Indian mathematics

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.

Fields of Indian mathematics

Some of the areas of mathematics studied by the Indians include the following:

• Decimal system — decimal units go back as far as the Indus Valley civilization.
• Infinity — see Yajur Veda
• Geometry — see Sulba Sutras
• Square roots, cube roots — see Baudhayana, Sulba Sutras, Bakhshali approximation, Mahavira
• Pythagorean triples — see Sulba Sutras; Baudhayana and Apastamba gave proofs of the Pythagorean theorem.
• Irrational numbers — see Sulba Sutras
• Quadratic equations — see Sulba Sutras
• Formal grammars, formal language theory, the Panini-Backus form — see Panini
• Binary number system — see Pingala
• Binary coding, early form of Morse code — see Pingala
• Pascal's triangle — see Pingala
• The Fibonacci series — see Pingala
• Transfinite numbers — see Jaina mathematics, Ancient Jaina Mathematics: an Introduction
• Logarithms — see Jaina mathematics
• Cubic equations, 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
• Trigonometry — see Aryabhata, Aryabhata the Elder, Varahamihira, Kerala School
• Algebra — see for example Aryabhata, Brahmagupta
• Algorithms — see Brahmagupta for example
• Floating point numbers — see Kerala School
• Calculus — see Bhaskara, Kerala School
• Mathematical analysis, including discoveries foundational to the development of calculus — see Madhava, Kerala School, Bhaskara, Jyeshtadeva
• Infinite series, trigonometric series, power series — see Kerala School

Harappan Mathematics (3300 BC - 1500 BC)

See also: Indus Valley Civilization

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 and measuring scales
• 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 achievements of the Harappan people of the Indus Valley Civilization include:

• Great accuracy in measuring length, mass, and time.
• The first system of uniform weights and measures.
• Extremely precise measurements. 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.
• The decimal division of measurement for all practical purposes, including the measurement of mass as revealed by their hexahedron weights.
• Brick sizes in a perfect ratio of 4:2:1.
• Decimal weights 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.
• Many of the weights uncovered have been produced in definite geometrical shapes, including cuboids, barrels, cones, and cylinders to name a few, which present knowledge of basic geometry, including the circle.
• This culture 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.
• 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)

See also: Vedic science and Ancient Vedic weights and measures

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 circling the square.
• A list of Pythagorean triples discovered algebraically.
• Statement and numerical proof of the Pythagorean theorem.
• 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.
• 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)

See also: Vedas

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.
• The earliest base 10 place-value number system (recognizably the ancestor of Hindu-Arabic numerals)
• The earliest known use of numbers up to a trillion (parardha) and numbers even larger upto 10⁵⁵.
• 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.

Lagadha (1350-1000 BC)

Lagadha composed the Jyotisha Vedanga, a work consisting of 49 verses, which contains:

• Descriptions of rules for tracking the motions of the sun and the moon.
• Procedures for calculating the time and position of the Sun and Moon in various naksatras (signs of the zodiac).
• The earliest known use of geometry and trigonometry for astronomy.

Much of Lagadha's works were later destroyed by foreign invaders of India.

Kalpa Vedanga (1200-900 BC)

The Kalpa Vedanga contains mathematical rules for rituals and ceremonials.

Samhitas (1200-500 BC)

The Taittiriya Samhita (1200-900 BC) contains:

• Rules for the construction of great fire altars.
• A rule implying knowledge of the Pythagorean theorem.

The other Samhitas contain:

• Fractions.
• Equations, such as 972x² = 972 + m for example.

Yajnavalkya (900-800 BC)

Yajnavalkya composed the Shatapatha Brahmana, which contains:

• Geometric, constructional, algebraic and computational aspects.
• Several computations of π, with the closest being correct to 2 decimal places, which remained the most accurate value of π for almost another 8 centuries.
• A rule implying knowledge of the Pythagorean theorem.
• References to the motions of the sun and the moon.
• A 95-year cycle to synchronize the motions of the sun and the moon.
• Compuation for the length of the year, correct to 2 decimal places.

Sulba Geometry (800-500 BC)

Further information: Sulba Sutras

Sulba Sutra means "Rule of Chords" in Vedic Sanskrit, and is another name for geometry. The Sulba Sutras were appendices to the Vedas giving rules for the construction of religious altars. The following discoveries found in these texts are mostly a result of altar construction:

• The first use of irrational numbers.
• The first use of quadratic equations of the form ax² = c and ax² + bx = c.
• indeterminate equations
• Diophantine equations
• A list of Pythagorean triples discovered algebraically
• A statement and numerical proof of the Pythagorean theorem predating Pythagoras (572 BC - 497 BC)
• Evidence of a number of geometrical proofs.
• Calculations for the square root of 2 found in three of the Sulba Sutras, which were correct to a remarkable five decimal places.

