Indian Mathematics

## History of Indian Mathematics

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

## Harappan Mathematics (3300 BC - 1500 BC)

The first appearance of evidence of the use of mathematics in the Indian subcontinent was in the Indus Valley Civilization, which dates back to around 3300 BC. Excavations at Harrapa, Mohenjo-daro and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The mathematics used by this early Harrapan civilisation was very much for practical means, and was primarily concerned with:

• Weights 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.
• 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.
• 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.

It has been suggested by some scholars that the Sulba Sutras, which are mathematical texts usually assigned to 800-500 BC in the Vedic period, were originally texts 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.

## Vedic Mathematics (1500 BC - 400 BC)

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

As a result of the mathematics required for the construction of religious altars, many rules and developments of geometry are found in Vedic works, along with many astronomical developments for religious purposes. These include:

• The use of geometric shapes, including triangles, rectangles, squares, trapezia and circles.
• Equivalence through numbers and area.
• 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

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

The Yajur-Veda (c. 1200-900 BC) 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 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 (c. 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 (fl. 1350-1000 BC) 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 BC) contains mathematical rules for rituals and ceremonials.

#### Samhitas

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

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

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

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

#### Yajnavalkya

Yajnavalkya (fl. 900-700 BC) 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.

#### Sulba Geometry (800-500 BC)

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.
• 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.
• 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.
• The earliest use of sine.
• The sine of π/4 (45°) correctly computed as 1/√2 in a procedure for circling the square.

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 BC) composed the Baudhayana Sulba Sutra, which contains:

• The earliest list of Pythagorean triples discovered algebraically.
• The earliest statement of the Pythagorean theorem.
• 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.
• The first use of irrational numbers.
• The earliest use of 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 up to four unknowns.

#### Manava

Manava (fl. 750-650 BC) 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 BC) 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

ini (c. 520-460 BC) 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:

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

#### 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 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.
• 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.
• 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 Prajnapti

Surya Prajnapti (c. 400 BC) 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.
• 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 BC) 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 (d. 298 BC) 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 BC) is a book that discusses the following in detail:

• The basic calculations of mathematics
• The numbers based on 10
• Fractions
• Square and cubes
• 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 BC - 200 CE) gave classifications of:

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

#### Katyayana

Though not a Jaina mathematician, Katyayana (c. 3rd century BC) 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 BC - 100 CE) described:

#### Yativrisham Acharya

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

#### Umasvati

Umasvati (c. 150 BC) 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:

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 BC - 400 CE)

This text 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).
• 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 geoemetry. 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 trignometry. It uses the following as trigonometric functions for the first time:

It also contains the earliest uses of:

• 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:

• The value of π, correct to 4 decimal places.
• Various other scientific problems.

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.

The words jya and kojya eventually became sine and cosine respectively after a mistranslation. (See Etymology of sine.)

Arithmetic:

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 using the kuttaka method.

Mathematical astronomy:

Calculus:

• 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.
• He expressed the near instantaneous motion of the moon in the form of a basic differential equation.
• He used the exponential function e in his differential equation of the near instantaneous motion of the moon.

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

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:

#### Chhedi calendar

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

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, though this was discovered by earlier Indian mathematicians.
• The integration of zero into the Indian numeral system (the modern number system used throughout the world).
• 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.

Other contributions in the Brahma Sphuta Siddhanta:

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.

#### 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), 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:

Mahavira also:

#### Shridhara

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

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

sina - sinb

could be expressed as

(a - b) cosb

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 (Beej Ganit).
• 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:

• General solution of the simultaneous indeterminate linear equation.

He was also the author of Dhikotidakarana, a work of twenty verses on:

The Dhruvamanasa is a work of 105 verses on:

#### Nemichandra Siddhanta Chakravati

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

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 inlcude:

Arithmetic:

• Interest computation.
• Arithmetical and geometrical progressions.
• 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.
• Operations with products of several unknowns.
• The solutions of:
• 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:

Calculus:

Trigonometry:

The trigonometric formulas:

Kerala Mathematics (1300 - 1600)

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) mostly due to subsequent political upheaval in Kerala. These astronomers, in attempting to solve problems, invented a number of important concepts including:

They achieved most of these results several centuries before European mathematicians. Jyeshtadeva consolidated the Kerala School's discoveries in the Yuktibhasa, the world's first calculus text. In many ways, the Kerala School represents the peak of mathematical knowledge in the middle ages.

#### 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 (c. 1340-1425) was 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.

Perhaps 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 also responsible for many other significant and original discoveries, including:

• Infinite series expansions of functions.
• The power series.
• The Taylor series.
• Trigonometric series.
• Rational approximations of infinite series.
• Taylor series of the sine and cosine functions (Madhava-Newton power series).
• Taylor series of the tangent function.
• Taylor series of the arctangent function (Madhava-Gregory series).
• 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).
• Power series of π/4 (Euler's series).
• Power series of the radius.
• Power series of the diameter.
• Power series of the circumference.
• Power series of angle θ (equivalent to the Gregory series).
• Infinite continued fractions.
• Term by term integration.
• The solution of transcendental equations by iteration.
• Approximation of transcendental numbers by continued fractions.
• Tests of convergence of infinite series.
• Correctly computed the value of π to 11 decimal places, the most accurate value of π after almost a thousand years.
• Sine and cosine tables to 9 decimal places of accuracy, which would remain the most accurate up to the 17th century.
• Laying the foundations for the development of calculus, which was then further developed by his successors at the Kerala School.

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 imporved 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:

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 (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.