satya anshu (i jitr) international journal of innovative

Satya Anshu* et al.(IJITR) INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND RESEARCH

Volume No.3, Issue No.2, February – March 2015, 1915 – 1924.

2320 –5547 @ 2013 http://www.ijitr.com All rights Reserved. Page | 1915

On the History of Indian MathematicsSATYAANSHU

Student, Computer Science & EngineeringR.V.College of Engineering, Mysore Road,

Bangalore-560 059, India

DR.N. SHIVAKUMARProfessor & Head, Department of Mathematics

R.V.College of Engineering, Mysore Road,Bangalore-560 059, India

Abstract: Indian mathematics has its roots in Vedic literature. Between 1000 B.C. and 1800 A.D. varioustreatises on mathematics were authored by Indian mathematicians in which were set forth for the firsttime, the concept of zero, numeral system, techniques of algebra and algorithm, square root and cuberoot. However, despite widely available, reliable information, there is a distinct and inequitable neglectoff the contributions of the sub-continent. Many of the developments of Indian mathematics remainalmost completely ignored, or worse, attributed to scholars of other nationalities, often European.However a few historians (mainly European) are reluctant to acknowledge the contributions of Indianmathematicians. They believe Indians borrowed the knowledge of mathematics from Greeks. It is a factthat Indians were way ahead than the Greeks and put forth beautiful concepts and theorems which werelater borrowed by the Greeks. Mathematics today, owes a huge debt to the outstanding contributionsmade by Indian mathematicians over many hundreds of years. Such a wonderful history of Indianmathematics cannot be masked by a few Eurocentric historians.

“India was the motherland of our raceAnd Sanskrit the mother of Europe’s languagesIndia was the mother of our philosophyOf much of our mathematics, of the ideas embodied inChristianity of self-government and democracyIn many ways, mother India is the mother of us all”

- Will Durant, American historian (1885-1981)

This basic chronology of the history ofmathematics was very simple, it had primarily beenthe invention of the ancient Greeks, whose workhad continued up to the middle of the firstmillennium A.D. Following which there was aperiod of almost 1000 years where no work ofsignificance was carried out until the Europeanrenaissance, which coincided with the‘reawakening’ of learning and culture in Europefollowing the so called dark ages.

Some historians made some concessions, byacknowledging the works of Egyptian, Babylonian,

Indian and Arabic mathematicians. Indian scholars,on the relatively rare occasions they werediscussed, were merely considered to be custodiansof ancient Greek learning. Mathematicaldevelopments of the Indian sub-continent is notonly neglected in histories of mathematics, but alsohas produced some of the most remarkable resultsof mathematics. These results, beyond being simplyremarkable because of the time in which they werederived, show that several ‘key’ mathematical tops,andsubsequent results, indubitably originate fromthe Indian sub-continent.

The book Vedang Jyotish (written 1000 B.C.)mentions the importance of mathematics asfollows:

Meaning: Just as branches of a peacock andjewel stone of a snake are placed at the highestplace of body (Forehead), similarly position ofmathematics is highest in all branches of Vedasand Shastras.

Famous Jain mathematician Mahaviracharya hassaid the following:

Meaning: What is the use of much speaking,whatever object exists in this moving and non-moving world, cannot be understood without thebase of mathematics.

Indian scholars made vast contributions to the fieldof mathematical astronomy and as a result,contributed mightily to the developments ofarithmetic, algebra, trigonometry and secondarily




geometry and combinatorics. Perhaps mostremarkable were developments in the fields ofinfinite series expansions of trigonometricexpressions and differential calculus. Surpassing allthese achievements however was the developmentof decimal numeration and his place value system,which without doubt stand together as the mostremarkable developments in the history ofmathematics, and possibly one the foremostdevelopments in the history of human kind. Thedecimal place value system allowed the subject ofmathematics to be developed in ways that simplywould not be possible otherwise. It also allowednumbers to be used more extensively and by vastlymore people than ever before.

Laplace said on the beautiful number systeminvented by the Indians:

“The ingenious method of expressing everypossible number using a set of ten symbols (eachsymbol having a place vale and an absolutevalue) emerged in India. The idea seems sosimple nowadays that its significance andprofound importance is no longer appreciated.Its simplicity lies in the way it facilitatedcalculation and invention is more readilyappreciated when one considers that it wasbeyond the two greatest men of antiquity,Archimedes and Apollonius”.

