Indian mathematics

The chronology of Indian mathematics spans from the Indus Valley civilization (3300-1500 BCE) and Vedic civilization (1500-500 BCE) to modern India (21st century CE).

Indian mathematicians have made major contributions to the development of mathematics as we know it today. One of the biggest contributions of Indian mathematics is the modern arithmetic and decimal notation of numbers used universally throughout the world (known as the Hindu-Arabic numerals). John Playfair, the famous Scottish mathematician published a dissertation titled "Remarks on the astronomy of Brahmins" in 1790. His following quotation shows the appreciation of the then European Scientific community on the achievements of ancient Indian mathematicians and scientists.

"The Constructions and these tables imply a great knowledge of geometry,arithmetic and even of the theoretical part of astronomy.But what, without doubt is to be accounted,the greatest refinement in this system, is the hypothesis employed in calculating the equation of the centre for the Sun,Moon and the planets that of a circular orbit having a double eccentricity or having its centre in the middle between the earth and the point about which the angular motion is uniform.If to this we add the great extent of the geometrical knowledge required to combine this and the other principles of their astronomy together and to deduce from them the just conclusion;the possession of a calculus equivalent to trigonometry and lastly their approximation to the quadrature of the circle, we shall be astonished at the magnitude of that body of science which must have enlightened the inhabitants of India in some remote age and which whatever it may have communicated to the Western nations appears to have received another from them...."

Albert Einstein in the 20th century also comments on the importance of Indian arithmetic: "We owe a lot to the Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made."

Said Laplace: "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of antiquity, Archimedes and Apollonius."

Other examples include zero, negative numbers, and the trigonometric functions of sine and cosine, which have all 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.

Fields of Indian mathematics

Some of the areas of mathematics studied in ancient and medieval India include the following:

• Arithmetic
  • Decimal system — decimal units go back as far as the Indus Valley civilization
  • Negative numbers — see Brahmagupta
  • Zero — see Hindu-Arabic numeral system
  • Hindu-Arabic numeral system, the modern positional notation numeral system now used universally
  • Floating point numbers — see Kerala School
• Number theory
  • Infinity — see Yajur Veda
  • Transfinite numbers — see
  • Irrational numbers — see Sulba Sutras
• Geometry
  • Square roots — see Sulba Sutras, Bakhshali approximation
  • Cube roots — see Mahavira
  • Pythagorean triples — see Sulba Sutras; Baudhayana and Apastamba gave proofs of the Pythagorean theorem
  • Transformation — see Panini
  • Pascal's triangle — see Pingala
• Algebra
  • Quadratic equations — see Sulba Sutras, Aryabhata, Brahmagupta
  • Cubic equations — see Mahavira, Bhaskara
  • Quartic equations (biquadratic equations) — see Mahavira, Bhaskara
• Mathematical logic
  • Formal grammars, formal language theory, the Panini-Backus form — see Panini
  • Recursion — see Panini
• Fibonacci numbers — see Pingala
• Earliest forms of Morse code — see Pingala
• Logarithms, indices — see Jaina mathematics
• Trigonometry
  • Trigonometric functions — see Surya Siddhanta, Aryabhata
  • Trigonometric series — see Madhava, Kerala School
• Algorithms
  • Algorism — see Aryabhata, Brahmagupta
• Calculus
  • Differential calculus — see Bhaskara, Madhava of Sangamagrama, Kerala School, Jyeshtadeva
  • Integral calculus — see Madhava of Sangamagrama, Kerala School
• Mathematical analysis, including discoveries foundational to the development of calculus
  • Infinite series — see Madhava
  • Power series, Taylor series — see Madhava, Kerala School

Harappan Mathematics (2600 BCE - 1700 BCE)

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 Harappa, 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 Harappan 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. [5]

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.

• 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 BCE - 400 BCE)

See also: Vedic science and Ancient Vedic weights and measures

Vedic Mathematics Overview

The geometry in Vedic mathematics was used for elaborate construction of religious and astronomical sites. Many aspects of practical mathematics are found in Vedic mathematics.

• Extensive use of geometric shapes, including triangles, rectangles, squares, trapezia and circles.
• Relation of sides to diagonals.
• Construction of equivalent square and rectangles.
• Squaring the circle
• Circling the square.
• A list of Pythagorean triples discovered algebraically.
• Statement and numerical proof of the Pythagorean theorem.
• Computations of π.
• All four arithmetical operators (addition, subtraction, multiplication and division).
• The invention of zero.
• Prime numbers.
• The rule of three.