Baudhayana (800-700 BC)

Baudhayana composed the Baudhayana Sulba Sutra, which contains:

• The earliest statement of the Pythagorean theorem.
• Geometrical proof for a special case of the Pythagorean theorem for a 45° right triangle.
• Geometric solutions of a linear equation in a single unknown.
• Several approximations of π, with the closest value being 3.114.
• The first use of irrational numbers.
• Quadratic equations of the forms ax² = c and ax² + bx = c.
• Calculation for the square root of 2, correct to a remarkable five decimal places.
• Indeterminate equations.
• Two sets of positive integral solutions to a set of simultaneous Diophantine equations.
• Uses simultaneous Diophantine equations with upto four unknowns.

Manava (750-650 BC)

Manava composed the Manava Sulba Sutra, which contains:

• Approximate constructions of circles from rectangles
• Squaring the circle
• Approximation of π, with the closest value being 3.125.

Apastamba (600 BC)

Apastamba composed the Apastamba Sulba Sutra, which:

• Gives methods for squaring the circle and also considers the problem of dividing a segment into 7 equal parts.
• Calculates the square root of 2 correct to five decimal places
• Solves the general linear equation
• Contains indeterminate equations and simultaneous Diophantine equations with upto five unknowns.
• The earliest 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.
• Gave shape to beezganit samikaran (algebraic equations).
• Used the word shunya meaning void to refer to zero. This word eventually became zero after a series of translations and transliterations.

Panini (500-400 BC)

Panini was a Sanskrit grammarian (the world's earliest known linguist) who's contributions to science and mathematics include:

• A comprehensive and scientific theory of phonetics, phonology, and morphology.
• The formulation of 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.
• The use of 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.
• He conceived the notion of modern formal language theory.
• The Panini-Backus form used by most modern programming languages is significantly similar to Panini's grammar rules.
• Paninian grammars have also been devised for non-Sanskrit languages.

Pingala (400-200 BC)

Pingala was a scholar who authored of the Chhandah-shastra. His contributions include:

• The formation of a matrix.
• Invention of the binary number system (while he was forming a matrix for musical purposes).
• The notion of a binary code (the precursor to Morse code).
• First use of the Fibonacci series
• First use of Pascal's triangle, which he refers to as Meru-prastaara.
• Used a dot (.) to denote zero
• 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
• Fractions
• Square and cubes
• 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.
• 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 (usually attributed to George Cantor).

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.
• 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 described:

• Four types of Pramaan (Measure).
• 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 (later mentioned by Brahmagupta).
• Division by factor (later found in the Trisatika of Shridhara).

Satkhandagama (100-200 CE)

The Satkhandagama contains:

• The first known use of logarithms.
• A theory of sets.

Various sets are operated upon by:

• Logarithmic functions to base two.
• Squaring and extracting square roots.
• Raising to finite or infinite powers.

These operations are repeated to produce new sets.

Bakhshali Manuscript (200 BC - 400 CE)

This text bridged the gap between the Jaina mathematics period and the 'Classical period' of Indian mathematics. The authorship of this text is unknown.

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
• Other advances in notation including the use of zero and negative sign.
• The earliest use of the modern place-value Hindu-Arabic numeral system now used universally (see also Hindu-Arabic numerals).
• An improved method for calculating square roots allowing 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. (See Bakhshali approximation and Chapter 6 - The Bakhshali manuscript)

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 broader and clearer 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.

Surya Siddhanta (400)

Though its authorship is unknown, the Surya Siddhanta contains the roots of modern trignometry. It contains:

• Jya (Sine), Otkram Jya (Inverse Sine) and Kozya (Cosine).
• Rules laid down to determine the true motions of the luminaries, which conforms to their actual positions in the sky.

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

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, correct to four decimal places.
• Various other scientific problems.

Aryabhata also wrote the Aryabhata-Siddhanta, which:

• Introduced the trigonometric functions.
• Defined the sine as the modern relationship between half an angle and half a chord.
• Defined the sine (jya), cosine (kojya), versine (ukramajya), and inverse sine (otkram jya).
• Gave methods of calculating their approximate numerical values.
• Contains the earliest tables of sine, cosine and versine values, in 3.75° intervals from 0° to 90°, correct to 3 decimal places.
• Contains the trigonometric formula sin(n + 1)x - sin nx = sin nx - sin(n - 1)x - (1/225)sin nx.