The chronology of the history of mathematics is notentirely linear. It is complicated than thediagrammatic representations where the work ofone group of people (or country) is proceeded bythe work of another group and so on.

G.Joseph states:

“…A variety of mathematical activity andexchange between a number of cultural areaswent on while Europe was in a deep slumber”.

Some of the prominent mathematicians andastronomers who made important contributions toIndian mathematics are:

Apasthamba, Aryabhata I, Aryabhata II,Varahamihira, Baudhayana, Mahavira,Jagannatha,Samrat, Sripati, Yajnavalka,Virasena, SrinivasaRamanujam,Pingala, Panini, Brahmagupta,Bhaskara I, Bhaskara II.

The history of Indian mathematics can be dividedinto five parts as:

1) Ancient Period

a) Vedic Period (around 3000 B.C. –1000 B.C.)

b) Post Vedic Period (1000 B.C. – 500B.C.)

2) Pre Middle period (500 B.C. – 400 A.D.)

3) Middle period or classic period (400 A.D.– 1200 A.D.)

4) End of classic period (1200 A.D. – 1800A.D.)

5) Current period (After 1800 A.D.)

1. Ancient Period

a) VEDIC PERIOD (3000 B.C. – 1000 B.C.)

The Vedic religion was followed by the Indo-Aryan people, who originated from the north of thesub-continent. It is through the works of Vedicreligion that we gain the literary evidence of Indianculture and mathematics. Written in Vedic Sanskrit,the Vedic works, Vedas and Vedangas areprimarily religious in content, but embody a largeamount of astronomical knowledge and hence asignificant knowledge of mathematics.

The works of this period which containedmathematics are:

Taittareya Samhita

Shatapata Brahmana

Yajurveda

Atharvaveda

Rig-Veda plus additional Samhitas

Numerals and decimals are clearly mentioned inthe Vedas. There is a Richa in Veda, which saysthe following:

In the above Richa, Dwadash (12), Treeni (3),Trishat (300) – numerals have been used. Thisindicates the use of writing numerals based on 10.In this age the discovery of ‘Zero’ and the ’10 placevalue method’ is a great contribution to the worldby India in the arena of mathematics.

In the second section of earlier portion of NaradVishnu Puran, written by Ved Vyas, mathematics isdescribed in the context of Triskandh Jyotish. Inthat, numbers have been described which are tentimes of each other, in a sequence of ’10 to thepower n’ i.e.10n

Many rules and developments of geometry arefound in Vedic works such as:

Use of geometric shapes, includingtriangles, rectangles, squares, trapeziumand circles.

Equivalence through numbers and area

Equivalence led to the problem ofsquaring the circle and vice versa

Early problems of Pythagoras theorem




Estimation for π – Three values for π arefound in Shatapata Brahmana.

It seems most probable that they arose fromtransformations of squares into circles and circlesto squares. The values are:1 = 25/8 (3.125)2 = 900/289 (3.11418685...)3 = 1156/361 (3.202216...)

All the four arithmetical operators (addition,subtraction, multiplication and division) are foundin Vedic works. This proves, at that time variousmathematical methods were not in concept stage,rather those were getting used in a methodical andexpanded manner.

b) POST VEDIC PERIOD (1000 B.C. – 500B.C.)

To perform rituals, altars were to be constructed. Ifthis ritual sacrifice wasto be successful, then thealtar had to conform to very precise measurements.To make those precise measurements, geometricalmathematics was developed. The rules wereavailable in the form of Shulv sutras (alsoSulbasutras). Shulv means rope. This rope wasused in measuring geometry while making altars.

SULBASUTRAS

The sulbasutras do not contain any proofs of therules which they describe. The most importantsulbasutras are the Baudhayana sulbasutra writtenabout 800 B.C., the Manava sulbasutra writtenabout 750 B.C. and the Katyayana sulbasutrawritten about 200 B.C.

The Baudhayana sulbasutra gave a special case ofPythagoras theorem explicitly:

“The rope which is stretched across the diagonalof a square produces an area double the size ofthe original square”.

The Katyayana sulbasutra gives a more generalversion:

“The rope which is stretched along the length ofthe diagonal of a rectangle produces an areawhich the vertical and the horizontal sides maketogether”.