Vedas

See also: Vedas

The Rig-Veda (c. 1500-1200 BCE) contains some rules for the construction of great fire altars.

The Yajur-Veda (c. 1200-900 BCE) contains:

• Sacrificial formulae for ceremonial occasions.
• Base 10 decimal numeral system (recognizably the ancestor of Hindu-Arabic numerals)
• The earliest known use of numbers up to a trillion (parardha) and numbers even larger up to 10.
• 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 (c. 1200-900 BCE) 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

Lagadha (fl. 1350-1000 BCE) 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

The Kalpa Vedanga (c. 1200-900 BCE) contains mathematical rules for rituals and ceremonials.

Samhitas

The Taittiriya Samhita (c. 1200-900 BCE) contains:

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

The other Samhitas (c. 1200-500 BCE) contain:

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

Yajnavalkya

Yajnavalkya (fl. 900-700 BCE) composed the astronomical text Shatapatha Brahmana, which contains:

• Geometric, constructional, algebraic and computational aspects.
• A rule implying knowledge of the Pythagorean theorem.
• Several computations of π, with the closest being correct to 2 decimal places, which remained the most accurate approximation of π anywhere in the world for another seven centuries.
• References to the motions of the Sun and the Moon.
• A 95-year cycle to synchronize the motions of the Sun and the Moon, which gives the average length of the tropical year as 365.24675 days, which is only 6 minutes longer than the modern value of 365.24220 days. This estimate for the length of the tropical year remained the most accurate anywhere in the world for over a thousand years.
• The distances of the Moon and the Sun from the Earth expressed as 108 times the diameters of these heavenly bodies. These are very close to the modern values of 110.6 for the Moon and 107.6 for the Sun, which were obtained using modern instruments.

Satapatha Brahmana (ca. 800 BCE)

The Satapatha Brahmana contains rules on geometry that are similar to the Sulba Sutras. The geometry of the Satapatha Brahmana predates Greek geometry.

Sulba Geometry (ca. 800-500 BCE)

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
• The Pythagorean theorem predating Pythagoras (572 BC - 497 BC)
• Evidence of a number of geometrical proofs.
• Squaring the circle.
• Circling the square.
• Calculations for the square root of 2 found in three of the Sulba Sutras, which were correct to a remarkable five decimal places.

It has been suggested by some scholars that the Sulba Sutras were written during the Harappan period. This is based on the evidence of advanced brick technology found in these texts, which was developed to a higher degree in the Harappan period than in the Vedic period (where it was limited to the bulding of religious altars). If the Sulba Sutras were not written during the Harappan period however, it is still possible that Harappan mathematics was at least as advanced as the Sulba Sutras, based on the evidence of superior brick technology in the Harappan period.

Baudhayana

Baudhayana (c. 8th century BCE) composed the Baudhayana Sulba Sutra, which contains:

• Examples of simple Pythagorean triples, such as: $(3,4,5)$, $(5,12,13)$, $(8,15,17)$, $(7,24,25)$, and $(12,35,37).$ (Note: Pythagorean triples are triples of integers $(a,b,c)$ with the property: $a^{2}+b^{2}=c^{2}$. Thus, $3^{2}+4^{2}=5^{2}$, $8^{2}+15^{2}=17^{2}$, $12^{2}+35^{2}=37^{2}$ etc.)
• A statement of the Pythagorean theorem in terms of the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square."
• All three Sulbasutras have a statement of the Pythagorean theorem in terms of the sides of a rectangle: "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together."
• Geometrical proof of the Pythagorean theorem for a 45° right triangle (the earliest proof of the Pythagorean theorem).
• Geometric solutions of a linear equation in a single unknown.
• Several approximations of π, with the closest value being 3.114.
• Approximations for irrational numbers. All three Sulbasutras give a formula for ${\sqrt {2}}$ given by:

${\sqrt {2}}=1+{\frac {1}{3}}+{\frac {1}{3\cdot 4}}-{\frac {1}{3\cdot 4\cdot 34}}\approx 1.4142156...$ The true value is $1.414213...$

Although this remarkable formula arose as a result of geometric measurements, with hindsight, it can be seen as a first order Taylor expansion in calculus:

${\sqrt {a^{2}+r}}={\sqrt {\left(a+{\frac {r}{2a}}\right)^{2}-\left({\frac {r}{2a}}\right)^{2}}}$ $\approx a+{\frac {r}{2a}}-{\frac {(r/2a)^{2}}{2(a+{\frac {r}{2a}})}},$ with $a=4/3$ and $r=2/9$.