The words jya and kojya eventually became sine and cosine respectively after a mistranslation after jya was transliterated into Arabic as jiba. European translators confused jiba for another word jaib, meaning "bay", by European translators. This was translated into the Latin word sinus, which means "bay" or "fold". This word eventually became sine in English.

Other contributions of Aryabhata include:

• Whole number solutions to linear equations by a method equivalent to the modern method.
• Accurate calculations for astronomical constants, such as the:
  • Solar eclipse.
  • Lunar eclipse.
  • The length of a day using integral calculus.
• Proposed for the first time, a heliocentric solar system with the planets following an elliptical orbit around the sun.

Aryabhata's works were translated into Arabic in the 8th century, and Latin in the 13th century. As a result, his works were very influential in Europe and the Middle East.

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:

• $\ sin^{2}(x)+cos^{2}(x)=1$

• $sin^{2}(x)=cos({\frac {\pi }{2}}-x)$

• ${\frac {1-cos(2x)}{2}}=sin^{2}(x)$

Chhedi calendar (594)

This document contains an early use of 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 produced:

• Solutions of indeterminate equations.
• A rational approximation of the sine function.
• A formula for calculating the sine of an acute angle without the use of a table, correct to 2 decimal places.

Brahmagupta (598-668)

Brahmagupta's famous work is his book titled Brahma-sphuta-siddhanta. Brahmgupta contributed:

• A method of calculating the volume of prisms and cones.
• Description of how to sum a geometric progression.
• The Brahmagupta interpolation formula to compute values of sines, up to second order of the Newton-Stirling interpolation formula.
• The method of solving indeterminate equations of the second degree.
• the first use of algebra to solve astronomical problems.

In the Brahma-sphuta-siddhanta:

• zero is clearly explained for the first time
• The modern place-value Hindu-Arabic numeral system is fully developed.
• Rules are given for manipulating both negative and positive numbers
• Methods are given for computing square roots
• methods are given for solving linear and quadratic equations
• Contains rules for summing series
• Brahmagupta's identity
• Brahmagupta's formula
• Brahmagupta theorem
• Methods are developed for calculations of:
  • The motions and places of various planets
  • Their rising and setting
  • Conjunctions
  • 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, such as the Hindu-Arabic numerals.

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
• 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.
• Plane geometry
• Solid geometry
• Problems relating to the casting of shadows.
• Formulae derived 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 eclipse.
• Lunar eclipse.

The Dhruvamanasa is a work of 105 verses on:

• Calculating planetary longitudes
• eclipses.
• 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.
• Anterest computation.

Arithmetical and geometrical progressions

• Plane geometry
• Solid geometry
• The shadow of the gnomon
• Methods to solve indeterminate equations
• solved combinations.

The Bijaganita ("Algebra") was a work in twelve chapters. It contained:

• The recognition of a positive number having two square roots.
• 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.
• Discovered the derivative.
• Developed Rolle's theorem.
• Developed Pell's equation.
• Gave a proof for the Pythagorean theorem.
• Gave a proof for division by zero being infinity.
• Computed π, correct to 5 decimal places.
• Calculated the time taken for the earth to orbit the sun to 9 decimal places.

His contibutions to trigonometry include:

• Spherical trigonometry
• The trigonometric formulas:
  • $\ sin(a+b)=sin(a)cos(b)+sin(b)cos(a)$
  • $\ sin(a-b)=sin(a)cos(b)-sin(b)cos(a)$

A number of his contributions were transmitted to Europe via the Arabs.

Kerala Mathematics (1300 CE - 1600 CE)

Main article: Kerala School

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

See also

• List of Indian mathematicians
• Indian science and technology
• Indian science
• History of mathematics

References

• The Crest of the Peacock: Non-European Roots of Mathematics by George Gheverghese Joseph, 1991
• A History of Mathematics: An Introduction by Victor J. Katz, 1998

External links

• History of Indian Mathematics
• An overview of Indian mathematics
• Indian Mathematics: Redressing the balance
• Bakhshali Manuscript
• Eurocentrism in Mathematics; Transmission of Calculus to Europe
• Indic Mathematics: India and the Scientific Revolution
• The Kerala School, European Mathematics and Navigation
• Ancient Jaina Mathematics: an Introduction
• History of Ganit (Mathematics)
• History of Mathematics -Indian Contribution