The diagram illustrates the result:

Baudhayana uses different approximations for πwhen constructing circular shapes. Constructionsare given which are equivalent to taking π equal to676/225 (where 676/225 = 3.004), 900/289 (where900/289 = 3.114) and to 1156/361 (where 1156/361= 3.202)

PANINI

Panini’s work ‘Ashtadhyay’ provided an exampleof a scientific notational model that could havepropelled later mathematicians to use abstractnotations in characterizingalgebraic equationspresenting algebraic theorems in a scientificformat.

JAIN MATHEMATICS

Jain Acharyas contributed extensively to thedevelopment of mathematics. There are elaborateddescriptions of mathematics in Jain literatures.Surya Pragyapti and Chandra Pragyapti are twofamous scriptures of Jaina branch of ancient India.These describe the use of mathematics.

The main topics of Jaina mathematics were:

The theory of numbers

Arithmetical operators

Geometry

Operations with fractions

Simple equations

Cubic and Quadratic equations

Permutation and Combination

Ellipse is cleverly described in Surya Pragyapti.Jainas developed a theory of the infinite containingfive levels of infinity: Infinite in one direction, intwo directions, in area, infinite everywhere andperpetually infinite. They had a primitiveunderstanding of indices and some notion oflogarithms to base 2.

Buddhist literature also gave due importance tomathematics. They divided mathematics under twocategories:

Garna (Simple mathematics)




Sankhyan (Higher mathematics)

They have described numbers under threecategories:

Sankheya (Countable)

Asankheya (Uncountable)

Anant (Infinite)

BAKSHALI MANUSCRIPT

The Bakshali manuscript, which was unearthed inthe 19th Century, does not appear to belong to anyspecific period. A few historians class it as a workof the early classical period and others suggest itmay be a work of Jaina mathematics. Howeverthere is still a debate surrounding the date of theBakshali manuscript.

The Bakshali manuscript was written on leaves ofBirch in Sarada characters and in the Gatha dialect,which is a combination of Sanskrit and Prakrit. TheBakshali manuscript highlights developments inarithmetic and algebra. The arithmetic containedwithin the work is of high quality. There are eightprincipal topics in the Bakshali manuscript:

Examples of the rules of three(and profit andloss and interest)

Solution to linear equations with as many asfive unknowns

Solutions of quadratic equations

Arithmetic and Geometric Progressions

Compound Series

Quadratic Indeterminate equations (origin oftype y = )

Simultaneous Equations

Advances in notation including use of zero andnegative sign took place. There was an improvedmethod for calculating square root, which allowedextremely accurate approximations to becalculated:

√A = a + r = a + r2a − ( r2a)2(a + r2a)The reason for the composition of the Bakshalimanuscript is unknown but it seems possible thatthe motivation was to bring out the developmentsof mathematics during the time period.

2. PRE MIDDLE PERIOD (500 B.C. – 400 A.D.)

Almost all the writings of this time are lost, exceptfor a few books and few pages of Vaychali Ganit,Surya Siddhanta and Ganita Anuyog. During thisperiod too, mathematics underwent sufficientdevelopment.

Sathnananga Sutra, Bhagavathi Sutra andAnuyogdwar Sutra are famous books of this time.Apart from these, the books titledTatvarthaadigyam Sutra Bhashya of Jainphilosopher Omaswati (135 B.C.) andTiloyapannati of Acharya Yativrisham (176B.C.)are famous writings of this time.

Vaychali Ganit discusses in detail the following:

Basic calculations of mathematics

Numbers based on 10

Fractions

Squares and Cubes of numbers

Rule of false position

Interest methods

Sathananga Sutra has mentioned five types ofinfinite and Anuyogdwar Sutra has mentioned fourtypes of measure and also describes permutationand combination and some rule of exponents. Rootsof modern trigonometry lie in the book SuryaSiddhanta.

By around 3rd century B.C., Brahmi numeralsbegan to appear. Here is one style of Brahminumerals.

BRAHMI NUMERALS

3. THE CLASSICAL PERIOD OR MIDDLEPERIOD (400 A.D. – 1200 A.D.)

Indian Mathematics suffered a kind of slumpfollowing the Bakshali manuscript. This wasprobably due to the massive communicationproblems, but also undoubtedly to the hugepolitical upheaval that took place between the 2nd

and 4th centuries A.D., prior to the capturing ofpower of most of India by the imperial Guptas.Following the establishment the ascent of a‘galaxy’ of mathematician-astronomers led byAryabhata. These men were, first and foremostastronomers, but the mathematical requirements ofastronomy (and no doubt further interest) led themto developing many areas of mathematics. The vastmajority of the works of this period were, however,in effect, Siddhantas.