• The earliest use of quadratic equations of the forms ax² = c and ax² + bx = c.
• Indeterminate equations.
• Two sets of positive integral solutions to a set of simultaneous Diophantine equations.
• Uses simultaneous Diophantine equations with up to four unknowns.

Manava

Manava (fl. 750-650 BCE) 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

Apastamba (c. 600 BCE) 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 up to five unknowns.
• The general numerical proof of the Pythagorean theorem, using an area computation (the earliest general proof of the Pythagorean theorem). According to historian Albert Burk, this is the original proof of the theorem, and Pythagoras copied it on his visit to India.

Panini

Pāṇini (c. 520-460 BCE) was a Sanskrit grammarian and is the world's earliest known linguist, and often considered the founder of linguistics. He also made contributions to mathematics, which include:

• The earliest 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, Pāṇini's constructions are similar to the way that a mathematical function is defined today.
• The earliest use of Boolean logic.
• The earliest use of the null operator.
• The earliest use of metarules, transformations and recursions, which were used with such sophistication that his grammar had the computing power equivalent to a Turing machine. In this sense Panini may be considered the father of computing machines.
• He conceived of formal language theory.
• He conceived of formal grammars.
• The Panini-Backus form used to describe most modern programming languages is significantly similar to Panini's grammar rules.
• Paninian grammars have also been devised for non-Sanskrit languages.

Pāṇini's grammar of Sanskrit was responsible the transition from Vedic Sanskrit to classical Sanskrit, hence marking the end of the Vedic period.

Critisism of Mathematics of the Vedic Period

According to J.J. O'Connor and E. F. Robertson, the Sulbasutras were appendices to the Vedas giving rules for constructing altars. "They contained quite an amount of geometrical knowledge, but the mathematics was being developed, not for its own sake, but purely for practical religious purposes."

As for the remarkable approximation formula for ${\sqrt {2}}$ and other approximation formulas for irrationals in the Sulbasutras, R. Cooke in A History of Mathematics: a brief course says:

It is not certain just how the early Hindu mathematicians concieved of irrational numbers, whether they had a name for them, or were merely content to find a number that would serve for practical purposes. ... Here we see an instance in which the Greek insistence on logical correctness was a hindrance. The Greek did not regard ${\sqrt {2}}$ as a number since they could not express it exactly as a ratio and they knew that they could not (i.e. after Euclid's proof that ${\sqrt {2}}$ is irrational). The Hindus may or may not have known of the impossibility of a rational expression for this number (they certainly knew that they did not have any rational expression for it); but, undeterred by the incompleteness of their knowledge, they proceeded to make what use they could of this number. This same "reckless" spirit served them well in the use of infinity and the invention of zero and negative numbers. They saw the usefulness of such numbers and either chose to live with, or did not notice, certain difficulties of a metaphysical character.

Jaina Mathematics (400 BCE - 200 CE)

Jainism is a religion and philosophy that predates Mahavira (6th century BC) a contemprory of Gautama Buddha who founded Buddhism. Followers of these religions played an important role in the future development of India. As most of the Jaina texts were composed after Mahavira, not much information is available prior to 6th century BC. 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 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.
• The binomial theorem.
• 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).
• Notations for squares, cubes and other exponents of numbers.
• Giving shape to beezganit samikaran (algebraic equations).
• Using the word shunya meaning void to refer to zero. This word eventually became zero after a series of translations and transliterations. (See Zero: Etymology.)

Jaina works also contained:

• 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 (to base 2).
• Sequences and progressions.
• 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 by Pingala many centuries before Pascal used it.

The Jaina work on number theory included:

• The earliest concept of infinite cardinal numbers.
• The earliest concept of transfinite numbers.
• A classification of all numbers into three groups: enumerable, innumerable and infinite.
• Each of these was in turn, subdivided into three orders:
  • Enumerable: lowest, intermediate and highest.
  • Innumerable: nearly innumerable, truly innumerable and innumerably innumerable.
  • Infinite: nearly infinite, truly infinite, infinitely infinite.
• The idea that all infinites were not the same or equal.
• The recognition of five different types of infinity:
  • Infinite in one direction (one dimension).
  • Infinite in two directions (one dimension).
  • Infinite in area (two dimensions).
  • Infinite everywhere (three dimensions)
  • Infinite perpetually (infinite number of dimenstions).
• The highest enumerable number (N) of the Jains corresponds to the modern concept of aleph-null $\aleph _{0}$ (the cardinal number of the infinite set of integers 1, 2, ..., N), the smallest transfinite cardinal number.
• A whole system of transfinite numbers, of which aleph-null is the smallest.

