Indian mathematics

The chronology of Indian mathematics spans from the Indus Valley Civilization (3300-1700 BC) and the Vedic period (1500-500 BC) to modern times.

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 impetus' to advances in the field. Concepts from ancient and medieval India were carried to the Middle East, where they were studied extensively. From there they made their way to Europe.

Indian contributions to mathematics

The Indian contributions to mathematics include the following:

• Decimal system — goes back as far as the Indus Valley civilization
• Hindu-Arabic numeral system, the modern positional notation numeral system now used universally — see Hindu-Arabic numerals
• Zero — see Hindu-Arabic numeral system
• Negative numbers — see Brahmagupta
• Infinity — see Yajur Veda
• Transfinite numbers — see Jaina mathematics, Ancient Jaina Mathematics: an Introduction
• Irrational numbers — see Sulba Sutras
• Geometry — see Sulba Sutras
• Square roots and cube roots — see Baudhayana
• Metarules, transformations, recursions — see Panini
• Turing machine — see Panini
• The Panini Backus Normal Form — see Panini
• Binary numbers — see Pingala
• Pascal's triangle — see Pingala
• The series popularly known as Fibonacci series — see Pingala
• One of the earliest statements of the Pythagorean theorem — see Sulba Sutras
• Pythagorean triples — see Sulba Sutras
• "James Gregory" series expansion — see Madhava
• Logarithms — see Jaina mathematics
• Quadratic equations and cubic equations — see Sulba Sutras
• Trigonometric functions and trigonometric tables — see Aryabhata, Aryabhata the Elder, Varahamihira
• Algebra — see for example Aryabhata, Brahmagupta
• Algorithms — see Ancient India - Mathematics
• Differential calculus — see Bhaskara, Kerala School
• Infinite series and Trigonometric series — see Kerala School
• Mathematical analysis, including developments foundational to the development of calculus — see Madhava, Bhaskara, Jyeshtadeva, Kerala School

Harappan Mathematics (3300 BC - 1700 BC)

Further information: 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, 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.

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.

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.

Vedic Mathematics (1500 BC - 500 BC)

See also: Vedic science

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 BC - 500 BC)

Rig-Veda (1500 BC -1200 BC)

The Rig-Veda contains rules for the construction of great fire altars.

Yajur-Veda (1200 BC - 900 BC)

The Yajur Veda contains the earliest known use of numbers up to a trillion (parardha). It even discusses the concept of numeric infinity (purna "fullness"), stating that if you subtract purna from purna, you are still left with purna.

Atharva-Veda (1200 BC - 900 BC)

Arithmetical sequences are found in the Atharva-Veda. According to Shri Bharati Krishna Tirthaji, his system of mental calculation also known as Vedic mathematics is based on a lost appendix of Atharva-Veda.

Samhitas (1500 BC - 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.

Taittiriya Samhita (1500 - 1000 BC)

The Taittiriya Samhita contains rules for the construction of great fire altars, and gives a rule implying knowledge of the Pythagorean theorem.

Lagadha (1350 BC - 800 BC)

Lagadha composed the Vedanga Jyotisha, which describes rules for tracking the motions of the sun and the moon. Lagadha is the only known mathematician to have used geometry and trigonometry for astronomy, much of whose works were destroyed by foreign invaders of India.

Yajnavalkya (1000 BC - 600 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 BC - 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 a x² = 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 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 BC - 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.

Ancient Period (500 BC - 400 CE)

Sathanang Sutra, Bhagvati Sutra and Anoyogdwar Sutra are famous books of this time. Apart from these the book titled Tatvarthaadigyam Sutra Bhashya by Jaina philosopher Omaswati (135 BC) and the book titled Tiloyapannati of Aacharya (Guru) Yativrisham (176 BC) are famous writings of this time.

Indian mathematicians during this period used notations for squares, cube and other exponents of numbers. They gave shape to Beezganit Samikaran (Algebraic Equations).

Panini (500 BC - 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 BC - 200 BC)

Author of the Chhandah-shastra. He is credited for the first use of binary numbers, Fibonacci series and Pascal's triangle, which he refers to as Meru-prastaara. He also uses a dot (.) to indicate zero for binary numbers.

Vaychali Ganit (300 BC - 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.

Jaina Mathematics (400 BC - 400 CE)

Jaina mathematicians played an important role in bridging the gap between earlier Indian mathematics and the so-called '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 an early 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 (400-298 BC)

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 BC)

Umaswati is known as a great writer on Jaina metaphysics but also wrote a work called Tattwarthadhigama-Sutra Bhashya, which contains mathematics.

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.

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. [2]

Surya Siddhanta (300 CE - 400 CE)

This text contains the roots of modern Trignometry. It mentions Zya (Sine), Otkram Zya (Inverse Sine) and Kotizya (Cosine).

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 (476-550)

He was a resident of Patna in India. 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 and cosine, and also contains the earliest tables of sine and cosine values (in 3.75-degree intervals from 0 to 90 degrees). Aryabhata also introduced the versine, and 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 produces the Pancasiddhantika (The Five Astronomical Canons). He makes important contributions to trigonometry, especially in terms of formulas relating sine and cosine functions.

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)

He expanded the work of Aryabhatta 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)

His 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)

He wrote books titled "Nav Shatika", "Tri Shatika", "Pati Ganit".

Mahavira Acharya (850)

He wrote the book titled "Ganit Saar Sangraha" on Numerical Mathematics. He derived formulae to calculate the area of ellipse and quadrilateral inside a circle.

Aryabhata II (920-1000)

He wrote a book titled Maha Siddhanta. This book discusses Numerical Mathematics (Ank Ganit) and Algebra.

Shripati Mishra (1019-1066)

He wrote the books titled Siddhanta Shekhar and Ganit Tilak. He worked mainly on permutations and combinations.

Nemichandra Siddhanta Chakravati (1100)

His book is titled Gome-mat Saar.

Bhaskara Acharya (Bhaskara II) (1114-1185)

He wrote a number of excellent treatises, namely the Siddhanta Shiromani, Lilavati Bizganitam, 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") is the first text to recognize that a positive number has two square roots. Bhaskara also conceives differential calculus, including the derivative, and also develops Rolle's theorem, Pell's equation, a proof for the Pythagorean Theorem, proves that division by zero is infinity, computes π correct to 5 decimal places, and calculates the time taken for the earth to orbit the sun to 9 decimal places. A number of his developments were transmitted to Europe via the Arabs.

Keralese Mathematics (1300 CE -1600 CE)

Further information: 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.

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 his 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.

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 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 was 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 aswell, 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: Kerala School § Possible transmission of Keralese mathematics to Europe

See also

• List of Indian mathematicians
• History of mathematics

External links

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