ARYABHATA

Aryabhata, who is occasionally known asAryabhata I, or Aryabhata the elder to distinguishhim from a 10th Century astronomer of the samename, stands a pioneer of the revival of Indianmathematics, and the so called ‘classical period’ or‘golden era’ of Indian mathematics.

LIFE AND WORK

Aryabhata was born in 476 A.D., as he writes thathe was 23 years old when he wrote his mostsignificant work Aryabhatiya in 499 A.D. He was amember of Kusumapura School.

The extant work of Aryabhata is his key work, the‘Aryabhata-siddhanta’, but more famously the‘Aryabhatiya’, a concise astronomical treatise of119 verses written in a poetic form, of which 33verses are concerned with mathematical rules. Therules didn’t contain any proofs.

In mathematical verses of Aryabhatiya, thefollowing topics are covered:

Arithmetic:

Method of Inversion

Various arithmetical operators, including thecube and cube root are thought to haveoriginated from Aryabhata’s work.

Aryabhata can also reliably be attributedwith credit for using relatively ‘new’functions of squaring and square rooting.

Algebra:

Formulas for finding the sum of severaltypes of series.

Rules for finding the number of terms of anarithmetical progression

Rule of three – improvement on Bakshalimanuscript

Rules for solving examples on interest –which led to the quadratic equation.

Trigonometry:

Tables of Sines. Gupta comments: “TheAryabhatia is the first historical work ofthe dated type, which uses some of these(trigonometric) functions and contains atable of sines”.

Spherical trigonometry.

Geometry:

Area of triangle

Similar triangles

Volume rules

We can see the use of ‘word numerals’ and‘alphabet numerals’ in Aryabhata’s work. This wasnot due to the absence of a satisfactory system ofnumeration but because it was helpful in poetry.The work of Aryabhata also affords a proof that‘the decimal system was well in vogue’.

Of mathematics contained within the Aryabhatia,the most remarkable is an approximation of π.

= 3.1416

He wrote that if 4 is added to 100, then multipliedby 8 then added to 62,000 the answer will be equalto circumference of a circle of diameter 20000.Aryabhata was aware that it was an irrationalnumber and that his value was an approximation,which shows his incredible insight.

In field of pure mathematics, his most significantcontribution was his solution to indeterminateequation: − = .

Aryabhata’s work on astronomy was alsopioneering, and was far less tinged with amythological flavour. He even computed thecircumference of the earth as 25835 miles which isclose to modern day calculation of 24900 miles.

The Aryabhatia was translated into Arabic by Abulal-Ahwazi (before 1000 A.D.) as Zij al-Arjabharand it is partly through this translation that Indiancomputational and mathematical methods wereintroduced to the Arabs, which had a significanteffect on the forward progress of mathematics. Thework of Aryabhata was also extremely influentialin India. Aryabhata died in 550 A.D. Thisremarkable man was a genius and continues tobaffle many mathematicians of today.

BHASKARA I

Bhaskara I continued where Aryabhata left off anddiscussed in detail further topics such as:

Longitudes of planets

Conjunctions of the planets with each otherand with bright stars

Rising and setting of the planets

Lunar crescent

He expanded on the trigonometric equationsprovided by Aryabhata. Bhaskara I correctlyassessed pi to be irrational. His most importantcontribution was his formula for calculating sinefunction which was 99% accurate. He also didpioneering work on indeterminate equations andconsidered for the first time quadrilaterals with allthe four sides unequal and none of the oppositesides parallel.




VAHAMIHIRA

A famous mathematician/astronomer during thisperiod was Varahamihira. He is thought to havelived from 505 A.D. till 587 A.D. and made onlyfairly small contributions to mathematics. Howeverhe increased the stature if the Ujjain school whileworking there, a legacy that was to last for a longperiod. Although his contributions to mathematicswere small, they were of some importance.

They included:

Several trigonometric formulas

Improvement of Aryabhata’s sine tables

Derivation of the Pascal triangle byinvestigating the problem of computingbinomial coefficients.

BRAHMAGUPTA

Following Aryabhata’s death around 550 A.D., thework of Brahmagupta resulted in Indianmathematics attaining an even greater level ofperfection.