In the Jaina work on set theory:

• Two basic types of transfinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between:
  • Rigidly bounded infinities (Asmkhyata).
  • Loosely bounded infinities (Ananata).
• With this distinction, the way was open for the Jains to develop a detailed classification of transfinite numbers and mathematical operations for handling transfinite numbers of different kinds. However, further research needs to be done on Jaina mathematics to understand more about their system of transfinite numbers.

Surya Prajnapti

Surya Prajnapti (c. 400 BCE) is a mathematical and astronomical text which:

• Classifies all numbers into three sets: enumerable, innumerable and infinite.
• Recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.
• First uses transfinite numbers.
• Measures the length of the lunar month (the orbital period of the Moon around the Earth) as 29.5161290 days, which is only 20 minutes longer than the modern measurement of 29.5305888 days.

Pingala

Pingala (fl. 400-200 BCE) was a scholar and musical theorist who authored of the Chhandah-shastra. His contributions to mathematics include:

• The formation of a matrix.
• Invention of the binary number system (while he was forming a matrix for musical purposes).
• The concept of a binary code, similar to Morse code.
• First use of the Fibonacci sequence
• 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.

Bhadrabahu

Bhadrabahu (d. 298 BCE) was the author of two astronomical works, the Bhadrabahavi-Samhita and a commentary on the Surya Prajinapti.

Vaishali Ganit

The Vaishali Ganit (c. 3rd century BCE) is a book that discusses the following in detail:

• The basic calculations of mathematics.
• The numbers based on 10.
• Fractions.
• Square and cubes.
• Rules of the false position method.
• Interest methods.
• Questions on purchase and sale.

The book has given the answers of the problems and also described testing methods.

Sthananga Sutra

The Sthananga Sutra (fl. 300 BCE - 200 CE) gave classifications of:

• The five types of infinities.
• Linear equation (yavat-tavat).
• Quadratic equation (varga).
• Cubic equation (ghana).
• Quartic equation (varga-varga or biquadratic).

Katyayana

Though not a Jaina mathematician, Katyayana (c. 3rd century BCE) is notable for being the last of the Vedic mathematicians. He wrote the Katyayana Sulba Sutra, which presented much geometry, including:

• The general Pythagorean theorem.
• An accurate computation of the square root of 2 correct to five decimal places.

Anoyogdwar Sutra

The Anoyogdwar Sutra (fl. 200 BCE - 100 CE) described:

• Four types of Pramaan (Measure).
• Permutations and combinations, which were termed as Bhang and Vikalp.
• The law of indices.
• The first use of logarithms.

Yativrisham Acharya

Yativrisham Acharya (c. 176 BCE) wrote a famous mathematical text called Tiloyapannati.

Umasvati

Umasvati (c. 150 BCE) 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

The Satkhandagama (c. 2nd century) contains:

• Operations with logarithms.
• A theory of sets.

Various sets are operated upon by:

• Logarithmic functions to base 2
• Squaring and extracting square roots.
• Raising to finite or infinite powers.

These operations are repeated to produce new sets.

Bakhshali Manuscript (200 BCE - 400 CE)

The Bakhshali Manuscript is a text that bridged the gap between the earlier Jaina mathematics and the 'Classical period' of Indian mathematics, though the authorship of this text is unknown. Perhaps the most important developments found in this manuscript are:

• The use of zero as a number.
• The use of negative numbers.
• The earliest use of the modern place-value Hindu-Arabic numeral system now used universally (see also Hindu-Arabic numerals).
• The development of syncopated algebra, evident in its algebraic notation, which using letters of the alphabet, and the . and + signs to represent zero and negative numbers respectively.

There are eight principal topics discussed in the Bakhshali Manuscript:

• Examples of the rule of three (and profit, loss and interest).
• Solutions of linear equations with as many as five unknowns.
• The solution of the quadratic equation (a 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.
• 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.)

Classical Period (400 - 1200)

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 and Bhaskara give a broader and clearer shape to almost all the branches of mathematics. The system of Indian mathematics used in this period was far superior to Hellenistic mathematics, in everything except geometry. Their important contributions to mathematics would spread throughout Asia and the Middle East, and eventually Europe and other parts of the world.

