Kerala School's discoveries in the Yuktibhasa, the world's first calculus text.

The floating point numbers were also first used by Keralese mathematicians and, using this system of numbers, they were able to investigate and rationalise about the convergence of series.

Narayana Pandit (1340-1400)

Narayana Pandit, the earliest of the notable Keralese mathematicians, had written two works, an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijganita Vatamsa. Narayana is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavati, titled Karmapradipika (or Karma-Paddhati).

Although the Karmapradipika contains very little original work, seven different methods for squaring numbers are found within it, a contribution that is wholly original to the author. Narayana's other major works contain a variety of mathematical developments, including a rule to calculate approximate values of square roots, using the second order indeterminate equation nq² + 1 = p² (Pell's equation). Mathematical operations with zero, several geometrical rules and discussion of magic squares and similar figures are other contributions of note. Evidence also exists that Narayana made minor contributions to the ideas of differential calculus found in Bhaskara II's work. Narayana has also made contributions to the topic of cyclic quadrilaterals.

Madhava of Sangamagramma (1340-1425)

Madhava is the founder of the Kerala School and considered to be one of the greatest mathematician-astronomers of the Middle Ages. It is vaguely possible that he may have written Karana Paddhati a work written sometime between 1375 and 1475 but all we really know of Madhava comes from works of later scholars.

His most significant contribution was in moving on from the finite procedures of ancient mathematics to treat their limit passage to infinity, which is considered to be the essence of modern classical analysis, and thus he is considered the father of mathematical analysis. Madhava was responsible for a number of discoveries, including the Madhava-Gregory series, Madhava-Newton power series, Euler's series, and the power series for π (usually attributed to Leibniz). Madhava is responsible for laying the foundations for the development of calculus, which was then further developed by his successors at the Kerala School.

Parameshvara (1370-1460)

Parameshvara wrote commentaries on the work of Bhaskara I, Aryabhata and Bhaskara II, and his contributions to mathematics include an outstanding version of the mean value theorem. Furthermore Paramesvara gave a mean value type formula for inverse interpolation of sine, and is thought to have been the first mathematician to give the radius of circle with inscribed cyclic quadrilateral, an expression that is normally attributed to Lhuilier (1782).

Nilakantha Somayaji (1444-1544)

In Nilakantha's most notable work Tantra Samgraha (which 'spawned' a later anonymous commentary Tantrasangraha-vyakhya and a further commentary by the name Yuktidipaika, written in 1501) he elaborates and extends the contributions of Madhava. Sadly none of his mathematical works are extant, however it can be determined that he was a mathematician of some note. Nilakantha was also the author of Aryabhatiya-bhasa a commentary of the Aryabhatiya. Of great significance is the presence of mathematical proof (inductive) in Nilakantha's work.

Jyesthadeva (1500-1575)

Jyesthadeva was another member of the Kerala School. His key work was the Yukti-bhasa (written in Malayalam, a regional language of Kerala). Similarly to the work of Nilakantha it is almost unique in the history of Indian mathematics, in that it contains both proofs of theorems and derivations of rules. He also studied various topics found in many previous Indian works, including integer solutions of systems of first degree equations solved using kuttaka.

Charges of Eurocentrism

Unfortunately, Indian contributions have not been given due acknowledgement in modern history, with many discoveries/inventions by Indian mathematicians now attributed to their western counterparts, due to Eurocentrism.

The historian Florian Cajori, one of the most celebrated historians of mathematics in the early 20th century, suggested that "Diophantus, the father of Greek algebra, got the first algebraic knowledge from India." This theory is supported by evidence of continous contact between India and the Hellenistic world from the late 4th century BC, and earlier evidence that the eminent Greek mathematician Pythagoras visited India, which further 'throws open' the Eurocentric ideal.

More recently, evidence has been unearthed that reveals that the foundations of calculus were laid in India, at the Kerala School. Some allege that calculus and other mathematics of India were transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries. Kerala was in continuous contact with China, Arabia, and from around 1500, Europe as well, thus transmission would have been possible. There is no evidence by way of relevant manuscripts but the evidence of methodological similarities, communication routes and a suitable chronology for transmission is hard to dismiss.

See also: Possible transmission of Keralese mathematics to Europe

References

The Crest of the Peacock: Non-European Roots of Mathematics by George Gheverghese Joseph

External links

@@ Line 30: / Line 30: @@
 * [[0 (number)|Zero]] &mdash; see [[Hindu-Arabic numeral system]], [http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch6.html Chapter 6 - The Bakhshali manuscript]
 * [[Hindu-Arabic numeral system]], the modern [[positional notation]] [[numeral system]] now used universally &mdash; see [[Hindu-Arabic numerals]]
-* [[Trigonometric functions]] and [[trigonometric]] tables &mdash; see [[Aryabhata]], [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aryabhata_I.html Aryabhata the Elder], [[Varahamihira]]
+* [[Trigonometric functions]] ([[sine]], [[cosine]], [[versine]], and inverse sine) and [[trigonometric]] tables &mdash; see [[Aryabhata]], [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aryabhata_I.html Aryabhata the Elder], [[Varahamihira]]
 * [[Algebra]]  &mdash; see for example [[Aryabhata]], [[Brahmagupta]]
 * [[Algorithm]]s  &mdash; see [[Brahmagupta]], [http://www.crystalinks.com/indiamathematics.html Ancient India - Mathematics]