LIFE AND WORK

Brahmagupta was born in in 598 A.D. in Bhinmalcity in the state of Rajasthan of northwest India. Hewas the head of the astronomical observatory atUjjain and during his tenure he wrote four texts onmathematics and astronomy:

The Cadamelaka in 624 A.D.

The Brahmasphutasiddhanta in 628 A.D.

The Khandakadyaka in 665 A.D.

The Durkeamynarda in 672 A.D.

The Brahmasphutasiddhanta (corrected treatise ofBrahma) is arguably his most famous work. Theworld’s mathematical content was of anexceptional quality.

Brahmagupta had a plethora of criticism directedtowards the work of rival astronomers, and in hisBrahmasphutasiddhant is found one of the earliestattested schisms among Indian mathematicians.The division was primarily about the application ofmathematics to the physical world rather than aboutthe mathematics itself. In Brahmagyota’s case, thedisagreements stemmed largely from the choice ofastronomical chapters and theories. Critiques ofrival theories appear throughout the first tenastronomical chapters and the eleventh chapter isentirely devoted the criticism of these theories,although no criticism appear in the twelfth tilleighteen chapters.

Brahmagupta’s Brahmasphutasiddhanta iscomposed in elliptic verse, as was common practicein Indian mathematics, and subsequently has a

poetic ring to it. As no proofs are given, it is notknown how Brahmagupta’s mathematics wasderived, In the Brahmasphutasiddhant among themajor developments are those in the areas of:

1) Arithmetic:

Brahmagupta possessed a greater understanding ofthe number system (and the place value system)than any one to that point. In the beginning ofchapter 12, entitled calculation, he detailsoperations on fractions. He explains how to find thecube and cube root of an integer and later givesrules facilitating the computation of squares andsquare roots.

He then gives rules for dealing 5 types ofcombination of fractions:

+ , . , + , + . = ( ) ,ac − bd . ac = a(d − b)cdBrahmagupta then goes on to give the sum of thesquares and cubes of the first n integers. He foundthe result in terms of the sum of the first n integersrather than in terms of n as is the modern practice.Brahmagupta made use of an important concept inmathematics, the number ‘zero’. TheBrahmasphutasiddhanta is the earliest known textto treat zero as a number in its own right, ratherthan as simply a placeholder digit in representinganother number as was done by the Babylonians oras a symbol for lack of quantity as was done byPtolemy and the Romans. In chapter 19, hedescribes operations on negative numbers. He alsodescribes addition, subtraction and multiplication.Brahmagupta was the first to attempt to divide byzero, and while his attempts of showingn0 = ∞were not ultimately successful they demonstrate anadvanced understanding of an extremely abstractconcept.

2) Algebra:

Brahmagupta gave the solution of the general linearequation ax + c = by in chapter 18. He further gavetwo equivalent solutions to the general quadraticequation which are respectively:x = −b ± √b − 4ac2aHe went on to solve systems of simultaneousindeterminate equations stating that the desiredvariable must first be isolated, and then theequation must be divided by the desired variable’scoefficient. The algebra of Brahmagupta issyncopated. In chapter 12, Brahmagupta finds




Pythagorean triplets for a given length m and anarbitrary multiplier x.

Let: a = mxb = m + mxx + 2Then: a and b form a Pythagorean triplet.

Brahmagupta went on to give a recurrence relationfor generating solutions to certain instances ofDiophantine equations of the second degree such as+ =by using the Euclidean algorithm. The Euclideanalgorithm was known to him as “Kuttaka”(Pulveriser) since it breaks numbers down intoeven smaller pieces. The key to his solution was theidentity:

(x12– Ny1

2 )(x22 – Ny2

2 ) = (x1x2+Ny1y2)2 –

N(x1y2+x2y1)2

Using this identity and the fact that if (x1,y1) and(x2, y2) are solutions to the equations x2-Ny2=k1

and x2-Ny2=k2, respectively, then (x1x2 +Ny1y2,x1y2 + x2y1) is a solution to x2-Ny2=k1k2. Hewas able to find integral solutions of the equationx2-Ny2=ki

3) Geometry:

Brahmagupta’s most famous result in geometry ishis formula for cyclic quadrilaterals. Given thelengths of the sides of any cyclic quadrilateral, hegave an approximate and an exact formula for thefigure’s area.