Surya Siddhanta

Though its authorship is unknown, the Surya Siddhanta (c. 400) contains the roots of modern trigonometry. It uses the following as trigonometric functions for the first time:

• Sine (Jya).
• Cosine (Kojya).
• Inverse sine (Otkram jya).

It also contains the earliest uses of:

• Tangent.
• Secant.

• The Hindu cosmological time cycles explained in the text, which was copied from an earlier work, gives:
  • The average length of the sidereal year as 365.2563627 days, which is only 1.4 seconds longer than the modern value of 365.2563627 days.
  • The average length of the tropical year as 365.2421756 days, which is only 2 seconds shorter than the modern value of 365.2421988 days.

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

Aryabhata I

Aryabhata (476-550) 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 4 decimal places.

Aryabhata also wrote the Arya Siddhanta, which is now lost. Aryabhata's contributions include:

Trigonometry:

• Introduced the trigonometric functions.
• Defined the sine (jya) as the modern relationship between half an angle and half a chord.
• Defined the cosine (kojya).
• Defined the versine (ukramajya).
• Defined the 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°, to 4 decimal places of accuracy.
• Contains the trigonometric formula sin (n + 1) x - sin nx = sin nx - sin (n - 1) x - (1/225)sin nx.
• Spherical trigonometry.

Arithmetic:

• Continued fractions.

Algebra:

• Solutions of simultaneous quadratic equations.
• Whole number solutions of linear equations by a method equivalent to the modern method.
• General solution of the indeterminate linear equation .

Mathematical astronomy:

• Proposed for the first time, a heliocentric solar system with the planets spinning on their axes and following an elliptical orbit around the Sun.
• Accurate calculations for astronomical constants, such as the:
  • Solar eclipse.
  • Lunar eclipse.
  • The length of a day using integral calculus.

Calculus:

• Infinitesimals:
  • In the course of developing a precise mapping of the lunar eclipse, Aryabhatta was obliged to introduce the concept of infinitesimals (tatkalika gati) to designate the near instantaneous motion of the moon.
• Differential equations:
  • He expressed the near instantaneous motion of the moon in the form of a basic differential equation.
• Exponential function:
  • He used the exponential function e in his differential equation of the near instantaneous motion of the moon.

Varahamihira

Varahamihira (505-587) produced the Pancha Siddhanta (The Five Astronomical Canons). He made important contributions to trigonometry, including sine and cosine tables to 4 decimal places of accuracy and the following formulas relating sine and cosine functions:

• $\sin ^{2}(x)+\cos ^{2}(x)=1$

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

• ${\frac {1-\cos(2x)}{2}}=\sin ^{2}(x)$

Chhedi calendar

This Chhedi calendar (594) contains an early use of the modern place-value Hindu-Arabic numeral system now used universally (see also Hindu-Arabic numerals).

Bhaskara I

Bhaskara I (c. 600-680) 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

Brahmagupta's (598-668) famous work is his book titled Brahma Sphuta Siddhanta, which contributed:

• The first lucid explanation of zero as both a place-holder and a decimal digit.
• The integration of zero into the Indian numeral system.
• A method of calculating the volume of prisms and cones.
• Description of how to sum a geometric progression.
• The method of solving indeterminate equations of the second degree.
• the first use of algebra to solve astronomical problems.

Other contributions 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.

Virasena

Virasena (8th century) was a Jaina mathematician who wrote the Dhavala, a commentary on Jaina mathematics, which:

• Deals with logarithms to base 2 (ardhaccheda) and describes its laws.
• First uses logarithms to base 3 (trakacheda) and base 4 (caturthacheda).

Virasena also gave:

• The derivation of the volume of a frustum by a sort of infinite procedure.

Mahavira

Mahavira Acharya (c. 800-870) from Karnataka, the last of the notable Jaina mathematicians, lived in the 9th century and was patronised by the Rashtrakuta king Amoghavarsha. 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

Mahavira also:

• 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.
• Solved cubic equations.
• Solved quartic equations.
• Solved quintic equations.
• Solved higher order polynomial equations.
• Gave the general solutions of the higher order polynomial equations:
  • $\ ax^{n}=q$
  • $a{\frac {x^{n}-1}{x-1}}=p$
• Solved indeterminate quadratic equations.
• Solved indeterminate cubic equations.
• Solved indeterminate higher order equations.

Shridhara

Shridhara (c. 870-930), who lived in Bengal, wrote the books titled Nav Shatika, Tri Shatika and Pati Ganita. He gave:

• A good rule for finding the volume of a sphere.
• The formula for solving quadratic equations.

The Pati Ganita is a work on arithmetic and mensuration. It deals with various operations, including:

• Elementary operations
• Extracting square and cube roots.
• Fractions.
• Eight rules given for operations involving zero.
• Methods of summation of different arithmetic and geometric series, which were to become standard references in later works.

Manjula

Aryabhata's differential equations were elaborated on by Manjula (10th century), who realised that the expression

$\ \sin w'-\sin w$

could be expressed as

$\ (w'-w)\cos w$

He understood the concept of differentiation after solving the differential equation that resulted from substituting this expression into Aryabhata's differential equation.

Aryabhata II

Aryabhata II (c. 920-1000) wrote a commentary on Shridhara, and an astronomical treatise Maha-Siddhanta. The Maha-Siddhanta has 18 chapters, and discusses:

• Numerical mathematics (Ank Ganit).
• Algebra.
• Solutions of indeterminate equations (kuttaka).

Shripati

Shripati Mishra (1019-1066) 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.
• General solution of the simultaneous indeterminate linear equation.

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

Nemichandra Siddhanta Chakravati (c. 1100) authored a mathematical treatise titled Gome-mat Saar.

Bhaskara II

Bhaskara Acharya (1114-1185) was a mathematician-astronomer who wrote a number of important treatises, namely the Siddhanta Shiromani, Lilavati, Bijaganita, Gola Addhaya, Griha Ganitam and Karan Kautoohal. A number of his contributions were later transmitted to the Middle East and Europe. His contributions include:

Arithmetic:

• Interest computation.
• Arithmetical and geometrical progressions.
• Plane geometry.
• Solid geometry.
• The shadow of the gnomon.
• Solutions of combinations.
• Gave a proof for division by zero being infinity.

Algebra:

• The recognition of a positive number having two square roots.
• Surds.
• Operations with products of several unknowns.
• The solutions of:
  • Quadratic equations.
  • Cubic equations.
  • Quartic equations.
  • Equations with more than one unknown.
  • Quadratic equations with more than one unknown.
  • The general form of Pell's equation using the chakravala method.
  • The general indeterminate quadratic equation using the chakravala method.
  • Indeterminate cubic equations.
  • Indeterminate quartic equations.
  • Indeterminate higher-order polynomial equations.

Geometry:

• Gave a proof of the Pythagorean theorem.

Calculus:

• Conceived of differential calculus.
• Discovered the derivative.
• Discovered the differential coefficient.
• Developed differentiation.
• Stated Rolle's theorem, a special case of the mean value theorem (one of the most important theorems of calculus and analysis).
• Derived the differential of the sine function.
• Computed π, correct to 5 decimal places.
• Calculated the length of the Earth's revolution around the Sun to 9 decimal places.

Trigonometry:

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

Kerala Mathematics (1300 - 1600)

Main article: Kerala School

The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala (South India) which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. It flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559-1632). In attempting to solve astronomical problems, the Kerala school astronomers independently created a number of important mathematics concepts. The most important results, series expansion for trigonometric functions, were given in Sanskrit verse in a book by Neelakanta called Tantrasangraha and a commentary on this work called Tantrasangraha-vakhya of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhasa (c.1500-c.1610), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha.

The results obtained by the Kerala school include:

• The (infinite) geometric series: ${\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\dots +\infty$ for $|x|<1$
• A semi-rigorous proof (see "induction" remark below) of the result: $1^{p}+2^{p}+\cdots +n^{p}\approx {\frac {n^{p+1}}{p+1}}$ for large n.
• Use of mathematical induction.
• Applications of ideas from (what was to become) differential and integral calculus to obtain (Taylor-Maclaurin) infinite series for $\sin x$, $\cos x$, and $\arctan x$ The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:

$r\arctan({\frac {y}{x}})={\frac {1}{1}}\cdot {\frac {ry}{x}}-{\frac {1}{3}}\cdot {\frac {ry^{3}}{x^{3}}}+{\frac {1}{5}}\cdot {\frac {ry^{5}}{x^{t}}}-\cdots ,$ where $y/x\leq 1.$

$\sin x=x-x\cdot {\frac {x^{2}}{(2^{2}+2)r^{2}}}+x\cdot {\frac {x^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {x^{2}}{(4^{2}+4)r^{2}}}-\cdot$