So, given the lengths p, q, r of a cyclicquadrilateral, the approx. area is ( )( ) while

letting t = ( )Exact Area = (t − p)(t − q)(t − r)(t − s)

One of his theorems on triangles states that whenthe base is divided by its altitude, the lengths oftwo segments are: b ± (c − a )b

He further gives a theorem on rational triangles. Atriangle with rational sides a, b, c and rational areais of the form

a = + v , b = + w , c = ( − v +− w)For some rational numbers u, v, w.

Brahmagupta’s theorem states that AF = FD.[10]

He also gives formulas for the lengths and areas ofgeometric figures, such as the circumradius of anisosceles trapezoid and a scalene quadrilateral, andthe lengths of diagonals in a scalene cyclicquadrilateral. In verse 40, he gives value for π.

4) Astronomy:

It was through Brahmasphutasiddhanta that theArabs learned of Indian astronomy. In chapter 7 ofhis Brahmasphutasiddhanta, entitled ‘Lunarcrescent’, Brahmagupta rebuts the idea that themoon is father from the earth than the sun, an ideawhich is maintained in scriptures. He does this byexplaining the illumination of the moon by the sun.He explains that since the moon is closer to theearth than the sun, the degree of the illuminatedpart of the moon depends on the relative positionsof the sun and the moon, and this can be computedfrom the size of the angle between the two bodies.He gave methods for calculating the position ofheavenly bodies over time, their rising and setting,conjunctions and the calculation of the solar andlunar eclipses.

About the earth’s gravity he said:“Bodies falltowards the earth as it is in the nature of theearth to attract bodies, just as it is in the natureof the water to flow”.

A translation of Brahmasphutasiddhanta wascarried out by Muhammad al-Fazari and had a farreaching influence on subsequent Arabic works. In860 A.D. an extensive commentary onBrahmasphutasiddhanta was written byPrthudakasvami. His work was extremely elaborateand unlike many Indian works did not ‘sufferbrevity’ of expression. Brahmagupta passed awayin 668 A.D.




MAHAVIRACHARYA

Mahaviracharya, a Jain by religion, is the mostcelebrated mathematician of the 9th century. Hismajor work Ganitasar Sangraha was written around850 A.D. Mahaviracharya was aware of the Jainamathematicians and also the works of Aryabhataand Brahmagupta, and refined and improved muchof their work. He was a member of themathematical school at Mysore in the south ofIndia. His major contributions to mathematicsinclude:

1) Arithmetic:

Detailed operations with fractions(and unitfractions)

Geometric progressions – He gave almost allthe required formulas

General formula for permutation andcombination.

2) Algebra:

Work on quadratic equations

Indeterminate equations

Simultaneous equations

3) Geometry:

Definitions for most of the geometric shapes

Repeated Brahmagupta’s construction forcyclic quadrilateral

He referred to the ellipse and gave itsperimeter.

Following Mahaviracharya, the most notablemathematician was Prthudakasvami (830 A.D. –890 A.D.), a prominent Indian algebraist who isbest known for his work on solving equations. Hewrote a commentary on Brahmagupta’sBrahmasphutasiddhanta.

SRIDHARA

In the early 10th century a mathematician by thename of Sridhara (870 A.D. – 930 A.D.) lived.However, it’s a fact that he wrote Patganita Sara, awork on arithmetic and mensuration. It containedexactly 300 verses and is hence also known by thename Trisatika. It contains the following topics:

Rules on extracting square and cube roots

Operations on functions

Eight rules for operations involving zero

Theory on cyclic quadrilaterals with rationalsides

A section concerning rational solutions ofvarious equations of the Pell’s type

Methods for summation of differentarithmetic and geometric series

It is thought that Sridhara also composed a text onalgebra. The legacy of Sridhara’s work was that ithad some influence on the work of BhaskaracharyaII.