$r-\cos x=r\cdot {\frac {x^{2}}{(2^{2}-2)r^{2}}}-r\cdot {\frac {x^{2}}{(2^{2}-2)r^{2}}}\cdot {\frac {x^{2}}{(4^{2}-4)r^{2}}}+\cdots ,$ where, for $r=1$, the series reduce to the standard power series for these trigonometric functions, for example:

• • $\sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots$ and
  • $\cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots$
• Use of rectification (computation of length) of the arc of a circle to give a proof of these results. (The later method of Leibniz, using quadrature (i.e. computation of area under the arc of the circle, was not used.)
• Use of series expansion of $\arctan x$ to obtain an infinite series expression (later known as Gregory series) for $\pi$:

${\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\ldots +\infty$

• A rational approximation of error for the finite sum of their series of interest. For example, the error, $f_{i}(n+1)$, (for n odd, and i = 1, 2, 3) for the series:

${\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-\cdots (-1)^{(n-1)/2}{\frac {1}{n}}+(-1)^{(n+1)/2}f_{i}(n+1)$

where $f_{1}(n)={\frac {1}{2n}},\ f_{2}(n)={\frac {n/2}{n^{2}+1}},\ f_{3}(n)={\frac {(n/2)^{2}+1}{(n^{2}+5)n/2}}.$

• Manipulation of error term to derive a faster converging series for $\pi$:

${\frac {\pi }{4}}={\frac {3}{4}}+{\frac {1}{3^{3}-3}}-{\frac {1}{5^{3}-5}}+{\frac {1}{7^{3}-7}}-\cdots \infty$

• Using the improved series to derive a rational expression, $104348/33215$ for $\pi$ correct up to nine decimal places, i.e. $3.141592653$
• Use of an intuitive notion of limit to compute these results.
• A semi-rigorous (see remark on limits above) method of differentiation of some trigonometric functions.

The works of the Kerala school were first written up for the Western world by Englishman C. M. Whish in 1835. According to Whish, the Kerala mathematicians had "laid the foundation for a complete system of fluxions" and these works abounded "with fluxional forms and series to be found in no work of foreign countries." However, Whish's results were almost completely neglected, until over a century later, when the discoveries of the Kerala school were investigated again by C. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series in Yuktibhasa given in two papers, a commentary on the Yuktibhasa's proof of the sine and cosine series and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary).

Narayana Pandit

Narayana Pandit (c. 1340-1400), the earliest of the notable Kerala 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 little original work, the following are found within it:

• Seven different methods for squaring numbers, 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.
• The second order indeterminate equation nq² + 1 = p² (Pell's equation).
• Solutions of indeterminate higher-order equations.
• Mathematical operations with zero.
• Several geometrical rules.
• Discussion of magic squares and similar figures.
• 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

Madhava (c. 1340-1425) was the founder of the Kerala School. Although it is possible that he wrote Karana Paddhati a work written sometime between 1375 and 1475, all we really know of his work comes from works of later scholars.

Little is known about Madhava, who lived near Cochin between the years 1340 and 1425. Nilkantha attributes the series for sine to Madhava. It is not known if Madhava discovered the other series as well, or whether they were discovered later by others in the Kerala school. Madhava's discoveries include:

• Taylor series for the sine.
• Second-order Taylor series approximations of the sine and cosine functions.
• Third-order Taylor series approximation of the sine function.
• Power series of π (usually attributed to Leibniz).
• The solution of transcendental equations by iteration.
• Approximation of transcendental numbers by continued fractions.
• Correctly computed the value of $\pi$ to 9 decimal places.
• Sine and cosine tables to 9 decimal places of accuracy.

He also extended some results found in earlier works, including those of Bhaskara.

Parameshvara

Parameshvara (c. 1370-1460) wrote commentaries on the works of Bhaskara I, Aryabhata and Bhaskara II. His Lilavati Bhasya, a commentary on Bhaskara II's Lilavati, contains one of his most important discoveries:

• An outstanding version of the mean value theorem, which is the most important result in differential calculus and one of the most important theorems in mathematical analysis. This result was later essential in proving the fundamental theorem of calculus.

The Siddhanta-dipika by Paramesvara is a commentary on the commentary of Govindsvamin on Bhaskara I's Maha-bhaskariya. It contains:

• Some of his eclipse observations in this work including one made at Navaksetra in 1422 and two made at Gokarna in 1425 and 1430.
• A mean value type formula for inverse interpolation of the sine.
• It presents a one-point iterative technique for calculating the sine of a given angle.
• A more efficient approximation that works using a two-point iterative algorithm, which is essentially the same as the modern secant method.

He was also the first mathematician to:

• Give the radius of circle with inscribed cyclic quadrilateral, an expression that is normally attributed to Lhuilier (1782).

Nilakantha Somayaji

In Nilakantha Somayaji's (1444-1544) 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 in Nilakantha's work includes:

• The presence of inductive mathematical proof.
• Proof of the Madhava-Gregory series of the arctangent.
• Improvements and proofs of other infinite series expansions by Madhava.
• An improved series expansion of π/4 that converges more rapidly.
• The relationship between the power series of π/4 and arctangent.

Citrabhanu

Citrabhanu (c. 1530) was a 16th century mathematician from Kerala who gave integer solutions to 21 types of systems of two simultaneous algebraic equations in two unknowns. These types are all the possible pairs of equations of the following seven forms:

$\ x+y=a,x-y=b,xy=c,x^{2}+y^{2}=d,x^{2}-y^{2}=e,x^{3}+y^{3}=f,x^{3}-y^{3}=g$

For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric.

Jyesthadeva

Jyesthadeva (c. 1500-1575) was another member of the Kerala School. His key work was the Yukti-bhasa (written in Malayalam, a regional language of Kerala), the world's first calculus text. It contained most of the developments of earlier Kerala School mathematicians, particularly Madhava. Similarly to the work of Nilakantha, it is almost unique in the history of Indian mathematics, in that it contains:

• Proofs of theorems.
• Derivations of rules and series.
• Proofs of most mathematical theorems and infinite series earlier discovered by Madhava and other Kerala School mathematicians.
• Proof of the series expansion of the arctangent function (equivalent to Gregory's proof), and the sine and cosine functions.

He also studied various topics found in many previous Indian works, including:

• Integer solutions of systems of first degree equations solved using kuttaka.
• Rules of finding the sines and the cosines of the sum and difference of two angles.

Jyesthadeva also gave:

• The earliest statement of Wallis' theorem.
• Geometric derivations of series.

References

• Bibhutibhusan Datta and Avadhesh Narayan Singh. History of Hindu Mathematics: A Source Book, Asia Publishing House, 1962.
• D. F. Almeida, J. K. John and A. Zadorozhnyy. 'Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications', Journal of Natural Geometry 20 (pages 77-104), 2001.
• Ebenezer Burgess. 'Surya Siddhanta: A Text Book of Hindu Astronomy', Journal of the American Oriental Society 6, New Haven, 1860.
• F. Nau. 'Notes d'astronomie indienne', Journal Asiatique 10 (pages 209 - 228), 1910.
• George Gheverghese Joseph. The Crest of the Peacock: Non-European Roots of Mathematics, 2nd Edition, Penguin Books, 2000.
• R. C. Gupta. 'Indian Mathematics Abroad up to the tenth Century A.D.', Ganita-Bharati 4 (pages 10-16), 1982.
• Victor J. Katz. A History of Mathematics: An Introduction, 2nd Edition, Addison-Wesley, 1998.

External links

• Ancient Jaina Mathematics: an Introduction, Infinity Foundation, 2001.
• The Kerala School, European Mathematics and Navigation, Infinity Foundation, 2001.
• Indic Mathematics: India and the Scientific Revolution, Infinity Foundation, 2000.
• An overview of Indian mathematics, MacTutor History of Mathematics Archive, St Andrews University, 2000.
• 'Index of Ancient Indian mathematics', MacTutor History of Mathematics Archive, St Andrews University, 2004.
• Indian Mathematics: Redressing the balance, Student Projects in the History of Mathematics]. Ian Pearce. MacTutor History of Mathematics Archive, St Andrews University, 2002.
• Possible transmission of Keralese mathematics to Europe. Student Projects in the History of Mathematics. Ian Pearce. St. Andrews University 2002.
• History of Indian Mathematics, 2002.
• Exegesis of Hindu Cosmological Time Cycles, 2003.
• History of Ganit (Mathematics), 2001.
• A Timeline of Mathematical Subjects, University of Georgia, 1998.
• Online course material for InSIGHT, a workshop on traditional Indian sciences for school children conducted by the Computer Science department of Anna University, Chennai, India.
• History of Indian Mathematics
• Bakhshali Manuscript
• Eurocentrism in Mathematics; Transmission of Calculus to Europe
• Mahavir Acharya still relevant

See also

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