ARYABHATTA II

A mathematical-astronomer, by name Aryabhata II(920 A.D. – 1000 A.D.) made importantcontributions to algebra. In his workMahasiddhanta he gives 20 verses of detailed rulesfor solving by = ax + c

SRIPATI

Sripati (1019 A.D. – 1066 A.D.) was a follower ofthe teaching of Lalla and in fact the most importantIndian mathematician of the 11th century. Heimpressively gave the identity:

(x + y) = x + x − y2 + x − x − y2BHASKARACHARYA II

Bhaskaracharya or Bhaskara II is regarded almostwithout question as the greatest Hindumathematician of all time and his contribution tonot just India, but world mathematics.[9]

LIFE AND WORK

He was born in 1114 A.D. in Vijayapura. Hebelonged to Bijjada Bida. He became the head ofthe Ujjain school of mathematical astronomy.C.Srinivasiengar claims he wrote SiddhantaSiromani in 1150 A.D., which contained 4 sections:

Lilavati (arithmetic)

Bijaganita (algebra)

Goladhyaya (sphere/celestial globe)

Grahaganita (mathematics of the planets)

Lilavati is divided into 13 chapters and coversmany branches of mathematics, arithmetic, algebra,geometry and a little trigonometry andmensuration. Lilavati is written in poetic form witha prose commentary. Tradition has it that Bhaskaranamed this work after daughter in order to consoleher. His astrological meddling coupled with anunfortunate twist of fate is said to have deprivedher of her only chance of marriage and happiness.

The contents of Lilavati include:

Properties of zero (including division)

Estimation of π

Methods of multiplication and squaring

Indeterminate equations




Integer solutions (first and second order)

Bijaganita is effectively a treatise on algebra andcontains the following topics:

Positive and negative numbers

Surds

Simple equations(indeterminate of second,third and fourth degree)

Indeterminate quadratic equations (ax +b =y )

Quadratic equations with more than oneunknown

Bhaskara derived a cyclic, ‘Cakraval’ method forsolving equations of the form ( + + = ).He investigated regular polygons up to thosehaving 384 sides, thus obtaining a good approx.value of π = 3.141666.

Bhaskara is thought to be the first to show that:( ) =Evidence suggests that Bhaskara was fullyacquainted with the principle of differentialcalculus. Bhaskara goes deeper into the differentialcalculus and suggests that the differentialcoefficient vanishes at an extremum value of thefunction, indicating knowledge of concept of‘infinitesimals’. He also gave the well-knownresults for ( + )and ( − ). There is alsoevidence for early form of Rolle’s Theorem:( ) = ( ) = , ’( ) =< x < b[15]

His work Siddhanta Siromani is an astronomicaltreatise and contains his observations ofconjunctions, geography and mathematicaltechniques etc.

4) END OF CLASSICAL PERIOD OR POSTMIDDLE PERIOD (1200 A.D – 1800 A.D.)

The work of Bhaskara was considered the highestpoint in Indian mathematics attained, and it waslong considered that Indian mathematics ceasedafter that point. Extreme political turmoil throughmuch of the sub-continent and Mongol invasionsshattered the atmosphere of discovery and learningand led to the stagnation of mathematicaldevelopments as scholars contented themselveswith duplicating earlier works.

There were occasional small developments andattempts to revive learning, but nothing of themagnitude of the previous millennium. Worth of abrief mention are Kamalakara (1616 A.D. – 1700A.D.) and Jagannatha Samrat (1690 A.D. – 1750A.D.). Both combined tradition ideas of Indianastronomy with Arabic concepts, Kamalakara gavetrigonometric results of interest and Samrat made

several Sanskrit translations of Arabic versions ofGreek works, including notably Euclid’s elements.Under the patronage of monarch Sawai JayasinhaRaja, Samrat, along with a group of scholars,attempted to ‘reinvigorate’ science and learning inIndia. Though the efforts were not whollysuccessful, they were in the greatest of faith andshould be applauded.

THEKERALA SCHOOL OF MATHEMATICS

Despite the political turmoil, mathematicscontinued to a high degree in the south of India upto the 16thcentury. The south of India avoided theworst of the political upheavals of the subcontinentand the Kerala School of mathematics flourishedfor some time, producing some truly remarkableresults.

Madhava (14th century A.D) gave a seriesexpansion of the cos and sine functions. Hiscontributions were instrumental in takingmathematics to the next stage, that of modernclassical analysis. Nilakantha (15th century A.D.)extended elaborations on the planetary theory andcommentaries on Nilakatha’s Tantrasangraha.Chirtabhanu (16th century A.D.) gave integersolutions to twenty types of systems of twoalgebraic equations, using both algebraic andgeometric methods in developing his results. Thereare seven forms:+ = , – = ,+ = , − = ,+ = , − ==For each case, Chitrabhanu gave an explanationand justification of his rule as well as anexample.[3]

Important discovery by Keralese mathematiciansinclude:

Infinite series

Expansions of trigonometric functions

Newton-Gauss interpolation formula

Sum of an infinite series and

It must be considered most unfortunate that acountry, which on reflection was unarguably aworld leader in the field of mathematics for severalthousands of years ceased to contribute in anysignificant way.

5) CURRENT PERIOD (1800 A.D. ONWARDS)

Bapudev Shastri (1813 A.D.) wrote books ongeometrical mathematics, numerical mathematicsand trigonometry. Sudhakar dwivedi (1831 A.D.)wrote books titled:




Deergha Vritta (dealing with ellipse)

Golaya Rekha Ganit (sphere linemathematics)

Samikaran Meemansa (analysis ofequations)

SRINIVASA RAMANUJAN

Srinivasa Ramanujan (1889 A.D.) is a modernmathematics scholar. He followed Vedic style ofwriting mathematical concepts in terms of formulaeand then proving it. His intellectuality is proved bythe fact; it took all mettle of current mathematiciansto prove a few out of his 50 theorems. Ramanujanalso showed that any big number can be written assum of not more than four prime numbers.

He further showed how to divide the number intotwo or more squares or cube. He also stated that1729 is the smallest number which can be writtenin the form of sum of cubes of two numbers in twoways, i.e. 1729 = 93 + 103 = 13 + 123. Since then thenumber 1729 is called Ramanujan’s Number.

Swami Bharti Krisnateerthaji (1884 A.D. – 1960A.D.)

Swami Bharti Krisnateerthaji wrote the book VedicGanit. He is the founder and father of VedicGanit.Bharati Krishnaji got the key to Ganita Sutracoded in the Atharva Veda and rediscovered VedicMathematics with the help of lexicographs. Hefound “Sixteen Sutras” or word formulas whichcover all the branches of Mathematics -Arithmetic, Algebra, Geometry, Trigonometry,Physics, plan and spherical geometry, conics,calculus- both differential and integral, appliedmathematics of all various kinds, dynamics,hydrostatics and all.[12]

SHAKUNTALA DEVI (1939A.D. – 2013 A.D.)

Sakuntala Devi has written many books onmathematics. She was also the personal astrologerof the President of India.Her feats include:In 1980,she gave the product of two, thirteen digit numberswithin 28 seconds, many countries have invited herto demonstrate her extraordinary talent.In Dallasshe competed with a computer to see who give thecube root of 188138517 faster, she won! Atuniversity of USA she was asked to give the23rd root of a 201 digit number. She answered in 50seconds. It took a UNIVAC 1108 computer, fullone minute to confirm that she was right after itwas fed with 13000 instructions

ACKNOWLEDGEMENTS

The authors thank Dr.S.Sridhar, Professor andDean, Cognitive & Central Computing ,R.V.College of Engineering, Bangalore, India forguiding them to present this history in the form ofa note by giving a good shape.

REFERENCES

[1]. C T Rajagopal and M S Rangachari: On anuntapped source of medieval Keralesemathematics, Arch. History Exact Sci. 18(1978), 89-102.

[2]. C T Rajagopal and M S Rangachari: Onmedieval Keralese mathematics, Arch.History Exact Sci. 35 (1986), 91-99.

[3]. A.K. Bag: Mathematics in Ancient andMedieval India (1979, Varanasi)

[4]. T.A. Saraswati: Geometry in Ancient andMedieval India (1979, Delhi)

[5]. R P Kulkarni: The value of pi known toSulbasutrakaras, Indian Journal Hist. Sci. 13(1) (1978), 32-41

[6]. B Datta: The science of the Sulba (Calcutta,1932)

[7]. L C Jain: System theory in Jaina school ofmathematics, Indian J. Hist. Sci. 14 (1)(1979), 31-65

[8]. http://www.esamskriti.com/essay-chapters/A-brief-history-of-Indian-Mathematics-1.aspx

[9]. http://www.storyofmathematics.com/indian.html

[10]. http://www.storyofmathematics.com/indian_brahmagupta.html

[11]. http://www-history.mcs.st-and.ac.uk/HistTopics/Indian_mathematics.html

[12]. http://www.vedicmathsindia.org/father-of-vedic-math/

[13]. corvallistoday.com/asia_pacific/mathindia.htm

[14]. www-history.mcs.st-and.ac.uk/Projects/Pearce/Chapters/Ch8_5.html

[15]. www.ijitr.com/index.php/ojs/article/viewFile/278/pdf

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